Paul Monk
1
I
~ Physical
Chemistry
UNDERSTANDING OUR CHEMICAL WORLD
(*)WILEY
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Physical Chemistry
Understanding our Chemical World
Physical Chemistry
Understanding our Chemical World
Paul Monk
Manchester Metropolitan University, UK
John Wiley & Sons, Ltd
Copyright © 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
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Library of Congress Cataloging-in-Publication Data
Monk, Paul M. S.
Physical chemistry : understanding our chemical world / Paul Monk,
p. cm.
Includes bibliographical references and index.
ISBN 0-471-49180-2 (acid-free paper) - ISBN 0-471-49181-0 (pbk. :
acid-free paper)
1. Chemistry, Physical and theoretical. I. Title.
QD453.3.M66 2004
S41 _ dc22
2004004224
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-471-49180-2 hardback
0-471-49181-0 paperback
Typeset in 10.5/12.5pt Times by Laserwords Private Limited, Chennai, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
This book is printed on acid-free paper responsibly manufactured from sustainable forestry
in which at least two trees are planted for each one used for paper production.
Contents
Preface xv
Etymological introduction xix
List of symbols xxiii
Powers of ten: standard prefixes xxviii
1 Introduction to physical chemistry 1
1.1 What is physical chemistry: variables, relationships and laws 1
Why do we warm ourselves by a radiator? 1
Why does water get hot in a kettle? 2
Are these two colours complementary? 2
Does my radio get louder if I vary the volume control? 3
Why does the mercury in a barometer go up when the air pressure
increases? 5
Why does a radiator feel hot to the touch when 'on', and cold when 'off? 7
1.2 The practice of thermodynamic measurement 9
What is temperature? 9
How long is a piece of string? 14
How fast is 'greased lightning'? 15
Why is the SI unit of mass the kilogram? 17
Why is 'the material of action so variable'? 18
1.3 Properties of gases and the gas laws 20
Why do we see eddy patterns above a radiator? 20
Why does a hot-air balloon float? 20
How was the absolute zero of temperature determined? 21
Why pressurize the contents of a gas canister? 23
Why does thunder accompany lightning? 25
How does a bubble-jet printer work? 26
/i CONTENTS
What causes pressure? 30
Why is it unwise to incinerate an empty can of air freshener? 32
1.4 Further thoughts on energy 33
Why is the room warm? 33
What do we mean by 'room temperature' ? 34
Why do we get warmed-through in front of a fire, rather than just our
skins? 35
2 Introducing interactions and bonds 37
2. 1 Physical and molecular interactions 37
What is 'dry ice' ? 37
How is ammonia liquefied? 38
Why does steam condense in a cold bathroom? 39
How does a liquid-crystal display work? 40
Why does dew form on a cool morning? 42
How is the three-dimensional structure maintained within the DNA double
helix? 44
How do we make liquid nitrogen? 47
Why is petrol a liquid at room temperature but butane is a gas? 49
2.2 Quantifying the interactions and their influence 50
How does mist form? 50
How do we liquefy petroleum gas? 52
Why is the molar volume of a gas not zero at K? 54
2.3 Creating formal chemical bonds 59
Why is chlorine gas lethal yet sodium chloride is vital for life? 59
Why does a bicycle tyre get hot when inflated? 59
How does a fridge cooler work? 60
Why does steam warm up a cappuccino coffee? 61
Why does land become more fertile after a thunderstorm? 63
Why does a satellite need an inert coating? 64
Why does water have the formula H 2 0? 66
Why is petroleum gel so soft? 67
Why does salt form when sodium and chlorine react? 69
Why heat a neon lamp before it will generate light? 69
Why does lightning conduct through air? 72
Why is argon gas inert? 74
Why is silver iodide yellow? 75
3 Energy and the first law of thermodynamics 77
3.1 Introduction to thermodynamics: internal energy 77
Why does the mouth get cold when eating ice cream? 77
Why is skin scalded by steam? 79
Why do we sweat? 81
Why do we still feel hot while sweating on a humid beach? 83
CONTENTS vii
Why is the water at the top of a waterfall cooler than the water at its base? 85
Why is it such hard work pumping up a bicycle tyre? 86
Why does a sausage become warm when placed in an oven? 87
Why, when letting down a bicycle tyre, is the expelled air so cold? 88
Why does a tyre get hot during inflation? 89
Can a tyre be inflated without a rise in temperature? 89
How fast does the air in an oven warm up? 90
Why does water boil more quickly in a kettle than in a pan on a stove? 91
Why does a match emit heat when lit? 94
Why does it always take 4 min to boil an egg properly? 95
Why does a watched pot always take so long to boil? 98
3.2 Enthalpy 99
How does a whistling kettle work? 99
How much energy do we require during a distillation? 102
Why does the enthalpy of melting ice decrease as the temperature
decreases? 104
Why does water take longer to heat in a pressure cooker than in an open
pan? 106
Why does the temperature change during a reaction? 107
Are diamonds forever? 109
Why do we burn fuel when cold? Ill
Why does butane burn with a hotter flame than methane? 114
3.3 Indirect measurement of enthalpy 118
How do we make 'industrial alcohol'? 118
How does an 'anti-smoking pipe' work? 120
Why does dissolving a salt in water liberate heat? 123
Why does our mouth feel cold after eating peppermint? 125
How does a camper's 'emergency heat stick' work? 127
4 Reaction spontaneity and the direction of thermodynamic change 129
4.1 The direction of physicochemical change: entropy 129
Why does the colour spread when placing a drop of dye in a saucer of
clean water? 129
When we spill a bowl of sugar, why do the grains go everywhere and
cause such a mess? 130
Why, when one end of the bath is hot and the other cold, do the
temperatures equalize? 131
Why does a room containing oranges acquire their aroma? 133
Why do damp clothes become dry when hung outside? 134
Why does crystallization of a solute occur? 137
4.2 The temperature dependence of entropy 139
Why do dust particles move more quickly by Brownian motion in warm
water? 139
Why does the jam of a jam tart burn more than does the pastry? 139
CONTENTS
4.3 Introducing the Gibbs function 144
Why is burning hydrogen gas in air (to form liquid water) a spontaneous
reaction? 144
How does a reflux condenser work? 144
4.4 The effect of pressure on thermodynamic variables 148
How much energy is needed? 148
Why does a vacuum 'suck'? 151
Why do we sneeze? 152
How does a laboratory water pump work? 153
4.5 Thermodynamics and the extent of reaction 156
Why is a 'weak' acid weak? 156
Why does the pH of the weak acid remain constant? 158
Why does the voltage of a battery decrease to zero? 159
Why does the concentration of product stop changing? 162
Why do chicken eggs have thinner shells in the summer? 165
4.6 The effect of temperature on thermodynamic variables 166
Why does egg white denature when cooked but remain liquid at room
temperature? 166
At what temperature will the egg start to denature? 170
Why does recrystallization work? 171
5 Phase equilibria 177
5.1 Energetic introduction to phase equilibria 177
Why does an ice cube melt in the mouth? 177
Why does water placed in a freezer become ice? 181
Why was Napoleon's Russian campaign such a disaster? 182
5.2 Pressure and temperature changes with a single-component system:
qualitative discussion 184
How is the 'Smoke' in horror films made? 184
How does freeze-drying work? 185
How does a rotary evaporator work? 188
How is coffee decaffeinated? 189
5.3 Quantitative effects of pressure and temperature change for a
single-component system 192
Why is ice so slippery? 192
What is 'black ice'? 193
Why does deflating the tyres on a car improve its road-holding on ice? 198
Why does a pressure cooker work? 199
5.4 Phase equilibria involving two-component systems: partition 205
Why does a fizzy drink lose its fizz and go flat? 205
How does a separating funnel work? 207
Why is an ice cube only misty at its centre? 208
How does recrystallization work? 209
Why are some eggshells brown and some white? 211
CONTENTS ix
5.5 Phase equilibria and colligative properties 212
Why does a mixed-melting-point determination work? 212
How did the Victorians make ice cream? 216
Why boil vegetables in salted water? 217
Why does the ice on a path melt when sprinkled with salt? 218
5.6 Phase equilibria involving vapour pressure 221
Why does petrol sometimes have a strong smell and sometimes not? 221
How do anaesthetics work? 222
How do carbon monoxide sensors work? 224
Why does green petrol smell different from leaded petrol? 224
Why do some brands of 'green' petrol smell different from others? 225
Why does a cup of hot coffee yield more steam than above a cup of
boiling water at the same temperature? 229
How are essential oils for aromatherapy extracted from plants? 229
6 Acids and Bases 233
6.1 Properties of Lowry-Br0nsted acids and bases 233
Why does vinegar taste sour? 233
Why is it dangerous to allow water near an electrical appliance, if water is
an insulator? 235
Why is bottled water 'neutral' ? 236
What is 'acid rain' ? 237
Why does cutting an onion make us cry? 239
Why does splashing the hands with sodium hydroxide solution make them
feel 'soapy' ? 239
Why is aqueous ammonia alkaline? 240
Why is there no vinegar in crisps of salt and vinegar flavour? 241
How did soldiers avoid chlorine gas poisoning at the Second Battle of
Ypres? 242
How is sherbet made? 244
Why do steps made of limestone sometimes feel slippery? 244
Why is the acid in a car battery more corrosive than vinegar? 245
Why do equimolar solutions of sulphuric acid and nitric acid have
different pHs? 250
What is the pH of a 'neutral' solution? 25 1
What do we mean when we say blood plasma has a 'pH of 7.4' ? 251
6.2 'Strong' and 'weak' acids and bases 253
Why is a nettle sting more painful than a burn from ethanoic acid? 253
Why is 'carbolic acid' not in fact an acid? 254
Why does carbonic acid behave as a mono-protic acid? 259
Why is an organic acid such as trichloroethanoic acid so strong? 260
6.3 Titration analyses 261
Why does a dock leaf bring relief after a nettle sting? 261
How do indigestion tablets work? 262
CONTENTS
6.4 pH buffers 267
Why does the pH of blood not alter after eating pickle? 267
Why are some lakes more acidic than others? 267
How do we make a 'constant-pH solution' ? 270
6.5 Acid-base indicators 273
What is 'the litmus test' ? 273
Why do some hydrangea bushes look red and others blue? 274
Why does phenolphthalein indicator not turn red until pH 8.2? 276
7 Electrochemistry 279
7.1 Introduction to cells: terminology and background 279
Why does putting aluminium foil in the mouth cause pain? 279
Why does an electric cattle prod cause pain? 281
What is the simplest way to clean a tarnished silver spoon? 282
How does 'electrolysis' stop hair growth? 283
Why power a car with a heavy-duty battery yet use a small battery in a
watch? 283
How is coloured ('anodized') aluminium produced? 285
How do we prevent the corrosion of an oil rig? 286
What is a battery? 288
Why do hydrogen fuel cells sometimes 'dry up' ? 289
Why bother to draw cells? 291
Why do digital watches lose time in the winter? 293
Why is a battery's potential not constant? 294
What is a 'standard cell' ? 295
Why aren't electrodes made from wood? 300
Why is electricity more dangerous in wet weather? 302
7.2 Introducing half-cells and electrode potentials 303
Why are the voltages of watch and car batteries different? 303
How do 'electrochromic' car mirrors work? 305
Why does a potential form at an electrode? 306
7.3 Activity 308
Why does the smell of brandy decrease after dissolving table salt in it? 308
Why does the smell of gravy become less intense after adding salt to it? 308
Why add alcohol to eau de Cologne? 309
Why does the cell em/ alter after adding LiCl? 312
Why does adding NaCl to a cell alter the emf, but adding tonic water
doesn't? 314
Why does MgCl 2 cause a greater decrease in perceived concentration than
KC1? 315
Why is calcium better than table salt at stopping soap lathering? 316
Why does the solubility of AgCl change after adding MgS0 4 ? 318
7.4 Half-cells and the Nernst equation 321
Why does sodium react with water yet copper doesn't? 321
CONTENTS xi
Why does a torch battery eventually 'go flat' ? 325
Why does E Ag ci,Ag change after immersing an SSCE in a solution of salt? 326
Why 'earth' a plug? 328
7.5 Concentration cells 333
Why does steel rust fast while iron is more passive? 333
How do pH electrodes work? 336
7.6 Transport phenomena 339
How do nerve cells work? 339
What is a 'salt bridge' ? 342
7.7 Batteries 343
How does an electric eel produce a current? 343
What is the earliest known battery? 345
8 Chemical kinetics 349
8.1 Kinetic definitions 349
Why does a 'strong' bleach clean faster than a weaker one does? 349
Why does the bleaching reaction eventually stop? 351
Why does bleach work faster on some greases than on others? 354
Why do copper ions amminate so slowly? 356
How fast is the reaction that depletes the ozone layer? 358
Why is it more difficult to breathe when up a mountain than at ground
level? 359
8.2 Qualitative discussion of concentration changes 361
Why does a full tank of petrol allow a car to travel over a constant
distance? 361
Why do we add a drop of bromine water to a solution of an alkene? 362
When magnesium dissolves in aqueous acid, why does the amount of
fizzing decrease with time? 364
8.3 Quantitative concentration changes: integrated rate equations 368
Why do some photographs develop so slowly? 368
Why do we often refer to a 'half-life' when speaking about radioactivity? 378
How was the Turin Shroud 'carbon dated' ? 382
How old is Otzi the iceman? 385
Why does the metabolism of a hormone not cause a large chemical change
in the body? 387
Why do we not see radicals forming in the skin while sunbathing? 388
8.4 Kinetic treatment of complicated reactions 393
Why is arsenic poisonous? 393
Why is the extent of Walden inversion smaller when a secondary alkyl
halide reacts than with a primary halide? 394
Why does 'standing' a bottle of wine cause it to smell and taste better? 397
Why fit a catalytic converter to a car exhaust? 399
Why do some people not burn when sunbathing? 400
How do Reactolite sunglasses work? 403
di CONTENTS
8.5 Thermodynamic considerations: activation energy, absolute reaction rates
and catalysis 408
Why prepare a cup of tea with boiling water? 408
Why store food in a fridge? 408
Why do the chemical reactions involved in cooking require heating? 409
Why does a reaction speed up at higher temperature? 411
Why does the body become hotter when ill, and get 'a temperature' ? 415
Why are the rates of some reactions insensitive to temperature? 416
What are catalytic converters? 420
9 Physical chemistry involving light: spectroscopy and
photochemistry 423
9.1 Introduction to photochemistry 423
Why is ink coloured? 423
Why do neon streetlights glow? 424
Why do we get hot when lying in the sun? 425
Why is red wine so red? 426
Why are some paints red, some blue and others black? 427
Why can't we see infrared light with our eyes? 429
How does a dimmer switch work? 433
Why does UV-b cause sunburn yet UV-a does not? 434
How does a suntan protect against sunlight? 436
How does sun cream block sunlight? 439
Why does tea have a darker colour if brewed for longer? 442
Why does a glass of apple juice appear darker when viewed against a
white card? 442
Why are some paints darker than others? 444
What is ink? 445
9.2 Photon absorptions and the effect of wavelength 446
Why do radical reactions usually require UV light? 446
Why does photolysis require a powerful lamp? 452
Why are spectroscopic bands not sharp? 453
Why does hydrogen look pink in a glow discharge? 455
Why do surfaces exposed to the sun get so dusty? 457
Why is microwave radiation invisible to the eye? 458
9.3 Photochemical and spectroscopic selection rules 459
Why is the permanganate ion so intensely coloured? 459
Why is chlorophyll green? 461
Why does adding salt remove a blood stain? 462
What is gold-free gold paint made of? 462
What causes the blue colour of sapphire? 463
Why do we get hot while lying in the sun? 464
What is an infrared spectrum? 467
Why does food get hot in a microwave oven? 469
Are mobile phones a risk to health? 471
CONTENTS xiii
9.4 Photophysics: emission and loss processes 472
How are X-rays made? 472
Why does metal glow when hot? 473
How does a light bulb work? 474
Why is a quartz-halogen bulb so bright? 474
What is 'limelight' ? 476
Why do TV screens emit light? 476
Why do some rotting fish glow in the dark? 478
How do 'see in the dark' watch hands work? 479
How do neon lights work? 480
How does a sodium lamp work? 481
How do 'fluorescent strip lights' work? 482
9.5 Other optical effects 483
Why is the mediterranean sea blue? 483
Do old-master paintings have a 'fingerprint' ? 485
10 Adsorption and surfaces, colloids and micelles 487
10.1 Adsorption and definitions 487
Why is steam formed when ironing a line-dried shirt? 487
Why does the intensity of a curry stain vary so much? 489
Why is it difficult to remove a curry stain? 492
Why is iron the catalyst in the Haber process? 494
Why is it easier to remove a layer of curry sauce than to remove a curry
stain! 496
How does water condense onto glass? 497
How does bleach remove a dye stain? 498
How much beetroot juice does the stain on the plate contain? 499
Why do we see a 'cloud' of steam when ironing a shirt? 503
10.2 Colloids and interfacial science 504
Why is milk cloudy? 504
What is an 'aerosol' spray? 505
What is 'emulsion paint' ? 506
Why does oil not mix with water? 508
10.3 Colloid stability 509
How are cream and butter made? 509
How is chicken soup 'clarified' by adding eggshells? 510
How is 'clarified butter' made? 510
Why does hand cream lose its milky appearance during hand rubbing? 511
Why does orange juice cause milk to curdle? 512
How are colloidal particles removed from waste water? 513
10.4 Association colloids: micelles 514
Why does soapy water sometimes look milky? 514
What is soap? 517
Why do soaps dissolve grease? 518
xiv CONTENTS
Why is old washing-up water oily when cold but not when hot? 519
Why does soap generate bubbles? 521
Why does detergent form bubbles? 522
Answers to SAQs 525
Bibliography 533
Index 565
Preface
This book
Some people make physical chemistry sound more confusing than it really is. One of
their best tricks is to define it inaccurately, saying it is 'the physics of chemicals'. This
definition is sometimes quite good, since it suggests we look at a chemical system and
ascertain how it follows the laws of nature. This is true, but it suggests that chemistry
is merely a sub-branch of physics; and the notoriously mathematical nature of physics
impels us to avoid this otherwise useful way of looking at physical chemistry.
An alternative and more user-friendly definition tells us that physical chemistry
supplies 'the laws of chemistry', and is an addition to the making of chemicals. This
is a superior lens through which to view our topic because we avoid the bitter aftertaste
of pure physics, and start to look more closely at physical chemistry as an applied
science: we do not look at the topic merely for the sake of looking, but because
there are real-life situations requiring a scientific explanation. Nevertheless, most
practitioners adopting this approach are still overly mathematical in their treatments,
and can make it sound as though the science is fascinating in its own right, but will
sometimes condescend to suggest an application of the theory they so clearly relish.
But the definition we will employ here is altogether simpler,
and also broader: we merely ask 'why does it happen?' as we
focus on the behaviour of each chemical system. Every example
we encounter in our everyday walk can be whittled down into
small segments of thought, each so simple that a small child can
understand. As a famous mystic of the 14th century once said, T
saw a small hazelnut and I marvelled that everything that exists could be contained
within it'. And in a sense she was right: a hazelnut looks brown because of the way
light interacts with its outer shell - the topic of spectroscopy (Chapter 9); the hazelnut
is hard and solid - the topic of bonding theory (Chapter 2) and phase equilibria
(Chapter 5); and the nut is good to eat - we say it is readily metabolized, so we think
Now published as Rev-
elations of Divine Love,
by Mother Julian of
Norwich.
xvi PREFACE
of kinetics (Chapter 8); and the energetics of chemical reactions (Chapters 2-4). The
sensations of taste and sight are ultimately detected within the brain as electrical
impulses, which we explain from within the rapidly growing field of electrochemistry
(Chapter 7). Even the way a nut sticks to our teeth is readily explained by adsorption
science (Chapter 10). Truly, the whole of physical chemistry can be encompassed
within a few everyday examples.
So the approach taken here is the opposite to that in most other books of physical
chemistry: each small section starts with an example from everyday life, i.e. both the
world around us and also those elementary observations that a chemist can be certain
to have pondered upon while attending a laboratory class. We then work backwards
from the experiences of our hands and eyes toward the cause of why our world is
the way it is.
Nevertheless, we need to be aware that physical chemistry is not a closed book in
the same way of perhaps classical Latin or Greek. Physical chemistry is a growing
discipline, and new experimental techniques and ideas are continually improving the
data and theories with which our understanding must ultimately derive.
Inevitably, some of the explanations here have been over-simplified because phys-
ical chemistry is growing at an alarming rate, and additional sophistications in theory
and experiment have yet to be devised. But a more profound reason for caution is
in ourselves: it is all too easy, as scientists, to say 'Now I understand!' when in fact
we mean that all the facts before us can be answered by the theory. Nevertheless, if
the facts were to alter slightly - perhaps we look at another kind of nut - the theory,
as far as we presently understand it, would need to change ever so slightly. Our
understanding can never be complete.
So, we need a word about humility. It is said, probably too often, that science is
not an emotional discipline, nor is there a place for any kind of reflection on the
human side of its application. This view is deeply mistaken, because scientists limit
themselves if they blind themselves to any contradictory evidence when sure they
are right. The laws of physical chemistry can only grow when we have the humility
to acknowledge how incomplete is our knowledge, and that our explanation might
need to change. For this reason, a simple argument is not necessary the right one; but
neither is a complicated one. The examples in this book were chosen to show how
the world around us manifests Physical Chemistry. The explanation of a seemingly
simple observation may be fiendishly complicated, but it may be beautifully simple. It
must be admitted that the chemical examples are occasionally artificial. The concept
of activity, for example, is widely misunderstood, probably because it presupposes
knowledge from so many overlapping branches of physical chemistry. The examples
chosen to explain it may be quite absurd to many experienced teachers, but, as
an aid to simplification, they can be made to work. Occasionally the science has
been simplified to the point where some experienced teachers will maintain that it is
technically wrong. But we must start from the beginning if we are to be wise, and
only then can we progress via the middle . . . and physical chemistry is still a rapidly
growing subject, so we don't yet know where it will end.
While this book could be read as an almanac of explanations, it provides students
in further and higher education with a unified approach to physical chemistry. As a
PREFACE xvii
teacher of physical chemistry, I have found the approaches and examples here to be
effective with students of HND and the early years of BSc and MChem courses. It has
been written for students having the basic chemical and mathematical skills generally
expected of university entrants, such as rearrangement of elementary algebra and a
little calculus. It will augment the skills of other, more advanced, students.
To reiterate, this book supplies no more than an introduction to physical chemistry,
and is not an attempt to cover the whole topic. Those students who have learned
some physical chemistry are invited to expand their vision by reading more special-
ized works. The inconsistencies and simplifications wrought by lack of space and
style in this text will be readily overcome by copious background reading. A com-
prehensive bibliography is therefore included at the end of the book. Copies of the
figures and bibliography, as well as live links can be found on the book's website at
http://www.wileyeurope.com/go/monkphysical.
A ckn o w ledge m en ts
One of the more pleasing aspects of writing a text such as this is the opportunity to
thank so many people for their help. It is a genuine pleasure to thank Professor Seamus
Higson of Cranfield University, Dr Roger Mortimer of Loughborough University,
and Dr Michele Edge, Dr David Johnson, Dr Chris Rego and Dr Brian Wardle from
my own department, each of whom read all or part of the manuscript, and whose
comments have been so helpful.
A particular 'thank you' to Mrs Eleanor Riches, formerly a high-school teacher,
who read the entire manuscript and made many perceptive and helpful comments.
I would like to thank the many students from my department who not only saw
much of this material, originally in the form of handouts, but whose comments helped
shape the material into its present form.
Please allow me to thank Michael Kaufman of The Campaign for a Hydrogen Econ-
omy (formerly the Hydrogen Association of UK and Ireland) for helpful discussions
to clarify the arguments in Chapter 7, and the Tin Research Council for their help in
constructing some of the arguments early in Chapter 5.
Concerning permission to reproduce figures, I am indebted to The Royal Society of
Chemistry for Figures 1.8 and 8.26, the Open University Press for Figure 7.10, Else-
vier Science for Figures 4.7 and 10.3, and John Wiley & Sons for Figures 7.19, 10.11
and 10.14. Professor Robin Clarke frs of University College London has graciously
allowed the reproduction of Figure 9.28.
Finally, please allow me to thank Dr Andy Slade, Commissioning Editor of Wiley,
and the copy and production editors Rachael Ballard and Robert Hambrook. A special
thank you, too, to Pete Lewis.
Paul Monk
Department of Chemistry & Materials
Manchester Metropolitan University
Manchester
Etymological
introduction
The hero in The Name of the Rose is a medieval English monk. He acts as sleuth,
and is heard to note at one point in the story how, 'The study of words is the
whole of knowledge'. While we might wish he had gone a little
"Etymology" means
the derivation of a
word's meaning.
further to mention chemicals, we would have to agree that many
of our technical words can be traced back to Latin or Greek roots.
The remainder of them originate from the principal scientists who
pioneered a particular field of study, known as etymology.
Etymology is our name for the science of words, and describes the sometimes-
tortuous route by which we inherit them from our ancestors. In fact, most words
change and shift their meaning with the years. A classic example describes how King
George III, when first he saw the rebuilt St Paul's Cathedral in London, described it
as 'amusing, artificial and awful', by which he meant, respectively, it 'pleased him',
was 'an artifice' (i.e. grand) and was 'awesome' (i.e. breathtaking).
Any reader will soon discover the way this text has an unusual etymological empha-
sis: the etymologies are included in the belief that taking a word apart actually helps us
to understand it and related concepts. For example, as soon as we know the Greek for
'green' is Mows, we understand better the meanings of the proper nouns c/z/orophyll
and chlorine, both of which are green. Incidentally, phyll comes from the Greek for
'leaf, and ine indicates a substance.
Again, the etymology of the word oxygen incorporates much historical informa-
tion: oxys is the Greek for 'sharp', in the sense of an extreme sensory experience,
such as the taste of acidic vinegar, and the ending gen comes from gignesthaw (pro-
nounced 'gin-es-thaw'), meaning 'to be produced'. The classical roots of 'oxygen'
reveal how the French scientists who first made the gas thought they had isolated the
distinguishing chemical component of acids.
The following tables are meant to indicate the power of this approach. There are
several dozen further examples in the text. The bibliography on p. 533 will enable
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List of Symbols
Symbols for variables
a Activity
a van der Waals constant
A optical absorbance
A Area
b van der Waals constant
B' virial coefficient
c concentration
c the intercept on the v-axis of a graph
c constant of proportionality
C virial coefficient
C heat capacity
C p heat capacity determined at constant
pressure
Cy heat capacity determined at constant
volume
E energy
E potential
E n activation energy
£( ea ) electron affinity
£j junction potential
£(ioad) potential of a battery or cell when
passing a current
emf potential of a cell, determined at zero
current
£o,R
electrode potential for the couple
+ Me" = R
c O,R
standard electrode potential for the
couple + ne~ = R
E(LHS)
electrode potential of the negative
electrode in a cell
E(RHS)
electrode potential of the positive
electrode in a cell
f
fugacity
f
frequency
F
force
G
Gibbs function
AG
change in Gibbs function
G°
standard Gibbs function
G*
Gibbs function of activation
h
height
H
enthalpy
AH
change in enthalpy
"(ads)
enthalpy of adsorption
H*
standard enthalpy
A // BE
bond enthalpy
A// c
enthalpy of combustion
AH f
enthalpy of formation
H*
enthalpy of activation
I
electrical current
I
intensity of light following absorption
LIST OF SYMBOLS
/o
intensity of incident light beam
m
/
ionic strength
M
/
ionization energy
n
J
rotational quantum number; rotational
quantum number of an excited
n
state
n m
J'
rotational quantum number of ground
state
N
k
force constant of a bond
P
k
proportionality constant
PU)
k
rate constant
P(i)
k'
pseudo rate constant
p«
kri
rate constant of an rath-order reaction
q
k—n
rate constant for the back reaction of
Q
an nth-order reaction
Q
k(n)
rate constant of the nth process in a
r
multi-step reaction
r
ka
rate constant of adsorption
r
kd
rate constant of desorption
r'
kn
Henry's law constant
K
equilibrium constant
r
K
correction constant of an ion-selective
R
electrode
s
K a
acidity constant ('acid dissociation'
s
constant)
S
Kn(n)
acidity constant for the nth
AS
dissociation reaction
5 e
K b
basicity constant
5*
K c
equilibrium constant formulated in
t
terms of concentration
t\_
K P
equilibrium constant formulated in
2
T
terms of pressure
T
K,
equilibrium constant of solubility
T
(sometimes called 'solubility
7k
product' or 'solubility constant')
U
K K
autoprotolysis constant of water
AU
K*
equilibrium constant of forming a
transition state 'complex'
V
I
length
v'
in
gradient of a graph
mass
relative molar mass
number of moles
number of electrons in a redox
reaction
amount of material in an adsorbed
monolayer
number
pressure
partial pressure of component i
vapour pressure of pure i
standard pressure of 10 Pa
heat energy
charge
reaction quotient
separation between ions
radius of a circle or sphere
bond length
bond length in an optically excited
species
equilibrium bond length
electrical resistance
solubility
stoichiometric ratio
entropy
change in entropy
standard entropy
entropy of activation
time
half life
temperature
optical transmittance
optical transmittance without a sample
Krafft temperature
internal energy
change in internal energy, e.g. during
reaction
quantum-number of vibration
quantum-number of vibration in an
excited-state species
LIST OF SYMBOLS
v" quantum-number of vibration in
a ground-state species
V volume
V voltage, e.g. of a power pack
V Coulomb potential energy
V m molar volume
w work
x controlled variable on the horizontal
axis of a graph
x deviation of a bond from its
equilibrium length
Xi mole fraction of i
y observed variable on the vertical
of a graph
z charge on ion (so z + for a cation
and z~ for an anion)
Z compressibility
p
a
a
4>
x
velocity
frequency (the reciprocal of the period
of an event)
frequency following transmission
(in Raman spectroscopy)
extent of reaction
density
electrical conductivity
standard deviation
electric field strength (electrostatic
interaction)
work function of a metal
primary quantum yield
quantum yield of a reaction
electronegativity
wavenumber of a vibration
(determined as co — k 4- c)
Y
Y±
Y
Y
S
9
A
K
^ (max)
11
v
activity coefficient
mean ionic activity coefficient
fugacity coefficient
surface tension
small increment
partial differential
change in a variable (so
Symbols for constants
AX = X
(final form)
x
(initial
form) )
extinction coefficient ('molar decadic
absorptivity')
relative permittivity
permittivity of free space
adsorption isotherm
angle
ionic conductivity
wavelength
the wavelength of a peak in a
spectrum
reduced mass
chemical potential of i
standard chemical potential of i
stoichiometric constant
A
Debye-Hiickel 'A' factor
c
the speed of light in vacuo
c*
standard concentration
e
charge on an electron, of value
1.6 x 10" 19 C
f
mathematical operator ('function of)
F
Faraday constant, of value
96485 CmoP 1
kB
Boltzmann constant, of value
1.38 x 1(T 23
L
Avogadro constant, of value
6.022 x 10 23 mor 1
N A
Avogadro number, of value
6.022 x 10 23 mor 1
g
acceleration due to gravity, of value
9.81 m s~ 2
h
Planck constant, of value
6.626 x 10" 34 J s
R
gas constant, of value
8.314 J K" 1 mol -1
LIST OF SYMBOLS
Symbols for units
A
A
bar
Hz
J
K
kg
m
mmHg
mol
N
Pa
s
S
V
w
yr
Q
ampere
angstrom, length of value 10~ 10 m
(non-IUPAC)
standard pressure of 10 5 Pa
(non-SI unit)
coulomb
centigrade (non-SI)
gram
hertz
joule
kelvin
kilogram
metre
millimetre of mercury (non-SI unit
of pressure)
mole
newton
pascal
second (SI unit)
siemen
volt
watt
year
ohm
Acronyms and abbreviations
CT
charge transfer
d
differential operator (which never
appears on its own)
HOM(
3 highest occupied molecular orbital
IQ
intelligence quotient
IR
infrared
IUPAC
1 International Union of Pure and
Applied Chemistry
IVF in vitro fertilization
LCD liquid crystal display
LHS left-hand side
LUMO lowest unoccupied molecular
orbital
MLCT metal-to-ligand charge transfer
MRI magnetic resonance imaging
NIR near-infra red
NMR nuclear magnetic resonance
O general oxidized form of a redox
couple
p mathematical operator,
— log 10 [variable], so
P H=-log 10 [H+]
PEM proton exchange membrane
R general reduced form of a redox
couple
RHS right-hand side
s.t.p. standard temperature and
pressure
SAQ self-assessment question
SCE saturated calomel electrode
SCUBA self-contained underwater breathing
apparatus
SHE standard hydrogen electrode
SHM simple harmonic motion
SI Systeme Internationale
Sn 1 unimolecular nucleophilic substitution
process
Sn2 bimolecular nucleophilic substitution
process
SSCE silver-silver chloride electrode
TS transition state
TV television
UPS UV-photoelectron spectroscopy
UV ultraviolet
UV-vis ultraviolet and visible
XPS X-ray photoelectron spectroscopy
LIST OF SYMBOLS
Standard subscripts (other
•
radical
than those where a word or
+ •
radical cation
phrase is spelt in full)
©
standard state
ads
aq
adsorption; adsorbed
aqueous
combustion
Chemicals and materials
c
eq
at equilibrium
A
general anion
f
formation
Bu
butyl
s
gas
CFC
chlorofluorocarbon
1
liquid
DMF
N, 7V-dimethylformamide
LHS
left-hand side of a cell
DMSO
dimethylsulphoxide
m
molar
DNA
deoxyribonucleic acid
P
at constant pressure
e~
electron
Pt
platinum (usually, as an electrode)
EDTA
ethylenediamine tetra-acetic acid
r
reaction
HA
general Lowry-Br0nsted acid
RHS
right-hand side of cell
LPG
liquid petroleum gas
s
solid
M
general cation
sat'd
saturated
MB
methylene blue
t
at time t (i.e. after a reaction or
MV
methyl viologen
process has commenced)
(1,1 ' -dime thy 1-4 , 4' -bipyridilium)
V
at constant volume
O
general oxidized form of a redox
initially (i.e. at time t — 0)
couple
oo
measurement taken after an infinite
PC
propylene carbonate
length of time
Ph
phenyl substituent
R
general alkyl substituent
R
general reduced form of a redox
Standard superscripts (other
couple
than those where a word or
SDS
sodium dodecyl sulphate
phrase is spelt in full)
TFA
tetrafluoroacetic acid
,
particle emitted during radioactive
disintegration of nucleus
t
activated quantity
UL
anion
P
particle emitted during radioactive
+
cation
disintegration of nucleus
*
p.Yritp.H state
y
high-energy photon (gamma ray)
c
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I I I I III I
O O O O OOO o
1
Introduction to physical
chemistry
Introduction
In this, our introductory chapter, we start by looking at the terminology of phys-
ical chemistry. Having decided what physical chemistry actually is, we discuss the
nature of variables and relationships. This discussion introduces the way relationships
underlying physical chemistry are formulated.
We also introduce the fundamental (base) units of the Systeme Internationale (SI),
and discuss the way these units are employed in practice.
We look at the simple gas laws to explore the behaviour of systems with no inter-
actions, to understand the way macroscopic variables relate to microscopic, molecular
properties. Finally, we introduce the statistical nature underlying much of the physical
chemistry in this book when we look at the Maxwell-Boltzmann relationship.
1.1 What is physical chemistry: variables,
relationships and laws
Why do we warm ourselves by a radiator?
Cause and effect
We turn on the radiator if we feel cold and warm ourselves in front of it. We become
warm because heat travels from the radiator to us, and we absorb its heat energy,
causing our own energy content to rise. At root, this explains why we feel more
comfortable.
While this example is elementary in the extreme, its importance lies in the way it
illustrates the concept of cause and effect. We would not feel warmer if the radiator
INTRODUCTION TO PHYSICAL CHEMISTRY
was at the same temperature as we were. We feel warmer firstly because the radiator
is warmer than us, and secondly because some of the heat energy leaves the radiator
and we absorb it. A transfer of energy occurs and, therefore, a change. Without the
cause, the effect of feeling warmer could not have followed.
We are always at the mercy of events as they occur around
us. The physical chemist could do nothing if nothing happened;
chemists look at changes. We say a physical chemist alters vari-
ables, such as pressure or temperature. Typically, a chemist causes
one variable to change and looks at the resultant response, if any.
Even a lack of a response is a form of result, for it shows us what is and what is not
a variable.
A variable is an exper-
imental parameter we
can change or 'tweak'.
The fearsome-looking
word 'physicochemi-
cal' means 'relating to
physical chemistry'.
Why does water get hot in a kettle?
Physicochemical relationships
Putting water into an electric kettle does not cause the water to get
hot. The water stays cold until we turn on the power to the kettle
element, which converts electrical energy from the mains into heat energy. The heat
energy from the kettle element is then absorbed by the water, which gets hot as a
direct consequence.
The temperature of the water does not increase much if a small amount of elec-
trical energy is consumed; conversely, the water gets hotter if a greater amount of
energy is consumed and thereafter passed to the water. A physi-
cal chemist says a 'relationship' exists (in this case) between heat
input and temperature, i.e. the temperature of the water depends on
the amount of energy consumed.
Mathematically, we demonstrate the existence of a relationship
by writing T = /(energy), where T is temperature and the small
/ means 'is a function of.
So the concept of variables is more powerful than just changing
parameters; nor do physical chemists merely vary one parameter and see what happens
to others. They search for 'physicochemical' relationships between the variables.
In words, the symbols
T = /"(energy) means T
is a function of energy'.
Note how variables are
usually printed in italic
type.
lese two colours complementary?
Qualitative and quantitative measurements
We often hear this question, either at the clothes shop or at a paint merchant. Either
someone wants to know if the pink colour of a sweatshirt matches the mauve of a
skirt, or perhaps a decorator needs to know if two shades of green will match when
painted on opposing bedroom walls.
WHAT IS PHYSICAL CHEMISTRY: VARIABLES, RELATIONSHIPS AND LAWS
Complementary means
'to make complete'.
But while asking questions concerning whether a series of
colours are complementary, we are in fact asking two questions
at once: we ask about the colour in relation to how dark or light
it is ('What is the brightness of the colour?'); but we also ask a
more subjective question, saying 'Is the pink more red or more white: what kind of
pink is it?' We are looking for two types of relationship.
In any investigation, we first look for a qualitative relationship. In effect, we ask
questions like, 'If I change the variable x, is there is a response in a different variable
yV We look at what kind of response we can cause - a scientist wants to know about
the qualities of the response, hence QUAL-itative. An obvious question relating to
qualitative relationships is, 'If I mix solutions of A and B, does a reaction occurV
Only after we know whether or not there is a response (and of what general kind)
does a physical chemist ask the next question, seeking a quantitative assessment. He
asks, 'How much of the response is caused?' In effect, physical chemists want to
know if the magnitude (or quantity) of a response is big, small or intermediate. We
say we look for a QUANT-itative aspect of the relationship. An obvious question
relating to quantitative relationships is, 'I now know that a reaction occurs when I
mix solutions of A and B, but to what extent does the reaction occur; what is the
chemical yield!'
Does my radio get louder if I vary the volume control?
Observed and controlled variables
We want to turn up the radio because it's noisy outside, and we want to hear what is
broadcast. We therefore turn the volume knob toward 'loud'. At its most basic, the
volume control is a variable resistor, across which we pass a current from the battery,
acting much like a kettle element. If we turn up the volume control then a larger
current is allowed to flow, causing more energy to be produced by the resistor. As
a listener, we hear a response because the sound from the speakers becomes louder.
The speakers work harder.
But we must be careful about the way we state these relationships. We do not 'turn
up the volume' (although in practice we might say these exact words and think in these
terms). Rather, we vary the volume control and, as a response, our ears experience
an increase in the decibels coming through the radio's speakers. The listener controls
the magnitude of the noise by deciding how far the volume-control knob needs to be
turned. Only then will the volume change. The process does not occur in reverse: we
do not change the magnitude of the noise and see how it changes
the position of the volume-control knob.
While the magnitude of the noise and the position of the volume
knob are both variables, they represent different types, with one
depending on the other. The volume control is a controlled variable
because the listener dictates its position. The amount of noise is the
observed variable because it only changes in response to variations
in the controlled variable, and not before.
We consciously, care-
fully, vary the magni-
tude of the controlled
variable and look at
the response of the
observed variable.
INTRODUCTION TO PHYSICAL CHEMISTRY
Relationships and graphs
Physical chemists often depict relationships between variables by
drawing graphs. The controlled variable is always drawn along the
x-axis, and the observed variable is drawn up the y-axis.
Figure 1.1 shows several graphs, each demonstrating a different
kind of relationship. Graph (a) is straight line passing through the
origin. This graph says: when we vary the controlled variable x,
the observed variable y changes in direct proportion. An obvious
example in such a case is the colour intensity in a glass of black-
currant cordial: the intensity increases in linear proportion to the
concentration of the cordial, according to the Beer-Lambert law
(see Chapter 9). Graph (a) in Figure 1.1 goes through the origin
because there is no purple colour when there is no cordial (its
concentration is zero).
Graph (b) in Figure 1.1 also demonstrates the existence of a relationship between
the variables x and y, although in this case not a linear relationship. In effect, the graph
tells us that the observed variable y increases at a faster rate than does the controlled
variable x. A simple example is the distance travelled by a ball as a function of time
t as it accelerates while rolling down a hill. Although the graph is not straight, we
still say there is a relationship, and still draw the controlled variable along the x-axis.
The x-axis (horizontal)
is sometimes called
the abscissa and the
/-axis (vertical) is the
ordinate. A simple way
to remember which
axis is which is to say,
'an expanse of road
goes horizontally along
the x-axis', and 'a Yo-
Yo goes up and down
the /-axis'.
CD
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CO
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Controlled var
able
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Controlled variable x
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(b)
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Controlled variable
X
Controlled variable x
(c)
(d)
Figure 1.1 Graphs of observed variable (along the y-axis) against controlled variable (along the
x-axis). (a) A simple linear proportionality, so y = constant x x; (b) a graph showing how y is not
a simple function of x, although there is a clear relationship; (c) a graph of the case where variable
y is independent of variable x ; (d) a graph of the situation in which there is no relationship between
y and x, although y does vary
WHAT IS PHYSICAL CHEMISTRY: VARIABLES, RELATIONSHIPS AND LAWS 5
Graph (c) in Figure 1.1 is a straight-line graph, but is horizontal. In other words,
whatever we do to the controlled variable x, the observed variable y will not change.
In this case, the variable y is not a function of x because changing x will not change
y. A simple example would be the position of a book on a shelf as a function of time.
In the absence of other forces and variables, the book will not move just because it
becomes evening.
Graph (d) in Figure 1.1 shows another situation, this time the
data do not demonstrate a straightforward relationship; it might
demonstrate there is no relationship at all. The magnitude of the
controlled variable x does not have any bearing on the observed
Data is plural; the sin-
gular is datum.
When two variables are
multiplied together, we
call them a compound
variable.
variable y. We say the observed variable y is independent of the
controlled variable x. Nevertheless, there is a range of results for
y as x varies. Perhaps x is a compound variable, and we are being
simplistic in our analysis: an everyday example might be a stu-
dent's IQ as x and his exam performance as y, suggesting that,
while IQ is important, there must be another variable controlling the magnitude of
the exam result, such as effort and commitment. Conversely, the value of y might be
completely random (so repeating the graphs with the same values of x would gen-
erate a different value of y - we say it is irreproducible). An example of this latter
situation would be the number of people walking along a main road as a function
of time.
Why does the mercury in a barometer go up when the
air pressure increases?
Relationships between variables
The pressure p of the air above any point on the Earth's surface relates ultimately
to the amount of air above it. If we are standing high up, for example on the top
of a tall mountain, there is less air between us and space for gravity to act upon.
Conversely, if we stand at the bottom of the Grand Canyon (one of the lowest places
on Earth) then more air separates us from space, causing the air pressure p to be
much greater.
A barometer is an instrument designed to measure air pressure p. It consists of
a pool of liquid mercury in a trough. A long, thin glass tube (sealed at one end)
is placed in the centre of the trough with its open-side beneath the surface of the
liquid; see Figure 1.2. The pressure of the air acts as a force on the surface of the
mercury, forcing it up and into the capillary within the tube. If the air pressure is
great, then the force of the air on the mercury is also great, causing much mer-
cury up the tube. A lower pressure is seen as a shorter length h of mercury in
the tube.
By performing experiments at different pressures, it is easy to prove the existence
of a relationship between the air pressure p and the height h of the mercury column
INTRODUCTION TO PHYSICAL CHEMISTRY
Vacuum.
r\
Thick-walled
glass tube *
Trough of mercury
Figure 1.2 A barometer is a device for measuring pressures. A vacuum-filled glass tube (sealed
at one end) is placed in a trough of mercury with its open end beneath the surface of the liquid
metal. When the tube is erected, the pressure of the external air presses on the surface and forces
mercury up the tube. The height of the mercury column h is directly proportional to the external
pressure p
In fact, the value
of the constant c in
Equation (1.1) com-
prises several natural
constants, including
the acceleration due
gravity g and the den-
sity p of the mercury.
in the tube. This relationship follows Equation (1.1):
h = c x p
(1-1)
where c is merely a proportionality constant.
In practice, a barometer is merely an instrument on which
we look at the length of the column of mercury h and, via
Equation (1.1), calculate the air pressure p. The magnitude of h is
in direct relation to the pressure p. We ascertain the magnitude of
h if we need to know the air pressure p.
While physical chemistry can appear to be horribly mathematical, in fact the mathe-
matics we employ are simply one way (of many) to describe the relationships between
variables. Often, we do not know the exact nature of the function until a later stage
of our investigation, so the complete form of the relationship has to be discerned
in several stages. For example, perhaps we first determine the existence of a linear
equation, like Equation (1.1), and only then do we seek to measure
an accurate value of the constant c.
But we do know a relationship holds, because there is a response.
We would say there was no relationship if there was no response.
For example, imagine we had constructed a poor-quality barometer
(meaning it does not follow Equation (1.1)) and gave it a test run. If
we could independently verify that the pressure p had been varied
over a wide range of values yet the length of the mercury h in
the barometer did not change, then we would say no relationship
existed between p and h.
We might see this
situation written math-
ematically as, h =£ f(p),
where the V means
'is not equal to'. In
other words, h is not a
function of p in a poor
barometer.
WHAT IS PHYSICAL CHEMISTRY: VARIABLES, RELATIONSHIPS AND LAWS
Why does a radiator feel hot to the touch when x on',
and cold when *off r ?
I
Laws and the minus-oneth law of thermodynamics
A 'law' in physical
chemistry relates to
a wide range of situa-
tions.
Feeling the temperature of a radiator is one of the simplest of
experiments. No one has ever sat in front of a hot radiator and
felt colder. As a qualitative statement, we begin with the excellent
generalization, 'heat always travels from the hotter to the colder
environment' . We call this observation a law because it is universal.
Note how such a law is not concerned with magnitudes of change
but simply relays information about a universal phenomenon: energy in the form of
heat will travel from a hotter location or system to a place which is colder. Heat
energy never travels in the opposite direction.
We can also notice how, by saying 'hotter' and 'colder' rather than just 'hot' and
'cold', we can make the law wider in scope. The temperature of a radiator in a living
room or lecture theatre is typically about 60 °C, whereas a human body has an ideal
temperature of about 37 °C. The radiator is hotter than we are, so heat travels to us
from the radiator. It is this heat emitted by the radiator which we absorb in order to
feel warmer.
Conversely, now consider placing your hands on a colder radiator having a tem-
perature of 20 °C (perhaps it is broken or has not been switched on). In this second
example, although our hands still have the same temperature of 37 °C, this time the
heat energy travels to the radiator from our hands as soon as we touch it. The direction
of heat flow has been reversed in response to the reversal of the relative difference
between the two temperatures. The direction in which the heat energy is transferred
is one aspect of why the radiator feels cold. We see how the movement of energy
not only has a magnitude but also a direction.
Such statements concerning the direction of heat transfer are
sometimes called the minus-oneth law of thermodynamics, which
sounds rather daunting. In fact, the word 'thermodynamics' here
may be taken apart piecemeal to translate it into everyday English.
First the simple bit: 'dynamic' comes from the Greek word
dunamikos, which means movement. We obtain the conventional
English word 'dynamic' from the same root; and a cyclist's
'dynamo' generates electrical energy from the spinning of a bicycle wheel, i.e. from a
moving object. Secondly, thermo is another commonly encountered Greek root, and
means energy or temperature. We encounter the root thermo incorporated into such
everyday words as 'thermometer', 'thermal' and 'thermos flask'. A 'thermodynamic'
property, therefore, relates to events or processes in which there are 'changes in heat
or energy'.
The 'minus-oneth law
of thermodynamics'
says, 'heat always
travels from hot to
cold'.
INTRODUCTION TO PHYSICAL CHEMISTRY
Aside
We need to explain the bizarre name of this law, which is really an accident of history.
Soon after the first law of thermodynamics was postulated in the mid nineteenth century,
it was realized how the law presupposed a more elementary law, which we now call
the zeroth law (see below). We call it the 'zeroth' because zero comes before one. But
scientists soon realized how even the zeroth law was too advanced, since it presupposed a
yet more elementary law, which explains why the minus-oneth law had to be formulated.
How does a thermometer work?
Thermal equilibrium and the zeroth law of thermodynamics
The word 'thermometer'
has two roots: meter
denotes a device to
measure something,
and thermo means
'energy' or 'temper-
ature'. Thus, a 'ther-
mometer' is a device for
measuring energy as
a function of tempera-
ture.
A fever is often the first visible sign of someone developing an
illness. The body's temperature rises - sometimes dramatically -
above its preferred value of 37 °C. As a good generalization, the
temperature is hotter when the fever is worse, so it is wise to
monitor the temperature of the sick person and thereby check the
progress of the illness. A thermometer is the ideal instrument for
this purpose.
When measuring a temperature with a thermometer, we place
the mercury-containing end into the patient's mouth or armpit and
allow the reading to settle. The mercury is encased within a thin-
walled glass tube, which itself is placed in contact with the patient.
A 'reading' is possible because the mercury expands with increas-
ing temperature: we take the length / of the mercury in the tube to be an accurate
function of its temperature T. We read the patient's temperature from the thermometer
scale only when the length of the mercury has stopped changing.
But how does the thermometer work in a thermodynamic sense, since at no time
can the toxic mercury be allowed to touch the patient?
Consider the flow of heat: heat energy first flows from the patient
to the glass, and thence flows through the glass into the mercury.
Only when all three - mercury, glass and patient - are at the same
temperature can the thermometer reading become steady. We say
we have thermal equilibrium when these three have the same tem-
perature; see Figure 1.3.
Although in some respects a trivial example, a thermometer helps us see a profound
truth: only when both (i) the mercury and the glass, and (ii) the glass and the patient
are at thermal equilibrium can the patient and the mercury truly be said to be at the
same temperature. By this means, we have measured the temperature of the patient by
Bodies together at the
same temperature are
said to be in 'thermal
equilibrium'.
THE PRACTICE OF THERMODYNAMIC MEASUREMENT
Patient's tongue
Mercury-in-glass thermometer
Figure 1.3 The zeroth law states, 'Imagine three bodies, A, B and C. If A and B are in thermal
equilibrium, and B and C are in thermal equilibrium, then A and C are also in thermal equilibrium'
(see inset). A medic would rephrase the law, 'If mercury is in thermal equilibrium with the glass
of a thermometer, and the glass of a thermometer is in thermal equilibrium with a patient, then the
mercury and the patient are also in thermal equilibrium'
utilizing a temperature-dependent property of the liquid metal inside the thermometer,
yet at no time do we need to expose the patient to the toxic mercury.
We begin to understand the power of thermodynamics when we
realize how often this situation arises: in effect, we have made
an indirect measurement - a frequent occurrence - so we need to
formulate another law of thermodynamics, which we call the zeroth
law. Imagine three bodies, A, B and C. If A and B are in thermal
equilibrium, and B and C are in thermal equilibrium, then A and
C are also in thermal equilibrium.
While sounding overly technical, we have in fact employed the
zeroth law with the example of a thermometer. Let us rephrase
the definition of the zeroth law and say, 'If mercury is in thermal
equilibrium with the glass of a thermometer, and the glass of a
thermometer is in thermal equilibrium with a patient, then the mercury and the patient
are also in thermal equilibrium'. A medic could not easily determine the temperature
of a patient without this, the zeroth law.
From now on we will assume the zeroth law is obeyed each time we use the phrase
'thermal equilibrium'.
The zeroth law of ther-
modynamics says:
imagine three bodies,
A, BandC. If Aand Bare
in thermal equilibrium,
and B and C are also
in thermal equilibrium,
then A and C will be in
thermal equilibrium.
1.2 The practice of thermodynamic
measurement
What is temperature?
Scientific measurement
Although the answer to the simple question 'what is temperature?' seems obvious at
first sight, it is surprisingly difficult to answer to everyone's satisfaction. In fact, it is
10
INTRODUCTION TO PHYSICAL CHEMISTRY
'Corollary' means a
deduction following on
from another, related,
fact or series of facts.
The word 'thermo-
chemistry' has two
roots: thermo, mean-
ing 'temperature or
energy', and chem-
istry, the science of
the combination of
chemicals. We see
how 'thermochemistry'
studies the energy and
temperature changes
accompanying chemi-
cal changes.
generally easier to state the corollary, 'a body has a higher temper-
ature if it has more energy, and a lower temperature if it has less
energy' .
We have been rather glib so far when using words such as
'heat' and 'temperature', and will be more careful in future. Heat
is merely one way by which we experience energy. Everything contains energy in
various amounts, although the exact quantity of the energy is not only unknown
but unknowable.
Much of the time, we, as physical chemists, will be thinking
about energy and the way energetic changes accompany chemical
changes (i.e. atoms, ions or whole groups of atoms combine, or
add, or are being lost, from molecules). While the total energy
cannot be known, we can readily determine the changes that occur
in tandem with chemical changes. We sometimes give the name
thermochemistry to this aspect of physical chemistry.
In practice, the concept of temperature is most useful when deter-
mining whether two bodies are in thermal equilibrium. Firstly, we
need to appreciate how these equilibrium processes are always
dynamic, which, stated another way, indicates that a body simulta-
neously emits and absorbs energy, with these respective amounts
of energy being equal and opposite. Furthermore, if two bodies
participate in a thermal equilibrium then we say that the energy
emitted by the first body is absorbed by the second; and the first
body also absorbs a similar amount of energy to that emitted by
the second body.
Temperature is most conveniently visualized in terms of the
senses: we say something is hotter or is colder. The first ther-
mometer for studying changes in temperature was devised in 1631
by the Frenchman Jean Rey, and comprised a length of water in a
glass tube, much like our current-day mercury-in-glass thermometers but on a much
bigger scale. The controlled variable in this thermometer was temperature T, and the
observed variable was the length I of the water in the glass tube.
Rey's thermometer was not particularly effective because the density of water is so
low, meaning that the volume of the tube had to be large. And the tube size caused
an additional problem. While the water expanded with temperature (as required for
the thermometer to be effective), so did the glass encapsulating it. In consequence
of both water and glass expanding, although the water expanded in a straightforward
way with increasing temperature, the visible magnitude of the expansion was not in
direct proportion to the temperature rise.
Although we could suggest that a relationship existed between
the length / and the temperature T (saying one is a function of the
other), we could not straightforwardly ascertain the exact nature
of the function. In an ideal thermometer, we write the mathemat-
ical relationship, / = f(T). Because Rey's thermometer contained
A body in 'dynamic equi-
librium' with another
exchanges energy with
it, yet without any net
change.
Scientists use the word
'ideal' to mean obeying
the laws of science.
THE PRACTICE OF THERMODYNAMIC MEASUREMENT
11
water, Rey was not able to observe a linear dependence of I on T for
so he could not write I = aT + b (where a and b are constants).
For a more dense liquid, such as mercury, the relationship bet-
ween I and T is linear - at least over a relatively narrow range
of temperatures - so a viable mercury-in-glass thermometer may
be constructed. But, because the temperature response is only lin-
ear over a narrow range of temperatures, we need to exercise
caution.
If we assume the existence of a linear response for such a ther-
mometer, then the thermometer is 'calibrated' by correlating the
readings of length / using the known properties of the standard, as
follows. First, the thermometer is placed in a trough of pure ice at
its melting temperature, and the end of the mercury bead marked
as °C. The same thermometer is then placed in water at its boil-
ing point and the end of the mercury bead marked as 100 °C. The
physical distance between these two extremes is subdivided into
100 equal portions, each representing a temperature increment of
1 °C. This centigrade scale is satisfactory for most purposes. (The
same scale is sometimes called Celsius after a Swedish physicist
who championed its use.)
This formulation of the centigrade scale presupposed a linear
relationship between length / and temperature T (i.e. the straight
line (a) on the graph in Figure 1.4), but we must be aware the
relationship might only be approximately linear (e.g. the curved
line (b) on the graph in Figure 1.4). The straight and the curved
lines only agree at the two temperatures 0°C and 100 °C merely
because they were defined that way.
his thermometer,
'Narrow
' in
this
:ase
means
50-
•70°C
at
most.
To 'calibrate' an instru-
ment such as a ther-
mometer, we correlate
a physico-chemical
property (such as the
length / of the mercury)
using the temperature-
dependent properties
of a known standard.
The 'centigrade' scale
was first proposed in
1694 by Renaldi. Centi
is a Latin prefix mean-
ing 'hundred'.
2 E
— o
2 E
-§ $
> >-
o|
(a) Ideal response
(b) More realistic response
0°C 100°C
Controlled variable (temperature, T)
Figure 1.4 In using a thermometer, we assume the existence of a linear response between the
length / of the mercury and the controlled variable temperature T. Trace (a) shows such a rela-
tionship, and trace (b) shows a more likely situation, in which there is a close approximation to a
linear relationship between length / and temperature T
12 INTRODUCTION TO PHYSICAL CHEMISTRY
This last paragraph inevitably leads to the questions, 'So how do we know what
the exact temperature is?' and 'How do I know if my thermometer follows profile
(a) or profile (b) in Figure 1.4?' Usually, we do not know the answer. If we had a
single thermometer whose temperature was always accurate then we could use it as a
primary standard, and would simply prepare a calibrated thermometer against which
all others are calibrated.
But there are no ideal (perfect) thermometers in the real world. In practice, we
generally experiment a bit until we find a thermometer for which a property X is as
close to being a linear function of temperature as possible, and call it a standard ther-
mometer (or 'ideal thermometer'). We then calibrate other thermometers in relation to
this, the standard. There are several good approximations to a standard thermometers
available today: the temperature-dependent (observed) variable in a gas thermometer
is the volume of a gas V. Provided the pressure of the gas is quite low (say, one-
hundredth of atmospheric pressure, i.e. 100 Pa) then the volume V and temperature
T do indeed follow a fairly good linear relationship.
A second, popular, standard is the platinum-resistance thermometer. Here, the elec-
trical resistance R of a long wire of platinum increases with increased temperature,
again with an essentially linear relationship.
Worked Example 1.1 A platinum resistance thermometer has a resistance R of 3.0 x
10" 4 12 at 0°C and 9.0 x 10" 4 12 at 100 °C. What is the temperature if the resistance R
is measured and found to be 4.3 x 10~ 4 12?
We first work out the exact relationship between resistance R and temperature T. We
must assume a linear relationship between the two to do so.
The change per degree centigrade is obtained as 'net change in
These discussions are
expressed in terms of
centigrade, although
absolute temperatures
are often employed -
see next section.
resistance 4- net change in temperature'. The resistance R increases
by 6.0 x 10~ 4 12 while the temperature is increased over the 100 °C
range; therefore, the increase in resistance per degree centigrade is
given by the expression
6.0 x 10~ 4 12 , ,
R per°C = = 6x 10" 6 12 "C" 1
100 °C
Next, we determine by how much the resistance has increased in going to the new
(as yet unknown) temperature. We see how the resistance increases by an amount (4.3 —
3.0) x 10~ 4 12 = 1.3 x 10- 4 12.
The increase in temperature is then the rise in resistance divided by the change in
resistance increase per degree centigrade.
We obtain
1.3 x 10" 4 12
6 x lO" 6 ^^- 1
so the new temperature is 21.7 °C.
THE PRACTICE OF THERMODYNAMIC MEASUREMENT 13
We should note, before proceeding, firstly how the units of £2 on both top and
bottom of this fraction cancel; and secondly, how °C _1 is in the denominator of the
fraction. As a consequence of it being on the bottom of the fraction, it is inverted
and so becomes °C. In summary, we see how a simple analysis of the units in this
sum automatically allows the eventual answer to be expressed in terms of °C. We are
therefore delighted, because the answer we want is a temperature, and the units tell
us it is indeed a temperature.
Aside
This manipulation of units is sometimes called dimensional analysis. Strictly speaking,
though, dimensional analysis is independent of the units used. For example, the units
of speed may be in metres per second, miles per hour, etc., but the dimensions of speed
are always a length [L] divided by a time [T]:
[speed] = [L] - [T]
Dimensional analysis is useful in two respects. (1) It can be used to determine the
units of a variable in an equation. (2) Using the normal rules of algebra, it can be used
to determine whether an equation is dimensionally correct, i.e. the units should balance
on either side of the equation. All equations in any science discipline are dimensionally
correct or they are wrong!
A related concept to dimensional analysis is quantity calculus, a method we find
particularly useful when it comes to setting out table header rows and graph axes.
Quantity calculus is the handling of physical quantities and their units using the normal
rules of algebra. A physical quantity is defined by a numerical value and a unit:
physical quantity = number x unit
e.g.
AH = 40.7 U mol" 1
which rearranges to
Atf/kJ mol -1 = 40.7
SAQ 1.1 A temperature is measured with the same platinum-resistance
thermometer used in Worked Example 1.1, and a resistance R = 11.4 x
10~ 4 Q, determined. What is the temperature?
14
INTRODUCTION TO PHYSICAL CHEMISTRY
The word 'philosoph-
ical' comes from the
Greek words philos
meaning 'love' and
sophia meaning 'wis-
dom'. Philosophy is
therefore the love of
wisdom. This same
usage of 'wisdom' is
seen with the initials
PhD, which means a
'philosophy doctorate'.
Some people might argue that none of the discussion above actu-
ally answers the philosophical question, 'What is temperature?'
We will never come to a completely satisfactory answer; but we
can suppose a body has a higher temperature if it contains more
energy, and that it has a lower temperature if it has less energy.
More importantly, a body will show a rise in temperature if its
energy content rises, and it will show a lower temperature if its
energy content drops. This is why we sit in front of a fire: we want
to absorb energy, which we experience as a higher temperature.
How long is a p/e<
tri
This definition of 'more
energy means hotter'
needs to be handled
with care: consider
two identical weights
at the same tempera-
ture. The higher weight
has a greater potential
energy.
The SI unit of length
A common problem in Anglo Saxon England, as well as much
of contemporary Europe, was the way cloth merchants could so
easily cheat the common people. At a market, it was all too easy
to ask for a yard of cloth, to see it measured against the merchant's
yardstick, and pay for the cloth only to get home to learn just how
short the merchant's stick was. Paying for 10 yards and coming
home with only 9 yards was common, it seems; and the problem
was not restricted to just cloth, but also to leather and timber.
According to legend, the far-sighted English King Edgar (ad 959-975) solved the
problem of how to stop such cheating by standardizing the length. He took 100 foot
soldiers and measured the length of the right foot of each, one after the other, as
they stood in line along the floor of his threshing hall. This overall length was then
subdivided into 100 equal parts to yield the standard length, the foot. The foot is still
commonly employed as a unit of length in Britain to this day. Three of these feet
made up 1 yard. The king was said to keep in his treasury a rod of gold measuring
exactly 1 yard in length. This is one theory of how the phrase 'yardstick' originated.
Any merchant accused of cheating was required to bring his yardstick and to compare
its length against that of the king. Therefore, a merchant whose stick was shorter was
a cheat and paid the consequences. A merchant whose stick was longer was an idiot.
While feet and yards are still used in Britain and other countries,
the usual length is now the metre. At the time of the French Rev-
olution in the 18th century and soon after, the French Academy
of Sciences sought to systemize the measurement of all scientific
quantities. This work led eventually to the concept of the Systeme
Internationale, or SI for short. Within this system, all units and
definitions are self-consistent. The SI unit of length is the metre.
The original metre rule was kept in the International Bureau
of Weights and Measures in Sevres, near Paris, and was a rod of
SI units are self-
consistent, with all
units being defined in
terms a basis of seven
fundamental units. The
SI unit of length / is the
metre (m).
THE PRACTICE OF THERMODYNAMIC MEASUREMENT
15
platinum -iridium alloy on which two deep marks were scratched
1 m apart. It was used in exactly the same way as King Edgar's
yardstick 10 centuries earlier.
Unfortunately, platinum -iridium alloy was a poor choice, for it
has the unusual property of shrinking (albeit microscopically) with
time. This SI metre rule is now about 0.3 per cent too short. King
Edgar's yardstick, being made of gold, would still be the same
length today as when it was made, but gold is too ductile, and
could have been stretched, bent or re-scored.
In 1960, the SI unit of length was redefined. While keeping
the metre as the unit of length, it is now defined as 1 650763.73
wavelengths of the light emitted in vacuo by krypton-86. This is a
sensible standard, because it can be reproduced in any laboratory
in the world.
'Ductile' means the
ability of a metal to
be drawn to form a
wire, or to be worked.
Ductile is the opposite
of 'brittle'.
In vacuo is Latin for
'in a vacuum'. Many
properties are mea-
sured in a vacuum to
avoid the complication
of interference effects.
How fast is "greased lightning'?
Other SI standards
In comic books of the 1950s, one of the favourite phrases of super-heroes such as
Superman was 'greased lightning!' The idea is one of extreme speed. The lightning we
see, greased or otherwise, is a form of light and travels very, very fast. For example,
it travels through a vacuum at 3 x 10 8 m s _1 , which we denote with the symbol c.
But while the speed c is constant, the actual speed of light may not be: in fact, it
alters very slightly depending on the medium through which it travels. We see how
a definition of time involving the speed of light is inherently risky, explaining why
we now choose to define time in terms of the duration (or fractions and multiples
thereof) between static events. And by 'static' we mean unchanging.
SI 'base units'
Time is one of the so-called 'base units' within the SI system, and so is length.
Whereas volume can be expressed in terms of a length (for example, a cube has a
volume P and side of area I 2 ), we cannot define length in terms of
something simpler. Similarly, whereas a velocity is a length per unit
time, we cannot express time in terms of something simpler. In fact,
just as compounds are made up of elements, so all scientific units
are made up from seven base units: length, time, mass, temperature,
current, amount of material and luminous intensity.
Table 1.1 summarizes the seven base (or 'fundamental') SI phys-
ical quantities and their units. The last unit, luminous intensity, will
not require our attention any further.
The SI unit of 'time' t is the second. The second was originally
defined as 1/86 400th part of a mean solar day. This definition is
There are seven base
SI units: length, time,
mass, temperature,
current, luminous
intensity and amount
of material.
The SI unit of 'time' r is
the second (s).
16 INTRODUCTION TO PHYSICAL CHEMISTRY
Table 1.1 The seven fundamental SI physical quantities and their units
Physical quantity
Symbol 3
SI unit
Abbreviation
Length
/
metre
m
Mass
m
kilogram
kg
Time
t
second
s
Electrical current
I
ampere
A
Thermodynamic temperature
T
kelvin
K
Amount of substance
n
mole
mol
Luminous intensity
/v
candela
cd
a Notice how the abbreviation for each quantity, being a variable, is always italicized,
whereas the abbreviation for the unit, which is not a variable, is printed with an upright
typeface. None of these unit names starts with a capital.
again quite sensible because it can be reproduced in any laboratory in the world.
While slight changes in the length of a solar year do occur, the word 'mean' in our
definition obviates any need to consider them. Nevertheless, it was felt necessary
to redefine the second; so, in the 1960s, the second was redefined as 9 192631 770
periods of the radiation corresponding to the transition between two of the hyperfine
levels in the ground state of the caesium- 133 atom. Without discussion, we note how
the heart of a so-called 'atomic clock' contains some caesium-133.
In a similar way, the Systeme Internationale has 'defined' other
The SI unit of 'tempera
ture' T is the kelvin (K)
common physicochemical variables. The SI unit of 'temperature'
T is the kelvin. We define the kelvin as 1/273. 16th part of the
thermodynamic temperature difference between absolute zero (see
Section 1.4) and the triple point of water, i.e. the temperature at which liquid water
is at equilibrium with solid water (ice) and gaseous water (steam) provided that the
pressure is 610 Pa.
The SI unit of 'current' / is the ampere (A). An ampere was
The SI unit of 'current'
I is the ampere (A).
first defined as the current flowing when a charge of 1 C (coulomb)
passed per second through a perfect (i.e. resistance-free) conductor.
The SI definition is more rigorous: 'the ampere is that constant
current which, if maintained in two parallel conductors (each of negligible resistance)
and placed in vacuo 1 m apart, produces a force between of exactly 2 x 10~ 7 N per
metre of length'. We will not employ this latter definition.
The SI unit of the 'amount of substance' n is the mole. Curi-
The SI unit of 'amount
of substance' n is the
mole (mol).
ously, the SI General Conference on Weights and Measures only
decided in 1971 to incorporate the mole into its basic set of funda-
mental parameters, thereby filling an embarrassing loophole. The
mole is the amount of substance in a system that contains as many
elementary entities as does 0.012 kg (12 g) of carbon- 12. The amount of substance
must be stated in terms of the elementary entities chosen, be they photons, electrons,
protons, atoms, ions or molecules.
The number of elementary entities in 1 mol is an experimentally determined quan-
tity, and is called the 'Avogadro constant' L, which has the value 6.022 x 10 23 mol -1 .
The Avogadro constant is also (incorrectly) called the 'Avogadro number'. It is
THE PRACTICE OF THERMODYNAMIC MEASUREMENT 17
Table 1.2 Several of the more common units that are not members of the Systeme
Internationale
Quantity
Non-SI unit
Abbreviation
Conversion from non-SI to SI
Energy
calorie
cal
1 cal = 4.184 J
Length
angstrom
A
1 A= 10 -10 m
Pressure
atmosphere
atm
1 atm = 101 325 Pa
Pressure
bar
bar
1 bar = 10 5 Pa
Volume
litre
dm 3
1 dm 3 = 1(T 3 m 3
increasingly common to see the Avogadro constant given a different symbol than
L. The most popular alternative symbol at present seems to be N&.
Non-SI units
It is important to be consistent with units when we start a calculation. An enormously
expensive spacecraft crashed on the surface of the planet Mars in 1999 because a
distance was calculated by a NASA scientist in terms of inches rather than centimetres.
Several non-SI units persist in modern usage, the most common being listed in
Table 1.2. A calculation performed wholly in terms of SI units will be self-consistent.
Provided we know a suitable way to interchange between the SI and non-SI units,
we can still employ our old non-SI favourites.
Aside
In addition to the thermodynamic temperature T there is also the Celsius temperature
f, defined as
t = T-T
where T = 273.15 K.
Sometimes, to avoid confusion with the use of t as the symbol for time, the Greek
symbol 9 (theta) is substituted for the Celsius temperature t instead.
Throughout this book we adopt T to mean temperature. The context will make clear
whether T is required to be in degrees Celsius or kelvin. Beware, though, that most
formulae require the use of temperature in kelvin.
Why is the SI unit of mass the kilogram?
Multiples and the SI unit of mass m
The definition of mass in the Systeme Internationale scheme departs from the stated
aim of formulating a rigorous, self-consistent set of standards. The SI unit of 'mass'
18
INTRODUCTION TO PHYSICAL CHEMISTRY
Table 1.3 Selection of a few physicochemical
parameters that comprise combinations of the
seven SI fundamental quantities
Quantity
Symbol
SI units
Acceleration
a
-2
ms
Area
A
m 2
Density
P
kgm~ 3
Force
F
kg m s~ 2
Pressure
P
kg ITT 1 s~ 2
Velocity
V
ms~'
Volume
V
m 3
The SI unit of 'mass' m
is the kilogram (kg).
In the SI system, 1 g
is defined as the mass
of 5.02 x 10 22 atoms of
carbon-12. This num-
ber comes from L/12,
where L is the Avo-
gadro number.
m is the 'kilogram'. Similar to the metre, the original SI standard
of mass was a block of platinum metal in Sevres, near Paris, which
weighted exactly 1 kg. The current SI definition is more compli-
cated: because 12.000 g in the SI system represents exactly 1 mol
of carbon-12, then 1 g is one-twelfth of a mole of carbon-12.
The problem with the SI base unit being a kilogram is the 'kilo'
part. The philosophical idea behind the SI system says any param-
eter (physical, chemical, mechanical, etc.) can be derived from
a suitable combination of the others. For example, the SI unit of
velocity is metres per second (m s _1 ), which is made up of the two
SI fundamental units of length (the metre) and time (the second).
A few of these combinations are cited in Table 1.3.
Why is 'the material of action so variable'?
Writing variables and phrases
The classical author Epictetus (ca 50-ca 138 ad) once said, 'The materials of action
are variable, but the use we make of them should be constant'. How wise.
When we build a house, we only require a certain number of
building materials: say, bricks, tubes and window panes. The quan-
tity surveyor in charge of the building project decides which materi-
als are needed, and writes a quantity beside each on his order form:
10000 bricks, 20 window panes, etc. Similarly, when we have a
velocity, we have the units of 'm' and 's -1 ', and then quantify
it, saying something like, 'The man ran fast, covering a distance
of 10 metres per second'. By this means, any parameter is defined
both qualitatively (in terms of its units) and quantitatively (in terms
of a number). With symbols, we would write v = 10 ms -1 .
We give the name
'compound unit' to
several units written
together. We leave a
space between each
constituent unit when
we write such a com-
pound unit.
THE PRACTICE OF THERMODYNAMIC MEASUREMENT
19
A variable (mass, length, velocity, etc.) is written in a standard format, according
to Equation (1.2):
Variable or physicochemical quantity = number x units
(1.2)
We sometimes call it a 'phrase'. Because some numbers are huge
and others tiny, the SI system allows us a simple and convenient
shorthand. We do not need to write out all the zeros, saying the
velocity of light c is 300000000 ms" 1 : we can write it as c = 3 x
10 8 ms -1 or as 0.3 Gms -1 , where the capital 'G' is a shorthand
for 'giga', or 1 000000000. The symbol G (for giga) in this context
is called a 'factor'. In effect, we are saying 300000000 ms -1 =
0.3 Gms -1 . The standard factors are listed on pp. xxviii-xxxi.
Most people find that writing 300000000 ms -1 is a bit long
winded. Some people do not like writing simple factors such as G
for giga, and prefer so-called scientific notation. In this style, we
write a number followed by a factor expressed as ten raised to an
appropriate power. The number above would be 3.0 x 10 8 ms -1 .
Worked Example 1.2 Identify the variable, number, factor and unit
in the phrase, 'energy = 12 kJmol -1 '.
'Giga' comes from the
Latin gigas, meaning
'giant' or 'huge'. We
also get the every-
day words 'giant' and
'gigantic' from this
root.
In physical chemistry,
a 'factor' is a number
by which we multiply
the numerical value of
a variable. Factors are
usually employed with
a shorthand notation.
Energy
12
J mol
Variable Number Factor Compound
unit
Reasoning,
Variable - in simple mathematical 'phrases' such as this, we almost always write
the variable on the left. A variable is a quantity whose value can be altered.
Number - the easy part! It will be made up of numbers 1, 2, 3, . . . , 0.
Factor - if we need a factor, it will always be written between the number and
the units (compound or single). A comprehensive list of the simple factors is
given on pp. xxviii-xxxi.
Units - the units are always written on the right of a phrase such as this. There
are two units here, joules (J) and moles (as 'mol~ ', in this case). We should
leave a space between them.
A factor is simply shorthand, and is dispensable. We could have dispensed with the
factor and written the number differently, saying energy = 12000 Jmol~ . This same
energy in scientific notation would be 12 x 10 3 Jmol~ . But units are not dispensable.
20 INTRODUCTION TO PHYSICAL CHEMISTRY
SAQ 1.2 Identify the variable, number, factor and unit in the phrase,
'length = 3.2 km'.
1.3 Properties of gases and the gas laws
Why do we see eddy patterns above a radiator?
The effects of temperature on density
The air around a hot radiator soon acquires heat. We explain this observation from
the 'minus oneth law of thermodynamics' (see Section 1.1), since heat travels from
hot to cold.
The density of a gas depends quite strongly on its temperature, so hot air has
a smaller density than does cold air; colder air is more dense than hot air. From
everyday experience, we know that something is dense if it tries to drop, which is
why a stone drops to the bottom of a pond and a coin sinks to the bottom of a pan of
water. This relative motion occurs because both the stone and the coin have higher
densities than does water, so they drop. Similarly, we are more dense than air and
will drop if we fall off a roof.
Just like the coin in water, cold air sinks because it is denser than warmer air.
We sometimes see this situation stated as warm air 'displaces' the cold air, which
subsequently takes its place. Alternatively, we say 'warm air rises', which explains
why we place our clothes above a radiator to dry them, rather than below it.
Light entering the room above the radiator passes through these pockets of warm
air as they rise through colder air, and therefore passes through regions of different
density. The rays of light bend in transit as they pass from region to region, much in
the same way as light twists when it passes through a glass of water. We say the light
is refracted. The eye responds to light, and interprets these refractions and twists as
different intensities.
So we see swirling eddy (or 'convective') patterns above a radiator because the
density of air is a function of temperature. If all the air had the same temperature, then
no such difference in density would exist, and hence we would see no refraction and
no eddy currents - which is the case in the summer when the radiator is switched off.
Then again, we can sometimes see a 'heat haze' above a hot road, which is caused
by exactly the same phenomenon.
Why does a hot-air balloon float?
The effect of temperature on gas volume
A hot-air balloon is one of the more graceful sights of summer. A vast floating
ball, powered only by a small propane burner, seems to defy gravity as it floats
effortlessly above the ground. But what is it causing the balloon to fly, despite its
considerable weight?
PROPERTIES OF GASES AND THE GAS LAWS
21
The small burner at the heart of the balloon heats the air within
the canvas hood of the balloon. The densities of all materials -
solid, liquid or gas - alter with temperature. Almost universally,
we find the density p increases with cooling. Density p is defined
as the ratio of mass m to volume V, according to
'Density' p is defined as
mass per unit volume.
density p
mass, m
volume, V
(1.3)
It is not reasonable to suppose the mass m of a gas changes by heating or cooling it
(in the absence of chemical reactions, that is), so the changes in p caused by heating
must have been caused by changes in volume V. On the other hand, if the volume
were to decrease on heating, then the density would increase.
So the reason why the balloon floats is because the air inside its voluminous hood
has a lower density than the air outside. The exterior air, therefore, sinks lower
than the less-dense air inside. And the sinking of the cold air and the rising of the
warm air is effectively the same thing: it is movement of the one relative to the
other, so the balloon floats above the ground. Conversely, the balloon descends back
to earth when the air it contains cools to the same temperature as the air outside
the hood.
How was the absolute zero of temperature
determined?
Charles's law
J. A. C. Charles (1746-1823) was an aristocratic amateur scientist of the 18th century
He already knew that the volume V of a gas increased with increasing temperature T
and was determined to find a relationship between these variables.
The law that now bears his name can be stated as, 'The ratio of
volume and temperature for a fixed mass of gas remains constant',
provided the external pressure is not altered.
Stated mathematically, Charles demonstrated
V
~T
constant
(1.4)
According to 'Charles's
law', a linear relation-
ship exists between
V and 7" (at constant
pressure p).
where the value of the constant depends on both the amount and
the identity of the gas. It also depends on the pressure, so the data
are obtained at constant pressure p.
This is one form of 'Charles's law'. (Charles's law is also called
'Gay-Lussac's law'.) Alternatively, we could have multiplied both
sides of Equation (1.4) by T, and rewritten it as
constant x T
(1.5)
A 'straight line' will
always have an equa-
tion of the type y =
mx + c, where m is the
gradient and c is the
intercept on the /-axis
(i.e. when the value of
x = 0).
22
INTRODUCTION TO PHYSICAL CHEMISTRY
Lord Kelvin (1824-1907) was a great thermodynamicist whom we shall meet quite
often in these pages. He noticed how the relationship in Equation (1.5) resembles the
equation of a straight line, i.e. takes the form
y = mx + c
observed gradient controlled constant
(1.6)
variable
variable
except without an intercept, i.e. c = 0. Kelvin obtained good-quality data for the
volume of a variety of gases as a function of temperature, and plotted graphs of
volume V (as y) against temperature T (as x) for each; curiously, however, he was
unable to draw a graph with a zero intercept for any of them.
Kelvin then replotted his data, this time extrapolating each graph
till the volume of the gas was zero, which he found to occur at a
temperature of —273.15 °C; see Figure 1.5. He then devised a new
temperature scale in which this, the coldest of temperatures, was
the zero. He called it absolute zero, and each subsequent degree
was equal to 1 °C. This new scale of temperature is now called
the thermodynamic (or absolute) scale of temperature, and is also
sometimes called the Kelvin scale.
The relationship between temperatures T on the centigrade and
the absolute temperature scales is given by
Note: degrees in the
Kelvin scale do not
have the degree sym-
bol. The units have a
capital K, but the noun
'kelvin' has a small
letter.
T in °C= T in K- 273.15
(1.7)
Equation (1.7) demonstrates how 1 °C = 1 K.
0.04
0.03
0.02
5
0.01
200 300
Temperature 77K
Figure 1.5 A graph of the volume V of a gas (as y) against temperature T (as x) is linear.
Extrapolating the gas's volume suggests its volume will be zero if the temperature is — 273.15°C
(which we call K, or absolute zero)
PROPERTIES OF GASES AND THE GAS LAWS
23
SAQ 1.3 What is the temperature T expressed in kelvin
if the temperature is 30 °C?
SAQ 1.4 What is the centigrade temperature correspon-
ding to 287.2 K?
SAQ 1.5 The data in the table below relate to gaseous
helium. Demonstrate the linear relationship between the
volume V and the temperature T.
We divide each tem-
perature, both kelvin
and centigrade, by
its respective unit
to obtain a number,
rather than the tem-
perature.
Temperature 77K
Volume V/m 3
280
0.023
300
0.025
320
0.027
340
0.028
360
0.030
380
0.032
400
0.033
420
0.035
440
0.037
Charles's law is often expressed in a slightly different form than
Equation (1.4), as
Tx T 2
(1.8)
which is generally regarded as superior to Equation (1.4) because
we do not need to know the value of the constant.
Equation (1.8) is also preferred in situations where the volume of
a fixed amount of gas changes in response to temperature changes
(but at constant pressure). The subscripts refer to the two situations;
so, for example, the volume at temperature T\ is V\ and the volume
at temperature T 2 is V 2 .
SAQ 1.6 The gas inside a balloon has a volume l/i of
1 dm 3 at 298 K. It is warmed to 350 K. What is the vol-
ume following warming? Assume the pressure remained
constant.
Note how we write
the controlled vari-
able along the top row
of a table, with the
observed following. (If
the table is vertical,
we write the controlled
variable on the far left.)
The subscripts written
to the right of a variable
are called 'descriptors'.
They are always written
as a subscript, because
a superscripted number
means a power, i.e. V 2
means V x V.
Why pressurize the contents of a gas canister?
The effect of pressure on gas volume: Boyle's law
It is easy to buy canisters of gas of many sizes, e.g. as fuel when we wish to camp
in the country, or for a portable welding kit. The gas will be w-butane if the gas is
for heating purposes, but might be oxygen or acetylene if the gas is to achieve the
higher temperatures needed for welding.
Typically, the components within the can are gaseous at most temperatures. The
typical volume of an aerosol can is about 0.3 dm 3 (3 x 10~ 4 m 3 ), so it could contain
very little gas if stored at normal pressure. But if we purchase a canister of gas and
release its entire contents at once, the gas would occupy a volume similar that of
24
INTRODUCTION TO PHYSICAL CHEMISTRY
Care: a small p indi-
cates pressure, yet a
big P is the symbol for
the element phospho-
rus. Similarly, a big V
indicates volume and a
small v is the symbol
for velocity.
an entire living room. To ensure the (small) can contains this (large)
amount of gas, we pressurize it to increase its capacity. We see
how volume and pressure are interrelated in a reciprocal way: the
volume decreases as the pressure increases.
Robert Boyle was the first to formulate a relationship between
p and V. Boyle was a contemporary of the greatest scientist the
world has ever seen, the 17th-century physicist Sir Isaac Newton.
Boyle's law was discovered in 1660, and states
pV = constant
(1.9)
where the numerical value of the constant on the right-hand side of the equation
depends on both the identity and amount of the gas, as well as its temperature T.
Figure 1.6 shows a graph of pressure p (as y) against volume V
(as x) for 1 mol of neon gas. There are several curves, each repre-
senting data obtained at a different temperature. The temperature
per curve was constant, so we call each curve an isotherm. The
word isotherm has two Greek roots: iso means 'same' and thermo
means temperature or energy. An isotherm therefore means at the
same energy.
The actual shape of the curves in Figure 1.6 are those of recipro-
cals. We can prove this mathematical form if we divide both sides
of Equation (1.9) by V, which yields
An 'isotherm' is a line
on a graph represent-
ing values of a variable
obtained at constant
temperature.
'Reciprocal' means to
turn a fraction upside
down. X can be thought
of as 'Xh- 1', so its
reciprocal is 1/X (i.e.
1- X).
1
p = — x constant
V
(1.10)
Figure 1.7 shows a graph of volume p (as y) against 1/volume
V (as x), and has been constructed with the same data as used for
100 000
80 000
"S. 60 000
CD
13
W
g 40 000
□I
20 000
Volume V/m 3
Figure 1.6 Graph of pressure p (as y) against volume V (as x) for 1 mol of an ideal gas as a
function of temperature: ( ) 200 K; ( ) 400 K; ( ) 600 K; ( ) 800 K
PROPERTIES OF GASES AND THE GAS LAWS
25
7.00E+05
1 .00E+05
0.00E+00
40 60
1/(l/7m 3 )
Figure 1.7 Graph of pressure p (as y) against reciprocal volume 1 -f- V (as x) for 1 mol of
an ideal gas as a function of temperature. The data are the same as those from Figure 1.6. The
temperatures are indicated. We need to appreciate how plotting the same data on a different set of
axes yields a linear graph, thereby allowing us to formulate a relationship between p and 1 -f- V
Figure 1.6. Each of the lines on the graph is now linear. Again, we find these data are
temperature dependent, so each has the gradient of the respective value of 'constant'.
At constant temperature T, an increase in pressure (so pi >
Pi) causes a decrease in volume (so V2 < ^i)- This observation
explains why the graph for the gas at the higher temperatures has
a smaller value for the constant.
An alternative way of writing Equation (1.9) is
P\V\ = P2V2
(1.11)
At constant temper-
ature T, an increase
in pressure (p 2 > Pi)
causes a decrease in
volume (V 2 < l/i).
SAQ 1.7 The usual choice of propellant within an aerosol of air freshener
is propane gas. What is the volume of propane following compression, if
1 dm 3 of gaseous propane is compressed from a pressure of 1 atm to a
pressure of 2.5 atm? Assume the temperature is kept constant during the
compression.
Why does thunder accompany lightning?
Effect of changing both temperature and pressure on gas
volume
Lightning is one of the most impressive and yet frightening manifestations of nature.
It reminds us just how powerful nature can be.
26 INTRODUCTION TO PHYSICAL CHEMISTRY
'Experiential' means
the way we notice
something exists fol-
lowing an experience
or sensation.
Lightning is quite a simple phenomenon. Just before a storm
breaks, perhaps following a period of hot, fine weather, we often
note how the air feels 'tense'. In fact, we are expressing an expe-
riential truth: the air contains a great number of ions - charged
particles. The existence of a large charge on the Earth is mirrored
by a large charge in the upper atmosphere. The only difference
between these two charges is that the Earth bears a positive charge and the atmosphere
bears a negative charge.
Accumulation of a charge difference between the Earth and the upper atmosphere
cannot proceed indefinitely. The charges must eventually equalize somehow: in prac-
tice, negative charge in the upper atmosphere passes through the air to neutralize
the positive charge on the Earth. The way we see this charge conducted between
the Earth and the sky is lightning: in effect, air is ionized to make it a conductor,
allowing electrons in the clouds and upper atmosphere to conduct through the air
to the Earth's surface. This movement of electrical charge is a current, which we
see as lightning. Incidentally, ionized air emits light, which explains why we see
lightning (see Chapter 9). Lightning comprises a massive amount of energy, so the
local air through which it conducts tends to heat up to as much as a few thousand
degrees centigrade.
And we have already seen how air expands when warmed, e.g. as described math-
ematically by Charles's law (Equation (1.6)). In fact, the air through which the
lightning passes increases in volume to an almost unbelievable extent because of
its rise in temperature. And the expansion is very rapid.
SAQ 1.8 Show, using the version of Charles's law in Equation (1.8), how
a rise in temperature from 330 K to 3300 K is accompanied by a tenfold
increase in volume.
We hear the sensation of sound when the ear drum is moved by compression waves
travelling through the air; we hear people because their speech is propagated by subtle
pressure changes in the surrounding air. In a similar way, the huge increase in air
volume is caused by huge changes in air pressure, itself manifested as sound: we hear
the thunder caused by the air expanding, itself in response to lightning.
And the reason why we see the lightning first and hear the thunder later is because
light travels faster than sound. The reason why thunder accompanies lightning, then,
is because pressure p, volume V and temperature T are interrelated.
How does a bubble-jet printer work?
The ideal-gas equation
A bubble-jet printer is one of the more useful and versatile inventions of the last
decade. The active component of the printer is the 'head' through which liquid ink
passes before striking the page. The head moves from side to side over the page. When
PROPERTIES OF GASES AND THE GAS LAWS
27
the 'head' is positioned above a part of the page to which an image is required, the
computer tells the head to eject a tiny bubble of ink. This jet of ink strikes the page
to leave an indelible image. We have printing.
The head is commonly about an inch wide, and consists of a row of hundreds of
tiny pores (or 'capillaries'), each connecting the ink reservoir (the cartridge) and the
page. The signals from the computer are different for each pore, allowing different
parts of the page to receive ink at different times. By this method, images or letters
are formed by the printer.
The pores are the really clever bit of the head. Half-way along each pore is a
minute heater surrounded by a small pocket of air. In front of the heater is a small
bubble of ink, and behind it is the circuitry of the printer, ultimately connecting the
heater to the computer. One such capillary is shown schematically in Figure 1.8.
Just before the computer instructs the printer to eject a bubble of ink, the heater
is activated, causing the air pocket to increase in temperature T at quite a rapid
rate. The temperature increase causes the air to expand to a greater volume V. This
greater volume increases the pressure p within the air pocket. The enhanced air
pressure p is sufficient to eject the ink bubble from the pore and onto the page. This
pressure-activated ejection is similar to spitting.
This ejection of ink from a bubble-jet printer ingeniously utilizes the interconnect-
edness of pressure p, volume V and temperature T. Experiments with simple gases
show how p, T and V are related by the relation
P V
= constant
(1.12)
which should remind us of both Boyle's law and Charles's law.
Inkjet nozzle
t> 5 us
t~ 10 us
f~20us
Figure 1.8 Schematic diagram of a capillary (one of hundreds) within the printing 'head' of a
bubble-jet printer. The resistor heats a small portion of solution, which boils thereby increasing the
pressure. Bubbles form within 5 (is of resistance heating; after 10 (jls the micro-bubbles coalesce
to force liquid from the aperture; and a bubble is ejected a further 10 p,s later. The ejected bubble
impinges on the paper moments afterwards to form a written image. Reproduced by permission of
Avecia
28 INTRODUCTION TO PHYSICAL CHEMISTRY
If there is exactly 1 mol of gas, the pressure is expressed in pascals (Pa), the
temperature is in kelvin and the volume is in cubic metres (both SI units), then the
value of the constant is 8.314 JK _1 mol -1 . We call it the gas constant and give it
the symbol R. (Some old books may call R the 'universal gas constant', 'molar gas
constant' or just 'the gas constant'. You will find a discussion about R on p. 54)
More generally, Equation (1.12) is rewritten as
Equation (1.13) tells us
the constant in Boyle's
law is y nRT' and the
(different) constant in
Charles's law is
V?R-p'.
P V = nRT (1.13)
where n is the number of moles of gas. Equation (1.13) is called
the ideal-gas equation (or, sometimes, in older books the 'universal
gas equation'). The word 'ideal' here usually suggests that the gas
in question obeys Equation (1.13).
Worked Example 1.3 What is the volume of 1 mol of gas at a room temperature of
25 °C at an atmospheric pressure of 10 5 Pa?
First, we convert the data into the correct SI units. In this example, only the temperature
needs to be converted. From Equation (1.7), the temperature is 298 K.
Secondly, we rearrange Equation (1.13) to make V the subject, by dividing both sides
by p:
nRT
V =
and then insert values:
V
1 mol x 8.314 JKT 1 mok 1 x 298 K
10 5 Pa
So the volume V = 0.0248 m 3 .
If we remember how there are 1000 dm in 1 m 3 , we see how 1 mol of gas at room
temperature and standard pressure has a volume of 24.8 dm .
SAQ 1.9 2 mol of gas occupy a volume V = 0.4 m 3 at a temperature
T — 330 K. What is the pressure p of the gas?
An alternative form of Equation (1.13) is given as
piVi p 2 V 2
T 2
(1.14)
and is used when we have to start with a constant number of moles of gas n housed in a
volume V\. Its initial pressure is p\ when the temperature is T\. Changing one variable
causes at least one of the two to change. We say the new temperature is T 2 , the new
PROPERTIES OF GASES AND THE GAS LAWS
29
pressure is p2 and the new volume is V2. Equation (1.14) then holds provided the number
of moles does not vary.
Worked Example 1.4 Nitrogen gas is housed in a sealed, hollow cylinder at a pressure
of 10 5 Pa. Its temperature is 300 K and its volume is 30 dm . The volume within the
cylinder is increased to 45 dm 3 , and the temperature is increased at the same time to
310 K. What is the new pressure, />2?
We first rearrange Equation (1.14) to make the unknown volume p 2 the subject, writing
PiViT 2
Pi
TiV 2
We insert values into the rearranged equation:
10 5 Pax 30 dm 3 x 310 K
Pi =
300 K x 45 dm 3
Note how the units of
volume cancel, mean-
ing we can employ any
unit of volume provided
the units of \A and V 2
are the same.
so P2 — 0.69 x 10 5 Pa. The answer demonstrates how the pressure drops by about a third
on expansion.
SAQ 1.10 The pressure of some oxygen gas is doubled from 1.2 x 10 5 Pa
to 2.4 x 10 5 Pa. At the same time, the volume of the gas is decreased
from 34 dm 3 to 29 dm 3 . What is the new temperature T2 if the initial
temperature 7~i was 298 K?
Justification Box 1.1
We start with n moles of gas at a temperature T\ , housed in a volume V\ at a pressure
of p\ . Without changing the amount of material, we change the volume and temperature
to V2 and T 2 respectively, therefore causing the pressure to change to p 2 .
The number of moles remains unaltered, so we rearrange Equation (1.13) to make n
the subject:
n = p l V l ~RT l
Similarly, the same number of moles n under the second set of conditions is
n = p 2 V 2 + RT 2
Again, although we changed the physical conditions, the number of moles n remains
constant, so these two equations must be the same. We say
RT,
P1V1
RT 2
30 INTRODUCTION TO PHYSICAL CHEMISTRY
As the value of R (the gas
multiplying both sides by R
constant) does
to obtain
P1V1
Ti
not vary,
p 2 V 2
T 2
we
can
simplify
the
equation
by
which is Equation (1.14).
What causes pressure?
Motion of particles in the gas phase
The question, 'What is pressure?' is another odd question, but is not too difficult
to answer.
The constituent particles of a substance each have energy. In practice, the energy
is manifested as kinetic energy - the energy of movement - and explains why all
molecules and atoms move continually as an expression of that kinetic energy. This
energy decreases as the temperature decreases. The particles only stop moving when
cooled to a temperature of absolute zero: K or —273.15 °C.
The particles are not free to move throughout a solid substance, but can vibrate
about their mean position. The frequency and amplitude of such vibration increases as
the temperature rises. In a liquid, lateral motion of the particles is possible, with the
motion becoming faster as the temperature increases. We call this energy translational
energy. Furthermore, as the particles acquire energy with increased temperature, so
the interactions (see Chapter 2) between the particles become comparatively smaller,
thereby decreasing the viscosity of the liquid and further facilitating rapid motion of
the particles. When the interactions become negligible (comparatively), the particles
can break free and become gaseous.
And each particle in the gaseous state can move at amazingly high speeds; indeed,
they are often supersonic. For example, an average atom of helium travels at a mean
speed of 1204 ms" 1 at 273.15 K. Table 1.4 lists the mean speeds of a few other gas
molecules at 273.15 K. Notice how heavier molecules travel more slowly, so carbon
dioxide has a mean speed of 363 ms -1 at the same temperature. This high speed
of atomic and molecular gases as they move is a manifestation of
their enormous kinetic energy. It would not be possible to travel
so fast in a liquid or solid because they are so much denser - we
call them condensed phases.
The separation between each particle in gas is immense, and
usually thousands of times greater than the diameter of a single
gas particle. In fact, more than 99 per cent of a gas's volume
is empty space. The simple calculation in Worked Example 1.5
demonstrates this truth.
The gas particles are
widely separated.
Particles of gas travel
fast and in straight
lines, unless they col-
lide.
PROPERTIES OF GASES AND THE GAS LAWS 31
Table 1.4 The average speeds of gas
molecules at 273.15 K, given in order of
increasing molecular mass. The speeds
c are in fact root-mean-square speeds,
obtained by squaring each velocity, tak-
ing their mean and then taking the square
root of the sum
Gas
Speed c/m s '
Monatomic gases
Helium
1204.0
Argon
380.8
Mercury
170.0
Diatomic gases
Hydrogen
1692.0
Deuterium
1196.0
Nitrogen
454.2
Oxygen
425.1
Carbon monoxide
454.5
Chlorine
285.6
Polyatomic gases
Methane
600.6
Ammonia
582.7
Water
566.5
Carbon dioxide
362.5
Benzene
272.8
Worked Example 1.5 What is the molar volume of neon, assuming it to be a straight-
forward solid?
We must first note how the neon must be extremely cold if it is to be
a solid - probably no colder than about 20 K.
We know that the radius of a neon atom from tables of X-ray crystal-
lographic data is about 10 -10 m, so the volume of one atom (from the
The 'molar volume' is
the name we give to
the volume 'per mole'.
equation of a sphere, V — ^rcr 3 ) is 4.2 x 10 -30 m 3 . If we assume the
neon to be a simple solid, then 1 mol of neon would occupy a volume of 4.2 x 10 -30 m 3
per atom x 6.022 x 10 23 atoms per mole = 2.5 x 10 -6 m 3 mol -1 . This volume represents
2.5 cm 3 mol -1 .
A volume of 2.5 cm 3 mol -1 is clearly much smaller than the value we calculated
earlier in Worked Example 1.3 with the ideal-gas equation, Equation (1.13). It is
also smaller than the volume of solid neon made in a cryostat,
suggesting the atoms in a solid are also separated by much empty
space, albeit not so widely separated as in a gas.
In summary, we realize how each particle of gas has enor-
mous kinetic energy and are separated widely. Yet, like popcorn
in a popcorn maker, these particles cannot be classed as wholly
independent, one from another, because they collide. They collide
By corollary, if the gas
particles move fast and
the gas is ideal, the gas
particles must travel in
straight lines between
collisions.
32
INTRODUCTION TO PHYSICAL CHEMISTRY
Newton's first law
states that every action
has an equal but oppo-
site reaction. His sec-
ond law relates the
force acting on an
object to the product of
its mass multiplied by
its acceleration.
The pressure of a
gas is a 'macroscopic'
manifestation of the
'microscopic' gas parti-
cles colliding with the
internal walls of the
container.
The surface area inside
a cylinder of radius r
and height h is 2nrh.
Don't forget to include
the areas of the two
ends, each of which is
Ttr 2 .
firstly with each other, and secondly with the internal walls of the
container they occupy.
Just like the walls in a squash court, against which squash balls
continually bounce, the walls of the gas container experience a
force each time a gas particle collides with them. From Newton's
laws of motion, the force acting on the wall due to this incessant
collision of gas particles is equal and opposite to the force applied
to it. If it were not so, then the gas particles would not bounce
following a collision, but instead would go through the wall.
We see how each collision between a gas particle and the internal
walls of the container causes the same result as if we had applied a
force to it. If we call the area of the container wall A and give the
symbol F to the sum of the forces of all the particles in the gas, then
the pressure p exerted by the gas-particle collisions is given by
pressure, p =
force, F
area, A
(1.15)
In summary, the pressure caused by a container housing a gas is
simply a manifestation of the particles moving fast and colliding
with the container walls.
SAQ 1.11 A cylindrical can contains gas. Its height is
30 cm and its internal diameter is 3 cm. It contains gas
at a pressure of 5 x 10 5 Pa. First calculate the area of the
cylinder walls (you will need to know that 1 m = 100 cm,
so 1 m 2 = 10 4 cm 2 ), and then calculate the force neces-
sary to generate this pressure.
Aside
A popular misconception says a molecule in the gas phase travels faster than when in
a liquid. In fact, the molecular velocities will be the same in the gas and liquid phases
if the temperatures are the same. Molecules only appear to travel slower in a liquid
because of the large number of collisions between its particles, causing the overall
distance travelled per unit time to be quite short.
Why is it unwise to incinerate an empty can of air
freshener?
The molecular basis of the gas laws
The writing printed on the side of a can of air freshener contains much information.
Firstly, it cites the usual sort of advertising prose, probably saying it's a better product
FURTHER THOUGHTS ON ENERGY
33
CFC stands for chlo-
rofluorocarbon. Most
CFCs have now been
banned because of
their ability to dam-
age the ozone layer in
the upper atmosphere.
than anyone else's, and smells nicer. Few people seem to bother reading these bits.
But in most countries, the law says the label on the can should also gives details of
the can's contents, both in terms of the net mass of air freshener it contains and also
perhaps a few details concerning its chemical composition. Finally, a few words of
instruction say how to dispose safely of the can. In this context, the
usual phrase printed on the can is, 'Do not incinerate, even when
empty'. But why?
It is common for the can to contain a propellant in addition to the
actual components of the air freshener mixture. Commonly, butane
or propane are chosen for this purpose, although CFCs were the
favoured choice in the recent past.
Such a can is thrown away when it contains no more air fresh-
ener, although it certainly still contains much propellant. Inciner-
ation of the can leads to an increase in the kinetic energy of the
remaining propellant molecules, causing them to move faster and
faster. And as their kinetic energy increases, so the frequency with
which they strike the internal walls of the can increases. The force
of each collision also increases. In fact, we rediscover the ideal gas
equation, Equation (1.13), and say that the pressure of the gas (in
a constant-volume system) increases in proportion to any increase
in its temperature. In consequence, we should not incinerate an old
can of air freshener because the internal pressure of any residual
propellant increases hugely and the can explodes. Also note the
additional scope for injury afforded by propane' s flammability.
Pressure increases with
increasing temperature
because the collisions
between the gas parti-
cles and the container
wall are more ener-
getic and occur more
frequently.
1.4 Further thoughts on energy
Why is the room warm?
The energy of room temperature
Imagine coming into a nice, warm room after walking outside in the snow. We
instantly feel warmer, because the room is warmer. But what exactly is the energy
content of the room? Stated another way, how much energy do we get from the air
in the room by virtue of it being at its own particular temperature?
For simplicity, we will consider only the molecules of gas. Each molecule of gas
will have kinetic energy (the energy of movement) unless the temperature is absolute
zero. This energy may be transferred through inelastic molecules collisions. But how
much kinetic energy does the gas have?
At a temperature T, 1 mol of gas has a kinetic energy of jRT, where T is the ther-
modynamic temperature and R is the gas constant. This energy is directly proportional
to the thermodynamic temperature, explaining why we occasionally call the kinetic
energy 'thermal motion energy'. This simple relationship says that temperature is
merely a measure of the average kinetic energy of gas molecules moving chaotically.
34
INTRODUCTION TO PHYSICAL CHEMISTRY
It is important to appreciate that this energy relates to the average energy of 1 mol
of gas molecules. The concept of temperature has no meaning when considering a
single molecule or atom. For example, the velocity (and hence the kinetic energy)
of a single particle changes with time, so in principle its temperature also changes.
Temperature only acquires any thermodynamic meaning when we consider average
velocities for a large number of particles.
Provided we know the temperature of the gas, we know its energy - the energy it
has simply by existing at the temperature T.
Worked Example 1.6 What is the energy of 1 mol of gas in a warm room at 310 K?
The energy per mole is | x R x T; so, inserting values, energy =
The Voom energy' §R7
derives from the kinetic
(movement) energy of
a gas or material.
| x 8.314 JK" 1 mol" 1 x 310 K.
Energy = 3866 JmoP 1 % 3.9 kJmok
The molar energy of these molecules is about 4 kJ moP ' , which is
extremely slight compared with the energy of the bonds connecting
the respective atoms within a molecule (see Chapters 2 and 3). There is little chance of
this room energy causing bonds to break or form.
SAQ 1.12 What is the room energy per mole on a cold winter's day, at
-8°C (265 K)?
What do we mean by 'room temperature'?
Standard temperature and pressure
Suppose two scientists work on the same research project, but one resides in the far
north of the Arctic Circle and the other lives near the equator. Even if everything
else is the same - such as the air pressure, the source of the chemicals and the
manufacturers of the equipment - the difference between the temperatures in the two
laboratories will cause their results to differ widely. For example, the 'room energy'
RT will differ. One scientist will not be able to repeat the experiments of the other,
which is always bad science.
An experiment should always be performed at known tempera-
ture. Furthermore, the temperature should be constant throughout
the course of the experiment, and should be noted in the labora-
tory notebook.
But to enable complete consistency, we devise what is called a set
of standard conditions. 'Standard pressure' is given the symbol p & ,
and has a value of 10 5 Pa. We sometimes call it '1 bar'. Atmospheric pressure has a
value of 101 325 Pa, so it is larger than p e . We often give atmospheric pressure the
symbol 'atm'.
An experiment should
always be performed at
a known, fixed temper-
ature.
FURTHER THOUGHTS ON ENERGY
35
'Standard temperature' has the value of 298 K exactly, which
equates to just below 25 °C. If both the pressure and the temper-
ature are maintained at these standard conditions, then we say the
measurement was performed at 'standard temperature and pres-
sure', which is universally abbreviated to 's.t.p.' If the scientists
at the equator and the Arctic Circle perform their work in thermo-
statically controlled rooms, both at s.t.p., then the results of their
experiments will be identical.
A 'thermostat' is a
device for maintain-
ing a temperature.
Thermo is Greek for
'energy' or 'tempera-
ture', and 'stat' derives
from the Greek root
statikos, meaning 'to
stand', i.e. not move or
alter.
Why do we get w 'armed-through in front of a fire,
rather than just our skins?
The Maxwell- Boltzmann distribution of energies
■
If no heat was distributed, then our faces and those parts closest to the fire
would quickly become unbearably hot, while the remainder of our flesh would
continue to feel cold. Heat conducts through the body principally by the fire
warming the blood on the surface of the skin, which is then
pumped to other parts of the body through the circulatory
system. The energy in the warmed blood is distributed within
cooler, internal tissues.
It is important to note how the heat energy is distributed around
the body, i.e. shared and equalized. Nature does not like diversity
in terms of energetic content, and provides many mechanisms by
which the energy can be 'shared'. We shall discuss this aspect of
thermochemistry in depth within Chapter 4.
We can be certain that molecules do not each have the same
energy, but a distribution of energies. The graph in Figure 1.9
concerns the energies in a body. The x-axis gives the range of
energies possible, and the y-axis represents the number of particles
in the body (molecules, atoms, etc.) having that energy. The graph
clearly shows how few particles possess a large energy and how
a few particles have a tiny energy, but the majority have lesser
energies. We call this spread of energies the 'Maxwell-Boltzmann
distribution'.
All speeds are found at all temperatures, but more molecules
travel at faster speeds at the higher temperatures.
The distribution law depicted in Figure 1.9 may be modelled
mathematically, to describe the proportions of molecules of molar
mass M with energies E in the range E to E + AE that exist in
We often see this rela-
tionship called merely
the 'Boltzmann dis-
tribution', after the
Austrian Physicist Lud-
wig Boltzmann (1844-
1906), who played a
pivotal role in marrying
thermodynamics with
statistical and molecu-
lar physics.
The thermodynamic
temperature is the sole
variable required to
define the Maxwell-
Boltzmann distribution:
raising the temperature
increases the spread of
energies.
36
INTRODUCTION TO PHYSICAL CHEMISTRY
500
1000
1500
Velocity Wm s
Figure 1.9 Molecular energies follow the Maxwell-Boltzmann distribution: energy distribution
of nitrogen molecules (as y) as a function of the kinetic energy, expressed as a molecular velocity
(as x). Note the effect of raising the temperature, with the curve becoming flatter and the maximum
shifting to a higher energy
thermal equilibrium with each other at a temperature T:
f(E) = An
M
ItzRT
3/2
E exp
Ms l \
2RT )
(1.16)
where / on the far left indicates the 'function' that is to be applied to the variable E :
the mathematical nature of this function is given by the right-hand side of the equation.
So, in summary, we feel warmer in front of a fire because energy is distributed
between those parts facing the flames and the more hidden tissues within.
2
Introducing interactions
and bonds
Introduction
We look first at deviations from the ideal-gas equation, caused by inter-particle
interactions. Having described induced dipoles (and hydrogen bonds) the interaction
strengths are quantified in terms of the van der Waals and virial equations of state.
Next, formal bonds are described, both covalent (with electrons shared between
participating atoms) and ionic (in which electrons are swapped to form charged ions;
these ions subsequently associate in response to electrostatic forces). Several under-
lying factors are expounded, such as ionization energy / and electron affinity E^.
The energy changes occurring while forming these interactions are alluded to, but are
treated properly in Chapter 3.
2.1 Physical and molecular interactions
What is 'dry ice'?
Deviations from the ideal-gas equation
We call solid carbon dioxide (CO2) 'dry ice'. To the eye, it looks
just like normal ice, although it sometimes appears to 'smoke';
see below. Carbon dioxide is a gas at room temperature and only
solidifies (at atmospheric pressure) if the temperature drops to about
—78 °C or less, so we make dry ice by cooling gaseous CO2 below
its freezing temperature. We call it dry ice because, unlike normal
ice made with water, warming it above its melting temperature
leaves no puddle of liquid, because the CO2 converts directly to a gas. We say
it sublimes.
Gases become denser as we lower their temperature. If CO2 was still a gas at
— 90 °C, then its molar volume would be 15 200 cm 3 . In fact, the molar volume of
Substances sublime if
they pass directly from
a solid to form a gas
without being a liquid
as an intermediate
phase; see Chapter 5.
38
INTRODUCING INTERACTIONS AND BONDS
solid CO2 at this temperature is about 30 cm 3 . We deduce that CO2 does not obey
the ideal-gas equation (Equation (1.13)) below its freezing temperature, for the very
obvious reason that it is no longer a gas.
SAQ 2.1 Show that the volume of 1 mol of C0 2 would be 15 200 cm 3 at
p^ and -90 °C (183 K). [Hint: use the ideal-gas equation. To express this
answer in cubic metres, you will need to remember that 1 m 3 = 10 3 dm 3
and 10 6 cm 3 .]
Although solidifying CO2 is an extreme example, it does show how deviations
from the ideal-gas equation occur.
How is ammonia liquefied?
We sometimes call
a solid or a liquid a
'condensed phase'.
'Intermolecular' means
'between molecules'.
Intermolecular forces
Compressing ammonia gas under high pressure forces the molecules into close
proximity. In a normal gas, the separation between each molecule is generally
large - approximately 1000 molecular diameters is a good
generalization. By contrast, the separation between the molecules in
a condensed phase (solid or liquid) is more likely to be one to two
molecular diameters, thereby explaining why the molar volume of
a solid or liquid is so much smaller than the molar volume of a gas.
As a direct consequence of the large intermolecular separations,
we can safely say no interactions form between the molecules in
ammonia gas. The molecules are simply too far apart. We saw
in the previous chapter how the property known as pressure is
a macroscopic manifestation of the microscopic collisions occurring between gas
particles and, say, a solid object such as a container's walls. But the gas particles can
also strike each other on the same microscopic scale: we say the resultant interactions
between molecules are intermolecular.
Intermolecular interactions only operate over relatively short dis-
tances, so we assume that, under normal conditions, each molecule
in a gas is wholly unaffected by all the others. By contrast, when
the gas is compressed and the particles come to within two or three
molecular diameters of each other, they start to 'notice' each other.
We say the outer-shell electrons on an atom are perturbed by the
charges of the electrons on adjacent atoms, causing an interaction.
We call these interactions bonds, even though they may be too
weak to be formal bonds such as those permanently connecting
the atoms or ions in a molecule.
The intermolecular interactions between molecules of gas are generally attractive;
so, by way of response, we find that, once atoms are close enough to interact, they
prefer to remain close - indeed, once a tentative interaction forms, the atoms or
A 'formal bond' involves
the permanent involve-
ment of electrons in
covalent or ionic bonds;
see p. 64. Interactions
between molecules
in a compressed gas
are temporary.
PHYSICAL AND MOLECULAR INTERACTIONS
39
Translational motion
is movement through
space, rather than a
vibration about a mean
point or a rotation
about an axis.
molecules generally draw themselves closer, which itself makes
the interaction stronger. We see a simple analogy with everyday
magnets: once two magnets are brought close enough to induce an
interaction, we feel the attractive force dragging them closer still.
As soon as the particles of a gas attract, the inertia of the
aggregate species increases, thereby slowing down all translational
motion. And slower particles, such as these aggregates, are an eas-
ier target for further collisions than fast-moving gas atoms and
molecules. The same principle explains why it is impossible to
catch someone who is running very fast during a playground game
of 'tig'. Only when the runners tire and slow down can they
be caught. In practice, as soon as an aggregate forms, we find
that other gas particles soon adhere to it, causing eventual coa-
lescence and the formation of a droplet of liquid. We say that
nucleation occurs.
With the same reasoning as that above, we can force the molecules of ammonia
still closer together by applying a yet larger pressure, to form a denser state such as
a solid.
Formation of an ag-
gregate facilitates fur-
ther coalescence (even-
tually forming a con-
densed phase). We say
'nucleation' occurs.
Why does steam condense in a cold bathroom?
Elastic and inelastic collisions
In the previous example, we looked at the interactions induced when changing the
external pressure, forcing the molecules into close proximity. We look here at the
effects of changing the temperature.
A bathroom mirror is usually colder than the temperature of the steam rising from
a hot bath. Each molecule of steam (gaseous water) has an enormous energy, which
comes ultimately from the boiler that heats the water. The particles of steam would
remain as liquid if they had less energy. In practice, particles evaporate from the
bath to form energetic molecules of steam. We see this energy as kinetic energy, so
the particles move fast (see p. 30). The typical speeds at which gas particles move
make it inevitable that steam molecules will collide with the mirror.
We say such a collision is elastic if no energy transfers during the
collision between the gas particle and the mirror; but if energy does
transfer - and it usually does - we say the collision is inelastic.
The energy transferred during an inelastic collision passes from
the hot molecule of steam to the cooler mirror. This energy flows
in this direction because the steam initially possessed more energy
per molecule than the mirror as a consequence of its higher temperature. It is merely a
manifestation of the minus-oneth law of thermodynamics, as discussed in Chapter 1.
But there are consequences to the collisions being inelastic: the molecules of steam
have less energy following the collision because some of their energy has transferred.
No energy is exchang-
ed during an 'elastic'
collision, but energy is
exchanged during an
'inelastic' collision.
40
INTRODUCING INTERACTIONS AND BONDS
Energy is never lost
or gained, only trans-
ferred or converted;
see Chapter 3.
We generally assume
that all particles in an
ideal gas do not inter-
act, meaning that the
gas obeys the ideal-gas
equation. This assump-
tion is sometimes poor.
We perceive this lower energy as a cooler temperature, meaning
that the water vapour in a steam-filled bathroom will cool down;
conversely, the mirror (and walls) become warmer as they receive
the energy that was previously possessed by the steam. These
changes in the temperatures of gas and mirror occur in a com-
plementary sense, so no energy is gained or lost.
These changes in temperature represent a macroscopic proof that
microscopic processes do occur. Indeed, it is difficult to envisage
a transfer of energy between the gas particles with the cold mirror
without these microscopic interactions.
We spent quite a lot of time looking at the concept of an ideal gas
in Chapter 1 . The simplest definition of an ideal gas is that it obeys
the ideal-gas equation (Equation (1.13)). Most gases can be con-
sidered as ideal most of the time. The most common cause of a gas
disobeying the ideal-gas equation is the formation of interactions,
and the results of intermolecular collisions.
How does a liquid-crystal display work?
Electronegativity and electropositivity
Liquid crystals are organic compounds that exhibit properties somewhere between
those of a solid crystal and a liquid. Compounds I and II in Figure 2.1 both form
liquid crystals at room temperature.
We observe that liquid crystals can flow like any other viscous liquid, but they also
possess some of the properties of crystalline solids, such as physical order, rather
than random chaos. Unlike most other liquids, liquid crystals have some properties
(l)
CiqH 2
(II)
Figure 2.1 Compounds that form room-temperature liquid crystals
PHYSICAL AND MOLECULAR INTERACTIONS
41
Light transmitted
Lower polarizer blocks
the transmission
of light
No voltage applied Voltage applied
Figure 2.2 The transparent electrodes in an LCD are coated with crossed polarizers. The liquid
crystals (depicted as slender lozenges) form helices, thereby 'guiding' polarized light from the upper
electrode through the LCD, enabling transmission through to the lower polarizer. This is why the
display has no colour. The helical structure is destroyed when a voltage is applied, because the polar
liquid crystals align with the electrodes' field. No light can transmit, so the display looks black
A physicist would say
the liquid crystal
adopted a twisted
nematic structure.
that depend on the direction of measurement, because of the alignment of their long,
rod-like structures.
In a liquid-crystal display (LCD) device, the two electrodes
are parallel and separated by a thin layer of liquid crystal (see
Figure 2.2). The liquid crystals in this layer naturally adopt a heli-
cal structure.
Light can be represented as a transverse electromagnetic wave
made up of fluctuating electric and magnetic fields, moving in
mutually perpendicular directions (see Chapter 9). Ordinary light is made up of waves
that fluctuate at all possible angles, which normally cannot be separated. A polarizer
is a material that allows only light with a specific angle of vibration to transmit.
We place a light polarizer on one side of either transparent electrode in the LCD,
each similar to one lens in a pair of polaroid sunglasses. The helix of the liquid
crystal twists the polarized light as it transmits through the LCD, guiding it from the
upper polarizer and allowing it unhindered passage through the 'sandwich' and lower
polarizer. The transmitting state of an LCD (at zero voltage) is thus 'clear'.
Applying a voltage to a pixel within the cell causes the mole-
cules to move, aligning themselves parallel with the electric field
imparted by the electrodes. This realignment destroys the helical
structure, precluding the unhindered transmission of light, and the
display appears black.
Molecules of this type are influenced by an external electric
field because they possess a dipole: one end of the molecule is
electron withdrawing while the other is electron attracting, with the result that one
end possesses a higher electron density than the other. As a result, the molecule
behaves much like a miniature bar magnet. Applying a voltage between the two
'Pixel' is short for 'pic-
ture element'. An LCD
image comprises many
thousands of pixels.
42
INTRODUCING INTERACTIONS AND BONDS
electrodes of the LCD causes the 'magnet' to reorientate in just the same way as a
magnet moves when another magnet is brought close to it.
These dipoles form because of the way parts of the molecule
attract electrons to differing extents. The power of an element
(when part of a compound) to attract electrons is termed its 'elec-
tronegativity' x- Highly electron-attracting atoms tend to exert
control over the outer, valence electrons of adjacent atoms. The
most electronegative elements are those placed near the right-hand
side of the periodic table, such as oxygen and sulphur in Group
VI(b) or the halogens in Group VII(b).
There have been a large number of attempts to quantify elec-
tronegativities x » either theoretically or semi-empirically, but none
has been wholly successful. All the better methods rely on bond
strengths or the physical dimensions of atoms.
Similar to the concept of electronegativity is the electropositivity of an element, which
is the power of its atoms (when part of a compound) to lose an electron. The most
electropositive elements are the metals on the far-left of the periodic table, particularly
Groups 1(a) and 11(a), which prefer to exist as cations. Being the opposite concept to
electronegativity, electropositivity is not employed often. Rather, we tend to say that an
atom such as sodium has a tiny electronegativity instead of being very electropositive.
Atoms or groups are
'electronegative' if they
tend to acquire neg-
ative charge at the
expense of juxtaposed
atoms or groups. Groups
acquiring a positive
charge are 'electropos-
itive'.
Why does dew form on a cool morning?
Van der Waals forces
Many people love cool autumn mornings, with the scent of the cool air and a rich
dew underfoot on the grass and paths. The dew forms when molecules of water
from the air coalesce, because of the cool temperature, to form minute aggregates
that subsequently nucleate to form visible drops of water. These water drops form a
stable colloid (see Chapter 10).
Real gases are never wholly ideal: there will always be some extent of non-ideality.
At one extreme are the monatomic rare gases such as argon and neon, which are non-
polar. Hydrocarbons, like propane, are also relatively non-polar, thereby precluding
stronger molecular interactions. Water, at the opposite extreme, is very polar because
some parts of the molecule are more electron withdrawing than
others. The central oxygen is relatively electronegative and the two
hydrogen atoms are electropositive, with the result that the oxygen
is more negative than either of the hydrogen atoms. We say it has
a slight excess charge, which we write as S~. Similar reasoning
shows how the hydrogen atoms are more positive than the oxygen,
with excess charges of <5 + .
These excess charges form in consequence of the molecule incorporating a variety
of atoms. For example, the magnitude of S~ on the chlorine of H-Cl is larger than
the excess charges in the F-Cl molecule, because the difference in electronegativity
The symbol 8 means 'a
small amount of . . .', so
\5 - ' is a small amount
of negative charge.
PHYSICAL AND MOLECULAR INTERACTIONS
43
Table 2.1 Values of electronegativity x for some main-group elements
H
2.1
Li
Be
B
C
N
O
F
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Na
Mg
Al
Si
P
S
CI
0.9
1.2
1.5
1.8
2.1
2.5
3.0
K
Ca
Ga
Ge
As
Se
Br
0.8
1.0
1.6
1.8
2.0
2.4
2.8
Rb
Sr
I
0.8
1.0
2.5
X between H and CI is greater than the difference between F and CI. There will be no
excess charge in the two molecules H-H or Cl-Cl because the atoms in both are the
same - we say they are homonuclear. Table 2.1 contains a few electronegativities.
SAQ 2.2 By looking at the electronegativities in Table 2.1, suggest wheth-
er the bonds in the following molecules will be polar or non-polar: (a) hydro-
gen bromide, HBr; (b) silicon carbide, SiC; (c) sulphur dioxide, 0=S=0; and
(d) sodium iodide, Nal.
The actual magnitude of the excess charge is generally unknown,
although we do know they are small. Whereas some calculations
suggest that S is perhaps as much as 0.1 of a full, formal charge,
others suggest about 0.01 or even less.
While debate persists concerning the magnitudes of each excess
charge within a molecule, it is certain that the overall charge on the
molecule is zero, meaning that the two positive charges in water
cancel out the central negative charge on the oxygen. We reason
this by saying that water is a neutral molecule.
Figure 2.3 shows the 'V shape of the water molecule. The top
of the molecule (as drawn) has a negative excess charge and the
bottom is positive. The S + and S~ charges are separated spatially,
which we call a dipole. Such dipoles are crucial when explain-
ing why water vapour so readily forms a liquid: those parts of
the molecule bearing a slight positive charge (<5 + ) attract those
parts of adjacent molecules that bear a slight negative charge (<5~).
The interaction is electrostatic, and forms in much a similar man-
ner to the north pole of a magnet attracting the south pole of
another magnet.
Electrostatic interactions of this type are called 'dipole-dipole
interactions', or 'van der Waals forces' after the Dutch physicist
Johannes Diderik van der Waals (1837-1923) who first postulated
their existence. A van der Waals force operates over a relatively
Water is a neutral
molecule, so the cen-
tral negative charge
in the water molecule
counteracts the two
positive charges.
A 'dipole' forms when
equal and opposite
charges are separated
by a short distance. 'Di'
means two, and 'pole'
indicates the two ends
of a magnet.
'Van der Waals forces'
are electrostatic inter-
actions between di-
poles. (Note how we
pronounce 'Waals' as
'vahls'.)
44
INTRODUCING INTERACTIONS AND BONDS
Figure 2.3 The water molecule has a 'V shape. Experiments show
that gaseous water has an O-H length of 0.957 18 A; the H-O-H angle
is 104.474°. Water is polar because the central oxygen is electronega-
tive and the two hydrogen atoms are electropositive. The vertical arrow
indicates the resultant dipole, with its head pointing toward the more
negative end of the molecule
-O— - H O-
\
H
\ I /O
\ H' K ,- H
/ H
H / H >— H
0' /
/ \ H H
H "
Figure 2.4 Water would be a gas rather than a liquid at room temperature if no van der Waals
forces were present to 'glue' them together, as indicated with dotted lines in this two-dimensional
representation. In fact, water coalesces as a direct consequence of this f/jree-dimensional network
of dipole-dipole interactions. Note how all the O-H • • • O bonds are linear
short distance because the influence of a dipole is not large. In practice, we find that
the oxygen atoms can interact with hydrogen atoms on an adjacent molecule of water,
but no further.
The interactions between the two molecules helps to 'glue' them together. It is a
sobering thought that water would be a gas rather than a liquid if hydrogen bonds
(which are merely a particularly strong form of van der Waals forces) did not promote
the coalescence of water. The Earth would be uninhabitable without them. Figure 2.4
shows the way that liquid water possesses a three-dimensional network, held together
with van der Waals interactions.
Each H2O molecule in liquid water undergoes at least one interaction with another
molecule of H2O (sometimes two). Nevertheless, the interactions are not particularly
strong - perhaps as much as 20 kJ mol~ .
Whereas the dipoles themselves are permanent, van der Waals interactions are
not. They are sufficiently weak that they continually break and re-form as part of a
dynamic process.
How is the three-dimensional structure maintained
within the DNA double helix?
Hydrogen bonds
DNA is a natural polymer. It was first isolated in 1869 by Meischer, but its role
in determining heredity remained unrecognized until 1944, by which time it was
PHYSICAL AND MOLECULAR INTERACTIONS
45
appreciated that it is the chromosomes within a cell nucleus that
dictate hereditary traits. And such chromosomes consist of DNA
and protein. In 1944, the American bacteriologist Oswald Avery
showed how it was the DNA that carried genetic information, not
the protein.
The next breakthrough came in 1952, when Francis Crick and
Donald Watson applied X-ray diffraction techniques to DNA and
elucidated its structure, as shown schematically in Figure 2.5. They
showed how its now famous 'double helix' is held together via
a series of unusually strong dipole-dipole interactions between
precisely positioned organic bases situated along the DNA poly-
mer's backbone.
There are four bases in DNA: guanine, thymine, cytosine and
adenine. Each has a ketone C=0 group in which the oxygen is quite
electronegative and bears an excess negative charge S~, and an
amine in which the electropositive hydrogen atoms bear an excess
The word 'theory' comes
from the Greek theoreo,
meaning 'I look at'. A
theory is something we
look at, pending accep-
tance or rejection.
The rules of 'base pair-
ing' (or nucleotide pair-
ing) in DNA are: ade-
nine (A) always pairs
with thymine (T); cyto-
sine (C) always pairs
with guanine (G).
3.4 nm
(a)
H
0--H-N N ^H
\ // \ / *C
c-c c-c ,
// \ // \\
H-C N — H --N C
\ / \ /
N-C C=N
** W /
DNA backbone 0--H — N
-N
%
Cytosine
H
DNA backbone
Guanine
H 3 C
H-
-N N H
\ //
\ / *c
c-c
C-C 7
// \
" * N
N C-\
\ / ^
H-C N-
-H
\ /
N-C
C = N DNA backbone
S* *
/
DNA backbone
H
Thymine
Adenine
(b)
Figure 2.5 (a) The structure of the 'double helix' at the heart of DNA. The slender 'rods' represent
the hydrogen bonds that form between the organic bases situated on opposing strands of the helix,
(b) Hydrogen bonds (the dotted lines) link adenine with thymine, and guanine with cytosine
46
INTRODUCING INTERACTIONS AND BONDS
Table 2.2 The energies of hydrogen bonds
Atoms in H-bond
Typical energy/kj mol
HandN
HandO
HandF
20
25
40
positive charge S + . Since the hydrogen atom is so small and so electropositive, its
excess charge leads to the formation of an unusually strong dipole, itself leading
to a strong van der Waals bond. The bond is usually permanent (unlike a typical
dipole-dipole interaction), thereby 'locking' the structure of DNA into its pair of
parallel helices, much like the interleaving teeth of a zip binding together two pieces
of cloth.
We call these extra-strong dipole-dipole bonds 'hydrogen bonds',
and these are defined by the IUPAC as 'a form of association between
an electronegative atom and a hydrogen atom attached to a second,
relatively electronegative atom'. All hydrogen bonds involve two
dipoles: one always comprises a bond ending with hydrogen; the
other terminates with an unusually electronegative atom. It is best
considered as an electrostatic interaction, heightened by the small
size of hydrogen, which permits proximity of the interacting dipoles
or charges. Table 2.2 contains typical energies for a few hydrogen
bonds. Both electronegative atoms are usually (but not necessarily)
from the first row of the periodic table, i.e. N, O or F. Hydrogen
bonds may be intermolecular or intramolecular.
Finally, as a simple illustration of how weak these forces are,
note how the energy required to break the hydrogen bonds in
liquid hydrogen chloride (i.e. the energy required to vaporize it)
is 16kJmol _1 , yet the energy needed break the chemical bond
between atoms of hydrogen and chlorine in H-Cl is almost 30
times stronger, at 431 kJmol -1 .
IUPAC is the Interna-
tional Union of Pure
and Applied Chemistry.
It defines terms, quan-
tities and concepts in
chemistry.
Strictly, the dipole-
dipole interactions dis-
cussed on p. 42 are
also hydrogen bonds,
since we discussed
the interactions arising
between H-O-bonded
species.
Aside
The ancient Greeks recognized that organisms often pass on traits to their offspring, but
it was the experimental work of the Austrian monk Gregor Mendel (1822-1884) that
led to a modern hereditary 'theory'. He entered the Augustinian monastery at Briinn
(now Brno in the Czech Republic) and taught in its technical school.
He cultivated and tested the plants at the monastery garden for 7 years. Starting in
1856, painstakingly analysing seven pairs of seed and plant characteristics for at least
28 000 pea plants. These tedious experiments resulted in two generalizations, which
were later called Mendel's laws of heredity. Mendel published his work in 1866, but
it remained almost unnoticed until 1900 when the Dutch botanist Hugo Marie de Vries
referred to it. The full significance of Mendel's work was only realized in the 1930s.
PHYSICAL AND MOLECULAR INTERACTIONS
47
His observations also led him to coin two terms that still persist in present-day
genetics: dominance for a trait that shows up in an offspring, and recessiveness for a
trait masked by a dominant gene.
How do we make liquid nitrogen?
London dispersion forces
A 'homonuclear' mole-
cule comprises atoms
from only one element;
homo is Greek for
'same'. Most molecules
are 'heteronuclear' and
comprise atoms from
several elements; het-
ero is Greek for 'other'
or 'different'.
Liquid nitrogen is widely employed when freezing sperm or eggs
when preparing for the in vitro fertilization (IVF). It is also essential
for maintaining the cold temperature of the superconducting magnet
at the heart of the NMR spectrometers used for structure elucidation
or magnetic resonance imaging (MRI).
Nitrogen condenses to form a liquid at — 196 °C (77 K), which
is so much lower than the temperature of 373.15 K at which water
condenses that we suspect a different physicochemical process is in
evidence. Below — 196 °C, molecules of nitrogen interact, causing
condensation. That there is any interaction at all should surprise us,
because the dipoles above were a feature of heteronuclear bonds,
but the di-nitrogen molecule (N=N) is homonuclear, meaning both
atoms are the same.
We need to invoke a new type of interaction. The triple bond
between the two nitrogen atoms in the di-nitrogen molecule incor-
porates a huge amount of electron density. These electrons are
never still, but move continually; so, at any instant in time, one
end of a molecule might be slightly more negative than the other.
A fraction of a second later and the imbalance departs. But a tiny
dipole forms during the instant while the charges are unbalanced:
we call this an induced dipole; see Figure 2.6.
The electron density changes continually, so induced dipoles
never last more than about 10~ n s. Nevertheless, they last suf-
ficiently long for an interaction to form with the induced dipole of another nitrogen
molecule nearby. We call this new interaction the London dispersion force after Fritz
London, who first postulated their existence in 1930.
London dispersion forces form between all molecules, whether polar or non-polar.
In a large atom or molecule, the separation between the nucleus and valence elec-
trons is quite large; conversely, the nucleus -electron separation in a lighter atom or
molecule is smaller, implying that the electrons are more tightly held. The tighter
binding precludes the ready formation of an induced dipole. For this reason, larger
(and therefore heavier) atoms and molecules generally exhibit stronger dispersion
forces than those that are smaller and lighter.
Dipoles are usually a
feature of heteronu-
clear bonds, although a
fuller treatment needs
to consider the elec-
tronic environment
of atoms and groups
beyond the bond of
interest.
48
INTRODUCING INTERACTIONS AND BONDS
Figure 2.6 Schematic diagram to show how an induced dipole forms when polarizable electrons
move within their orbitals and cause a localized imbalance of charge (an 'induced dipole' in which
the negative electrons on one atom attract the positive nucleus on another). The dotted line represents
the electrostatic dipole interaction
Aside
The existence of an attractive force between non-polar molecules was first recognized
by van der Waals, who published his classic work in 1873. The origin of these forces
was not understood until 1930 when Fritz London (1900-1954) published his quantum-
mechanical discussion of the interaction between fluctuating dipoles. He showed how
these temporary dipoles arose from the motions of the outer electrons on the two
molecules.
We often use the term 'dispersion force' to describe these attractions. Some texts
prefer the term 'London-van der Waals' forces.
The ease with which
the electron distribu-
tion around an atom
or molecule can be
distorted is called its
'polarizability'.
The weakest of all the
intermolecular forces in
nature are always Lon-
don dispersion forces.
Polarizability
The electrons in a molecule's outer orbitals are relatively free to
move. If we could compare 'snapshots' of the molecule at two
different instants in time then we would see slight differences in
the charge distributions, reflecting the changing positions of the
electrons in their orbitals. The ease with which the electrons can
move with time depends on the molecule's polarizability, which
itself measures how easily the electrons can move within their
orbitals.
In general, polarizability increases as the orbital increases in size:
negative electrons orbit the positive nucleus at a greater distance
in such atoms, and consequently experience a weaker electrostatic
interaction. For this reason, London dispersion forces tend to be
stronger between molecules that are easily polarized, and weaker
between molecules that are not easily polarized.
PHYSICAL AND MOLECULAR INTERACTIONS
49
Why is petrol a liquid at room temperature but butane
a gas? ^^|^^^^^^^^^^^H
The magnitude of London dispersion forces
The major component of the petrol that fuels a car is octane, present as a mixture of
various isomers. It is liquid at room temperature because its boiling point temperature
7(boii) is about 125.7 °C. The methane gas powering an oven has a T^oii) = —162.5 °C
and the butane propellant in a can of air freshener has r^oii) = —0.5 °C. Octadecane
is a gel and paraffin wax is a solid. Figure 2.7 shows the trend in T(boii) for a series of
straight-chain alkanes Each hydrocarbon experiences exactly the same intermolecular
forces, so what causes the difference in T^oU)?
Interactions always form between molecules because London forces cannot be erad-
icated. Bigger molecules experience greater intermolecular forces. These dispersion
forces are weak, having a magnitude of between 0.001 and 0.1 per cent of the strength
of a typical covalent bond binding the two atoms in diatomic molecule, H2. These
forces, therefore, are so small that they may be ignored within molecules held together
by stronger forces, such as network covalent bonds or large permanent dipoles or ions.
The strength of the London dispersion forces becomes stronger with increased
polarizability, so larger molecule (or atoms) form stronger bonds. This observation
helps explain the trends in physical state of the Group VII(b) halogens: L is a solid,
Br2 is a liquid, and CI2 and F2 are gases.
But the overall dispersion force strength also depends on the total number of elec-
trons in the atom or molecule. It is a cumulative effect. Butane contains 14 atoms and
58 electrons, whereas octane has 26 atoms and 114 electrons. The greater number of
electrons increases the total number of interactions possible and, since both melting
150
100
BO-
'S -50
-100
-150
-200
20
O CsH,
O C 7 H 16
O C 6 H 14
O C 5 H 12
—I "I 1 -
40 60C 4 H 10 80
O C 3 H 8
100
OC,H 6
O CH 4
Molecular mass/g mol 1
120
1'0
Figure 2.7 The boiling temperature of simple linear hydrocarbons increases as a function of
molecular mass, as a consequence of a greater number of induced dipoles
50
INTRODUCING INTERACTIONS AND BONDS
and boiling points depend directly on the strength of intermolecular bonds, the overall
strength of the London forces varies as the molecule becomes larger.
Aside
When is a dispersion force sufficiently strong that we can safely call it a hydrogen bond?
Hydrogen bonds are much stronger than London dispersion forces for two princi-
pal reasons:
(1) The induced dipole is permanent, so the bond is permanent.
(2) The molecule incorporates a formal H-X covalent bond in which X is a
relatively electronegative element (see p. 42).
We call an interaction 'a hydrogen bond' when it fulfils both criteria.
2.2 Quantifying the interactions
and their influence
How does mist form?
Condensation and the critical state
Why is it that no dew forms if the air pressure is low, however cool the air tempe-
rature?
To understand this question, we must first appreciate how molecules come closer
together when applying a pressure. The Irish physical chemist Thomas Andrews
(1813-1885) was one of the first to study the behaviour of gases as they liquefy:
most of his data refer to CO2. In his most famous experiments, he observed liquid CO2
at constant pressure, while gradually raising its temperature. He readily discerned a
clear meniscus between condensed and gaseous phases in his tube at low temperatures,
but the boundary between the phases vanished at temperatures of about 31 °C. Above
this temperature, no amount of pressure could bring about liquefaction of the gas.
Andrews suggested that each gas has a certain 'critical' tem-
perature, above which condensation is impossible, implying that
no liquid will form by changes in pressure alone. He called this
temperature the 'critical temperature' ^critical)-
Figure 2.8 shows a Boyle' s-law plot of pressure p (as y) against
volume V (as x) for carbon dioxide. The figure is drawn as a
function of temperature. Each line on the graph represents data
obtained at a single, constant temperature, and helps explain why we call each line
an isotherm. The uppermost isotherm represents data collected at 31.5 °C. Its shape is
essentially straightforward, although it clearly shows distortion. The middle trace (at
The 'critical temper-
ature' 7" (C riticai) is that
temperature above
which it is impossible
to liquefy a gas.
QUANTIFYING THE INTERACTIONS AND THEIR INFLUENCE
51
32 36 40 44 48 52
Molar volume/cm 3 moh 1
56 60 64
Figure 2.8 Isotherms of carbon dioxide near the critical point of 3 1.013 °C. The shaded parabolic
region indicates those pressures and volumes at which it is possible to condense carbon dioxide
the cooler temperature of 30.4 °C) is more distorted - it even shows a small region
where the pressure experienced is independent of temperature, which explains why
the graph is linear and horizontal. And in the bottom trace the isotherm at 29.9 °C is
even more distorted still.
The area enclosed by the parabola at the bottom of Figure 2.8 represents those
values of pressure and temperature at which CO2 will condense. If the temperature
is higher than that at the apex of the parabola (i.e. warmer than
31.013 °C) then, whatever the pressure, CO2 will not liquefy. For
this reason, we call 31.013 °C the critical temperature T( cr iticai) of
CO2. Similarly, the gas will not liquefy unless the pressure is above
a minimum that we call the 'critical pressure' /? (critical) •
Critical fluids are dis-
cussed in Chapter 5,
where values of 7" (crit icai)
are listed.
52
INTRODUCING INTERACTIONS AND BONDS
As the temperature rises above the critical temperature and the pressure drops
below the critical pressure, so the gas approximates increasingly to an ideal gas,
i.e. one in which there are no interactions and which obeys the ideal-gas equation
(Equation (1.13)).
How do we liquefy petroleum gas?
We often call a gas that
is non-ideal, a real gas.
Quantifying the non-ideality
Liquefied petroleum gas (LPG) is increasingly employed as a fuel. We produce it by
applying a huge pressure (10-20 x p & ) to the petroleum gas obtained from oil fields.
Above a certain, critical, pressure the hydrocarbon gas condenses; we say it reaches
the dew point, when droplets of liquid first form. The proportion of the gas liquefying
increases with increased pressure until, eventually, all of it has liquefied.
Increasing the pressure forces the molecules closer together, and the intermolecular
interactions become more pronounced. Such interactions are not particularly strong
because petroleum gas is a non-polar hydrocarbon, explaining why it is a gas at room
temperature and pressure. We discuss other ramifications later.
The particles of an ideal gas (whether atoms or molecules) do
not interact, so the gas obeys the ideal-gas equation all the time.
As soon as interactions form, the gas is said to be non-ideal,
with the result that we lose ideality, and the ideal-gas equation
(Equation (1.13)) breaks down. We find that pV ^ nRT.
When steam (gaseous water) is cooled below a certain tem-
perature, the molecules have insufficient energy to maintain their
high-speed motion and they slow down. At these slower speeds,
they attract one another, thereby decreasing the molar volume.
Figure 2.9 shows a graph of the quotient pV 4- nRT (as y)
against pressure p (as x). We sometime call such a graph an
Andrews plot. It is clear from the ideal-gas equation (Equa
tion (1.13)) that if pV = nRT then pV 4 nRT should always
equal to one: the horizontal line drawn through y = 1, therefore,
indicates the behaviour of an ideal gas.
But the plots in Figure 2.9 are not horizontal. The deviation of a
trace from the y = 1 line quantifies the extent to which a gas devi-
ates from the ideal-gas equation; the magnitude of the deviation
depends on pressure. The deviations for ammonia and ethene are
clearly greater than for nitrogen or methane: we say that ammo-
nia deviates from ideality more than does nitrogen. Notice how
the deviations are worse at high pressure, leading to the empiri-
cal observation that a real gas behaves more like an ideal gas at
lower pressures.
Figure 2.10 shows a similar graph, and displays Andrews plots
for methane as a function of temperature. The graph clearly
The physical chem-
istry underlying the
liquefaction of a gas
is surprisingly compli-
cated, so we shall not
return to the question
until Chapter 5.
We form a 'quotient'
when dividing one thing
by another. We meet
the word frequently
when discussing a
person's IQ, their 'intel-
ligence quotient', which
we define as: (a per-
son's score in an intel-
ligence test h- the aver-
age score) x 100.
QUANTIFYING THE INTERACTIONS AND THEIR INFLUENCE
53
N
o
o
CO
E
o
O
2.0
1.5
1.0
0.5
H 2
V N 2
CH 4
/ C2H4
NH 3
200 400 600 800
Applied pressure/p 6 "
1000
1200
Figure 2.9 An Andrews plot of P V -h nRT (as y) against pressure p (as x) for a series of real
gases, showing ideal behaviour only at low pressures. The function on the y-axis is sometimes
called the compressibility Z
2.5
N
o 20
>, 1.5
1.0
o
O
0.5
0.0
200 K
300 K
400 K
— 600 K
_L
_L
200 400 600
Applied pressure/p®
800
1000
Figure 2.10 An Andrews plot of PV -H nRT (as y) against pressure p (as x) for methane gas as
a function of temperature. Methane behaves more like an ideal gas at elevated temperatures
demonstrates how deviations from ideality become less severe with increasing
temperature. In fact, we should expect the deviations to decrease as the temperature
increases, because a higher temperature tells us how the particles have more energy,
decreasing the likelihood of interparticle interactions being permanent.
Drawing graphs such as Figure 2.10 for other gases suggests a
second empirical law, that gases behave more like ideal gases as
the temperature rises. The ideal-gas equation (Equation (1.13)) is
so useful that we do not want to lose it. Accordingly, we adapt it
Gases behave more like
ideal gases at higher
temperatures.
54
INTRODUCING INTERACTIONS AND BONDS
somewhat, writing it as
We sometimes call the
function "pV + nRT'
the 'compressibility'
or 'compressibility fac-
tor' Z.
pV = Z x (nRT)
(2.1)
where Z is the compressibility or compressibility factor. The value
of Z will always be one for an ideal gas, but Z rarely has a value
of one for a real gas, except at very low pressures. As soon as p
increases, the gas molecules approach close enough to interact and
pV 7^ nRT. The value of Z tells us a lot about the interactions
between gas particles.
Aside
The gas constant R is generally given the value 8.314 JK _1 mol _1 , but in fact this
numerical value only holds if each unit is the SI standard, i.e. pressure expressed in
pascals, temperature in kelvin and volume in cubic metres.
The value of R changes if we express the ideal-gas equation (Equation (1.13)) with
different units. Table 2.3 gives values of R in various other units. We must note an
important philosophical truth here: the value of the gas constant is truly constant, but
the actual numerical value we cite will depend on the units with which we express it.
We met a similar argument before on p. 19, when we saw how a standard prefix (such as
deca, milli or mega) will change the appearance of a number, so V = 1 dm = 10 3 cm 3 .
In reality, the number remains unaltered.
We extend this concept here by showing how the units themselves alter the numerical
value of a constant.
Table 2.3 Values of the gas constant R
expressed with various units 3
8.3145 jr'mor 1
2 calKT'mor 1
0.083 145 dm 3 barmor 1 Kr 1
l v-i
83.145 cm J barmor'K
0.082058 dm 3 atmmor 1 K
i v-i
82.058 cm J atmmor 1 K
-l
a l bar = p 6 = 10 5 Pa. 1 atm = 1.013 25 x 10 5
Pa. The 'calorie' is a wholly non-SI unit of energy;
1 cal = 4.157 J.
Why is the molar volume of a gas not zero at K?
The van der Waals equation
In Chapter 1 we recalled how Lord Kelvin devised his temperature scale after cooling
gases and observing their volumes. If the simplistic graph in Figure 1.5 was obeyed,
QUANTIFYING THE INTERACTIONS AND THEIR INFLUENCE
55
then a gas would have a zero volume at — 273.15°C. In fact, the molar volume of
a gas is always significant, even at temperatures close to absolute zero. Why the
deviation from Kelvin's concept?
Every gas consists of particles, whether as atoms (such as neon) or as molecules
(such as methane). To a relatively good first approximation, any atom can be regarded
as a small, incompressible sphere. The reason why we can compress a gas relates to
the large separation between the gas particles. The first effect of compressing a gas
is to decrease these interparticle distances.
Particles attract whenever they approach to within a minimum distance. Whatever
the magnitude of the interparticle attraction, energetic molecules will separate and
continue moving after their encounter; but, conversely, molecules of lower energy do
not separate after the collision because the attraction force is enough to overwhelm the
momentum that would cause the particles to bounce apart. The process of coalescence
has begun.
Compressing a gas brings the particles into close proximity, thereby increasing the
probability of interparticle collisions, and magnifying the number of interactions. At
this point, we need to consider two physicochemical effects that operate in opposing
directions. Firstly, interparticle interactions are usually attractive, encouraging the
particles to get closer, with the result that the gas has a smaller molar volume than
expected. Secondly, since the particles have their own intrinsic volume, the molar
volume of a gas is described not only by the separations between particles but also
by the particles themselves. We need to account for these two factors when we
describe the physical properties of a real gas.
The Dutch scientist van der Waals was well aware that the ideal-
gas equation was simplistic, and suggested an adaptation, which
we now call the van der Waals equation of state:
2
n a
P + ^2
(V - nb) = nRT
(2.2)
The a term reflects
the strength of the
interaction between
gas particles, and the
b term reflects the
particle's size.
where the constants a and b are called the 'van der Waals constants', the values
of which depend on the gas and which are best obtained experimentally. Table 2.4
contains a few sample values. The constant a reflects the strength of the interaction
between gas molecules; so, a value of 18.9 for benzene suggests a strong interaction
whereas 0.03 for helium represents a negligible interaction. Incidentally, this latter
value reinforces the idea that inert gases are truly inert. The magnitude of the constant
b reflects the physical size of the gas particles, and are again seen
to follow a predictable trend. The magnitudes of a and b dictate
the extent to which the gases deviate from ideality.
Note how Equation (2.2) simplifies to become the ideal-gas equa-
tion (Equation (1.13)) if the volume V is large. We expect this
result, because a large volume not only implies a low pressure,
but also yields the best conditions for minimizing all instances of
interparticle collisions.
Equation (2.2) simpli-
fies to become the
ideal-gas equation
(Equation (1.13))
whenever the volume
V is large.
56
INTRODUCING INTERACTIONS AND BONDS
Table 2.4 Van der Waals constants for various gases
Gas
a/(mol dm 3 ) 2 bar
fc/(mol dm" 3 )- 1
Monatomic
gases
Helium
0.034 589
0.023 733
Neon
0.21666
0.017 383
Argon
1.3483
0.031830
Krypton
2.2836
0.038650
Diatomic gases
Hydrogen
0.24646
0.026665
Nitrogen
1.3661
0.038577
Oxygen
1.3820
0.031860
Carbon monoxide
1.4734
0.039523
Polyatomic
gases
Ammonia
4.3044
0.037 847
Methane
2.3026
0.043 067
Ethane
5.5818
0.065 144
Propane
9.3919
0.090494
Butane
13.888
0.11641
Benzene
18.876
0.11974
The value of p calcu-
lated with the ideal-gas
equation is 3.63 x
10 5 Pa, or 3.63 bar.
Worked Example 2.1 0.04 mol of methane gas is enclosed within
a flask of volume 0.25 dm 3 . The temperature is 0°C. From Table 2.4,
a = 2.3026 dm 6 bar mol" 2 and b = 0.043 067 dm 3 mol - ' . What is the
pressure p exerted?
We first rearrange Equation (2.2), starting by dividing both sides by
the term (V — nb), to yield
Calculations with the
van der Waals equation
are complicated be-
cause of the need to
convert the units to
accommodate the SI
system. The value of/?
comes from Table 2.3.
P+T7T
nRT
V 2 (V-nb)
We then subtract (n 2 a) H- V 2 from both sides:
nRT
P =
(V-nb)
($)'
Next, we insert values and convert to SI units, i.e. 0°C is expressed
as 273.15 K.
P =
0.04 molx(8.314xl0" 2 dm 3 bar K" 1 mol _1 )x273.15 K
(0.25 dm 3 - 0.04 mol x 0.043 067 dm 3 mol" 1 )
0.04 molY
0.25 dm 3 /
l\2
x 2.3026 (dm J mol" 1 ) 2 bar
QUANTIFYING THE INTERACTIONS AND THEIR INFLUENCE
57
p = 3.65887 bar - 0.05895 bar
p = 3.59992 bar
The pressure calculated with the ideal-gas equation (Equation (1.13)) is 3.63 bar, so
the value we calculate with the van der Waals equation (Equation (2.2)) is 1 per cent
smaller. The experimental value is 3.11 bar, so the result with the van der Waals equation
is superior.
The lower pressure causes coalescence of gas particles, which decreases their kinetic
energy. Accordingly, the impact between the aggregate particle and the container's
walls is less violent, which lowers the observed pressure.
The virial equation
An alternative approach to quantifying the interactions and deviations from the ideal-
gas equation is to write Equation (1.13) in terms of 'virial coefficients':
V
RT (1 + B'p + C'p 2 + ...)
(2.3)
The word 'virial' comes
from the Latin for force
or powerful.
where the V 4- n term is often rewritten as V m and called the molar volume.
Equation (2.3) is clearly similar to the ideal-gas equation,
Equation (1.13), except that we introduce additional terms, each
expressed as powers of pressure. We call the constants, B' ', C etc.,
'virial coefficients', and we determine them experimentally. We
call B' the second virial coefficient, C" the third, and so on.
Equation (2.3) becomes the ideal-gas equation if both B' and C are tiny. In fact,
these successive terms are often regarded as effectively 'fine-tuning' the values of p or
V m . The C coefficient is often so small that we can ignore it; and D' is so minuscule
that it is extremely unlikely that we will ever include a fourth virial coefficient in any
calculation. Unfortunately, we must exercise care, because B' constants are themselves
a function of temperature.
Worked Example 2.2 What is the molar volume V m of oxygen gas
at 273 K and p^l Ignore the third and subsequent virial terms, and
take B' = -4.626 x 10" 2 bar" 1 .
From Equation (2.3)
V RT
x (1 + B'p)
Care: the odd-looking
units of B' require us
to cite the gas con-
stant R in SI units with
prefixes.
58
INTRODUCING INTERACTIONS AND BONDS
The value of V m cal-
culated with the ideal-
gas equation (Equation
(1.13)) is 4.4 per cent
higher.
Inserting numbers (and taking care how we cite the value of R) yields
83.145 cm 3 barmor' KT 1 x 273 K
1 bar
x (1 - 4.626 x 1(T 2 bar" 1 x 1 bar)
V m = 22 697 x 0.954 cm 3
V m = 21647 cm 3
1 cm 3 represents a volume of 1 x 10~ 6 m 3 , so expressing this value
of V m in SI units yields 21.6 x 10~ 3 m 3 .
SAQ 2.3 Calculate the temperature at which the molar volume of oxygen
is 24 dm 3 . [Hint: you will need some of the data from Worked Example 2.2.
Assume that B' has not changed, and be careful with the units, i.e.
V m = 24 000 cm 3 .]
The relationship bet-
ween 6 and 6' is B' =
(6- RT).
An alternative form of the virial equation is expressed in terms
of molar volume V m rather than pressure:
A positive virial coef-
ficient indicates repul-
sive interactions
between the particles.
The magnitude of B
indicates the strength
of these interaction.
B C
PV m = RT\\ + — + — +
(2.4)
Note that the constants in Equation (2.4) are distinguishable from
those in Equation (2.3) because they lack the prime symbol. For
both Equations (2.3) and (2.4), the terms in brackets represents
the molar compressibility Z. Table 2.5 lists a few virial coeffi-
cients.
SAQ 2.4 Calculate the pressure of 1 mol of gaseous argon housed within
2.3 dm 3 at 600 K. Take B= 11.9 cm 3 mol _1 , and ignore the third virial
term, C. [Hint:
volume to m 3 .]
take care with all units; e.g. remember to convert the
Table 2.5 Virial coefficients B for real gases as a function
of temperature, and expressed in units of cm 3 mol -1
Gas
100 K
273 K
373 K
Argon
-187.0
-21.7
-4.2
Hydrogen
-2.0
13.7
15.6
Helium
11.4
12.0
11.3
Nitrogen
-160.0
-10.5
6.2
Neon
-6.0
10.4
12.3
Oxygen
-197.5
-22.0
-3.7
CREATING FORMAL CHEMICAL BONDS
59
The word 'chlorine'
derives from the Greek
chloros, meaning
'green'.
2.3 Creating formal chemical bonds
Why is chlorine gas lethal yet sodium chloride is vital
for life?
The interaction requires electrons
Chlorine gas is very reactive, and causes horrific burns to the eyes
and throat; see p. 243. The two atoms are held together by means of
a single, non-polar covalent bond. Q2 has a yellow -green colour
and, for a gas, is relatively dense at s.t.p. Conversely, table salt
(sodium chloride) is an ionic solid comprising Na + and Cl~ ions,
held together in a three-dimensional array. What is the reason for
their differences in behaviour?
The outer shell of each 'atom' in CI2 possesses a full octet of electrons: seven
electrons of its own (which explains why it belongs to Group VII(b) of the periodic
table) and an extra electron from covalent 'sharing' with the other atom in the CI2
molecule. The only other simple interactions in molecular chlorine are the inevitable
induced dipolar forces, which are too weak at room temperature to allow for the
liquefying of Cl2( g ) .
Each chloride ion in NaCl also has eight electrons: again, seven electrons come
from the element prior to formation of a chloride ion, but the extra eighth electron
comes from ionizing the sodium counter ion. This extra electron resides entirely on
the chloride ion, so no electrons are shared. The interactions in solid NaCl are wholly
ionic in nature. Induced dipoles will also exist within each ion, but their magnitude
is utterly negligible when compared with the strength of the formal charges on the
Na + and Cl~ ions. We are wise to treat them as absent.
So, in summary, the principal differences between Cl2( g ) and NaCl( s ) lie in the
location and the interactions of electrons in the atoms' outer shells. We say these
electrons reside in an atom's frontier orbitals, meaning that we can ignore the inner
electrons, which are tightly bound to the nucleus.
Why does a bicycle tyre get hot when inflated?
Bonds and interactions involve energy changes
A bicycle tyre gets quite hot during its inflation. The work of
inflating the tyre explains in part why the temperature increases, but
careful calculations (e.g. see pp. 86 and 89) show that additional
factors are responsible for the rise in temperature.
On a macroscopic level, we say we compress the gas into the
confined space within the tyre; on a microscopic level, interparti-
cle interactions form as soon as the gas particles come into close
proximity.
We look on p. 86 at the
effect of performing
'work' while inflating a
bicycle tyre, and the
way work impinges on
the internal energy of
the gas.
60
INTRODUCING INTERACTIONS AND BONDS
All matter seeks to minimize its energy and entropy; see Chapter 4. This concept
explains, for example, why a ball rolls down a hill, and only stops when it reaches
its position of lowest potential energy. These interparticle interactions form for a
similar reason.
When we say that two atoms interact, we mean that the outer electrons on the two
atoms 'respond' to each other. The electrons within the inner orbitals are buried too
deeply within the atom to be available for interactions or bonding. We indicate this
situation by saying the electrons that interact reside within the 'frontier' orbitals.
And this interaction always occurs in such as way as to minimize the energy. We
could describe the interaction schematically by
A + B
product + energy
(2.5)
where A and B are particles of gas which interact when their frontier orbitals are
sufficiently close to form a 'product' of some kind; the product is generally a molecule
or association complex. (A less naive view should also accommodate changes in
entropy; see Chapter 4.)
We saw earlier (on p. 33) that measuring the temperature is the
simplest macroscopic test for an increased energy content. There-
fore, we understand that the tyre becomes warmer during inflation
because interactions form between the particles with the concurrent
release of energy (Equation (2.5)).
Energy is liberated
when bonds and inter
actions form.
How does a fridge cooler work?
Introduction to the energetics of bond formation
At the heart of a fridge's cooling mechanism is a large flask containing volatile
organic liquids, such as alkanes that have been partially fluorinated and/or chlorinated,
which are often known as halons or chlorofluorocarbons (CFCs). We place this flask
behind the fridge cabinet, and connect it to the fridge interior with a thin-walled pipe.
The CFCs circulate continually between the fridge interior and the rear, through a
heat exchanger.
Now imagine placing a chunk of cheese in the refrigerator. We need to cool the
cheese from its original temperature to, say, 5 °C. Because the cheese is warmer
than the fridge interior, energy in the form of heat transfers from
the cheese to the fridge, as a consequence of the zeroth law of
thermodynamics (see p. 8). This energy passes ultimately to the
volatile CFCs in the cooling system.
The CFC is initially a liquid because of intermolecular interac-
tions (of the London dispersion type). Imagine that the interactions
involves 4 kJ of energy but cooling the cheese to 5 °C we liberate
about 6 kJ of energy: it should be clear that more energy is liber-
ated than is needed to overcome the induced dipoles. We say that
Converting the liquid
CFC to a gas (i.e. boil
ing) is analogous to
putting energy into a
kettle, and watching
the water boil off as
steam.
CREATING FORMAL CHEMICAL BONDS
61
absorption of the energy from the cheese 'overcomes' the interactions - i.e. breaks
them - and enables the CFC to convert from its liquid form to form a gas:
CFC(i) + energy
CFC
(g)
(2.6)
The fridge pump circulates the CFC, so the hotter (gaseous) CFC is removed from
the fridge interior and replaced with cooler CFC (liquid). We increase the pressure of
the gaseous CFC with a pump. The higher pressure causes the CFC to condense back
to a liquid. The heat is removed from the fridge through a so-called heat exchanger.
Incidentally, this emission of heat also explains why the rear of a
fridge is generally warm - the heat emitted is the energy liberated
when the cheese cooled.
In summary, interactions form with the liberation of energy, but
adding an equal or greater amount of energy to the system can
break the interactions. Stated another way, forming bonds and inter-
actions liberates energy; breaking bonds and interactions requires
the addition of energy.
A similar mechanism operates at the heart of an air-conditioning mechanism in a
car or office.
Forming bonds and
interactions liberates
energy; breaking bonds
and interactions re-
quires the addition
of energy.
Why does steam warm up a cappuccino coffee?
Forming a bond releases energy: introducing calorimetry
To make a cappuccino coffee, pass high-pressure steam through a cup of cold milk
to make it hot, then pour coffee through the milk froth. The necessary steam comes
from a kettle or boiler.
A kettle or boiler heats water to its boiling point to effect the process:
H2OQ + heat energy
H 2
2 u (g)
(2.7)
Water is a liquid at room temperature because cohesive forces bind together the
molecules; the bonds in this case are hydrogen bonds - see p. 44. To effect the phase
transition, liquid — > gas, we overcome the hydrogen bonds, which explains why we
must put energy into liquid water to generate gaseous steam. Stated another way,
steam is a high-energy form of water.
Much of the steam condenses as it passes through the cool milk. This condensation
occurs in tandem with forming the hydrogen bonds responsible for
the water being a liquid. These bonds form concurrently with the
liberation of energy. This energy transfers to the milk, explaining
why its temperature increases.
Calorimetry is the measurement of energy changes accompany-
ing chemical or physical changes. We usually want to know how
much energy is liberated or consumed per unit mass or mole of
The word 'calorimetry'
comes from the Latin
calor, which means
heat. We get the word
'calorie' from the same
root.
62
INTRODUCING INTERACTIONS AND BONDS
substance undergoing the process. Most chemists prefer data to be
presented in the form of energy per mole. In practice, we measure
accurately the amount of heat energy liberated or consumed by a
known amount of steam while it condenses.
A physical chemist reading from a data book learns that 40.7 kJ
mol -1 of energy are liberated when 1 mol of water condenses and
will 'translate' this information to say that when 1 mol (18 g) of
steam condenses to form liquid water, bonds form concurrently
with the liberation of 40 700 J of energy.
As 40.7 kJmol - is the molar energy (the energy per mole), we
can readily calculate the energy necessary, whatever the amount of water involved.
In fact, every time the experiment is performed, the same amount of energy will be
liberated when 18 g condense.
Worked Example 2.3 How much energy is liberated when 128 g of water con-
denses?
Strictly, this amount
of energy is liber-
ated only when the
temperature remains
at 100°C during the
condensation process.
Any changes in tem-
perature need to be
considered separately.
Note the way the units
of 'g' cancel, to leave n
expressed in the units
of moles.
Firstly, we calculate the amount of material n involved using
mass in grams
amount of material n — (2.8)
molar mass in grams per mole
so, as 1 mol has a mass of 18 gmol -1
128 g
n — r
18 gmol" 1
n — 7.11 mol
Secondly, the energy liberated per mole is 40.7 kJ mol ' , so the overall amount of energy
given out is 40.7 kJmor 1 x 7.11 mol = 289 kJ.
SAQ 2.5 How much energy will be liberated when 21 g
of water condense?
Cappuccino coffee is
named after Marco
d'Aviano, a 'Capuchin'
monk who was recently
made a saint. He
entered a looted Turk-
ish army camp, and
found sacks of roasted
coffee beans. He mixed
it with milk and honey
to moderate its bit-
ter flavour.
A physical chemist will go one stage further, and say that this
energy of 40.7 kJmol -1 relates directly to processes occurring dur-
ing the condensation process. In this case, the energy relates to the
formation of hydrogen bonds.
As each water molecule forms two hydrogen bonds, so 1 mol of
water generates 2 mol of hydrogen bonds. The energy per hydrogen
bond is therefore (40.7 kJmol - 4-2); so the energy of forming a
hydrogen bond is 20.35 kJmol -1 .
In summary, the macroscopic changes in energy measured in an
experiment such as this are a direct reflection of microscopic energy
changes occurring on the molecular level. The milk of a cappuccino
coffee is warmed when steam passes through it because the steam
CREATING FORMAL CHEMICAL BONDS
63
condenses to form liquid water; and the water is a liquid because of the formation of
intermolecular forces in the form of hydrogen bonds.
The reaction of elemen-
tal nitrogen to form
compounds that can
be readily metabolized
by a plant is termed
'fixing'. All the principal
means of fixing nitro-
gen involve bacteria.
Why does land become more fertile after a
thunderstorm ?
Breaking bonds requires an input of energy
A plant accumulates nutrients from the soil as it grows. Such accumulation depletes
the amount of nutrient remaining in the soil; so, harvesting an arable crop, such as
maize, barley or corn, removes nutrients from the field. A farmer
needs to replenish the nutrients continually if the land is not to
become 'exhausted' after a few seasons.
In the context here, 'nutrients' principally comprise compounds
of nitrogen, most of which come from bacteria that employ natu-
rally occurring catalysts (enzymes) which feed on elemental nitro-
gen - a process known as fixing. An example is the bacterium
Rhizobium which lives on beans and peas. The bacteria convert
atmospheric nitrogen into ammonia, which is subsequently avail-
able for important biological molecules such as amino acids, pro-
teins, vitamins and nucleic acids.
Other than natural fixing, the principal sources of nutrients are the man-made fertil-
izers applied artificially by the farmer, the most common being inorganic ammonium
nitrate (NH4NO3), which is unusually rich in nitrogen.
But lightning is also an efficient fertilizer. The mixture of gases
we breathe comprises nitrogen (78 per cent), oxygen (21 per cent)
and argon (1 per cent) as its principal components. The nitro-
gen atoms in the N2 molecule are bound together tightly via a
triple bond, which is so strong that most reactions occurring dur-
ing plant growth (photosynthesis) cannot cleave it: N2 is inert. But
the incredible energies unleashed by atmospheric lightning are able
to overcome the N=N bond.
The actual mechanism by which the N=N molecule cleaves is
very complicated, and is not fully understood yet. It is nevertheless
clear that much nitrogen is oxidized to form nitrous oxide, NO. This NO dissolves in
the water that inevitably accompanies lightning and forms water-soluble nitrous acid
HNO2, which further oxidizes during the storm to form nitric acid, HNO3. Nitric acid
functions as a high-quality fertilizer. It has been estimated that a thunderstorm can
yield many tonnes of fertilizer per acre of land.
To summarize, the NsN bond in the nitrogen molecule is very
strong and cannot be cleaved unless a large amount of energy is
available to overcome it. Whereas bacteria can fix nitrogen, the
biological processes within crops, such as corn and maize, can-
not provide sufficient energy. But the energy unleashed during a
Notice the difference
between the two words
'princiPAL' (meaning
'best', 'top' or 'most
important') and 'prin-
ciPLE' (meaning 'idea',
'thought' or 'concept').
We require energy to
cleave bonds: bond
energies are discussed
on p. 114.
64
INTRODUCING INTERACTIONS AND BONDS
thunderstorm easily overcomes the N=N bond energy, fixing the nitrogen without
recourse to a catalyst.
Aside
In the high mountains of Pashawa in Pakistan, near the border with Afghanistan, thun-
derstorms are so common that the soil is saturated with nitrates deriving from the
nitric acid formed by lightning. The soil is naturally rich in potassium compounds.
Ion-exchange processes occur between the nitric acid and potassium ions to form large
amounts of potassium nitrate, KNO3, which forms a thick crust of white crystals on the
ground, sometimes lending the appearance of fresh snow.
High concentrations of KNO3 are relatively toxic to plant growth because the ratio
of K + to Na + is too high, and so the soil is not fertile.
The hydrogen atoms in
space form a 'hydride'
with the materials on
the surface of the satel-
lite.
Why does a satellite need an inert coating?
Covalency and bond formation
A satellite, e.g. for radio or TV communication, needs to be robust to withstand its
environment in space. In particular, it needs to be protected from the tremendous
gravitational forces exerted during take off, from the deep vacuum of space, and from
atoms in space.
Being a deep vacuum, there is a negligible 'atmosphere' sur-
rounding a satellite as it orbits in space. All matter will exist solely
as unattached atoms (most of them are hydrogen). These atoms
impinge on the satellite's outer surface as it orbits. On Earth, hydro-
gen atoms always seek to form a single bond. The hydrogen atoms
in space interact similarly, but with the satellite's tough outer skin.
Such interactions are much stronger than the permanent hydrogen
bonds or the weaker, temporary induced dipoles we met in Section 2.1. They form a
stronger interaction, which we call a covalent bond.
The great American scientist G. N. Lewis coined the word covalent, early in the
20th century. He wanted to express the way that a bond formed by means of electron
sharing. Each covalent bond comprises a pair of electrons. This pairing is permanent,
so we sometimes say a covalent bond is a formal bond, to distinguish it from weak
and temporary interactions such as induced dipoles.
The extreme strength of the covalent bond derives from the way electrons accumu-
late between the two atoms. The space occupied by the electrons as they accumulate
is not random; rather, the two electrons occupy a molecular orbital that is orientated
spatially in such a way that the highest probability of finding the electronic charges
is directly between the two atomic nuclei.
As we learn about the distribution of electrons within a covalent bond, we start
with a popular representation known as a Lewis structure. Figure 2.11 depicts the
CREATING FORMAL CHEMICAL BONDS
65
Atomic nucleus
• X Electrons
Figure 2.11 Lewis structure of the covalent hydrogen molecule in
which electrons are shared
Lewis structure of the hydrogen molecule, in which each atom of
hydrogen (atomic number 1) provides a single electron. The resul-
tant molecule may be defined as two atoms held together by means
of sharing electrons. Incidentally, we note that the glue holding the
two atoms together (the 'bond') involves two electrons. This result
is common: each covalent bond requires two electrons.
We often call these
Lewis structures 'dot-
cross diagrams'.
Aside
Why call it a 'molecule'?
Colloids are discussed
in Chapter 10.
The word 'molecule' has a long history. The word itself comes from the old French
molecule, itself derived from the Latin molecula, the diminutive of moles, meaning
'mass'.
One of the earliest cited uses of the word dates from 1794, when Adams wrote,
'Fermentation disengages a great quantity of air, that is disseminated among the fluid
molecules'; and in 1799, Kirwen said, 'The molecules of solid abraded and carried
from some spots are often annually recruited by vegetation'. In modern parlance, both
Kirwen and Adams meant 'very small particle'.
Later, by 1840, Kirwen' s small particle meant "micro-
scopic particle' . For example, the great Sir Michael Fara-
day described a colloidal suspension of gold, known then
as Purple of Cassius, as comprising molecules which
were 'small particles'. The surgeon William Wilkinson said in 1851, 'Molecules are
merely indistinct granules; but under a higher magnifying power, molecules become
[distinct] granules'.
Only in 1859 did the modern definition come into being, when the Italian scientist
Stanislao Cannizarro (1826-1910) defined a molecule as 'the smallest fundamental unit
comprising a group of atoms of a chemical compound'. This statement arose while
Cannizarro publicized the earlier work of his compatriot, the chemist and physicist
Amedeo Avogadro (1776-1856).
This definition of a molecule soon gained popularity. Before modern theories of
bonding were developed, Tyndall had clearly assimilated Cannizarro' s definition of a
molecule when he described the way atoms assemble, when he said, 'A molecule is a
group of atoms drawn and held together by what chemists term affinity'.
66 INTRODUCING INTERACTIONS AND BONDS
Why does water have the formula H 2 OP
Covalent bonds and valence
The water molecule always has a composition in which two hydrogen atoms combine
with one oxygen. Why?
The Lewis structure in Figure 2.1 1 represents water, H2O. Oxygen is element num-
ber eight in the periodic table, and each oxygen atom possesses six electrons in its
outer shell. Being a member of the second row of the periodic table, each oxygen
atom seeks to have an outer shell of eight electrons - we call this trend the 'octet
rule'. Each oxygen atom, therefore, has a deficiency of two electrons. As we saw
immediately above, an atom of hydrogen has a single electron and, being a row 1
element, requires just one more to complete its outer shell.
The Lewis structure in Figure 2.12 shows the simplest way in
The concept of the full
outer shell is crucial if
we wish to understand
covalent bonds.
which nature satisfies the valence requirements of each element:
each hydrogen shares its single electron with the oxygen, meaning
that the oxygen atom now has eight electrons (six of its own - the
crosses in the figure) and two from the hydrogen atoms. Looking
now at each hydrogen atom (the two are identical), we see how
each now has two electrons: its own original electron (the dot in the diagram) together
with one extra electron each from the oxygen (depicted as crosses).
We have not increased the number of electrons at all. All we have done is shared
them between the two elements, thereby enabling each atom to have a full outer shell.
This approach is known as the electron-pair theory.
Valence bond theory
The valence bond theory was developed by Linus Pauling (1901-1992) and others in
the 1930s to amalgamate the existing electron-pair bonding theory of G. N. Lewis and
new data concerning molecular geometry. Pauling wanted a single, unifying theory.
He produced a conceptual framework to explain molecular bonding, but in practice
it could not explain the shapes of many molecules.
^ Atomic nucleus
• X Electrons
Figure 2.12 Lewis structure of the covalent water molecule. The inner shell of the oxygen atom
has been omitted for clarity
CREATING FORMAL CHEMICAL BONDS
67
Nevertheless, even today, we often discuss the bonding of organic compounds in
terms of Lewis structures and valence bond theory.
Why is petroleum gel so soft?
Properties of covalent compounds
Clear petroleum gel is a common product, comprising a mixture of simple hydro-
carbons, principally n-octadecane (III). It is not quite a solid at room temperature;
neither is it really a liquid, because it is very viscous. We call it a gel. Its principal
applications are to lubricate (in a car) or to act as a water-impermeable barrier (e.g.
between a baby and its nappy, or on chapped hands).
(Ill)
We saw on p. 52 how methane is a gas unless condensed by
compression at high pressure or frozen to low temperatures. But
octadecane is neither a solid nor a gas. Why?
There are several, separate types of interaction in III: both cova-
lent bonds and dipoles. Induced dipoles involve a partial charge,
which we called <5 + or <5~, but, by contrast, covalent bonds involve
whole numbers of electrons. A normal covalent bond, such as that
between a hydrogen atom and one of the carbon atoms in the back-
bone of III, requires two electrons. A 'double bond' consists simply
of two covalent bonds, so four electrons are shared. Six electrons
are incorporated in each of the rare instances of a covalent 'triple
bond'. A few quadruple bonds occur in organometallic chemistry,
but we will ignore them here.
Most covalent bonds are relatively non-polar. Some are com-
pletely non-polar: the diatomic hydrogen molecule is held together
with two electrons located equidistantly from the two hydrogen
nuclei. Each of the two atoms has an equal 'claim' on the elec-
trons, with the consequence that there is no partial charge on the
atoms: each is wholly neutral. Only homonuclear molecules such
as H2, F2, O2 or N2 are wholly non-polar, implying that the major-
ity of covalent bonds do possess a slight polarity, arising from an
unequal sharing of the electrons bound up within the bond.
We see the possibility of a substance having several types of
bond. Consider water for example. Formal covalent bonds hold
together the hydrogen and oxygen atoms, but the individual water
molecules cohere by means of hydrogen bonds. Conversely, paraf-
fin wax (n-C^f^) is a solid. Each carbon is bonded covalently
Molecules made of only
one element are called
'homonuclear', since
homo is Greek for
'same'. Examples of
homonuclear molecules
are H 2 , N 2 , S 8 and ful-
lerene C6o-
Even a covalent bond
can possess a perma-
nent induced dipole.
Covalent compounds
tend to be gases or liq-
uids. Even when solid,
they tend to be soft.
But many covalent
compounds are only
solid at lower temper-
atures and/or higher
pressures, i.e. by max-
imizing the incidence of
induced dipoles.
68
INTRODUCING INTERACTIONS AND BONDS
Table 2.6 Typical properties covalent compounds
Property
Example
Low melting point
Low boiling point
Physically soft
Malleable, not brittle
Low electrical conductivity
Dissolve in non-polar solvents
Insoluble in polar solvents
Ice melts in the mouth
Molecular nitrogen is a gas at room
temperature
We use petroleum jelly as a lubricant
Butter is easily spread on a piece of bread
We insulate electrical cables with plastic"
We remove grease with methylated spirit b
Polyurethane paint protects the window frame
from rain
"The polythene coating on an electrical wire comprises a long-chain alkane.
b 'Methylated spirit' is the industrial name for a mixture of ethanol and methanol.
^^M4r^
Figure 2.13 Diamond has a giant macroscopic structure in which
each atom is held in a rigid three-dimensional array. Other covalent
solids include silica and other p-block oxides such as AI2O3
to one or two others to form a linear chain; the hydrogen atoms are bound to this
backbone, again with covalent bonds. But the wax is a solid because dispersion
forces 'glue' together the molecules. Table 2.6 lists some of the common properties
of covalent compounds.
Finally, macromolecular covalent solids are unusual in comprising atoms held
together in a gigantic three-dimensional array of bonds. Diamond and silica are
the simplest examples; see Figure 2.13. Giant macroscopic structures are always
solid.
Aside
The word covalent was coined in 1919 when the great American Chemist Irving Lang-
muir said, 'it is proposed to define valence as the number of pairs of electrons which
a given atom shares with others. In view of the fact . . . that 'valence' is very often
used to express something quite different, it is recommended that the word covalence
be used to denote valence defined as above.' He added, 'In [ionic] sodium chloride, the
covalence of both sodium and chlorine is zero'.
The modern definition from IUPAC says, 'A covalent bond is a region of rela-
tively high electron density between nuclei which arises (at least partially) from shar-
ing of electrons, and gives rise to an attractive force and characteristic inter-nuclear
distance'.
CREATING FORMAL CHEMICAL BONDS
69
Why does salt form when sodium and chlorine react?
Bond formation with ions
Ionic interactions are electrostatic by nature, and occur between ions of opposite charge.
The overwhelming majority of ionic compounds are solids, although a few biological
exceptions do occur. Table 2.7 lists a few typical properties of ionic compounds.
It is generally unwise to think of ionic compounds as holding together with physical
bonds; it is better to think of an array of point charges, held together by the balance of
their mutual electrostatic interactions. (By 'mutual' here, we imply equal numbers of
positive and negative ions, which therefore impart an overall charge of zero to the solid.)
Ionic compounds generally form following the reaction of metal-
lic elements; non-metals rarely have sufficient energy to provide
the necessary energy needed to form ions (see p. 123).
The structure in Figure 2.14 shows the result of an ionic reac-
tion: sodium metal has reacted with chlorine gas to yield white
crystalline sodium chloride, NaCl. Each Na atom has lost an elec-
tron to form an Na + cation and each chlorine atom has gained an
electron and is hence a Cl~ anion. In practice, the new electron
possessed by the chloride came from the sodium atom.
The electron has transferred and in no way is it shared. Sodium
chloride is a compound held together with an ionic bond, the
strength of the bond coming from an electrostatic interaction bet-
ween the positive and negative charges on the ions.
Care: chlorlNE is an
elemental gas; chlo-
rlDE is a negatively
charged anion.
The chloride ion has
a negative charge
because, following ion-
ization, it possesses
more electrons than
protons.
Why heat a neon lamp before it will generate light?
Ionization energy
Neon lamps generate a pleasant pink-red glow. Gaseous neon within the tube (at
low pressure) is subjected to a strong electric discharge. One electron per neon atom
Table 2.7 Typical properties of ionic compounds
Property
Example
High melting point
High boiling point
Physically hard
Often physically brittle
High electrical conductivity in
solution
Dissolve in polar solvents
Insoluble in non-polar solvents
We need a blast furnace to melt metals
A lightning strike is needed to volatilize some substances
Ceramics (e.g. plates) can bear heavy weights
Table salt can be crushed to form a powder
Using a hair dryer in the bath risks electrocution
Table salt dissolves in water
We dry an organic solvent by adding solid CaCl2
70
INTRODUCING INTERACTIONS AND BONDS
Sodium cation, Na +
Chloride anion, CI
^ Atomic nucleus
• X Electrons
Figure 2.14 Lewis structure of ionic sodium chloride. Note how the outer shell of the sodium
ion is empty, so the next (inner) shell is full
is lost, forming positively charged Ne + ions:
Ne
(g)
-> Ne + (g) + e"
(g)
(2.9)
Generally, the flask
holding the neon gas
contains a small amount
of sodium to catalyse
('kick start') the ion-
ization process - see
p. 481.
'Monatomic' is an ab-
breviation for 'mono-
atomic', meaning the
'molecule' contains
only one atom. The
word generally applies
to the Group VIII(a)
rare gases.
We 'ionized' the neon atoms to form Ne + cations, i.e. each bears a
positive charge. (On p. 480 we discuss in detail the photochemical
processes occurring at the heart of the neon lamp.)
We generally need quite a lot of energy to ionize an atom or
molecule. For example, 2080 kJ of energy are required to ionize
1 mol of monatomic neon gas. This energy is large and explains
the need to heat the neon strongly via a strong electric discharge.
We call this energy the ionization energy, and give it the symbol
/ (some people symbolize it as I e ). Ionization energy is defined
formally as the minimum energy required to ionize 1 mol of an
element, generating 1 mol of electrons and 1 mol of positively
charged cations.
The energy required will vary slightly depending on the condi-
tions employed, so we need to systematize our terminology. While
the definition of / is simple enough for neon gas, we need to
be more careful for elements that are not normally gaseous. For
example, consider the process of ionizing the sodium catalyst at
the heart of the neon lamp. In fact, there are two energetically
distinct processes:
(1) Vaporization of the sodium, to form a gas of sodium atoms: Na( S )
(2) Ionization of gaseous atoms to form ions: Na( g ) — »■ Na + ( g ) + e~(g).
Na
(g)-
To remove any possible confusion, we further refine the definition of ionization
energy, and say that / is the minimum energy required to ionize 1 mol of a gaseous
element. The ionization energy / relates to process (2); process (1) is additional.
CREATING FORMAL CHEMICAL BONDS
71
Table 2.8 Ionization energies / ( „). For convenience, the figures in the table are given in MJ mol
rather than the more usual kj moP 1 to emphasize their magnitudes
Element
/,
(i)
/,
(2)
'(3)
/,
(4)
/,
(5)
h6
(6,
'(7)
'(H)
'(<■»
'(10)
Hydrogen
1.318
Helium
2.379
5.257
Lithium
0.526
7.305
11.822
Beryllium
0.906
1.763
14.855
21.013
Boron
0.807
2.433
3.666
25.033
32.834
Carbon
1.093
2.359
4.627
6.229
37.838
47.285
Nitrogen
1.407
2.862
4.585
7.482
9.452
53.274
64.368
Oxygen
1.320
3.395
5.307
7.476
10.996
13.333
71.343
84.086
Fluorine
1.687
3.381
6.057
8.414
11.029
15.171
17.874
92.047
106.443
Neon
2.097
3.959
6.128
9.376
12.184
15.245
20.006
23.076
115.389 131.442
2500
40 60
Atomic number
Figure 2.15
(as x)
The first ionization energies / of the first 105 elements (as y) against atomic number
Table 2.8 lists several ionization energies: notice that all of them are positive.
Figure 2.15 depicts the first ionization energies 1^ (as y) for the elements hydrogen
to nobelium (elements 1-102) drawn as a function of atomic number (as x).
It is clear from Figure 2.15 that the rare gases in Group VIII(b) have the highest
values of /, which is best accounted for by noting that they each have a full outer shell
of electrons and, therefore, are unlikely to benefit energetically from being ionized.
Similarly, the halogens in Group VII(b) have high values of / because their natural
tendency is to accept electrons and become anions X~, rather than to lose electrons.
The alkali metals in Group 1(a) have the lowest ionization energies, which is again
expected since they always form cations with a +1 valence. There is little variation
in / across the d-block and f-block elements, with a slight increase in / as the atomic
number increases.
72
INTRODUCING INTERACTIONS AND BONDS
Why does lightning conduct through air?
Electron affinity
Lightning is one of the more impressive manifestations of the power in nature: the
sky lights up with a brilliant flash of light, as huge amounts of electrical energy pass
through the air.
As an excellent generalization, gases may be thought of as electrical insulators, so
why do we see the lightning travel through the air? How does it conduct? Applying a
huge voltage across a sample of gas generates an electric discharge, which is apparent
by the appearance of light. In fact, the colour of the light depends on the nature of the
gas, so neon gives a red colour, krypton gives a green colour and helium is invisible
to the eye, but emits ultraviolet light.
The source of the light seen with an electric discharge is the plasma formed by
the electricity, which is a mixture of ions and electrons, and unionized atoms. If,
for example, we look solely at nitrogen, which represents 78 per cent of the air, an
electric discharge would form a plasma comprising N2" 1 ", N + , electrons e~, nitrogen
radicals N*, as well as unreacted N2. Incidentally, the formation of these ions explains
how air may conduct electricity.
Very soon after the electric discharge, most of the electrons and nitrogen cations
reassemble to form uncharged nitrogen, N2. The recombination produces so much
energy that we see it as visible light - lightning. Some of the electrons combine with
nitrogen atoms to form nitrogen anions N~, via the reaction
N (g) + e (g)
N"
(g)
(2.10)
Care: do not con-
fuse the symbols for
electron affinity E (ea)
and activation energy
E a from kinetics (see
Chapter 8).
and, finally, some, N2" 1 " cations react with water or oxygen in the
air to form ammonium or hydroxy lamine species.
The energy exchanged during the reaction (in Equation (2.10))
is called the 'electron affinity' £( ea )- This energy (also called the
electron attachment energy) is defined as the change in the inter-
nal energy that occurs when 1 mol of atoms in the gas phase are
converted by electron attachment to form 1 mol of gaseous ions.
The negative ions formed in Equation (2.10) are called anions.
Most elements are sufficiently electronegative that their electron
affinities are negative, implying that energy is given out during
the electron attachment. For example, the first electron affinity
of nitrogen is only 7 kJmol -1 , but for chlorine £ (ea ) =
—364 kJmol - . Table 2.9 lists the electron affinities of gaseous
halogens, and Figure 2.16 depicts the electron affinities for the first
20 elements (hydrogen-calcium).
The electron affinity measures the attractive force between the incoming electron
and the nucleus: the stronger the attraction, the more energy there is released. The
factors that affect this attraction are exactly the same as those relating to ionization
The negative value of
£ ( ea) illustrates how the
energy of the species
X~ (g) is lower than that
of its precursor, X (g) .
CREATING FORMAL CHEMICAL BONDS
73
100
3=
03
C
O
V,
LU
-100
-200
-300
-400
Table 2.9 The first electron affini-
ties of the Group VII(a) elements
Gas
^(eaj/kJ mol
F 2
Cl 2
Br 2
h
-348
-364
-342
-314
Note: there is much disagreement in the
literature about the exact values of elec-
tron affinity. These values are taken from
the Chemistry Data Book by Stark and
Wallace. If we use a different data source,
we may find slightly different numbers.
The trends will be the same, whichever
source we consult.
Atomic number
Figure 2.16 Graph of the electron affinities E (ea) of the first 30 elements (as y) against atomic
number (as x)
energies - nuclear charge, the distance between the nucleus and the electron, as
well as the number of electrons residing between the nucleus and the outer,
valence electrons.
Aside
We must be careful with the definition above: in many older textbooks, the electron
affinity is defined as the energy released when an electron attaches to a neutral atom.
This different definition causes £ (ea ) to change its sign.
74
INTRODUCING INTERACTIONS AND BONDS
Krypton, xenon and
radon will form a
very limited number
of compounds, e.g.
with fluorine, but only
under quite excep-
tional conditions.
Why is argon
First electron affinity and reactivity
Gases such as helium, neon and argon are so unreactive that we
call them the inert gases. They form no chemical compounds, and
their only interactions are of the London dispersion force type.
They cannot form hydrogen bonds, since they are not able to bond
with hydrogen and are not electronegative.
The outer shell of the helium atom is full and complete: the
shell can only accept two electrons and, indeed, is occupied by
two electrons. Similarly, argon has a complete octet of electrons
in its outer shell. Further reaction would increase the number of
electrons if argon were to undergo a covalent bond or become an anion, or would
decrease the number of electrons below the 'perfect' eight if a cation were to form.
There is no impetus for reaction because the monatomic argon is already at its position
of lowest energy, and we recall that bonds form in order to decrease the energy.
Sodium atoms always seek to lose a single electron to form the Na + monocation,
because the outer valence shell contains only one electron - that is why we assign
sodium to Group 1(a) of the periodic table. This single electron helps us explain why
it is so favourable, energetically, to form the Na + cation: loss of the electron empties
the outer shell, to reveal a complete inner shell, much like removing the partial skin
of an onion to expose a perfectly formed inner layer. So, again, removal of sodium's
single outer electron occurs in order to generate a full shell of electrons.
But if we look at an element like magnesium, there are several ionization pro-
cesses possible:
(1) Formation of a monocation: Mg ( •. —> Mg + ( g) + e ( g ).
(2) Formation of a dication: Mg" 1
(g)
M:
,2+
(g) + e (g ).
The energy change in reaction (1) is called the first ionization
energy and the energy associated with reaction (2) is the second
ionization energy. We symbolize the two processes as I^ and Iq)
respectively.
The second ionization energy is always larger than the first,
because we are removing a negative electron from a positively
charged cation, so we need to overcome the attractive force bet-
ween them. The value of 7(i) for a magnesium atom is 734 kJ
mol -1 , but 1(2) for removing an electron from the Mg + monoca-
tion is 1451 kJmol -1 . Both ionization energies are huge, but 7(2) is clearly much the
larger. Table 2.8 contains many other ionization energies for elements 1-10.
It is clear from Table 2.8 that each ionization energy is larger than the one before.
Also note that the last two ionization energies of an element are always larger than the
others. The sudden rise follows because the last two energies represent the removal
of the two 1 s electrons: removal of electrons from the 2s and 2p orbitals is easier.
Care: do not con-
fuse the symbols for
molecular iodine I 2 and
the second ionization
energy 7 (2) . Hint: note
carefully the use of
italic type.
CREATING FORMAL CHEMICAL BONDS
75
Why is silver iodide yellow?
Mixed bonding
Silver chloride is white; silver bromide is pale yellow; and silver iodide has a rich
yellow colour. We might first think that the change in colour was due to Agl incor-
porating the iodide anion, yet Nal or HI are both colourless, so the colour does not
come from the iodide ions on their own. We need to find a different explanation.
Silver iodide also has other anomalous properties: it is physically soft - it can
even be beaten into a sheet, unlike the overwhelming majority of ionic compounds.
More unusual still, it is slightly soluble in ethanol. Clearly, silver iodide is not a
straightforward ionic compound. In fact, its properties appear to overlap between
covalent (see Table 2.6) and ionic (see Table 2.7).
Silver iodide is neither wholly covalent nor wholly ionic; its bonding shows con-
tributions from both. In fact, most formal chemical bonds comprise a contribution
from both covalent and ionic forces. The only exceptions to this general rule are
homonuclear molecules such as hydrogen or chlorine, in which the bonding is 100
per cent covalent. The extent of covalency in compounds we prefer
to think of as ionic will usually be quite small: less than 0.1 per
cent in NaCl. For example, each C-H bond in methane is about 4%
ionic, but the bonding can be quite unusual in compounds compris-
ing elements from the p- and d-blocks of the periodic table. For
example, aluminium chloride, AICI3, has a high vapour pressure
(see p. 221); tungsten trioxide will sublime under reduced pres-
sures to form covalent W3O9 trimers; sulphur trioxide is a gas but
will dissolve in water. Each, therefore, demonstrates a mixture of
ionic and covalent bonding.
In other words, the valence bonds approach is suitable for com-
pounds showing purely ionic or purely covalent behaviour; we
require molecular orbitals for a more mature description of the
bonding in such materials. So the yellow colour of silver iodide
reflects the way the bonding is neither ionic nor covalent. We find,
in fact, that the charge clouds of the silver and iodide ions overlap
to some extent, allowing change to transfer between them. We will
look at charge transfer in more detail on p. 459.
A 'trimer' is a species
comprising three com-
ponents (the Latin
tri means 'three').
The W3O9 trimer has
a triangle structure,
with a WO3 unit at
each vertex.
We require 'molec-
ular orbitals' for a
more mature descrip-
tion of the bonding in
such materials.
Oxidation numbers
Valency is the number of electrons lost, borrowed or shared in a
chemical bond. Formal charges are indicated with Arabic numerals,
so the formal charge on a copper cation is expressed as Cu 2+ ,
meaning each copper cation has a deficiency of two electrons. In
this system of thought, the charge on the central carbon of methane
is zero.
Numbers written as
1,2,3,..., etc. are called
'Arabic
numerals'.
76 INTRODUCING INTERACTIONS AND BONDS
Table 2.10 Rule for assigning oxidation numbers
1 . In a binary compound, the metal has a positive oxidation number and, if a
non-metal, it has a negative oxidation number.
2. The oxidation number of a free ion equals the charge on the ion, e.g. in
Na + the sodium has a +1 oxidation number and chlorine in the Cl~ ion
has an oxidation number of —I. The oxidation number of the MnOzt~ ion
is —I, oxide 2 ~ is —II and the sulphate SOzt 2 ~ ion is —II.
3. The sum of the oxidation numbers in a polyatomic ion equals the
oxidation number of the ions incorporated: e.g. consider MnOzt~ ion.
Overall, its oxidation number is —I (because the ion's charge is —1). Each
oxide contributes —II to this sum, so the oxidation number of the central
manganese must be +VII.
4. The oxidation number of a neutral compound is zero. The oxidation
number of an uncombined element is zero.
5. Variable oxidation numbers:
H = +1 (except in the case of hydrides)
CI = —I (except in compounds and ions containing oxygen)
O = —II (except in peroxides and superoxides)
Unfortunately, many compounds contain bonds that are a mixture of ionic and
covalent. In such a case, a formal charge as written is unlikely to represent the
actual number of charges gained or lost. For example, the complex ferrocyanide anion
[Fe(CN) 6 ] 4 ~ is prepared from aqueous Fe 2+ , but the central iron atom in the complex
definitely does not bear a +2 charge (in fact, the charge is likely
to be nearer +1.5). Therefore, we employ the concept of oxidation
Numbers written as I,
II, III, . .. etc. are called
'Roman numerals'.
number. Oxidation numbers are cited with Roman numbers, so the
oxidation number of the iron atom in the ferrocyanide complex
is +11. The IUPAC name for the complex requires the oxidation
number: we call it hexacyanoferrate (II).
Considering the changes in oxidation number during a reaction can dramatically
simplify the concept of oxidation and reduction: oxidation is an increase in oxidation
number and reduction is a decrease in oxidation number (see Chapter 7). Be aware,
though, oxidation numbers rarely correlate with the charge on an ion. For example,
consider the sulphate anion S0 4 ~ (IV).
O
= S =
O
(IV)
The central sulphur has eight bonds. The ion has an overall charge of —2. The
oxidation number of the sulphur is therefore 8 — 2 = +6. We generally indicate oxi-
dation numbers with roman numerals, though, so we write S(VI). Table 2.10 lists the
rules required to assign an oxidation number.
3
Energy and the first law
of thermodynamics
Introduction
In this chapter we look at the way energy may be converted from one form to another,
by breaking and forming bonds and interactions. We also look at ways of measuring
these energy changes.
While the change in internal energy AU is relatively easy to visualize, chemists
generally concentrate on the net energy AH, where H is the enthalpy. AH relates to
changes in AU after adjusting for pressure -volume expansion work, e.g. against the
atmosphere and after transfer of energy q into and out from the reaction environment.
Finally, we look at indirect ways of measuring these energies. Both internal energy
and enthalpy are state functions, so energy cycles may be constructed according to
Hess's law; we look also at Born-Haber cycles for systems in which ionization
processes occur.
3.1 Introduction to thermodynamics:
internal energy
Why does the mouth get cold when eating ice cream?
Energy
Eating ice cream soon causes the mouth to get cold, possibly to
the extent of making it feel quite uncomfortable. The mouth of a
normal, healthy adult has a temperature of about 37 °C, and the ice
cream has a maximum temperature of °C, although it is likely to
be in the range —5 to —10 °C if it recently came from the freezer.
A large difference in temperature exists, so energy transfers from
the mouth to the ice cream, causing it to melt.
Ice cream melts as it
warms in the mouth
and surpasses its nor-
mal melting tempera-
ture; see Chapter 5.
78
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
The evidence for such a transfer of energy between the mouth and the ice cream is
the change in temperature, itself a response to the minus-oneth law of thermodynamics
(p. 7), which says heat travels from hot to cold. Furthermore, the zeroth law (p. 8)
tells us energy will continue to transfer from the mouth (the hotter object) to the
ice cream (the colder) until they are at the same temperature, i.e. when they are in
thermal equilibrium.
We cannot know how
much energy a body
or system has Mocked'
within it. Experimen-
tally, we can only study
changes in the internal
energy, AU.
Internal energy U
Absolutely everything possesses energy. We cannot 'see' this energy directly, nor do
we experience it except under certain conditions. It appears to be invisible because it
is effectively 'locked' within a species. We call the energy possessed by the object
the 'internal energy', and give it the symbol U.
The internal energy U is defined as the total energy of a body's
components. Unfortunately, there is no way of telling how much
energy is locked away. In consequence, the experimentalist can
only look at changes in U.
The energy is 'locked up' within a body or species in three prin-
cipal ways (or 'modes'). First, energy is locked within the atomic
nuclei. The only way to release it is to split the nucleus, as hap-
pens in atomic weapons and nuclear power stations to yield nuclear
energy. The changes in energy caused by splitting nuclei are mas-
sive. We will briefly mention nuclear energy in Chapter 8, but the
topic will not be discussed otherwise. It is too rare for most physical
chemists to consider further.
This second way in which energy is locked away is within chem-
ical bonds. We call this form of energy the chemical energy, which
is the subject of this chapter. Chemical energies are smaller than
nuclear energies.
And third, energy is possessed by virtue of the potential energy,
and the translational, vibrational, rotational energy states of the
atoms and bonds within the substance, be it atomic, molecular or ionic. The energy
within each of these states is quantized, and will be discussed in greater detail in
Chapter 9 within the subject of spectroscopy. These energies are normally much
smaller than the energies of chemical bonds.
As thermodynamicists, we generally study the second of these
modes of energy change, following the breaking and formation of
bonds (which are held together with electrons), although we occa-
sionally consider potential energy. The magnitude of the chemical
energy will change during a reaction, i.e. while altering the number
and/or nature of the bonds in a chemical. We give the name calorimetry to the study
of energy changes occurring during bond changes.
The energy E locked
into the atomic nucleus
is related to its mass
m and the speed of
light c, according to
the Einstein equation,
E = mc 2 .
Strictly, the bonds
are held together with
'outer-shell' electrons.
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
79
Chemists need to understand the physical chemistry underlying
these changes in chemical energy. We generally prefer to write in
shorthand, so we don't say 'changes in internal energy' nor the
shorter phrase 'changes in [/', but say instead 'At/'. But we need
to be careful: the symbol A does not just mean 'change in'. We
define it more precisely with Equation (3.1):
AU t
(overall)
u,
(final state)
u,
(initial state)
(3.1)
where the phrases 'initial state' and 'final state' can refer to a single
chemical or to a mixture of chemicals as they react. This way, AC/
has both a magnitude and a sign.
Placing the Greek letter
A (Delta) before the
symbol for a parameter
such as U indicates
the change in U while
passing from an initial
to a final state. We
define the change in a
parameter X as AX =
-^(flnal state) — -^(initial state)-
Aside
In some texts, Equation (3.1) is assumed rather than defined, so we have to work out
which are the final and initial states each time, and remember which comes first in
expressions like Equation (3.1). In other texts, the final state is written as a subscript
and the initial state as a superscript. The value of AU for melting ice cream would
be written as AU
(final state)
(initial state) '
• Le - A ^("e 1 crea^ e befor ) melting)- Zt ma Y even be abbreviated to
AI/ S , where s = solid and 1 = liquid.
To further complicate matters, other books employ yet another notation. They retain
the sub- and super- scripts, but place them before the variable, so the last expression in
the previous paragraph would be written as A\U.
We will not use any of these notation styles in this book.
Why is skin scalded by steam?
Exothermic reactions
Water in the form of a gas is called 'steam'. Two things hap-
pen concurrently when human skin comes into close contact with
steam - it could happen, for example, when we get too close to
a boiling kettle. Firstly, the flesh in contact with the steam gets
burnt and hurts. Secondly, steam converts from its gaseous form
to become liquid water. We say it condenses. We summarize the
condensation reaction thus:
H,0
2^(g)
H,0
The 'condensation'
reaction is one of the
simplest forms of a
'phase change', which
we discuss in greater
depth in Chapter 5.
2^(1)
(3.2)
80
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
We will use the word
'process' here to mean
any physical chemistry
requiring a change in
energy.
From Equation (3.1), which defines the changes to internal energy,
AC/ for the process in Equation (3.2) is A C/ (condensation) = £/ (water , i)
C/ (water, g) •
As we saw in Chapter 2, the simplest way of telling whether some-
thing gains energy is to ascertain whether its temperature goes up.
The temperature of the skin does increase greatly (so it feels hot); its
energy increases following the condensation reaction. Conversely,
the temperature of the water decreases - indeed, its temperature decreases to below its
boiling temperature, so it condenses. The water has lost energy. In summary, we see
how energy is transferred, with energy passing/rom the steam to the skin.
When energy passes from one body to another, we say the
process is thermodynamic. The condensation of water is a ther-
modynamic process, with the energy of the water being lower fol-
lowing condensation. Stated another way, the precursor steam had
more energy than the liquid water product, so C/(fi na i) is lower than
^(initial) - Figure 3.1 represents this situation visually, and clearly
shows how the change in internal energy AC/ during steam con-
densation is negative. We say the change in U is exothermic.
The energy lost by the steam passes to the skin, which therefore
gains energy. We experience this excess energy as burning: with the
skin being an insulator, the energy from the steam remains within
the skin and causes damaging thermal processes. Nerve endings in the skin report the
damage to the brain, which leads to the experience of pain.
But none of the energy is lost during condensation, so exactly the same amount of
energy is given out by the steam as is given to the skin. (In saying this, we assume
no other thermodynamic processes occur, such as warming of the surrounding air.
Even if other thermodynamic processes do occur, we can still say confidently that no
energy is lost. It's just more difficult to act as an 'energy auditor', and thereby follow
where it goes.)
The word 'exother-
mic' comes from two
Greek roots: thermo,
meaning 'energy' or
'temperature', and exo
meaning 'outside' or
'beyond'. An exother-
mic process therefore
gives out energy.
E?
CD
c
CD
c
Initially
(before reaction)
Finally
(after reaction)
Figure 3.1 In an exothermic process, the final product has less energy than the initial starting
materials. Energy has been given out
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
81
Worked Example 3.1 Use Equation (3.1) to demonstrate that At/
is negative for the condensation of steam if, say, t/(fi na i) = 12 J and
^(initial) = 25 J.
Inserting values into Equation (3.1):
At/ = (/(fi na l) — (/(initial)
At/ = (12-25) J
At/ = -13 J
Important: although
we have assigned
numerical values to
(/(final) and (/(initial)/ it
is, in fact, impossible
to know their values.
In reality, we only
know the difference
between them.
So we calculate the value of A U as — 13 J. The change in U is negative
and, therefore, exothermic, as expected.
We see that A(/ is negative. We could have reasoned this result
by saying t/( nn ai) < (/(initial), and subtracting a larger energy from a
smaller one generates a deficit.
The symbol 'J' here
means joule, which is
the SI unit of energy.
Why do we sweat?
Endothermic reactions
We all sweat at some time or other, e.g. after running hard, living
in a hot climate or perhaps during an illness when our temperature
is raised due to an infection (which is why we sometimes say, we
have 'got a temperature').
Producing sweat is one of the body's natural ways of cooling
itself, and it operates as follows. Sweat is an aqueous solution
of salt and natural oils, and is secreted by glands just below the
surface of the skin. The glands generate this mixture whenever the
body feels too hot. Every time air moves over a sweaty limb, from
a mechanical fan or natural breeze, the skin feels cooler following
evaporation of water from the sweat.
When we say the water evaporates when a breeze blows, we
mean it undergoes a phase transition from liquid to vapour, i.e. a
phase transition proceeding in the opposite direction to that in the
previous example, so Equation (3.2) occurs backwards. When we
consider the internal energy changes, we see (/(final) = (/(water, g)
and (/(initial) = t^(water, i) . so the final state of the water here is more
energetic than was its initial state. Figure 3.2 shows a schematic
representation of the energy change involved.
We need the salt in
sweat to decrease the
water's surface tension
in order to speed up the
evaporation process
(we feel cooler more
quickly). The oils in
sweat prevents the
skin from drying out,
which would make it
susceptible to sunburn.
Evaporation is also
called 'vaporization'.
It is a thermodynamic
process, because en-
ergy is transferred.
82
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
>>
en
i—
<D
c
<D
"ctj
c
1—
0)
c
Initially
(before reaction)
Finally
(after reaction)
Figure 3.2 In an endothermic process, the final product has more energy than the initial starting
materials. Energy has been taken in
A process is endo-
thermic if the final
state has more energy
than does the ini-
tial state. The word
derives from the Greek
roots thermo (mean-
ing 'energy' or 'tem-
perature') and endo
(meaning 'inside' or
'within'). An endother-
mic process takes in
energy.
Worked Example 3.2 What is the change in internal energy dur-
ing sweating?
The definition of At/ in Equation (3.1) is AU — U t
(final)
the value of AU,
(evaporation)
is obtained as U,
(water, g)
u.
U(innml), SO
. We
(water, 1)
already know that the final state of the water is more energetic than
its initial state, so the value of AU is positive. We say such a process
is endothermic.
We feel cooler when sweating because the skin loses energy
by transferring it to the water on its surface, which then evapo-
rates. This process of water evaporation (sweating) is endothermic
because energy passes from the skin to the water, and a body
containing less energy has a lower temperature, which is why we
feel cooler.
Aside
Heat is absorbed from the surroundings while a liquid evaporates. This heat does not
change the temperature of the liquid because the energy absorbed equates exactly to
the energy needed to break intermolecular forces in the liquid (see Chapter 2). Without
these forces the liquid would, in fact, be a gas.
At constant temperature, the heat absorbed during evaporation is often called the
latent heat of evaporation. This choice of words arises from the way evaporation occurs
without heating of the liquid; 'latent' is Latin for 'hidden', since the energy added to is
not 'seen' as a temperature rise.
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY 83
Why do we still feel hot while sweating
on a humid beach?
State functions
Sometimes we feel hot even when sweating, particularly in a humid environment
like a beach by the sea on a hot day. Two processes occur in tandem on the skin:
evaporation (liquid water — »■ gaseous water) and condensation (gaseous water — »■
liquid water). It is quite possible that the same water condenses on our face as
evaporated earlier. In effect, then, a cycle of 'liquid — »■ gas — >■ liquid' occurs. The
two halves of this cycle operate in opposite senses, since both exo- and endo-thermic
processes occur simultaneously. The net change in energy is, therefore, negligible,
and we feel no cooler.
These two examples of energy change involve water. The only difference between
them is the direction of change, and hence the sign of At/. But these two factors are
related. If we were to condense exactly 1 mol of steam then the amount of energy
released into the skin would be 40700 J. The change in internal energy AU (ignoring
volume changes) is negative because energy is given out during the condensation
process, so AU = -40700 J.
Conversely, if we were to vaporize exactly 1 mol of water from the skin of a
sweaty body, the change in internal energy would be +40 700 J. In other words, the
magnitude of the change is identical, but the sign is different.
While the chemical substance involved dictates the magnitude of AU (i.e. the
amount of it), its sign derives from the direction of the thermodynamic process. We
can go further: if the same mass of substance is converted from state A to state B,
then the change in internal energy is equal and opposite to the same process occurring
in the reverse direction, from B to A. This essential truth is depicted schematically
in Figure 3.3.
The value of AU when condensing exactly 1 mol of water is termed the molar
change in internal energy. We will call it AU m (condensation) » where the small 'm'
indicates that a mole is involved in the thermodynamic process. Similarly, the molar
^(reaction)
Figure 3.3 The change in internal energy when converting a material from state A to state B is
equal and opposite to the change in U obtained when performing the same process in reverse, from
B to A
84
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
We
often omit
the
sma
II 'm'; so,
from
now
on, we assume
that
changes in
inter-
nal
energy are
molar
quantities.
change in energy during vaporization can be symbolized as
AU m (vaporization)- If we compare the molar energies for these two
similar processes, we see the following relation:
AU,
m (condensation)
-AU,
in (vaporization)
(3.3)
The two energies are equal and opposite because one process occurs
in the opposite direction to the other, yet the same amount of
material (and hence the same amount of energy) is involved in both.
Following from Equation (3.3), we say that internal energy is a
state function. A more formal definition of state function is, 'A ther-
modynamic property (such as internal energy) that depends only on
the present state of the system, and is independent of its previous
history'. In other words, a 'state function' depends only on those
variables that define the current state of the system, such as how
much material is present, whether it is a solid, liquid or gas, etc.
The concept of a state function can be quite difficult, so let us
consider a simple example from outside chemistry. Geographical
position has analogies to a thermodynamic state function, insofar as it does not matter
whether we have travelled from London to New York via Athens or flew direct.
The net difference in position is identical in either case. Figure 3.4 shows this truth
diagrammatically. In a similar way, the value of AU for the process A — > C is the
same as the overall change for the process A — > B — > C. We shall look further at the
consequences of U being a state function on p. 98.
Internal energy U is
a 'state function' be-
cause: (1) it is a ther-
modynamic property;
and (2) its value
depends only on the
present state of the
system, i.e. is indepen-
dent of the previous
history.
Direct
London
Athens
Figure 3.4 If geographical position were a thermodynamic variable, it would be a state function
because it would not matter if we travelled from London to New York via Athens or simply
flew direct. The net difference in position would be identical. Similarly, internal energy, enthalpy,
entropy and the Gibbs function (see Chapter 4) are all state functions
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
85
Furthermore, because internal energy is a state function, the over-
all change in U is zero following a series of changes described by
a closed loop. As an example, imagine three processes: a change
from A — > B, then B — > C and finally from C — > A. The only rea-
son why the net value of AU for this cycle is zero is because we
have neither lost nor picked up any energy over the cycle. We can
summarize this aspect of physical chemistry by saying, 'energy can-
not be created or destroyed, only converted' - a vital truth called
the first law of thermodynamics .
If we measure AU over a thermodynamic cycle and obtain a non-zero value,
straightaway we know the cycle is either incomplete (with one or more processes not
accounted for) or we employed a sloppy technique while measuring AU.
The 'first law of ther-
modynamics' says
energy can neither be
created nor destroyed,
only converted from
one form to another.
Aside
William Rankine was the first to propose the first law of thermodynamics explicitly, in
1853 (he was famous for his work on steam engines). The law was already implicit in the
work of other, earlier, thermodynamicists, such as Kelvin, Helmholtz and Clausius. None
of these scientists sought to prove their theories experimentally; only Joule published
experimental proof of the first law.
op of a waterfall cooler
Why is the water at the
the water at its base?
The mechanical equivalence of work and energy
Two of the architects of modern thermodynamics were William Thompson (better
known as Lord Kelvin) and his friend James Prescott Joule - a scientist of great
vision, and a master of accurate thermodynamic measurement, as well as being some-
thing of an English eccentric. For example, while on a holiday in Switzerland in 1847,
Thompson met Joule. Let Thompson describe what he saw:
I was walking down from Chamonix to commence a tour of Mont Blanc, and whom
should I meet walking up but Joule, with a long thermometer in his hand and a carriage
with a lady in, not far off. He told me that he had been married since we parted in
Oxford [two weeks earlier] and that he was going to try for the elevation of temperature
in waterfalls.
Despite it being his honeymoon, Joule possessed a gigantic thermometer fully 4 to
5 feet in length (the reports vary). He spent much of his spare time during his honey-
moon in making painstaking measurements of the temperature at the top and bottom
of elongated Swiss waterfalls. He determined the temperature difference between the
86 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
water at the bottom and top of the waterfall, finding it to be about 1 °F warmer at
the bottom. In Joule's own words, 'A [water]fall of 817 feet [249 m] will generate
one degree [Fahrenheit] of temperature' . This result is not attributable to colder air at
the top of the waterfall, nor due to friction or viscous drag, or other effects occurring
during the water's descent, but is wholly due to a change in internal energy. The
water was simply changing its altitude.
The potential energy of a raised object is given by the expression
potential energy = mgh (3.4)
where m is the mass, g is the acceleration due to gravity and h the height by which it
is raised. The potential energy of the water decreases during descent because its height
decreases. This energy is liberated; and, as we have noted several times already, the
simplest way to tell if the internal energy has increased is to determine its temperature.
Joule showed the temperature of the water of the waterfalls had indeed increased.
We could summarize by saying that thermodynamic work w is energetically equiv-
alent to the lowering or raising of a weight (like the water of the waterfall, above),
as discussed below.
Why is it such hard work pumping up a bicycle tyre?
Thermodynamic work
No one who has pumped up a bicycle tyre says it's easy. Pumping a car tyre is harder
still. It requires a lot of energy, and we really have to work at it.
We saw in Chapter 1 how increasing the amount of a gas causes
its volume to increase. This increase in volume is needed to oppose
any increases in pressure. It also explains why blowing into a party
balloon causes it to get bigger. By contrast, a car tyre cannot expand
greatly during pumping, so increasing the amount of gas it contains
will increase its internal pressure. In a fully inflated car tyre, the
internal pressure is about 10 times greater than 'standard pressure'
p & , where p & has a value of 10 5 Pa.
The first law of thermodynamics states that energy may be con-
verted between forms, but cannot be created or destroyed. Joule
was a superb experimentalist, and performed various types of work,
each time generating energy in the form of heat. In one set of exper-
iments, for example, he rotated small paddles immersed in a water
trough and noted the rise in temperature. This experiment was
apparently performed publicly in St Anne's Square, Manchester.
Joule discerned a relationship between energy and work (symbol w). We have to
perform thermodynamic work to increase the pressure within the tyre. Such work
is performed every time a system alters its volume against an opposing pressure or
force, or alters the pressure of a system housed within a constant volume.
The pressure inside a
party balloon is higher
than the external,
atmospheric pressure,
as evidenced by the
way it whizzes around a
room when punctured.
Work is a form of ener-
gy. The word 'energy'
comes from the Greek
en ergon, meaning
'from work'.
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
87
Work and energy can be considered as interchangeable: we per-
form work whenever energy powers a physical process, e.g. to
propel a car or raise a spoon to the mouth. The work done on
a system increases its energy, so the value of U increases, itself
causing AU to be positive). Work done by a system corresponds
to a negative value of AU.
Work done on a system
increases its energy,
so AU is positive. Work
done by a system cor-
responds to a negative
value of AU.
Why does a sausage become warm when placed
in an oven?
Isothermal changes in heat and work
At first sight, the answer to our title question is obvious: from the minus-oneth law
of thermodynamics, heat travels from the hot oven to the cold(er) item of food we
place in it. Also, from the zeroth law, thermal equilibrium is attained only when the
sausage and the oven are at the same temperature. So the simplest answer to why
a sausage gets hot is to say the energy content of the sausage (in the form of heat)
increases, causing its internal energy to rise. And, yet again, we see how the simplest
test of an increasing internal energy is an increased temperature.
We can express this truth by saying the sausage gets warmer as
the magnitude of its internal energy increases; so, from Equation
(3.1), AC/ = C/(fi na i) — C/(i n i t i a i), hence AU = [/(after heating) _
[^(before heating)- We see how the value of AU is positive since
U (after heating) ^ U (before heating) •
But we can now be more specific. The internal energy U changes
in response to two variables, work w and heat energy q, as defi-
ned by
AU = q + w (3.5)
We have already met the first law of thermodynamics. Equa-
tion (3.5) here is the definitive statement of this law, and is ex-
pressed in terms of the transfer of energy between a system and its
environment. In other words, the magnitude of AU is the sum of
the changes in the heat q added (or extracted) from a system, and
the work w performed by (or done to) it.
The internal energy can increase or decrease even if one or other
of the two variables, q and w, remains fixed. Although the sausage
does no work w in the oven, the magnitude of AU increases
because the food receives heat energy q from the oven.
Worked Example 3.3 What is the energy of the sausage after heat-
ing, if its original energy is 4000 J, and 20 000 J is added to it?
Care: in the past,
Equation (3.5) was
often written as AU =
q- w, where the minus
sign is intended to
show how the internal
energy decreases fol-
lowing work done by
a system. We will use
Equation (3.5), which
is the more usual form.
Care: The symbol of
the joule is J. A small 'j'
does not mean joules;
it represents another
variable from a com-
pletely different branch
of physical chemistry.
No work is done, so w — 0.
88
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
The definition of AC/ is given by Equation (3.1):
Care: this is a highly
artificial calculation and
is intended for illus-
trative purposes only.
In practice, we never
know values of U, only
changes in U, i.e. AU.
The word 'isothermal'
can be understood by
looking at its Greek
roots. Iso means 'same'
and thermo means
'energy' or 'tempera-
ture', so a measure-
ment is isothermal
when performed at a
constant temperature.
AU — (/(fi na i) — (/(initial)
Rearranging to make (/( nna i) the subject, we obtain
^(final) = ^(initial) + AC/
Equation (3.5) is another expression for AC/. Substituting for AC/ in
Equation (3.1) allows us to say
t/(final) = ^/(initial) + 1 + W
Inserting values into this equation, we obtain
C/(tinai) = 4000 J + 20000 J = 24000 J
The example above illustrates how energy flows in response
to the minus-oneth law of thermodynamics, to achieve thermal
equilibrium. The impetus for energy flow is the equalization of
temperature (via the zeroth law), so we say that the measurement
is isothermal.
We often want to perform thermodynamic studies isothermally
because, that way, we need no subsequent corrections for inequal-
ities in temperature; isothermal measurements generally simplify
our calculations.
Why, when letting down a bicycle tyre, is the expelled
air so cold?
Thermodynamic work
When a fully inflated car tyre is allowed to deflate, the air streaming through the
nozzle is cold to the touch. The pressure of the air within the tyre is fairly high, so
opening the tyre valve allows it to leave the tyre rapidly - the air movement may even
cause a breeze. We could feel a jet of cold air on our face if we were close enough.
As it leaves the tyre, this jet of air pushes away atmospheric air, which requires
an effort. We say that work is performed. (It is a form of pres-
sure-volume work, and will be discussed in more depth later, in
Section 3.2.)
The internal energy of the gas must change if work is performed,
because AC/ = q + w. It is unlikely that any energy is exchanged
so, in this simplistic example, we assume that q = 0.
Energy is consumed because work w is performed by the gas,
causing the energy of the gas to decrease, and the change in internal
Energy added to, or
work done on, a sys-
tem is positive. Energy
removed from, or work
done by, a system is
negative.
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
89
energy is negative. If AU is negative and q = 0, then w is also negative. By corollary,
the value of w in Equation (3.5) is negative whenever the gas performs work.
From Chapter 2, we remember again that the simplest way to tell whether the
internal energy decreases is to check whether the temperature also decreases. We see
that the gas coming from the tyre is cold because it performs work, which decreases
its internal energy.
Why does a tyre get hot during inflation?
The temperature of
a tyre also increases
when inflated, and is
caused by interparticle
interactions forming;
see p. 59.
Adiabatic changes
Anyone inflating a tyre with a hand pump will agree that much
hard work is needed. A car or bicycle tyre usually gets hot during
inflation. In the previous example, the released gas did thermody-
namic work and the value of w was negative. In this example, work
is done to the gas in the tyre, so the value of w is positive. Again,
we assume that no energy is transferred, which again allows us to
take q as zero.
Looking again at Equation (3.5), AU = q + w, we see that if
q = and w increases (w is positive), then AU increases. This increase in AU
explains why the temperature of the gas in the tyre increases.
Let us return to the assertion that q is zero, which implies that the system is ener-
getically closed, i.e. that no energy can enter or leave the tyre. This statement is
not wholly true because the temperature of the gas within the tyre will equilibrate
eventually with the rubber of the tyre, and hence with the outside air, so the tyre
becomes cooler in accordance with the minus-oneth and zeroth laws of thermody-
namics. But the rubber with which tyre is made is a fairly good thermal insulator,
and equilibration is slow. We then make the good approximation that the system is
closed, energetically. We say the change in energy is adiabatic.
Energetic changes are adiabatic if they can be envisaged to occur
while contained within a boundary across which no energy can
pass. In other words, the energy content within the system stays
fixed. For this reason, there may be a steep temperature jump in
going from inside the sealed system to its surroundings - the gas
in the tyre is hot, but the surrounding air is cooler.
In fact, a truly adiabatic system cannot be attained, since even
the most insulatory materials will slowly conduct heat. The best
approximations are devices such as a Dewar flask (sometimes called a 'vacuum
flask').
A thermodynamic
pro-
cess is adiabatic
if
it occurs within
a
(conceptual) bound-
ary across which
no
energy can flow.
Can a tyre be inflated without a rise in temperature?
Thermodynamic reversibility
A tyre can indeed be inflated without a rise in temperature, most simply by filling
it with a pre-cooled gas, although some might regard this 'adaptation' as cheating!
90
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
'Infinitesimal' is the
reciprocal of infinite,
i.e. incredibly small.
Alternatively, we could consider inflating the tyre with a series of, say, 100 short
steps - each separated by a short pause. The difference in pressure before and after
each of these small steps would be so slight that the gas within the tyre would be
allowed to reach equilibrium with its surroundings after adding each increment, and
before the next. Stated a different way, the difference between the pressure of the gas
in the hand pump and in the tyre will always be slight.
We can take this idea further. We need to realize that if there is
no real difference in pressure between the hand pump and the gas
within the tyre, then no work would be needed to inflate because
there would never be the need to pump against a pressure. Alter-
natively, if the inflation were accomplished at a rate so slow that it
was infinitesimally slow, then there would never be a difference in pressure, ensuring
w was always zero. And if w was zero, then U would stay constant per increment. (We
need to be aware that this argument requires us to perform the process isothermally.)
It should be clear that inflating a tyre under such conditions is never going to occur
in practice, because we would not have the time, and the inflation would never be
complete. But as a conceptual experiment, we see that working at an infinitesimally
slow rate does not constitute work in the thermodynamic sense.
It is often useful to perform thought experiments of this type,
changing a thermodynamic variable at an infinitely slow rate: we
say we perform the change reversibly. (If we perform a process in a
non-reversible manner then we say it is 'irreversible'.) As a simple
definition, a process is said to be reversible if the change occurs
at an infinitesimal rate, and if an infinitesimal change in an exter-
nal variable (such as pressure) could change the direction of the
thermodynamic process. It is seen that a change is only reversible
if it occurs with the system and surroundings in equilibrium at
all times. In practice this condition is never attained, but we can
sometimes come quite close.
Reversibility can be a fairly difficult concept to grasp, but it
is invaluable. In fact, the amount of work that can be performed
during a thermodynamic process is maximized when performing
it reversibly.
The discussion here has focused on work done by changes in
pressure, but we could equally have discussed it in terms of volume
changes, electrical work (see Chapter 7) or chemical changes (see
Chapter 4).
A thermodynamic pro-
cess is reversible if an
infinitesimal change in
an external variable
(e.g. pressure) can
change the direction
in which the process
occurs.
The amount of work
that can be performed
during a thermody-
namic process is maxi-
mized by performing it
reversibly.
How fast does the air in an oven warm up?
Absorbing energy
The air inside an oven begins to get warm as soon as we switch it on. We can
regard the interior of the oven as a fixed system, so the internal energy U of the
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
91
gases increases as soon as energy is added, since AU = q + w (Equation (3.5)).
For simplicity, in this argument we ignore the expansion of the atmosphere inside
the oven.
But what is the temperature inside the oven? And by how much does the temperature
increase? To understand the relationship between the temperature and the amount of
heat entering the system, we must first appreciate that all energies are quantized. The
macroscopic phenomenon of temperature rise reflects the microscopic absorption of
energy. During absorption, quanta of energy enter a substance at the lowest energy
level possible, and only enter higher quantal states when the lower energy states are
filled. We see the same principle at work when we fill a jar with marbles: the first
marbles fall to the jar bottom (the position of lowest potential energy); and we only
see marbles at the top of the jar when all the lower energy levels are filled. Continuing
the analogy, a wide jar fills more slowly than does a narrower jar, even when we add
marbles at a constant rate.
On a macroscopic level, the rate at which the quantal states are filled as a body
absorbs energy is reflected by its heat capacity C. We can tell how quickly the
quantum states are occupied because the temperature of a body is in direct proportion
to the proportion of states filled. A body having a large number of quantum states
requires a large number of energy quanta for the temperature to increase, whereas a
body having fewer quantum states fills more quickly, and becomes hot faster.
Why does water boil more quickly in a kettle
than in a pan on a stove?
Heat capacity
The SI unit of power is
the watt (W). A heater
rated at a power of 1 W
emits 1 Js _1 .
Most modern kettles contain a powerful element (the salesman's word for 'heater')
operating at a power of 1000 W or more. A heater emits 1 W if it gives out 1 Js _1
so, a heater rated at 1000 W emits 1000 Js _1 . We may see this
power expressed as 1 kW (remember that a small 'k' is shorthand
for kilo, meaning 1000). By contrast, an electrical ring on the stove
will probably operate between 600 and 800 W, so it emits a smaller
amount of heat per second. Because the water absorbs less heat
energy per unit time on a stove, its temperature rises more slowly.
The amount of energy a material or body must absorb for its
temperature to increase is termed its 'heat capacity' C. A fixed
amount of water will, therefore, get warmer at a slower rate if the
amount of heat energy absorbed is smaller per unit time.
Equation (3.6) expresses the heat capacity C in a mathematical
form:
c *= df )„ a6)
The heat capacity C
of a material or body
relates the amount of
energy absorbed when
raising its temperature.
92
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
The expression in Equation (3.6) is really a partial differential: the value of U depends
on both T and V, the values of which are connected via Equation (1.13). Accordingly,
we need to keep one variable constant if we are unambiguously to attribute changes in
Cv to the other. The two subscript ' V terms tell us C is measured while maintaining
the volume constant. When the derivative is a partial derivative, it is usual to write
the 'd' as '3'.
We call Cv 'the heat capacity at constant volume'. With the
volume constant, we measure Cv without performing any work
(so w = 0), so we can write Equation (3.6) differently with Aq
rather than dU.
Unfortunately, the value of Cv changes slightly with tempera-
ture; so, in reality, a value of Cy is obtained as the tangent to
the graph of internal energy (as v) against temperature (as x); see
Figure 3.5.
If the change in temperature is small, then we can usually assume
that Cv has no temperature dependence, and write an approximate
form of Equation (3.6), saying
We also call C v the
isochoric heat capacity.
A tangent is a straight
line that meets a curve
at a point, but not
does cross it. If the
heat capacity changes
slightly with temper-
ature, then we obtain
the value of Cv as the
gradient of the tangent
to a curve of AU (as y)
against T (as x).
C v =
AU
(3.7)
Analysing Equations (3.6) and (3.7) helps us remember how the
SI unit of heat capacity Cv is J K _1 . Chemists usually cite a heat
capacity after dividing it by the amount of material, calling it the specific heat capacity,
either in terms of J K _1 mol~ orJ K _1 g _1 . As an example, the heat capacity of water
is 4.18 JK _1 g _1 , which means that the temperature of 1 g of water increases by 1 K
for every 4.18 J of energy absorbed.
SAQ 3.1 Show that the molar heat capacity of water is 75.24 J K 1 mol 1
if Cv — 4.18 J K _1 g _1 . [Hint: first calculate the molar mass of H2O.]
Tangential gradient = heat capacity, C v
Figure 3.5 The value of the heat capacity at constant volume Cv changes slightly with temper-
ature, so its value is best obtained as the gradient of a graph of internal energy (as y) against
temperature (as x)
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
93
Worked Example 3.4 An electrical heater warms 12 g of water. Its initial temperature
is 35.0 °C. The heater emits 15 W for 1 min. What is the new temperature of the water?
Answer strategy. Firstly, we will calculate the energy produced by the
heater, in joules. Secondly, knowing the heat capacity of the water C,
we divide this energy by C to obtain the temperature rise.
(I) To calculate the energy produced by the heater. Remember that
1 W = Us -1 , so a wattage of 15 W means 15 Js _1 . The heater oper-
ates for 1 min (i.e. 60 s), so the energy produced is 15 Js x 60s =
900 J.
This amount of energy is absorbed by 12 g of water, so the energy
absorbed per gram is
The word 'strategy'
comes from the Greek
stratos meaning 'army'.
Strategy originally con-
cerned military
manoeuvres.
900 J
75 J g"
(2) To calculate the temperature rise. The change in temperature AT is sufficiently small
that we are justified in assuming that the value of Cy is independent of temperature. This
assumption allows us to employ the approximate equation, Equation (3.7). We rearrange
it to make AT the subject:
At/
AT =
C v
Inserting values:
yielding
AT
75 J g- 1
4.18 JK-'g" 1
A! = 17.9 K
As 1 °C = 1 K, the final temperature of the water is (25.0 + 17.9) °C = 42.9 °C.
SAQ 3.2 How much energy must be added to 1.35 kg of water in a pan if
it is to be warmed from 20 °C to its boiling temperature of 100 °C? Assume
Cv does not vary from 4.18 J K mol .
The heat capacity Cy is an extensive quantity, so its value depends on how much
of a material we want to warm up. As chemists, we usually want a value of Cy
expressed per mole of material. A molar heat capacity is an intensive quantity.
Aside
Another heat capacity is C p , the heat capacity measured at constant pressure (which is
also called the isobaric heat capacity). The values of C p and Cy will differ, by perhaps
as much as 5-10 per cent. We will look at C p in more depth in the next section.
94
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
Why does a match emit heat when lit?
Reintroducing calorimetry
'Lighting' a match means initiating a simple combustion reaction. Carbohydrates
in the wood combine chemically with oxygen in the air to form water and carbon
dioxide. The amount of heat liberated is so great that it catches fire (causing the water
to form as steam rather than liquid water).
Heat is evolved because the internal energy of the system changes during the
combustion reaction. Previously, the oxygen was a gaseous element characterized by
0=0 bonds, and the wood was a solid characterized by C-C, C-H and C-0 bonds.
The burning reaction completely changes the number and type of bonds, so the internal
energies of the oxygen and the wood alter. This explains the change in At/.
We know from Equation (3.5) that AU = q + w. Because AU changes, one or
both of q and w must change. It is certain that much energy is liberated because we
feel the heat, so the value of q is negative. Perhaps work w is also performed because
gases are produced by the combustion reaction, causing movement of the atmosphere
around the match (i.e. w is positive).
The simplest way to measure the change in internal energy AU is to perform a
reaction in a vessel of constant volume and to look at the amount of heat evolved.
We perform a reaction in a sealed vessel of constant volume called a calorimeter. In
practice, we perform the reaction and look at the rise in temperature. The calorimeter is
completely immersed in a large reservoir of water (see Figure 3.6) and its temperature
is monitored closely before, during, and after the reaction. If we know the heat
Water stirrer
Oxygen
inlet
Electrical
contacts
Resistance
thermometer
Water
Bomb Sample
Figure 3.6 Schematic representation of the bomb calorimeter for measuring the changes in internal
energy that occur during combustion. The whole apparatus approximates to an adiabatic chamber,
so we enclose it within a vacuum jacket (like a Dewar flask)
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
95
capacity C of both the calorimeter itself and the surrounding water, then we can
readily calculate the change in energy AC/ accompanying the reaction.
Why does it always take 4 min to boil an egg properly?
Thermochemistry
Most people prefer their eggs to be lightly boiled, with the yellow yolk still liquid
and the albumen solid and white. We say the egg white has been 'denatured'. The
variation in egg size is not great. An average egg contains essentially a constant
amount of yolk and albumen, so the energy necessary to heat both the yolk and
albumen (and to denature the albumen) is, more or less, the same for any egg.
If the energy required to cook an egg is the same per egg, then the simplest way
to cook the egg perfectly every time is to ensure carefully that the same amount of
energy is absorbed. Most people find that the simplest way to do this is to immerse
an egg in boiling water (so the amount of energy entering the egg per unit time is
constant), and then to say, 'total energy = energy per second x number of seconds'.
In practice, it seems that most people prefer an egg immersed in boiling water for
about 240 s, or 4 min.
This simple example introduces the topic of thermochemistry.
In a physical chemist's laboratory, we generally perform a similar
type of experiment but in reverse, placing a sample in the calorime-
ter and measuring the energy released rather than absorbed. The
most commonly performed calorimetry experiment is combustion
inside a bomb calorimeter (Figure 3.6). We place the sample in the
calorimeter and surround it with oxygen gas at high pressure, then
seal the calorimeter securely to prevent its internal contents leaking
away, i.e. we maintain a constant volume. An electrical spark then
ignites the sample, burning it completely. A fearsome amount of energy is liberated
in consequence of the ignition, which is why we call this calorimeter a 'bomb'.
The overall heat capacity of the calorimeter is a simple function of the amount of
steel the bomb comprises and the amount of water surrounding it. If the mass is m
Thermochemistry is
the branch of thermo-
dynamics concerned
with the way energy
is transferred, released
or consumed during a
chemical reaction.
and the heat capacity is C, then the overall heat capacity is expressed by
C (overall) = \ m (steel) x C m (steel)) + \ m (water) x C m (water))
(3.8)
If the amount of compound burnt in the calorimeter is n, and
remembering that no work is done, then a combination of Equa-
tions (3.7) and (3.8) suggests that the change in internal energy
occurring during combustion is given by
Qoveraii) is the heat
capacity of the reac-
tion mixture and the
calorimeter.
AC/,
C
m (combustion)
(overall)
AT
n
(3.9)
96
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
where the 'm' means 'molar'. The negative sign arises from the conventions above,
since heat is given out if the temperature goes up, as shown by A T being positive.
It is wise first to calibrate the calorimeter by determining an
accurate value of C( ove raii)- This is achieved by burning a com-
pound for which the change in internal energy during combustion
is known, and then accurately warming the bomb and its reservoir
with an electrical heater. Benzoic acid (I) is the usual standard of
choice when calibrating a bomb calorimeter.
Al/ (C ombustion) for ben-
zoic acid is -3.2231 MJ
mor 1 at 298 K.
(I)
The electrical energy passed is q, defined by
q = V x I x t
(3.10)
where V is the voltage and / the current of the heater, which operates for a time of
t seconds.
SAQ 3.3 A voltage of 10 V produces a current of 1.2 A when applied
across a heater coil. The heater is operated for 2 min and 40 s. Show that
the energy produced by the heater is 1920 J.
We can assume C p is
constant only if AT is
small. For this reason,
we immerse the 'bomb'
in a large volume of
water. This explains
why we need to oper-
ate the heater for a
long time.
so
Worked Example 3.5 A sample of glucose (10.58 g) is burnt com-
pletely in a bomb calorimeter. What is the change in internal energy
At/ if the temperature rises by 1.224 K? The same heater as that in
SAQ 3.3 is operated for 11 240 s to achieve a rise in temperature of
1.00 K.
Firstly, we calculate the energy evolved by the reaction. From
Equation (3.10), the energy given out by the heater is q — 10 V x
1.2Ax 11240s = 134880J.
Secondly, we determine the value of C( 0ve raii) for the calorimeter,
saying from Equation (3.9)
C
energy released
134 880 J
change in temperature 1.00 K
C= 134 880 J K"
INTRODUCTION TO THERMODYNAMICS: INTERNAL ENERGY
97
Thirdly, we determine the amount of glucose consumed n. We obtain the value of n
as 'amount = mass 4- molar mass'. The molar mass of glucose is 180 gmor
number of moles is 5.88 x 10~ 2 mol.
Finally, we calculate the value of At/ from Equation (3.9). Insert-
ing values:
so the
AU,
(combustion)
SO
AU
(combustion)
134 880 JKT 1 x 1.224K
0.0588 mol
= -2.808 MJmol"
Notice how this value of A U is negative. As a good generalization, the
change in internal energy AU liberated during combustion is negative,
which helps explain why so many fires are self-sustaining (although
see Chapter 4).
The value of A f/( Com b us tion) for glucose is huge, but most values of
AU are smaller, and are expressed in kilo joules per mole.
SAQ 3.4 A sample of anthracene (C14H10, II) was burnt
in a bomb calorimeter. A voltage of 10 V and a current
of 1.2 A were passed for exactly 15 min to achieve the
same rise in temperature as that caused by the burning
of 0.40 g. Calculate the molar energy liberated by the
anthracene.
The minus sign is a
consequence of the
way Equation (3.9) is
written.
An energy change of MJ
mor 1 is exceptional.
Most changes in AU are
smaller, of the order of
kJ mor 1 .
We often calculate a
volume of AU but
cite the answer after
adjusting for pres-
sure-volume work;
see p. 102.
(II)
Aside
The large value of At/ in Worked Example 3.5 helps explain why sweets, meals and
drinks containing sugar are so fattening. If we say a single spoonful of sugar comprises
5 g of glucose, then the energy released by metabolizing it is the same as that needed
to raise a 3.5 kg weight from the ground to waist level 7000 times.
(We calculate the energy per lift with Equation (3.4), saying E = m x g x h, where
m is the mass, g is the acceleration due to gravity and h is the height through which
the weight is lifted.)
98
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
Why does a watched pot always take so long to boil?
Introduction to Hess's law
We sometimes say, 'A watched pot never boils'. This empirical observation - that
we get bored waiting a long time for the pot to boil - follows because we need to put
a lot of energy (heat) in order for the water to boil. The amount
of energy we can put into the water per unit time was always low
in the days of coal and wood fires. Accordingly, a long time was
required to boil the water, hence the long wait.
Imagine we want to convert 1 mol of water starting at a room
temperature of, say, 25 °C to steam. In fact we must consider
two separate thermodynamic processes: we first consider the heat
needed to warm the water from 25 °C to its boiling temperature
of 100 °C. The water remains liquid during this heating process.
Next, we convert 1 mol of the liquid water at 100 °C to gaseous
water (i.e. we boil it), but without altering the temperature.
We will at the moment ignore once more the problems caused
by volume changes. The change in internal energy AC/( 0V eraii) f° r
The popular saying
A watched pot never
boils' arose when most
fires were wood or
coal, neither of which
generates heat as fast
as, say, a modern 1 kW
kettle.
This argument relies
only on words. In real-
ity, the situation is
somewhat more com-
plicated because water
expands slightly on
heating, and greatly on
boiling.
the overall process H20(i) at 25 C
separated into two components:
H 2 (g) at 100 C can be
Energy AC/i relates to the process
H 2 (1) at 25 °C ► H 2 0(i) at 100 °C
Energy A C/ 2 relates to the process
H 2 ( i) at 100 °C ► H 2 (g) at 100 °C
so AUi relates to warming the water until it reaches the boiling temperature, and
AC/ 2 relates to the actual boiling process itself.
We can obtain A[/( overa ii) algebraically, according to
We can obtain the
answer in several dif-
ferent ways because
internal energy is a
'state function'.
A [/ (0V e r all) = AC/] + AC/ 2
(3.11)
Hess's law states that
the value of an energy
obtained is indepen-
dent of the number of
intermediate reaction
steps taken.
In practice, we could have measured A [/(overall) directly in the
laboratory. Alternatively, we could have measured AU\ or AC/ 2
in the laboratory and found the AC/ values we did not know in a
book of tables. Either way, we will get the same answer from these
two calculation routes.
Equation (3.11) follows directly from AC/ being a state function,
and is an expression of Hess's law. The great German fhermody-
namicist Hess observed in 1840 that, 'If a reaction is performed in
more than one stage, the overall enthalpy change is a sum of the
enthalpy changes involved in the separate stages'.
We shall see shortly how the addition of energies in this way
provides the physical chemist with an extremely powerful tool.
ENTHALPY 99
Hess's law is a restatement if the first law of thermodynamics. We do not need to
measure an energy change directly but can, in practice, divide the reaction into several
constituent parts. These parts need not be realizable, so we can actually calculate the
energy change for a reaction that is impossible to perform in the laboratory. The only
stipulation is for all chemical reactions to balance.
The importance of Hess's law lies in its ability to access information about a
reaction that may be difficult (or impossible) to obtain experimentally, by looking at
a series of other, related reactions.
3.2 Enthalpy
How does a whistling kettle work?
Pressure-volume work
The word 'work' in the question above could confuse. In common parlance, we say
a kettle works or does not work, meaning it either functions as a kettle or is useless.
But following the example in the previous section, we now realize how the word
'work' has a carefully defined thermodynamic meaning. 'Operate' would be a better
choice in this context. In fact, a kettle does not perform any work at all, since it has
no moving parts and does not itself move.
In a modern, automatic kettle, an electric heater warms the water inside the ket-
tle - we call it the 'element'. The electric circuit stops when the water reaches 100 °C
because a temperature-sensing bimetallic strip is triggered. But the energy for a more
old-fashioned, whistling kettle comes from a gas or a coal hob. The water boils on
heating and converts to form copious amounts of gas (steam), which passes through
a small valve in the kettle lid to form a shrill note, much like in a football ref-
eree's whistle.
The whistle functions because boiling is accompanied by a change in volume, so
the steam has to leave the kettle. And the volume change is large: the volume per
mole of liquid water is 18 cm 3 (about the size of a small plum) but the volume of a
mole of gaseous water (steam) is huge.
SAQ 3.5 Assuming steam to be an ideal gas, use the ideal-gas equation
(Equation (1.13)) to prove that 1 mol of steam at 100°C (373 K) and
standard pressure (p s = 10 5 Pa) has a volume of is 0.031 m 3 .
A volume of 0.031 m 3 corresponds to 31 dm 3 , so the water increases its volume by
a factor of almost 2000 when boiled to generate steam. This staggering result helps
us realize just how great the increase in pressure is inside the kettle when water boils.
The volume inside a typical kettle is no more than 2 dm 3 . To avoid a rapid build
up of pressure within the kettle (which could cause an explosion), the steam seeks
to leave the kettle, exiting through the small aperture in the whistle. All the vapour
passes through this valve just like a referee blowing 'time' after a game. And the
100 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
large volume ensures a rapid exit of steam, so the kettle produces an intense, shrill
whistle sound.
The steam expelled from the kettle must exert a pressure against the air as it leaves
the kettle, pushing it aside. Unless stated otherwise, the pressure of the air surrounding
the kettle will be 10 5 Pa, which we call 'standard pressure' p^. The value of p^ is
10 5 Pa. The steam must push against this pressure when leaving the kettle. If it
does not do so, then it will not move, and will remain trapped within the kettle. This
pushing against the air represents work. Specifically, we call it pressure-volume work,
because the volume can only increase by exerting work against an external pressure.
The magnitude of this pressure- volume work is w, and is expressed by
w = -A( P V) (3.12)
where A(pV) means a change in the product of p x V. Work is done to the gas
when it is compressed at constant pressure, i.e. the minus sign is needed to make
w positive.
Equation (3.12) could have been written as Ap x A V if both the pressure and the
volume changed at the same time (an example would be the pushing of a piston
in a car engine, to cause the volume to decrease at the same time as the pressure
increases). In most of the physicochemical processes we will consider here, either
p or V will be constant so, in practice, there is only one variable. And with one
variable, Equation (3.12) becomes either w = p x AV or w = AV x p, depending
on whether we hold p or V constant.
Most chemists perform experiments in which the contents of our
For most purposes, a
chemist can say w =
p x AV.
beaker, flask or apparatus are open to the air - obvious examples
include titrations and refluxes, as well as the kinetic and electro-
chemical systems we consider in later chapters. The pressure is
the air pressure (usually p & ), which does not change, so any pres-
sure-volume work is the work necessary to push back the atmosphere. For most
purposes, we can say w = pAV.
It should be obvious that the variable held constant - whether p or V - cannot be
negative, so the sign of w depends on which of the variables we change, so the sign
of w in Equation (3.12) depends on the sign(s) of Ap or AV. The sign of w will be
negative if we decrease the volume or pressure while performing work.
Worked Example 3.6 We generate 1 mol of water vapour in a kettle by boiling liquid
water. What is the work w performed by expansion of the resultant steam?
We have already seen how 1 mol of water vapour occupies a volume of 0.031 m 3 (see
SAQ 3.5). This volume of air must be pushed back if the steam is to leave the kettle. The
external pressure is p e , i.e. 10 5 Pa.
The change in volume
AV — V(final) — ^(initial)
SO
AV — V(„ a ter, g) — V( water , 1)
ENTHALPY
101
Inserting numbers yields
AV =
(
0.031
/
volume per mole
of steam
18 x 10~ 6
J m 3 = 0.
031m 3
\
volume per mole
of liquid water
Each aliquot of 1 cm 3
represents a volume of
1 x 10 6 m 3 .
Inserting values into Equation (3.12), w — pAV:
w = 10 5 Pax 0.031m 3
io = 3.1x 10 3 J
so the pressure -volume work is 3100 J.
We often encounter energies of the order of thousands of joules.
As a shorthand, we often want to abbreviate, so we rewrite the
answer to Worked Example 3.6, and say w = 3.1 x 1000 J. Next,
we substitute for the factor of 1000 with an abbreviation, generally
choosing a small letter 'k'. We rewrite, saying w = 3.1 kJ.
SAQ 3.6 What is the work done when the gas from a
party balloon is released? Assume the inflated balloon
has a volume of 2 dm 3 , and a volume of 10 cm 3 when
deflated. Assume there is no pressure change, so p = p & .
[Hint: 1 1 = lx KHm 3 ].
An energy expressed
with a letter 'k' in the
answer means "thou-
sands of joules'. We
say kilojoules. The
choice of 'k' comes
from the Greek word
for thousand, which is
kilo.
Worked Example 3.7 What is the work performed when inflating a car tyre from p f
to 6 x p e . Assume the volume inside the tyre stays constant at 0.3 m 3 .
Firstly, we calculate the change in pressure, from an equation like Equation (3.1), Ap =
P(flnal) - P(initial), SO Ap = (6 - 1) X p e , i.e.
Ap — 5 x p e
A P = 5x 10 5 Pa
Then, inserting values into Equation (3.5), w — Ap x V:
w = 5 x 10 5 Pax 0.3m 3
yielding
w = 15000 J = 15kJ
102 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
How much energy do we require during a distillation?
The effect of work on AU: introducing enthalpy H
Performing a simple distillation experiment is every chemist's delight. We gently
warm a mixture of liquids, allowing each component to boil off at its own charac-
teristic temperature (the 'boiling temperature' 7(b ii))- Each gaseous component cools
and condenses to allow collection. Purification and separation are thereby effected.
Although we have looked already at boiling and condensation, until now we have
always assumed that no work was done. We now see how invalid this assumption
was. A heater located within the distillation apparatus, such as an isomantle, supplies
heat energy q to molecules of the liquid. Heating the flask increases the internal
energy U of the liquids sufficiently for it to vaporize and thence become a gas.
But not all of the heater's energy q goes into raising U. We need some of it to
perform pressure -volume work, since the vapour formed on boiling works to push
back the external atmosphere. The difference between the internal energy U and the
available energy (the enthalpy) is given by
AH = AU + pAV (3.13)
H is a state function since p, V and U are each state functions. As a state function,
the enthalpy is convenient for dealing with systems in which the pressure is constant
but the volume is free to change. This way, an enthalpy can be equated with the
energy supplied as heat, so q = AH.
Worked Example 3.8 A mole of water vaporizes. What is the change in enthalpy, A HI
Take pressure as p^ .
We have already seen in the previous section that AU — +40.7 kJ per mole of water,
and from SAQ 3.5 the volume of 1 mol of water vapour is 0.031 m 3 per mole of water.
Inserting values into Equation (3.13):
AH--
40700 J mol" 1 \+( 10 5 Pa x 0.031m 3 mol" 1 J
/ / /
AU p AV
so
AH = 40700 Jmol" 1 + (SlOOJmor 1 )
and
AH = 45.8 kJmol" 1
In this example, the difference between AU and AH is about 11 per cent.
The magnitude of the difference will increase as the values of AH and AU get
smaller.
ENTHALPY
103
Justification Box 3.1
We saw above how the work w performed by a gas is pV. Because performing work
will decrease the internal energy, we say
w = —pV
(3.14)
Substitution of this simple relationship into the definition of internal energy in Equation
(3.5) yields
U = q-pV (3.15)
and rearranging Equation (3.15) yields q = U + pV .
This combination of variables occurs so often in phys-
ical chemistry, that we give it a name: we call it the
enthalpy, and give it the symbol H. Accordingly, we
rewrite Equation (3.15) as:
We often call a col-
lection of variables a
'compound variable'.
H = U + pV
(3.16)
The change in enthalpy AH during a thermodynamic process is defined in terms of
internal energy and pressure-volume work by
AH = AU + A(pV)
(3.17)
Because it is usual to perform a chemical experiment with the top of the beaker
open to the open air, the pressure p during most chemical reactions and thermodynamic
processes is the atmospheric pressure p e ', Furthermore, this pressure will not vary. In
other words, we usually simplify A(pV) saying pAV because only the volume changes.
Accordingly, Equation (3.17) becomes
AH = AU + pAV
(3.18)
The equation in the form of Equation (3.18) is the usual form we use. Changes in U
are not equal to the energy supplied as heat (at constant pressure p) because the system
employs some of its energy to push back the surroundings as they expand. The pV term
for work is, therefore, a correction for the loss of energy as work. Because many, if
not most, physicochemical measurements occur under conditions of constant pressure,
changes in enthalpy are vitally important because it automatically corrects for the loss
of energy to the surroundings.
Aside
One of the most common mistakes we make during calculations of this kind is forgetting
the way 'k' stands for '1000'. Think of it this way: a job advertisement offers a salary
of £14 k. We would be very upset if, at the end of the first year, we were given just
£14 and the employer said he 'forgot' the 'k' in his advert!
104
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
Why does the enthalpy of melting ice decrease
as the temperature decreases?
Temperature dependence of enthalpy
The enthalpy of freezing water is —6.00 kJmol -1 at its normal freezing temperature
of 273.15 K. The value is negative because energy is liberated during freezing. But
the freezing temperature of water changes if the external pressure
is altered; so, for example, water freezes at the lower temperature
of 253 K when a pressure of about 100 x p & is applied. This high
pressure is the same as that along the leading edge of an aeroplane
wing. At this lower temperature, AH( me \ t ) =5.2 kJmol - .
The principal cause of A//( me it) changing is the decreased temperature. The magni-
tude of an enthalpy depends on the temperature. For this reason, we need to cite the
temperature at which an enthalpy is determined. If the conditions are not cited, we
assume a temperature of 298 K and a pressure of p* . We recognize these conditions
The word 'normal' in
this context means 'at
a pressure of p*'.
as s.t.p. Values of enthalpy are often written as A//,
r 298 K
for this reason.
The temperature dependence of the standard enthalpy is related by Kirchhoff's law:
AH* = AH* +
/;
AC p (T)dT
(3.19)
Reminder. The 'curly d'
symbols 3 tells us the
bracketed term in the
equation is a 'partial
differential'.
C P (T) means C p as a
function of thermody-
namic temperature.
where C p is the molar heat capacity at constant pressure of the
substance in its standard state at a temperature of T . We define C p
according to
'BH\
(3.20)
C p =
dT
The value of C p is itself a function of temperature (see p. 140),
which explains why we integrate C P {T) rather than C p alone.
The Kirchhoff law is a direct consequence of the heat capacity at
constant pressure being the derivative of enthalpy with respect to
temperature. It is usually sufficient to assume that the heat capacity
C p is itself independent of temperature over the range of temperatures required, in
which case Equation (3.19) simplifies to
AH t \ = AH; Ti + AC P (T 2 - TO
(3.21)
The experimental scientist should ensure the range of temperatures is slight if calcu-
lating with Equation (3.21).
Worked Example 3.9 The standard enthalpy of combustion AH* for benzoic acid
(I) is —3223.1 kJmol -1 at 20°C. What is AH C 298 K ? The change in C p during the
ENTHALPY 105
reaction is 118.5 JKr'moE 1 . Assume this value is temperature independent over this
small temperature interval.
Inserting data into Equation (3.21):
A// r*298 k = -3223.7 kJmor 1 + 118.5 JKT 1 mol _1 (298 - 293) K
A// r ° 29g K = -3223.7 kJmor 1 + 592.5 JmoP 1
so
A/f r ° 298 K = -3223.1 kJmoP'or- 3.2231 MJmol" 1
SAQ 3.7 Ethane burns completely in oxygen to form carbon dioxide and
water with an enthalpy of AH & = -1558.8 kJ mol" 1 at 25 °C. What is AH*
at 80 °C? First calculate the change in heat capacity C p from the data in
the following table and Equation (3.22).
Substance
C p at 80°C/J K^mol -1
C2H6(g) 02(g) C02(g) H 2 ( |)
52.6 29.4 37.1 75.3
ac p = J2 vC p- J2 vC p (3 - 22 ^
products reactants
where the upper-case Greek letter Sigma £ means 'sum of, and the
lower-case Greek letters v (nu) represent the stoichiometric number of
each species, which are the numbers of each reagent in a fully balanced
equation. In the convention we adopt here, the values of v are positive for
products and negative for reactants.
Justification Box 3.2
Starting with the definition of heat capacity in Equation
(3.20):
\w) p = Cp
This equation represents C p for a single, pure substance.
Separating
; the variables
yields
dH = C p dT
Then we integrate between limits, saying the enthalpy is H\ at T\
and #2
at T 2 :
/ dH = / C p dT
106
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
Integrating yields
(H 2 - HO = CJT 2 - TO
(3.23)
where the term on the left-hand side is AH. Equation (3.23) relates to a single, pure sub-
stance.
If we consider a chemical reaction in which several chemicals combine, we can write
an expression like this for each chemical. Each chemical has a unique value of AH and
C p , but the temperature change (T 2 — 7i) remains the same for each.
We combine each of the AH terms to yield AH r T (i.e. AH r e at T 2 ) and AH T T .
Combining the C p terms according to Equation (3.22) yields AC p . Accordingly, Equa-
tion (3.23) then becomes Equation (3.21), i.e.:
AH,
r T 2
Aff* + AC„(72 - 7i)
Why does water take longer to heat in a pressure
cooker than in an open pan?
The differences between C v and C p
■
A pressure cooker is a sealed cooking pan. Being sealed, as soon as boiling occurs,
the pressure of steam within the pan increases dramatically, reaching a maximum
pressure of about 6 x p & , causing the final boiling temperature to
increase (see Fig. 5.12 on p. 200). Unlike other pans, the internal
volume is fixed and the pressure can vary; the pressure in most
pans is atmospheric pressure (~ p & ), but the volume of the steam
increases continually.
The heat capacity of the contents in a pressure cooker is Cy
because the internal volume is constant. By contrast, the heat capacity of the food
or whatever inside a conventional pan is C p . The water is a pressure cooker warms
slower because the value of C p is always smaller than Cy. And being smaller, the
temperature increases faster per unit input of energy.
In fact, the relationship between Cy and C p is given by
See p. 199 to see why
a pressure cooker can
cook faster than a con-
ventional, open pan.
It is relatively rare
that we need C v val-
ues; most reactions are
performed at constant
pressure, e.g. refluxing
a flask at atmospheric
pressure.
Cy — C r
nR
(3.24)
where we have met all terms previously.
Worked Example 3.10 What is the heat capacity Cy of 1 mol of
water? Take the value of C p from SAQ 3.7.
Rearranging Equation (3.24) slightly yields
C v — nR + C B
ENTHALPY 107
Inserting values:
C v = (1 x 8.3 + 75.3) J KT'mor 1
so C v = 83.6 JK-'mor 1 .
The value of Cv is 11 per cent higher than C p , so the water in the pressure cooker
will require 1 1 per cent more energy than if heated in an open pan.
Justification Box 3.3
Starting with the definition of enthalpy in Equation (3.16):
H = U + pV
The pV term can be replaced with 'nRT' via the ideal-gas equation (Equation (1.13)),
giving
H = U + nRT
The differential for a small change in temperature is
The values of n and R
dH = dU + nRdT are constants and do
not change.
dividing throughout by dr yields
™) = ( 9 JL) +nR
dT J \dT J
The first bracket equals CV and the second bracket equals C p , so
Cy = C p + nR
which is just Equation (3.24) rearranged. Dividing throughout by n yields the molar
heat capacities:
C v = C P + R (3.25)
Why does the temperature change during a reaction?
Enthalpies and standard enthalpies of reaction: AH r and AH*
One of the simplest definitions of a chemical reaction is 'changes in the bonds'. All
reactions proceed with some bonds cleaving concurrently with others forming. Each
108 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
bond requires energy to form, and each bond liberates energy when breaking (see
p. 63 ff). Typically, the amount of energy consumed or liberated is characteristic of
the bond involved, so each C-H bond in methane releases about 220 kJmol -1 of
energy. And, as we have consistently reported, the best macroscopic indicator of a
microscopic energy change is a change in temperature.
Like internal energy, we can never know the enthalpy of a reagent; only the change
in enthalpy during a reaction or process is knowable. Nevertheless, we can think of
changes in H. Consider the preparation of ammonia:
N 2(g ) + 3H 2(g) ► 2NH 3(g) (3.26)
We obtain the standard enthalpy change on reaction AH^ as a sum of the molar
enthalpies of each chemical participating in the reaction:
products reactants
The values of v for the reaction in Equation (3.26) are V(nh 3 ) = +2, V(h 2 ) = —3 and
v (N2 ) = — 1. We obtain the standard molar enthalpy of forming ammonia after inserting
values into Equation (3.27), as
AH; = 2/ONH 3 ) " t#m (N 2 ) + 3/OH 2 )]
SAQ 3.8 Write out an expression for AHf for the reaction 2NO + 2 ->
2N0 2 in the style of Equation (3.27).
Unfortunately, we do not know the enthalpies of any reagent. All we can know is a
change in enthalpy for a reaction or process. But what is the magnitude of this energy
change? As a consequence of Hess's law (see p. 98), the overall change in enthalpy
accompanying a reaction follows from the number and nature of the bonds involved.
We call the overall enthalpy change during a reaction the 'reaction enthalpy' AH V ,
and define it as 'the change in energy occurring when 1 mol of reaction occurs'. In
consequence, its units are J mol -1 , although chemists will usually want to express
AH in kJ mol" 1 .
In practice, we generally prefer to tighten the definition of AH V above, and look
at reagents in their standard states. Furthermore, we maintain the temperature T at
298 K, and the pressure p at p e . We call these conditions standard temperature and
pressure, or s.t.p. for short. We need to specify the conditions because temperature and
pressure can so readily change the physical conditions of the reactants and products.
As a simple example, elemental bromine is a liquid at s.t.p., so we say the standard
state of bromine at s.t.p. is Br 2 (i). If a reaction required gaseous bromine Br 2 ( g ) then
we would need to consider an additional energy - the energy of vaporization to effect
the process Br 2 (i) — »■ Br 2 ( g ). Because we restricted ourselves to s.t.p. conditions, we
no longer talk of the reaction enthalpy, but the 'standard reaction enthalpies' A// r e ,
where we indicate the standard state with the plimsoll sign ' e '.
ENTHALPY
109
In summary, the temperature of a reaction mixture changes because energy is
released or liberated. The temperature of the reaction mixture is only ever constant in
the unlikely event of AH^ being zero. (This argument requires an adiabatic reaction
vessel; see p. 89.)
Some standard enthalpies have special names. We consider below some of the more
important cases.
Are diamonds forever?
Some elements exist in
several different crys-
tallographic forms. The
differing crystal forms
are called allotropes.
Enthalpies of formation
We often hear it said that 'diamonds are forever'. There was even a James Bond novel
and film with this title. Under most conditions, a diamond will indeed last forever,
or as near 'for ever' as makes no difference. But is it an absolute statement of fact?
Diamond is one of the naturally occurring allotropes of carbon,
the other common allotrope being graphite. (Other, less common,
allotropes include buckminster fullerine.) If we could observe a
diamond over an extremely long time scale - in this case, several
billions of years - we would observe a slow conversion from bril-
liant, clear diamond into grey, opaque graphite. The conversion
occurs because diamond is slightly less stable, thermodynamically,
than graphite.
Heating graphite at the same time as compressing it under enormous pressure
will yield diamond. The energy needed to convert 1 mol of graphite to diamond is
2.4 kJmol - . We say the 'enthalpy of formation' AHf for the diamond is +2.4 kJ
mol -1 because graphite is the standard state of carbon.
We define the 'standard enthalpy of formation' AH^ as the
enthalpy change involved in forming 1 mol of a compound from
its elements, each element existing in its standard form. Both T
and p need to be specified, because both variables influence the
magnitude of AH. Most books and tables cite AH^ at standard
pressure p e and at a temperature of 298 K. Table 3.1 cites a few
representative values of AH f .
It will be immediately clear from Table 3.1 that several val-
ues of AHf are zero. This value arises from the definition we
chose, above: as AHf relates to forming a compound from its con-
stituent elements, it follows that the enthalpy of forming an element
can only be zero, provided it exists in its standard state. Inci-
dentally, it also explains why A//f(Br2, 1) = but A//f(Br2, g) =
29.5 kJmol -1 , because the stable form of bromine is liquid at s.t.p.
For completeness, we stipulate that the elements must exist in
their standard states. This sub-clause is necessary, because whereas
most elements exist in a single form at s.t.p. (in which case their
enthalpy of formation is zero), some elements, such as carbon
The 'standard enthalpy
of formation' A/-/ f & is
the enthalpy change
involved in forming
1 mol of a compound
or non-stable allotrope
from its elements, each
element being in its
standard form, at s.t.p.
We define the enthalpy
of formation of an ele-
ment (in its normal
state) as zero.
110 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
Table 3.1 Standard enthalpies of formation AH f at 298 K
Compound AH f /kj moP 1
Organic
Hydrocarbons
methane (CH4, g)
ethane (CH3CH3, g)
propane (CH 3 CH 2 CH 3 , g)
w-butane (C4H10, g)
ethane (CH 2 =CH 2 , g)
ethyne (CH=CH, g)
c/i-2-butene (C4H8, g)
?ra«.s-2-butene (C4H8, g)
w-hexane (Cf,Hi4, 1)
cyclohexane (C6H12, 1)
Alcohols
methanol (CH 3 OH, 1)
ethanol (C 2 H 5 OH, 1)
Aromatic s
benzene (C6H 6 , 1)
benzene (CeHj,, g)
toluene (CH 3 C 6 H 5 ,1)
Sugars
a-D-glucose (C6H 12 06, s)
jS-D-glucose^gH^Oe, s)
sucrose (Ci 2 H 22 0n, s)
Elements
bromine (Br 2 , 1)
bromine (Br 2 , g)
chlorine (Cl 2 , g)
chlorine (CI, g)
copper (Cu, s)
copper (Cu, g)
fluorine (F 2 , g)
fluorine (F, g)
iodine (I 2 , s)
iodine (I 2 , g)
iodine (I, g)
nitrogen (N, g)
phosphorus (P, white, s)
phosphorus (P, red, s)
sodium (Na, g)
sulphur (S, rhombic, s)
sulphur (S, monoclinic, s)
Inorganic
carbon (diamond, s)
carbon monoxide (CO, g)
— ;t.o
-84.7
-103.9
-126.2
54.3
226.7
-7.00
-11.2
-198.7
-156
-238.7
-277.7
49.0
82.9
50.0
1274
1268
2222
0.00
30.9
0.00
121.7
0.00
338.3
0.00
78.99
0.00
62.4
106.8
472.7
0.00
15.9
107.3
0.00
0.33
2.4
-110.5
ENTHALPY 111
Table 3.1 (continued)
Compound
AH*/kJ moP 1
carbon dioxide (CO2, g)
-393.0
copper oxide (CuO, s)
-157.3
hydrogen oxide (H 2 0, 1)
-285.8
hydrogen oxide (H2O, g)
-241.8
hydrogen fluoride (HF, g)
-271.1
hydrogen chloride (HC1, g)
-92.3
nitrogen hydride (NH 3 , g)
-46.1
nitrogen hydride (NH3, aq)
-80.3
nitrogen monoxide (NO, g)
90.3
nitrogen dioxide (NO2, g)
33.2
phosphine (PH 3 , g)
5.4
silicon dioxide (SiC>2, s)
-910.9
sodium hydroxide (NaOH, s)
-425.6
sulphur dioxide (SO2, g)
-296.8
sulphur trioxide (SO3, g)
-395.7
sulphuric acid (H2SO4, 1)
-909.3
(above), sulphur or phosphorus, have allotropes. The enthalpy of formation for the
stable allotrope is always zero, but the value of A//f for the non-stable allotropes will
not be. In fact, the value of A//f for the non-stable allotrope is cited with respect to
the stable allotrope. As an example, A//f for white phosphorus is zero by definition
(it is the stable allotrope at s.t.p.), but the value of A//f for forming red phosphorus
from white phosphorus is 15.9 kJmol -1 .
If the value of A//f is determined within these three constraints of standard T,
standard p and standard allotropic form, we call the enthalpy a standard enthalpy,
which we indicate using the plimsoll symbol '*' as A// f & .
To conclude: are diamonds forever? No. They convert slowly into graphite, which
is the stablest form of carbon. Graphite has the lowest energy for any of the allotropes
of carbon, and will not convert to diamond without the addition of energy.
Why do we burn fuel when cold?
Enthalpies of combustion
A common picture in any book describing our Stone Age forebears shows short, hairy
people crouched, warming themselves round a flickering fire. In fact, fire was one of
the first chemical reactions discovered by our prehistoric ancestors. Primeval fire was
needed for warmth. Cooking and warding off dangerous animals with fire was a later
'discovery' .
But why do they burn wood, say, when cold? The principal reactions occurring
when natural materials burn involve chemical oxidation, with carbohydrates
combining with elemental oxygen to yield water and carbon dioxide. Nitrogen
112
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
compounds yield nitrogen oxide, and sulphur compounds yield sulphur dioxide, which
itself oxidizes to form SO3.
Let us simplify and look at the combustion of the simplest hydrocarbon, methane.
CH4 reacts with oxygen according to
CH4(g) + 20
2(g)
> C0 2 ( £) + 2H 2
(g)
(3.28)
The reaction is very exothermic, which explains why much of the developed world
employs methane as a heating fuel. We can measure the enthalpy change accompany-
ing the reaction inside a calorimeter, or we can calculate a value with thermochemi-
cal data.
This enthalpy has a special name: we call it the enthalpy of
combustion, and define it as the change in enthalpy accompanying
the burning of methane, and symbolize it as A//( com b ust ion) or just
AH C . In fact, we rarely perform calculations with AH C but with the
standard enthalpy of combustion AH^, where the plimsoll symbol
' e ' implies s.t.p. conditions.
Table 3.2 contains values of A// c & for a few selected organic
compounds. The table shows how all value of AH^ are nega-
tive, reminding us that energy is given out during a combustion reaction. We say
combustion is exothermic, meaning energy is emitted. All exothermic reactions are
characterized by a negative value of A// c °.
But we do not have to measure each value of A// c & : we can
calculate them if we know the enthalpies of formation of each
chemical, product and reactant, we can adapt the expression in
eq. (3.27), saying:
Most authors abbre-
viate 'combustion' to
just V, and symbolize
the enthalpy change
as AHf . Others write
A "(comb)-
We can use equations
like Equation (3.27) for
any form of enthalpy,
not just combustion.
AH*
= J2 vA// f e - Yl vAH ?
(3.29)
products
We could not perform
cycles of this type
unless enthalpy was
a stare function.
The word 'calorific'
means heat contain-
ing, and comes from
the Latin calor, mean-
ing 'heat'.
where each AH term on the right-hand side of the equation is a
molar enthalpy of formation, which can be obtained from tables.
Worked Example 3.11 The wood mentioned in our title question
is a complicated mixture of organic chemicals; so, for simplicity, we
update the scene. Rather than prehistoric men sitting around a fire, we
consider the calorific value of methane in a modern central-heating
system. Calculate the value of AH C for methane at 25 °C using molar
enthalpies of formation A H f .
The necessary values of AH f are:
Species (all as gases)
AHf/kJmoT 1
CH 4
-74.81
o 2
co 2
-393.51
H 2
-285.83
ENTHALPY 113
Table 3.2 Standard enthalpies of combustion AH C
for a few organic compounds (all values are at 298 K)
Substance
AH*/kJ moP 1
Hydrocarbons
methane (CH 4 , g)
-890
ethane (CH 3 CH 3 , g)
-1560
propane (CH 3 CH 2 CH 3 , g)
-2220
n-butane (C4H10, g)
-2878
cyclopropane (C 3 H(,, g)
-2091
propene (C 3 H 6 , g)
-2058
1-butene (C 4 H 8 , g)
-2717
«'.?-2-butene (C4H8, g)
-2710
fra«.?-2-butene (C4H8, g)
-2707
Alcohols
methanol (CH 3 OH, 1)
-726
ethanol (C 2 H 5 OH, 1)
-1368
Aromatic s
benzene (C^Hs, 1)
-3268
toluene (CH 3 C 6 H 5 ,1)
-3953
naphthalene (CioHs, s)
-5147
Acids
methanoic (HC0 2 H, 1)
-255
ethanoic (CH 3 C0 2 H,1)
-875
oxalic (HC0 2 • C0 2 H, s)
-254
benzoic (C 6 H 5 ■ C0 2 H, s)
-3227
Sugars
a-D-glucose (CgH 12 06, s)
-2808
yS-D-glucose (C 6 H 12 6 , s)
-2810
sucrose (Ci 2 H 22 0n, s)
-5645
O2 is an element, so its value of AH f is zero. The other values of A// f are exothermic.
Inserting values into Equation (3.29):
A// c & = [(-393.51) + (2 x -285.83)] - [(1 x -74.81) + (2 x 0)] kJmor 1
A// c & = -965.17 - (-74.81) kJmor 1
A// C e = -886.36 kJmor 1
which is very close to the experimental value of —890 kJmol -1 in
Table 3.2.
SAQ 3.9 Calculate the standard enthalpy of combustion
AHc for burning ^S-D-glucose, C 6 Hi 2 06. The required val-
ues of AHf may be found in Table 3.1.
The massive value
of AHf for glucose
explains why ath-
letes consume glucose
tablets to provide them
with energy.
114
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
We defined the value of AH* during combustion as ff (final) — H (initial)) so a neg-
ative sign for AH* suggests the final enthalpy is more negative after combustion.
In other words, energy is given out during the reaction. Our Stone Age forebears
absorbed this energy by their fires in the night, which is another way of saying 'they
warmed themselves'.
Why does butane burn with a hotter flame
than methane?
Bond enthalpies
Methane is easily bottled for transportation because it is a gas. It burns with a
clean flame, unlike coal or oil. It is a good fuel. The value of AH* for methane is
—886 kJmol -1 , but AH* for ra-butane is —2878 kJmol -1 . Burning butane is clearly
far more exothermic, explaining why it burns with a hotter flame. In other words,
butane is a better fuel.
The overall enthalpy change during combustion is AH*. An alternative way of
calculating an enthalpy change during reaction dispenses with enthalpies of forma-
tion AH* and looks at the individual numbers of bonds formed and broken. We
saw in Chapter 2 how we always need energy to break a bond, and release energy
each time a bond forms. Its magnitude depends entirely on the enthalpy change
for breaking or making the bonds, and on the respective numbers of each. For
example, Equation (3.28) proceeds with six bonds cleaving (four C-H bonds and
two 0=0 bonds) at the same time as six bonds form (two C=0 bonds and four H-0
bonds).
A quick glance at Worked Example 3.11 shows how the energy released during
combustion is associated with forming the CO2 and H2O. If we
could generate more CO2 and H2O, then the overall change in
AH would be greater, and hence the fuel would be superior. In
fact, many companies prefer butane to methane because it releases
more energy per mole.
We can calculate an enthalpy of reaction with bond enthalpies by
assuming the reaction consists of two steps: first, bonds break, and
then different bonds form. This approach can be simplified further
if we consider the reaction consists only of reactive fragments,
and the products form from these fragments. The majority of the
molecule can remain completely unchanged, e.g. we only need to
consider the hydroxyl of the alcohol and the carboxyl of the acid
during a simple esterification reaction.
To simplify the cal-
culation, we pretend
the reaction proceeds
with all bonds break-
ing at once; then, an
instant later, different
bonds form, again all at
once. Such an idea is
mechanistic nonsense
but it simplifies the
calculation.
Worked Example 3.12 What fragments do we need to consider during the esterification
of 1-butanol with ethanoic acid?
ENTHALPY
115
We first draw out the reaction in full:
CH-,
The butyl ethanoate
produced by Equation
(3.30) is an ester, and
smells of pear drops.
(3.30)
,0, , .0.
^CH 3 +H H
O
Second, we look for those parts that change and those that remain unchanged. In this
example, the bonds that cleave are the O— H bond on the acid and the C-0 bond on the
alcohol. Such cleavage will require energy. The bonds that form are an O-H bond (to
yield water) and a C-0 bond in the product ester. All bonds release energy as they form.
In this example, the bonds outside the box do not change and hence do not change their
energy content, and can be ignored.
The value of A// r relates to bond changes. In this example, equal numbers of O-H
and C-O bonds break as form, so we expect an equal amount of energy to be released as
is consumed, leading to an enthalpy change of zero. In fact, the value of AH r is tiny at
-12kJmol _1 .
In some texts, A/-/g E is
written simply as 'BE'.
Worked Example 3.12 is somewhat artificial, because most reac-
tions proceed with differing numbers of bonds breaking and form-
ing. A more rigorous approach quantifies the energy per bond - the
'bond enthalpy' AH BE (also called the 'bond dissociation energy').
AH BE is the energy needed to cleave 1 mol of bonds. For this reason, values of AH BE
are always positive, because energy is consumed.
The chemical environment of a given atom in a molecule will influence the mag-
nitude of the bond enthalpy, so tabulated data such as that in Table 3.3 represent
average values.
We can calculate a value of A// r & with an adapted form of Equation (3.29):
AW
/products
E yA// i
BE
E VAH BE
(3.31)
where the subscripted i means those bonds that cleave or form within each reactant
or product species during the reaction. We need the minus sign because of the way
we defined the bond dissociation enthalpy. All the values in Table 3.3 are positive
because AH BE relates to bond dissociation.
The stoichiometric numbers v here can be quite large unless the molecules are small.
The combustion of butane, for example, proceeds with the loss of 10 C-H bonds. A
116 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
Table 3.3 Table of mean bond enthalpies AH — as a function of bond order and atoms. All
r BE
values cited in kj moP 1 and relate to data obtained (or corrected) to 298 K
c
N
S
O
I
Br
CI
F
H
H-
414
389
368
464
297
368
431
569
435
F-
490
280
343
213
280
285
255
159
Cl-
326
201
272
205
209
218
243
Br-
272
163
209
-
176
192
I
218
-
-
-
151
0-
326
230
423
142
0=
803 a
590 b
523
498
0=
1075
-
-
-
s-
289
-
247
s=
582
-
-
N-
285
159
N=
515
473
N=
858
946
C-
331
c=
590 c
c=
812
a 728 if -C=0.
b 406if-NO 2 ; 368 if-N0 3 .
c 506 if alternating - and =.
moment's thought suggests an alternative way of writing Equation (3.31), i.e.:
Values of &H? can
vary markedly from
experimental values if
calculated in terms of
AH*.
/bonds formed bonds broken
A// r° = " E VA <E- E VA// B*E (3-32)
Note again the minus sign, which we retain for the same reason as
for Equation (3.31).
Each of these bond enthalpies is an average enthalpy, measured from a series of
similar molecules. Values of AH^ E for, say, C-H bonds in hydrocarbons are likely
to be fairly similar, as shown by the values in Table 3.3. The bond energies of C-H
bonds will differ (sometimes quite markedly) in more exceptional molecules, such as
those bearing ionic charges, e.g. carbocations. AH^ E values differ for the OH bond
in an alcohol, in a carboxylic acid and in a phenol.
These energies relate to bond rearrangement in gaseous molecules, but calculations
are often performed for reactions of condensed phases, by combining the enthalpies
of vaporization, sublimation, etc. We can calculate a value without further correction
if a crude value of AH T is sufficient, or we do not know the enthalpies of phase
changes.
ENTHALPY
117
Worked Example 3.13 Use the bond enthalpies in Table 3.3 to calculate the enthalpy
of burning methane (Equation (3.28)). Assume all processes occur in the gas phase.
Strategy. We start by writing a list of the bonds that break and form.
Broken: 4 x C-H and 2 x 0=0
Formed: 2 x C=0 and 4 x O-H
so
AH; = -[2 x A< E(C=0) + 4 x A< E(0 _ H) ] - [4 x A/f* E(C _ H)
Inserting values of AH^ E from Table 3.3 into Equation (3.32):
AH* = -[(2 x 803) + (4 x 464)]
_ [(4 x 414) + (2 x 498)] klmoP 1
AH* = -(3462 - 2652) kJmof 1
+ 2x AH
BE(0=0)J
Reminder: all &H* E
values are positive
because they relate to
dissociation of bonds.
so
AH* = -810kJmol"
which is similar to the value in Worked Example 3.11, but less exothermic.
Aside
Calculations with bond enthalpies AH BE tend to be relatively inaccurate because each
energy is an average. As a simple example, consider the sequential dissociation of
ammonia.
(1) NH3 dissociates to form NH" and H", and re-
quires an energy of 449 kj mol~ ' .
(2) NHj dissociates to form NH" and H", and re-
quires an energy of 384 kJmol -1 .
The symbol V means
a radical species, i.e.
with unpaired elec-
tron^).
(3) NH* dissociates to form N" and H*, and requires an energy of 339 kJmol .
The variations in AH m are clearly huge, so we usually work with an average bond
enthalpy, which is sometimes written as BE or AH — The average bond enthalpy for
BE
the three processes above is 390.9 kJmol - .
118
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
3.3 Indirect measurement of enthalpy
How do we make "industrial alcohol'?
Enthalpy Cycles from Hess's Law
Industrial alcohol is an impure form of ethanol made by hydrolysing ethene,
CH 2 =CH 2 :
(3.33)
We pass ethene and water (as a vapour) at high pressure over a suitable catalyst,
causing water to add across the double bond of the ethene molecule. The industrial
alcohol is somewhat impure because it contains trace quantities of ethylene glycol
(1,2-dihydroxyethane, III), which is toxic to humans. It also contains unreacted water,
and some dissolved ethene.
We may rephrase
Hess's law, saying The
standard enthalpy of an
overall reaction is the
sum of the standard
enthalpies of the indi-
vidual reactions into
which the reaction may
be divided'.
OH
H
H
H OH
(III)
But what is the enthalpy of the hydration reaction in Equation
(3.33)? We first met Hess's law on p. 98. We now rephrase it by
saying 'The standard enthalpy of an overall reaction is the sum of
the standard enthalpies of the individual reactions into which the
reaction may be divided.'
Accordingly, we can obtain the enthalpy of reaction by drawing
a Hess cycle, or we can obtain it algebraically. In this example, we
will use the cycle method.
Worked Example 3.14 What is the enthalpy change AH r of the reaction in Equation
(3.33)?
We start by looking up the enthalpies of formation AH t for ethene,
ethanol and water. Values are readily found in books of data; Table 3.1
contains a suitable selection.
Notice that each of
these formation reac-
tions is highly exother-
mic, explaining why
energy is needed to
obtain the pure ele-
ments.
A// f(1) [CH 2 =CH 2 ] = -52 kJmol -1
A# f(2) [CH 3 CH 2 OH] = -235 kJmol"
A// f( 3)[H 2 0] = -286 kJmol"
INDIRECT MEASUREMENT OF ENTHALPY 119
(We have numbered these three (1) to (3) simply to avoid the necessity of rewriting
the equations.)
To obtain the enthalpy of forming ethanol, we first draw a cycle. It is usual to start
by writing the reaction of interest along the top, and the elements parallel, along the
bottom. Remember, the value of A// r is our ultimate goal.
Step 1
AH T
CH 2 = CH 2 + H 2 -* CH 3 CH 2 OH Reaction of interest
2C + 3H 2 + i-0 2 Elements
The next three stages are inserting the three enthalpies AHf (1) to (3). Starting on
the left-hand side, we insert AHf^y.
AH r
CH 2 =CH 2 + H 2 ► CH 3 CH 2 OH
Step 2
A// f (i)
2C + 3H 2 + i0 2
We then put in the enthalpy of forming water, AHf^y.
AH r
CH 2 =CH 2 + H 2 CH 3 CH 2 OH
Step 3
^H f(1) \ AH^
2C + 3H 2 + \0 2
And finally we position the enthalpy of forming the product ethanol, A//f( 2 ):
AH r
CH 2 =CH 2 + H 2 ► CH 3 CH 2 OH
Step 4
AH f (i) \ ZiH f(3 ) \ / AH i( 2)
2C + 3H 2 + i-0 2
We can now determine the value of A// r . Notice how we only draw one arrow per
reaction. The rules are as follows:
120
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
We are only allowed
to make a choice of
route like this
because
enthalpy is a
state
function.
(1) We wish to go from the left-hand side of the reaction to the
right-hand side. We can either follow the arrow labelled
AH V , or we pass via the elements (along the bottom line)
and thence back up to the ethanol.
(2) If we go along an arrow in the same direction as the arrow
is pointing, then we use the value of AH as it is written.
(3) If we have to go along an arrow, but in the opposite direction to the
direction in which it points, then we multiply the value of AH by '— 1'.
In the example here, to go from the left-hand side to the right-hand side via the
elements, we need to go along two arrows AHf^ and AHfQ) in the opposite directions
to the arrows, so we multiply the respective values of AH and multiply each by —1.
We then go along the arrow AHfQ), but this time we move in the same direction as
the arrow, so we leave the sign of the enthalpy unaltered.
And then we tie the threads together and say:
Note there are three
arrows, so there are
three AH terms within
AH r .
AH r = (-1 x AH m ) + (-1 x AH m ) + AH i(2)
Inserting values into this equation:
so
AH T = (-1 x -52 kJmol" 1 ) + (-1 x -286 kJmol _1 ) + (-235 kJmol -1 )
AH T = 52 kJmol -1 + 286 kJmol -1 + (-235 kJinoP 1 )
AH r = 103 kJmol" 1
We obtained this value of AH r knowing the other enthalpies in the cycle, and
remembering that enthalpy is a state function. Experimentally, the value of AH T =
99 kJmol -1 , so this indirect measurement with Hess's law provides relatively good
data.
Sometimes, these cycles are considerably harder than the example here. In such
cases, it is usual to write out a cycle for each reaction, and then use the results from
each cycle to compile another, bigger cycle.
How does an 'anti-smoking pipe' work?
Hess's Law Cycles with Enthalpies of Combustion
Smoking causes severe damage to the heart, lungs and respiratory system. The tobacco
in a cigarette or cigar is a naturally occurring substance, and principally comprises
the elements carbon, oxygen, hydrogen and nitrogen.
Unfortunately, because the tobacco is contained within the bowl of a pipe or a paper
wrapper, complete combustion is rare, meaning that the oxidation is incomplete. One
INDIRECT MEASUREMENT OF ENTHALPY 121
of the worse side effects of incomplete combustion during smoking is the formation
of carbon monoxide (CO) in relatively large quantities. Gaseous CO is highly toxic,
and forms an irreversible complex with haemoglobin in the blood. This complex helps
explain why people who smoke are often breathless.
A simple way of overcoming the toxic effects of CO is to oxidize it before the
smoker inhales the tobacco smoke. This is where the 'anti-smoking' pipe works. (In
fact, the name is a misnomer: it does not stop someone smoking, but merely makes
the smoke less toxic.) The cigarette is inserted into one end of a long, hollow tube
(see Figure 3.7) and the smoker inhales from the other. Along the tube's length are
a series of small holes. As the smoker inhales, oxygen enters the holes, mixes with
the CO and combines chemically with it according to
CO (g) + \0 2 ► C0 2(g) (3.34)
The C02( g ) produced is considerably less toxic than CO( g ), thereby averting at least
one aspect of tobacco poisoning.
We might wonder: What is the enthalpy change of forming the CO in Equation
(3.34)? It is relatively easy to make CO in the laboratory (for example by dehydrating
formic acid with concentrated sulphuric acid), so the enthalpy of oxidizing CO to CO2
is readily determined. Similarly, it is easy to determine the enthalpy of formation
of CO2, by burning elemental carbon; but it is almost impossible to determined
AH C for the reaction C + 5O2 —*■ CO, because the pressure of oxygen in a bomb
calorimeter is so high that all the carbon is oxidized directly to CO2 rather than CO.
Therefore, we will employ Hess's law once more, but this time employing enthalpies
of combustion AH C .
The enthalpies of combustion of carbon and CO are obtained
readily from books of data. We can readily find out the following
from such data books or Table 3.2:
A// c( i)[C (s) + 2( g) ► C0 2( g)] = -393.5 kJmor
A// c(2) [CO (g) + ±0 2(g) ► C0 2(g) ] = -283.0 kJmol"
1
The enthalpy AH C(1) is
huge, and helps explain
why we employ coke
and coal to warm a
house; this reaction
occurs when a coal fire
burns.
Again, we have numbered the enthalpies, to save time.
Once more, we start by drawing a Hess-law cycle with the elements at the bottom
of the page. This time, it is not convenient to write the reaction of interest along the
Mouth
\
Cigarette
Small holes to allow in oxygen
Figure 3.7 An anti-smoking device: the cigarette is inserted into the wider end. Partially oxidized
carbon monoxide combines chemically with oxygen inside the device after leaving the end of the
cigarette but before entering the smoker's mouth; the oxygen necessary to effect this oxidation
enters the device through the small circular holes positioned along its length
122 ENERGY AND THE FIRST LAW OF THERMODYNAMICS
top, so we have drawn it on the left as AH r .
CO + -L0 2 C0 2 Compounds
Step 1
C + O2 Constituent elements
Next we insert the enthalpies for the reactions of interest; we first insert AH c ^y.
CO + \0 2 C0 2
c + o 2
and finally, we insert AH C (2):
. AH C(2)
co + ^o 2 co 2
Step 3
/\M r \ / /iti Q n\
C + 2
We want to calculate a value of AH T . Employing the same laws as before, we see
that we can either go along the arrow for AH r directly, or along A// C (i) in the same
direction as the arrow (so we do not change its sign), then along the arrow AH C ^)-
Concerning this last arrow, we go in the opposite direction to the arrow, so we
multiply its value by —1.
The value of AH V is given as
A// r = AH cm + (-1 x AH C(2) )
Inserting values:
AH r = -393.5 kJmol _1 + (+283 kJmol -1 )
so
AH r = -110.5 kJmol" 1
SAQ 3.10 Calcite and aragonite are both forms of calcium carbonate,
CaC03. Calcite converts to form aragonite. If AHf (ca | Cite) = -1206.92 kJ
INDIRECT MEASUREMENT OF ENTHALPY 123
mor 1 and Atf f & (aragonite) =
for the transition process:
1207.13 kJ mol 1 , calculate the value of A/-/ r
CaC03( S , calcite) > CaC03( S , aragoni
te)
(3.35)
Why does dissolving a salt in water liberate heat?
The 'lattice enthalpy' is
defined as the standard
change in enthalpy
when a solid sub-
stance is converted
from solid to form
gaseous constituent
ions. Accordingly, val-
ues of AH ( iattice) are
always positive.
Hess's Law Applied to Ions: Constructing Born-Haber Cycles
Dissolving an ionic salt in water often liberates energy. For example, 32.8 kJmol -1
of energy are released when 1 mol of potassium nitrate dissolves in water. Energy is
released, as experienced by the test tube getting warmer.
Before we dissolved the salt in water, the ions within the crys-
tal were held together by strong electrostatic interactions, which
obeyed Coulomb's law (see p. 313). We call the energetic sum
of these interactions the lattice enthalpy (see p. 124). We need
to overcome the lattice enthalpy if the salt is to dissolve. Stated
another way, salts like magnesium sulphate are effectively insolu-
ble in water because water, as a solvent, is unable to overcome the
lattice enthalpy.
But what is the magnitude of the lattice enthalpy? We cannot
measure it directly experimentally, so we measure it indirectly, with
a Hess's law energy cycle. The first scientists to determine lattice
enthalpies this way were the German scientists Born and Haber:
we construct a Born-Haber cycle, which is a form of Hess's-
law cycle.
Before we start, we perform a thought experiment; and, for con-
venience, we will consider making 1 mol of sodium chloride at
25 °C. There are two possible ways to generate 1 mol of gaseous
Na + and Cl~ ions: we could start with 1 mol of solid NaCl and
vaporize it: the energy needed is A// ( ^ ttice) . Alternatively, we could
start with 1 mol of sodium chloride and convert it back to the ele-
ments (1 mol of metallic sodium and 0.5 mol of elemental chlorine gas (for which
the energy is — A// f °) and, then vaporize the elements one at a time, and ionize each
in the gas phase. The energies needed to effect ionization are / for the sodium and
£(ea) for the chlorine.
In practice, we do not perform these two experiments because we can calculate a
value of lattice enthalpy A //(lattice) with an energy cycle. Next, we appreciate how
generating ions from metallic sodium and elemental chlorine involves several pro-
cesses. If we first consider the sodium, we must: (i) convert it from its solid state to
gaseous atoms (for which the energy is A// ( ^ ublimation) ); (ii) convert the gaseous atoms
to gaseous cations (for which the energy is the ionization energy /). We next consider
the chlorine, which is already a gas, so we do not need to volatilize it. But: (i) we
must cleave each diatomic molecule to form atoms (for which the energy is AH^ E );
It is common to see
values of AH ( i at t iC e)
called 'lattice energy'.
Strictly, this latter term
is only correct when the
temperature T is K.
124
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
(ii) ionize the gaseous atoms of chlorine to form anions (for which the energy is the
electron affinity £( ea )). Finally, we need to account for the way the sodium chloride
forms from elemental sodium and chlorine, so the cycle must also include A// f s .
Worked Example 3.15 What is the lattice enthalpy A//(i att i ce )
25 °C?
of sodium chloride at
Strategy. (1) We start by compiling data from tables. (2) We construct an energy cycle.
(3) Conceptually, we equate two energies: we say the lattice enthalpy is the same as the
sum of a series of enthalpies that describe our converting solid NaCl first to the respective
elements and thence the respective gas-phase ions.
(1) We compile the enthalpies:
These energies are
huge. Much of this
energy is incorporated
into the lattice; other-
wise, the value of AH*
would be massive.
for sodium chloride
Na (s) + ±Cl 2(g)
for the sodium
Na (s)
Na (g)
-* NaCl (s
> Na (g)
► Na+ (g) + e"
(s)
AH
J(Na)
Aff f * =
-411.15 kJmol"
(sublimation)
= 107.32 kJmol"
502.04 kJ mol"
for the chlorine
ci (g
-> ci;
C1 ( g ) + e ~
(?)
► cr
5 A// BE
121.68 kJmol" 1 (i.e. half of 243.36 kJmor 1 )
(g)
-tea)
-354.81 kJmol"
(2) We construct the appropriate energy cycle; see Figure 3.8. For simplicity, it is usual
to draw the cycle with positive enthalpies going up and negative enthalpies going down.
(3) We obtain a value of A H, ]mi . , equating it to the energy needed
to convert solid NaCl to its elements and thence the gaseous ions. We
construct the sum:
We multiply AH f w by
'-1' because we con-
sider the reverse pro-
cess to formation. (We
travel in the opposite
direction to the arrow
representing &Hf in
Figure 3.8.)
AH,
(lattice)
= -AH t + AH {
(sublimation) + 7 (Na) + J A// BE + £ (ea)
Inserting values:
A/ Vtice) = -(-411.153) + 107.32 + 502.04
+ 121.676 + (-354.81) kJ moP 1
so
^(Lice) = 787.38 kJmor
SAQ 3.11 Calculate the enthalpy of formation AH* for calcium fluoride.
Take
lattice) = "2600 kJ mor
AH
(sublimation)
= 178 kJmol -1 ;/
l(Ca^Ca + )
INDIRECT MEASUREMENT OF ENTHALPY 125
k
1 mol gaseous Na + and CI ions
a
F
v (ea)
AH*
1 V
'(Na)
)
^(lattice)
i
^"(sublimatior
1 mol elements (in standard
states)
Figure 3.8 Born-Haber cycle constructed to obtain the lattice enthalpy AH f
(lattice)
of sodium chlo-
ride. All arrows pointing up represent endothermic processes and arrows pointing down represent
exothermic processes (the figure is not drawn to scale)
= 596kJmor 1 and I 2(Ca+ _^ Ca 2 +) = 1152 kJ mor 1 ; AWg E = 157 kJ mor 1
and f ( ea) = -334 kJ mol" 1 .
Why does our mouth feel cold after eating
peppermint?
Enthalpy of Solution
Natural peppermint contains several components that, if ingested, lead to a cold sen-
sation in the mouth. The best known and best understood is (— )-menthol (IV), which
is the dominant component of the peppermint oil extracted from Mentha piperiia and
M. arvensia.
CH 3
'OH
CH3 CH3
(IV)
126
ENERGY AND THE FIRST LAW OF THERMODYNAMICS
The cause of the cooling sensation is the unusually positive enthalpy of solution.
Most values of AH,
(solution)
are positive, particularly for simple inorganic solutes.
Pure IV is a solid at s.t.p. Dissolving IV in the mouth disrupts its molecular
structure, especially the breaking of the hydrogen bonds associated with the hydroxy 1
group. These bonds break concurrently with new hydrogen bonds forming with the
water of the saliva. We require energy to break the existing bonds, and liberate energy
as new bonds form. Energetically, dissolving (— )-menthol is seen to be endothermic,
meaning we require energy. This energy comes from the mouth and, as we saw earlier,
the macroscopic manifestation of a lower microscopic energy is a lower temperature.
Our mouth feels cold.
The other substance sometimes added to foodstuffs to cause cooling of the mouth
is xylitol (V). It is added as a solid to some sweets, chewing gum, toothpastes and
mouth-wash solutions.
CH2 I CH J CHg
I OH I OH I
OH OH OH
(V)
Measuring values of Ar^ solution
with a
It is quite difficult to measure an accurate enthalpy of solution A//, solutk , ni
calorimeter, but we can measure it indirectly. Consider the example of sodium chlo-
ride, NaCl. The ions in solid NaCl are held together in a tight array by strong ionic
bonds. While dissolving in water, the ionic bonds holding the constituent ions of
Na + and Cl~ in place break, and new bonds form between the ions and molecules
of water to yield hydmted species. Most simple ions are surrounded with six water
molecules, like the [Na(H20)6] + ion (VI). Exceptions include the proton with four
water molecules (see p. 235) and lanthanide ions with eight.
The positive charge
does not reside on
the central sodium
alone. Some charge
is distributed over the
whole ion.
OH 2
H 2 0,, I ^OH 2
' Na'~
H 2 0*^ [ ^OH 2
OH 2
(VI)
Each hydration bond is partially ionic and partially covalent.
Each oxygen atom (from the water molecules) donates a small
amount of charge to the central sodium; hence the ionicity. The
orbitals also overlap to impart covalency to the bond.
INDIRECT MEASUREMENT OF ENTHALPY 127
Energy is needed to break the ionic bonds in the solid salt and energy is liberated
forming hydration complexes like VI. We also break some of the natural hydrogen
bonds in the water. The overall change in enthalpy is termed the enthalpy of solu-
tion, A// ( ^ olmion) . Typical values are —207 kJmol - for nitric acid; 34 kJmol - for
potassium nitrate and —65.5 kJmol -1 for silver chloride.
One of the most sensitive ways of determining a value of AH? Mion) is to measure
the temperature T at which a salt dissolves completely as a function of its solubility
s. A plot of In s (as y) against 1 4- T (as x) is usually linear. We obtain a value
of A// (solution N by multiplying the gradient of the graph by —R, where R is the gas
constant (as described in Chapter 5, p. 210).
low does a camper's ^emergency heat stick' work?
Enthalpies of Complexation
'Exposure' is a condi-
tion of being exposed
to the elements, lead-
ing to hypothermia,
and can lead to death.
A camper is in great danger of exposure if alone on the moor or in
the desert when night falls and the weather becomes very cold. If
a camper has no additional heating, and knows that exposure is not
far off, then he can employ an 'emergency heat stick'. The stick
is long and thin. One of its ends contains a vial of water and, at
the other, a salt such as anhydrous copper sulphate, CUSO4. Both
compartments are housed within a thin-walled glass tube, itself encased in plastic.
Bending the stick breaks the glass, allowing the water to come into contact with the
copper sulphate and effect the following hydration reaction:
CuS0 4(s) + 5H 2 0(i) ► CuS0 4 • 5H 2 (S) (3.36)
The reaction in Equation (3.36) is highly exothermic and releases 134 kJmol -1 of
energy. The camper is kept warm by this heat. The reaction in Equation (3.36)
involves complexation. In this example, we could also call it 'hydration' or 'adding
water of crystallization' . We will call the energy released the 'energy of complexation'
^ ^(complexation) •
Heat is liberated when adding water to anhydrous copper sulphate because a new
crystal lattice forms in response to strong, new bonds forming between the water and
Cu + and S0 4 ~ ions. As corroborative evidence of a change in the crystal structure,
note how 'anhydrous' copper sulphate is off-white but the pentahydrate is blue.
4
Reaction spontaneity and
the direction of
thermodynamic change
Introduction
We start by introducing the concept of entropy S to explain why some reactions
occur spontaneously, without needing additional energy, yet others do not. The sign
of A S for a thermodynamic universe must be positive for spontaneity. We explore
the temperature dependence of A S.
In the following sections, we introduce the concept of a thermodynamic universe
(i.e. a system plus its surroundings). For a reaction to occur spontaneously in a
system, we require the change in Gibbs function G to be negative. We then explore
the thermodynamic behaviour of G as a function of pressure, temperature and reaction
composition.
Finally, we investigate the relationship between AG and the respective equilibrium
constant K, and outline the temperature interdependence of AG and K.
4.1 The direction of physicochemical
change: entropy
Why does the colour spread when placing a drop
of dye in a saucer of clean water?
Reaction spontaneity and the direction of change
*
However gently a drop of dye solution is added to a saucer of clean, pure water,
the colour of the dye soon spreads into uncoloured regions of the water. This mixing
occurs inevitably without warming or any kind of external agitation - the painter with
watercolour would find his art impossible without this effect. Such mixing continues
130
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Mixing occurs spon-
taneously, but we
never see the reverse
process, with dye sud-
denly concentrating
into a coloured blob
surrounded by clear,
uncoloured water.
A reaction is 'spon-
taneous' if it occurs
without any additional
energy input.
It used to be thought
reactions were spon-
taneous if AH was
negative. This sim-
plistic idea is incorrect.
until the composition of the solution in the saucer is homogeneous,
with the mixing complete. We never see the reverse process, with
dye suddenly concentrating into a coloured blob surrounded by
clear, uncoloured water.
In previous chapters we looked at the way heat travels from hot
to cold, as described by the so called 'minus-oneth' law of ther-
modynamics, and the way net movements of heat cease at thermal
equilibrium (as described by the zeroth law). Although this transfer
of heat energy was quantified within the context of the first law, we
have not so far been able to describe why such chemical systems
occur. Thermodynamic changes only ever proceed spontaneously
in one direction, but not the other. Why the difference?
In everyday life, we say the diffusion of a dye 'just happens'
but, as scientists, we say the process is spontaneous. In years past,
it was thought that all spontaneous reactions were exothermic, with
non-spontaneous reactions being endothermic. There are now many
exceptions to this overly simplistic rule; thus, we can confidently
say that the sign of AH does not dictate whether the reaction is
spontaneous or not, so we need a more sophisticated way of looking
at the problem of spontaneity.
When we spill a bowl of sugar, why do the grains go
everywhere and cause such a mess?
Changes in the extent of disorder
The granules of spilt
sugar have a range of
energies.
Surely everyone has dropped a bowl of sugar, flour or salt, and caused a mess! The
powder from the container spreads everywhere, and seems to cover the maximum
area possible. Spatial distribution of the sugar granules ensures a
range of energies; so, for example, some particles reside on higher
surfaces than others, thereby creating a range of potential energies.
And some granules travel faster than others, ensuring a spread of
kinetic energies.
The mess caused by dropping sugar reflects the way nature always seeks to max-
imize disorder. Both examples so far, of dye diffusing in water and sugar causing
a mess, demonstrate the achievement of greater disorder. But if we are specific, we
should note how it is the energetic disorder that is maximized spontaneously.
It is easy to create disorder; it is difficult to create order. It requires effort to clean up
the sugar when re-establishing order, showing in effect how reversing a spontaneous
process requires an input of energy. This is why the converse situation - dropping
a mess of sugar grains and creating a neat package of sugar - does not happen
spontaneously in nature.
THE DIRECTION OF PHYSICOCHEMICAL CHANGE: ENTROPY
131
Why, when one end of the bath is hot and the other
cold, do the temperatures equalize?
I
The word 'entropy'
comes from the Greek
en tropa, meaning
'in change' or 'during
transformation'.
Entropy and the second law of thermodynamics
Quite often, when running a bath, the water is initially quite cold. After the hot
water from the tank has had time to travel through the pipes, the water from the tap
is hot. As a result, one end of the bath is hotter than the other. But a short time
later, the temperature of the water is the same throughout the bath, with the hot
end cooler and the cold end warmer. Temperature equilibration occurs even without
stirring. Why?
We saw in Chapter 1 how the simplest way to gauge how much energy a molecule
possesses is to look at its temperature. We deduce through a reasoning process such as
this that molecules of water at the cold end of the bath have less energy than molecules
at the hot end. Next, by combining the minus-oneth and zeroth laws of thermody-
namics, we say that energy (in the form of heat) is transferred from molecules of
water at the hot end of the bath to molecules at the cold end. Energy transfers until
equilibrium is reached. All energy changes are adiabatic if the bath
is lagged (to prevent energy loss), in accordance with the first law
of thermodynamics.
As no chemical reactions occur, we note how these thermo-
dynamic changes are purely physical. But since no bonds form or
break, what is the impetus - the cause - of the transfer of energy?
We have already seen the way processes occur with an attendant
increase in disorder. We now introduce the concept of entropy. The
extent of energetic disorder is given the name entropy (and has the
symbol S). A bigger value of S corresponds to a greater extent of
energetic disorder.
We now introduce the second law of thermodynamics : a physic-
ochemical process only occurs spontaneously if accompanied by
an increase in the entropy S. By corollary, a non-spontaneous
process - one that we can force to occur by externally adding
energy - would proceed concurrently with a decrease in the ener-
getic disorder.
We can often think of entropy merely in terms of spatial disorder,
like the example of the sugar grains above; but the entropy of a
substance is properly the extent of energetic disorder. Molecules of
hot and cold water in a bath exchange energy in order to maximize
the randomness of their energies.
Figure 4.1 depicts a graph of the number of water molecules
having the energy E (as y) against the energy of the water mole-
cules E (as x). Trace (a) in Figure 4.1 shows the distribution of
energies in a bath where half the molecules have one energy while
the other half has a different energy, which explains why the graph
contains two peaks. A distribution of energies soon forms as energy
The 'second law of ther-
modynamics' says a
process occurs sponta-
neously only when the
concomitant energetic
disorder increases. We
can usually approxi-
mate, and talk in terms
of 'disorder' alone.
The energy is trans-
ferred via random,
inelastic collisions bet-
ween the molecules of
water. Such molecular
movement is some-
times called Brownian
motion; see p. 139.
132 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Colder end
a of bath
LU
C
Hotter end
of bath
Energy of water molecules £
(a)
LU
C
Energy of water molecules E
(b)
Figure 4.1 Graph of the number of water molecules of energy E against energy, (a) Soon after
running the bath, so one end is hotter and the other cooler; and (b) after thermal equilibration. The
(average) energy at the peak relates to the macroscopic temperature
is transferred from one set of water molecules to other. Trace (b) in Figure 4.1 shows
the distribution of energies after equilibration. In other words, the energetic disorder
S increases. The reading on a thermometer placed in the bath will represent an
average energy.
The spread of energies in Figure 4.1 is a direct indication of entropy, with a wider
spread indicating a greater entropy. Such energetic disorder is the consequence of
having a range of energies. The spread widens spontaneously; an example of a non-
spontaneous process would be the reverse process, with the molecules in a bath at,
say, 50 °C suddenly reverting to one having a temperature of 30 °C at one end and a
temperature of 70 °C at the other.
The German scientist Rudolf Clausius (1822-1888) was the
first to understand the underlying physicochemical principles dic-
tating reaction spontaneity. His early work aimed to understand
the sky's blue colour, the red colours seen at sunrise and sun-
set, and the polarization of light. Like so many of the 'greats'
of early thermodynamics, he was a mathematician. He was inter-
ested in engines, and was determined to improve the efficiency of
steam-powered devices, such as pumping engines used to remove
water from mines, and locomotives on the railways. Clausius was
the first to introduce entropy as a new variable into physics and
chemistry.
In the thermodynamic
sense, an 'engine' is a
device or machine for
converting energy into
work. Clausius himself
wanted to devise an effi-
cient machine to con-
vert heat energy (from
a fuel) into mechani-
cal work.
THE DIRECTION OF PHYSICOCHEMICAL CHANGE: ENTROPY
133
Why does a room containing oranges acquire
their aroma?
Spontaneity and the sign of AS
When a bowl containing fresh oranges is placed on the dining room
table, the room acquires their fragrance within a few hours. The
organic substance we smell after its release from the oranges is the
organic terpene (+)-limonene (I), each molecule of which is small
and relatively non-polar. I readily evaporates at room temperature
to form a vapour.
We sometimes say
these compounds
volatilize.
The process we detect when we note the intensifying smell of the oranges, is:
limonene
(i)
limonene
(g)
(4.1)
Gaseous materials
have greater entropy
than their respective
liquids.
so the concentration of volatile limonene in the gas phase increases with time. But
why does it evaporate in this way?
Liquids can flow (and hence transfer energy by inelastic colli-
sions), so they will have a distribution of energies. Molecules in
the liquid state possess a certain extent of energetic disorder and,
therefore, have a certain extent of entropy S. By contrast, molecules
in the gas phase have a greater freedom to move than do liquids,
because there is a greater scope for physical movement: restrictions
arising from hydrogen bonds or other physicochemical interactions are absent, and
the large distances between each molecule allow for wider variations in speed, and
hence in energy. Gas molecules, therefore, have greater entropy than do the liquids
from which they derive. We deduce the simple result S( g ) > S(i).
We could obtain this result more rigorously. We have met the symbol 'A' several
times already, and recall its definition 'final state minus initial state', so the change
in entropy AS for any process is given by the simple equation
AS,
(process)
5(fl]
nal state)
^(initial state)
(4.2)
If the final disorder of a spontaneous process is greater than
the initial disorder, then we appreciate from Equation (4.2) how a
spontaneous process is accompanied by AS of positive sign. This
will remain our working definition of spontaneity.
Ultimately, the sign of AS explains why the smell of the oranges increases with time.
A spontaneous process
is accompanied by a
positive value of AS.
134
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Worked Example 4.1 Show mathematically how the entropy of a gas is higher than
the entropy of its respective liquid.
If S (fina i state) is 5(g) and S (Mtia i state) is S(i) , then AS — 5(g) - S (l) . Because the volatilizing
of the compound is spontaneous, the sign of A S must be positive.
The only way to make AS positive is when S (g ) > Sq).
Why do damp clothes become dry when hung outside?
Reaction spontaneity by inspection
Everyone knows damp clothes become dry when hung outside on the washing line.
Any residual water is lost by evaporation from the cloth. In fact, moisture evap-
orates even if the damp clothes hang limp in the absence of a breeze. The water
spontaneously leaves the fabric to effect the physicochemical process H2O0) — »■
H 2 (g ).
The loss of water occurs during drying in order to increase the
overall amount of entropy, because molecules of gaseous water
have a greater energy than do molecules of liquid, merely as a
result of being gas and liquid respectively. In summary, we could
have employed our working definition of entropy (above), which
leads to a prediction of the clothes becoming dry, given time,
as a result of the requirement to increase the entropy. Inspection
alone allows us a shrewd guess at whether a process will occur
spontaneously or not.
We obtain the energy
for evaporating the
water by lowering the
internal energy of the
garment fibres, so the
clothes feel cool to the
touch when dry.
Worked Example 4.2 By inspection alone, decide whether the sublimation of iodine
(Equation (4.3)) will occur spontaneously or not:
2(s)
-> I
2(g)
(4.3)
Molecules in the gas phase have more entropy than molecules in the liquid phase; and
molecules in the liquid phase have more entropy than molecules in the solid state. As an
excellent generalization, the relative order of the entropies is given by
This argument says
nothing about the rare
of sublimation. In fact,
we do not see subli-
mation occurring sig-
nificantly at room tem-
perature because it is
so slow.
5(g) » 5(1) > 5 (s
>(s)
(4.4)
The product of sublimation is a gas, and the precursor is a solid.
Clearly, the product has greater entropy than the starting material,
so A S increases during sublimation. The process is spontaneous
because A S is positive.
In a bottle of iodine, the space above the solid I2 always shows
a slight purple hue, indicating the presence of iodine vapour.
THE DIRECTION OF PHYSICOCHEMICAL CHANGE: ENTROPY
135
SAQ 4.1 By inspection alone, decide whether the condensation of water,
H 2
2^(g)
H 2 ( i) is spontaneous or not.
Worked Example 4.3 Now consider the chemical process
SOCl 2 (o + H 2
igj
2HCl (g) + SO
2(g)
(4.5)
The reaction occurs spontaneously in the laboratory without recourse
to heating or catalysis. The sight of 'smoke' above a beaker of SOCl2(i)
is ample proof of reaction spontaneity.
We see that the reaction in Equation (4.5) consumes 1 mol of gas
(i.e. water vapour) and 1 mol of liquid, and generates 3 mol of gas.
There is a small change in the number of moles: principally, the
amount of gas increases. As was seen above, the entropy of a gas
is greater than its respective liquid, so we see a net increase in the
entropy of the reaction, making AS positive.
This increase in the
entropy S of a gas
explains why an open
beaker of thionyl chlo-
ride SOCI 2 in the lab-
oratory appears to be
'smoking'.
Worked Example 4.4 By inspection alone, decide whether the formation of ammonia
by the Haber process (Equation (4.6)) is spontaneous or not.
N 2 ( g ) + 3H 2(g) > 2NH
3(g)
(4.6)
All the species in Equation (4.6) are gases, so we cannot use the simple method of looking
to see the respective phases of reactants and products (cf. Equation (4.4)).
But we notice the consumption of 4 mol of reactant to form 2 mol of product. As a
crude generalization, then, we start by saying, '4 mol of energetic disorder are consumed
during the process and 2 mol of energetic disorder are formed'. Next, with Equation (4.1)
before us, we suggest the overall, crude entropy change AS is roughly —2 mol of disorder
per mole of reaction, so the amount of disorder decreases. We suspect the process will
not be spontaneous, because AS is negative.
In fact, we require heating to produce ammonia by the Haber pro-
cess, so the reaction is definitely not spontaneous.
SAQ 4.2 By inspection alone, decide whether the oxi-
dation of sulphur dioxide is thermodynamically sponta-
neous or not. The stoichiometry of the reaction is 7202(g)
+ S02(g) -» S03(g).
The word 'stoichiom-
etry' comes from the
Greek stoicheion, mean-
ing 'a part'.
Worked Example 4.5 By inspection alone, decide whether the reaction Cl2( g ) + F 2(g )
2FCl( g ) should occur spontaneously.
Occasionally we need to be far subtler when we look at reaction spontaneity. The
reaction here involves two molecules of diatomic gas reacting to form two molecules of
a different diatomic gas. Also, there is no phase change during reaction, nor any change
in the numbers of molecules, so any change in the overall entropy is likely to be slight.
136 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Before we address this reaction, we need to emphasize how all these are equilibrium
reactions: at completion, the reaction vessel contains product as well as unconsumed
reactants. In consequence, there is a mixture at the completion of the reaction.
This change in A S arises from the mixing of the elements between the two reacting
species: before reaction, all atoms of chlorine were bonded only to other chlorine atoms
in elemental CI2. By contrast, after the reaction has commenced, a choice arises with
some chlorine atoms bonded to other chlorine atoms (unreacted CI2) and others attached
to fluorine in the product, FC1.
In fact, the experimental value of AS is very small and positive.
Aside
Entropy as 'the arrow of time'
The idea that entropy is continually increasing led many philosophers to call entropy
'the arrow of time' . The argument goes something like this. From the Clausius equality
(see p. 142), entropy is the ratio of a body's energy to its temperature. Entropy is
generally understood to signify an inherent tendency towards disorganization.
It has been claimed that the second law means that the universe as a whole must
tend inexorably towards a state of maximum entropy. By an analogy with a closed
system, the entire universe must eventually end up in a state of equilibrium, with the
same temperature everywhere. The stars will run out of fuel. All life will cease. The
universe will slowly peter out in a featureless expanse of nothingness. It will suffer a
'heat death'.
The idea was hugely influential. For example, it inspired the poet T. S. Eliot to write
his poem The Hollow Men with perhaps his most famous lines
This the way the world ends
not with a bang but a whimper.
He wrote this in 1925. Eliot's poem, in turn, inspired others. In 1927, the astronomer
Sir Arthur Eddington said that if entropy was always increasing, then we can know the
direction in which time moves by looking at the direction in which it increases. The
phrase 'entropy is the arrow of time' gripped the popular imagination, although it is
rather meaningless.
In 1928, the English scientist and idealist Sir James Jean revived the old 'heat death'
argument, augmented with elements from Einstein's relativity theory: since matter and
energy are equivalents, he claimed, the universe must finally end up in the complete
conversion of matter into energy:
The second law of thermodynamics compels materials in the universe to move
ever in the same direction along the same road which ends only in death and
annihilation.
THE DIRECTION OF PHYSICOCHEMICAL CHANGE: ENTROPY
137
The extent of solute
disorder decreases dur-
ing crystallization.
Why does crystallization of a solute occur?
Thermodynamic systems and universes
Atoms or ions of solute leave solution during the process of crystallization to form a
regular repeat lattice.
The extent of solute disorder is high before crystallization, be-
cause each ion or molecule resides in solution, and thereby expe-
riences the same freedom as a molecule of liquid. Conversely,
the extent of disorder after crystallization will inevitably be much
smaller, since solute is incorporated within a solid comprising a
regular repeat lattice.
The value of AS can only be negative because the symbol 'A' means 'final state
minus initial state', and the extent of disorder during crystallization clearly follows the
order 'solute disorder (initial) > solute disorder (nna i)'. We see how the extent of disorder
in the solute decreases during crystallization in consequence of forming a lattice and,
therefore, do not expect crystallization to be a spontaneous process.
But crystallization does occur, causing us to ask, 'Why does
crystallization occur even though AS for the process is negative?'
To answer this question, we must consider all energetic consid-
erations occurring during the process of crystallization, possibly
including phenomena not directly related to the actual processes
inside the beaker.
Before crystallization, each particle of solute is solvated. As
a simple example, a chloride ion in water is attached to six water molecules, as
[Cl(H20)fi]~. Being bound to a solute species limits the freedom of solvent molecules,
that is, when compared with free, unbound solvent.
Crystallization releases these six waters of solvation; see Figure 4.2:
A process is thermody-
namically spontaneous
only if the'overall' value
of AS is positive.
[Cl(H 2 0) 6 r
(aq)
CI"
(in solid lattice) ~T~ t>ri2U(f r ee, not solvating)
(4.7)
J 2 ° H2 °
H 2 0— + Na— H 2
H2(/ A 2 o
Mobile aquo ion
Na +6H 2 0(f re e)
/\
Ion immobilized
within a 3-D
repeat lattice
Mobile water
molecules
Figure 4.2 Schematic representation of a crystallization process. Each solvated ion, here Na + ,
releases six waters of solvation while incorporating into its crystal lattice. The overall entropy of
the thermodynamic universe increases by this means
138
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
The word 'universe' in
this context is com-
pletely different from
a 'universe' in astron-
omy, so the two should
not be confused. A
thermodynamic 'uni-
verse' comprises both
a 'system' and its 'sur-
roundings'.
We define a thermo-
dynamic universe as
'that volume large
enough to enclose
all the changes'; the
size of the surround-
ings depends on the
example.
After their release from solvating this chloride ion, each water
molecule has as much energetic disorder as did the whole chlo-
ride ion complex. Therefore, we expect a sizeable increase in the
entropy of the solvent during crystallization because many water
molecules are released.
When we look at the spontaneity of the crystallization pro-
cess, we need to consider two entropy terms: (i) the solute (which
decreases during crystallization) and (ii) the concurrent increase as
solvent is freed. In summary, the entropy of the solute decreases
while the entropy of the solvent increases.
The crystallization process involves a system (which we are inter-
ested in) and the surroundings. In terms of the component entropies
in this example, we say AS( system ) is the entropy of the solute crys-
tallizing and that A»S( Sun -oundings) represents the entropy change of
the solvent molecules released.
We call the sum of the system and its surroundings the ther-
modynamic universe (see Figure 4.3). A thermodynamic universe
is described as 'that volume large enough to enclose all the ther-
modynamic changes'. The entropy change of the thermodynamic
universe during crystallization is A5( to tai)> which equates to
L-*-^ (total) — ^ ^(system) ~f~ ^^(surroundings)
(4.8)
The value of A System) is negative in the example of crystallization. Accordingly, the
value of AS(surroundings) must be so much larger than AS( S y Ste m) that AS( to tai) becomes
positive. The crystallization is therefore spontaneous.
Thermodynamic
universe
Figure 4.3 We call the sum of the system and its surroundings the 'thermodynamic universe'.
Energy is exchanged between the system and its surroundings; no energy is exchanged beyond
the surrounds, i.e. outside the boundaries of the thermodynamic universe. Hence, the definition 'a
universe is that volume large enough to enclose all the thermodynamic changes'
THE TEMPERATURE DEPENDENCE OF ENTROPY
139
This result of A5( to tai) being positive helps explain how consid-
ering the entropy of a system's surroundings can obviate the appar-
ent problems caused by only considering the processes occurring
within a thermodynamic system. It also explains why crystallization
is energetically feasible.
The concept of a thermodynamic system is essentially macro-
scopic, and assumes the participation of large numbers of
molecules. Indeed, the word 'system' derives from the Greek
susiema, meaning to set up, to stay together.
As a new criterion for
reaction spontaneity,
we say AS (to tai) must be
positive. We must con-
sider the surroundings
if we are to understand
how the overall extent
of energetic disorder
increases during a
process.
4.2 The temperature dependence of entropy
Why do dust particles move more quickly by Brownian
motion in warm water?
Entropy is a function of temperature
Brownian motion is the random movement of small, solid particles sitting on the
surface of water. They are held in position by the surface tension y of the meniscus.
When looking at the dust under a microscope, the dust particles appear to 'dance'
and move randomly. But when the water is warmed, the particles, be they chalk or
house dust, move faster than on cold water.
The cause of the Brownian motion is movement of water mole-
cules, several hundred of which 'hold' on to the underside of each
dust particle by surface tension. These water molecules move and
jostle continually as a consequence of their own internal energy.
Warming the water increases the internal energy, itself causing the
molecules to move faster than if the water was cool.
The faster molecules exhibit a greater randomness in their mo-
tion than do slower molecules, as witnessed by the dust particles,
which we see 'dancing' more erratically. The Brownian motion is
more extreme. The enhanced randomness is a consequence of the water molecules
having higher entropy at the higher temperature. Entropy is a function of temperature.
'Brownian motion' is a
macroscopic observa-
tion of entropy.
Entropy is a function of
temperature.
Why does the jam of a jam tart burn more than does
the pastry?
Relationship between entropy and heat capacity
When biting into a freshly baked jam tart, the jam burns the tongue but the pastry
(which is at the same temperature) causes relatively little harm. The reason why the
jam is more likely to burn is its higher 'heat capacity' C.
140
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
The heat capacity
39 JK _1 mol
-p(lce)
Sections 3.1 and 3.2 describe heat capacity and explain how it
may be determined at constant pressure C p or at constant volume
Cy Most chemists need to make calculations with C p , which repre-
sents the amount of energy (in the form of heat) that can be stored
within a substance - the measurement having been performed at
constant pressure p. For example, the heat capacity of solid water
(ice) is 39 J K _1 mol -1 . The value of C p for liquid water is higher,
at 75 JK _1 mol~ , so we store more energy in liquid water than when it is solid;
stated another way, we need to add more energy to H20(i) if its temperature is to
increase. C p for steam (H20( g )) is 34 JK _1 mol -1 . C p for solid sucrose (II) - a major
component of any jam - is significantly higher at 425 JK _1 mol~ .
tells us that adding 39 J
of energy increases the
temperature of 1 mol
of water by 1 K.
H OH/
H \/ \/ CH 2 OH
The heat capacity of a liquid is always greater than the heat capacity of the respec-
tive solid because the liquid, having a greater amount of energetic disorder, has a
greater entropy according to
AS = S 2 - Si
f" C T
dT
(4.9)
More energy is 'stored' within a liquid than in its respective solid, as gauged by
the relative values of C p implied by the connection between the heat capacity and
entropy S (of a pure material). This is to be expected from everyday experience:
to continue with our simplistic example, when a freshly baked jam tart is removed
from the oven, the jam burns the mouth and not the pastry, because the (liquid)
jam holds much more energy, i.e. has a higher C p than does the solid pastry, even
though the two are at the same temperature. The jam, in cooling to the same tem-
perature as the tongue, gives out more energy. The tongue cannot absorb all of this
energy; the energy that is not absorbed causes other processes in the mouth, and hence
the burn.
THE TEMPERATURE DEPENDENCE OF ENTROPY 141
Aside
This argument is oversimplified because it is expressed in terms of jam:
(1) Jam, in comprising mainly water and sugar, will contain more moles per gram
than does the pastry, which contains fats and polysaccharides, such as starch in
the flour. Jam can, therefore, be considered to contain more energy per gram
from a molar point of view, without even considering its liquid state.
(2) The jam is more likely to stick to the skin than does the pastry (because it is
sticky liquid), thereby maximizing the possibility of heat transferring to the
skin; the pastry is flaky and/or dusty, and will exhibit a lower efficiency in
transferring energy.
Worked Example 4.6 Calculate the entropy change AS caused by heating 1 mol of
sucrose from 360 K to 400 K, which is hot enough to badly burn the mouth. Take C p —
425JK~ 1 mor 1 .
Because the value of C p has a constant value, we place it outside the integral, which
allows us to rewrite Equation (4.9), saying
AS = C p ln(^) (4.10)
7i
We insert data into Equation (4.10) to obtain
400 K
360 K
AS = 425 JKT'mor 1 x In
so
AS = 425 JKT 1 mof 1 x ln(l.ll)
and
AS = 44.8 JKT'mor 1
SAQ 4.3 We want to warm the ice in a freezer from a temperature of
-15 °C to 0°C. Calculate the change in entropy caused by the warming
(assuming no melting occurs). Take C p for ice as 39 J K _1 mol -1 . [Hint:
remember to convert to K from °C]
Aside
C p is not independent of temperature, but varies slightly. For this reason, the approach
here is only valid for relatively narrow temperature ranges of, say, 30 K. When determin-
ing AS over wider temperature ranges, we can perform a calculation with Equation (4.9)
142 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
provided that we know the way C p varies with temperature, expressed as a mathematical
power series in T . For example, C p for liquid chloroform CHCI3 is
CyjKT'mor 1 =91.47 + 7.5 x 10~ 2 T
Alternatively, because Equation (4.9) has the form of an integral, we could plot a
graph of C p -7- T (as y) against T (as x) and determine the area beneath the curve. We
would need to follow this approach if C p -7- T was so complicated a function of T that
we could not describe it mathematically.
Justification Box 4.1
Entropy is the ratio of a body's energy to its temperature according to the Clausius
equality (as defined in the next section). For a reversible process, the change in entropy
is defined by
dq
dS=y (4.11)
where q is the change in heat and T is the thermodynamic temperature. Multiplying
the right-hand side of Equation (4.11) by AT /AT (which clearly equals one), yields
do dr
dS= — x — (4.12)
T dr
If no expansion work is done, we can safely assume that q = H. Substituting H for q,
and rearranging slightly yields
/&H\ 1
dS = x -dr (4.13)
\dT J T
where the term in brackets is simply C p . We write
dS=-?-dT (4.14)
Solution of Equation (4.14) takes two forms: (a) the case where C p is considered not
to depend on temperature (i.e. determining the value of A S over a limited range of
temperatures) and (b) the more realistic case where C p is recognized as having a finite
temperature dependence.
(a) C p is independent of temperature (over small temperature ranges).
f 2 dS = C p f
J Si Jt
dr (4.15)
r, 1
So
AS = S 2 -Si = Cp[lnT]| (4.16)
THE TEMPERATURE DEPENDENCE OF ENTROPY 143
and hence
AS =
q,m(|)
(4.17)
(b) C p is not independent of temperature
ranges). We employ a similar approach to that above
into the integral, yielding
(over
, except
larger
that C p
temperature
is incorporated
/ dS
= C'^dT
(4.18)
which, on integration, yields Equation
(4.9).
Worked Example 4.7 What is the increase in entropy when warming 1 mol of chlo-
roform (III) from 240 K to 330 K? Take the value of C p for chloroform from the Aside
box on p. 142.
H
CI CI
(III)
We start with Equation (4.9), retaining the position of C p within the
integral; inserting values:
AS — Sao K — ^240
Rearranging:
/■330 K
K= /
J 240 K
91.47 + 7.5 x 10~ 2 T
dT
AS
/■330 K
^240 K
91.47
+ 7.5 x 10" 2 d7;
Performing the integration, we obtain
AS = [91.47 In T]^ 7.5 x 10 '\T\^£
Then, we insert the variables:
AS = 91.47 In
330 K
The T' on top and
bottom cancel in the
second term within the
integral.
+ 7.5 x 10 _2 (330 K-240 K)
to yield
so
,240 K,
AS = (29.13 + 6.75) J KT 1 moP 1
AS = 35.9 J K" 1 mol" 1
144 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
SAQ 4.4 1 mol of oxygen is warmed from 300 K to 350 K. Calculate the
associated rise in entropy AS if C p( o 2 )/J K _1 moT 1 = 25.8 + 1.2 x 10 _2 7/K.
4.3 Introducing the Gibbs function
Why is burning hydrogen gas in air (to form liquid
water) a spontaneous reaction?
Reaction spontaneity in a system
The 'twin' subscript
of 'I and g' arises
because the reaction
in Equation (4.19) is so
exothermic that most
of the water product
will be steam.
Equation (4.19) describes the reaction occurring when hydrogen
gas is burnt in air:
02(g) + 2H 2(g) ► 2H 2 (1 and g) (4.19)
We notice straightaway how the number of moles decreases from
three to two during the reaction, so a consideration of the system
alone suggests a non-spontaneous reaction. There may also be a
concurrent phase change from gas to liquid during the reaction,
which confirms our original diagnosis: we expect AS to be negative, and so we
predict a non-spontaneous reaction.
But after a moment's reflection, we remember that one of the simplest tests for
hydrogen gas generation in a test tube is to place a lighted splint nearby, and hear the
'pop' sound of an explosion, i.e. the reaction in Equation (4.19) occurs spontaneously.
The 'system' in this example comprises the volume within which chemicals com-
bine. The 'surroundings' are the volume of air around the reaction vessel or flame;
because of the explosive nature of reaction, we expect this volume to be huge. The
surrounding air absorbs the energy liberated during the reaction; in this example, the
energy is manifested as heat and sound. For example, the entropy of the air increases
as it warms up. In fact, AS( surroim dings) is sufficiently large and positive that the value
of ASftotai) is positive despite the value of AS( systsm ) being negative. So we can now
explain why reactions such as that in Equation (4.19) are spontaneous, although at
first sight we might predict otherwise.
But, as chemists, we usually want to make quantitative predictions, which
are clearly impossible here unless we can precisely determine the magnitude of
AS( Sun - oun dings)> i- e - quantify the influence of the surroundings on the reaction, which
is usually not a trivial problem.
How does a reflux condenser work?
Quantifying the changes in a system: the Gibbs function
All preparative chemists are familiar with the familiar Liebig condenser, which we
position on top of a refluxing flask to prevent the flask boiling dry. The evaporating
INTRODUCING THE GIBBS FUNCTION
145
solvent rises up the interior passage of the condenser from the flask, cools and thence
condenses (Equation (4.20)) as it touches the inner surface of the condenser. Con-
densed liquid then trickles back into the flask beneath.
solvent
(g)
solvent(i) + energy
(4.20)
The energy is transferred to the glass inner surface of the condenser. We maintain
a cool temperature inside the condenser by running a constant flow of water through
the condenser's jacketed sleeve. The solvent releases a large amount of heat energy
as it converts back to liquid, which passes to the water circulating within the jacket,
and is then swept away.
Addition of heat energy to the flask causes several physicochemi-
cal changes. Firstly, energy allows the chemical reaction to proceed,
but energy is also consumed in order to convert the liquid solvent
into gas. An 'audit' of this energy is difficult, because so much of
the energy is lost to the escaping solvent and thence to the sur-
rounding water. It would be totally impossible to account for all
the energy changes without also including the surroundings as well
as the system.
So we see how the heater beneath the flask needs to provide
energy to enable the reaction to proceed (which is what we want to
happen) in addition to providing the energy to change the surround-
ings, causing the evaporation of the solvent, the extent of which
we do not usually want to quantify, even if we could. In short,
we need a simple means of taking account of all the surroundings
without, for example, having to assess their spatial extent. From
the second law of thermodynamics, we write
The 'Gibbs function' G
is named after Josiah
Willard Gibbs (1839-
1903), a humble Amer-
ican who contributed
to most areas of phys-
ical chemistry. He also
had a delightful sense
of humour: 'A math-
ematician may say
anything he pleases,
but a physicist must be
at least partially sane'.
AG
(system)
AH
(system)
TAS,
(system)
(4.21)
(see Justification Box 4.2) where the H, T and S terms have their
usual definitions, as above, and G is the 'Gibbs function'. G is
important because its value depends only on the system and not on
the surroundings. By convention, a positive value of AH denotes
an enthalpy absorbed by the system.
AH here is simply the energy given out by the system, i.e. by the
reaction, or taken into it during endothermic reactions. This energy
transfer affects the energy of the surroundings, which respectively
absorb or receive energy from the reaction. And the change in the
energy of the surroundings causes changes in the entropy of the
surroundings. In effect, we can devise a 'words-only' definition
of the Gibbs function, saying it represents 'The energy available
for reaction (i.e. the net energy), after adjusting for the entropy
changes of the surroundings'.
As well as calling G
the Gibbs function, it is
often called the 'Gibbs
energy' or (incorrectly)
'free energy'.
The Gibbs function is
the energy available for
reaction after adjusting
for the entropy changes
of the surroundings.
146 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Justification Box 4.2
The total change in entropy is A5'( tota i), which must be positive for a spontaneous
process. From Equation (4.8), we say
A O (total) = Ao (system) ~r ^ ^(surroundings) - > "
We usually know a value for A5( sy stem) from tables. Almost universally, we do not know
a Value for A S(surroundings) •
The Clausius equality says that a microscopic process is at equilibrium if dS = dq/T
where q is the heat change and T is the thermody-
namic temperature (in kelvin). Similarly, for a macro-
scopic process, AS = Aq/T. In a chemical reaction,
the heat energy emitted is, in fact, the enthalpy change
of reaction A //(system) , and the energy gained by the
surroundings of the reaction vessel will therefore be
-A //( Sys tem)- Accordingly, the Value of A Surroundings)
is -AH 4- T.
Rewriting Equation (4.8) by substituting for
A ^(surroundings) gives
This sign change occur-
ring here follows since
energy is absorbed
by the surroundings if
energy has been emit-
ted by the reaction,
and vice versa.
AH
AS(total) = AS( S y S tem) — (4.22a)
The right-hand side must be positive if the process is spontaneous, so
A^tem) - Aif T em) > (4.22b i
^ -^(system)
o > Ajf(system) - AS,
rp (system)
Multiplying throughout by T gives
> AZ/( S ystem) — TASfsystem) (4.23)
So the compound variable A//( Sys tem) — T AS( Sys tem) must be negative if a process is
spontaneous.
This compound variable occurs so often in chemistry that we will give it a symbol
of its own: G, which we call the Gibbs function. Accordingly, a spontaneous process
in a system is characterized by saying,
> AG(system) (4.24)
In words, the Gibbs energy must be negative if a change occurs spontaneously.
INTRODUCING THE GIBBS FUNCTION
147
The sign of AG
Equation (4.24) in Justification Box 4.2 shows clearly that a pro-
cess only occurs spontaneously within a system if the change in
Gibbs function is negative, even if the sign of AS( S y Ste m) is slightly
negative or if A//( Syste m) is slightly positive. Analysing the reaction
in terms of our new variable AG represents a great advance: pre-
viously, we could predict spontaneity if we knew that A5 , ( to tai) was
positive - which we now realize is not necessarily a useful crite-
rion, since we rarely know a value for AS( sun - oim di ng s)- It is clear
from Equation (4.21) that all three variables, G, H and S, each
relate to the system alone, so we can calculate the value of AG by
looking up values of A S and AH from tables, and without needing
to consider the surroundings in a quantitative way.
A process occurring in a system is spontaneous if AG is nega-
tive, and it is not spontaneous if AG is positive, regardless of the
sign of A5( syste m)- The size of AG (which is negative) is maxi-
mized for those processes and reactions for which AS is positive
and which are exothermic, with a negative value of AH.
Worked Example 4.8 Methanol (IV) can be prepared in the
gas phase by reacting carbon monoxide with hydrogen, according
to Equation (4.25). Is the reaction feasible at 298 K if A// e =
The Gibbs energy must
be negative if a change
occurs spontaneously.
We see the analytical
power of AG when we
realize how its value
does not depend on
the thermodynamic
properties of the sur-
roundings, but only on
the system.
90.7 kJmor 1 and AS* = -219 JKT 1 mol
A process occurring in a
system is spontaneous
if AG is negative, and is
not spontaneous when
AG is positive.
CO(g) + 2H 2( g)
CH 3 OH
(g)
(4.25)
■OH
H
(IV)
We shall use Equation (4.21), AG 9 = A// s - TAS i
values (and remembering to convert from kJ to J):
Inserting
©
= (-90700 J mol"
l ) - (298 K
X
e-
= (-90700 + 65 262) JmoP 1
= -25.4 kJ mol -1
-219 J K" 1 mol -1 )
This value of AG e is negative, so the reaction will indeed be
spontaneous in this example. This is an example of where
The Gibbs function is
a function of state,
so values of AG G
obtained with the van't
Hoff isotherm (see
p. 162) and routes
such as Hess's law
cycles are identical.
148
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
the negative value of AH overcomes the unfavourable positive — AS term. In
fact, although the reaction is thermodynamically feasible, the rate of reaction (see
Chapter 8) is so small that we need to heat the reaction vessel strongly to about 550 K
to generate significant quantities of product to make the reaction viable.
Some zoologists be-
lieve this reaction
inspired the myth of
fire-breathing dragons:
some large tropical
lizards have glands
beside their mouths to
produce both H 2 S and
S0 2 - Under advanta-
geous conditions, the
resultant sulphur reacts
so vigorously with the
air that flames form as
a self-defence mecha-
nism.
SAQ 4.5 Consider the reaction 2H 2 S ( g) + SC>2(g) ->
2H 2 ( g) + 3S (S) , where all species are gaseous except the
sulphur. Calculate AG? for this reaction at 298 K with the
thermodynamic data below:
AH?/kl mol -1
AS? /J KT 1 mor 1
H 2 S
S0 2
H 2
-22.2
205.6
-296.6
247.9
-285.8
70.1 31.9
[Hint: it is generally easier first to determine values of AH r
and AS r by constructing separate Hess's-law-type cycles.]
SAQ 4.6 The thermodynamic quantities of charge-
transfer complex formation for the reaction
methyl viologen + hydroquinone
charge-transfer complex (4.26)
-i
are AH r = -22.6 kJ mor 1 and AS r = -62.1 J K _i mol" 1 at 298 K. Such val-
ues have been described as 'typical of weak charge-transfer complex
interactions'. Calculate the value of AG r .
4.4 The effect of pressure on thermodynamic
variables
How much energy is needed?
The Gibbs-Duhem equation
'How much energy is needed?' is a pointless question. It is too imprecise to be useful
to anyone. The amount of energy needed will depend on how much material we wish
to investigate. It also depends on whether we wish to perform a chemical reaction
or a physical change, such as compression. We cannot answer the question until we
redefine it.
The total amount we need to pay when purchasing goods at a shop depends both
on the identity of the items we buy and how many of each. When buying sweets and
THE EFFECT OF PRESSURE ON THERMODYNAMIC VARIABLES
149
apples, the total price will depend on the price of each item, and the amounts of each
that we purchase. We could write it as
d(money) = (price of item 1 x number of item 1)
+ (price of item 2 x number of item 2)
(4.27)
While more mathematical in form, we could have rewritten Equa-
tion (4.27)
3 (money) 3 (money)
d(money) = — — x JV(1) + — — x N(2) + ■■■
J 3(1) 3(2)
(4.28)
where N is merely the number of item (1) or item (2), and each
bracket represents the price of each item: it is the amount of money
per item. An equation like Equation (4.28) is called a total differ-
ential.
In a similar way, we say that the value of the Gibbs function
changes in response to changes in pressure and temperature. We
write this as
G = f( P , T) (4.29)
and say G is a function of pressure and temperature.
So, what is the change in G for a single, pure substance as the
temperature and pressure are altered? A mathematician would start
answering this question by writing out the total differential of G:
dG =
dG\ (dG
dp
37
dT
(4.30)
We must use the sym-
bol 3 ('curly d') in a dif-
ferential when several
terms are changing.
The term in the first
bracket is the rate of
change of one variable
when all other variables
are constant.
The value of G for
a single, pure mate-
rial is a function of
both its temperature
and pressure.
The small subscripted
p on the first bracket
tells us the differential
must be obtained at
constant pressure. The
subscripted T indicates
constant temperature.
which should remind us of Equation (4.28). The first term on the
right of Equation (4.30) is the change in G per unit change in
pressure, and the subsequent dp term accounts for the actual change in pressure.
The second bracket on the right-hand side is the change in G per unit change in
temperature, and the final dT term accounts for the actual change in temperature.
Equation (4.30) certainly looks horrible, but in fact it's simply a statement of the
obvious - and is directly analogous to the prices of apples and sweets we started by
talking about, cf. Equation (4.27).
We derived Equation (4.30) from first principles, using pure mathematics. An alter-
native approach is to prepare a similar equation algebraically. The result of the
algebraic derivation is the Gibbs-Duhem equation:
dG = Vdp- SdT
(4.31)
150
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Justification Box 4.3
We start with Equation (4.21) for a single phase, and write G = H — TS . Its differen-
tial is
dG = dH -TdS- SdT (4.32)
From Chapter 2, we recall that H = U + pV , the differential of which is
dH = dU + pdV + Vdp (4.33)
Substituting for dH in Equation (4.32) with the expres-
Equation (4.35) com-
bines the first and
second laws of ther-
sion for dH in Equation (4.33), we obtain
modynamics: it derives
from Equation (3.5)
dG = (dU + pdV +Vdp)-TdS- SdT (4.34)
and says, in effect,
till = dq + tiw. The ptiV
For a closed system (i.e. one in which no expansion
term relates to expan-
sion work and the TtiS
work is possible)
term relates to the
adiabatic transfer of
dU = TdS-pdV (4.35)
heat energy.
Substituting for the dU term in Equation (4.34) with
the expression for dU in Equation (4.35) yields
dG = (T dS - p dV) + p dV + V dp - T dS - S dT
The Gibbs-Duhem
equation is also com-
monly (mis-)spelt
'Gibbs-Duheme '.
(4.36)
The TdS and p dV terms will cancel, leaving the
Gibbs-Duhem equation, Equation (4.31).
We now come to the exciting part. By comparing the total differential of Equa-
tion (4.30) with the Gibbs-Duhem equation in Equation (4.31) we can see a pat-
tern emerge:
dG =
dG =
dG
dp
V
dp +
dp
dG
~df~
dT
dT
So, by direct analogy, comparing one equation with the other, we can say
dG
V =
dp
(4.37)
THE EFFECT OF PRESSURE ON THERMODYNAMIC VARIABLES
151
and
dG
97
(4.38)
These two equations
are known as the
Maxwell relations.
Equations (4.37) and (4.38) are known as the Maxwell relations.
The second Maxwell relation (Equation (4.38)) may remind us of
the form of the Clausius equality (see p. 142). Although the first Maxwell relation
(Equation (4.37)) is not intuitively obvious, it will be of enormous help later when
we look at the changes in G as a function of pressure.
Why does a vacuum l suck'?
The value of G as a function of pressure
Consider two flasks of gas connected by a small tube. Imagine also that a tap separates
them, as seen by the schematic illustration in Figure 4.4. One flask contains hydrogen
gas at high pressure p, for example at 2 atm. The other has such a low pressure of
hydrogen that it will be called a vacuum.
As soon as the tap is opened, molecules of hydrogen move spon-
taneously from the high-pressure flask to the vacuum flask. The
movement of gas is usually so rapid that it makes a 'slurp' sound,
which is why we often say the vacuum 'sucks'.
Redistributing the hydrogen gas between the two flasks is essen-
tially the same phenomenon as a dye diffusing, as we discussed
at the start of this chapter: the redistribution is thermodynamically
favourable because it increases the entropy, so A S is positive.
We see how the spontaneous movement of gas always occurs
from high pressure to low pressure, and also explains why a balloon
will deflate or pop on its own, but work is needed to blow up
the balloon or inflate a bicycle tyre (i.e. inflating a tyre is not
spontaneous).
The old dictum, 'nature
abhors a vacuum' is not
just an old wives tale, it
is also a manifestation
of the second law of
thermodynamics.
Gases move spon-
taneously from high
pressure to low.
Before
After
Figure 4.4 Two flasks are connected by a tap. One contains gas at high pressure. As soon as the
tap separating the two flasks is opened, molecules of gas move spontaneously from the flask under
higher pressure to the flask at lower pressure. (The intensity of the shading represents the pressure
of the gas)
152 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Why do we sneeze?
The change in Gibbs function during gas movement: gas
molecules move from high pressure to low
When we sneeze, the gases contained in the lungs are ejected through the throat so
violently that they can move at extraordinary speeds of well over a hundred miles
an hour. One of the obvious reasons for a sneeze is to expel germs, dust, etc. in the
nose - which is why a sneeze can be so messy.
But a sneeze is much more sophisticated than mere germ removal: the pressure of
the expelled gas is quite high because of its speed. Having left the mouth, a partial
vacuum is left near the back of the throat as a result of the Bernoulli effect described
below. Having a partial vacuum at the back of the throat is thermodynamically unsta-
ble, since the pressure in the nose is sure to be higher. As in the example above,
gas from a region of high pressure (the nose) will be sucked into the region of low
pressure (the back of the mouth) to equalize them. The nose is unblocked during this
process of pressure equilibration, so one of the major reasons why we sneeze is to
unblock the nose.
It is relatively easy to unblock a nose by blowing, but a sneeze is a superb means
for unblocking the nose from the opposite direction.
Aside
The Bernoulli effect
Hold two corners of a piece of file paper along its narrow side. It will droop under its
own weight because of the Earth's gravitational pull acting on it. But the paper will
rise and stand out almost horizontally when we blow gently over its upper surface, as
if by magic (see Figure 4.5). The paper droops before blowing. Blowing induces an
additional force on the paper to counteract the force of gravity.
The air pressure above the upper side of the paper decreases because the air moves
over its surface faster than the air stream running past the paper's underside. The
Blow over face of
paper from here
Partial vaccum here
causes paper to lift
(b)
Figure 4.5 The Bernoulli effect occurs when the flow of fluid over one face of a body is
greater than over another, leading to pressure inequalities. Try it: (a) hold a pace of paper in
both hands, and feel it sag under its own weight, (b) Blow over the paper's upper surface,
and see it lift
THE EFFECT OF PRESSURE ON THERMODYNAMIC VARIABLES
153
disparity in air speed leads to a difference in pressure. In effect, a partial vacuum forms
above the paper, which 'sucks' the paper upwards. This is known as the Bernoulli effect.
A similar effect enables an aeroplane to fly: the curve on a plane's wing is carefully
designed such that the pressure above the wing is less than that below. The air flows over
the upper face of the wing with an increased speed, leading to a decrease in pressure.
Because the upward thrust on the underside of the wing is great (because of the induced
vacuum), it counterbalances the downward force due to gravity, allowing the plane to
stay airborne.
How does a laboratory water pump work?
Gibbs function of pressure change
The water pump is another example of the Bernoulli effect, and is
an everyday piece of equipment in most laboratories, for example
being used during Biichner filtration. It comprises a piece of rubber
tubing to connect the flask to be evacuated to a pump. Inside the
pump, a rapid flow of water past one end of a small aperture inside
the head decreases the pressure of the adjacent gas, so the pressure
inside the pump soon decreases.
Gas passes from the flask to the pump where the pressure is
lower. The change in Gibbs function associated with these pressure
changes is given by
AG = RT lnf-^-)
V /? (initial) /
(4.39)
where the AG term represents the change in G per mole of gas. We
will say here that gas enters the pump at pressure p( nn ai) from a flask
initially at pressure /7 (in itiai)- Accordingly, since p (final) < p(Mm)
and the term in brackets is clearly less than one, the logarithm term
is negative. AG is thus negative, showing that gas movement from
a higher pressure p(Mtiai) to a lower pressure P( nn ai) is spontaneous.
It should be clear from Equation (4.39) that gas movement in
the opposite direction, from low pressure (/?( nn ai)) to high (/"(initial))
would cause AG to be positive, thereby explaining why the process
of gas going from low pressure to high never occurs naturally.
Stated another way, compression can only occur if energy is put
into the system; so, compression involves work, which explains
why pumping up a car tyre is difficult, yet the tyre will deflate of
its own accord if punctured.
These highly oversim-
plified explanations
ignore the effects of
turbulent flow, and the
formation of vortices.
The minimum pres-
sure achievable with
a water pump equals
the vapour pressure of
water, and has a value
of about 28 mmHg.
AG is negative for the
physical process of gas
moving from higher to
lower pressure.
Since Equation (4.39)
relies on a ratio of pres-
sures, we say that a
gas moves from 'higher'
to 'lower', rather than
'high' to 'low'.
Compression involves
work.
154
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Equation (4.39) in-
volves a ratio of pres-
sures, so, although
mmHg (millimetres of
mercury) is not an SI
unit of pressure, we
are permitted to use
it here.
The pressure of vapour
above a boiling liquid
is the same as the
atmospheric pressure.
Worked Example 4.9 The pressure inside a water pump is the same
as the vapour pressure of water (28 mmHg). The pressure of gas inside
a flask is the same as atmospheric pressure (760 mmHg). What is
the change in Gibbs function per mole of gas that moves? Take T —
298 K.
Inserting values into Equation (4.39) yields
, i 28 mmHg
AG = 8.414 JKT 1 moH 1 x 298 K x In '
760 mmHg
AG = 2477 JmoP 1 x ln(3.68 x 10~ 2 )
AG = 2477 JmoP 1 x (-3.301)
AG = -8.2kJmol _1
SAQ 4.7 A flask of methyl-ethyl ether (V) is being evaporated. Its boiling
temperature is 298 K (the same as room temperature) so the vapour
pressure of ether above the liquid is the same as atmospheric pressure,
i.e. at 100 kPa. The source of the vacuum is a water pump, so the pressure
is the vapour pressure of water, 28 mmHg.
Cri2 ,Cri3
CH 3 ^ ^O
(V)
(1) Convert the vacuum pressure P( V acuum) into an SI pressure,
remembering that 1 atm = 101 325 kPa = 760 mmHg.
(2) What is the molar change in Gibbs function that occurs when
ether vapour is removed, i.e. when ether vapour goes from the
flask at p s into the water pump at p (va cuum)?
Justification Box 4.4
We have already obtained the first Maxwell relation (Equation (4.37)) by comparing the
Gibbs-Duhem equation with the total differential:
8G
— = V
dp
The ideal-gas equation says pV = nRT ', or, using a
molar volume for the gas (Equation (1.13)):
We obtain the molar
volume l/m as V h- n.
pV m
RT
THE EFFECT OF PRESSURE ON THERMODYNAMIC VARIABLES 155
Substituting for V m from Equation (4.37) into Equation (1.13)
gives
"(f) = Rr
(4.40)
And separation of the variables gives
dG 1
= RT-
dp p
(4.41)
so
1
dG = RT-dp
P
(4.42)
Integration, taking G, at p<MM) and G 2 at P( nna i), yields
G 2 G X = RT lnf /?(final) )
V P (initial) /
(4.43)
Or, more conveniently, if G 2 is the final value of G and G\
the initial value of G, then
the change in Gibbs function is
AG = RT ln( — )
i.e. Equation (4.39).
Aside
In practice, it is often found that compressing or de-
compressing a gas does not follow closely to the ideal-
gas equation, particularly at high p or low T, as exem-
plified by the need for equations such as the van der
Waals equation or a virial expression. The equation
above is a good approximation, though.
A more thorough treatment takes one of two courses:
The fugacity f can be
regarded as an 'effec-
tive' pressure. The
'fugacity coefficient' y
represents the devia-
tion from ideality. The
value of y tends to one
asp tends to zero.
(1) Utilize the concept of virial coefficients; see
p. 57.
(2) Use fugacity instead of pressure.
Fugacity / is defined as
f = pxy (4.44)
The word 'fugacity'
comes from the Latin
fugere, which means
'elusive' or 'difficult to
capture'. The modern
word 'fugitive' comes
from the same source.
156
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
where p is conventional pressure and y is the fugacity coefficient representing the
deviation from ideality. Values of y can be measured or calculated.
We employ both concepts to compensate for gas non-ideality.
4.5 Thermodynamics and the extent
of reaction
Incomplete reactions and extent of reaction
As chemists, we should perhaps re-cast the question 'Why is a 'weak' acid weak?' by
asking 'How does the change in Gibbs function relate to the proportion of reactants
that convert during a reaction to form products?'
An acid is defined as a proton donor within the Lowry-Br0nsted theory (see
Chapter 6). Molecules of acid ionize in aqueous solution to form an anion and a
proton, both of which are solvated. An acid such as ethanoic acid (VI) is said to
be 'weak' if the extent to which it dissociates is incomplete; we call it 'strong' if
ionization is complete (see Section 6.2).
H O
H O-H
(VI)
Ionization is, in fact, a chemical reaction because bonds break and form. Consider
the following general ionization reaction:
HA + H 2 ► H 3 + + A"
(4.45)
We give the extent of the reaction in Equation (4.45) the Greek
symbol £. It should be clear that § has a value of zero before
the reaction commences. By convention, we say that f = 1 mol
if the reaction goes to completion. The value of % can take any
value between these two extremes, its value increasing as the reac-
tion proceeds. A reaction going to completion only stops when no
reactant remains, which we define as £ having a value of 1 mol,
although such a situation is comparatively rare except in inorganic
redox reactions. In fact, to an excellent approximation, all preparative organic reac-
tions fail to reach completion, so < £ < 1.
The value of £ only stops changing when the reaction stops, although the rate at
which % changes belongs properly to the topic of kinetics (see Chapter 8). We say
We give the Greek
symbol £ Cxi') to the
extent of reaction.
f is commonly mis-
pronounced as 'ex-
eye'.
THERMODYNAMICS AND THE EXTENT OF REACTION
157
it reaches its position of equilibrium, for which the value of £ has
its equilibrium value §( e q). We propose perhaps the simplest of the
many possible definitions of equilibrium: 'after an initial period of
reaction, no further net changes in reaction composition occur'.
So when we say that a carboxylic acid is weak, we mean that
§(eq) is small. Note how, by saying that £( eq ) is small at equilibrium,
we effectively imply that the extent of ionization is small because
[H 3 + ] and [A - ] are both small.
But we need to be careful when talking about the magnitudes
of £. Consider the case of sodium ethanoate dissolved in dilute
mineral acid: the reaction occurring is, in fact, the reverse of that
in Equation (4.45), with a proton and carboxylate anion associat-
ing to form undissociated acid. In this case, £ = 1 mol before the
reaction occurs, and its value decreases as the reaction proceeds.
In other words, we need to define our reaction before we can speak
knowledgeably about it. We can now rewrite our question, asking
'Why is £ <C 1 for a weak acid?'
The standard Gibbs function change for reaction is AG e , and
represents the energy available for reaction if 1 mol of reactants
react until reaching equilibrium. Figure 4.6 relates AG and f , and
clearly shows how the amount of energy available for reaction
AG decreases during reaction (i.e. in going from left to right as %
increases). Stated another way, the gradient of the curve is always
negative before the position of equilibrium, so any increases in %
cause the value of AG to become more negative.
We assume such an
equilibrium is fully
reversible in the sense
of being dynamic - the
rate at which products
form is equal and oppo-
site to the rate at which
reactants regenerate
via a back reaction.
Reminder: the energy
released during reac-
tion originates from
the making and break-
ing of bonds, and the
rearrangement of sol-
vent. The full amount
of energy given out
is AH e , but the net
energy available is
less that AH*, being
AH* -TAS*.
CD
CD
Extent of reaction f
Figure 4.6 The value of the Gibbs function AG decreases as the extent of reaction £ until, at
£ ( eq), there is no longer any energy available for reaction, and AG = 0. £ = represents no reaction
and f = 1 mol represents complete reaction
158 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
The amount of energy liberated per incremental increase in reaction is quite large
at the start of reaction, but decreases until, at equilibrium, a tiny increase in the extent
of reaction would not change AG( tota i). The graph has reached a minimum, so the
gradient at the bottom of the trough is zero.
The minimum in the graph of AG against § is the reaction's position of equilib-
rium - we call it £( eq ). The maximum amount of energy has already been expended
at equilibrium, so AG is zero.
Any further reaction beyond £( eq ) would not only fail to liberate any further energy,
but also would in fact consume energy (we would start to go 'uphill' on the right-
hand side of the figure). Any further increment of reaction would be character-
ized by AG > 0, implying a non-spontaneous process, which is why the reaction
Stops at £ (eq) .
Why does the pH of the weak acid remain constant?
The law of mass action and equilibrium constants
The amount of ethanoic acid existing as ionized ethanoate anion and solvated proton
is always small (see p. 253). For that reason, the pH of a solution of weak acid is
always higher than a solution of the same concentration of a strong acid. A naive
view suggests that, given time, all the undissociated acid will manage to dissociate,
with the dual effect of making the acid strong, and hence lowering the pH.
We return to the graph in Figure 4.6 of Gibbs function (as y) against extent of
reaction % (as x). At the position of the minimum, the amounts of free acid and
ionized products remain constant because there is no longer any energy available for
reaction, as explained in the example above.
The fundamental law of chemical equilibrium is the law of mass action, formulated in
1 864 by Cato Maximilian Guldberg and Peter Waage. It has since been redefined several
times. Consider the equilibrium between the four chemical species A, B, C and D:
a A + bB = cC + dD (4.46)
where the respective stoichiometric numbers are —a, —b, c and d. The law of mass
action states that, at equilibrium, the mathematical ratio of the concentrations of
the two reactants [A]" x [B] h and the product of the two product concentrations
[C] c x [D] d , is equal. We could, therefore, define one of two possible fractions:
[Af[B]" [Cf[DY
7 or (4.47)
[C] e [D] rf [A] a [B]''
This ratio of concentrations is called an equilibrium constant, and is symbolized as K.
The two ratios above are clearly related, with one being the reciprocal of the other.
Ultimately, the choice of which of these two we prefer is arbitrary, and usually relates
to the way we write Equation (4.46). In consequence, the way we write this ratio is
dictated by the sub-discipline of chemistry we practice. For example, in acid-base
THERMODYNAMICS AND THE EXTENT OF REACTION
159
chemistry (see Chapter 6) we write a dissociation constant, but in complexation equi-
libria we write a formation constant.
In fact, an equilibrium constant is only ever useful when we have carefully defined
the chemical process to which it refers.
The reaction quotient
It is well known that few reactions (other than inorganic redox reactions) ever reach
completion. The value of £( eq ) is always less than one.
The quotient of products to reactants during a reaction is
Q
Y\ [products] v
\\ [reactants] v
(4.48)
The mathematical sym-
bol n means the 'pi
product', meaning the
terms are multiplied
together, so \\{2, 3, 4)
is 2 x 3 x 4 = 24.
which is sometimes called the reaction quotient. The values of v
are the respective stoichiometric numbers. The mathematical value
of Q increases continually during the course of reaction because
of the way it relates to concentrations during reaction. Initially,
the reaction commences with a value of Q = 0, because there is no product (so the
numerator is zero).
Aside
A statement such as 'K = 0.4 moldm" ' is wrong, although we find examples in a
great number of references and textbooks. We ought, rather, to say K = 0.4 when the
equilibrium constant is formulated (i) in terms of concentrations, and (ii) where each
concentration is expressed in the reference units of mol dm" . Equilibrium constants
such as K c or K p are mere numbers.
ery decrease to zero.
Relationships between K and AG e
The voltage of a new torch battery (AA type) is about 1 .5 V. After
the battery has powered the torch for some time, its voltage drops,
which we see in practice as the light beam becoming dimmer. If
further power is withdrawn indefinitely then the voltage from the
battery eventually drops to zero, at which point we say the battery
is 'dead' and throw it away.
A battery is a device for converting chemical energy into electri-
cal energy (see p. 344), so the discharging occurs as a consequence
of chemical reactions inside the battery. The reaction is complete
The battery produces
'power' W (energy per
unit time) by passing
a current through a
resistor. The resister in
a torch is the bulb
filament.
160
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
The relationship be-
tween AG and volt-
age is discussed in
Section 7.1.
when the battery is 'dead', i.e. the reaction has reached its equilib-
rium extent of reaction £( eq ) .
The battery voltage is proportional to the change in the Gibbs
function associated with the battery reaction, call it A G (battery)-
Therefore, we deduce that AG (battery) must decrease to zero because
the battery voltage drops to zero. Figure 4.7 shows a graph of battery voltage (as y)
against time of battery discharge (as x); the time of discharge is directly analogous
to extent of reaction £. Figure 4.7 is remarkably similar to the graph of AG against
£ in Figure 4.6.
The relationship between the energy available for reaction AG r and the extent of
reaction (expressed in terms of the reaction quotient Q) is given by
AG r = AG? + RT In Q
(4.49)
where AG is the energy available for reaction during chemical changes, and AG^
is the standard change of Gibbs function AG S , representing the change in Gibbs
function from £ = to £ = £( eq) .
Equation (4.49) describes the shape of the graph in Figure 4.6.
Before we look at Equation (4.46) in any quantitative sense, we
note that if RT In Q is smaller than AG?, then AG r is positive.
The value of AG r only reaches zero when AG? is exactly the same
as RT In Q. In other words, there is no energy available for reaction
when AG r = 0: we say the system has 'reached equilibrium'. In fact, AG r = is
one of the best definitions of equilibrium.
In summary, the voltage of the battery drops to zero because the value of AG r is
zero, which happened at f = £( eq) .
AG r =
is one
of the
best defi
nitions of equi-
librium.
E 0.4
20 30 40 50
Extent of discharge f
Figure 4.7 Graph of battery emf (as y) against extent of discharge (as x). Note the remarkable
similarity between this figure and the left-hand side of Figure 4.6, which is not coincidental because
emf on AG, and extent of discharge is proportional to £. The trace represents the ninth discharge of
a rechargeable lithium-graphite battery, constructed with a solid-state electrolyte of polyethylene
glycol containing LiC104. The shakiness of the trace reflects the difficulty in obtaining a reversible
measurement. Reprinted from S. Lemont and D. Billaud, Journal of Power Sources 1995; 54: 338.
Copyright © 1995, with permission from Elsevier
THERMODYNAMICS AND THE EXTENT OF REACTION
161
Justification Box 4.5
Consider again the simple reaction of Equation (4.46):
aA + bB = cC + dD
We ascertain the Gibbs energy change for this reaction. We start by saying
AG = 2 , V G (products) — / t V G (reactants)
where v is the respective stoichiometric number; so
AG = cG c + dG D -aG A -bG B (4.50)
From an equation like Equation (4.43), G = 6 s + RT ln(p/p & ), so each G term in
Equation (4.50) may be converted to a standard Gibbs function by inserting a term like
Equation (4.43):
AG =
cG° + cRT In ( -^ j + dG^ + dRT In ( -^
-a^rin I —
bGl -bRT\n( —
. p^ J ' \/>°
We can combine the G e terms as AG e by saying
AG S = cG c + <f G D — «G A — bG
So Equation (4.51) simplifies to become:
AG = AG* + cRT\n{ —) + dRT in' /l!
(4.51)
(4.52)
florin
Pa
P*
Then, using the laws of logarithms, we can simplify further:
' (Pc/P*Y(PD/P*y l
AG = AG" + RTln
(Pa/p*)"(Pb/p°)
(4.53)
(4.54)
The bracketed term is the reaction quotient, expressed
in terms of pressures, allowing us to rewrite the equation
in a less intimidating form of Equation (4.49):
AG r = AGf + RT\nQ
A similar proof may be used to derive an expression
relating to AG® and K c .
We changed the posi-
tioned of each stoi-
chiometric number via
the laws of logarithms,
saying b x Ina = Ina 6 .
162
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Aside
A further complication arises from the AG term in Equation (4.49). The diagram above
is clearer than the derivation: in reality, the differential quantity 3G/3f only corresponds
to the change in Gibbs function AG under certain, well defined, and precisely controlled
experimental conditions.
This partial differential is called the reaction affinity in older texts and in newer texts
is called the reaction free energy.
Why does the concentration of product stop
changing?
The van't Hoff isotherm
Because we only ever
write K (rather than
Q) at equilibrium, it is
tautologous but very
common to see K writ-
ten as K (eq) or K e .
The descriptor 'iso-
therm' derives from
the Greek iso meaning
'same' and thermos
meaning 'temperature'.
Jacobus van't Hoff
was a Dutch scientist
(1852-1911). Notice
the peculiar arrange-
ment of the apostro-
phe, and small and
capital letters in his
surname.
It would be beneficial if we could increase the yield of a chem-
ical reaction by just leaving it to react longer. Unfortunately, the
concentrations of reactant and product remain constant at the end
of a reaction. In other words, the reaction quotient has reached a
constant value.
At equilibrium, when the reaction stops, we give the reaction
quotient the special name of equilibrium constant, and re-symbolize
it with the letter K. The values of K and Q are exactly the
same at equilibrium when the reaction stops. The value of Q is
always smaller than K before equilibrium is reached, because some
product has yet to form. In other words, before equilibrium, the
top line of Equation (4.48) is artificially small and the bottom is
artificially big.
Q and K only have the same value when the reaction has reached
equilibrium, i.e. when AG r = 0. At this extent of reaction, the rela-
tionship between £ and AG S is given by the van't Hoff isotherm:
AG'
-RTXnK
(4.55)
where R and T have their usual thermodynamics meanings. The
equation shows the relationship between AG° and K, indicat-
ing that these two parameters are interconvertible when the tem-
perature is held constant.
SAQ 4.8 Show that the van't Hoff isotherm is dimensionally self-
consistent.
Worked Example 4.10 Consider the dissociation of ethanoic (acetic) acid in water to
form a solvated proton and a solvated ethanoate anion, CH3COOH + H2O -¥ CH3COO
THERMODYNAMICS AND THE EXTENT OF REACTION
163
+ H30 + . This reaction has an equilibrium constant K of about 2 x
10~ 5 at room temperature (298 K) when formulated in the usual units
of concentration (mol dm~ ). What is the associated change in Gibbs
function of this reaction?
Inserting values into the van't Hoff isotherm (Equation (4.55)):
AG* = -8.314 JKT 1 moP 1 x 298 K x ln(2 x 1(T 5 )
AG S = -2478 Jmor 1 x -10.8
AG*
+26 811 Jmor
so
AG* = +26.8 kJmor 1
Note how AG 9 is positive here. We say it is endogenic.
The correct use of the
van't Hoff isotherm
necessitates using the
thermodynamic tem-
perature (expressed
in kelvin).
A process occurring
with a negative value
of AG is said to be
exogenic. A process
occurring with a posi-
tive value of AG is said
to be endogenic.
Justification Box 4.6
We start with Equation
(4.49):
AG = AG* + RT In Q
At equilibrium, the value of AG is zero. Also, the value of Q is
called K:
= AG* + RTlnK
(4.56)
Subtracting the '— RTlnK' term from both sides yields the
(Equation (4.55)):
AG* = -RTlnK
van't Hoff isotherm
This derivation proves that equilibrium constants do exist. The value of AG^
on T , so the value of K should be independent of the total pressure.
depends
We sometimes want to know the value of K from a value of AG*, in which case
we employ a rearranged form of the isotherm:
K = exp
-AG*
RT
(4.57)
so a small change in the Gibbs function means a small value of K. Therefore, a weak
acid is weak simply because AG S is small.
164
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Care: we must always
convert from kJ to J
before calculating with
Equation (4.57).
Worked Example 4.11 Consider the reaction between ethanol and
ethanoic acid to form a sweet- smelling ester and water:
CH 2 CH 2 OH + CH3COOH ► CH 2 CH 2 C0 2 CH 3 + H 2 (4.58)
What is the equilibrium constant K at room temperature (298 K) if
the associated change in Gibbs function is exogenic at —3.4 kJmol -1 ?
Care: if we calculate
a value of K that is
extremely close to
one, almost certainly
we forgot to convert
from kJ to J, mak-
ing the fraction in the
bracket a thousand
times too small.
Inserting values into eq. (4.57):
K — exp
-(-3400 J moP 1 )
SJWJKT'mor 1 x 298 K
K = exp(+ 1.372)
K = 3.95
A value of K greater than one corresponds to a negative value of
AG S , so the esterincation reaction is spontaneous and does occur
to some extent without adding addition energy, e.g. by heating.
A few values of AG 9 are summarized as a function of K in
Table 4.1 and values of K as a function of AG 9 are listed in
Table 4.2. Clearly, K becomes larger as AG becomes more neg-
ative. Conversely, AG** is positive if K is less than one.
Justification Box 4.7
We start with Equation (4.55):
AG e = -RTlnK
Both sides are divided by — RT , yielding
/-AG & \
\ RT )
(4.59)
Then we take the exponential of both sides to generate Equation (4.57).
Table 4.1 The relationship between AG S and equilib-
rium constant K: values of AG ^ as a function of K
We see from Table 4.1
that every decade
increase in K causes
AG° to become more
negative by 5.7 kJ mor 1
per tenfold increase
in K.
K
AG^/kJ mok
1
10
l() 2
10 3
10 4
10- 1
10- 2
10- 3
-5.7
-11.4
-17.1
-22.8
+5.7
+ 11.4
+ 17.1
THERMODYNAMICS AND THE EXTENT OF REACTION 165
Table 4.2 The relationship between AG e
and equilibrium constant K: values of K as
a function of AG S
AG e
/kJ
moP 1
K
1
-1
1.50
-10
56.6
-10 2
3.38 x 10 17
-10 3
00
+ 1
0.667
+ 10
0.0177
+ 10 2
2.96 x 10~ 18
SAQ 4.9 What is the value of AC corresponding to AGf 98 K = -12 kJ mo
Why do chicken eggs have thinner shells
in the summer?
The effect of altering the concentration on £
Egg shells are made of calcium carbonate, CaCC>3. The chicken ingeniously makes
shells for its eggs by a process involving carbon dioxide dissolved in its blood,
yielding carbonate ions which combine chemically with calcium ions. An equilibrium
is soon established between these ions and solid chalk, according to
Ca 2+ (aq) + C0 2 - 3( a q ) = CaC0 3(s , she ii) (4.60)
Unfortunately, chickens have no sweat glands, so they cannot perspire. To dissipate
any excess body heat during the warm summer months, they must pant just like a
dog. Panting increases the amount of carbon dioxide exhaled, itself decreasing the
concentration of CO2 in a chicken's blood. The smaller concentration [C0 3 7 J during
the warm summer causes the reaction in Equation (4.60) to shift further toward the
left-hand side than in the cooler winter, i.e. the amount of chalk formed decreases.
The end result is a thinner eggshell.
Chicken farmers solve the problem of thin shells by carbonating the chickens'
drinking water in the summer. We may never know what inspired the first farmer to
follow this route, but any physical chemist could have solved this problem by first
writing the equilibrium constant K for Equation (4.60):
[CaC0 3(s )]
^(shell formation) = — ^ 2 - , ( 4 " 61 )
|Ca (aq)J|C0 3(aq) J
The value of ^( S h e ii formation) will not change provided the temperature is fixed. There-
fore, we see that if the concentration of carbonate ions (see the bottom line) falls then
166 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
the amount of chalk on the top line must also fall. These changes must occur in tandem
if K is to remain constant. In other words, decreasing the amount of CO2 in a chicken's
blood means less chalk is available for shell production. Conversely, the same reason-
ing suggests that increasing the concentration of carbonate - by adding carbonated
water to the chicken's drink - will increases the bottom line of Equation (4.61), and
the chalk term on the top increases to maintain a constant value of K.
Le Chatelier's principle
Arguments of this type illustrate Le Chatelier's principle, which was formulated in
1888. It says:
Le Chatelier's prin-
ciple is named after
Henri Louis le Chatelier
(1850-1937). He also
spelt his first name the
English way, as'Henry'.
Any system in stable chemical equilibrium, subjected to the influence
of an external cause which tends to change either its temperature or
its condensation (pressure, concentration, number of molecules in unit
volume), either as a whole or in some of its parts, can only undergo such
internal modifications as would, if produced alone, bring about a change
of temperature or of condensation of opposite sign to that resulting from
the external cause.
The principle represents a kind of 'chemical inertia', seeking to minimize the changes
of the system. It has been summarized as, 'if a constraint is applied to a system in
equilibrium, then the change that occurs is such that it tends to annul the constraint'.
It is most readily seen in practice when:
(1) The pressure in a closed system is increased (at fixed temperature) and
shifts the equilibrium in the direction that decreases the system's volume,
i.e. to decrease the change in pressure.
(2) The temperature in a closed system is altered (at fixed pressure), and the
equilibrium shifts in such a direction that the system absorbs heat from its
surroundings to minimize the change in energy.
4.6 The effect of temperature on
thermodynamic variables
Why does egg white denature when cooked but
remain liquid at room temperature?
Effects of temperature on AG & : the Gibbs-Helmholtz equation
Boiling an egg causes the transparent and gelatinous albumen ('egg white') to modify
chemically, causing it to become a white, opaque solid. Like all chemical reactions,
THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES
167
denaturing involves the rearrangement of bonds - in this case, of
hydrogen bonds. For convenience, in the discussion below we say
that bonds change from a spatial arrangement termed '1' to a dif-
ferent spatial arrangement '2'.
From everyday experience, we know that an egg will not dena-
ture at room temperature, however long it is left. We are not saying
here that the egg denatures at an almost infinitesimal rate, so the
lack of reaction at room temperature is not a kinetic phenomenon;
rather, we see that denaturation is energetically non-spontaneous
at one temperature (25 °C), and only becomes spontaneous as the
temperature is raised above a certain threshold temperature, which
we will call 7( cr jticai) (about 70 °C for an egg).
The sign of the Gibbs function determines reaction spontaneity,
so a reaction will occur if AG e is negative and will not occur if
AG 9 is positive. When the reaction is 'poised' at r( cr iti ca i) between
spontaneity and non-spontaneity, the value of AG S = 0.
The changes to AG e with temperature may be quantified with
the Gibbs-Helmholtz equation:
AG
AG
A/T
1
1
(4.62)
where AG 9 is the change in Gibbs function at the temperature Tj_
and AGf is the change in Gibbs function at temperature T\. Note
how values of T must be expressed in terms of thermodynamic
temperatures. AH is the standard enthalpy of the chemical pro-
cess or reaction, as determined experimentally by calorimetry or
calculated via a Hess's-law-type cycle.
The value of AH V for denaturing egg white is likely to be
quite small, since it merely involves changes in hydrogen bonds.
For the purposes of this calculation, we say A// r has a value of
35 kJmol" 1 .
Additionally, we can propose an equilibrium constant of reaction,
although we must call it apseudo constant ^( pS eudo) because we can-
not in reality determine its value. We need a value of A"(p Seu do) in
order to describe the way hydrogen bonds change position during
denaturing. We say that ^( pS eudo, i) relates to hydrogen bonds in the
pre-reaction position '1' (i.e. prior to denaturing) and ^( pseu do, 2)
relates to the number of hydrogen bonds reoriented in the post-
reaction position '2' (i.e. after denaturing). We will say here that
^(pseudo) is '1/10' before the denaturing reaction, i.e. before boiling
the egg at 298 K. From the van't Hoff isotherm, ^"(pseudo) equates
to a Gibbs function change of +5.7 kJmol -1 .
This argument here has
been oversimplified
because the reaction
is thermodynamically
irreversible - after all,
you cannot 'unboil an
egg'!
Some reactions
are
spontaneous
at
one
temperature
but
not
at others.
The temperature de-
pendence of the Gibbs
function change is
described quantita-
tively by the Gibbs-
Helmholtz equation.
The word 'pseudo'
derives from the Greek
stem pseudes meaning
'falsehood', which is
often taken to mean
having an appearance
that belies the actual
nature of a thing.
Denaturing albumen
is an 'irreversible'
process, yet the deriva-
tions below assume
thermodynamic re-
versibility. In fact,
complete reversibility
is rarely essential; try
to avoid making calcu-
lations if a significant
extent of irreversibility
is apparent.
168 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
Worked Example 4.12 The white of an egg denatures while immersed in water boiling
at its normal boiling temperature of 373 K. What is the value of AG at this higher
temperature? Take AH — 35 kJmol - .
It does not matter
which temperature is
chosen as T± and which
as 7" 2 so long as 7"i
relates to AGi and T 2
relates to AG 2 .
The value of AG S at 298 K is +5.7 kJmol , the positive sign
explaining the lack of a spontaneous reaction. Inserting values into
the Gibbs-Helmholtz equation, Equation (4.62), yields
AG373 k 5700 J mor 1 , / 1 1
+35 000 J moP
373 K 298 K V 373 K 298 K
Note that T is a thermodynamic temperature, and is cited in kelvin. All energies have
been converted from kJmoP 1 to Jmol" .
P^ = (19.13 jr'mor')+ (35 000 Jmol" 1 ) x (-6.75 x 10" 4 K" 1 )
373 K
A C &
^Ji = (19.13 jk" 1 mor 1 ) - (23.62 JKT 1 moP 1 )
373 K
A ^373 K ^_ 44JK -l mol -l
373 K
so
and
AG 3 e 73 K = -4.4 JK -1 mol -1 x 373 K
AG 373 K = —1.67 kJmol
AG 373 K has a negative value, implying that the reaction at this new, elevated temperature
is now spontaneous. In summary, the Gibbs-Helmholtz equation quantifies a qualitative
observation: the reaction to denature egg white is not spontaneous at room temperature,
but it is spontaneous at elevated temperatures, e.g. when the egg is boiled in water.
SAQ 4.10 Consider the reaction in Equation (4.63), which occurs wholly
in the gas phase:
NH 3 + |0 2 >NO + |H 2 (4.63)
The value of AG? for this reaction is -239.9 kJ moT 1 at 298 K. If the
enthalpy change of reaction A/-/ r = -406.9 kJ mor 1 , then
(1) Calculate the associated entropy change for the reaction in
Equation (4.63), and comment on its sign.
(2) What is the value of the Gibbs function for this reaction when the
temperature is increased by a further 34 K?
THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES
169
Justification Box 4.8
We will use the quantity 'G -r T' for the purposes of
this derivation. Its differential is obtained by use of the
product rule. In general terms, for a compound function
ab, i.e. a function of the type y = f(a, b):
(4.64)
(4.65)
dx
do da
= ax hDX —
dx dx
so here
d(G H- T)
dr
1 dG /
= - x \-Gx\-
T dT I
1
The function G-
-T
occurs so
often
in
thermodynamics
that
we call it
the Planck
function.
All standard signs ©
have been omitted for
clarity
dG
Note that the term is —S, so the equation becomes
dr H
d(G -r T) _ S G
df ~ ~T ~ f 1
(4.66)
Recalling the now-familiar relationship G = H — TS , we may substitute for the —S
term by saying
G- H
-S =
d(G^T) (G-H\ 1 G
d7
T T 2
The term in brackets on the right-hand side is then split up; so
d(G -r T)
G
H
G
dT
J2
T 2
T 2
(4.67)
(4.68)
(4.69)
On the right-hand side, the first and third terms cancel, yielding
d(G H- T)
dT
H
(4.70)
Writing the equation in this way tells us that if we
know the enthalpy of the system, we also know the
temperature dependence of G 4- T . Separating the vari-
ables and defining G\ as the Gibbs function change at
T\ and similarly as the value of G2 at T 2 , yields
/ d(G/T) = H - —
dT
(4.71)
This derivation as-
sumes that both H
and S are tempera-
ture invariant - a safe
assumption if the vari-
ation between 7~i and
7" 2 is small (say, 40 K
or less).
170 REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
so
|~Gl
G 2 /T 2
r 1 1
T 2
J .
= H
Gi/Tt
J -
n
(4.72)
So, for a single chemical:
G 1 _G 1 = H n_l\
T 2 T { \T 2 TJ
(4.73)
And, for a chemical reaction we have Equation (4.62):
AG 2 AG] / 1 1 \
= AH {
T 2 T x \T 2 Tj
We call this final equation the Gibbs-Helmholtz equation.
At what temperature will the egg start to denature?
Reactions 'poised' at the critical temperature
Care: the nomenclature
7"(criticai) is employed
in many other areas
of physical chemistry
(e.g. see pp. 50 and
189).
The reaction is 'poised'
at the critical tempera-
ture with AG= 0.
If AG 9 goes from positive to negative as the temperature alters,
then clearly the value of AG 9 will transiently be zero at one unique
temperature. At this 'point of reaction spontaneity', the value of
AG 9 = 0. We often call this the 'critical temperature' T^criticai) -
The value of T( cr iticai)» i- e - the temperature when the reaction first
becomes thermodynamically feasible, can be determined approxi-
mately from
AH
Critical) = — (4.74)
Worked Example 4.13
egg albumen 'poised' ?
AS
At what temperature is the denaturation of
We will employ the thermodynamic data from Worked Example 4.12. Inserting
values into Equation (4.74):
This method yields only
an approximate value
of 7" (cr iticai) because AS
and AH are themselves
functions of tempera-
ture.
AH
35 000 J mol"
' (critical)
AS 98.3 JK" 1 mol"
356 K or 83 C
We deduce that an egg will start denaturing above about T =
83 °C, confirming what every cook knows, that an egg cooks in
boiling water but not in water that is merely 'hot'.
THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES 171
SAQ 4.11 Water and carbon do not react at room temperature. Above
what temperature is it feasible to prepare synthesis gas (a mixture of CO
and H2)? The reaction is
'(S)
2H 2
(g)
CO
(g) "I" H 2 (g)
(4.75)
Take AH & = 132 kJ mol" 1 and AS* = 134 J K _1 mol -1 .
Justification Box 4.9
We start with Equation (4.21):
AG e =0= AH - TAS
When just 'poised', the value of AG* is equal to zero. Accordingly, =
= AH
= TAS.
Rearranging slightly, we obtain
AH = TAS
(4.76)
which, after dividing both sides by 'AS', yields Equation (4.74).
Why does recrystallization work?
The effect of temperature on K: the van't Hoff isochore
To purify a freshly prepared sample, the preparative chemist will crystallize then
recrystallize the compound until convinced it is pure. To recrystallize, we first dissolve
the compound in hot solvent. The solubility s of the compound depends on the
temperature T. The value of s is high at high temperature, but it decreases at lower
temperatures until the solubility limit is first reached and then surpassed, and solute
precipitates from solution (hopefully) to yield crystals.
The solubility s relates to a special equilibrium constant we call
the 'solubility product' K s , defined by
K s = [solute]
(solution)
(4.77)
We say the value of
[solute]( S ) = 1 because
its activity is unity; see
Section 7.3.
The [solute] term may, in fact, comprise several component parts
if the solute is ionic, or precipitation involves agglomeration. This
equilibrium constant is not written as a fraction because the 'effec-
tive concentration' of the undissolved solute [solute] ( s ) can be taken
to be unity.
Like all equilibrium constants, the magnitude of the equilibrium
constant K s depends quite strongly on temperature, according to
The word 'isochore'
implies constant pres-
sure, since iso is Greek
for 'same' and the root
chore means pressure.
172
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
the van't Hoff isochore:
In
\^s(l)/
-AH,
(cryst)
R
1
1
(4.78)
where R is the gas constant, AH, t , is the change in enthalpy associated with
crystallization, and the two temperatures are expressed in kelvin, i.e. thermodynamic
temperatures.
Worked Example 4.14 The solubility s of potassium nitrate is 140 g per 100 g of water
at 70.9 °C, which decreases to 63.6 g per 100 g of water at 39.9 °C. Calculate the enthalpy
of crystallization, AH, t .
Strategy. For convenience, we will call the higher temperature Ti and the lower temper-
ature T\. (1) The van't Hoff isochore, Equation (4.78), is written in terms of a ratio, so
we do not need the absolute values. In other words, in this example, we can employ the
solubilities s without further manipulation. We can dispense with the units of s for the
same reason. (2) We convert the two temperatures to kelvin, for the van't Hoff isochore
requires thermodynamic temperatures, so T-i — 343.9 K and T\ — 312.0 K. (3) We insert
values into the van't Hoff isochore (Equation (4.78)):
Note how the two
minus signs on the
right will cancel.
In
140
6X6
-AH,
(cryst)
8.314 JKT'mol
1
1
343.9 K 312.0 K
ln(2.20) =
-AH,
(cryst)
8.314 JK" 1 mol"
In 2.20 = 0.7889
x (-2.973 x 10" 4 K _1 )
We then divide both sides by '2.973 x 10 4 K , so:
0.7889
AH,
(cryst)
Only when the differ-
ence between T 2 and
7"i is less than ca.
40 K can we assume
the reaction enthalpy
AH 9 is independent of
temperature. We oth-
erwise correct for the
temperature depen-
dence of AH e with
the Kirch hoff equation
(Equation (3.19)).
2.973 x 10- 4 K"
8.314 JKT'mor 1
The term on the left equals 2.654 x 10 3 K. Multiplying both sides by
R then yields:
A/7* ryst) = 2.654 x 10 3 K x 8.314 JK ' mol" 1
Atf ( : ry st) = 22.1 kJ mol" 1
SAQ 4.12 The simple aldehyde ethanal (VII) reacts with
the di-alcohol ethylene glycol (VIII) to form a cyclic
acetal (IX):
THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES
173
CH 3 ^
(VII)
OH.
= +
OH'
XH 2
I
,CH 2
(VIM)
CHo 0-CH 2
H 0-CH 2
(IX)
Calculate the enthalpy change AH & for the reaction if the equilibrium constant for
the reaction halves when the temperature is raised from 300 K to 340 K.
Justification Box 4.10
We start with the Gibbs-Helmholtz equation (Equation (4.62)):
AG*
AG7 „/l 1
1 = Afl e
Tj
T 2 Ti
Each value of AG e can be converted to an equilibrium constant via the van't Hoff
isotherm AG*
-RTlnK, Equation (4.55). We say, AGf = -RT { \nK { at 7i; and
ag;
-RT 2 \nK 2 at T 2 .
Substituting for each value of AG S yields
-RT 2 lnK 2 -RTilnKi
AH i
1
T 2 Ti \T 2 :
which, after cancelling the T\ and T 2 terms on the right-hand side, simplifies to
{-R\nK 2 )-(-R\nK0 MI»[±--±-
J 2 1\
(4.79)
(4.80)
Next, we divide throughout by '-R' to yield
AH*
In K 2 - In Ki =
R
1 1
¥ 2 ~Y t
(4.81)
and, by use of the laws of logarithms, In a — \nb = In (a -=- b), so the left-hand side of
the equation may be grouped, to generate the van't Hoff isochore. Note: it is common
(but incorrect) to see A// e written without its plimsoll sign ' e '.
The isochore, Equation (4.81), was derived from the integrated
form of the Gibbs-Helmholtz equation. It is readily shown that the
van't Hoff isochore can be rewritten in a slightly different form, as:
We
often
talk about
'the
isochore' when we
mean the
'van't Hoff
isochore'.
174
REACTION SPONTANEITY AND THE DIRECTION OF THERMODYNAMIC CHANGE
InK
AH®
1
x ^+
T
R
m
X
constant
(4.82)
which is known as either the linear or graphical form of the equation. By analogy
with the equation for a straight line (y = mx + c), a plot of In K (as y) against 1 4- T
(as x) should be linear, with a gradient of — AH®/R.
Worked Example 4.15 The isomerization of 1-butene (X) to form trans -2-butsne. (XI).
The equilibrium constants of reaction are given below. Determine the enthalpy of reaction
AH® using a suitable graphical method.
(X)
(XI)
(4.83)
77K
K
686
1.72
702
1.63
733
1.49
779
1.36
826
1.20
Strategy. To obtain AH®: (1) we plot a graph of In K (as y) against values of 1 4- T
(as x) according to Equation (4.82); (2) we determine the gradient; and (3) multiply the
gradient by — R.
(1) The graph is depicted in Figure 4.8. (2) The best gradient is 1415 K. (3) AH® —
'gradient x — R', so
AH® = 1415 K x (-8.314 JK" 1 mol -1 )
AH*
-11.8 kJmob
SAQ 4.13 The following data refer to the chemical reaction between
ethanoic acid and glucose. Obtain AH® from the data using a suitable
graphical method. (Hint: remember to convert all temperatures to kelvin.)
7/ °C
K
25
21.2 x
10 4
36
15.1 x
10 4
45
8.56 x
10 4
55
5.46 x
10 4
THE EFFECT OF TEMPERATURE ON THERMODYNAMIC VARIABLES 175
0.0012
0.0013
0.0014
0.0015
1*(7VK)
Figure 4.8 The equilibrium constant K for the isomerization of 1-butene depends on the temper-
ature: van't Hoff isochore plot of In K (as y) against 1 -h T (as x) from which a value of A// e
can be calculated as 'gradient x — R'
Aside
Writing table headers
Each term in the table for SAQ 4.13 has been multiplied by 10 4 , which is repetitious,
takes up extra space, and makes the table look messy and cumbersome.
Most of the time, we try to avoid writing tables in this way, by incorporating the
common factor of 'xlO 4 ' into the header. We accomplish this by making use of the
quantity calculus concept (see p. 13). Consider the second value of K. The table says
that, at 36 °C, K = 15.1 x 10 4 . If we divide both sides of this little equation by 10 4 ,
we obtain, K -r 10 4 = 15.1. This equation is completely correct, but is more usually
written as 10~ 4 K = 15.1.
According, we might rewrite the first two lines of the table as:
77°C
lO" 4 ^
25
21.2
The style of this latter version is wholly correct and probably more popular than the
style we started with. Don't be fooled: a common mistake is to look at the table heading
and say, 'we need to multiply K by 10~ 4 '. It has been already!
5
Phase equilibria
Introduction
Phase equilibrium describes the way phases (such as solid, liquid and/or gas) co-exist
at some temperatures and pressure, but interchange at others.
First, the criteria for phase equilibria are discussed in terms of single-component
systems. Then, when the ground rules are in place, multi-component systems are
discussed in terms of partition, distillation and mixing.
The chapter also outlines the criteria for equilibrium in terms of the Gibbs function
and chemical potential, together with the criteria for spontaneity.
5.1 Energetic introduction to phase equilibria
Why does an ice cube melt in the mouth?
Introduction to phase equilibria
The temperature of the mouth is about 37 °C, so an overly simple explanation of
why ice melts in the mouth is to say that the mouth is warmer than the transition
temperature T( me \ t ). And, being warmer, the mouth supplies energy to the immobilized
water molecules, thereby allowing them to break free from those bonds that hold them
rigid. In this process, solid H2O turns to liquid H2O - the ice melts.
Incidentally, this argument also explains why the mouth feels cold after the ice has
melted, since the energy necessary to melt the ice comes entirely from the mouth. In
consequence, the mouth has less energy after the melting than before; this statement
is wholly in accord with the zeroth law of thermodynamics, since heat energy travels
from the hot mouth to the cold ice. Furthermore, if the mouth is considered as an
adiabatic chamber (see p. 89), then the only way for the energy to be found for
melting is for the temperature of the mouth to fall.
178
PHASE EQUILIBRIA
A phase is a compo-
nent within a system,
existing in a precisely
defined physical state,
e.g. gas, liquid, or a
solid that has a single
crystallographic form.
Further thermodynamic background: terminology
In the thermodynamic sense, a phase is defined as part of a chemical system in
which all the material has the same composition and state. Appropriately, the word
comes from the Greek phasis, meaning 'appearance'. Ice, water and steam are the
three simple phases of H2O. Indeed, for almost all matter, the three
simple phases are solid, liquid and gas, although we must note that
there may be many different solid phases possible since H20( s )
can adopt several different crystallographic forms. As a related
example, the two stable phases of solid sulphur are its monoclinic
and orthorhombic crystal forms.
Ice is a solid form of water, and is its only stable form below
°C. The liquid form of H2O is the only stable form in the tem-
perature range < T < 100 °C. Above 100 °C, the normal, stable
phase is gaseous water, 'steam'. Water's normal melting temperature T( me i t ) is 0°C
(273.15 K). The word 'normal' in this context implies 'at standard pressure p & ' . The
pressure p & has a value of 10 5 Pa. This temperature T( me it) is often called the melting
point because water and ice coexist indefinitely at this temperature and pressure, but
at no other temperature can they coexist. We say they reside together at equilibrium.
To melt the ice, an amount of energy equal to A// (melt) must be
added to overcome those forces that promote the water adopting
a solid-state structure. Such forces will include hydrogen bonds.
Re-cooling the melted water to re-solidify it back to ice involves
the same amount of energy, but this time energy is liberated, so
. The freezing process is often called fusion.
Concerning transi-
tions
between
the
two
phases '1'
and
'2', Hess's Law
states
that
AH ( i-2) =
-1 X
AW<2
-*•!)■
AH,
(melt)
= -AH,
(freeze)
(Strictly, we ought to define the energy by saying that no pres-
sure-volume work is performed during the melting and freezing
processes, and that the melting and freezing processes occur without any changes in
temperature.)
Table 5.1 gives a few everyday examples of phase changes, together with some
useful vocabulary.
Two or more phases can coexist indefinitely provided that we maintain certain
conditions of temperature T and pressure p. The normal boiling temperature of water
is 100 °C, because this is the only temperature (at p = p & ) at which both liquid and
Table 5.1 Summary of terms used to describe phase changes
Phase transition
Name of transition
Everyday examples
Solid — >• gas
Solid -*■ liquid
Liquid — ► gas
Liquid — >• solid
Gas — y liquid
Gas — y solid
Sublimation
Melting
Boiling or vaporization
Freezing, solidification or fusion
Condensation or liquification
Condensation
'Smoke' formed from dry ice
Melting of snow or ice
Steam formed by a kettle
Ice cubes formed in a fridge; hail
Formation of dew or rain
Formation of frost
ENERGETIC INTRODUCTION TO PHASE EQUILIBRIA
179
A phase diagram is a
graph showing values
of applied pressure and
temperature at which
equilibrium exists.
gaseous H2O coexist at equilibrium. Note that this equilibrium is dynamic, because
as liquid is converted to gas an equal amount of gas is also converted back to liquid.
However, the values of pressure and temperature at equilibrium
depend on each other; so, if we change the pressure, then the tem-
perature of equilibrium shifts accordingly (as discussed further in
Section 5.2). If we plotted all the experimental values of pressure
and temperature at which equilibrium exists, to see the way they
affect the equilibrium changes, then we obtain a graph called a
phase diagram, which looks something like the schematic graph in
Figure 5.1.
We call each solid line in this graph a phase boundary. If the val-
ues of p and T lie on a phase boundary, then equilibrium between
two phases is guaranteed. There are three common phase bound-
aries: liquid-solid, liquid-gas and solid-gas. The line separating
the regions labelled 'solid' and 'liquid', for example, represents
values of pressure and temperature at which these two phases coex-
ist - a line sometimes called the 'melting-point phase boundary'.
The point where the three lines join is called the triple point,
because three phases coexist at this single value of p and T. The
triple point for water occurs at T = 273.16 K (i.e. at 0.01 °C) and
p = 610 Pa (0.006/5°). We will discuss the critical point later.
Only a single phase is stable if the applied pressure and tem-
perature do not lie on a phase boundary, i.e. in one of the areas
between the phase boundaries. For example, common sense tells is
that on a warm and sunny summer's day, and at normal pressure,
the only stable phase of H2O is liquid water. These conditions of
p and T are indicated on the figure as point 'D'.
A phase boundary is a
line on a phase diagram
representing values of
applied pressure and
temperature at which
equilibrium exists.
The triple point on a
phase diagram rep-
resents the value of
pressure and temper-
ature at which three
phases coexist at equi-
librium.
.2 0.006 p*=
273.16 K
Temperature T
Figure 5.1 Schematic phase diagram showing pressures and temperatures at which two phases
are at equilibrium. Phase boundary (a) represents the equilibrium between steam and ice; boundary
(b) represents equilibrium between water and ice; and boundary (c) represents equilibrium between
water and steam. The point D represents p and T on a warm, sunny day. Inset: warming an ice
cube from —5 °C to the mouth at 37 °C at constant pressure causes the stable phase to convert from
solid to liquid. The phase change occurs at 0°C at p e
180
PHASE EQUILIBRIA
When labelling a phase
diagram, recall how the
only stable phase at
high pressure and low
temperature is a solid;
a gas is most stable
at low pressure and
high temperature. The
phase within the crook
of the 'Y' is therefore
a liquid.
We can predict whether an ice cube will melt just by looking
carefully at the phase diagram. As an example, suppose we take
an ice cube from a freezer at —5 °C and put it straightaway in
our mouth at a temperature of 37 °C (see the inset to Figure 5.1).
The temperature of the ice cube is initially cooler than that of the
mouth. The ice cube, therefore, will warm up as a consequence
of the zeroth law of thermodynamics (see p. 8) until it reaches the
temperature of the mouth. Only then will it attain equilibrium. But,
as the temperature of the ice cube rises, it crosses the phase bound-
ary, as represented by the bold horizontal arrow, and undergoes a
phase transition from solid to liquid.
We know from Hess's law (see p. 98) that it is often useful to
consider (mentally) a physical or chemical change by dissecting it
into its component parts. Accordingly, we will consider the melting of the ice cube
as comprising two processes: warming from —5 °C to 37 °C, and subsequent melting
at 37 °C. During warming, the water crosses the phase boundary, implying that it
changes from being a stable solid (when below 0°C) to being an unstable solid
(above °C). Having reached the temperature of the mouth at 37 °C, the solid ice
converts to its stable phase (water) in order to regain stability, i.e. the ice cube melts
in the mouth. (It would be more realistic to consider three pro-
cesses: warming to 0°C, melting at constant temperature, then
warming from to 37 °C.)
Although the situation with melting in two stages appears a little
artificial, we ought to remind ourselves that the phase diagram is
made up of thermodynamic data alone. In other words, it is possible
to see liquid water at 105 °C, but it would be a metastable phase,
i.e. it would not last long!
The Greek root meta
means 'adjacent to' or
'near to'. Something
metastable is almost
stable . .. but not quite.
Aside
The arguments in this example are somewhat simplified.
Remember that the phase diagram's y-axis is the applied pressure. At room tempera-
ture and pressure, liquid water evaporates as a consequence of entropy (e.g. see p. 134).
For this reason, both liquid and vapour are apparent even at s.t.p. The pressure of the
vapour is known as the saturated vapour pressure (s.v.p.), and can be quite high.
The s.v.p. is not an applied pressure, so its magnitude is generally quite low. The
s.v.p. of water will certainly be lower than atmospheric pressure. The s.v.p. increases
with temperature until, at the boiling temperature, it equals the atmospheric pressure.
One definition of boiling says that the s.v.p. equals the applied pressure.
The arguments in this section ignore the saturated vapour pressure.
ENERGETIC INTRODUCTION TO PHASE EQUILIBRIA
181
Wh
y does water placed in a freezer become ice?
Spontaneity of phase changes
It will be useful to concentrate on the diagram in Figure 5.2 when considering why
a 'phase change' occurs spontaneously. We recall from Chapter 4 that one of the
simplest tests of whether a thermodynamic event can occur is to ascertain whether
the value of AG is negative (in which case the change is indeed spontaneous) or
positive (when the change is not spontaneous).
The graph in Figure 5.2 shows the molar Gibbs function G m as a function of
temperature. (G m decreases with temperature because of increasing entropy.) The
value of G m for ice follows the line on the left-hand side of the graph; the line in
the centre of the graph gives values of G m for liquid water; and the line on the
right represents G m for gaseous water, i.e. steam. We now consider the process of
an ice cube being warmed from below T( me i t ) to above it. The molar Gibbs functions
of water and ice become comparable when the temperature reaches T( me \ t j . At T( me i t )
itself, the two values of G m are the same - which is one definition of equilibrium.
The two values diverge once more above 7(meit)-
Below 7( me i t ), the two values of G m are different, implying that the two forms of
water are energetically different. It should be clear that if one energy is lower than
the other, then the lower energy form is the stablest; in this case,
the liquid water has a higher value of G m and is less stable than
solid ice (see the heavy vertical arrow, inset to Figure 5.2). Liquid
water, therefore, is energetically unfavourable, and for that reason it
is unstable. To attain stability, the liquid water must release energy
and, in the process, undergo a phase change from liquid to solid,
i.e. it freezes.
Remember how the
symbol A means 'final
state minus initial
state', so AG m =
G m (final state) —
Gm (initial state)'
'melt
Temperature T
Figure 5.2 Graph of molar Gibbs function G m as a function of temperature. Inset: at temperatures
below r (rae i t ) the phase transition from liquid to solid involves a negative change in Gibbs function,
so it is spontaneous
182
PHASE EQUILIBRIA
These arguments rep-
resent a simple ex-
ample of phase equi-
libria. This branch of
thermodynamics tells
us about the direction
of change, but says
nothing about the rare
at which such changes
occur.
temperature if
It should be clear from the graph in Figure 5.2 that AG m is
negative (as required for a spontaneous change) only if the final
state is solid ice and the initial state is liquid water. This sign of
AG m is all that is needed to explain why liquid water freezes at
temperatures below 7( me i t ).
Conversely, if an ice cube is warmed beyond T (me i t ) to the tem-
perature of the mouth at 37 °C, now it is the solid water that
has excess energy; to stabilize it relative to liquid water at 37 °C
requires a different phase change to occur, this time from ice to
liquid water. This argument again relies on the relative magnitudes
of the molar Gibbs function, so AG m is only negative at this higher
the final state is liquid and the initial state is solid.
Why was Napoleon's Russian campaign
such a disaster?
Solid-state phase transitions
A large number of French soldiers froze to death in the winter of 1812 within a
matter of weeks of their emperor Napoleon Bonaparte leading them into Russia. The
loss of manpower was one of the principal reasons why Napoleon withdrew from the
outskirts of Moscow, and hence lost his Russian campaign.
But why was so ruthless a general and so obsessively careful a tactician as Napoleon
foolhardy enough to lead an unprepared army into the frozen wastes of Russia? In
fact, he thought he was prepared, and his troops were originally well clothed with
thick winter coats. The only problem was that, so the story goes, he chose at the last
moment to replace the brass of the soldiers' buttons with tin, to save money.
Metallic tin has many allotropic forms: rhombic white tin (also called /3-tin) is
stable at temperatures above 13 °C, whereas the stable form at lower temperatures is
cubic grey tin (also called a-tin). A transition such as tin (w hit e ) —*■ tm (grey) is called a
solid-state phase transition.
Figure 5.3 shows the phase diagram of tin, and clearly shows the transition from
t m (white) to tin( grey ). Unfortunately, the tin allotropes have very
The transition from
white tin to grey was
first noted in Europe
during the Middle Ages,
e.g. as the pipes of
cathedral organs dis-
integrated, but the
process was thought
to be the work of
the devil.
different densities p, so p^n, grey) =5.8 gem -3 but p (tin> white ) =
7.3 gem -3 . The difference in p during the transition from white to
grey tin causes such an unbearable mechanical stress that the metal
often cracks and turns to dust - a phenomenon sometimes called
'tin disease' or 'tin pest'.
The air temperature when Napoleon entered Russia was appar-
ently as low as —35 °C, so the soldiers' tin buttons converted from
white to grey tin and, concurrently, disintegrated into powder. So,
if this story is true, then Napoleon's troops froze to death because
they lacked effective coat fastenings. Other common metals, such
ENERGETIC INTRODUCTION TO PHASE EQUILIBRIA
183
Q_
CD
6
Solid \
(grey) \
tin^-*"-
4
2
Solid N.
(white) \
tin \
Liquid tin
-2
-4
-6
_^-"-^ Tin vapour
200 400 600 800
Temperature/°C
1000 1200
Figure 5.3 Phase diagram of tin computed from thermodynamic data, showing the transition
from grey tin from white tin at temperatures below 13 °C. Note the logarithmic y-axis. At
/> e , Tf w hite^ grey) = 13 °C, and T( me ] t) = 231. 9°C. (Figure constructed from data published in Tin
and its Alloys and Compounds, B. T. K. Barry and C. J. Thwaits, Ellis Horwood, Chichester, 1983)
as copper or zinc, and alloys such as brass, do not undergo phase changes of this sort,
implying that the troops could have survived but for Napoleon's last-minute change
of button material.
The kinetics of phase changes
Like all spontaneous changes, the rate at which the two forms of tin interconvert is a
function of temperature. Napoleon's troops would have survived if they had entered
Russia in the summer or autumn, when the air temperature is similar to the phase-
transition temperature. The rate of conversion would have been slower in the autumn,
even if the air temperature had been slightly less than T( trans j t j on ) - after all, the tin
coating of a can of beans does not disintegrate while sitting in a cool cupboard! The
conversion is only rapid enough to noticeably destroy the integrity of the buttons when
the air temperature is much lower than ^transition) » i.e. when the difference between
r (air) and r (transition) is large.
Phase changes involving liquids and gases are generally fast, owing to the high
mobility of the molecules. Conversely, while phase changes such as tin( wmte ) — »■
i m (grey) can and do occur in the solid state, the reaction is usually very much slower
because it must occur wholly in the solid state, often causing any thermodynamic
instabilities to remain 'locked in'; as an example, it is clear from the phase diagram
of carbon in Figure 5.4 that graphite is the stable form of carbon (cf. p. 109), yet the
phase change carbon
(diamond)
carbon
(graphite)
is so slow that a significant extent of
conversion requires millions of years.
We consider chemical kinetics further in Chapter 8.
184
PHASE EQUILIBRIA
CD
CD
Q.
▲
4
Diamond
10 6 -
Liquid
10 4 -
10 2 _
Graphite
I
Vapour
I i
2000
4000
Temperature/K
6000
Figure 5.4 The phase diagram of carbon showing the two solid-state extremes of diamond and
graphite. Graphite is the thermodynamically stable form of carbon at room temperature and pressure,
but the rate of the transition C(diamond) — * C( grap hite) is virtually infinitesimal
5.2 Pressure and temperature changes
with a single-component system:
qualitative discussion
How is the ' Smoke' in horror films made?
Effect of temperature on a phase change: sublimation
Horror films commonly show scenes depicting smoke or fog billowing about the
screen during the 'spooky' bits. Similarly, smoke is also popular during pop concerts,
perhaps to distract the fans from something occurring on or off
Dry ice is solid car
bon dioxide.
stage. In both cases, it is the adding of dry ice to water that produces
the 'smoke'.
Dry ice is carbon dioxide (CO2) in its solid phase. We call it
'dry' because it is wholly liquid-free at p & : such solid CO2 looks similar to normal ice
(solid water), but it 'melts' without leaving a puddle. We say it sublimes, i.e. undergoes
a phase change involving direct conversion from solid to gas, without liquid forming
as an intermediate phase. CC>2(i) can only be formed at extreme pressures.
Solid CO2 is slightly denser than water, so it sinks when placed in a bucket of
water. The water is likely to have a temperature of 20 °C or so at room temperature,
while typically the dry ice has a maximum temperature of ca — 78 °C (195 K). The
stable phase at the temperature of the water is therefore gaseous CC>2- We should
understand that the C02( s ) is thermodynamically unstable, causing the phase transition
CO
2(s)
C02( g ) on immersion in the water.
PRESSURE AND TEMPERATURE CHANGES WITH A SINGLE-COMPONENT SYSTEM
185
-78.2 56.6 31
Temperature 77°C
Figure 5.5 Phase diagram of a system that sublimes at room temperature: phase diagram of carbon
dioxide. (Note that the y-'dxis here is logarithmic)
Incidentally, the water in the bucket is essential for generating the effect of theatrical
'smoke' because the large volumes of C02( g ) entrap minute particles of water (which
forms a colloid; see Chapter 10.2). This colloidal water is visible because it creates
the same atmospheric condition known as fog, which is opaque.
Look at the phase diagram of CO2 in Figure 5.5, which is clearly similar in general
form to the schematic phase diagram in Figure 5.1. A closer inspection shows that
some features are different. Firstly, notice that the phase boundary between solid and
liquid now has a positive gradient; in fact, water is almost unique in having a negative
gradient for this line (see Section 5.1). Secondly, the conditions of room temperature
(T = 298 K and p = p & ) relate to conditions of the solid-gas phase boundary rather
than the liquid-gas phase boundary.
By drawing a horizontal line across the figure at p = p B , we see how the line cuts
the solid-gas phase boundary at — 78.2°C. Below this temperature, the stable form
of CO2 is solid dry ice, and C02( g ) is the stable form above it. Liquid CO2 is never
the stable form at p & ; in fact, Figure 5.5 shows that CO20) will not form at pressures
below 5.1 x p e . In other words, liquid CO2 is never seen naturally on Earth; which
explains why dry ice sublimes rather than melts under s.t.p. conditions.
How does freeze-drying work?
Effect of pressure change on a phase change
Packets of instant coffee proudly proclaim that the product has been 'freeze-dried'.
In practice, beans of coffee are ground, boiled in water and filtered to remove the
depleted grounds. This process yields conventional 'fresh' coffee, as characterized
by its usual colour and attractive smell. Finally, water is removed from the coffee
solution to prepare granules of 'instant' coffee.
In principle, we could remove the water from the coffee by just boiling it off, to
leave a solid residue as a form of 'instant coffee'. In fact, some early varieties of
instant coffee were made in just this way, but the flavour was generally unpleasant as
186
PHASE EQUILIBRIA
a result of charring during prolonged heating. Clearly, a better method of removing
the water was required.
We now look at the phase diagram of water in Figure 5.6, which will help us follow
the modern method of removing the water from coffee to yield anhydrous granules.
A low temperature is desirable to avoid charring the coffee. Water vapour can be
removed from the coffee solution at any temperature, because liquids are always sur-
rounded by their respective vapour. The pressure of the vapour is the saturated vapour
pressure, s.v.p. The water is removed faster when the applied pressure decreases.
Again, a higher temperature increases the rate at which the vapour is removed. The
fastest possible rate occurs when the solution boils at a temperature we call r^oii).
Figure 5.6 shows the way in which the boiling temperature alters,
with boiling becoming easier as the applied pressure decreases or
the temperature increases, and suggests that the coffee solution will
boil at a lower temperature when warmed in a partial vacuum. At
Freeze-drying is a
layman's description,
and acknowledges that
external conditions
may alter the
conditions of a phase
change, i.e. the drying
process (removal of
water) occurs at a
temperature lower than
100°C.
a pressure of about -j4j x p , water is removed from the coffee
by warming it to temperatures of about 30 C, when it boils. We
see that the coffee is dried and yet is never subjected to a high
temperature for long periods of time.
It is clear that decreasing the external pressure makes boiling
easier. It is quite possible to remove the water from coffee at or
near its freezing temperature - which explains the original name
of freeze-drying.
In many laboratories, a nomograph (see Figure 5.7) is pinned to
the wall behind a rotary evaporator. A nomograph allows for a simple estimate of
the boiling temperature as a function of pressure. Typically, pressure is expressed in
the old-fashioned units of atmospheres (atm) or millimetres of mercury (mmHg).
1 atm = 760 mmHg. (The curvature of the nomograph is a consequence of the
mathematical nature of the way pressure and temperature are related; see Section 5.2).
Atmospheric
pressure
Temperature T
Figure 5.6 Freeze-drying works by decreasing the pressure, and causing a phase change; at higher
pressure, the stable form of water is liquid, but the stable form at lower pressures is vapour.
Consequently, water (as vapour) leaves a sample when placed in a vacuum or low-pressure chamber:
we say the sample is 'freeze-dried'
PRESSURE AND TEMPERATURE CHANGES WITH A SINGLE-COMPONENT SYSTEM 187
"C °F
400-
-1200
-1100
-1000
-900
-800
-700
-600
-500
-400
-300
-200
Observed
boiling point
(a)
Boiling point
corrected to
760 mmHg
(b)
Pressure
p/mmHg
(c)
Figure 5.7 A typical nomograph for estimating the temperature at which a pure liquid boils when
the pressure is decreased
This is how a boiling temperature at reduced pressure is estimated with a nomo-
graph: place a straight ruler against the applied pressure as indicated on the curved
right-hand scale (c). The ruler must also pass through the 'normal' boiling temperature
on the middle scale (b). The reduced-pressure boiling temperature is then read off
the left-hand scale (a). As an example, if the normal boiling temperature is 200 °C,
then the reduced boiling temperature may be halved to 100 °C if the applied pressure
is approximately 20 mmHg.
SAQ 5.1 A liquid has a normal boiling temperature of 140 °C. Use the
nomograph to estimate the applied pressure needed to decrease the
boiling temperature to 90°C.
188
PHASE EQUILIBRIA
How does a rotary evaporator work?
Thermodynamics of phase changes
Rotary evaporators are a common feature in most undergraduate laboratories. Their
primary purpose is to remove solvent following a reflux, perhaps before crystallization
of a reaction product.
To operate the evaporator, we place the reaction solution in a round-bottomed flask
while the pressure inside the evaporator is decreased to about ^ x p & . The flask
is then rotated. The solvent evaporates more easily at this low pressure than at p & .
The solvent removed under vacuum is trapped by a condenser and collected for easy
re-use, or disposal in an environmentally sensitive way.
But molecules need energy if they are to leave the solution during boiling. The
energy comes from the solution. The temperature of the solution would decrease
rapidly if no external supply of energy was available, as a reflection of its depleted
energy content (see p. 33). In fact, the solution would freeze during evaporation, so
the rotating bulb is typically immersed in a bath of warm water.
An atmosphere of vapour always resides above a liquid, whether
the liquid is pure, part of a mixture, or has solute dissolved within
it. We saw on p. 180 how the pressure of this gaseous phase is
called its saturation vapour pressure, s.v.p. The s.v.p. increases with
increased temperature until, at the boiling point r(t,oii), it equals the
external pressure above the liquid. Evaporation occurs at tempera-
tures below T(boii), and only above this temperature will the s.v.p.
exceed p & . The applied pressure in a rotary evaporator is less than
p & , so the s.v.p. of the solvent can exceed the applied pressure
(and allow the liquid to boil) at pressures lower than p e .
We see this phenomenon in a different way when we look back at
the phase diagram in Figure 5.6. The stable phase is liquid before
applying a vacuum. After turning on the water pump, to decrease
the applied pressure, the s.v.p. exceeds /?( ap piied), and the solvent
boils at a lower pressure. The bold arrow again indicates how a
phase change occurs during a depression of the external pressure.
We see how decreasing the pressure causes boiling of the solvent
at a lower temperature than at its normal boiling temperature, i.e.
if the external pressure were p & . Such a vacuum distillation is
desirable for a preparative organic chemist, because a lower boiling
temperature decreases the extent to which the compounds degrade.
Coffee, for example, itself does not evaporate even at low pressure, since it is
a solid. Solids are generally much less volatile than liquids, owing to the stronger
interactions between the particles. In consequence, the vapour pressure of a solid is
several orders of magnitude smaller than that above a liquid.
Strictly, the term s.v.p.
applies to pure liq-
uids. By using the term
s.v.p., we are implying
that all other com-
ponents are wholly
involatile, and the
s.v.p. relates only to
the solvent.
Normal in the con-
text of phase equilibria
means 'performed at a
pressure of 1 bar, p s '.
The rotary evaporator
is a simple example of
a vacuum distillation.
PRESSURE AND TEMPERATURE CHANGES WITH A SINGLE-COMPONENT SYSTEM
189
How is coffee decaffeinated?
Critical and supercritical fluids
We continue our theme of 'coffee'. Most coffees contain a large amount of the
heterocyclic stimulant caffeine (I). Some people prefer to decrease the amounts of
caffeine they ingest for health reasons, or they simply do not like to consume it at
all, and they ask for decaffeinated coffee instead.
H 3 C
The modern method of removing I from coffee resembles the operation of a coffee
percolator, in which the water-soluble chemicals giving flavour, colour and aroma
are leached from the ground-up coffee during constant irrigation with a stream of
boiling water.
Figure 5.8 shows such a system: we call it a Soxhlet apparatus. Solvent is passed
continually through a porous cup holding the ground coffee. The solvent removes the
caffeine and trickles through the holes at the bottom of the cup, i.e. as a solution of
caffeine. The solvent is then recycled: solvent at the bottom of the flask evaporates
to form a gas, which condenses at the top of the column. This pure, clean solvent
then irrigates the coffee a second time, and a third time, etc., until all the caffeine
has been removed.
Water is a good choice of solvent in a standard kitchen percolator because it removes
all the water-soluble components from the coffee - hence the flavour. Clearly, how-
ever, a different solvent is required if only the caffeine is to be removed. Such a
solvent must be cheap, have a low boiling point to prevent charring of the coffee
and, most importantly, should leave no toxic residues. The presence of any residue
would be unsatisfactory to a customer, since it would almost certainly leave a taste;
and there are also health and safety implications when residues persist.
The preferred solvent is supercritical CO2. The reasons for this
choice are many and various. Firstly, the CO2 is not hot (CO2 first
becomes critical at 31 °C and 73 atm pressure; see Figure 5.5), so
no charring of the coffee occurs during decaffeination. Furthermore,
at such a low temperature, all the components within the coffee that
impart the flavour and aroma remain within the solid coffee - try
soaking coffee beans in cold water and see how the water tastes afterwards! Caffeine
is removed while retaining a full flavour.
C0 2 is supercritical
at temperatures and
pressures above the
critical point.
190
PHASE EQUILIBRIA
Condenser
Cool, pure
fluid
Coffee
beans to be
decaffeinated
Hot, pure
fluid
Reservoir, where
extracted
caffeine accumulates
| Heat
Figure 5.8 Coffee is decaffeinated by constantly irrigating the ground beans with supercritical
carbon dioxide: schematic representation of a Soxhlet apparatus for removing caffeine from coffee
Secondly, solid CO2 is relatively cheap. Finally, after caffeine removal, any occluded
CO2 will vaporize from the coffee without the need to heat it or employ expensive
vacuum technology. Again, we retain the volatile essential oils of the coffee. Even
if some CO2 were to persist within the coffee granules, it is chemically inert, has no
taste and would be released rapidly as soon as boiling water was added to the solid,
decaffeinated coffee.
What is a critical or supercritical fluid?
We look once more at the phase diagram of CO2 in Figure 5.5. The simplest way of
obtaining the data needed to construct such a figure would be to take a sample of
CO2 and determine those temperatures and pressures at which the liquid, solid and
gaseous phases coexist at equilibrium. (An appropriate apparatus
involves a robust container having an observation window to allow
us to observe the meniscus.) We then plot these values of p (as
'y') against T (as 'x').
We first looked at criti-
cal fluids on p. 50.
PRESSURE AND TEMPERATURE CHANGES WITH A SINGLE-COMPONENT SYSTEM
191
It is impossible to dis-
tinguish between the
liquid and gaseous
phases of C0 2 at tem-
peratures and pres-
sures at and above the
critical point.
Let us consider more closely what happens as the conditions become more extreme
inside the observation can. As heating proceeds, so the amount of CO20) convert-
ing to form gas increases. Accordingly, the amount of CO2 within the gaseous
phase increases, which will cause the density p of the vapour to increase. Con-
versely, if we consider the liquid, at no time does its density alter
appreciably, even though its volume decreases as a result of liquid
forming vapour.
From a consideration of the relative densities, we expect the
liquid phase to reside at the bottom of the container, with the
less-dense gaseous phase 'floating' above it. The 'critical' point is
reached when the density of the gas has increased until it becomes
the same as that of the liquid. In consequence, there is now no
longer a lighter and a heavier phase, because P(Uq U id) = P(vapour)-
Accordingly, we no longer see a meniscus separating liquid at the
bottom of the container and vapour above it: it is impossible to see
a clear distinction between the liquid and gas components. We say
that the CO2 is critical.
Further heating or additional increases in pressure generate
supercritical CO2. The pressure and temperature at which the fluid
first becomes critical are respectively termed r( cr iticai) and /? (critical) •
Table 5.2 contains a few examples of ^(critical) and p (critical)-
The inability to distinguish liquid from gaseous CO2 explains
why we describe critical and supercritical systems as fluids - they
are neither liquid nor gas.
It is impossible to distinguish between the liquid and gaseous
phases of CO2 at and above the critical point, which explains
why a phase diagram has no phase boundary at temperatures and
pressures above ^critical). The formation of a critical fluid has an
unusual corollary: at temperatures above ^critical), we cannot
cause the liquid and gaseous phases to separate by decreasing or
increasing the pressure alone. The critical temperature, therefore,
represents the maximum values of p and T at which liquification
The intensive prop-
erties of the liquid
and gas (density, heat
capacity, etc.) become
equal at the criti-
cal point, which is
the highest temper-
ature and pressure at
which both the liquid
and gaseous phases
of a given compound
can coexist.
Table 5.2 Critical constants T^ticd) an d /?(criticai) for
some common elements and bi-element compounds
IUPAC defines super-
critical chromatogra-
phy as a separation
technique in which
the mobile phase is
kept above (or rel-
atively close to) its
critical temperature
and pressure.
Substance
^(critical) /**■
/'(critical)//'
H 2
33.2
12.97
He
5.3
2.29
o 2
154.3
50.4
Cl 2
417
77.1
C0 2
304.16
73.9
S0 2
430
78.7
H 2
647.1
220.6
NH 3
405.5
113.0
192
PHASE EQUILIBRIA
of the gas is possible. We say that there cannot be any C02(i) at temperatures above
-» (critical) ■
Furthermore, supercritical CO2 does not behave as merely a mixture of liquid
and gaseous CO2, but often exhibits an exceptional ability to solvate molecules in
a specific way. The removal of caffeine from coffee relies on the chromatographic
separation of caffeine and the other organic substances in a coffee bean; supercritical
fluid chromatography is a growing and exciting branch of chemistry.
5.3 Quantitative effects of pressure
and temperature change
for a single-component system
Wh
n
.7
y is ice so slippery
The coefficient of fric-
tion [i (also called
'friction factor') is the
quotient of the fric-
tional force and the
normal force. In other
words, when we apply a
force, is there a resis-
tance to movement
or not?
Effect of p and T on the position of a solid -liquid equilibrium
We say something is 'as slippery as an ice rink' if it is has a tiny
coefficient of friction, and we cannot get a grip underfoot. This is
odd because the coefficient of friction /x for ice is quite high - try
dragging a fingernail along the surface of some ice fresh from the
ice box. It requires quite a lot of effort (and hence work) for a
body to move over the surface of ice.
At first sight, these facts appear to represent a contradiction in
terms. In fact, the reason why it is so easy to slip on ice is that ice
usually has a thin layer of liquid water covering its surface: it is
this water-ice combination that is treacherous and slippery.
But why does any water form on the ice if the weather is
sufficiently cold for water to have frozen to form ice? Consider the ice directly
beneath the blade on a skater's ice-shoe in Figure 5.9: the edge of the blade is
so sharp that an enormous pressure is exerted on the ice, as indicated by the grey
tints.
We now look at the phase diagram for water in Figure 5.10.
Ice melts at 0°C if the pressure is p s (as represented by 7\ and
Px respectively on the figure). If the pressure exerted on the ice
increases to P2, then the freezing temperature decreases to T2. (The
freezing temperature decreases in response to the negative slope
of the liquid-solid phase boundary (see the inset to Figure 5.10),
which is most unusual; virtually all other substances show a posi-
tive slope of dp/dT.)
If the temperature Ti is lower than the freezing temperature of the water - and it
usually is - then some of the ice converts to form liquid water; squeezing decreases
the freezing temperature of the water. The water-on-ice beneath the skater's blade is
slippery enough to allow effortless skating.
The sign of dp/dT for
the liquid-solid line
on a phase diagram
is almost always posi-
tive. Water is the only
common exception.
QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE 193
Skater's blade
Water formed by pressure
Figure 5.9 Skaters apply an enormous pressure beneath the blades of their skates. This pressure
causes solid ice to melt and form liquid water
Temperature
Figure 5.10 Phase diagram of water. Inset: applying a high pressure from p\ (here p e ) to p 2
causes the melting temperature of the ice to decrease from temperature T\ (here 0°C) to T 2
What is ' black ice'?
The Clapeyron equation
We give the name 'black ice' to the phenomenon of invisible ice on a road. In
practice, anything applying a pressure to solid ice will cause a similar depression of
the freezing temperature to that of the skater, so a car or heavy vehicle travelling
over ice will also cause a momentary melting of the ice beneath its wheels. This
water-on-ice causes the car to skid - often uncontrollably - and leads to many deaths
every year. Such ice is particularly dangerous: whereas an ice skater wants the ice to
be slippery, a driver does not.
194
PHASE EQUILIBRIA
We move from the qualitative argument that T( me \ t ) decreases as p increases, and
next look for a quantitative measure of the changes in melt temperature with pressure.
We will employ the Clapeywn equation:
Ap_
AT
AH*
TAV
(5.1)
In fact, it does not
matter whether AH
relates to the direction
of change of solid ->
liquid or of liquid ->
solid, provided that AV
relates to the same
direction of change.
The molar change in
volume AV m has units
of m 3 mol -1 . Values
typically lie in the
range irr 5 -lCT 6 m 3
mol -1 .
The minus sign of AV m
reflects the way water
expands on freez-
ing. This expansion
explains why a car
radiator cracks in cold
weather (if it contains
no 'de-icer'): the water
freezes and, in expand-
ing, exerts a huge a
pressure on the metal.
where T is the normal melting temperature, AT is the change in
the melting temperature caused by changing the applied pressure
by an amount of dp (in SI units of pascals), where 1 Pa is the
pressure exerted by a force of 1 N over an area of 1 m 2 . AH is
the enthalpy change associated with the melting of water and A V
is the change in volume on melting. Strictly, both AH and AV are
molar quantities, and are often written as AH m and A V m , although
the 'm' is frequently omitted.
The molar change in volume A V m has SI units of m 3 mol - .
We should note how these volumes are molar volumes, so they
refer to 1 mol of material, explaining why AV is always very
small. The value of AV m is usually about 10~ 6 to 10~ 5 m 3 mol _1
in magnitude, equating to 1 to 10 cm 3 mol -1 respectively.
We recall from Chapter 1 how the symbol A means 'final state
minus initial state', so a positive value of AVm during melting
(which is y m (ii qu id> — V m (solid)) tells us that the liquid has a slightly
larger volume than the solid from which it came. AV m (melt) is
positive in the overwhelming majority of cases, but for water
A V m (melt) = — 1.6 x 10~ 6 m 3 mol~ . This minus sign is extremely
unusual: it means that ice is less dense than water. This explains
why an iceberg floats in water, yet most solids sink when immersed
in their respective liquid phases.
The enthalpy A^ elt) is the energy required to melt 1 mol of
material at constant pressure. We need to be careful when obtain-
ing data from tables, because many books cite the enthalpy of
fusion, which is the energy released during the opposite process of
solidification. We do not need to worry, though, because we know
from Hess's law that AH,
(melt)
-AH.
(fusion)
The molar enthalpy
of melting water is +6.0 kJmol
Care: following Hess's
law, we say:
AH'
(melt)"
-AH
(fusion) '
Worked Example 5.1 Consider a car weighing 1000 kg (about
2200 lbs) parked on a sheet of ice at 273.15 K. Take the area under
wheels in contact with the ice as 100 cm 2 i.e. 10~ 2 m 2 . What is the
'(melt) —
new melting temperature of the ice - call it T^nai)? Take AH*.
6.0 kJmol and water AV,
m (melt)
= -1.6 x 10 -6 m J mor
QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE
195
Strategy. (1) Calculate the pressure exerted and hence the pressure change. (2) Insert
values into the Clapeyron equation (Equation (5.1)).
(1) The pressure exerted by the car is given by the equation
'force 4- area'. The force is simply the car weight expressed
in newtons (N): force = 10000 N, so we calculate the
pressure exerted by the wheels as 10 000 N -f- 10~ 2 m 2 ,
which is 10 6 Pa. We see how a car exerts the astonishing
pressure beneath its wheels of 10 6 Pa (about 10 bar).
(2) Before inserting values into the Clapeyron equation, we
rearrange it slightly, first by multiplying both sides by
T AV , then dividing both sides by A// e , to give
At
sea
level,
a mass
of
1 kg
has a weight
(i-
2. exerts a
force) of
approxi
mately 10 N.
dr =
dT
dp TAV
AH*
10 6 Pax 273.15Kx (-1.6 x 10
6 ) m 3 mol
Next, we recall that the symbol
Accordingly, we say
6.0 x 10 3 Jmor
dT = -0.07 K
'A' means 'final state — initial state'
AT = T,
(final)
' (initial)
Notice how the free-
zing temperature of
water decreases when
a pressure is applied.
This decrease is directly
attributable to the
minus sign of AV.
where the temperatures relate to the melting of ice. The normal melting temperature of
ice r(i n j t i a i) is 273.15 K. The final temperature T(fiDai) of the ice with the car resting on it
is obtained by rearranging, and saying
AT + r^tiai) = -0.07 K + 273.15 K
%nal) = 273.08 K
The new melting temperature of the ice T^.^ is 273.08 K. Note how we performed
this calculation with the car parked and immobile on the ice. When driving rather than
parked, the pressure exerted beneath its wheels is actually considerably greater. Since
Equation (5.1) suggests that dp oc dT, the change in freezing temperature dT will be
proportionately larger (perhaps as much as —3 K), so there will be a layer of water on
the surface of the ice even if the ambient temperature is —3 °C. Drive
with care!
SAQ 5.2 Paraffin wax has a normal melting temperature
7~(meit) of 320 K. The temperature of equilibrium is raised
Care: the word 'normal'
here is code: it means
, atp = p°'.
196 PHASE EQUILIBRIA
by 1.2 K if the pressure is increased fivefold. Calculate AV m for the wax as
it melts. Take AW (me it) = 8.064 kJ mol" 1 .
Justification Box 5.1
Consider two phases (call them 1 and 2) that reside together in thermodynamic equi-
librium. We can apply the Gibbs-Duhem equation (Equation 4.31) for each of the two
phases, 1 and 2.
For phase 1 : dG a) = (V m( i) dp) - (S m(i) dT) (5.2)
For phase 2 : dG (2 ) = (V m(2 ) dp) - (S m(2) dT) (5.3)
where the subscripts 'm' imply molar quantities, i.e. per mole of substance in each phase.
Now, because equilibrium exists between the two phases 1 and 2, the dG term in
each equation must be the same. If they were different, then the change from phase 1 to
phase 2 (Gp) — G(i)) would not be zero at all points; but at equilibrium, the value of
AG will be zero, which occurs when G(2) = G(i). In fact, along the line of the phase
boundary we say dG(i) = dG(2).
In consequence, we may equate the two equations, saying:
(V m( i) x dp) - (5 m( i) x dT) = (V m{2 ) x dp) - (5 m(2) x dT)
Factorizing will group together the two S and V terms to yield
(5 m (2) - Sm(l)) dT = (Vm(2) - V m (i)) dp
which, after a little rearranging, becomes
dp A5 m (i_ > .2)
dT AV,
m(l->-2)
(5.4)
Finally, since AG & = Afl* - TAS* (Equation (4.21)), and since AG S = at
equilibrium:
Afl*
T = AS m( i^ 2 )
Inserting this relationship into Equation (5.4) yields the Clapeyron equation in its famil-
iar form.
dp _ A #m(1^2)
dT ~ TAV m(l ^ 2)
In fact, Equation (5.4) is also called the Clapeyron equation. This equation holds for
phase changes between any two phases and, at heart, quantitatively defines the phase
QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE 197
boundaries of a phase diagram. For example:
for the liquid — > solid line
for the vapour — > solid line
dp AH,
m (l->s)
dT TAV
m (l->-s)
d/> AH m ( v -«)
d7 TA'V,
v *m (v-*-s)
for the vapour — > liquid line — =
dp AH* (V _
dr taLv,
v *m (v->-s)
Approximations to the Clapeyron equation
We need to exercise a little caution with our terminology: we performed the calculation
in Worked Example 5.1 with Equation (5.1) as written, but we should have written Ap
rather than dp because 10 6 Pa is a very large change in pressure.
Similarly, the resultant change in temperature should have been
written as AT rather than dT, although 0.07 K is not large. To
accommodate these larger changes in p and T, we ought to be
rewrite Equation (5.1) in the related form:
Ap_
AT
TAVr
The 'd' in Equation
(5.1) means an infini-
tesimal change,
whereas the 'A' symbol
here means a large,
macroscopic, change.
We are permitted to assume that dp is directly proportional to dT because AH and
A V are regarded as constants, although even a casual inspection of a phase diagram
shows how curved the solid-gas and liquid-gas phase boundaries are. Such curvature
clearly indicates that the Clapeyron equation fails to work except
over extremely limited ranges of p and T . Why?
We assumed in Justification Box 5.1 that A// (melt) is indepen-
dent of temperature and pressure, which is not quite true, although
the dependence is usually sufficiently slight that we can legiti-
mately ignore it. For accurate work, we need to recall the Kirchhoff
equation (Equation (3.19)) to correct for changes in AH.
Also, we saw on p. 23 how Boyle's Law relates the volume of
a gas to changes in the applied pressure. Similar expressions apply
for liquids and solids (although such phases are usually much less
compressible than gases). Furthermore, we assumed in the deriva-
tion of Equation (5.1) that AV m does not depend on the pressure
changes, which implies that the volumes of liquid and solid phases
each change by an identical amount during compression. This
approximation is only good when (1) the pressure change is not
extreme, and (2) we are considering equilibria for the solid-liquid
The Clapeyron equa-
tion fails to work for
phase changes involv-
ing gases, except
over extremely limited
ranges of p and T.
It is preferable to
analyse the equilib-
ria of gases in terms
of the related Clau-
sius-Clapeyron equa-
tion; see Equation (5.5).
198 PHASE EQUILIBRIA
phase boundary, which describes melting and solidification. For these reasons, the
Clapeyron equation is most effective when dp is relatively small, i.e. 2-10 atm
at most.
The worst deviations from the Clapeyron equation occur when one of the phases
is a gas. This occurs because the volume of a gas depends strongly on temperature,
whereas the volume of a liquid or solid does not. Accordingly, the value of A V m is
not independent of temperature when the equilibrium involves a gas.
Why does deflating the tyres on a car improve its
road-holding on ice?
The Clapeyron equation, continued
We saw from the Clapeyron equation, Equation (5.1), how the decrease in freezing
temperature AT is proportional to the applied pressure dp, so one of the easiest ways
of avoiding the lethal conversion of solid ice forming liquid water is to apply a smaller
pressure - which will decrease AT in direct proportion.
The pressure change dp is caused by the additional weight of,
for example, a car, lorry or ice skater, travelling over the surface
of the ice. We recall our definition of 'pressure' as 'force -r area'.
There is rarely a straightforward way of decreasing the weight of
a person or car exerting the force, so the best way to decrease
the pressure is to apply the same force but over a larger area.
An elementary example will suffice: cutting with a sharp knife
is easier than with a blunt one, because the active area along
the knife-edge is greater when the knife is blunt, thus causing p
to decrease.
In a similar way, if we deflate slightly the tyres on a car, we see the tyre bulge
a little, causing it to 'sag', with more of the tyre in contact with the road surface.
So, although the weight of the car does not alter, deflating the tyre increases the area
over which its weight (i.e. force) is exerted, with the result that we proportionately
decrease the pressure.
In summary, we see that letting out some air from a car tyre decreases the value of
dp, with the result that the change in melting temperature dT of the ice, as calculated
with the Clapeyron equation (Equation (5.1)), also decreases, thereby making driving
on ice much safer.
The pressure beneath
the blades of an ice-
skater's shoe is enor-
mous - maybe as much
as 100 atm when the
skater twists and turns
at speed.
SAQ 5.3 A man is determined not to slip on the ice, so instead of wearing
skates of area 10 cm 2 he now wears snow shoes, with the underside of
each sole having an extremely large area to spread his 100 kg mass
(equating to 1000 N). If the area of each snow shoe is 0.5 m 2 , what is the
depression of the freezing temperature of the ice caused by his walking
over it?
Use the thermodynamic data for water given in Worked Example 5.1.
QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE
199
Aside
If water behaved in a similar fashion to most other materials and possessed a positive
value of AV m , then water would spontaneously freeze when pressure was applied,
rather than solid ice melting under pressure. Furthermore, a positive value of AV m
would instantly remove the problems discussed above, caused by vehicles travelling
over 'black' ice, because the ice would remain solid under pressure; and remember that
the slipperiness occurs because liquid water forms on top of solid ice.
Unfortunately, a different problem would present itself if AV m was positive! If AV m
was positive, then Equation (5.1) shows that applying a pressure to liquid water would
convert it to ice, even at temperatures slightly higher than °C, which provides a
different source of black ice.
Why does a pressure cooker work?
The Clausius-Clapeyron equation
A pressure cooker is a sealed saucepan in which food cooks faster than it does in a
simple saucepan - where 'simple', in this context means a saucepan that is open to
the air. A pressure cooker is heated on top of a cooker or hob in the conventional way
but, as the water inside it boils, the formation of steam rapidly causes the internal
pressure to increase within its sealed cavity; see Figure 5.11. The internal pressure
inside a good-quality pressure cooker can be as high 6 atm.
The phase diagram in Figure 5.12 highlights the pressure- temp-
erature behaviour of the boiling (gas-liquid) equilibrium. The
normal boiling temperature 7(b ii) of water is 100 °C, but 7(boii)
increases at higher pressures and decreases if the pressure de-
creases. As a simple example, a glass of water would boil instantly at the cold
temperature of 3 K in the hard vacuum of deep space. The inset to Figure 5.12
Remember that all
equilibria are dynamic.
Condensation
Vaporization
t
Heat
Figure 5.11 A pressure cooker enables food to cook fast because its internal pressure is high,
which elevates the temperature at which food cooks
200
PHASE EQUILIBRIA
2 p^
CD
i—
Q.
T3
CD
Q.
Q.
<
The normal boiling point of water
Solid
Gas
100 °C
Temperature T
Figure 5.12 Phase diagram to show how a pressure cooker works. Inset: applying a high pressure
from p & to p2 causes the boiling temperature of the water to increase from temperature 100 °C
to T 2
shows why the
consequence of
The Clausius -Clapey-
ron equation quantifies
the way a boiling tem-
perature changes as a
function of the applied
pressure. At the boil-
ing points of 7"i and
T 2 , the external pres-
sures Pi and pi are the
same as the respective
vapour pressures.
water inside the pressure cooker boils at a higher temperature as a
the pan's large internal pressure.
Having qualitatively discussed the way a pressure cooker facil-
itates rapid cooking, we now turn to a quantitative discussion.
The Clapeyron equation, Equation (5.1), would lead us to suppose
that dp oc AT , but the liquid -gas phase boundary in Figure 5.12
is clearly curved, implying deviations from the equation. There-
fore, we require a new version of the Clapeyron equation, adapted
to cope with the large volume change of a gas. To this end, we
introduce the Clausius- Clapeyron equation:
In
p 2 at T 2
Pi at T x
AH,
(boil)
R
1
~T 2
1
(5.5)
It does not matter
which of the values we
choose as '1' and
'2' provided that 7~i
relates to pi and T 2
relates to p 2 . It is per-
missible to swap 7~i
for 7" 2 and p\ for p 2
simultaneously, which
amounts to multiply-
ing both sides of the
equation by '-!'.
where R is the familiar gas constant, and AH? n) is the enthalpy
of vaporization. AHf [X) is always positive because energy must be
put in to a liquid if it is to boil. T 2 here is the boiling temperature
when the applied pressure is p 2 , whereas changing the pressure to
p\ will cause the liquid to boil at a different temperature, T\.
We need to understand that the Clausius -Clapeyron equation is
really just a special case of the Clapeyron equation, and relates to
phase changes in which one of the phases is a gas.
Worked Example 5.2 What is the boiling temperature of pure water
inside a pressure cooker? Let 7i be the normal boiling temperature
r (boi i) of water (i.e. 100 °C, 373 K, at p°) and let p 2 of 6 x p & be the
pressure inside the pan. The enthalpy of boiling water is 50.0 kJmol - .
QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE
201
In this example, it is simpler to insert values into Equation (5.5) and
to rearrange later. Inserting values gives
In
6x p*
-50 000 J mo!"
8.314 JK-'mol"
1
1
373 K
We can omit the units of the two pressures on the left-hand side
because Equation (5.5) is written as a ratio, so the units cancel: we
require only a relative change in pressure.
Notice how the ratio
within the bracket on
the left-hand side of the
Clausius-Clapeyron
equation permits us
to dispense with abso-
lute pressures.
In 6.0= -6104 Kx
1
1
T 2 373 K
where In 6.0 has a value of —1.79. Next, we rearrange slightly by dividing both sides by
6104 K, to yield:
-1.79 1
1
so
and
6104 K T 2 373 K
1
1
= -2.98 x 10" 4 K _1 +
T 2 373 K
— = 2.38 x 10" 3 K" 1
T 2
We obtain the temperature at which water boils by taking the reciprocal of both side.
T 2 , is 420 K, or 147 °C at a pressure of 6 x p & , which is much higher than the normal
boiling temperature of 100 °C.
SAQ 5.4 A mountaineer climbs Mount Everest and wishes to make a
strong cup of tea. He boils his kettle, but the final drink tastes lousy
because the water boiled at too low a temperature, itself because the
pressure at the top of the mountain is only 0.4 xp e . Again taking the
enthalpy of boiling the water to be 50 kJmor 1 and the normal boiling
temperature of water to be 373 K, calculate the temperature of the water
as it boils at the top of the mountain.
The form of the Clausius-Clapeyron equation in Equation (5.5) is called the inte-
grated form. If pressures are known for more than two temperatures, an alternative
form may be employed:
\np
AH,
(boil)
1
R
+ constant
(5.6)
202
PHASE EQUILIBRIA
so a graph of the form 'y = mx + c' is obtained by plotting In p (as 'y') against
1/r (as 'x'). The gradient of this Clapeyron graph is '— AH,. ir . 4- R', so we obtain
AH,
(boil)
We employ the inte-
grated form of the
Clausius-Clapeyron
equation when we
know two tempera-
tures and pressures,
and the graphical form
for three or more.
(boil)
as 'gradient x — 1 x R\
The intercept of a Clapeyron graph is not useful; its value may
best be thought of as the pressure exerted by water boiling at infinite
temperature. This alternative of the Clausius-Clapeyron equation
is sometimes referred to as the linear (or graphical) form.
Worked Example 5.3 The Clausius-Clapeyron equation need not
apply merely to boiling (liquid-gas) equilibria, it also describes sub-
limation equilibria (gas-solid).
Consider the following thermodynamic data, which concern the sub-
limation of iodine:
-* (sublimation) / -^
P(h)/ Pa
270 280 290 300 310 320 330 340
50 133 334 787 1755 3722 7542 14 659
We obtain the value
of AH as 'gradientx
-1 xR'.
Figure 5.13 shows a plot of In p (h) (as 'y') against 1 / r (sublimation) (as
V). The enthalpy AH*
(sublimation)
is obtained via the gradient of the
graph 62 kJ mol (note the positive sign).
O.
0.0029
0.0037
Figure 5.13 The linear form of the Clausius-Clapeyron equation: a graph of In p (as 'y') against
1/r (as 'x') should be linear with a slope of —AH
(vap)
R
QUANTITATIVE EFFECTS OF PRESSURE AND TEMPERATURE CHANGE 203
Aside
Why does food cook faster at higher pressures?
The process of cooking involves a complicated series of chemical reactions, each of
which proceeds with a rate constant of k. When boiling an egg, for example, the rate-
limiting process is denaturation of the proteins from which albumen is made. Such
denaturation has an activation energy E a of about 40 kj mol~ .
The rate constant of reaction varies with tempera-
We consider the Arrhe-
nius equation in appro-
priate detail in Chap-
ter 8.
ture, with k increasing as the temperature increases, k
is a function of T according to the well-known Arrhe-
nius equation:
\katTj R \T 2 TJ
We saw in Worked Example 5.2 how the temperature of the boiling water increases
from 100 °C to 147 °C in a pressure cooker. A simple calculation with the Arrhenius
equation (Equation (5.7)) shows that the rate constant of cooking increases by a little
over fourfold at the higher temperature inside a pressure cooker.
Boiling an egg takes about 4 min at 100 °C, so boiling an egg in a pressure cooker
takes about 1 min.
Justification Box 5.2
The Clapeyron equation, Equation (5.1), yields a quantitative description of a phase
boundary on a phase diagram. Equation (5.1) works quite well for the liquid- solid
phase boundary, but if the equilibrium is boiling or sublimation - both of which involve
a gaseous phase - then the Clapeyron equation is a poor predictor.
For simplicity, we will suppose the phase change is the boiling of a liquid: liquid — >
gas. We must make three assumptions if we are to derive a variant that can accommodate
the large changes in the volume of a gas:
Assumption 1: we assume the enthalpy of the phase change is independent of tem-
perature and pressure. This assumption is good over limited ranges of both p and T,
although note how the Kirchhoff equation (Equation (3.19)) quantifies changes in AH .
Assumption 2: we assume the gas is perfect, i.e. it obeys the ideal-gas equation,
Equation (1.13), so
pV = nRT or pV m = RT
where V m is the molar volume of the gas.
204
PHASE EQUILIBRIA
Assumption 3: AV m is the molar change in volume during the phase change. The
value of A V m = V m ( g ) — V m (i) , where V m (i) is typically 20 cm mol
22.4 dm 3 mol" 1 (at s.t.p.), i.e. 22400 cm 3 mol"*
between V m ( g ) and V m (i), we assume that AV m
and V,
m(g)
In response to the vast discrepancy
i.e. that V m (i) is negligible by
V,
m(g)
comparison. This third approximation is generally good, and will only break down at
very low temperatures.
First, we rewrite the Clapeyron equation in response to approximation 2:
AH<
dp
dT TV,
mi f i
Next, since we assume the gas is ideal, we can substitute for the V m term via the
ideal-gas equation, and say V m = RT -4- p:
dp_ = Afl^ x _p_
dT T K RT
Next, we multiply together the two T terms, rearrange and separate the variables, to
give:
1 . AH* 1
■dp
— — x — dT
R T 2
We place the'AHZ + R'
term outside the right-
hand integral because
its value is constant.
Integrating with the limits p2 at T2 and p\ at T\ gives
f' 2 1 AH* f Tl 1
/ -dp = —^ —dT
J P , p R Jt, t 2
Subsequent integration yields
[lnp]«
AK
R
-| ? 2
-I T,
Next, we insert limits:
In
AH* ( 1 1 \
And, finally, we group together the two logarithmic terms to yield the Clausius- Clap-
eyron equation:
> 2 \ AH*/\ 1
Pi) R \T 2 Ti
In
PHASE EQUILIBRIA INVOLVING TWO-COMPONENT SYSTEMS: PARTITION
205
5.4 Phase equilibria involving two-component
systems: partition
Why does a fizzy drink lose its fizz and go flat?
Equilibrium constants of partition
Drinks such as lemonade, orangeade or coke contain dissolved CO2 gas. As soon as
the drink enters the warm interior of the mouth, CO2 comes out of solution, imparting
a sensation we say is 'fizzy'.
The CO2 is pumped into the drink at the relatively high pressure of about 3 bar.
After sealing the bottle, equilibrium soon forms between the gaseous CO2 in the space
above the drink and the CO2 dissolved in the liquid drink (Figure 5.14). We say the
CO2 is partitioned between the gas and liquid phases.
The proportions of CO2 in the space above the liquid and in the liquid are fixed
according to an equilibrium constant, which we call the partition constant:
K
(partition)
amount of CO2 in phase 1
amount of CO2 in phase 2
We need to note how the identities of phases 1 and 2 must be
defined before K can be cited. We need to be aware that ^(partition)
is only ever useful if the identities of phases 1 and 2 are defined.
On opening the drink bottle we hear a hissing sound, which
occurs because the pressure of the escaping CO2 gas above the
liquid is greater than the atmospheric pressure. We saw in Chapter 4
that the molar change in Gibbs function for movement of a gas is
given by
AG = RTln(-^^-)
V /'(initial)/
(5.8)
This equilibrium con-
stant is often incor-
rectly called a 'partition
function' - which is in
fact a term from statis-
tical mechanics.
(5.9)
Carbon dioxide
in the gas phase
Carbon dioxide
dissolved in solution
Figure 5.14 In a bottle of fizzy drink, carbon dioxide is partitioned between the gas and the
solution phases
206
PHASE EQUILIBRIA
The value of AG is only ever negative, as required by a thermodynamically sponta-
neous process, if the initial pressure p (initial) is greater than the final pressure p(finai)>
i.e. the fraction is less than one. In other words, Equation (5.9) shows why AG is
negative only if the pressure of the CO2 in the space above the liquid has a pressure
that is greater than p & .
We disrupted the equilibrium in the bottle when we allowed out much of the CO2
gas that formerly resided within the space above the liquid; conversely, the CO2
dissolved in the liquid remains in solution.
After drinking a mouthful of the drink, we screw on the bottle top to stop any
more CO2 being lost, and come back to the bottle later when a thirst returns. The
CO2 re-equilibrates rapidly, with some of the CO2 in the liquid phase passing to the
gaseous phase. Movement of CO2 occurs in order to maintain the constant value of
^(partition): we call it 're-partitioning' .
Although the value of ^(partition) does not alter, the amount of
CO2 in each of the phases has decreased because some of the CO2
was lost on opening the bottle. The liquid, therefore, contains less
CO2 than before, which is why it is perceived to be less fizzy. And
after opening the bottle several times, and losing gaseous CO2 each
time, the overall amount of CO2 in the liquid is so depleted that the
drink no longer sparkles, which is when we say it has 'gone flat'.
A fizzy drink goes 'flat'
after opening it sev-
eral times because the
water is depleted of
C0 2 .
Worked Example 5.4 A bottle of fizzy pop contains CO2. What are the relative amounts
of CO2 in the water and air if ^( part i t i on ) = 4?
Firstly, we need to note that stating a value of ^"(partition) is useless unless we know how
the equilibrium constant A"( part ition) was written, i.e. which of the phases '1' and '2' in
Equation (5.8) is the air and which is the liquid?
In fact, most of the CO2 resides in the liquid, so Equation (5.8) would be written as
K
(partition) —
concentration of CO2 in the drink
concentration of CO2 in the air above the liquid
A bottle of fizzy drink
going flat is a fairly triv-
ial example of partition,
but the principle is vital
to processes such as
reactions in two-phase
media or the operation
of a high-performance
liquid chromatography
column.
This partition constant has a value of 4, which means that four times
as much CO2 resides in the drink as in the liquid of the space above
the drink. Stated another way, four-fifths of the CO2 is in the gas
phase and one-fifth is in solution (in the drink).
SAQ 5.5 An aqueous solution of sucrose is prepared. It
is shaken with an equal volume of pure chloroform. The
two solutions do not mix. The sucrose partitions between
the two solutions, and is more soluble in the water. The
value of /((partition) for this water-chloroform system is 5.3.
What percentage of the sucrose resides in the chloro-
form?
PHASE EQUILIBRIA INVOLVING TWO-COMPONENT SYSTEMS: PARTITION
207
How does a separating funnel work?
Partition as a function of solvent
The operation of a separating funnel depends on partition. A solvent contains some
solute. A different solvent, which is immiscible with the first, contains no compound.
Because the two solvents are immiscible - which means they do
not mix - the separating funnel will show two distinct layers (see
Figure 5.15). After shaking the funnel vigorously, and allowing its
contents to settle, some of the solute will have partitioned between
the two solvents, with some sample passing from the solution into
the previously pure solvent 1.
We usually repeat this procedure two or three times during the
practice of solvent extraction, and separate the two layers after each
vigorous shake (we call this procedure 'running off the heavier,
lower layer of liquid). Several extractions are needed because ^(partition) is usually
quite small, which implies that only a fraction of the solute is removed from the
solution during each partition cycle.
Immiscible solutions
do not mix. The words
'miscible' and its con-
verse 'immiscible'
derive from the Latin
word miscere, meaning
'to mix'.
Solvent 1
Meniscus (across which
partition occurs)
Solvent 2
Figure 5.15 A separating funnel is a good example of partition: solute is partitioned between two
immiscible liquids
208
PHASE EQUILIBRIA
Aside
The reason we need to shake the two solutions together when partitioning is because
the solute only passes from one solvent to the other across the interface between them,
i.e. across the meniscus.
The meniscus is quite small if the funnel is kept still, and partitioning is slow.
Conversely, shaking the funnel generates a large number of small globules of solvent,
which greatly increases the 'active' surface area of the meniscus. Therefore, we shake
the funnel to increase the rate of partitioning.
A fish would not be
able to 'breathe' in
water if it contained no
oxygen gas.
Why is an ice cube only misty at its centre?
The temperature dependence of partition
Most ice cubes look misty at their centre, but are otherwise quite clear. The ice from
which the ice cubes are made is usually obtained from the tap, so it contains dissolved
impurities such as chlorine (to ensure its sterility) and gases from the atmosphere. The
mist at the centre of the ice cube comprises millions of minute air bubbles containing
these gases, principally nitrogen and oxygen.
Gaseous oxygen readily partitions with oxygen dissolved in solu-
tion, in much the same way as the partitioning of CO2 in the
fizzy-drink example above. The exact amount of oxygen in solu-
tion depends on the value of ^(partition), which itself depends on the
temperature.
Tap water is always saturated with oxygen, the amount depend-
ing on the temperature. The maximum concentration of oxygen in
water - about 0.02 moldm -3 - occurs at a temperature of 3 °C.
The amount of oxygen dissolved in water will decrease below
this temperature, since ^partition) decreases. Accordingly, much dis-
solved oxygen is expelled from solution as the water freezes, merely
to keep track of the constant decreasing value of ^(partition)-
The tap water in the ice tray of our fridge undergoes some interesting phase changes
during freezing. Even cold water straight from a tap is warmer than the air within a
freezer. Water is a poor thermal conductor and does not freeze evenly, i.e. all at once;
rather, it freezes progressively. The first part of the water to freeze is that adjacent to
the freezer atmosphere; this outer layer of ice gradually becomes thicker with time,
causing the amount of liquid water at the cube's core to decrease during freezing.
But ice cannot contain much dissolved oxygen, so air is expelled from solution
each time an increment of water freezes. This oxygen enters any liquid water nearby,
which clearly resides near the centre of the cube. We see how the oxygen from the
water concentrates progressively near the cube centre during freezing.
Eventually, all the oxygen formerly in the water resides in a small volume of water
near the cube centre. Finally, as the freezing process nears its completion and even
Like all other equilib-
rium constants, the
value Of /((partition)
depends strongly on
temperature.
PHASE EQUILIBRIA INVOLVING TWO-COMPONENT SYSTEMS: PARTITION
209
this last portion solidifies, the amount of oxygen in solution exceeds ^(partition) and
leaves solution as gaseous oxygen. It is this expelled oxygen we see as tiny bubbles
of gas.
Aside
Zone refining is a technique for decreasing the level of impurities in some metals, alloys,
semiconductors, and other materials; this is particularly so for doped semiconductors, in
which the amount of an impurity must be known and carefully controlled. The technique
relies on the impurities being more soluble in a molten sample (like oxygen in water,
as noted above) than in the solid state.
To exploit this observation, a cylindrical bar of material is passed slowly through
an induction heater and a narrow molten 'zone' is moved along its length. This causes
the impurities to segregate at one end of the bar and super-pure material at the other.
In general, the impurities move in the same direction as the molten zone moves if the
impurities lower the melting point of the material (see p. 212).
How does recrystallization work?
Partition and the solubility product
We say the solution is saturated if solute is partitioned between a liquid-phase solution
and undissolved, solid material (Figure 5.16). In other words, the solution contains
as much solute as is feasible, thermodynamically, while the remainder remains as
solid. The best way to tell whether a solution is saturated, therefore, is to look for
undissolved solid. If ^(partition) is small then we say that not much of the solute resides
in solution, so most of the salt remains as solid - we say the salt is not very soluble.
Conversely, most, if not all, of the salt enters solution if ^(partition) is large.
Like all equilibrium constants, the value of ^(partition) depends on temperature,
sometimes strongly so. It also depends on the solvent polarity. For example, ^T(partition)
Solution saturated
with solute
Solid crystals
of solute
Figure 5.16 In a saturated solution, the solute is partitioned between the solid state and solute
in solution
210
PHASE EQUILIBRIA
of sodium chloride (NaCl) in water is large, so a saturated solution has a concentration
of about 4 moldm -3 ; a saturated solution of NaCl in ethanol contains less than
0.01 moldm -3 of solute.
An alternative way of expressing the partition constant of a spar-
ingly soluble salt is to define its 'solubility product' K sp (also called
the 'solubility constant' K s ). K s is defined as the product of the ion
activities of an ionic solute in its saturated solution, each raised to
its stoichiometric number V;. K s is expressed with due reference to
the dissociation equilibria involved and the ions present.
We saw above how the extent of partition is temperature depen-
dent; in that example, excess air was expelled from solution during freezing, since
the solubility of air was exceeded in a cold freezer box, and the gas left solution in
Strictly, we should
speak in terms of ionic
activities rather than
concentrations; see
p. 312 ff.
order for the value of K
(partition)
to be maintained.
Like all equilibrium constants, ^(partition) is a function of temper-
ature, thereby allowing the preparative chemist to recrystallize a
freshly made compound. In practice, we dissolve the compound in
a solvent that is sufficiently hot so that ^(partition) is large, as shown
by the high solubility. Conversely, ^(partition) decreases so much on
cooling that much of the solute undergoes a phase change from
the solution phase to solid in order to maintain the new, lower
value of X" (partition) • The preparative chemist delights in the way
that the precipitated solid retrieved is generally purer than that initially added to the
hot solvent.
The energy necessary to dissolve 1 mol of solute is called the 'enthalpy of solution'
(cf. p. 125). A value of AH can be estimated by analysing the solubility
The improved purity
of precipitated solute
implies that /C (partition)
for the impurities is
different from that for
the major solute.
AH,
(solution)
s of a solute (which is clearly a function of K (partition)) with temperature T.
The value of ^(partition) changes with temperature; the temperature dependence of
an equilibrium constant is given by the van't Hoff isochore:
In
^(partition)2 \ AH,
K
(partition) 1
)
* (solution) / ^
R
1
(5.10)
AW (s iution) is sometimes
called 'heat of solu-
tion', particularly in
older books. The word
'heat' here can mislead,
and tempts us to ignore
the possibility of pres-
sure-volume work.
so an approximate value of AH^ olutio . may be obtained from the
gradient of a graph of an isochore plot of In s (as 'y') against
1/r (as 'x'). Since s increases with increased T, we predict that
A Solution) wil1 be Positive.
Worked Example 5.5 Calculate the enthalpy of solution AH, olution)
from the following solubilities 5 of potassium nitrate as a function of
temperature T. Values of s were obtained from solubility experiments.
77K 354 347.6 342 334 329 322 319 317
slg per 100 g of water 140.0 117.0 100.0 79.8 68.7 54.6 49.4 46.1
PHASE EQUILIBRIA INVOLVING TWO-COMPONENT SYSTEMS: PARTITION
211
01
Q_
CO
0.0028
0.0032
Figure 5.17 The solubility s of a partially soluble salt is related to the equilibrium constant
^(partition) an d obeys the van't Hoff isochore, so a plot of In s (as 'y') against \/T (as 'x') should
be linear, with a slope of '— A//- luti . -=- R' . Note how the temperature is expressed in kelvin; a
graph drawn with temperatures expressed in Celsius would have produced a curved plot. The label
KIT on the x-axis comes from \/T -f- 1/K
The solubility s is a function of A" (partition) ; so, from Equation (5.10), a plot of In s
(as 'v') against \IT (as 'x') yields the straight-line graph in Figure 5.17. A value of
A//,
(solution)
= 34 kJmol is obtained by multiplying the gradient '—1 x R' .
Why are some eggshells brown and some white?
Partition between two solid solutes
The major component within an eggshell is calcium carbonate (chalk). Binders and
pigments make up the remainder of the eggshell mass, accounting for about 2-5%.
Before a hen lays its egg, the shell forms inside its body via a complicated series
of precipitation reactions, the precursor for each being water soluble. Sometimes the
hen's diet includes highly coloured compounds, such as corn husk. The chemicals
forming the colour co-precipitate with the calcium carbonate of the shell during shell
formation, which we see as the egg shell's colour. Any substantial change in the hen's
diet causes a different combination of chemicals to precipitate during shell formation,
explaining why we see differently coloured eggs.
In summary, this simple example illustrates the partition of solutes during pre-
cipitation: the colour of an egg shell results from the partitioning of chemicals,
some coloured, between the growing shell and the gut of the hen during shell
growth.
212 PHASE EQUILIBRIA
5.5 Phase equilibria and colligative properties
Why does a mixed-melting-point determination work?
Effects of impurity on phase equilibria
in a two-component system
The best 'fail-safe' way of telling whether a freshly prepared compound is identical
to a sample prepared previously is to perform a mixed-melting-point experiment.
In practice, we take two samples: the first comprises material
A 'mixed melting point
is the only absolutely
fail-safe way of deter-
mining the purity of
a sample.
whose origin and purity we know is good. The second is fresh
from the laboratory bench: it may be pure and identical to the
first sample, pure but a different compound, or impure, i.e. a mix-
ture. We take the melting point of each separately, and call them
respectively T (mdu pure) and T imAt , unknown)- We know for sure that
the samples are different if these two melting temperatures differ.
Ambiguity remains, though. What if the melting temperatures are the same but, by
some strange coincidence, the new sample is different from the pure sample but has
the same melting temperature? We therefore determine the melting temperature of a
mixture. We mix some of the material known to be pure into the sample of unknown
compound. If the two melting points are still the same then the two materials are
indeed identical. But any decrease in T( me i ti i mpure ) means they are not the same. The
value of 7( me i t m i xtU re) will always be lower than 7( me i t] pure ) if the two samples are
different, as evidenced by the decrease in 7( me i t ). We call it a depression of melting
point (or depression of freezing point).
Introduction to colligative properties: chemical potential
The depression of a melting point is one of the simplest manifestations of a colliga-
tive property. Other everyday examples include pressure, osmotic pressure, vapour
pressure and elevation of boiling point.
'Colligative properties'
depend on the num-
ber, rather than the
nature, of the chem-
ical particles (atoms
or molecules) under
study.
For simplicity, we will start by thinking of one compound as the
'host' with the other is a 'contaminant'. We find experimentally
that the magnitude of the depression AT depends only on the
amount of contaminant added to the host and not on the identity of
the compounds involved - this is a general finding when working
with colligative properties. A simple example will demonstrate how
this finding can occur: consider a gas at room temperature. The
ideal-gas equation (Equation (1.13)) says pV = nRT, and holds
reasonably well under s.t.p. conditions. The equation makes it clear that the pressure
p depends only on n, V and T, where V and T are thermodynamic variables, and
n relates to the number of the particles but does not depend on the chemical nature
of the compounds from which the gas is made. Therefore, we see how pressure is a
colligative property within the above definition.
PHASE EQUILIBRIA AND COLLIGATIVE PROPERTIES
213
Earlier, on p. 181, we looked at the phase changes of a single-component system
(our examples included the melting of an ice cube) in terms of changes in the molar
Gibbs function AG m . In a similar manner, we now look at changes in the Gibbs
function for each component within the mixture; and because several components
participate, we need to consider more variables, to describe both the host and the
contaminant.
We are now in a position to understand why the melting point
of a mixture is lower than that of the pure host. Previously, when
we considered the melting of a simple single-component system,
we framed our thinking in terms of the molar Gibbs function G m -
In a similar way, we now look at the molar Gibbs function of each
component i within a mixture. Component i could be a contami-
nant. But because i is only one part of a system, we call the value of G m for material
i the partial molar Gibbs function. The partial molar Gibbs function is also called
the chemical potential, and is symbolized with the Greek letter mu, /x.
We define the 'mole fraction' x, as the number of moles of component i expressed
as a proportion of the total number of moles present:
For a pure substance,
the chemical potential
11 is merely another
name for the molar
Gibbs function.
number of moles of component i
Xi =
total number of moles
The value of /x, - the molar Gibbs function of the contaminant -
decreases as x, decreases. In fact, the chemical potential //, of
the contaminant is a function of its mole fraction within the host,
according to Equation (5.11):
(5.11)
fit = [i: + RT \r\Xi
(5.12)
where x,- is the mole fraction of the species i, and fx i is its
standard chemical potential. Equation (5.12) should remind us of
Equation (4.49), which relates AG and AG e .
Notice that the mole fraction x has a maximum value of unity.
The value of x decreases as the proportion of contaminant in-
creases. Since the logarithm of a number less than one is always
negative, we see how the RT In x, term on the right-hand side of
Equation (5.12) is zero for a pure material (implying /x, = fx t ). At
x, < 1, causing the term RT In x, to be negative. In other words,
will always decrease from a maximum value of \x i as the amount
in creases.
Figure 5.18 depicts graphically the relationship in Equa-
tion (5.12), and shows the partial molar Gibbs function of the host
material as a function of temperature. We first consider the heavy
bold lines, which relate to a pure host material, i.e. before con-
tamination. The figure clearly shows two bold lines, one each for
the material when solid and another at higher temperatures for the
The mole fraction x of
the host DEcreases as
the amount of contam-
inant INcreases. The
sum of all the mole
fractions must always
equal one; and the
mole fraction of a pure
material is also one.
Strictly, Equation (5.12)
relates to an ideal
mixture at constant p
and T.
all other times,
the value of /x
of contaminant
Remember: in this type
of graph, the lines
for solid and liquid
intersect at the melting
temperature.
214
PHASE EQUILIBRIA
C5
I
Solid
\
\. Liquid
*
V
►
en as
E E
_ ©
(0 J=
■cS.
(0
Q_
' (melt) mixture ' (melt) pure solvent (pure)
Temperature T
Figure 5.18 Adding a chemical to a host (mixing) causes its chemical potential [i to decrease,
thereby explaining why a melting-point temperature is a good test of purity. The heavy solid
lines represent the chemical potential of the pure material and the thin lines are those of the host
containing impurities
respective liquid. In fact, when we remember that the chemical potential for a pure
material is the same as the molar Gibbs function, we see how this graph (the bold
line for the pure host material) is identical to Figure 5.2. And we recall from the start
of this chapter how the lines representing G m for solid and G m for liquid intersect
at the melting temperature, because liquid and solid are in equilibrium at r( me i t ), i.e.
G m (li qu id) = G m ( so lid) at T( me i t ).
We look once more at Figure 5.18, but this time we concentrate on the thinner
lines. These lines are seen to be parallel to the bold lines, but have been displaced
down the page. These thin lines represent the values of G m of the host within the
mixture (i.e. the once pure material following contamination). The line for the solid
mixture has been displaced to a lesser extent than the line for the
liquid, simply because the Gibbs function for liquid phases is more
sensitive to contamination.
The vertical difference between the upper bold line (representing
/x^) and the lower thin line (which is /x) arises from Eq. (5.12):
it is a direct consequence of mixing. In fact, the mathematical
composition of Eq. (5.12) dictates that we draw the line for an
impure material (when x, < 1) lower on the page than the line for
the pure material.
It is now time to draw all the threads together, and look at
the temperature at which the thin lines intersect. It is clear from
Figure 5.18 that the intersection temperature for the mixture occurs
at a cooler temperature than that for the pure material, showing
why the melting point temperature for a mixture is depressed rel-
ative to a pure compound. The depression of freezing point is a
direct consequence of chemical potentials as defined in Equation
(5.12).
As the mole fraction of
contaminant increases
(as Xj gets larger), so
we are forced to draw
the line progressively
lower down the figure.
A mixed-melting-point
experiment is an ideal
test of a material's
purity since 7" (me i t)
never drops unless the
compound is impure.
PHASE EQUILIBRIA AND COLLIGATIVE PROPERTIES 215
We now see why the melting-point temperature decreases following contamination,
when its mole fraction deviates from unity. Conversely, the mole fraction does not
change at all if the two components within the mixed-melting-point experiment are
the same, in which T( me it) remains the same.
Justification Box 5.3
When we formulated the total differential of G (Equation (4.30)) in Chapter 4, we only
considered the case of a pure substance, saying
/3G\ /3G\
We assumed then the only variables were temperature and pressure. We must now
rewrite Equation (4.30), but we add another variable, the amount of substance «, in a
mixture:
We append an additional subscript to this expression for dG to emphasize that we refer
to the material i within a mixture. As written, Equation (5.13) could refer to either the
host or the contaminant - so long as we define which is i .
The term 3G,/3«j occurs so often in second law of thermodynamics that it has its
own name: the 'chemical potential' /x, which is defined more formally as
V 9 ".' / p,T,nj
where the subscripts to the bracket indicate that the variables p, T ', and the amounts of
all other components tij in the mixture, each remain constant. The chemical potential
is therefore seen to be the slope on a graph of Gibbs function G (as 'y') against the
amount of substance m, (as 'x'); see Figure 5.19. In general, the chemical potential
varies with composition, according to Equation (5.12).
The chemical potential fi can be thought of as the constant of proportionality between
a change in the amount of a species and the resultant change in the Gibbs function of
a system.
The way we wrote 3G in Equation (5.13) suggests the chemical potential /x is the
Gibbs function of 1 mol of species i mixed into an infinite amount of host material.
For example, if we dissolve 1 mol of sugar in a roomful of tea then the increase in
Gibbs function is /X( SUg ar)- An alternative way to think of the chemical potential /x is to
consider dissolving an infinitesimal amount of chemical i in 1 mol of host.
216
PHASE EQUILIBRIA
Composition n,
Figure 5.19 The chemical potential //,- (the partial molar Gibbs function) of a species in a
mixture is obtained as the slope of a graph of Gibbs function G as a function of composition
We need to employ 'mental acrobatics' of this type merely to ensure that our definition
of ji is watertight - the overall composition of the mixture cannot be allowed to change
significantly.
How did the Victorians make ice cream?
Cryoscopy and the depression of freezing point
The people of London and Paris in Victorian times (the second half of the nineteenth
century) were always keen to experience the latest fad or novelty, just like many rich
and prosperous people today. And one of their favourite 'new inventions' was ice
cream and sorbets made of frozen fruit.
The ice cream was made this way: the fruit and/or cream to be frozen is packed
into a small tub and suspended in an ice bath. Rock salt is then added to the ice,
which depresses its freezing temperature (in effect causing the ice to melt). Energy
is needed to melt the ice. A//( me i t ) = 6.0 kJmol - for pure water. This energy comes
from the fruit and cream in the tub. As energy from the cream and fruit passes through
the tub wall to the ice, it freezes. Again, we see how a body's temperature is a good
gauge of its internal energy (see p. 34).
The first satisfactory theory to explain how this cooling process works was that of
Francois -Marie Raoult, in 1878. Though forgotten now, Raoult already knew 'Blag-
den's law': a dissolved substance lowers the freezing point of a
solvent in direct proportion to the concentration of the solute. In
practice, this law was interpreted by saying that an ice-brine mix-
ture (made with five cups of ice to one of rock salt) had a freezing
point at about —2.7 °C. Adding too much salt caused the tempera-
ture to fall too far and too fast, causing the outside of the ice
Dissolving a solute in
a solvent causes a
depression of freezing
point, in the same way
as mixing solids.
PHASE EQUILIBRIA AND COLLIGATIVE PROPERTIES
217
cream to freeze prematurely while the core remained liquid. Adding
too little salt meant that the ice did not melt, or remained at a tem-
perature close to °C, so the cream and fruit juices remained liquid.
This depression of the freezing point occurs in just the same
way as the lower melting point of an impure sample, as discussed
previously. This determination of the depression of the freezing
point is termed crysoscopy.
The word 'cryoscopy'
comes from the Greek
kryos, which literally
means 'frost'.
Why boil vegetables in salted water?
Ebullioscopy and the elevation of boiling point
We often boil vegetables in salted water (the concentration of table salt is usually
in the range 0.01-0.05 moldm -3 ). The salt makes the food taste nicer, although we
should wash off any excess salt water if we wish to maintain a healthy blood pressure.
But salted water boils at a higher temperature than does pure
water, so the food cooks more quickly. (We saw on p. 203 how
a hotter temperature promotes faster cooking.) The salt causes an
elevation of boiling point, which is another colligative property. We
call the determination of such an elevation ebullioscopy.
Look at Figure 5.20, the left-hand side of which should remind us
of Figure 5.18; it has two intersection points. At the low-tempera-
ture end of the graph, we see again why the French ice-cream
makers added salt to the ice, to depress its freezing point. But,
when we look at the right-hand side of the figure, we see a second
intersection, this time between the lines for liquid and gas: the temperature at which
the lines intersect gives us the boiling point T(boii).
The word 'ebullioscopy'
comes from the Latin
(e)bulirre, meaning
'bubbles' or 'bubbly'.
In a related way, we
say that someone is
'ebullient' if they have a
'bubbly' personality.
Freezing point
of water + solute
Freezing point Boiling point Boiling point of
of pure water °* P ure water water + solute
Figure 5.20 Salt in water causes the water to boil at a higher temperature and freeze at a lower
temperature; adding a solute to a solvent decreases the chemical potential ji of the solvent. The
bold lines represent pure water and the thinner lines represent water-containing solute
218
PHASE EQUILIBRIA
The figure shows how adding salt to the water has caused both the lines for liquid
and for gas to drop down the page, thus causing the intersection temperature to change.
Therefore, a second consequence of adding salt to water, in addition to changing its
chemical potential, is to change the temperature at which boiling occurs. Note that
the boiling temperature is raised, relative to that of pure water.
Why does the ice on a path melt when sprinkled
with salt?
Quantitative cryoscopy
The ice on a path or road is slippery and dangerous, as we saw when considering
black ice and ice skaters. One of the simplest ways to make a road or path safer is to
sprinkle salt on it, which causes the ice to melt. In practice, rock salt is preferred to
table salt, because it is cheap (it does not need to be purified) and because its coarse
grains lend additional grip underfoot, even before the salt has dissolved fully.
The depression of freezing temperature occurs because ions from the salt enter the
lattice of the solid ice. The contaminated ice melts at a lower temperature than does
pure ice, and so the freezing point decreases. Even at temperatures below the normal
melting temperatures of pure ice, salted water remains a liquid - which explains why
the path or road is safer.
We must appreciate, however, that no chemical reaction occurs
between the salt and the water; more or less, any ionic salt, when
put on ice, will therefore cause it to melt. The chemical identity of
the salt is irrelevant - it need not be sodium chloride at all. What
matters is the amount of the salt added to the ice, which relates
eventually to the mole fraction of salt. So, what is the magnitude
of the freezing-point depression?
Let the depression of the freezing point be AT, the magnitude
of which depends entirely on the amount of solute in the solvent.
Re-interpreting Blagden's law gives
The 'molaLity' m is the
number of moles of
solute dissolved per
unit mass of solvent;
'molaRity' (note the
different spelling) is
the number of moles
of solute dissolved per
unit volume.
We prefer 'molaL-
ity' m to 'molaRity'
(i.e. concentration c)
because the volume
of a liquid or solution
changes with temper-
ature, whereas that
of a mass does not.
Accordingly, molal-
ity is temperature
independent whereas
concentration is not.
AT oc molality
(5.15)
The amount is measured in terms of the molality of the solute.
Molality (note the spelling) is defined as the amount of solute
dissolved per unit mass of solvent:
molality, m
moles of solute
mass of solvent
(5.16)
where the number of moles of solute is equal to 'mass of solute 4-
molar mass of solute'. The proportionality constant in Equation
(5.15) is the cryoscopic constant AT(cryoscopic) ■ Table 5.3 contains a
few typical values of ^(cryoscopic)* from which it can be seen that
PHASE EQUILIBRIA AND COLLIGATIVE PROPERTIES 219
Table 5.3 Sample values of boiling and freezing points, and cryoscopic and ebullioscopic constants
Boiling
Freezing
■^-(ebullioscopic)
**■ (cryoscopic)
Substance
point/ °C
point/ °C
/K kg mof 1
/KkgrnoP 1
Acetic acid
118.5
16.60
3.08
3.59
Acetone
56.1
-94.7
1.71
-
Benzene
80.2
5.455
2.61
5.065
Camphor
208.0
179.5
5.95
40
Carbon disulphide
46.3
-111.5
2.40
3.83
Carbon tetrachloride
76.5
-22.99
5.03
29.8
Chloroform
61.2
-65.5
3.63
4.70
Cyclohexane
80.74
6.55
2.79
20.0
Ethanol
78.3
-114.6
1.07
1.99
Ethyl acetate
77.1
-83.6
2.77
-
Ethyl ether
34.5
-116.2
2.02
1.79
Methanol
64.7
-97.7
0.83
-
Methyl acetate
57
-98.1
2.15
-
n-Hexane
68.7
-95.3
2.75
-
n-Octane
125.7
-56.8
4.02
-
Naphthalene
217.9
80.3
6.94
5.80
Nitrobenzene
210.8
5.7
5.24
8.1
Phenol
181.8
40.9
3.56
7.27
Toluene
110.6
-95.0
3.33
-
Water
100
0.512
1.858
camphor as a solvent causes the largest depression. Note that K has the units of
K kgmol -1 , whereas mass and molar mass are both expressed with the units in units
of grammes, so any combination of Equations (5.15) and (5.16) requires a correction
term of 1000 gkg -1 . Accordingly, Equation (5.15) becomes
(mass of solute \ 1
— ; ;— ; — x ;— ; — < 5 - 17 )
molar mass of solute/ mass of solvent
where the term in parentheses is n, the number of moles of solute.
Worked Example 5.6 10 g of pure sodium chloride is dissolved in 1000 g of water.
By how much is the freezing temperature depressed from its normal melting temperature
of T — 273.15 K? Take A"( cryoscop i C ) from Table 5.3 as 1.86 Kkgmol -1 .
Inserting values into Equation (5.17) yields
AT = 1.86 KkgrnoP 1 x lOOOgkg" 1 x
10e 1
58.5 gmol -1 1000 g
so AT = 0.32 K
220
PHASE EQUILIBRIA
This value of AT repre-
sents the depression of
the freezing tempera-
ture, so it is negative
showing that the water will freeze at the lower temperature of
(273.16-0.32) K.
SAQ 5.6 Pure water has a normal freezing point
of 273.15 K. What will be the new normal freezing point
of water if 11 g of KCI is dissolved in 0.9 dm 3 of water?
The cryoscopic constant of water is 1.86 Kkg 1 mol -1 ; assume the density
of water is 1 g cm' 3 , i.e. molality and molarity are the same.
An almost identical equation relates the elevation of boiling point to the molality:
AT.
"(elevation) = -K(ebuiiioscopic) x 1000 x molality of the salt (5.18)
where -K"( e buiiioscopic) relates to the elevation of boiling temperature. Table 5.3 con-
tains a few sample values of ^(ebuilioscopic) • It can be seen from the relative val-
ues of AT(ebuiiioscopic) and AT( C ryoscopic) in Table 5.3 that dissolving a solute in a sol-
vent has a more pronounced effect on the freezing temperature than on the boiling
temperature.
Aside
The ice on a car windscreen will also melt when squirted with de-icer. Similarly, we add
anti- freeze to the water circulating in a car radiator to prevent it freezing; the radiator
would probably crack on freezing without it; see the note on p. 194.
Windscreen de-icer and engine anti-freeze both depress the freezing point of water
via the same principle as rock salt depressing the temperature at which ice freezes on a
road. The active ingredient in these cryoscopic products is ethylene glycol (II), which
is more environmentally friendly than rock salt. It has two physicochemical advantages
over rock salt: (1) being liquid, it can more readily enter between the microscopic
crystals of solid ice, thereby speeding up the process of cryoscopic melting; (2) rock
salt is impure, whereas II is pure, so we need less II to effect the same depression of
freezing point.
CH2 CH2
/ \
OH OH
(II)
Ethylene glycol is also less destructive to the paintwork of a car than rock salt is,
but it is toxic to humans.
PHASE EQUILIBRIA INVOLVING VAPOUR PRESSURE
221
5.6 Phase equilibria involving
vapour pressure
Why does petrol sometimes have a strong smell
and sometimes not?
Dalton's law
The acrid smell of petrol on a station forecourt is sometimes overpoweringly strong,
yet at other times it is so weak as to be almost absent. The smell is usually stronger
on a still day with no wind, and inspection shows that someone has spilled some
petrol on the ground nearby. At the other extreme, the smell is weaker when there
is a breeze, which either blows away the spilt liquid or merely dilutes the petrol in
the air.
The subjective experience of how strong a smell is relates to the amount of petrol
in the air; and the amount is directly proportional to the pressure of gaseous petrol.
We call this pressure of petrol the 'partial pressure' /?( pe troi)-
And if several gases exist together, which is the case for petrol in air, then the total
pressure equals the sum of the partial pressures according to Dalton's law:
P (total)
X>
(5.19)
In the case of a petrol smell near a station forecourt, the smell is strong when the
partial pressure of the petrol vapour is large, and it is slight when p( pe troi) is small.
These differences in /?( pe troi) need not mean any difference in the overall pressure
P (total) » merely that the composition of the gaseous mixture we breathe is variable.
SAQ 5.7 What is the total pressure of 10 g of nitrogen gas and 15 g
of methane at 298 K, and what is the partial pressure of nitrogen in the
mixture? [Hint: you must first calculate the number of moles involved.]
Justification Box 5.4
The total number of moles equals the sum of its constituents, so
"(total) = «A + «B + • • ■
The ideal-gas equation (Equation (1.13)) says pV = nRT; thus P( pe troi)^ = n(petroi)RT,
SO M(petrol) = P(petrol) V ~ RT .
Accordingly, in a mixture of gases such as petrol, oxygen and nitrogen:
P(total)V _ ,P(petrol)V P (oxygen) V p (nitrogen) V
RT
RT
RT
RT
222 PHASE EQUILIBRIA
We can
cancel the
gas
constant R, the volume and
temperature,
which
are
all constant,
to yield
P(total) =
P(petrol) t P(oxygen
~r P (nitrogen)
which is
Dalton 's
law,
Equation
(5.19).
How do anaesthetics work?
^H
Gases dissolving in liquids: Henry's law
'Anaesthesia' is the
science of making
someone unconscious.
The word comes from
the Greek aesthesis,
meaning sensation
(from which we get
the modern English
word 'aesthetic', i.e. to
please the sensations).
The initial s ana' makes
the word negative, i.e.
without sensation.
An anaesthetist administers chemicals such as halothane (III) to a
patient before and during an operation to promote unconsciousness.
Medical procedures such as operations would be impossible for
the surgeon if the patient were awake and could move; and they
would also be traumatic for a patient who was aware of what the
surgery entailed.
?:ci
x
F H
(III)
A really deep, chem-
ically induced sleep
is termed 'narcosis',
from the Greek narke,
meaning 'numbness'.
Similarly, we similarly
call a class-A drug a
'narcotic'.
Henry's law is named
after William Henry
(1775-1836), and says
that the amount of gas
dissolved in a liquid or
solid is in direct pro-
portion to the partial
pressure of the gas.
Although the topic of anaesthesia is hugely complicated, it is
clear that the physiological effect of the compounds depends on
their entrapment in the blood. Once dissolved, the compounds pass
to the brain where they promote their narcotic effects. It is now
clear that the best anaesthetics dissolve in the lipids from which
cell membranes are generally made. The anaesthetic probably alters
the properties of the cell membranes, altering the rates at which
neurotransmitters enter and leave the cell.
A really deep 'sleep' requires a large amount of anaesthetic and a
shallower sleep requires less material. A trained anaesthetist knows
just how much anaesthetic to administer to induce the correct depth
of sleep, and achieves this by varying the relative pressures of the
gases breathed by the patient.
In effect, the anaesthetist relies on Henry's law, which states
that the equilibrium amount of gas that dissolves in a liquid is
proportional to the mole fraction of the gas above the liquid. Henry
published his studies in 1803, and showed how the amount of gas
dissolved in a liquid is directly proportional to the pressure (or
PHASE EQUILIBRIA INVOLVING VAPOUR PRESSURE
223
Table 5.4 Henry's law constants
&h for gases in water at 25 °C
Gas & H /moldm~ 3 bar -1
C0 2
o 2
CH 4
N 2
3.38 x 1(T 2
1.28 x 1(T 3
1.34 x 1(T 3
6.48 x 1(T 4
partial pressure) of the gas above it. Stated in another form, Henry's
law says:
[z'(soln)] = knpi
(5.20)
where pt is the partial pressure of the gas i, and [z'( SO in)] is the
concentration of the material i in solution. The constant of propor-
tionality ku is the respective value of Henry's constant for the gas,
which relates to the solubility of the gas in the medium of choice.
Table 5.4 lists a few Henry's law constants, which relate to the
solubility of gases in water.
Worked Example 5.7 What is the concentration of molecular oxy-
gen in water at 25 °C? The atmosphere above the water has a pressure
of 10 5 Pa and contains 21 per cent of oxygen.
Strategy. (1) We calculate the partial pressure of oxygen /?(o,)- (2) We
calculate the concentration [02( aq )] using Henry's law, Equation
(5.20), [0 2 (aq)] = P(0 2 ) X £ H (0 2 )-
One of the simplest
ways of removing
gaseous oxygen from
water is to bubble
nitrogen gas through
it (a process called
'sparging').
Strictly, Henry's law
only holds for dilute
systems, typically in
the mole-fraction range
0-2 per cent. The law
tends to break down
as the mole fraction
x increases.
(1) From the partial of oxygen p(o 2 ) — X{0 2 ) x the total pressure P( to tai)> where x is
the mole fraction:
p ( 2 ) = 0.21 x 10 5 Pa
P(Oi)
2.1 x 10 4 Pa or 0.21 bar
(2)
To obtain the concentration of oxygen, we insert values into Henry's law,
Equation (5.20):
-,-3
,-3
[0 2( aq)] = 0.21 x p a x 1.28 x 10 _J moldnT^ bar"
q)l
[0 2 (aq)] = 2.69 x 10~ 4 moldm" 3
We need to be aware that ku is an equilibrium constant, so
its value depends strongly on temperature. For example, at 35 °C,
water only accommodates 7.03 mg of oxygen per litre, which ex-
plains why fish in warm water sometimes die from oxygen
starvation.
This relatively high
concentration of oxy-
gen helps explain why
fish can survive in
water.
224
PHASE EQUILIBRIA
How do carbon monoxide sensors work?
Henry's law and solid-state systems
Small, portable sensors are now available to monitor the air we breathe for such
toxins as carbon monoxide, CO. As soon as the air contains more than a critical
concentration of CO, the sensor alerts the householder, who then opens a window or
identifies the source of the gas.
At the 'heart' of the sensor is a slab of doped transition-metal oxide. Its mode
of operation is to detect the concentration of CO within the oxide slab, which is in
direct proportion to the concentration of CO gas in the air surrounding it, according
to Henry's law.
A small voltage is applied across the metal oxide. When it contains no CO, the
electrical conductivity of the oxide is quite poor, so the current through the sensor is
minute (we argue this corollary from Ohm's law). But increasing the concentration of
CO in the air causes a proportionate increase in the amount of CO incorporating into
the solid oxide, which has a profound influence on electrical conductivity through the
slab, causing the current through the slab to increase dramatically. A microchip within
the sensor continually monitors the current. As soon as the current
increases above its minimum permissible level, the alarm sounds.
So, in summary, CO gas partitions between the air and carefully
formulated solid oxides. Henry's law dictates the amount of CO in
the oxide.
In general, Henry's
law only applies over
relatively small ranges
of gas pressure.
Why does green petrol smell different from
leaded petrol?
Effects of amount of material on vapour pressure
Petrol is only useful in
a car engine because it
is volatile.
A car engine requires petrol as its source of fuel. Such petrol has
a low boiling temperature of about 60 °C. Being so volatile, the
liquid petrol is always surrounded with petrol vapour. We say it has
a high vapour pressure (also called 'saturated vapour pressure'),
which explains why we smell it so readily.
Once started, the engine carburettor squirts a mixture of air and volatile petrol
into a hot engine cylinder, where the mixture is ignited with a spark. The resultant
explosion (we call it 'firing') provides the ultimate source of kinetic energy to propel
the car.
A car engine typically requires four cylinders, which fire in a carefully synchro-
nized manner. Unfortunately, these explosions sometimes occur prematurely, before
PHASE EQUILIBRIA INVOLVING VAPOUR PRESSURE
225
the spark has been applied, so the explosions cease to be synchro-
nized. It is clearly undesirable for a cylinder to fire out of sequence,
since the kinetic energy is supplied in a jerky, irreproducible man-
ner. The engine sounds dreadful, hence the word 'knock'.
Modern petrol contains small amounts of additives to inhibit this
knocking. 'Leaded' petrol, for example, contains the organometallic
compound lead tetraethyl, PbEt4. Although PbEt4 is excellent at
stopping knocking, the lead by-products are toxic. In fact, most
EU countries now ban PbEt/t.
So-called 'green' petrol is a preferred alternative to leaded petrol:
it contains about 3 per cent of the aromatic hydrocarbon benzene
(CgHg, IV) as an additive, the benzene acting as a lead -free alter-
native to PbEt 4 as an 'anti-knocking' compound.
We experience knock-
ing (which we collo-
quially call 'pinking')
when explosions within
a car engine are not
synchronized.
Lead tetraethyl is the
most widely made
organometallic com-
pound in the world.
It is toxic, and killed
over 40 chemical work-
ers during its early
development.
The PbEt4 in petrol does not smell much because it is not volatile. By contrast,
benzene is much more volatile - almost as volatile as petrol. The vapour above 'green'
petrol, therefore, contains quite a high proportion of benzene (as detected by its
cloying, sweet smell) as well as gaseous petrol. That is why green petrol has a
sweeter smell than petrol on its own.
Why do some brands of ' green' petrol smell different
from others?
I
Rao u It's law
The 'petrol' we buy comprises a mixture of naturally occurring
hydrocarbons, a principal component of which is octane; but the
mixture also contains a small amount of benzene. Some brands of
petrol contain more benzene than others, both because of varia-
tions in the conditions with which the crude oil is distilled into
fractions, and also variations in the reservoir from which the crude
oil is obtained. The proportion varies quite widely: the average is
presently about 3 per cent.
Petrol containing a lot of benzene smells more strongly of ben-
zene than petrol containing less of it. In fact, the intensity of
the smell is in direct proportion to the amount of benzene in the
petrol: at equilibrium, the pressure of vapour above a liquid mixture
In the countries of
North America, petrol is
often called 'gas', which
is short for gasoline'.
Raoult's law is merely a
special form of Henry's
law.
226
PHASE EQUILIBRIA
depends on the liquid's composition, according to Raoult's law.
P (benzene) — P (benzene) X (benzene)
(5.21)
Raoult's law states
that (at constant tem-
perature) the partial
pressure of component
/ in the vapour residing
at equilibrium above
a liquid is proportional
to the mole fraction
x, of component in
the liquid.
where X(b en zene) is the mole fraction of the benzene in the liquid.
If we assume that liquid benzene and petrol have the same den-
sities (which is entirely reasonable), then petrol containing 3 per
cent of benzene represents a mole fraction X(benzene) = 0.03; the
mole fraction of the petrol in the liquid mixture is therefore 0.97
(or 97 per cent). The vapour above the petrol mixture will also
be a mixture, containing some of each hydrocarbon in the petrol.
We call the pressure due to the benzene component its partial pres-
sure ^(benzene)- The constant of proportionality in Equation (5.21) is
P (benzene) ' which represents the pressure of gaseous benzene above
pure (i.e. unmixed) liquid benzene.
Calculations with Raoult's law
If a two-component
system of A and B
forms an ideal mixture,
then we can calcu-
late xa if we know xb
because x A + x B = 1, so
x B = (l-x A ).
If we know the mole fraction of a liquid i (via Equation (5.11))
and the vapour pressures of the pure liquids p, , then we can ascer-
tain the total vapour pressure of the gaseous mixture hovering at
equilibrium above the liquid.
The intensity of the benzene smell is proportional to the amount
of benzene in the vapour, /? (benzene)- According to Equation (5.21),
P(benzene) is a simple function of how much benzene resides within
the liquid petrol mixture. Figure 5.21 shows a graph of the partial
pressures of benzene and octane above a mixture of the two liq-
uids. (For convenience, we assume here that the mixture comprises only these two
components.)
The extreme mole fractions, and 1, at either end of the graph relate to pure petrol
(x = 0) and pure benzene (x = 1) respectively. The mole fractions between these
values represent mixtures of the two. The solid, bold line represents the total mole
fraction while the dashed lines represent the vapour pressures of the two constituent
vapours. It is clear that the sum of the two dashed lines equals the bold line, and
represents another way of saying Dalton's law: the total vapour pressure above a
mixture of liquids is the sum of the individual vapour pressures.
Benzene is more vola-
tile than bromobenzene
because its vapour
pressure is higher.
Worked Example 5.8 The two liquids benzene and bromobenzene
are mixed intimately at 298 K. At equilibrium, the pressures of the
gases above beakers of the pure liquids are 100.1 kPa and 60.4 kPa
respectively. What is the vapour pressure above the mixture if 3 mol
of benzene are mixed with 4 mol of bromobenzene?
PHASE EQUILIBRIA INVOLVING VAPOUR PRESSURE
227
re
Q.
"35
re
o
o
o
70 000 -
60 000 -
50 000 -
40 000 -
30 000 -
20 000 -
10 000-
C
(
,.o"'
Total vapour pressure^----^'''^ ..O
^^-^^ ,o""
^^ a''
^ ,.--'' Vapour pressu
re
re
D.
a5
c
<B
N
C
O
o
</>
<D
1—
Q.
«
'■E
03
Q.
,.0 * of benzene
Vapour pressure h*"~*
of octane "^^^
3
O)
05
2
re
re
)
i i i i
0.2 0.4 0.6 0.8
1
Mole fraction of benzene
Figure 5.21 Petrol ('gasoline') is a mixture of liquid hydrocarbons. The partial pressure of ben-
zene is nearly twice that of octane, making it much more volatile. The bold line represents the
total pressure of vapour above a basin of petrol, and comprises the sum of two partial pressures:
benzene (open circles) and octane (filled circles). Each partial pressure is proportional to the mole
fraction of the respective liquid in the petrol mixture
From Dalton's law, the total vapour pressure is simply the sum of the individual vapour
pressures:
/'(total) — "(benzene) ' /'(bromobenzene)
so, from Raoult's law, these partial pressures may be obtained by
substituting each p term with p i x x-, :
/"(total) - (/? (benzene) X ^(benzene)) + (P
(bromobenzene) X -'•(bromobenzene))
(5.22)
Care: do not confuse
p & (the standard pres-
sure of 10 5 Pa) withpf ,
the vapour pressure of
pure /".
We know from the question that there are 7 mol of liquid. We obtain the respective mole
fractions x from Equation (5.11): the mole fraction of benzene is | and the mole fraction
of bromobenzene is ^ .
Substituting values of x-, and p t into Equation (5.22) yields the total pressure /? (total) as
/'(total) = (100.1 kPa x f) + (60.4 kPa x f)
Pdotai) = (42.9 kPa) + (34.5 kPa)
so
P (total) = 77.4 kPa
228
PHASE EQUILIBRIA
An ideal mixture com-
prises a pair (or more)
of liquids that obey
Raoult's law.
Because these two liquids, when mixed, obey Raoult's law, we
say they form an ideal mixture. In fact, relatively few pairs of
liquids form ideal mixtures: a few examples include benzene and
bromobenzene, benzene and toluene, bromobenzene and chloroben-
zene, re-pentane and i-pentane. Note how each set represents a pair
of liquids showing a significant extent of similarity.
SAQ 5.8 Benzene and toluene form an ideal mixture, i.e. they obey
Raoult's law. At 20 °C, the pressure p e of benzene and toluene are 0.747 x
p^ and 0.223 xp e respectively. What is the pressure above a mixture of
these two liquids that contains 12 mol% of benzene?
Worked Example 5.9 (Continuing from Worked Example 5.8.) What are the mole frac-
tions of benzene and bromobenzene in the vapour!
From the definition of mole fraction x in Equation (5.11) above, we say
moles of benzene in the vapour
■^(benzene, vapour)
total number of moles in the vapour phase
The numbers of moles «, are directly proportional to the partial pressures /?, if we assume
that each vapour behaves as an ideal gas (we assume here that T, R and V are constant).
Accordingly, we can say
-^-(benzene)
pressure of benzene
Note how the units
cancel to yield a dimen-
sionless mole fraction.
total pressure
Substituting numbers from Worked Example 5.8:
42.9 kPa
-^(benzene)
77.4 kPa
•^-(benzene) — U.JJ4-
The mole fraction of benzene in the vapour is 0.554, so it contains 55.4
per cent benzene. The remainder of the vapour comprises the second
component bromobenzene, so the vapour contains (100 — 55.4)% =
44.6% of bromobenzene.
Note how the liquid comprises 43 per cent benzene and 57 per
cent bromobenzene, but the vapour contains proportionately more of
the volatile benzene. We should expect the vapour to be richer in the
more volatile component.
SAQ 5.9 Continuing with the system in SAQ 5.8, what is the mole fraction
of toluene in the vapour above the mixture?
In fact, most liquid mixtures do not obey Raoult's law particularly well, owing to
molecular interactions.
We need four mole
fractions to define this
two-component sys-
tem - two for the liquid
phases and two for the
vapour phases.
PHASE EQUILIBRIA INVOLVING VAPOUR PRESSURE
229
Why does a cup of hot coffee yield more steam than
above a cup of boiling water at the same temperature?
The rate of steam pro-
duction decreases with
time as the water cools
down because energy
is lost from the cup as
water molecules enter
the gas phase.
The effects of poor mixing (immiscibility)
Prepare two cups: put boiling water into one and boiling coffee in the other. The
temperature of each is the same because the water comes from the same kettle, yet
the amount of steam coming from the coffee is seen to be greater. (We obtain a better
view of the steam by placing both cups on a sunny window sill, and looking at the
shadows cast on the opposing wall as the light passes through the vapour as it rises
from the cups.)
When performing this little experiment, we will probably notice
how the steam above the coffee has an extremely strong smell of
coffee, although the smell dissipates rapidly as the rate of steam
production decreases.
This experiment is a simple example of steam distillation.
Adding steam promotes the volatilization of otherwise non-volatile
components, simplifying their extraction. For simplicity, we will
say that the smell derives from a single sweet-smelling chemical
'coffee'. Coffee and water are not wholly miscible, with some of
the essential oils from the coffee existing as tiny globules - we call the mixture a
colloid (see Chapter 10). We have generated a two-phase system. Both phases, the
water and the coffee, are saturated with each other. In fact, these globules would
cause strong coffee to appear slightly misty, but for its strong colour blocking all
light. We never see phase separation in the coffee cup, with a layer of oil floating
above a layer of water, because the coffee's concentration is never high enough.
We say a pure liquid boils when its vapour pressure equals the
external, atmospheric pressure (see p. 188). Similarly, when boiling
a mixture, boiling occurs when the sum of the partial pressures
(P(water) + /'(coffee)) equals p* . It is for this reason that the steam
above the coffee cup smells strongly of coffee, because the vapour
contains the essential oils (e.g. esters) that impart the smell. But the
water generates steam at a pressure of p & when the water added
to the cup is boiling, so the partial pressure of the coffee P( co ttee)
is additional. For this reason, we produce more steam than above
the cup containing only water.
The boiling of such a
mixture requires the
sum of the pressures,
not just the pressure
of one component, to
equal p e .
How are essential oils for aromatherapy extracted
from plants?
Steam distillation
■
The 'essential oils' of a plant or crop usually comprise a mixture of esters. At its
simplest, the oils are extracted from a plant by distillation, as employed in a standard
230
PHASE EQUILIBRIA
undergraduate laboratory. Since plants contain such a small amount of this precious
oil, a ton of plant may be needed to produce a single fluid ounce. Some flowers, such
as jasmine or tuberose, contain very small amounts of essential oil, and the petals
are very temperature sensitive, so heating them would destroy the blossoms before
releasing the essential oils.
To add to the cost further, many of these compounds are rather sensitive to tem-
perature and would decompose before vaporizing. For example, oil of cloves (from
Eugenia caryophyllata) is rich in the phenol eugenol (V), which has a boiling point
of 250 °C). We cannot extract the oils via a conventional distillation apparatus.
Heat-sensitive or
water-immiscible
compounds are purified
by steam distillation
at temperatures
considerably lower
than their usual boiling
temperatures.
Solvent extraction of
essential oils tends to
generate material that
is contaminated with
solvent (and cannot be
sold); and mechanical
pressing of a plant usu-
ally generates too poor
a yield to be economi-
cally viable.
The most common method of extracting essential oils is steam
distillation. The plant is first crushed mechanically, to ensure a
high surface area, and placed in a closed still. High-pressure steam
is forced through the still, with the plant pulp becoming hot as the
steam yields its heat of vaporization (see p. 79). The steam forces
the microscopic pockets holding the essential oils to open and to
release their contents. Tiny droplets of essential oil evaporate and
mix in the gas-phase mixture with the steam. The mixture is then
swept through the still before condensing in a similar manner to a
conventional distillation.
Such 'steam heating' is even, and avoids the risk of overheating
and decomposition that can occur in hot spots when external heat-
ing is used. The steam condenses back into water and the droplets
coagulate to form liquid oil. Esters and essential oils do not mix
with water, so phase separation occurs on cooling, and we see
a layer of oil forming above a layer of condensed water. The
oil is decanted or skimmed off the surface of the water, dried,
and packaged.
The only practical problem encountered when collecting organic
compounds by steam distillation is that liquids of low volatility
will usually distil slowly, since the proportion of compound in the
vapour is proportional to the vapour pressure, according to
P(oil)
"(oil)
/'(water) "(water)
(5.23)
PHASE EQUILIBRIA INVOLVING VAPOUR PRESSURE 231
In practice, we force water vapour (steam) at high pressure through the clove pulp to
obtain a significant partial pressure of eugenol (V).
Justification Box 5.5
When considering the theory behind steam distillation, we start with the ideal-gas
equation (Equation (1.13)), pV = nRT . We will consider two components: oil and
water. For the oil, we say P( i\)V = n^n^RT, and for the water /'(water)V' = "(water) RT .
Dividing the two equations by R and V (which are both constant) yields
/'(oil) = "(oil) x T for the oil
P(water) = "(water) x T for the water
We then divide each pressure by the respective number of moles «, , to obtain
P(oii) -r "(oil) = T for the oil
P(water) ^- "(water) = T for the water
The temperature of the two materials will be T, which is the same for each as they are
in thermal equilibrium. We therefore equate the two expressions, saying
/'(oil) -H "(oil) = P(water) "T "(water)
Dividing both sides by />( wa ter) and multiplying both sides by n( aj yields Equation (5.23):
/>(oil) "(oil)
P (water) "(water)
so we see how the percentage of each constituent in the vapour depends only on its
vapour pressure at the distillation temperature.
To extract a relatively involatile oil such as eugenol (V) without charring requires a
high pressure of steam, although the steam will not be hotter than 100 °C, so we generate
a mixture of vapours at a temperature lower than that of the less volatile component.
6
Acids and Bases
Introduction
Equilibria involving acids and bases are discussed from within the Lowry-Br0nsted
theory, which defines an acid as a proton donor and a base as a proton acceptor (or
'abstracter'). The additional concept of pH is then introduced. 'Strong' and 'weak'
acids are discussed in terms of the acidity constant K a , and then conjugate acids and
bases are identified.
Acid -base buffers comprise both a weak acid or base and its respective salt. Cal-
culations with buffers employing the Henderson- Has selbach equation are introduced
and evaluated, thereby allowing the calculation of the pH of a buffer. Next, titrations
and pH indicators are discussed, and their modes of action placed into context.
6.1 Properties of Lowry-Bronsted acids
and bases
Why does vinegar taste sour?
The Lowry-Br0nsted theory of acids
We instantly experience a sour, bitter taste when consuming anything containing
vinegar. The component within the vinegar causing the sensation is ethanoic acid,
CH3COOH (I) (also called 'acetic acid' in industry). Vinegar con-
tains between 10 and 15% by volume of ethanoic acid, the remain-
der being water (85-90%) and small amounts of other components
such as caramel, which are added to impart extra flavour.
The Latin word for
'sour' is acidus.
H
H — C-
H
O
(I)
O — H
234
ACIDS AND BASES
The German chemist
Liebig, in 1838, was
the first to suggest
mobile, replaceable,
hydrogen atoms being
responsible for acidic
properties. Arrhenius
extended the idea in
1887, when he said the
hydrogen existed as a
proton.
The ethanoic acid molecule is essentially covalent, explaining
why it is liquid when pure at room temperature. Nevertheless, the
molecule is charged, with the O-H bond characterized by a high
percentage of ionic character. Because water is so polar a sol-
vent, it strongly solvates any solute dissolved within it. In aqueous
solutions, water molecules strongly solvate the oxygen- and proton-
containing ends of the O-H bond, causing the bond to break in a
significant proportion of the ethanoic acid molecules, according to
the following simplistic reaction:
CH 3 COOH (aq)
CH 3 COO"
(aq)
+ H" 1
(aq)
(6.1)
The O-H bond in an
acid is sometimes said
to be 'labile', since it is
so easily broken. The
word derives from the
Latin labi, to lapse (i.e.
to change).
The 'Lowry-Br0nsted
theory' says an acid is
a proton donor.
Acid property
We say the acid dissociates. The bare proton is very small, and has
a large charge density, causing it to attract the negative end of the
water dipole. The proton produced by Equation (6.1) is, therefore,
hydrated in aqueous solutions, and is more accurately represented
by saying H+ (aq) .
We see how solvated protons impart the subjective impression
of a sour, bitter flavour to the ethanoic acid in vinegar. In fact, not
only the sour flavour, but also the majority of the properties we
typically associate with an acid (see Table 6.1) can be attributed to
an acidic material forming one or more solvated protons H + (aq ) in
solution.
This classification of an acid is called the Lowry-Br0nsted theory
after the two scientists who (independently) proposed this definition
of an acid in 1923. More succinctly, their theory says an acid is
Table 6.1 Typical properties of Lowry-Br0nsted acids
Example from everyday life
Acids dissolve a metal to form a salt plus
hydrogen
Acids dissolve a metal oxide to form a salt
and water
Acids react with metal carbonates to form a
salt and carbon dioxide
Acids are corrosive
Acids react with a base to form a salt and
water ('neutralization')
Metallic sodium reacts with water, and
'fizzes' as hydrogen gas evolves
The ability of vinegar to clean tarnished
silver by dissolving away the coloured
coating of Ag 2
The fizzing sensation in the mouth when
eating sherbet (saliva is acidic, with a pH
of 6.5); sherbet generally contains an
organic acid, such as malic or ascorbic
acids
Teeth decay after eating sugar, and one of the
first metabolites from sugar is lactic acid
Rubbing a dock leaf (which contains an
organic base) on the site of a nettle sting
(which contains acid) will neutralize the
acid and relieve the pain
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES 235
a substance capable of donating a proton. Therefore, we describe ethanoic acid as a
'Lowry-Br0nsted acid'.
Care: the symbol of
conductivity is not K
but the Greek letter
kappa, k.
Why is it dangerous to allow water near an electrical
appliance, if water is an insulator?
The solvated proton
We all need to know that electricity and water are an extremely dangerous combi-
nation, and explains why we are taught never to sit in the bath at the same time as
shaving with a plug-in razor or drying our hair. Electrocution is almost inevitable,
and is often fatal.
The quantity we call 'electricity' is a manifestation of charge Q
passing through a suitable conductor. The electrical conductivity k
of water (be it dishwater, rainwater, bathwater, etc.) must be rela-
tively high because electricity can readily conduct through water.
Nevertheless, the value of k for water is so low that we class water
as an insulator. Surely there is a contradiction here?
'Super-pure water' has been distilled several times, and is indeed an insulator:
its conductivity k is low at 6.2 x 10 -8 Scm -1 at 298 K, and lies midway between
classic insulators such as Teflon, with a conductivity of about 10 -15 Scm -1 , and
semiconductors such as doped silicon, for which k = 10 -2 Scm -1 . The conductivity
of metallic copper is as high as 10 6 Scm -1 .
The value of k cited above was for super-pure water, i.e. the
product of multiple distillations, but 'normal' water from a tap
will inevitably contain solutes (hence the 'furring' inside a kettle
or pipe). Inorganic solutes are generally ionic salts; most organic
solutes are not ionic. The conductivity of super-pure water is low
because the molecules of water are almost exclusively covalent,
with the extent of ionicity being very slight. But as soon as a salt
dissolves, the extent of ionicity in the water increases dramatically,
causing more extensive water dissociation.
Ignoring for the moment the solute in solution, the water dis-
sociation involves the splitting of water itself in a process called
autoprotolysis. The reaction is usually represented as
Pure water is a mixture
of three components:
H 2 0, and its two dis-
sociation products, the
solvated proton (H 3 + )
and the hydroxide ion
(OH").
2H 2
H 3 cr
(aq)
+ OH"
(aq)
(6.2)
where the H30 + ( aq ) species is often called a solvated proton. It also
has the names hydroxonium ion and hydronium ion. The complex
ion H30 + ( aq ) is a more accurate representation of the proton respon-
sible for acidic behaviour than the simplistic 'H + (aq )' we wrote in
Equation (6.1). Note how the left-hand side of Equation (6.2) is
covalent and the right-hand side is ionic.
It is safer in many
instances to assume
the solvated proton
has the formula unit
[H(H 2 0) 4 ]+, with four
water molecules ar-
ranged tetrahedrally
around a central pro-
ton, the proton being
stabilized by a lone
pair from each oxygen
atom.
236
ACIDS AND BASES
Dissolving a solute generally shifts the reaction in Equation (6.2) from left to
right, thereby increasing the concentration of ionic species in solution. This increased
number of ions causes the conductivity k of water to increase, thereby making it a
fatally efficient conductor of charge.
Why is bottled water ^neutral'?
Autoprotolysis
The word 'criterion' is
used of a principle or
thing we choose to use
as a standard when
judging a situation.
The plural or crite-
rion is 'criteria', not
'criterions'.
'Autoprotolysis' comes
from proto- indicating
the proton, and lysis,
which is a Greek root
meaning 'to cleave
or split'. The prefix
auto means 'by self
or 'without external
assistance'.
The labels of many cosmetic products, as well as those on most
bottles of drinking water, emphasize how the product is 'neutral',
implying how it is neither acidic nor alkaline. This stipulation is
deemed to show how healthy the water is. But how do they know?
And, furthermore, what is their criterion for testing?
A better way of defining 'neutral' is to say equal numbers of
protons and hydroxide ions reside in solution (both types of ion
being solvated). How does this situation arise? Autoprotolysis, as
mentioned above, represents the .^//-production of protons, which
is achieved by the splitting of water according to Equation (6.2).
It is clear from Equation (6.2) how the consequence of such auto-
protolytic splitting is a solution with equal numbers of protons and
hydroxide ions.
When water contains no dissolved solutes, the concentrations of
the solvated protons and the hydroxide ions are equal. Accord-
ingly, from our definition of 'neutral' above, we see why pure
water should always be neutral, since [H30 + ( aq )] = [OH~( aq )].
As with all physicochemical processes, the extent of Equation
(6.2) may be quantified by an equilibrium constant K. We call it
the autoprotolysis constant, as defined by
K =
[H3O ( aq )][OH ( aq )]
[H 2 0] 2
(6.3)
The water term in the denominator of Equation (6.3) is always large when com-
pared with the other two concentrations on the top, so we say it remains constant.
This assumption explains why it is rare to see the autoprotolysis constant written as
Equation (6.3). Rather, we usually rewrite it as
K„
[H3O (aq)][OH ( aq )]
(6.4)
where K in Equation (6.3) = K w x [H2O] 2 in Equation (6.4). We will only employ
Equation (6.4) from now on. We call K w the autoprotolysis constant or ionic product
of water.
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES 237
Table 6.2 Values of the autoprotolysis constant A" w as a function of
temperature
Temperature 77 °C 18 25 34 50
K w x 10 14 0.12 0.61 1.04 2.05 5.66
Source: Physical Chemistry, W. J. Moore (4th Edn), Longmans, London, 1962,
p. 365.
The value of K w is 1.04 x 10~ 14 at 298 K when expressed in
concentration units of mol dm -3 . Like all equilibrium constants, its
value depends on the temperature. Table 6.2 lists a few values of
K w as a function of temperature. Note how K w increases slightly
as the temperature increases.
It should now be clear from Equation (6.4) how water splits
(dissociates) to form equal number of protons and hydroxide ions,
hence its neutrality, allowing us to calculate the numbers of each
from the value of K w .
Note how K„ has units.
K„ is often re-ex-
pressed as p/C w , where
the 'p' is a mathe-
matical operator (see
p. 246). p/C w has a
value of 14 at 298 K.
Worked Example 6.1 What is the concentration of the solvated proton in super-pure
water?
Since [H30 + ( aq )] = [OH~( aq )], we could rewrite Equation (6.4) as
K w — [H 3 (aq )]
Taking the square root of both sides of this expression, we obtain
[H 3 + (aq) ]/mol dm" 3 = /I^ = [10" 14 ] 1/2
so
[H 3 + (aq) ] = 1(T 7 mol dm" 3
The concentration of solvated protons in super-pure water is clearly very small.
What is 'acid rain'?
Hydrolysis
Acid rain is one of the worst manifestations of the damage we, as humans, inflict on
our planet. Chemicals combine with elemental oxygen during the burning of fossil
fuels, trees and rubbish to generate large amounts of 'acidic oxides' such as nitrogen
monoxide (NO), carbon dioxide (CO2) and sulphur dioxide (SO2).
Natural coal and oil contain many compounds of nitrogen. One of the worst products
of their combustion is the acidic oxide of nitrogen, NO. At once, we are startled by
this terminology, because the Lowry-Br0nsted definition of an acid involves the
release of a proton, yet nitrogen monoxide NO has no proton to give.
238
ACIDS AND BASES
As long ago as the
18th century, French
chemists appreciated
how burning elemen-
tal carbon, nitrogen
or sulphur generated
compounds which,
when dissolved in
water, yielded an acidic
solution.
The nitric acid in acid
rain forms by a more
complicated mecha-
nism: 4NO (g) + 2H 2 ( D
+ 2( g) ► 4HN0 3 (aq)
'Hydrolysis' means to
split water, the word
coming from the two
Greek roots hydro
meaning water, and
lysis meaning 'to cleave
or split'.
To understand the acidity of pollutants such as NO and CO2, we
need to appreciate how the gas does not so much dissolve in water
as react with it, according to
C0 2(g) + 2H 2 O fl ) ► HCO- 3 (a q) + H 3 H
(aq)
(6.5)
Carbonic acid, H2C03( aq ), never exists as a pure compound; it only
exists as a species in aqueous solution, where it dissociates in
just the same way as ethanoic acid in Equation (6.1) to form a
solvated proton and the HCCC -, ion. Note how we form a sol-
vated proton H30 + ( aq ) by splitting a molecule of water, rather
than merely donating a proton. Carbonic acid is, nevertheless, a
Lowry-Br0nsted acid.
The carbonic acid produced in Equation (6.5) is a proton donor,
so the solution contains more solvated protons than hydroxide
ions, resulting in rain that is (overall) an acid. To make the risk
of pollution worse, 'acid rain' in fact contains a mixture of sev-
eral water-borne acids, principally nitric acid, HNO3 (from nitrous
oxide in water), and sulphurous acid, H2SO3 (an aqueous solution
of sulphur dioxide).
In summary, we see how the concentrations of H3O" 1 " and OH~
are the same if water contains no dissolved solutes, but dissolving
a solute such as NO increases the concentration of H3O" 1 "; in a
similar way, the concentration of OH~ will increase if the water
contains any species capable of consuming protons.
It is time to introduce a few new words. We say carbonic acid
forms by hydrolysis, i.e. by splitting a molecule of water. We
describe the extent of hydrolysis in Equation (6.5) by the following
equilibrium constant:
K
[HCQ3-][H 3 + ]
[C0 2 ][H 2 0] 2
(6.6)
Care: the values of K
from these equations
are only meaningful
for concentrations at
equilibrium.
We sometimes call Equation (6.6) the hydrolysis constant of carbon
dioxide. In fact, the water term in the 'denominator' (the bottom
line) is so large compared with all the other terms that it remains
essentially constant. Therefore, we write Equation (6.6) in a differ-
ent form:
[HC0 3 -][H 3 0+]
K'
[C0 2 ]
(6.7)
Note how the two K terms, K in Equation (6.6) and K' in Equation
(6.7), will have different values.
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
239
Why does cutting an onion make us cry?
Other aqueous acids in the environment
The reason why our eyes weep copiously when peeling an onion is because the onion
contains minute pockets of sulphur trioxide, S03( g ). Cutting the onion releases this
gas. A mammalian eye is covered with a thin film of water-based liquid ('tears') to
minimize friction with the eyelid. The tears occur in response to SO3 dissolving in
this layer of water to form sulphuric acid:
S0 3(g ) + H 2 0(1)
H 2 SO
4(aq)
(6.8)
The sulphuric acid produced dissociates in the water to form SO4 2 and two protons.
The eyes sting as a direct consequence of contact with this acid.
Why does splashing the hands
with sodium hydroxide solution make
them feel x soapy'?
Proton abstraction
Sodium hydroxide in solution dissociates to yield solvated cations
and anions, Na + and the hydroxide ion 0H~ respectively:
NaOH
(s)
Na"*
(aq)
+ 0H"
(aq)
(6.9)
The solvated hydroxide ion in Equation (6.9) is formed in addi-
tion to the hydroxide ions produced during water autoprotolysis,
so there are more hydroxide ions in solution than solvated pro-
tons, yielding excess hydroxide in solution. We say the solution
is alkaline. As an alternative name, we say hydroxide is a base
(see p. 241).
Oils in the skin react readily with the hydroxide ions via the
same chemical process occurring when spray-on oven cleaner 'eats'
into the grime in an oven, reacting principally by the OH~( aq )
ion consuming protons. Let us start, for example, with a molecule
possessing a proton capable of being donated; call it HA, where
'A' is merely an anion of some sort. This proton must be labile.
The hydroxide ion removes this labile proton to generate water,
according to
HA + OH"
(aq)
> A (aq) + H 2
(6.10)
This proton-removing ability characterizes the reactions of hy-
droxide ions in aqueous solutions, and of bases in general. We
Care: It
is
a common
mistake
to
call the
Ol-T ion
a
'hydroxyl'.
It is not:
a
hydroxyl is
correctly
a
covalently
bound -
DH
group, for
example
in
an alcohol.
Fullers' earth is a type
of clay named after a
fuller, whose job was to
clean cloth, e.g. strip-
ping wool of its grease.
Fullers' earth removes
oils and grease from
cloth because of its
alkalinity, just like an
oven cleaner solution.
The Lowry-Br0nsted
theory says a base is a
proton remover.
240
ACIDS AND BASES
A basic chemical con-
sumes protons.
go further by defining hydroxide as a base because it reacts with
(i.e. consumes) labile protons. And any chemical capable of remov-
ing protons is said to be basic.
Saponification
Aside
The word 'saponify'
comes from the Latin
sapo, meaning soap.
Hydroxide ions react to split ('hydrolyse') natural esters
in the skin to form glycerol (II) and palmitic or stearic
acid - a reaction called saponification. Palmitic and stea-
ric acids subsequently react with the base to form the
respective long-chain carboxy late anions - which is soap.
H
Q
H2C j CH2
OH ° H OH
(II)
These cleansing properties of bases were appreciated in antiquity. For example, in a
portion of the Bible probably written in about 1200 BC, a character called Job declares his
desire to be clean, saying, 'If I washed myself with soap and snow, and my hands with
washing soda . . .' (snow was thought to be especially pure and soda (Na2CC>3 • IOH2O)
is alkaline and has long been used as a soap). This quote may be found in full in the
Bible, see Job 9:30.
The Jewish Prophet Jeremiah writing in about 700 BC says much the same thing:
look at Jeremiah 2:21-23 in the Hebrew Bible.
Why is aqueous ammonia alkaline'-.
Lowry-Br0nsted bases
All aqueous solutions naturally contain hydroxide ions in consequence of the auto-
protolytic reaction in Equation (6.2). As we have seen, there will be equal numbers
of solvated protons and solvated hydroxide ions unless we add an acid or base to
it. A solution containing more solvated protons than hydroxide ions is said to be an
'acid' within the Lowry-Br0nsted theory, and a solution comprising more hydroxide
ions than solvated protons is said to be a base.
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
241
But a word of caution: species other than metal hydroxides can
act as bases. Ammonia is such an example, since it can abstract
protons in aqueous solution according to
NH
3(aq)
+ H 2 ► NH"
4(aq)
+ OH"
(aq)
(6.11)
We say a proton is
abstracted when re-
moved selectively.
Similarly, we call a
selective summary or
precis of a piece of
prose 'an abstract'.
To abstract a proton is to remove only the proton. The substan-
tial extent of dissociation in Equation (6.11) helps explain why
'aqueous ammonia' is more properly called 'ammonium hydrox-
ide', NH4OH. We generate the solvated hydroxide ion OH~( aq ) by abstracting a proton
from water. The OH~( aq ) ion in Equation (6.1 1) is chemically and physically identical
to the solvated hydroxide ion generated by dissolving NaOH or KOH in water.
Why is there no vinegar in crisps of salt
and vinegar flavour?
Conjugate acids and bases
Potato crisps come in many flavours, perhaps the most popular being 'salt and vine-
gar'. Curiously, a quick glance at the packet's list of ingredients reveals how the
crisps contain unhealthy amounts of salt, but no vinegar (ethanoic acid) at all. In fact,
the manufacturer dusts the crisps with powdered sodium ethanoate (NaCC^CHs),
because 'real' vinegar would soon make the crisps limp and soggy. Inside the mouth,
acid from the saliva reacts with the ethanoate anion to form ethanoic acid:
CH 3 CO- aq) +H 3 0^
(aq)
-> CH 3 C0 2 H (aq) + H 2
(6.12)
This reaction proceeds inside the mouth, rapidly reaching its position of equilibrium
and allowing the ethanoic acid to impart its distinctive vinegary flavour.
The solvated proton on the left of Equation (6.12) acts as an
acid, since it donates a proton at the same time as the ethanoate
ion behaves as a base, because it accepts a proton. To complicate
the situation, the reaction is one half of a dynamic equilibrium,
i.e. it proceeds in both the forward and backward directions. In the
backward direction, we notice how this time the ethanoic acid acts
as an acid and the water acts as a base.
The reaction in Equation (6.12) illustrates the coexistence of two
acids and two bases. We say the ethanoate ion and ethanoic acid
represent a conjugate pair, and the solvated proton and the water
form a second conjugate pair. Within the ethanoic-ethanoate pair,
the ethanoic acid is the conjugate acid and the ethanoate anion
is the conjugate base. Similarly, H3O" 1 " is a conjugate acid to the
The word 'conjugate'
comes from the Latin
conjugare, meaning
'to yoke together'
(the prefix con means
'together' and jugare
is 'to yoke'). Simi-
larly, the English word
'conjugal' relates to
marriage and concerns
the joining of husband
and wife.
242
ACIDS AND BASES
conjugate base of H2O. Other examples of conjugate acid-base pairs include nitric
acid and nitrate ion, and ammonium ion and ammonia (the acid being cited first in
each case).
We must treat with caution one further aspect of the Br0nsted theory: multiple
proton-donation reactions. Consider the example of the bicarbonate ion HC03~ in
water. When titrating bicarbonate with a base such as hydroxide, the ion behaves as
an acid to form the carbonate anion and water:
A substance like bicar-
bonate, which can react
as either an acid or as
a base, is said to be
amphoteric. The word
comes from the Greek
amphoteros, meaning
'both'.
HCO3" + OH"
C0 3 2 " + H 2
(6.13)
But, conversely, when titrating ions with an acid, the bicarbonate
behaves as a base, losing its proton to form carbonic acid:
HC0 3 "+H 3 H
-> H2CO3 + H 2
(6.14)
We see how the same ion acts as an acid or as a base, depending
on the other reagents in solution. We say the bicarbonate ion is
amphoteric, since it reacts either as an acid or as a base.
SAQ 6.1 Consider the following pairs, and for each decide which is
the conjugate acid and which the base: (a) carbonate and bicarbonate;
(b) H 2 EDTA 2 - and H3EDTA ; (c) HN0 2 and N0 2 .
Aside
Related models of acids and bases
The concept of acid and base can be generalized in several ways. In liquid ammonia, for
example, the ammonium and amide ions (NH4 + and NH2~ respectively) coexist. The
roles of these ions are directly comparable with H30 + and OH~ in water. In ammonia,
the species NH4CI and NaNH2 can be considered to be the respective acid and base
conjugates, just as HC1 and NaOH are an acid-base pair in water. This solvent-based
classification of acids and bases derived from Franklin, in 1905. His ideas are worth
careful thought, although we no longer use his terminology.
Br0nsted's definition of acids and bases (see p. 234 and 240) emphasizes the com-
plementary nature of acids and bases, but it is broader than Franklin's model because
it does not require a solvent, and can even be applied to gas-phase reactions, e.g.
HCl( g ) + NH :
3(g)
NH 4 C1 (S) .
How did soldiers avoid chlorine gas poisoning
at the Second Battle of Ypres?
Neutralization reactions with acids and bases
The bloody Second Battle of Ypres was fought in France on 22 April 1915, and was
the first time in modern warfare when poison gases were employed. At a crucial
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
243
stage in the battle, the German forces filled the air above the enemy trenches with
chlorine gas.
Elemental chlorine CI2 dissolves slightly in water, and hydrolyses
some of the water to yield hypochlorous acid, HOC1, according to
Cl2(aq) + H2O
(1)
> HCbaal + HOC1
l(aq)
(aq)
(6.15)
The reaction in Equation (6.15) occurs readily in the lungs and
eyes (the sensitive tissues of which are lined with water) to cause
irreparable damage. Troops exposed to chlorine apparently experi-
enced a particularly slow and nasty death.
The German troops did not advance, because they were not sure
if the gas masks issued to their own troops could withstand the
chlorine. They were also deterred by the incursion of a Canadian
regiment. But one of the young Canadian soldiers knew a little
chemistry: sniffing the gas, he guessed its identity correctly, and
ordered the soldiers to cover their faces with handkerchiefs (or
bandages) soaked in their own urine. The idea spread quickly, and
the Canadians, together with two Yorkshire territorial battalions,
were able to push back the German troops.
One of the major constituents of urine is the di-amine, urea
(III). Each amine group in urea should remind us of ammonia in
Equation (6.11). Solutions of urea in water are basic because the
two amine moieties each abstract a proton from water, to generate
an ammonium salt and a hydroxide ion:
O
H?N
NHc
(III)
Hypochlorous acid,
HOCI, is one of the
active components in
household bleach.
After this battle, both
sides showed reluc-
tance to employ poi-
sonous gases again,
being afraid it would
drift back and poi-
son their own troops.
Cl 2 gas also caused
extensive corrosion
of rifles and artillery
breech blocks, making
them unusable.
Reminder, to a chem-
ist, the word basic does
not mean 'elementary'
or 'fundamental', but
'proton abstracting'.
O
H,N
+ 2H 2
NH 2
H^N
O
©
NH 3
20H"
(6.16)
The two OH~ ions formed during Equation (6.16) explain why aqueous solutions of
urea are alkaline.
As we saw above, chlorine forms hypochlorous acid, HOCI. The hydroxide ions
generated from urea react with the hypochlorous acid in a typical acid-base reaction,
244
ACIDS AND BASES
to form a salt and water:
HOCl( a q) + OH ( aq )
CIO"
(aq)
+ H 2
(6.17)
where C10~ is the hypochlorite ion.
Equation (6.17) is an example of a neutralization reaction, a topic we discuss in
more depth in Section 6.3.
How is sherbet made?
Effervescence and reactions of acids
Malic acid (IV) occurs
naturally, and is the
cause of the sharp taste
in over-ripe apples.
Sherbet and sweets yielding a fizzy sensation in the mouth gen-
erally contain two components, an acid and a simple carbonate
or bicarbonate. A typical reaction of an acid with a carbonate is
effervescence: the generation of gaseous carbon dioxide. In a well-
known brand of British 'fizzy lolly', the base is sodium bicarbonate
and the acid is malic acid (IV). Ascorbic acid (vitamin C) is another
common acid included within sherbet.
HO
X H
HOOC
CH 2 COOH
(IV)
The acid and the bicarbonate dissolve in saliva as soon as the 'fizzy lolly' is placed
in the mouth. If we abbreviate the malic acid to HM (M being the maliate anion),
the 'fizzing' reaction in the mouth is described by
HM (aq) + HCO
3(aq)
> M" (aq) + C0 2(g) f +H 2
M)
(6.18)
where in this case the subscript 'aq' means aqueous saliva. The subjective sensation
of 'fizz' derives from the evolution of gaseous carbon dioxide.
This reaction between an acid and a carbonate is one of the oldest
chemical reactions known to man. For example, it says in a portion
of the Hebrew Bible written about 1000 bc, '. . . like vinegar poured
on soda, is one who sings songs to a heavy heart', i.e. inappropriate
and lighthearted singing can lead to a dramatic response ! The quote
may be found in full in the book of Proverbs 25:20.
Soda is an
old fash-
ioned name for sodium
carbonate,
Na 2 C0 3 -
10H 2 O.
Ossified means 'to
make rigid and hard.'
The word comes from
the Latin ossis, mean-
ing bone.
Why do steps made of limestone
sometimes feel slippery?
The typical reactions of an alkali
Limestone is an ossified form of calcium carbonate, CaC03. Lime-
stone surfaces soon become slippery and can be quite dangerous if
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
245
Table 6.3 Typical properties of Lowry-Br0nsted bases
Base property
Example from everyday life
Bases react with an acid to form a salt and
water ('neutralization')
Bases react with esters to form an alcohol
and carboxylic acid ('saponification')
Bases can be corrosive
Rubbing a dock leaf (which contains an
organic base) on the site of a nettle sting
(which contains acid) will neutralize the
acid and relieve the pain
Aqueous solutions of base feel 'soapy' to the
touch
Oven cleaner comprising caustic soda
(NaOH) can cause severe burns to the skin
pools of stagnant rainwater collect. Although calcium carbonate is essentially insoluble
in water, minute amounts do dissolve to form a dilute solution, which is alkaline.
The alkali in these water pools reacts with organic matter such as algae and moss
growing on the stone. The most common of these reactions is saponification (see
p. 240), which causes naturally occurring esters to split, to form the respective car-
boxylic acid and an alcohol. Once formed, this carboxylic acid
reacts with more alkaline rainwater to form a metal carboxylate,
according to
2RCOOH (aa) + CaCO
>3(aq)
,2+
> Ca z+ (RCOO-) 2(aq)
+ C0 2 ( aq ) + H 2
(6.19)
The carbon dioxide generated by Equation (6.19) generally remains
in solution as carbonic acid, although the rainwater can look a little
cloudy because minute bubbles form.
Like most other metal carboxylates, the calcium carboxylate
(Ca(RCOO) 2 ) formed during Equation (6.19) readily forms a
'soap', the name arising since its aqueous solutions feel slippery
and soapy to the touch. The other commonly encountered metal
carboxylates are the major components of household soap, which
is typically a mixture of potassium stearate and potassium palmitate
(the salts of stearic and palmitic acids).
In summary, steps of limestone become slippery because the
stagnant water on their surface is alkaline, thereby generating a
solution of an organic soap. Other typical properties of bases and
alkalis are listed in Table 6.3.
Metal carboxylates are
called soaps because
they saponify oils in
the skin and decrease
the surface tension y of
water, which makes the
surfaces more slippery.
The serial TV programs
known as 'soap operas'
earned their name in
the USA at a time when
much of a program-
maker's funding came
from adverts for house-
hold soap.
Why is the acid in a car battery more corrosive
than vinegar?
pH
Car batteries generally contain sulphuric acid at a concentration of about 10 mol dm -3 .
It is extremely corrosive, and can generate horrific chemical burns. By contrast, the
246
ACIDS AND BASES
Care: the 'H' in pH
derives from the sym-
bol for hydrogen, and
is always given a big
letter. The 'p' is a math-
ematical operator, and
is always small.
concentration of the solvated protons in vinegar lies in the range
10- 4 -10" 5 mol dm" 3 .
Between these two acids, there is up to a million-fold differ-
ence in the number of solvated protons per litre. We cannot cope
with the unwieldy magnitude of this difference and tend to talk
instead in terms of the logarithm of the concentration. To this end,
we introduce a new concept: the pH. This is defined mathemati-
cally as 'minus the logarithm (to the base ten) of the hydrogen ion
concentration' :
An acid's pH is defined
as minus the loga-
rithm (to the base ten)
of the hydrogen ion
concentration.
pH = -log 10 [H+/mol dm" 3 ]
(6.20)
The 'p' in Equation
(6.20) is the mathe-
matical operator
'-log 10 ' of something.
pH means we have
applied the operator 'p'
to [H + ]. The p is short
for potenz, German for
power.
The concentrations of bench acids in an undergraduate labora-
tory are generally less than 1 mol dm~ , so by corollary the minus
sign to Equation (6.20) suggests we generally work with positive
values of pH. Only if the solution has a concentration greater than
1 mol dm -3 will the pH be negative. Contrary to popular belief,
a negative pH is not impossible. (Try inserting a concentration of
2.0 mol dm -3 into Equation (6.20) and see what happens!)
Notice how we generally infer the solvated proton, H3O" 1 ", each
time we write a concentration as [H + ], which helps explain why
the concept of pH is rarely useful when considering acids dis-
solved in non-aqueous solvents. When comparing the battery acid
with the bench acid, we say that the battery acid has a lower pH
than does the bench acid, because the number of solvated protons
is greater and, therefore, it is more acidic. Figure 6.1 shows the
relationship between the concentration of the solvated protons and
pH. We now appreciate why the pH increases as the concentration
decreases.
Apart from the convenience of the logarithmically compressed
scale, the concept of pH remains popular because one of the most
popular methods of measuring the acidity of an aqueous solution is the glass electrode
(see p. 336), the measurement of which is directly proportional to pH, rather than to
[H 3 0+].
We need to introduce a word of caution. Most modern calculators cite an answer
with as many as ten significant figures, but we do not know the concentration to more
than two or three significant figures. In a related way, we note how the pH of blood
is routinely measured to within 0.001 of a pH unit, but most chemical applications
The lower the pH, the
more concentrated the
acid.
[H 3 + ]/mol dm" 3 10 1 1CT 1 1CT 2 10 -3 . . . 1CT 7 1CT 10 1Cr 11
pH -10 1 2 3 ... 7 ... 10 11
Figure 6.1 The relationship between concentrations of strong acids and the solution pH
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
247
do not require us to cite the pH to more than 0.05 of a pH unit. In fact, we can rarely
cite pH to a greater precision than 0.01 for most biological applications.
Worked Example 6.2 What is the pH of bench nitric acid having a
concentration of 0.25 moldnT 3 ?
Inserting values into Equation (6.20):
pH = -log[0.25 moldm~ 3 /mol dm" 3 ] = 0.6
The acid has a pH of 0.6.
SAQ 6.2 What is the pH of hydrochloric acid having a
concentration of 0.2 moldrrT 3 ?
SAQ 6.3 To highlight the point made above concerning
numbers of significant figures, determine the (negligible)
difference in [H 3 + ] between two acid solutions, one hav-
ing a pH of 6.31 and the other 6.32.
Sometimes we know the pH and wish to know the concentration
of the solvated protons. Hence, we need to rewrite Equation (6.20),
making [H + ] the subject, to obtain
The concentration of
the solvated protons in
Equation (6.18) needs
to be expressed in the
familiar (but non-SI)
units of mol dm 3 ; the
SI unit of concentration
is mol rrr 3 .
We divide the concen-
tration by its units to
yield a dimensionless
number.
[H+/mol dm" 3 ] = 10 _pH
(6.21)
We are assuming the
concentration of the
proton H + is the same
as the concentration of
H3O ( aq).
Worked Example 6.3 What is the concentration of nitric acid having a pH of 3.5?
Inserting values into Equation (6.21):
[H+] = 10
-3.5
so
[H+] = 3.16 x 10~ 4 mol dm" 3
SAQ 6.4 What is the concentration of nitric acid of pH = 2.2?
Occasionally, we can merely look at a pH and say straightaway
what is its concentration. If the pH is a whole number - call it
x - the concentration will take the form 1 x 10 - * mol dm -3 . As
examples, if the pH is 6, the concentration is 10~ 6 mol dm -3 ; if
the concentration is 10~ 3 mol dm -3 then the pH is 3, and so on.
Sometimes, we refer to
a whole number as an
integer.
Worked Example 6.4 Without using a calculator, what is the concentration of HNO3
solution if its pH is 4?
248 ACIDS AND BASES
If the pH is x, then the concentration will be lO - * moldirT , so the acid concentration
is 10~ 4 moldirT 3 .
SAQ 6.5 Without using a calculator, what is the pH of hydrochloric acid
of concentration 10~ 5 moldm" 3 ?
Aside
Units
We encounter problems when it becomes necessary to take the logarithm of a con-
centration (which has units), since it contravenes one of the laws of mathematics. To
overcome this problem, we implicitly employ a 'dodge' by rewriting the equation as
, .'[H 3 + ]
pH = - log 10
where the term c e is the standard state, which generally has the value of 1 moldm -3 .
The c° term is introduced merely to allow the units within the bracketed term above
to cancel. Throughout this chapter, concentrations will be employed relative to this
standard, thereby obviating the problem inherent with concentrations having units.
S0renson introduced
this definition of pH in
1909.
Justification Box 6.1
The definition of pH is given in Equation (6.20) as
P H=-log 10 [H+]
First we multiply both sides by '— 1':
- P H = log 10 [H+]
and then take the antilog, to expose the concentration term. Correctly, the function
'antilog' means a mathematical operator, which performs the opposite job to the original
function. The opposite function to log is a type of exponential. As the log is written in
base 10, so the exponent must also be in base 10.
If y = log| JC, then x = 10 > . This way we obtain Equation (6.21).
Sometimes we need to know the pH of basic solutions.
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
249
PK W =
-log 10 K w
pH =
-log 10 [H 3 O+ q) ]
pOH =
-log 10 [OH- (aq) ]
Worked Example 6.5 What is the pH of a solution of sodium hydroxide of concentra-
tion 0.02 moldirT 3 ? Assume the temperature is 298 K.
At first sight, this problem appears to be identical to those in previ-
ous Worked Examples, but we soon appreciate how it is complicated
because we need first to calculate the concentration of the free protons
before we can convert to a pH. However, if we know the concentration
of the alkali, we can calculate the pH thus:
pK w = pH + pOH (6.22)
where pA" w and pH have their usual definitions, and we define pOH as
pOH=-log 10 [Oir] (6.23)
Inserting the concentration [NaOH] = 0.02 mol dm - into Equation
(6.22) yields the value of pOH = 2. The value of pA" w at 298 K is 14
(see Table 6.2). Therefore:
pH = pK w - pOH
pH = 14 - 2
pH= 12
SAQ 6.6 What is the pH of a solution of potassium hydroxide of concen-
tration 6 x 10 3 mol dm 3 . Again, assume the temperature is 298 K.
We must look up the
value of K w from Table
6.2 if the temperature
differs from 298 K,
and then calculate a
different value using
Equation (6.26).
Justification Box 6.2
Water dissociates to form ions according to Equation (6.2). The ionic
concentrations is the autoprotolysis constant K w , according to Equation
Taking logarithms of Equation (6.4) yields
product
(6.4).
of the
log 10 K w = log 10 [H 3 O + ] + log 10 [OH-]
(6.24)
Next, we multiply each term by '— 1' to yield
- log 10 K w = - log 10 [H 3 O + ] - log 10 [OH-
"J
(6.25)
The term '—
according to
log 10 [H3O + ]' is the solution pH and the term
pK w = - log 10 K w
'- lo gio
K w ' is
defined
(6.26)
We give the
tion (6.22).
name pOH to the third term '— log 10 [OH - ]'. We thereby obtain
Equa-
250
ACIDS AND BASES
Why do equimolar solutions of sulphuric acid
and nitric acid have different pHs?
Mono-, di- and tri- basic acids
Equimolar means 'of
equal molarity', so
equimolar solutions
have the same concen-
tration.
Nitric acid, HNO3, readily dissolves in water, where it dissociates
according to
HN0 3 (a q) +H 2 ► lH 3 0+ (aq) +NO
3(aq)
(6.27)
The stoichiometry illustrates how each formula unit generates a sin-
gle solvated proton. By contrast, sulphuric acid, H2SO4, dissociates
in solution according to
H 2 S0 4( a q ) + 2H 2
2H,0 H
(aq)
+ SO
4(aq)
(6.28)
so each formula unit of sulphuric acid generates two solvated protons. In other words,
each mole of nitric acid generates only 1 mol of solvated protons but each mole of
sulphuric acid generates 2 mol of solvated protons. We say nitric acid is a mono-protic
acid and sulphuric acid is a di-protic acid. Tri-protic acids are rare. Fully protonated
ethylene diamine tetra-acetic acid H4EDTA (V) is a tetra-protic acid.
OH
O
N-CH 2 -CH 2 -N
OH
O
OH
(V)
Equation (6.27) demonstrated how the concentration of the solvated protons equates
to the concentration of a mono-protic acid from which it derived; but, from Equation
(6.28), the concentration of the solvated protons will be twice the concentration if the
parent acid is di-protic. These different stoichiometries affect the pH, as demonstrated
now by Worked Examples 6.6 and 6.7.
Worked Example 6.6 Nitric acid of concentration 0.01 mol dm" is dissolved in water.
What is its pH?
Since one solvated proton is formed per molecule of acid, the concentration [H + ( aq )] is
also 0.01 mol dm -3 .
The pH of this acidic solution is obtained by inserting values into Equation (6.20):
pH = -log 10 [0.01]
pH = 2
PROPERTIES OF LOWRY-B RONS TED ACIDS AND BASES
251
Worked Example 6.7 What is the pH of sulphuric acid having the same concentration
in water as the nitric acid in Worked Example 6.6?
This time, two solvated protons are formed per molecule of acid, so the concentration of
(aq).
will be 0.02 mol dm
The pH of this acidic solution is obtained by inserting values into Equation (6.20):
P H=-log 10 [2x0.01]
pH= 1.68
A pH electrode immersed in turn into these two solutions would
register a different pH despite the concentrations of the parent acids
being the same.
We need to be careful with these calculations, because the extent
of dissociation may also differ; see p. 255 ff.
pH electrodes and pH
meters are discussed in
Chapter 7.
All neutral solutions
have a pH of 7 at
298 K.
What is the pH of a y neutral' solution?
pH and neutrality
A medicine or skin lotion is often described as 'pH neutral' as though it was obvi-
ously a good thing. A solution is defined as neutral if it contains neither an excess
of solvated protons nor an excess of hydroxide ions. Equation (6.4) tells us the auto-
protolysis constant K w of super-pure water (water containing no
additional solute) is 10 -14 (mol dm -3 ) 2 . Furthermore, we saw in
Worked Example 6.1 how the concentration of the solvated protons
was 10" 7 mol dm" 3 at 298 K.
By considering both the definition of pH in Equation (6.20)
and the concentration of the solvated protons from Worked
Example 6.1, we see how a sample of super-pure water - which is
neutral - has a pH of 7 at 298 K. We now go further and say all
neutral solutions have a pH of 7. By corollary, we need to appre-
ciate how an acidic solution always has a pH less than 7. If the pH
is exactly 7, then the solution is neutral.
The pH of a Lowry-Br0nsted acid decreases as its concentration
increases. Bench nitric acid of concentration 1 mol dm -3 has a
pH = 0. An acid of higher concentration will, therefore, have a
negative pH (the occasions when we need to employ such solutions
are, thankfully, rare).
What do we mean when we say blood plasma
has a l pHof7.4'?
The pH of alkaline solutions
Table 6.4 lists the pH of many natural substances, and suggests human blood plasma,
for example, should have a pH in the range 7.3-7.5. The pH of many natural
The
maximum pH
of
an acid
will be just less
than
7,
at 298 K.
The pH of a Lowry-
Brpnsted acid DEcre-
ases as its concentra-
tion INcreases.
252
ACIDS AND BASES
Table 6.4 The pHs of naturally occurring substances, listed in order
of decreasing acidity
Human body
Common foodstuffs
Gastric juices
1.0-3.0
Limes
1.8-2.0
Faeces
4.6-8.4
Rhubarb
3.1-3.2
Duodenal contents
4.8-8.2
Apricots
3.6-4.0
Urine
4.8-8.4
Tomatoes
4.0-4.4
Saliva
6.5-7.5
Spinach
5.1-5.7
Milk
6.6-7.6
Salmon
6.1-6.3
Bile
6.8-7.0
Maple syrup
6.5-7.0
Spinal fluid
7.3-7.5
Tap water
6.5-8.0
Blood plasma
7.3-7.5
Egg white (fresh)
7.6-8.0
The word 'product'
is used here in its
mathematical sense of
'multiplied by'.
Source: Handbook of Chemistry and Physics (66th Edition), R. C. Weast (ed.),
CRC Press, Boca Raton, Florida, 1985, page D-146.
substances is higher than 7, so we cannot call them either 'acidic' or 'neutral'. The
pHs range from 1.8 for limes (which explains why they taste so sour) to 7.8 or so
for fresh egg white (albumen).
Blood 'plasma' is that part of the blood remaining after removal
of the haemoglobin cells that impart a characteristic 'blood-red'
colour. According to Table 6.4, most people's plasma has a pH
in the range 7.3-7.5. So, what is the concentration of solvated
protons in such plasma? We met the autoprotolysis constant A" w
in Equation (6.4). Although we discussed it in terms of super-pure
water, curiously the relationship still applies to any aqueous system. The product of
the concentrations of solvated protons and hydroxide ions is always 10~ 14 at 298 K.
If Equation (6.4) applies although the water contains dissolved solute, then we can
calculate the concentration of solvated protons and the concentration of hydroxide
ions, and hence ascertain what a pH of more than '7' actually means.
Worked Example 6.8 What is the concentration [OH~( aq) ] in blood plasma of pH 7.4?
Answer strategy. (1) We first calculate the concentration of solvated protons from the pH,
via Equation (6.20). (2) Second, we compare the concentration [H30 + ( aq )] with that of a
neutral solution, via Equation (6.4).
(1) The concentration [H 3 + (aq )] is obtained from Equation (6.20). Inserting values:
[H 3 H
(aq)J
10"
so
[H 3 + (aq) ] = 4x 10~ 8 moldm" 3
(2) We see how the concentration obtained in part (1) is in fact less than the 1 x
10~ 7 moldnT we saw for pure water, as calculated in Worked Example 6.1.
'STRONG' AND 'WEAK' ACIDS AND BASES
253
Equation (6.4) says A" w = [H30 + ( aq )] x [OH ( aq )], where the product of the two
concentration has a value of 10~ 14 (moldm~ ) 2 at 298 K. Knowing the values of K w and
[H30 + ( aq )], we can calculate the concentration of hydroxide ions in the blood plasma.
Rearranging to make [OH~( aq) ] the subject, we obtain
Inserting values,
[OH'
K»
(aq)J
[H 3 0+ (aq) ]
3,2
[OH- (aq) ]
10" 14 (mol dm" 3 )
4 x 10" 8 mol dm"
so
[OH" (aq) ] = 2.5 x 1(T 7 mol dm"
Aqueous solutions in
which the concen-
tration of hydroxide
exceeds the concentra-
tion of solvated protons
show a pH higher
than 7.
We see how the pH of blood plasma is higher than 7, so the con-
centration of hydroxide ions exceeds their concentration in super-
pure water. We derive the generalization: aqueous solutions in
which the concentration of hydroxide is greater than the concen-
tration of solvated protons have a pH higher than 7. The pH is
lower than 7 if the concentration of hydroxide is less than the
concentration [H30 + ( aq )].
SAQ 6.7 A solution of ammonia in water has a pH of 9. Without using a
calculator, what is the concentration of solvated protons and hence what
is the concentration of hydroxide ions?
SAQ 6.8 What is the pH of sodium hydroxide solution of concentration
lO" 2 moldrrr 3 ?
6.2 "Strong' and "weak' acids and bases
Why is a nettle sting more painful than a bur
from ethanoic acid?
Introducing 'strong' and 'weak' acids
Brushing against a common nettle Urtica dioica can cause a pain-
ful sting. The active component in a nettle sting is methanoic acid
(VI), also called 'formic acid'. The sting of a nettle also contains
natural additives to ensure that the methanoic acid stays on the
skin, thereby maximizing the damage to its sensitive underlying
tissue known as the epidermis.
The sting of a red ant
also contains meth-
anoic acid.
254
ACIDS AND BASES
The word 'epidermis'
derives from two Greek
words, derma meaning
skin, and epi meaning
'at', 'at the base of, or
'in additional to'. The
same root epi occurs
in 'epidural', a form of
pain relief in which an
injection is made at
the base of the dura,
located in the spine.
We say an acid is
strong if the extent
of its ionization is
high, and weak if the
extent of its ionization
is small.
Care: do not confuse
the words strong and
weak acids with every-
day usage, where we
usually say something
is 'strong' if its con-
centration is large, and
'weak' if its concentra-
tion is small.
H
O-H
(VI)
The chemical structures of I and VI reveal the strong similarities
between ethanoic and methanoic acids, yet the smaller molecule is
considerably nastier to the skin. Why? Methanoic acid dissociates
in water to form the solvated methanoate anion HCOO
(aq)
and
a solvated proton in a directly analogous fashion to ethanoic acid
dissolving in water; Equation (6.1). In methanoic acid of concen-
tration 0.01 moldm -3 , about 0.14 per cent of the molecules have
dissociated to yield a solvated proton. By contrast, in ethanoic acid
of the same concentration, only 0.04 per cent of the molecules
have dissociated. We say the methanoic acid is a stronger acid
than ethanoic since it yields more protons per mole. Conversely,
ethanoic acid is weaker.
We might rephrase this statement, and say an acid is strong if its
extent of ionization is high, and weak if the extent of ionization is
small. Within this latter definition, both I and VI are weak acids.
In summary, the word 'acid' is better applied to methanoic acid
than to ethanoic acid, since it is more acidic, and so methanoic acid
in a nettle sting is more able to damage the skin than the ethanoic
acid in vinegar.
But we need to be careful. In everyday usage, we say often some-
thing is 'strong' when we mean its concentration is large; similarly,
we say something is 'weak' if its concentration is small. As a good
example, when a strong cup of tea has a dark brown colour (because
the compounds imparting a colour are concentrated) we say the tea
is 'strong'. To a chemist, the words 'strong' and 'weak' relate only
to the extent of ionic dissociation.
To a chemist, the
words 'strong' and
'weak' relate only to
the extent of ionic dis
sociation.
Why is 'carbolic acid' not in fact an acid?
Acidity constants
'Carbolic acid' is the old-fashioned name for hydroxybenzene
(VII), otherwise known as phenol. It was first used as an anti-
septic to prevent the infection of post-operative wounds. The British surgeon Joseph
(later 'Lord') Lister (1827-1912) discovered these antiseptic qualities in 1867 while
working as Professor of Medicine at the Glasgow Royal Infirmary. He squirted a
'STRONG' AND 'WEAK' ACIDS AND BASES
255
dilute aqueous solution of VII directly onto a post-operative wound
and found that the phenol killed all the bacteria, thereby yielding
the first reliable antiseptic in an era when medical science was in
its infancy.
(VII)
The antibacterial properties of VII are no longer utilized in
modern hospitals because more potent antiseptics have now been
formulated. But its memory persists in the continued use of 'car-
bolic soap', which contains small amounts of phenol.
Phenol in water is relatively reactive, thereby explaining its
potency against bacteria. But phenol dissolved in water contains
relatively few solvated protons, so it is not particularly acidic. But
its old name is carbolic acid\
Phenol (VII) can dissociate according to
PhOH (aq) + H 2
(i)
->PhO- (aq) + H 3 4
(aq)
(6.29)
where Ph is a phenyl ring and PhO ( aq ) is the phenolate anion. An
equilibrium constant may be written to describe this reaction:
K =
[PhO"(a q )][H 3 + (aq) ]
[PhOH (aq) ][H 2 0]
(6.30)
In fact, the water term in the denominator remains essentially con-
stant, since it is always huge compared with all the other terms.
Accordingly, we usually write a slightly altered version of K, cross-
multiplying both sides of the equation with the concentration of
water to yield
[PhO- (aq) ][H 3 + (aq) ]
^a
[PhOH (aq) ]
(6.31)
The resultant (modified) equilibrium constant is called the acidity
constant of phenol, and has the new symbol K a , which has a value
is 10~ 10 for phenol. K & is also called the acid constant, the acid
dissociation constant or just the dissociation constant. The value
of K a for phenol is clearly tiny, and quantifies just how small the
extent is to which it dissociates to form a solvated proton.
Phenol is a rare exam-
ple of a stable enol
(pronounced 'ene-ol'),
with a hydroxyl bonded
to a C=C bond. Most
enols tautomerize to
form a ketone.
The word 'antiseptic'
comes from the Latin
prefix anti meaning
'before' or 'against',
and 'septic' comes
from the Latin sep-
tis, meaning a bacterial
infection. An 'antisep-
tic', therefore, prevents
the processes or sub-
stances causing an
infection.
Historically, carbolic
acid was so called
because solid phenol
causes nasty chemical
burns to the skin. The
root carbo comes from
the French for 'coal'.
Care: Do not con-
fuse 'Ph' (a common
abbreviation for a
phenyl ring) with 'pH'
(which is a mathemat-
ical operator meaning
-log 10 [H + (aq) ]).
K a for phenol is 10 10
when expressing the
concentrations with the
units of mol dm 3 .
256
ACIDS AND BASES
A strong acid has
a
large value
of K a ,
and
a weak acid
has a
low
value of K a .
The value of K a for ethanoic acid is a hundred thousand times larger at 1.8 x 10~ 5 ,
and K a for methanoic acid is ten times larger still, at 1.8 x 10~ 4 ; so methanoic acid
generates more solvated protons per mole of acid than either phenol or ethanoic acid.
We discover the relative differences in K a when walking in the country, for a nettle
can give a nasty sting (i.e. a chemical burn) but vinegar does not burn the skin. We
say methanoic acid is a stronger acid than ethanoic acid because its value of K a is
larger. A mole of phenol yields few protons, so we say it is a weak acid, because its
value of K a is tiny.
These descriptions of 'strong' and 'weak' acid are no longer
subjective, but depend on the magnitude of K a : a strong acid has
a large value of K a and a weak acid has a low value of K a . Stated
another way, the position of the acid-dissociation equilibrium lies
close to the reactants for a weak acid but close to the products for
a strong acid, as shown schematically in Figure 6.2.
Carboxylic acids such as ethanoic acid are generally weak be-
cause their values of K a are small (although see p. 261). By con-
trast, so called mineral acids such as sulphuric or nitric are classed
as strong because their respective values of K a are large. Although
there is little consensus, a simplistic rule suggests we class an acid
as weak if its value of K a drops below about 10~ 3 . The acid is
strong if K a > 10~ 3 .
Table 6.5 contains a selection of K a values. Acids characterized
by large values of K a are stronger than those with smaller values of
K a . Each K a value in Table 6.5 was obtained at 298 K. Being an
equilibrium constant, we anticipate temperature-dependent values
of K a , with K a generally increasing slightly as T increases.
A crude generalization
suggests that inorganic
acids are strong and
organic acids are weak.
The values of K a gen-
erally increase with
increasing tempera-
ture, causing the acid to
be stronger at high T.
C3
Reactants, i.e. HA + H 2 Products, i.e. H 3 + + A"
Extent of reaction f
Figure 6.2 Graph of Gibbs function G (as 'y') against the extent of reaction % (as 'x'). The
minimum of the graph corresponds to the position of equilibrium: the position of equilibrium for
a weak acid, such as ethanoic acid, lies near the un-ionized reactants; the position of equilibrium
for a strong acid, like sulphuric acid, lies near the ionized products
'STRONG' AND 'WEAK' ACIDS AND BASES
257
Table 6.5 Acidity ('dissociation') constants K a for inorganic Lowry-Br0nsted acids in water
at 298 K. Values of K & are dimensionless: all values presuppose equilibrium constants such as
Equation (6.35), and were calculated with concentrations expressed in mol dnT 3
Acid
K
a(l)
K
a(2)
K
a(3)
Hypochlorous, HOC1
Hydrochloric, HC1
Nitrous, HN0 2
Sulphuric, H2SO4
Sulphurous, H2SO3
Carbonic, H 2 C0 3
Phosphoric, H3PO4
4.0 x 10~ 8
1.0 x 10 7
4.6 x 10~ 4
1.0 x 10 2
1.4 x 10~ 2
4.3 x 10~ 7
7.53 x 10~ 3
1.2 x 10~ 2
1.02 x 10~ 7
5.61 x 10-"
6.23 x 10~ 8
2.2 x 10-
Table 6.6 As for Table 6.5, but for inorganic acids
and showing the effects of various structural changes
Acid
10 5 K,
Effect of extent of halogenation
CH3COOH
1.75
CICH2COOH
136
CI2CHCOOH
5530
CI3COOH
23 200
Effect of halide
FCH2COOH
260
CICH2COOH
136
BrCH 2 COOH
125
ICH2COOH
67
Effect of chain length
HCOOH
17.7
CH3COOH
1.75
CH3CH2COOH
1.35
CH3CH2CH2COOH
1.51
Effect of substituent in benzoic acids
C 6 H 5 COOH
6.3
p-N0 2 -C 6 H 4 COOH
36.0
p-CH 3 0-C 6 H 4 COOH
3.3
p-NH 2 -C 6 H 4 COOH
1.4
In summary, carbolic acid (phenol, VII) is an extremely weak
acid because its value of K a is 10~ 10 , quantifying how small is the
concentration of solvated protons in its solutions.
Basicity constants
Having categorized acids into 'strong' and 'weak' via the concept
of acidity constants K a , we now look at the strengths of various
The cause of phenol's
corrosive properties
does not relate to its
ability to form solvated
protons (as indicated
by the value of K a ) but
its ability to penetrate
the skin and disrupt
the chemical processes
occurring within the
epidermis, to painful
effect.
258
ACIDS AND BASES
bases. It is possible to write an equilibrium constant K to describe the hydrolysis of
bases such as ammonia (see Equation (6.12)). We write the appropriate equilibrium
constant in just the same way as we wrote an expression for K to describe the acidic
behaviour of phenol:
[NH 4 +][OH-]
(6.32)
We sometimes call the
equilibrium constant
in Equation (6.33) a
basicity constant, and
symbolize it as K b .
K =
[NH 3 ][H 2 0]
As with the expression in Equation (6.6), this equilibrium constant
can be simplified by incorporating the water term into K, thereby
yielding a new constant which we will call ^b, the basicity con-
stant:
[NH^HOH-l
[NH 3 ]
where K\, in Equation (6.33) is quite different from the K in Equation (6.32). The
value of K\, for ammonia is 1.74 x 10~ 5 , which is quite small, causing us to say
ammonia is a weak base. The value of K b for sodium hydroxide is much larger at
0.6, so we say NaOH is a strong base.
But, curiously, this new equilibrium constant K\, is redundant because we could
have calculated its value from known values of K a according to
K„ x Kh = K„
(6.34)
where K w is the autoprotolysis constant of water from p. 236. Older textbooks some-
times cite values of K b , but we really do not need to employ two separate K constants.
SAQ 6.9 What is the value of K a for the ammonium ion, NH 4 +? Take K^
from the paragraphs immediately above, and K w = 10~ 14 .
Justification Box 6.3
Consider a weak acid, HA
, dissociating: HA — » H 3 + + A~
. Its
acidity constant A" a is
given by
K,
[A"][H 3 + ]
[HA]
(6.35)
and then consider a weak base (the
conjugate of the weak
acid)
forming a
hydroxide
ion in solution, H2O + A~
-> OH"
+ HA. Its basicity constant is given by
K b
[HA][OH"]
[A"]
(6.36)
'STRONG' AND 'WEAK' ACIDS AND BASES 259
Multiplying the
expressions for K a and K\, yields
K*XK h
[A-][H 3 0+]
[HA]
[HA][OH"
[A-]
The HA and A"
terms clearly
cancel to yield [H:
+ ][OH"],
which is
K w .
Why does carbonic acid behave as a mono-protic acid?
Variations in the value of K a
Carbonic acid, H2CO3, is naturally occurring, and forms when carbon dioxide from
the air dissolves in water. From its formula, we expect it to be a di-protic acid, but
it is generally classed as mono-protic. Why?
In water at 298 K, the ionization reaction follows the equation
H2C03( aq ) + H 2
HC0 3 - (aq) +H 3 0^
(aq)
(6.37)
The value of K a for the reaction in Equation (6.37) is 4.3 x 10~ 7 , so carbonic acid is
certainly a very weak acid. The hydrogen carbonate anion HC03~ could dissociate
further, according to
HCO"
3(aq)
+ H 2 ► CO^ aq) + H 3
(aq)
(6.38)
but its value of K a is low at 5.6 x 10 -11 , so we conclude that the HCO^~ ion is too
weak an acid to shed its proton under normal conditions. Thus, carbonic acid has two
protons: the loss of the first one is relatively easy, but the proportion of molecules
losing both protons is truly minute. Only one of the protons is labile.
This situation is relatively common. If we look, for example, at the values of
K a in Table 6.5, we see that phosphoric acid is a strong acid insofar as the loss of
the first proton occurs with K a = 7.5 x 10~ 3 , but the loss of the second proton, to
form HP0 4 ~, is difficult, as characterized by K a = 6.2 x 10~ 8 . In other words, the
dihydrogen phosphate anion H2PO4 is a very weak acid. And the hydrogen phosphate
di-anion HP0 4 ~ has a low a value of K a = 2.2 x 10 -11 , causing us to say that the
P0 4 ~ anion does not normally exist. Even the loss of the second proton of sulphuric
acid is characterized by a modest value of K a = 10~ 2 .
This formal definition of K a can be extended to multi-protic
acids. We consider the dissociation to occur in a step-wise manner,
the acid losing one proton at a time. Consider, for example, the
two-proton donation reactions of sulphuric acid:
(1)
(2)
H2SO4
HSCV
> H + + HSO4
■* H+ + SO* _
Multi-protic acids have
a different value of K a
for each proton dona-
tion step, with the
values of K a decreas-
ing with each proton
donation step.
260
ACIDS AND BASES
The subscript (1) tells
us we are considering
the first proton to be
lost.
The equilibrium constant for the dissociation of H2SO4:
[HS0 4 -][H+]
K.
a(l)
Similarly,
Ka ( i) is always bigger
than K a(2) .
K.
a(2)
[H 2 S0 4 ]
[SOf][H+]
[HSO4-]
In fact, we can also extend this treatment to bases, looking at
the step-base addition of protons.
SAQ 6.10 The tetra-protic acid H 4 EDTA (V) has four possible proton
equilibrium constants. Write an expression for each, for K a{1) to K a(4) .
Why is an organic acid such as trichloroethanoic acid
so strong?
Effect of structure on the K a of a weak acid
The value of K a for trichloroacetic acid CCI3COOH (VIII) is very large at 0.23.
Indeed, it is stronger as an acid than the HSO4 ion - quite remarkable for an organic
acid!
CI
CI
^
J
CI OH
(VIII)
Let us return to the example of ethanoic acid (I). The principal structural difference
between I and VIII is the way we replace each of the three methyl protons in ethanoic
acid with chlorine atoms.
The three methyl protons in I are slightly electropositive, implying that the central
carbon of the -CH3 group bears a slight negative charge. This excess charge is not
large, but it is sufficient to disrupt the position of the acid-dissociation equilibrium, as
follows. Although the undissociated acid has no formal charge, the ethanoate anion
has a full negative charge, which is located principally on the carboxyl end of the
anion. It might be easier to think of this negative charge residing on just one of the
oxygen atoms within the anion, but in fact both oxygen atoms and
the central carbon each bear some of the charge. We say the charge
is delocalized, according to structure IX, which is a more accurate
representation of the carboxylate anion than merely -COO~. The
Derealization is a
means of stabilizing
an ion.
TITRATION ANALYSES
261
right-hand structure of IX is effectively a mixture of the two structures to its left.
Note how the name resonance implies charge derealization.
O
O
O
(IX)
o
o
A double-headed arrow
'+*' indicates reso-
nance.
Ions are more likely to
form if they are stable,
and less likely to form
if unstable.
To reiterate, the hydrogen atoms in the methyl group are slightly electropositive,
with each seeking to relocate their own small amounts of charge onto the central
carboxyl carbon. In consequence, the ethanoic anion (cf. structure
IX with R = CH3) has a central carbon bearing a larger negative
charge - both from the ionization reaction but also from the hydro-
gen atoms of the -CH3 group. In consequence, the central carbon
of the ethanoate anion is slightly destabilized; and any chemical
species is less likely to form if it is unstable.
Next we look at the structure of trichloroethanoic acid (VIII). In contrast to the
hydrogen atoms of ethanoic acid, the three chlorine atoms are powerfully electron
withdrawing. The chlorine atoms cause extensive derealization of the negative charge
on the C^COO - anion, with most of the negative charge absorbed by the three
chlorine atoms and less on the oxygen atoms of the carboxyl. Such a relocation of
charge stabilizes the anion; and any chemical species is more likely to form if it
is stable.
Statistically, we find fewer ethanoate anions than trichloroethanoate anions in the
respective solutions of the two acids. And if there are fewer ethanoate anions in
solution per mole of ethanoic acid, then there will be fewer solvated protons. In other
words, the extent to which ethanoic acid dissociates is less than the corresponding
extent for trichloroethanoic acid. I is a weak acid and VIII is strong; dipping a simple
pH electrode into a solution of each of the two acids rapidly demonstrates this truth.
This sort of derealization stabilizes the ion; in fact, the ClsCOO - anion is more
stable than the parent molecule, CI3COOH. For this reason, the solvated anion resides
in solution in preference to the acid. K a is therefore large, making trichloroacetic acid
one of the strongest of the common organic acids.
Trifluoroethanoic acid (probably better known as trifluoroocef/c acid, TFA) is
stronger still, with a value of K a = 1.70.
6.3 Titration analyses
iy does a dock leaf bring relief after
a nettle sting?
Introducing titrations
We first met nettle stings on p. 253, where methanoic ('formic')
acid was identified as the active toxin causing the pain. Like its
The common dock leaf,
Rumex obtusifolia, and
the yellow dock leaf,
Rumex crispus, are in
fact equally common.
262
ACIDS AND BASES
The naturally occurring
substance histamine
causes blood capillaries
to dilate and smooth
muscle to contract.
Most cells release it
in response to wound-
ing, allergies, and most
inflammatory condi-
tions. Antihistamines
block the production of
this substance, thereby
combating a painful
swelling.
structurally similar sister, ethanoic acid (I), methanoic acid disso-
ciates in water to yield a solvated proton, H30 + ( aq ).
Rubbing the site of the sting with a crushed dock leaf is a sim-
ple yet rapid way of decreasing the extent of the pain. In common
with many other weeds, the sap of a dock leaf contains a mix-
ture of natural amines (e.g. urea (III) above), as well as natural
antihistamines to help decrease any inflammation. The amines are
solvated and, because the sap is water based, are alkaline. Being
alkaline, these amines react with methanoic acid to yield a neutral
salt, according to
H
O
H
H
:n — R
H
H
O
O"
H-,N — R
(6.39)
where R is the remainder of the amine molecule. We see how the
process of pain removal involves a neutralization process.
Notice how the lone
pair on nitrogen of the
amine attracts a proton
from the carboxylic
acid.
How do indigestion tablets won
Calculations concerning neutralization
Excess acid in the stomach is one of the major causes of indiges-
tion, arising from a difficulty in digesting food. The usual cause
of such indigestion is the stomach simply containing too much hydrochloric acid, or
the stomach acid having too high a concentration (its pH should be about 3). These
failures cause acid to remain even when all the food has been digested fully. The
excess acid is not passive, but tends to digest the lining of the stomach to cause an
ulcer, or reacts by alternative reaction routes, generally resulting in 'wind', the gases
of which principally comprises methane.
Most indigestion tablets are made of aluminium or magnesium
hydroxides. The hydroxide in the tablet removes the excess stom-
ach acid via a simple acid-base neutralization reaction:
Some indigestion
tablets contain chalk
(CaC0 3 ) but the large
volume of C0 2 pro-
duced (cf. Equation
(6.19)) can itself cause
dyspepsia.
3HC1
(stomach)
+ Al(OH) 3 (table t ) ► 3H 2 + A1C1 3
(aq)
(6.40)
The cause of the indigestion is removed because the acid is con-
sumed. Solid (unreacted) aluminium hydroxide is relatively insolu-
ble in the gut, and does not dissolve to generate an alkaline solution.
Rather, the outer layer of the tablet dissolves slowly, with just sufficient entering solu-
tion to neutralize the acid. Tablet dissolution stops when the neutralization reaction
is complete.
TITRATION ANALYSES
263
A similar process occurs when we spread a thick paste of zinc and
castor oil on a baby's bottom each time we change its nappy. The
'zinc' is in fact zinc oxide, ZnO, which, being amphoteric, reacts
with the uric acid in the baby's urine, thereby neutralizing it.
Worked Example 6.9 But how much stomach acid is neutralized
by a single indigestion tablet? The tablet contains 0.01 mol of MOH,
where 'M' is a monovalent metal and M + its cation.
We first consider the reaction in the stomach, saying it proceeds with
1 mol of hydrochloric acid reacting with 1 mol of alkali:
Some campaigners
believe the AICI 3 pro-
duced by Equation
(6.40) hastens the
onset of Alzheimer's
disease. Certainly, the
brains of people with
this nasty condition
contain too much alu-
minium.
MOH (s) + H 3 H
faq)
► M+ (aq) + 2H 2
(6.41)
M + is merely a cation. We say Equation (6.41) is a 1:1 reaction, occurring with a 1:1
stoichiometry. Such a stoichiometry simplifies the calculation; the 3:1 stoichiometry in
Equation (6.40) will be considered later.
From the stoichiometry of Equation (6.41), we say the neutralization is complete
after equal amounts of acid and alkali react. In other words, we neutralize an amount
n of hydrochloric acid with exactly the same amount of metal hydroxide, i.e. with
1 x 10" 2 mol.
The tablet can neutralize 0.01 mol of stomach acid.
This simple calculation illustrates the fundamental truth under-
lying neutralization reactions: complete reaction requires equal
amounts of acid and alkali. In fact, the primary purpose of a
titration is to measure an unknown amount of a substance in
a sample, as determined via a chemical reaction with a known
amount of a suitable reagent. We perform the titration to ascer-
tain when an equivalent amount of the reagent has been added to
the sample. When the amount of acid and alkali are just equal,
we have the equivalence point, from which we can determine the
unknown amount.
In a typical titration experiment, we start with a known volume
of sample, call it V( sam pie)- If we know its concentration c (samp i e ),
we also know the amount of it, as V( sam pie) x C( sam pie) • During the
course of the titration, the unknown reagent is added to the solu-
tion, usually drop wise, until the equivalence point is reached (e.g.
determining the endpoint by adding an indicator; see p. 273ff).
At equivalence, the amounts of known and unknown reagents are
the same, so n( sam pie) = w (unknown)- Knowing the amount of sample
and the volume of solution of the unknown, we can calculate the
concentration of the unknown.
The experimental tech-
nique of measuring out
the amount of acid and
alkali needed for neu-
tralization is termed a
titration.
The amounts of acid
and alkali are equal at
the equivalence point.
The linguistic similar-
ity between these two
words is no coinci-
dence!
We need equal num-
bers of moles of acid
and alkali to effect
neutralization.
264
ACIDS AND BASES
Worked Example 6.10 The methanoic acid from a nettle sting is extracted into 50 cm 3
of water and neutralized in the laboratory by titrating with sodium hydroxide solution.
The concentration of NaOH is 0.010 mol dm -3 . The volume of NaOH solution needed to
neutralize the acid is 34.2 cm 3 . What is the concentration c of the acid?
Unlike the Worked Example 6.9, we do not know the number of moles n of either reactant,
we only know the volumes of each. But we do know one of the concentrations.
Answer strategy. (1) First, we calculate the amount n of hydroxide required to neutralize
the acid. (2) We equate this amount n with the amount of acid neutralized by the alkali.
(3) Knowing the amount of acid, we finally calculate its concentration.
(1) To determine the amount of alkali, we first remember the defi-
nition of concentration c as
concentration, c ■■
amount, n
volume, V
(6.42)
We could have achiev-
ed this conversion
with quantity calcu-
lus: knowing there are
1000 cm 3 per dm 3 (so
10 3 dm 3 cnrr 3 ). In SI
units, we write the
volume as 34.2 cm 3 x
10- 3 dm 3 crrr 3 . The
units of cm 3 and cnrr 3
cancel to yield V =
0.0342 dm 3 .
Notice now the units of
dm 3 and drrr 3 cancel
out here.
Notice how we con-
verted the volume of
acid solution (50 cm 3 )
to 0.05 dm 3 .
and rearrange Equation (6.41) to make amount n the subject, i.e.
n — c x V
The volume V of alkali is 34.2 cm . As there are 1000 cm in a litre,
this volume equates to (34.2 4- 1000) dm 3 = 0.0342 dm 3 . Accordingly
n = 0.0342 dm 3 x 0.010 mol dm -3
3.42 x 10" 4 mol
(2) The reaction between the acid and alkali is a simple 1:1 reac-
tion, so 3.42 x 10~ 4 mol of alkali reacts with exactly 3.42 x 10~ 4 mol
of acid.
(3) The concentration of the acid is given by Equation (6.42) again.
Inserting values:
concentration c
3.42 x 10" 4 mol
0.05 dm 3
c = 6.8x 10~ 3 mol dm" 3
After extraction, the concentration of the methanoic acid is 0.068 mol
dm" 3 .
An altogether simpler and quicker way of calculating the concentration of an acid
during a titration is to employ the equation
C(acid) X V(acid) = C(alkali) X V(alkali) (6.43)
where the V terms are volumes of solution and the c terms are concentrations.
TITRATION ANALYSES
265
Worked Example 6.11 A titration is performed with 25 cm of NaOH neutralizing
29.4 cm 3 of nitric acid. The concentration of NaOH is 0.02 moldm
concentration of the acid.
. Calculate the
We rearrange Equation (6.43), to make C( ac id) the subject:
C(alkali) ^(alkali)
C(acid) — ~
v(acid)
We then insert values into Equation (6.44):
0.02 moldm 3 x 25cm 3
(6.44)
C(acid)
C(acid)
29.4 cm 3
0.017 moldm -3
We obtain here a ratio
of volumes (l/ (a ikaii) h-
l/(add)), enabling us to
cancel the units of the
two volumes. Units
are irrelevant if both
volumes have the same
units.
Justification Box 6.4
A definition of the point of 'neutralization' in words says, 'at the neutralization point,
the number of moles of acid equals the number of moles of hydroxide'. We re-express
the definition as
«(acid) = "(alkali) (6.45)
Next, from Equation (6.42), we recall how the concentration of a solution c when
multiplied by its respective volume V equals the number of moles of solute: n = c x V.
Clearly, «( ac ;d) = C( ac id) x V( ac id), and n(aikaii) = C( a ik a u) x V( a ikaii) •
Accordingly, substituting for «( ac ;d)
tion (6.43).
and n
(alkali!
into Equation (6.45) yields Equa-
SAQ 6.11 What volume of NaOH (of concentration 0.07 moldm ) is
required to neutralize 12 cm 3 of nitric acid of concentration 0.05 mol
drrr 3 ?
Aside
Equation (6.43) is a simplified version of a more general equation:
C (alkali) ^(alkali)
C(acid) = S -
v,
(acid)
(6.46)
where s is the so-called stoichiometric ratio.
For the calculation of a mono-protic acid with a mono-basic base, the stoichiometry
is simply 1 : 1 because 1 mol of acid reacts with 1 mol of base. We say the stoichiometric
ratio s = 1 . The value of s will be two if sulphuric acid reacts with NaOH since 2 mol
of base are required to react fully with 1 mol of acid. For the reaction of NaOH with
citric acid, s = 3; and s = 4 if the acid is H4EDTA.
266
ACIDS AND BASES
The value of s when Ca(OH) 2 reacts with HNO3 will be j, and the value when citric
acid reacts with Ca(OH) 2 will be |.
.-3-1
SAQ 6.12 What volume of Ca(OH) 2 (of concentration 0.20 moldrrr J )
is required to neutralize 50 cm 3 of nitric acid of concentration 0.10 mol
dm" 3 ?
An alternative way of determining the endpoint of a titration
is to monitor the pH during a titration, and plot a graph of pH
(as 'y') against volume V of alkali added (as 'x'). Typically, the
concentration of the acid is unknown, but we know accurately the
concentration of alkali. Figure 6.3 shows such as graph - we call
it a pH curve - in schematic form. The shape is sigmoidal, with
the pH changing very rapidly at the end point.
In practice, we obtain the end point by extrapolating the two
linear regions of the pH curve (the extrapolants should be parallel).
A third parallel line is drawn, positioned exactly midway between
the two extrapolants. The volume at which this third line crosses the
pH curve indicates the end point. Knowing the volume V( en d p0 mt)>
we can calculate the concentration of the acid via a calculation similar to Worked
Example 6.11.
Incidentally, the end point also represents the volume at which the pH changes most
dramatically, i.e. the steepest portion of the graph. For this reason, we occasionally
plot a different graph of gradient (as 'y') against volume V (as 'x'); see Figure 6.4.
We obtain the gradient as 'ApH -j- AV. The end point in Figure 6.4 relates to the
graph maximum.
Sigmoidal literally
means 'shaped like
a Greek sigma g'. The
name derives from the
Greek word sigmoides,
meaning 'sigma-like'.
(There are two Greek
letters called sigma,
used differently in word
construction. The other
has the shape a.)
Volume of alkali added V
Figure 6.3 A schematic pH curve for the titration of a strong acid with a strong base. At the
equivalence point, the amount of alkali added is the same as the amount of acid in solution
initially, allowing for an accurate calculation of the acid's concentration. Note how the end
point is determined by extrapolating the linear regions, and drawing a third parallel line between
them
pH BUFFERS 267
?l^
Volume of alkali added V
Figure 6.4 A schematic of the first derivative of the pH curve in Figure 6.3. The end-point volume
is determined as the volume at the peak. A first derivative plot such as this can yield a more accurate
end point than drawing parallel lines on Figure 6.3
6.4 pH buffers
Why does the pH of blood not alter after eating pickle?
Introduction to buffers
A 'pickle' is a food preserved in vinegar (ethanoic acid). Pickles generally have a sharp,
acidic flavour in consequence of the acid preservative. Many systems - especially living
cells - require their pH to be maintained over a very restricted range in order to prevent
catastrophic damage to the cell. Enzymes and proteins denature, for example, if the
pH deviates by more than a fraction. Traces of the food we eat are readily detected in
the blood quite soon after eating, so why does the concentration in the blood remain
constant, rather than dropping substantially with the additional acid in our diet?
Before we attempt an answer, look again at Figure 6.3, which clearly shows an
almost invariant pH after adding a small volume of alkali. Similarly, at the right-
hand side of the graph the pH does not vary much. We see an insensitivity of the
solution pH to adding acid or alkali; only around the end point does the pH alter
appreciably. The parts of the titration graph having an invariant pH are termed the
buffer regions, and we call the attendant pH stabilization a buffer action.
In a similar way, blood does not change its pH because it contains suitable concen-
trations of carbonic acid and bicarbonate ion, which act as a buffer, as below.
Why are some lakes more acidic than
others?
Buffer action
Acid rain is the major cause of acidity in open-air lakes and ponds
(see p. 237). Various natural oxides such as CO2 dissolve in water
Pollutant gases include
S0 2 , S0 3 , NO and N0 2 .
It is now common to
write SO x and NO x
to indicate this vari-
able valency within the
mixture.
268
ACIDS AND BASES
Remember: 'weak' in
this sense indicates the
extent to which a weak
acid dissociates, and
does not relate to its
concentration.
to generate an acid, so the typical pH of normal rainwater is about 5.6; but rainwater
becomes more acidic if pollutants, particularly SO v and NOjc, in addition to natural
CO2, dissolve in the water. As an example, the average pH of rain in the eastern United
States of America (which produces about one-quarter of the world's pollution) lies in
the range 3.9-4.5. Over a continental landmass, the partial pressure of SO2 can be
as high as 5 x 10~ 9 x p & , representing a truly massive amount of pollution.
After rainfall, the pH of the water in some lakes does not change,
whereas others rapidly become too acidic to sustain aquatic life.
Why? The difference arises from the buffering action of the water.
Some lakes resist gross change in pH because they contain other
chemicals that are able to take up or release protons into solution
following the addition of acid (in the rain). These chemicals in
the lake help stabilize the water pH, to form a buffer. Look at
Figure 6.5, which shows a pH curve for a weak acid titrated with
an alkali. Figure 6.5 is clearly similar to Figure 6.3 after the end-point volume, but it
has a much shallower curve at lower volumes. In fact, we occasionally have difficulty
ascertaining a clear end point because the curvature is so pronounced.
A buffer comprises (1) a weak acid and a salt of that acid, (2) a weak base and a
salt of that base, or (3) it may contain an acid salt. We define an acid-base buffer as
'a solution whose pH does not change after adding (small amounts
of) a strong acid or base'. Sodium ascorbate is a favourite buffer
in the food industry.
We can think of water entering the lake in terms of a titration.
A solution of alkali enters a fixed volume of acid: the alkaline
solution is water entering from the lake's tributary rivers, and the
acid is the lake, which contains the weak acid H2CO3 (carbonic
acid) deriving from atmospheric carbon dioxide. The alkali in the
tributary rivers is calcium hydroxide Ca(OH) 2 , which enters the
A buffer is a solution
of constant pH, which
resists changes in pH
following the addition
of small amounts of
acid or alkali.
X
Q.
Half volume
at the end point
Volume of alkali added V
Figure 6.5 A typical pH curve for the titration of carbonic acid (a weak acid) with a strong base.
The concentration of H2CO3 and HCO^ are the same after adding half the neutralization volume
of alkali. At this point, pH = pK 3
pH BUFFERS
269
water as the river passes over the limestone floor of river basins.
Calcium hydroxide is a fairly strong base.
Figure 6.5 shows a buffering action since the pH does not change
particularly while adding alkali to the solution. In fact, as soon as
the alkali mixes with the acid in the lake, its hydroxide ions are
neutralized by reaction with solvated protons in the lake, thereby
resisting changes in the pH. Figure 6.5 shows how little the lake pH
changes; we term the relatively invariant range of constant pH the
buffer region of the lake water. The mid pH of the buffer region cor-
responds quite closely to the pK a of the weak acid (here H2CO3),
where the pK a is a mathematical function of K a , as defined by
Limestone or chalk dis-
solve in water to a lim-
ited extent. The CaC0 3
decomposes naturally
to form Ca(OH) 2 ,
thereby generating
alkaline water.
pK a = - log 10 K a
(6.47)
The natural buffers in
the lake 'mop up' any
additional alkali enter-
ing the lake from the
tributary rivers, thereby
restricting any changes
to the pH.
A buffer is only really
effective at restrict-
ing changes if the pH
remains in the range
p/C a ±l.
As a good generalization, the buffer region extends over the range
0fp^ a ±l.
Only when all the acid in the lake has been consumed will the pH
rise significantly. In fact, the end point of such a titration is gauged
when the pH rises above pH 7, i.e. the pH of acid-base neutrality.
The pH of the lake water fluctuates when not replenished by the
alkaline river water in the tributary rivers. In fact, the pH of the
lake water drops significantly each time it rains (i.e. when more
H2CO3 enters the lake). If the amounts of acid and alkali in the
water remain relatively low, then the slight fluctuations in water pH will not be great
enough to kill life forms in the lake.
The system above describes the addition of alkali to a lake containing a weak
acid. The reverse process also occurs, with acid being added to a base, e.g. when the
tributary rivers deliver acid rain to a lake and the lake basin is made of limestone or
chalk. In such a case, the lake pH drops as the acid rain from the rivers depletes the
amounts of natural Ca(OH) 2 dissolved in the lake.
As a further permutation, adding a strong acid to a weak base also yields a buffer
solution, this time with a buffer region centred on the pK a of the base. The pH at the
end point will be lower than 7.
Buffers
Each species within a buffer solution participates in an equilibrium reaction, as char-
acterized by an equilibrium constant K. Adding an acid (or base) to a buffer solution
causes the equilibrium to shift, thereby preventing the number of
protons from changing, itself preventing changes in the pH. The
change in the reaction's position of equilibrium is another mani-
festation of Le Chatelier's principle (see p. 166).
One of the most common buffers in the laboratory is the so-
called 'phosphate buffer', which has a pH of 7.0. It comprises salts
A buffer is a solution
of a weak acid mixed
with its conjugate base,
which restricts changes
to the pH.
270
ACIDS AND BASES
of hydrogen phosphate and dihydrogen phosphate, in the following equilibrium:
H 2 P0 4 2 -
— >
HPO4-
+ H+
(6.48)
conjugate acid
conjugate base
If this example were to proceed in the reverse direction, then the hydrogen phosphate
(on the right) would be the base, since it receives a proton, and the dihydrogen
phosphate (on the left) would be the conjugate the acid.
The equilibrium constant of the reaction in Equation (6.48) is given by
K„ =
[H+][HPOf]
[H 2 P07]
(6.49)
Notice how the equilibrium constant K in Equation (6.49) is also an acidity constant,
hence the subscripted 'a'. The value of K remains constant provided the temperature
is not altered.
Now imagine adding some acid to the solution - either by mistake or deliberately.
Clearly, the concentration of H + will increase. To prevent the value of K A changing,
some of the hydrogen phosphate ions combine with the additional protons to form di-
hydrogen phosphate (i.e. Equation (6.48) in reverse). The position of the equilibrium
adjusts quickly and efficiently to 'mop up' the extra protons in the buffer solution. In
summary, the pH is prevented from changing because protons are consumed.
How do we make a 'constant-pH solution'?
The Henderson - Hasselbach equation
We often need to prepare a solution having a constant pH. Such solutions are vital in
the cosmetics industry, as well as when making foodstuffs and in the more traditional
experiments performed by the biologist and physical chemist.
To make such a solution, we could calculate exactly how many moles of acid to add
to water, but this method is generally difficult, since even small errors in weighing
the acid can cause wide fluctuations in the pH. Furthermore, we cannot easily weigh
out one of acid oxides such as NO. Anyway, the pH of a weak acid does not clearly
follow the acid's concentration (see p. 254).
The Henderson-Hasselbach equation, Equation (6.50), relates
the pH of a buffer solution to the amounts of conjugate acid and
conjugate base it contains:
In some texts, Equa-
tion (6.50) is called the
Henderson-
HasselbaLch equation.
pH = pK a + log
10
[A~]
[HA]
(6.50)
We follow the usual pattern here by making a buffer with a weak acid HA and a
solution of its conjugate base, such as the sodium salt of the respective anion, A - .
pH BUFFERS 271
We can prepare a buffer of almost any pH provided we know the pK a of the acid; and
such values are easily calculated from the K d values in Table 6.5 and in most books
of physical chemistry and Equation (6.50). We first choose a weak acid whose pK^
is relatively close to the buffer pH we want. We then need to measure out accurately
the volume of acid and base solutions, as dictated by Equation (6.50).
Worked Example 6.12 We need to prepare a buffer of pH 9.8 by mixing solutions of
ammonia and ammonium chloride solution. What volumes of each are required? Take
the K & of the ammonium ion as 6 x 10~ 10 . Assume the two solutions have the same
concentration before mixing.
Strategy: (1) We calculate the pAT a of the acid. (2) We identify which component is the
acid and which the base. (3) And we calculate the proportions of each according to
Equation (6.50).
(1) From Equation (6.50), we define the pK. A as — log 10 K^. Inserting values, we obtain
a p# a of 9.22.
(2) The action of the buffer represents the balanced reaction, NH4CI — > NH3 + HC1,
so NH4CI is the acid and NH3 is the base.
(3) To calculate the ratio of acid to base, we insert values into Equation (6.50):
, [NH
= 9.2 + log 10
0.6 = log 10
\[NHJ]
[NHJ
[NH 3 ]
Taking antilogs of both sides to remove the logarithm, we obtain
10°
,,,.„ _ [NH 3 ]
" [NH+]
[NH 3 ]
[NH+]
= 4
We are permitted to
calculate a ratio like
this if the concen-
trations of acid and
conjugate base are the
same.
So, we calculate the buffer requires four volumes of ammonia solution
to one of ammonium (as the chloride salt, here).
SAQ 6.13 What is the pH of ammonia-ammonium buffer if three vol-
umes of NH4CI are added to two volumes of NH 3 ?
We want a buffer solution because its pH stays constant after adding small amounts
of acid or base. Consider the example of adding hydrochloric acid to a buffer, as
described in the following Worked Example.
Worked Example 6.13 Consider the so-called 'acetate buffer', made with equal vol-
umes of sodium ethanoate and ethanoic acid solutions. The concentration of each solu-
tion is 0.1 moldnT . A small volume (10 cm ) of strong acid (HC1 of concentration
1 moldm^ 3 ) is added to a litre of this buffer. The pH before adding HC1 is 4.70. What
is its new pH?
272
ACIDS AND BASES
Strategy: we first calculate the number of moles of hydrochloric
acid added. Second, we calculate the new concentrations of ethanoic
acid and ethanoate. And third, we employ the Henderson-Hasselbach
equation once more.
(1) 10 cm 3 represents one-hundredth of a litre. From
Equation (6.42), the number of moles is 0.01 mol.
(2) Before adding the hydrochloric acid, the concentrations
of ethanoate and ethanoic acid are constant at 0.1 mol dm" 3 . The
hydrochloric acid added reacts with the conjugate base in the buffer
(the ethanoate anion) to form ethanoic acid. Accordingly, the concentration [CH^COO - ]
decreases and the concentration [CH3COOH] increases. (We assume the reaction is
quantitative.) Therefore, the concentration of ethanoate is (0.1 — 0.01) mol dnT =
0.09 mol dm" 3 . The concentration of ethanoic acid is (0.1 + 0.01) mol dnT 3 =
0.11 mol dm" 3 .
(3) Inserting values into Equation (6.50):
The acetate buffer is
an extremely popu-
lar choice in the food
industry. The buffer
might be described on
a food packet as an
acidity regulator.
pH = 4.70 + log 10
0.09
oTTT
so
pH = 4.70 + log (0.818)
pH = 4.70 + (-0.09)
pH = 4.61
So, we see how the pH shifts by less than one tenth of a pH unit after adding quite
a lot of acid. Adding this same amount of HC1 to distilled water would change the
pH from 7 to 2, a shift of five pH units.
SAQ 6.14 Consider the ammonia -ammonium buffer in Worked Exam-
ple 6.12. Starting with 1 dm 3 of buffer solution containing 0.05 mol dm" 3
each of NH 3 and NH 4 CI, calculate the pH after adding 8 cm 3 of NaOH
solution of concentration 0.1 mol dm" 3 .
Justification Box 6.5
We start by
writing
the equilibrium constant for a weak acid HA dissociating in water,
HA + H 2
-> H 3
*" + A" , where each ion is solvated. The dissociation constant for
the acid K a
is given
by Equation (6.35):
„ [H 3 0+][A-]
K a =
LHAj
where, as usual, we
ignore the water term. Taking logarithms of
Equation (6.35) yields
v , [H 3 0+][A-]
lOglQ ^a = lOg 10
(6.51)
ACID-BASE INDICATORS
273
We can split the fraction term in Equation (6.51) by employing
the laws of logarithms,
to yield
+ [A - ]
logio K * = log 10 [H 3 CT] + log 10
(6.52)
The term Tog| K^' should remind us of pA" a (Equation 6.52), and the term log 10 [H3O + ]
will remind us of
pH in Equation (6.20), so we rewrite Equation (6.52) as
[A - ]
p^ a = pH + log 10
(6.53)
which, after a
little rearranging, yields the Henderson-
-Hasselbach
equation,
Equation (6.50).
6.5 Acid- base indicators
What is l the litmus test'?
pH indicators
Litmus is a naturally occurring substance obtained from lichen. It
imparts an intense colour to aqueous solutions. In this sense, the
indicator is a dye whose colour is sensitive to the solution pH.
If the solution is rich in solvated protons (causing the pH to be
less than 7) then litmus has an intense red colour. Conversely, a
solution rich in hydroxide ions (with a pH greater than 7) causes
the litmus to have a blue colour.
To the practical chemist, the utility of litmus arises from the way
its colour changes as a function of pH. Placing a single drop of
litmus solution into a beaker of solution allows us an instant test of
the acidity (or lack of it). It indicates whether the pH is less than
7 (the litmus is red, so the solution is acidic), or the pH is greater
than 7 (the litmus is blue, so the solution is alkaline). Accordingly,
we call litmus a pH indicator.
In practical terms, we generally employ litmus during a titration.
The flask will contain a known volume of acid of unknown con-
centration, and we add alkali from a burette. We know we have
reached neutralization when the Litmus changes from red (acid
in excess) and just starts changing to blue. We know the pH of
the solution is exactly 7 when neutralization is complete, and then
note the volume of the alkali, and perform a calculation similar to
Worked Example 6.11.
The great English scientist Robert Boyle (1627-1691) was the
first to document the use of natural vegetable dyes as acid-base indicators.
The name 'litmus'
comes from the Old
Norse litmosi, which
derives from litr and
mosi, meaning dye and
moss respectively.
Much of the litmus
in a laboratory is
pre-impregnated on
dry paper.
Litmus is an indicator.
To avoid ambiguity,
we shall call it an
'acid -base indicator'
or a "pH indicator'.
Litmus often looks
purple-grey at the
neutralization point.
This colour tells us we
have a mixture of both
the red and blue forms
of litmus.
274
ACIDS AND BASES
Why do some hydrangea bushes look red and others
blue?
The chemical basis of acid -base indicators
■
The name 'hydrangea'
derives from classi-
cal Greek mythol-
ogy, in which the
'hydra' was a beast
with many heads.
Hydrangeas (genus Hydrangea) are beautiful bushy plants having
multiple flower heads. In soils comprising much compost the flow-
ers have a blue colour, but in soils with much lime or bone meal
the heads are pink or even crimson-pink in colour. Very occa-
sionally, the flowers are mauve. 'Lime' is the old-fashioned name
for calcium oxide, and is alkaline; bone meal contains a lot of
phosphate, which is also likely to raise the soil pH. The colour of
the hydrangea is therefore an indication of the acid content of the
soil: the flower of a hydrangea is blue in acidic soil because the
plant sap is slightly acidic; red hydrangeas exist in alkaline soil
because the sap transports alkali from the soil to the petals. The
rare mauve hydrangea indicates a soil of neutral pH. We see how
the chromophore in the flower is an acid -base indicator.
The chromophore in hydrangeas is delphinidin (X), which is a
member of the anthrocyanidin class of compounds. Compound X
reminds us of phenol (VII), indicating that delphinidin is also a
weak acid. In fact, all pH indicators are weak acids or weak bases,
and the ability to change colour is a visible manifestation of the
indicator's ability to undergo reversible changes in structure. In the
laboratory, only a tiny amount of the pH indicator is added to the
titration solution, so it is really just a probe of the solution pH. It
does not participate in the acid -base reaction, except insofar as its
own structure changes with the solution pH.
As an example, whereas the anthracene-based core of molecular X is relatively inert,
the side-chain 'X' is remarkably sensitive to the pH of its surroundings (principally,
to the pH of the solution in which it dissolves).
The word 'chromo-
phore' comes from two
Greek words, ^khro-
mos' meaning colour
and 'prtoro', which
means 'to give' or 'to
impart'. A chromophore
is therefore a species
imparting colour.
All pH indicators are
weak acids or weak
bases.
Figure 6.6 shows the structure of the side substituent as a function of pH.
The hydroxyl group placed para to the anthracene core is protonated in acidic
solutions (i.e. when the hydrangea sap is slightly acidic). The proton is abstracted in
alkaline sap, causing molecular rearrangement to form the quinone moiety.
ACID-BASE INDICATORS
275
Red
Blue
Mauve
Figure 6.6 Anthrocyanidins impart colour to many natural substances, such as strawberries and
cherries. The choice of side chains can cause a huge change in the anthrocyanidin's colour. If the
side chain is pH sensitive then the anthrocyanidin acts as an acid-base indicator: structures of an
anthrocyanidin at three pHs (red in high acidity and low pH, blue in low acidity and high pH and
mauve in inter-midiate pHs)
It is also astonishing how the rich blue of a cornflower (Centaurea cyanus) and
the majestic red flame of the corn poppy (Papaver rheas) each derive from the same
chromophore - again based on an anthrocyanidin. The pH of cornflower and poppy
sap does not vary with soil composition, which explains why we see neither red
cornflowers nor blue corn poppies.
Aside
It is fascinating to appreciate the economy with which nature produces colours (ele-
mentary colour theory is outlined in Chapter 9). The trihydroxyphenyl group of the
anthrocyanidin (X) imparts a colour to both hydrangeas and delphiniums. The dihydrox-
yphenyl group (XI) is remarkably similar, and imparts a red or blue colour to roses,
cherries and blackberries. The singly hydroxylated phenyl ring in XII is the chromophore
giving a red colour to raspberries, strawberries and geraniums, but it is not pH sensitive.
OH
(XI)
(XII)
276
ACIDS AND BASES
Why does phenolphthalein indicator not turn red until
pH 8.2?
Which acid -base indicator to use?
Litmus was probably the most popular choice of acid-base indicator, but it is not a
good choice for colour-blind chemists. The use of phenolphthalein as an acid-base
indicator comes a close second. Phenolphthalein (XIII) is another weak organic acid.
It is not particularly water soluble, so we generally dissolve it in aqueous ethanol.
The ethanol explains the pleasant, sweet smell of phenolphthalein solutions.
OH
^^
Phenolphthalein is colourless and clear in acidic solutions, but imparts an intense
puce pink colour in alkaline solutions of higher pH, with A( max ) = 552 nm. The
coloured form of phenolphthalein contains a quinone moiety; in fact, any chromophore
based on a quinone has a red colour. But if a solution is prepared at pH 7 (e.g. as
determined with a pH meter), we find the phenolphthalein indicator is still colour-
less, and the pink colour only appears when the pH reaches 8.2. Therefore, we have
a problem: the indicator has not detected neutrality, since it changes colour at too
Table 6.7 Some common pH indicators, their useful pH ranges and
the changes in colour occurring as the pH increases. An increasing
pH accompanies a decreasing concentration of the solvated proton
Indicator
pH range
Colour change
Methyl violet
0.0-1.6
Yellow — ¥ blue
Crystal violet
0.0-1.8
Yellow -*■ blue
Litmus
6.5-7.5
Red -*■ blue
Methyl orange
3.2-4.4
Red — ► yellow
Ethyl red
4.0-5.8
Colourless — ► red
Alizarin red S
4.6-8.0
Yellow — ► red
3-Nitrophenol
6.8-8.6
Colourless — > yellow
Phenolphthalein
8.2-10.0
Colourless — > pink
ACID-BASE INDICATORS 277
high a value of pH. However, the graph in Figure 6.3 shows how the pH of the
titration solution changes dramatically near the end point: in fact, only a tiny incre-
mental addition of alkali solution is needed to substantially increase the solution pH
by several pH units. In other words, a fraction of a drop of alkali solution is the only
difference between pH 7 (at the true volume at neutralization) and pH 8 when the
phenolphthalein changes from colourless to puce pink.
Table 6.7 lists the pH changes for a series of common pH indicators. The colour
changes occur over a wide range of pHs, the exact value depending on the indicator
chosen. Methyl violet changes from yellow to blue as the pH increases between and
1.6. At the opposite extreme, phenolphthalein responds to pH changes in the range
8.2 to 10.
7
Electrochemistry
Introduction
This chapter commences by describing cells and redox chemistry. Faraday's laws of
electrolysis describe the way that charge and current passage necessarily consume
and produce redox materials. The properties of each component within a cell are
described in terms of potential, current and composition.
Next, the nature of half-cells is explained, together with the necessary thermody-
namic backgrounds of the theory of activity and the Nernst equation.
In the final sections, we introduce several key electrochemical applications, such
as the pH electrode (a type of concentration cell), nerve cells (which rely on junction
potentials) and batteries.
7.1 Introduction to cells: terminology
and background
Why does putting aluminium foil in the mouth cause
pain?
I
Introduction to electrochemistry
Most people have at some time experienced a severe pain in their teeth after acci-
dentally eating a piece of sweet wrapper. Those teeth that hurt are usually nowhere
near the scrap of wrapper. The only people who escape this nasty sensation are those
without metal fillings in their teeth.
The type of sweet wrapper referred to here is generally made of aluminium metal,
even if we call it 'silver paper'. Such aluminium dissolves readily in acidic, conductive
electrolytes; and the pH of saliva is about 6.5-7.2.
The dissolution of aluminium is an oxidative process, so it gener-
ates several electrons. The resultant aluminium ions stay in solution
next to the metal from which they came. We generate a redox cou-
ple, which we define as 'two redox states of the same material'.
Two redox states of
the same material are
called a redox couple.
280
ELECTROCHEMISTRY
The word 'amalgam'
probably comes from
the Greek malagma
meaning 'to make
soft', because a metal
becomes pliable when
dissolved in mercury.
Another English word
from the same root is
'malleable'.
A cell comprises two
or more half-cells in
contact with a common
electrolyte. The cell is
the cause of the pain.
While it feels as though all the mouth fills with this pain, in fact
the pain only manifests itself through those teeth filled with metal,
the metal being silver dissolved in mercury to form a solid - we
call it a silver amalgam. Corrosion of the filling's surface causes it
to bear a layer of oxidized silver, so the tooth filling also represents
a redox couple, with silver and silver oxide coexisting.
An electrochemical cell is defined as 'two or more half-cells in
contact with a common electrolyte'. We see from this definition
how a cell forms within the mouth, with aluminium as the more
positive pole (the anode) and the fillings acting as the more negative
pole (the cathode). Saliva completes this cell as an electrolyte. All
the electrochemical processes occurring are contained within the
boundaries of the cell.
Oxidation proceeds at the anode of the cell according to
Al
(s)
Al
3+
(aq)
+ 3e"
(7.1)
Oxidation reactions
occur at the anode.
Reduction reactions
occur at the cathode.
and occurs concurrently with a reduction reaction at the cathode:
Ag 2 (s) + 2e" ► 2Ag° (s) + O 2 " (7.2)
The origins of the words 'anode' and 'cathode' tell us much.
'Anode' comes from the Greek words ana, meaning 'up', and
hodos means 'way' or 'route', so the anode is the electrode to which electrons travel
from oxidation, travelling to higher energies (i.e. energetically 'uphill'). The word
'cathode' comes from the Greek hodos (as above), and cat meaning 'descent'. The
English word 'cascade' comes from this same source, so a cathode is the electrode
to which the electrons travel (energetically downhill) during reduction.
The oxidation and reduction reactions must occur concurrently because the electrons
released by the dissolution of the aluminium are required for the reduction of the
silver oxide layer on the surface of the filling. For this reason, we need to balance
the two electrode reactions in Equations (7.1) and (7.2) to ensure the same number of
electrons appear in each. The pain felt at the tooth's nerve is a response to this flow
of electrons. The paths of electron flow are depicted schematically in Figure 7.1.
Each electron has a 'charge' Q. When we quantify the number of electrons produced
or consumed, we measure the overall charge flowing. Alternatively, we might measure
the rate at which the electrons flow (how many flow per unit time, t): this rate is
termed the current /. Equation (7.3) shows the relationship between current / and
charge Q:
/.«
dt
(7.3)
So ultimately the pain we feel in our teeth comes from a flow of current.
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
281
Aluminium metal
Aluminium ions
Electrons
Silver metal ► Tooth interior
and nerve
- electrons
(oxidation)
+ electrons
(reduction)
Silver ions
Aluminium
('silver foil')
Electrolyte
(saliva)
Silver
(tooth filling)
Figure 7.1 Schematic illustration of the electron cycles that ultimately cause a sensation of pain
in the teeth in people who have metallic fillings and who have inadvertently eaten a piece of
aluminium ('silver') foil, e.g. while eating sweets
Why does an electric cattle prod cause pain?
The magnitude of a current
An electric cattle prod looks a little like a walking stick with an attached battery. A
potential from the tip of the stick is applied to a cow's flank, and the induced current
hurts the animal. The cow moves where prompted to avoid a reapplication of the
pain, thereby simplifying the job of cowherding.
Ohm's law says that applying a potential V across an electrical resistor R induces
a proportional current /. We can state this relationship mathematically as
V = IR
(7.4)
It is reported that the great Victorian scientist Michael Faraday
discovered a variation of Ohm's law some 20 years before Ohm
himself published the law that today bears his name. Faraday, with
his typical phlegm and ingenuity, grasped a resistor between his
two hands, immersed them both in a bowl of tepid water, and
applied a voltage between them. It hurt. He found the empirical
relationship 'pain oc I x R\
The pain Faraday induced was in direct proportion to the magni-
tude of the current passed. He discovered the principle underlying
the action of an electric cattle prod. It is sobering to realize how
Faraday's result is today employed through most of the world as
the basis of torture. Despite the explicit banning of torture by the
UN Charter on Human Rights (of 1948), it is common knowledge that giving an
electric shock is one of the most effective known means of causing pain.
In summary, the cattle prod causes pain because of the current formed in response
to applying a voltage.
The word 'empirical'
implies a result derived
from experiment rather
than theory. Similarly,
a chemist calculates a
compound's 'empirical
formula' while fully
aware its value is based
on experiment rather
than theory.
282
ELECTROCHEMISTRY
What is the simplest way to clean a tarnished silver
spoon?
Electrochemistry: the chemistry of electron transfer
■
Oxidation is loss of
electrons. Reduction is
gain of electrons.
Cutlery or ornaments made of silver tarnish and become black;
this is a shame, because clean, shiny silver is very attractive. The
'tarnish' comprises a thin layer of silver that has oxidized following
contact with the air to form black silver(I) oxide:
In fact, the tarnish on
silver usually comprises
both silver(I) oxide
and a little silver(I)
sulphide.
4Ag (s) + 2( g) ► 2Ag 2 (s)
(7.5)
Such silver can be rather difficult to clean without abrasives
(which wear away the metal). The following is a simple electro-
chemical means of cleaning the silver: immerse the tarnished silver
in a saucer of electrolyte, such as salt solution or vinegar, and wrap
it in a piece of aluminium foil. Within a few minutes the silver is cleaner and bright,
whereas the aluminium has lost some of its shininess.
The shine from the aluminium is lost as atoms on the surface of the foil are
oxidized to form Al 3+ ions (Equation (7.1)), which diffuse into solution. Because
the aluminium touches the silver, the electrons generated by Equation (7.1) enter the
silver and cause electro-reduction of the surface layer of Ag 2 (Equation (7.2)).
In summary, we construct a simple electrochemical cell in which the silver to be
cleaned is the cathode (Equation (7.2)) and aluminium foil as the anode supplies the
electrons via Equation (7.1).
The salt or vinegar acts as an electrolyte, and is needed since the product Al 3+
requires counter ions to ensure electro-neutrality (so aluminium ethanoate forms).
The oxide ions combine with protons from the vinegar to form water. Figure 7.2
illustrates these processes occurring in schematic form.
Aluminium metal
Aluminium ions
- electrons
(oxidation)
Silver metal
+ electrons
(reduction)
Silver ions
Aluminium foil
Electrolyte
(vinegar)
Silver spoon
Figure 7.2 Illustration of the electron cycles that allow for the trouble-free cleaning of silver: we
immerse the tarnished silver in an electrolyte, such as vinegar, and touch the silver with aluminium
foil
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND 283
And finally a word of caution: aluminium ethanoate is toxic, so wash the silver
spoon thoroughly after removing its tarnish.
How does Electrolysis' stop hair growth?
Electrochemical reactions and electrolysis
To many people, particularly the image conscious, electrolysis
means removing hairs from the arms and legs - a practice some-
times called 'electrology'. The purpose of such electrolysis is to
remove the hair follicles temporarily, thereby avoiding the need to
shave. We 'treat' each individual hair by inserting a tiny surgical
'probe' (in reality, an electrode) into the hair follicle. Applying a
voltage to the hair root for a fraction of a second kills the root. The
electrolysed hair is then removed, and will not regrow for some
time. This procedure is performed repeatedly until the desired area
is cleared. But how does it work?
A voltage is applied to the electrode: we say it is polarized. A
current flows in response to the voltage, and electrons are consumed
by electrochemical reactions around the electrodes. Electrolysis
occurs. Each electron that flows through the electrode must be
involved in a redox reaction, either oxidation or reduction. The
electrons entering or leaving the electrode move as a result of
reactions occurring in the immediate vicinity of the electrode. Con-
versely, electrons can travel in the opposite direction (leaving the
electrode) to facilitate reduction reactions.
In summary, this form of electrolysis is effective because the
charge passing through the electrode generates chemicals inside
the hair follicle. The resultant trauma kills the hair root. A leg or
arm treated in this way remains hairless until a new, healthy root
regrows later in the previously damaged follicle.
Electrochemical means
the chemistry of the
electron.
It is wrong to claim
a permanent method
of hair removal: after
electrolysis, the hair
will grow back, albeit
thinner and finer.
The word 'electroly-
sis' derives from the
Greek words lysis,
meaning 'splitting' or
'cleavage', and the
root electro, meaning
'charge' or 'electric-
ity'. Strictly, then,
electrolysis involves
electrochemical bond
cleavage.
Why power a car with a heavy-duty battery yet use a
small battery in a watch?
Faraday's laws of electrolysis
A battery is a device for converting chemical energy into electrical
energy. The amount of energy required by the user varies according
to the application in mind. For example, a watch battery only pow-
ers the tiny display on the face of a digital watch. For this purpose,
Batteries are described
in more detail in
Section 7.7.
284 ELECTROCHEMISTRY
Table 7.1 Faraday's laws of electrolysis
Faraday' s first law
The number of moles of a species formed at an electrode during electrolysis is proportional to the
electrochemical charge passed: Q = I x t
Faraday's second law
A given charge liberates different species in the ratio of their relative formula masses, divided by
the number of electrons in the electrode reaction
it only needs to deliver a tiny current of a micro-amp or so. Conversely, a car battery
(usually a 'lead-acid cell', as described on p. 347) is bulky and heavy because it
must deliver a massive amount of electrical energy, particularly when starting the car.
Other batteries generate currents of intermediate magnitude, such as those needed in
torches, mobile phones and portable cassette and CD players.
The amount of charge generated or consumed by a battery is in direct proportion
to the number of electrons involved, according to Faraday 's laws, which are given in
Table 7.1. Both electrons and ions possess charge. When a current is drawn through
a cell, the charged electrons move through the conductive electrodes (as defined on
p. 300) concurrently with charged ions moving through the electrolyte. The ions are
anions (which bear a negative charge) and cations (which are positive).
Underlying both of Faraday's laws lies the fundamental truth that each electron
possesses the same charge.
Worked Example 7.1 What is the charge on 1 mol of electrons?
The charge e on a single electron is 1.6 x 10" 19 C and there are
6.022 x 10 23 electrons per mole (the Avogadro number L), so the
charge on a mole of electrons is given by the simple expression
The coulomb, C, is the
SI unit of charge.
The charge on 1 mol of
electrons is termed 'a
faraday' F.
charge on one electron = L x e (7.6)
Inserting numbers into Equation (7.6), we obtain
charge on 1 mole of electrons
= 1.6 x 10" 19 C x 6.026 x 10 23 mol" 1
We see that 1 mol has a charge of 96487 Cmol" . This quantity of
charge is known as the 'Faraday' F.
SAQ 7.1 An electrolysis needle (i.e. an electrode) delivers 1 nrmol of
electrons to a hair root. How many faraday's of charge are consumed, and
how many coulombs does it represent?
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
285
Worked Example 7.2 How much silver is generated by reductively
passing 1 F of charge through a silver-based watch battery?
We will answer this question in terms Faraday's first law, which was
first formulated in 1834 (see Table 7.1).
Each electron is required to effect the reduction reaction Ag + (aq ) +
e~ — > Ag (s) , so one electron generates one atom of Ag, and IF of
charge (i.e. 1 mol of electrons) generates 1 mol of Ag atoms. The
metal forms at the expense of 1 mol of Ag + ions. Similarly, 1 mol
of electrons, if passed oxidatively, would generate 1 mol of Ag + ion
from Ag metal.
We see a direct proportionality between the charge passed and the
amount of material formed during electrolysis, as predicted by Fara-
day's first law.
Silver has the symbol
'Ag' because its Latin
name is argentium,
itself derived from the
Greek for money, argu-
rion. The Spanish col-
onized parts of South
America in the 16th
century. They named
it Argentina (dog-Latin
for 'silver land') when
they discovered its vast
reserves of silver.
SAQ 7.2 10~ 10 F of charge are inserted into a hair pore through a fine nee-
dle electrode. Each electron generates one molecule of a chemical to poison
the hair root. How much of the chemical is formed?
Worked Example 7.3 A Daniell cell (see p. 345) is constructed, and IF of charge is
passed through a solution containing copper(II) ions. What mass of copper is formed?
Assume that the charge is only consumed during the reduction reaction, and is performed
with 100 percent efficiency.
We will answer this question by introducing Faraday's second law (see Table 7.1).
The reduction of one copper(II) ion requires two electrons according to the reaction
Cu z
(aq)
+ 2e~
-* Cu (s
(s)
(7.7)
We need 2F of charge to generate 1 mol of Cu( S ), and IF of charge will form 0.5 mol
of copper.
1 mol of copper has a mass of 64 g, so 1 F generates (64 4- 2) g of copper. The mass
of copper generated by passing 1 F is 32 g.
SAQ 7.3 How much aluminium metal is formed by passing 2F of charge
through a solution of Al 3+ ions? [Hint: assume the reaction at the electrode
is AI J
(aq)
■3e"
Al ( s).]
low is coloured Canodized') aluminium produced?
Currents generate chemicals: dynamic electrochemistry
Saucepans and other household implements made of aluminium often have a brightly
coloured, shiny coating. This outer layer comprises aluminium oxide incorporating a
small amount of dye.
286
ELECTROCHEMISTRY
AI2O3 is also called
alumina.
The layer is deposited with the saucepan immersed in a vat of
dye solution (usually acidified to pH 1 or 2), and made the positive
terminal of a cell. As the electrolysis proceeds, so the aluminium
on the surface of the saucepan is oxidized:
2A1 (S) + 3H 2 ► A1 2 3 ( S ) + 6H^
(aq)
(7.8)
We say the dye occlu-
des within a matrix of
solid aluminium oxide.
The aluminium is white and shiny before applying the potential. A critical potential
exists below which no electro-oxidation will commence. At more extreme potentials,
the surface atoms of the aluminium oxidize to form Al + ions, which combine with
oxide ions from the water to form AI2O3. This electro-precipitation of solid alu-
minium oxide is so rapid that molecules of dye get trapped within it, and hence its
coloured aspect.
The dye resides inside the layer of alumina. Its colour persists
because it is protected from harmful UV light, as well as mechan-
ical abrasion and chemical attack.
But the chemical reaction forming this coloured layer of oxide
represents only one part of the cell. A cell contains a minimum of
two electrodes, so a cell comprises two reactions - we call them
half-reactions: one describes the chemical changes at the positive
electrode (the anode) and the other describes the changes that occur
at the negative cathode.
The same number of electrons conducts through (i.e. are conducted by) each of the
two electrodes. If we think in terms of charge flowing per unit time, we would say
the same 'current' / flows through each electrode. The electrons travel in opposite
directions, insofar as they leave or enter an electrode, which explains why the current
through the anode is oxidative and the current through the cathode is reductive.
We say
A anode) = — ^(cathode) (7.9)
A cell must comprise a
minimum of two elec-
trodes.
where the minus sign reminds us that the electrons either move in or out of the
electrode.
Because these two currents are equal (and opposite), the same amount of reaction
will occur at either electrode. We see how an electrode reaction must also occur at
the cathode as well as the desired oxidative formation of alumina at the anode. (The
exact nature of the reaction at the anode will depend on factors such as the choice of
electrode material.)
How do we prevent the corrosion of an oil rig?
Introduction to electrochemical equilibrium
Oil rigs are often built to survive in some of the most inhospitable climates in the
world. For example, the oil rigs in the North Sea between the UK and Scandinavia
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
287
frequently withstand force- 10 gales. Having built the rig, we appreciate how impor-
tant is the need of maximizing its lifetime. And one of the major limits to its life
span is corrosion.
Oil rigs are made of steel. The sea in which they stand contains vast quantities of
dissolved salts such as sodium chloride, which is particularly 'aggressive' to ferrous
metals. The corrosion reaction generally involves oxidative dissolution of the iron, to
yield ferric salts, which dissolve in the sea:
Fe
(s)
-> Fe
3+
(aq)
+ 3e"
(7.10)
In reality, several of
the iron compounds
are solid, such as rust.
This clever method of
averting corrosion can
also arrest the corro-
sion of rails and the
undersides of boats.
If left unchecked, dissolution would cause thinning and hence weakening of the legs
on which the rig stands.
One of the most ingenious ways in which corrosion is inhibited
is to strap a power pack to each leg (just above the level of the
sea) and apply a continuous reductive current. An electrode couple
would form when a small portion of the iron oxidizes. The couple
would itself set up a small voltage, itself promoting further disso-
lution. The reductive current coming from the power pack reduces
any ferric ions back to iron metal, which significantly decreases
the rate at which the rig leg corrodes.
Clearly, we want the net current at the iron to be zero (hence no
overall reaction). The rate of corrosion would be enhanced if the
power pack supplied an oxidative current, and wasteful side reac-
tions involving the seawater itself would occur if the power pack
produced a large reductive current. The net current through the
iron can be positive, negative or zero, depending on the potential
applied to the rig's leg. The conserver of the rig wants equilibrium,
implying no change.
All the discussions of electrochemistry so far in this chapter concern current - the
flow of charged electrons. We call this branch of electrochemistry dynamic, implying
that compositions change in response to the flow of electrons. Much of the time,
however, we wish to perform electrochemical experiments at equilibrium.
One of our best definitions of 'equilibrium electrochemistry' says
the net current is zero; and from Faraday's laws (Table 7.1), a zero
current means that no material is consumed and no products are
formed at the electrode.
But this equilibrium at the oil rig is dynamic: the phrase
'dynamic equilibrium' implies that currents do pass, but the cur-
rent of the forward reaction is equal and opposite to the current of the back reaction,
according to
' (forward, eq) = ' (backward, eq) \ ' • * *)
The simplest definition
of equilibrium in an
electrochemistry cell is
that no concentrations
change.
Electrochemical mea-
surements at equilib-
rium are made at zero
current.
and the overall (net) current is the sum of these two:
J(net) = I*
(forward)
+ /,
(backward)
(7.12)
288
ELECTROCHEMISTRY
Equation (7.11) is important, since it emphasizes how currents flow even at equi-
librium.
But the value of 7( net ) is only ever zero at equilibrium because /(forward) = — /(backward)*
which can only happen at one particular energy, neither too reductive nor too oxidative.
The voltage around the legs of the oil rig needs to be chosen carefully.
What is a battery?
The emf of cells
A battery is a device
for converting chemical
energy into electrical
energy.
The word 'cell' comes
from the Latin for'smal
room', which explains
why a prisoner is kept
in a 'cell'.
A battery is an electrochemical cell, and is defined as 'a device
comprising two or more redox couples' (where each couple com-
prises two redox states of the same material). An oxidation reaction
occurs at the negative pole of the battery in tandem with a reduc-
tion reaction at the positive pole. Both reactions proceed with the
passage of current. The two redox couples are separated physically
by an electrolyte.
The battery requires two redox couples because it is a cell. Each
couple could be thought of as representing half of a complete cell.
This sort of reasoning explains why the two redox couples are
called half-cells. We could, therefore, redefine a cell as a device
comprising two half-cells separated with an electrolyte.
In practice, the voltage of a battery is measured when its two ends are connected
to the two terminals of a voltmeter, one contact secured to the positive terminal of
the battery and the other at the negative. But a voltmeter is a device to measure
differences in potential, so we start to see how the 'voltage' cited on a battery label
is simply the difference in potential between the two poles of the battery.
While the voltage of the cell represents the potential difference
between the two 'terminals' of the battery, in reality it relates to
the separation in energy between the two half-cells. We call this
separation the emf where the initials derive from the archaic phrase
electromotive force. An emf is defined as always being positive.
We have already seen from Faraday's laws how a zero current
implies that no redox chemistry occurs. Accordingly, we stipulate
that the meter must draw absolutely no current if we want to mea-
sure the battery's emf at equilibrium. Henceforth, we will assume
that all values of emf were determined at zero current.
The cell's emf is a pri-
mary physicochemical
property, and is mea-
sured with a voltmeter
or potentiometer.
An emf is always
defined as being posi
tive.
Aside
The term
'emf
follows from the archaic term 'electromotive force'. Physicists prefer
to call the
emf
a 'potential difference' or symbolize it as a
p.d.'.
Confusingly, potential also has the symbols of U, V and E
, depending on
the context.
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
289
Why do hydrogen fuel cells sometimes *dry up'?
Cells and half-cells
Hydrogen fuel cells promise to fuel prototype cars in the near future. We define such
a fuel cell as a machine for utilizing the energies of hydrogen and oxygen gases,
hitherto separated, to yield a usable electric current without combustion or explosion.
Unlike the simple batteries above, the oxygen and hydrogen gases fuelling these cells
are transported from large, high-pressure tanks outside the cell. The gases then feed
through separate pipes onto the opposing sides of a semi-permeable membrane (see
Figure 7.3), the two sides of which are coated with a thin layer of platinum metal,
and represent the anode and cathode of the fuel cell. This membrane helps explain
why such cells are often called PEM fuel cells, where the acronym stands for 'proton
exchange membrane'.
When it reaches the polymer membrane, hydrogen gas is
oxidized at the negative side of the cell (drawn on the left of Figure
7.3), forming protons according to
H
2(g)
-* 2H+ + 2e"
(Pt)
(7.13)
The subscripted 'Pt' helps emphasize how the two electrons con-
duct away from the membrane through the thin layer of platinum
The energy necessary
to cleave the H-H bond
is provided by the
energy liberated when
forming the two H-Pt
bonds after molecular
dissociation.
Load
Hydrogen gas
Permeable
polymer -
membrane
1
Oxygen gas
\f
Water
Thin electrodes
(layers of platinum)
Figure 7.3 A hydrogen-oxygen fuel cell. The water formed at the cathode on the right-hand side
of the cell condenses and collects at the bottom of the cell, and drains through a channel at the
bottom right-hand side
290
ELECTROCHEMISTRY
metal surrounding the electrolyte, and enter into the external circuit where they per-
form work. The platinum also catalyses the dissociation of diatomic H2 gas to form
reactive H" atoms.
Once formed, the protons diffuse through the platinum layer and
enter deep into the layer of semi-permeable membrane. They travel
from the left-hand side of the membrane to its right extremity in
response to a gradient in concentration. (Movement caused by a
concentration gradient will remind us of dye diffusing through a
saucer of water, as described on p. 129.)
At the positive side of the cell (drawn here on the right), oxygen
gas is reduced to oxide ions, according to
The membrane, made
from a perfluorinated
polymer bearing sul-
phonic acid groups, is
known in the trade as
Nafion™.
Note: the electron
count in Equations
(7.13) and (7.14)
should balance in
reality
02(g) + 4e"
(Pt)
20'
(aq)
(7.14)
The electrons necessary to effect the reduction of gaseous oxy-
gen come from the external circuit, and enter the oxygen half-cell
through the layer of platinum coating the cathode, thereby explain-
ing why the electrons in Equation (7.14) are subscripted with 'Pt'.
The 2 ~ ions combine chemically with protons that have traversed the Nafion
membrane and form water, which collects at the foot of the cell.
Because the overall cell reaction is exothermic, the value of the cell emf decreases
with increasing temperature, so the temperature is generally kept relatively low at
about 200 °C. The cell emf is 1.23 V at this temperature.
One of the main advantages of this hydrogen fuel cell is the rapid
rate at which hydrogen is oxidized at the platinum surface. Most
of the cell's operational difficulties relate to the oxygen side of the
device. Firstly, the reduction of gaseous oxygen in Equation (7.14)
is relatively slow, so the rate at which the cell operates is somewhat
limited. But more serious is the way the cell requires a continual
flow of gas, as below. The hydrogen half-cell comprises an elec-
trode couple because two redox states of the same material coexist
there (H2 and H + ). In a similar way, the oxygen half-cell also comprises an electrode
couple, but this time of O2 and O ~.
If the flow of oxygen falters, e.g. when the surface of the cathode is covered with
water, then no gaseous O2 can reach the platinum outer layer. In response, firstly no
electrons are consumed to yield oxide ions, and secondly the right-hand side of the
cell 'floods' with the excess protons that have traversed the polymer membrane and
not yet reacted with O 2- . Furthermore, without the reduction of oxygen, there is no
redox couple at the cathode. The fuel cell ceases to operate, and can produce no more
electrical energy.
This simple example helps explain why a cell requires no fewer than two half-
cells. A half-cell on its own cannot exchange electrons, and cannot truly be termed
a cell.
The rate of oxygen
reduction can be accel-
erated by finely divid-
ing the platinum cata-
lyst, thereby increasing
its effective area.
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
291
Why bother to draw cells?
Cells and the 'cell schematic'
Having denned a cell, we now want to know the best way of representing it. Undoubt-
edly, the simplest is to draw it diagrammatically - no doubt each picture of a cell
being a work of art in miniature.
But drawing is laborious, so we generally employ a more sensible alternative: we
write a cell schematic, which is a convenient abbreviation of a cell. It can be 'read'
as though it was a cross-section, showing each interface and phase. It is, therefore,
simply a shorthand way of saying which components are incorporated in the cell as
cathode, anode, electrolyte, etc., and where they reside.
Most people find that a correct understanding of how to write a cell schematic also
helps them understand the way a cell works. Accordingly, Table 7.2 contains a series
of simple rules for constructing the schematic.
Worked Example 7.4 We construct a cell with the copper(II) | copper and zinc(II) |
zinc redox couples, the copper couple being more positive than the zinc couple. What is
the cell schematic?
Answer strategy: we will work sequentially through the rules in Table 7.2.
First, we note how the copper couple is the most positive, so we
write it on the right. The zinc is, therefore, the more negative and
we write it on the left. We commence the schematic by writing,
ezn (s) ...cu (s) e.
For convenience, we
often omit the subscript
descriptors and the 'e'
and '©' signs.
Table 7.2 Rules for constructing a cell schematic
1. We always write the redox couple associated with the positive electrode on the
right-hand side
2. We write the redox couple associated with the negative electrode on the left-hand side
3. We write a salt bridge as a double vertical line: ||
4. If one redox form is conductive and can function as an electrode, then we write it on one
extremity of the schematic.
5. We represent the phase boundary separating this electrode and the solution containing the
other redox species by a single vertical line: a |
6. If both redox states of a couple reside in the same solution (e.g. Pb + and Pb + ), then they
share the same phase. Such a couple is written conventionally with the two redox states
separated by only a comma: Pb 4+ , Pb 2+
7. Following from 6: we see that no electrode is in solution to measure the energy at
equilibration of the two redox species. Therefore, we place an inert electrode in solution;
almost universally, platinum is the choice
a We write a single line | or, better, a dotted vertical line, if the salt bridge is replaced by a simple porous
membrane.
292
ELECTROCHEMISTRY
There is a phase bound-
ary between the Zn
and Zn 2+ because the
Zn is solid but the Zn 2+
is dissolved within a
liquid electrolyte. A
similar boundary exists
in the copper half-cell.
An alternative way of
looking at the schematic
is to consider it as
'the path taken by a
charged particle dur-
ing a walk from one
electrode to the other'.
To be a redox couple, the zinc ions will be in contact with the
zinc electrode, which we write as Zn 2+ ( aq )|Zn( S ), the vertical line
emphasizing that there is a phase boundary between them. We can
write the other couple as Cu 2+ ( aq )|Cu (s ), with similar reasoning. Note
that if the two electrodes are written at the extreme ends of the
cell schematic, then the redox ionic states must be located some-
where between them. The schematic now looks like ©Zn( S ) |Zn 2+ ( aq ) . . .
Cu
2+
(aq)
|Cu (s
(s)t
Finally, we note that the two half-cells must 'communicate' some-
how - they must be connected. It is common practice to assume that
a salt bridge has been incorporated, unless stated otherwise, so we
join the notations for the two half-cell with a double vertical line, as
0Zn (s) I Zn 2+ (aq) 1 1 Cu 2+ (aq) | Cu (s) © .
SAQ 7.4 Write the cell schematic for a ce
the Fe 3+ c ~ ' — "■'"-'* ■"-* r ~- 2
,Fe (positive) and Co^
comprising
Co (negative) couples.
Worked Example 7.5 Write a cell schematic for a cell comprising
the couples Br2, Br - and H + , H2. The bromine | bromide couple is
the more positive. Assume that all solutions are aqueous.
This is a more complicated cell, because we have to consider the involvement of more
phases than in the previous example.
Right-hand side: the bromine couple is the more positive couple, so we write it on the
right of the schematic. Neither Br2 nor Br~ is metallic, so we need an inert electrode. By
convention, we employ platinum if no other choice is stipulated. The electrode at the far
right of the schematic is therefore Pt, as . . . Pt( S )©.
Br2 and Br - are both soluble in water - indeed, they are mutually
soluble, forming a single-phase solution. Being in the same phase, we
cannot write a phase boundary (as either 'Br2|Br~' or as 'Br~|Br2'),
We write the oxidized
form first if both redox
states reside in the
same phase, sepa-
rating them with a
comma.
so we write it as 'Br2, Br~ (aq )'. Note how we write the oxidized form
first and separate the two redox states with a comma. The right-hand
side of the schematic is therefore 'Br2, Br~( aq )|Pt( S )©'.
Left-hand side : neither gaseous hydrogen nor aqueous protonic solu-
tions will conduct electrons, so again an inert electrode is required on
the extremity of the schematic. We again choose platinum. The left-hand side of the cell
is: ePt( S ).
Hydrogen gas is in immediate contact with the platinum inert electrode. (We bub-
ble it through an acidic solution.) Gas and solution are different phases, so we write the
hydrogen couple as H2( g )|H + ( aq ), and the left-hand side of the schematic becomes '©Pt( S )|
H2( g )|H + (aq) '.
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
293
Finally, we join the two half-cells via a salt bridge, as
Pt (s) | H 2(g) | H+ (aq) || Br 2 , Br" (aq) | Pt (s) © .
From now on, we will omit both the symbols and ©, and merely
assume that the right-hand side is the positive pole and the left-hand
side the negative.
As an extra check,
we note how the salt
bridge dips into the
proton solution, so the
term for H + needs to
be written adjacent to
the symbol Y-
Why do digital watches lose time in the winter?
The temperature dependence of emf
A digital watch keeps time by applying a tiny potential (voltage)
across a crystal of quartz, causing it to vibrate at a precise fre-
quency of v cycles per second. The watch keeps time by counting
off 1 s each time the quartz vibrates v times, explaining why the
majority of the components within the watch comprise a count-
ing mechanism.
Unfortunately, the number of vibrations of the crystal per second
is dictated by the potential applied to the quartz, so a larger voltage
makes the frequency v increase, and a smaller voltage causes v
to slow. For this reason, the potential of the watch battery must
be constant.
In Section 4.6, we saw how the value of AG is never indepen-
dent of temperature, except in those rare cases when A5 , ( ce ii) = 0.
Accordingly, the value of AG( ce ii) for a battery depends on whether
someone is wearing the watch while playing outside in the cold
snow or is sunbathing in the blistering heat of a tropical summer.
And the emf of the watch battery is itself a function of the
change in Gibbs function, AG( ce ii) according to
AG ( C eii) = —nF x emf
(7.15)
where F is the Faraday constant and n is the number of moles of
electrons transferred per mole of reaction. The value of AG( ce ii) is
negative if the reaction proceeds reversibly (see Section 4.3), so the
emf is defined as positive to ensure that AG( ce ii) is always negative.
In other words, the value of AG( ce ii) relates to the spontaneous
cell reaction.
So we understand that as the emf changes with temperature, so
the quartz crystal vibrates at a different frequency - all because
Frequency v has the SI
unit of hertz (Hz). 1 Hz
represents 1 cycle or
vibration per second,
so a frequency of v
Hz means v cycles per
second.
The voltage from the
battery induces a minute
mechanical strain in
the crystal, causing it
to vibrate - a property
known as the piezo-
electric effect.
Care: The output volt-
age of a battery is only
an emf when mea-
sured at zero current,
i.e. when not operating
the watch.
294
ELECTROCHEMISTRY
AG ( C eii) is a function of temperature. Ultimately, then, a digital watch loses time in
the winter as a simple result of the cold.
Worked Example 7.6 The emf of a typical 'lithium' watch battery (which is a cell) is
3.0 V. What is AG ^11)?
Look at the units and
note how 1 J = 1 C x
IV.
The number of electrons transferred n in a lithium battery is one,
since the redox couple is Li + , Li. Inserting values into Equation
(7.15) yields:
AG (
cell)
-1 x 96485 CmoP 1 x 3.0 V
AG (C eii) = -289 500 Jmol -1 = -290 kJmoi" 1
Notice how the molar energy released by a simply battery is enormous.
SAQ 7.5 A manganese dioxide battery has an emf of 1.5 V and n = 2.
Calculate AG (C eii)-
In the absence of any pressure- volume work, the value of AG( ce ii) is equal to the
work needed to transfer charge from the negative end of the cell to the positive. In
practice, AG( ce ii) equates to the amount of charge passed, i.e. the number of charged
particles multiplied by the magnitude of that charge.
hy Is a battery's potential no
Non-equilibrium measurements of emf
We define the emf
as having a positive
value and, strictly, it is
always determined at
equilibrium.
The two currents (for
anode and cathode)
are generally differ-
ent. Neither of them is
related to potential in a
linear way.
A healthy battery for powering a Walkman or radio has a voltage
of about 1.5 V. In the terminology of batteries, this value is called
its open-circuit potential, but an electrochemist talking in terms of
cells will call it the emf. This voltage is read on a voltmeter when
we remove the battery from the device before measurement. But
the voltage would be different if we had measured it while the
battery was, for example, powering a torch.
We perform work whenever we connect the two poles of the
battery across a load. The 'load' in this respect might be a torch,
calculator, car, phone or watch - anything which causes a current
to pass. And this flow of current causes the voltage across the
battery or cell to decrease; see Figure 7.4. We call this voltage the
'voltage under load'.
A similar graph to Figure 7.4 could have been drawn but with
the x axis being the resistance between the two electrodes: if the
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
295
0'
Current drawn/A
Figure 7.4 Schematic diagram showing how a cell's potential decreases with current. We call the
cell potential the emf only when the current is zero
resistance between the two electrodes is zero, which is clearly the case if they should
touch, then the cell potential is zero - we say the cell has 'shorted'.
Ohm's law, Equation (7.16), describes the difference between the emf and a voltage
under load:
V=IR (7.16)
where / is the current flowing, R is the resistance of the load and V is the decrease
in the voltage of the cell. When a current is drawn, the potential of the cell decreases
by the amount V in Equation (7.16). We will call this new (smaller) voltage E(\ oad ),
and its magnitude is given by
£(k>ad) = emf - IR
In summary, we say the voltage of a cell is the same as a cell's
emf if determined at zero current. From Faraday's laws of elec-
trolysis, this criterion implies that none of the compositions within
the cell can change. In other words, a cell emf is an equilib-
rium quantity.
For this reason, it is not wise to speak of terms such as 'anode'
of 'cathode' for a cell at equilibrium, because these terms relate
to electrodes that give or receive charge during current flow; and
our definition of equilibrium implies that no current does flows.
We therefore adopt the convention: the terms 'anode' or 'cath-
ode ' will no longer be employed in our treatment of equilibrium
electrochemistry .
(7.17)
The emf can only ever
be determined at zero
current.
Why a
battery's emf
decreases permanently
after a
current has
flowed
is explained on
p. 328.
What is a 'standard cell'?
The thermodynamics of cells
A standard cell produces a precise voltage and, before the advent of reliable volt-
meters, was needed to calibrate medical and laboratory equipment. It is generally
agreed that the first standard cell was the Clark cell (see p. 299), but the most popular
was the Weston saturated cadmium cell, patented in 1893.
296
ELECTROCHEMISTRY
Edward Weston (1850-1936) was a giant in the history of electrical measuring
instruments. In the field of measurement, he developed three important components:
the standard cell, the manganin resistor and the electrical indicating instrument.
The main advantage of Weston's cell was its insensitivity to temperature, and the
emf of almost 1 V: to be precise, 1.0183 V at 20 °C. It is usually constructed in an H-
shaped glass vessel. One arm contains a cadmium amalgam electrode beneath a paste
of hydrated cadmium sulphate (3CdSC>4 • 5H2O) cadmium sulphate and mercury(I) sul-
phate. The other arm contains elemental mercury. Its schematic is Cd(Hg)|CdS04( aq ),
Hg 2 S0 4 |Hg.
The Weston saturated cadmium cell became the international
standard for emf in 1911. Weston waived his patent rights shortly
afterward to ensure that anyone was allowed to manufacture it.
Weston's cell was much less temperature sensitive than the previ-
ous standard, the Clark cell. We recall how the value of AG changes
with temperature according to Equation (4.38). In a similar way, the
value of AG ( C eii) for a cell relates to the entropy change AS( ce n) such
that the change of emf with temperature follows
The Clark cell was
patented by Latimer
Clark in the 1880s, and
was the first standard
cell.
Remember.
the small
subscripted
'p' indi-
cates that the quantity
is measurec
at con-
stant pressure. It does
not mean 'multiplied
byp'.
The temperature volt-
age coefficient has
several names: 'tem-
perature coefficient',
'voltage coefficient' or
'temperature coef-
ficient of voltage'.
Table 7.3 contains a
few values of
(d (emO/dT).
A 5,
(cell)
nF
d(gm/)
dr
(7.18)
the value of (d(emf)/dT) p is virtually zero for the Weston cell.
If we assume the differential (d (emf)IdT) is a constant, then
Equation (7.18) has the form of a straight line, y = mx, and a
graph of emf (as 'v') against T (as 'x') should be linear. Figure 7.5
shows such a graph for the Clark cell, Hg|HgS0 4 , ZnS04(sat'd)|
Zn. Its gradient represents the extent to which the cell emf varies
with temperature, and is called the temperature voltage coefficient.
The gradient may be either positive or negative depending on the
cell, and typically has a magnitude in the range 10~ 5 to 10~ 4
VK" 1 . We want a smaller value of (d (emf)IdT) if the emf is
to be insensitive to temperature.
Having determined the temperature dependence of emf as the
gradient of a graph of emf against temperature, we obtain the value
of AS( C eii) as 'gradient x n x F' .
That this value of
d(emf)/dT is neg-
ative tells us that
the emf DEcreases
when the temperature
INcreases.
Worked Example 7.7 The temperature voltage coefficient for a sim-
ple alkaline torch battery is —6.0 x 10~ 4 VK - '. What is the entropy
change associated with battery discharge? The number of electrons
transferred in the cell reaction n = 1.
Inserting values into Equation (7.18):
A 5,
(cell)
= 2x 96485 CmoP 1 x -6.0 x 10" 4 VK"
AS( C eii) = -116 J K" 1 mor 1
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND 297
1.45
1.44-
>
^1.43-
1.42-
Gradient = 'temperature
voltage coefficient'
1.41
280
290
~T
T~
300 310
Temperature T/K
320
330
Figure 7.5 Graph of cell emf against temperature for the Clark cell Hg|HgS0 4 , ZnS04(sat'd)|Zn.
We call the gradient of this graph the 'temperature voltage coefficient'
Table 7.3 Temperature voltage coefficients for various
cells and half cells
Cell 3
(d(em/)/dr)„/VK-
Standard hydrogen electrode
Clark standard cell
Saturated calomel electrode
Silver- silver chloride
Silver- silver bromide
Weston standard cell
(by definition) 13
6.0 x 1(T 4
+7 x 1CT 4
-8.655 x 1CT 5
-4.99 x 1(T 4
-5 x irr 5
"Individual electrodes are cited with the SHE as the second elec-
trode of the cell.
b The potential of the SHE is defined as zero at all temperatures.
SAQ 7.6 The emf of a lithium watch battery is exactly 3.000 V at 298 K,
but the value decreases to 2.985 V at 270 K. Calculate the temperature
voltage coefficient and hence the change in entropy AS (C eii) during cell
discharge. (Take n = 1.)
Occasionally, the temperature voltage coefficient is not expressed as a simple number,
but as a power series in T (we generally call it a virial series, or expansion). For example,
Equation (7.19) cites such a series for the cell Pt( s )|H2( g )|HBr( aq )|AgBr (s JAg (s) :
emf IN = 0.071 31 - 4.99 x 10 _4 (r/K - 298)
3.45 x 10 _6 (r/K - 298) 2 (7.19)
We insert values of temperature T into the expression to obtain a value for emf. Values
of AS( ce ii) are obtained by performing two calculations, inserting first one temperature
298
ELECTROCHEMISTRY
7\ to obtain the emf at T\, and then a second T2 to obtain another value of emf. We
calculate a value of (d(emf)/dT) using
d(emf) emf at T2 — emf at T\
AT ~ T 2 -Ti
The value of A5( ce ii) is then determined in the usual way via Equation (7.18)
(7.20)
SAQ 7.7 Insert values of T = 310 K into Equation (7.19) to calculate the
potential of the cell Pt (S )|H2(g)|HBr (aq) |AgBr (s) |Ag (s) .
SAQ 7.8 Repeat the calculation in SAQ 7.7, this time with 7 = 360 K,
and hence determine AS (ce ii).
Justification Box 7.1
The relationship between changes in Gibbs function and temperature (at constant pres-
sure p) is defined using Equation (4.38):
-AS
9AG\
dT ),
We know from Equation (7.15) that the change in AG( ce ii) with temperature is '— nF x
emf . The entropy change of the cell AS( Ce ii) is then obtained by substituting for AG( ce ii)
in Equation (7.18):
-A5 ( .
cell)
d(—nF x emf)
dT
(7.18)
Firstly, the two minus signs cancel; and, secondly, n and F are both constants. Taking
them out of the differential yields Equation (7.18) in the form above.
These values of AG,
AH and AS relate to a
complete cell, because
thermodynamic data
cannot be measured
experimentally for half-
cells alone.
To obtain the change in enthalpy during the cell reaction, we
recall from the second law of thermodynamics how AH = AG +
TAS (Equation (4.21)). In this context, each term relates to the
cell. We substitute for AG( ce ii) and A5( ce n) via Equations (7.15)
and (7.18) respectively, to yield
AH,
(cell)
= — nF x emf + TnF
d(em/)
dT
(7.21)
so, knowing the emf as a function of temperature, we can readily obtain a value of
A //(cell)-
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
299
Worked Example 7.8 The Clark cell Zn|Zn 2+ , Hg 2 S04|Hg is often employed as a
standard cell since its emf is known exactly as a function of temperature. The cell emf is
1.423 V at 298 K and its temperature coefficient of voltage is —1.2 x 10~ 4 VK" 1 . What
are AG( ce ii), AS( ce ii) and thence A//( ce u) at 298 K?
Before we commence, we note that the spontaneous cell reaction is
Zn + Hg 2 S0 4 + 7H 2 ► ZnS0 4 ■ 7H 2 + 2Hg°
so the cell reaction is a two-electron process.
Next, we recall from Equation (7.15) that AG( Ce u) = — nF x emf. Inserting values for
the cell at 298 K gives
AG (cell) = -2 x 96485 Cmol" 1 x 1.423 V
AG ( C eii) = —275 kJnmT 1
Then, from Equation (7.18), we recall that
AS( C eii) = nF
d(em/)
dT
Inserting values:
AS (ce ii) = 2x 96485 CmoP 1 x (-1.2 x 10" 4 VK~')
A5 (ce ii) = -23.2 JK" 1 mor 1
Finally, from Equation (4.21), we say that A//( ce ii) = AG (ce n) + TAS( ce \i). We again
insert values:
AH,
(cell)
(-275 kJmol" 1 ) + (298 K x -23.2 JKT 1 moP 1 )
so
AH,
(cell)
-282 kJ mor
SAQ 7.9 A different cell has an emf of 1.100 V at 298 K.
The temperature voltage coefficient is +0.35 mVK -1 . Cal-
culate AG ( ceii), AS ( ceii) and hence AW (ce n) for the cell at
298 K. Take n = 2.
Remember: 5 mV (5
m/7//-volts) is the same
as 5 x 10- 3 V.
Having performed a few calculations, we note how values of AG( ce ii) and for
A //(cell) tend to be rather large. Selections of AG( ce ii) values are given in Table 7.4.
300
ELECTROCHEMISTRY
Table 7.4 Table of values of AG (CE ii) as a function of
emf and n
AG
cell)/kJ
mor 1
AG (
C eii)/kJ mol '
emf IV
(when n -
= D
(when n = 2)
1.1
106
212
1.2
116
232
1.3
125
251
1.4
135
270
1.5
145
289
1.6
154
309
1.7
164
328
1.8
174
347
1.9
183
367
2.0
193
386
Aside
While electrochemical methods are experimentally easy,
the practical difficulties of obtaining accurate thermo-
dynamic data are so severe that the experimentally deter-
mined values of AG( Ce ii), A//( ce u) and A5( ce ii) can only
be regarded as 'approximate' unless we perform a daunt-
ing series of precautions. The two most common errors
are: (1) allowing current passage to occur, causing the
value of the cell emf to be too small; and (2) not per-
forming the measurement reversibly.
The most common fault under (2) is changing the
temperature of the cell too fast, so the temperature inside the cell is not the same as the
temperature of, for example, the water bath in which a thermometer is placed.
None of these ther-
modynamic equations
is reliable if we fail
to operate the cell
reversibly, since the
emf is no longer an
exact thermodynamic
quantity.
Why aren't electrodes made from wood?
Electrodes: redox, passive and amalgam
Electrochemical cells comprise a minimum of two half-cells, the
energetic separation between them being proportional to the cell
emf. Since this energy is usually expressed as a voltage, we see
that the energy needs to be measured electrically as a voltage.
We have seen already how a cell's composition changes if a
charge flows through it - we argued this phenomenon in terms
of Faraday's laws. We cause electrochemical reactions to occur
whenever a cell converts chemical energy into electrical energy.
The impetus for elec-
tronic motion is the
chance to lose energy
(see p. 60), so elec-
trons move from high
energy to low.
INTRODUCTION TO CELLS: TERMINOLOGY AND BACKGROUND
301
Currents conduct
through an electrode
by means of electrons.
A redox electrode acts
as a reagent as well as
an electron conductor,
as the metal of an
electrode can also be
one component part of
a redox couple.
For this reason, we say a battery or cell discharges during operation, with each
electron from the cell flowing from high energy to low.
All electrochemical cells (including batteries) have two poles:
one relates to the half-cell that is positively charged and the other
relates to the negatively charged pole. Negatively charged electrons
are produced at the anode as one of the products of the electro-
chemical reaction occurring at there. But if the electrons are to
move then we need something through which they can conduct to and from the
terminals of the cell: we need an electrode.
The phenomenon we call electricity comprises a flow of charged electrons. Wood
is a poor conductor of electricity because electrons are inhibited from moving freely
through it: we say the wood has a high electrical 'resistance' R. By
contrast, most metals are good conductors of charge. We see how
an electrode needs to be electrically conductive if the electrons are
to move.
Most electrodes are metallic. Sometimes the metal of an electrode
can also be one component part of a redox couple. Good examples
include metallic iron, copper, zinc, lead or tin. A tin electrode forms
a couple when in contact with tin(IV) ions, etc. Such electrodes are
called redox electrodes (or non-passive). In effect, a redox electrode
has two roles: first, it acts as a reagent; and, secondly, it measures
the energy of the redox couple of which it forms one part when
connected to a voltmeter.
Some metals, such as aluminium or magnesium, cannot function
as redox electrodes because of a coating of passivating oxide. Oth-
ers, such as calcium or lithium, are simply too reactive, and would
dissolve if immersed in solution.
But it is also extremely common for both redox states of a redox
couple to be non-conductive. Simple examples might include dis-
solving bromine in an aqueous solution of bromide ions, or the
oxidation of hydrogen gas to form protons, at the heart of a hydro-
gen fuel cell; see Equation (7.13). In such cases, the energy of
the couple must be determined through a different sort of elec-
trode, which we call an inert electrode. Typical examples of inert
electrodes include platinum, gold, glassy carbon or (at negative
potentials) mercury. The metal of an inert electrode itself does not
react in any chemical sense: such electrodes function merely as a
probe of the electrode potential for measurements at zero current,
and as a source, or sink, of electrons for electrolysis processes if
current is to flow.
The final class of electrodes we encounter are amalgam elec-
trodes, formed by 'dissolving' a metal in elemental (liquid) mer-
cury, generally to yield a solid. We denote an amalgam with brack-
ets, so the amalgam of sodium in mercury is written as Na(Hg). The
properties of such amalgams can be surprisingly different from their
Metallic mercury is a
poor choice of inert
electrode at positive
potentials because it
oxidizes to form Hg(II)
ions.
We require an inert
electrode when both
parts of a redox couple
reside in solution, or
do not conduct: the
electrode measures the
energy of the couple.
We denote an amal-
gam by writing the 'Hg'
in brackets after the
symbol of the dissolved
element; cobalt amal-
gam is symbolized as
Co(Hg).
302
ELECTROCHEMISTRY
constituent parts, so Na(Hg) is solid and, when prepared with certain concentrations
of sodium, does not even react with water.
Currents conduct
through an electrolyte
by means of ions.
Why is electricity more dangerous in wet weather?
Electrolytes for cells, and introducing ions
Most electrical apparatus is safe when operated in a dry environment, but everyone
should know that water and electricity represent a lethal combination. Only a minimal
amount of charge conducts through air, so cutting dry grass with an electric mower
is safe. Cutting the same grass during a heavy downpour risks electrocution, because
water is a good conductor of electricity.
But water does not conduct electrons, so the charges must move
through water by a wholly different mechanism than through a
metallic electrode. In fact, the charge carriers through solutions -
aqueous or otherwise - are solvated ions. The 'mobility' /x of an
ion in water is sufficiently high that charge conducts rapidly from
a wet electrical appliance toward the person holding it: it behaves
as an electrolyte.
All cells comprise half-cells, electrodes and a conductive elec-
trolyte; the latter component separates the electrodes and conducts
ions. It is usually, although not always, a liquid and normally has
an ionic substance dissolved within it, the solid dissociating in
solution to form ions. Aqueous electrolytes are a favourite choice
because the high 'dielectric constant' e of water imparts a high
'ionic conductivity' k to the solution.
Sometimes electrochemists are forced to construct electrochem-
ical cells without water, e.g. if the analyte is water sensitive or
merely insoluble. In these cases, we construct the cell with an
organic solvent, the usual choice being the liquids acetonitrile,
propylene carbonate (I), A^.A^-dimethylformamide (DMF) or di-
methylsulphoxide (DMSO), each of which is quite polar because
of its high dielectric constant e.
Ultimately, the word
'ion' derives from the
Greek eimi 'to go',
implying the arrival of
someone or something.
We get the English
word 'aim' from this
root.
Ionic conductivity is
often given the Greek
symbol kappa (k)
whereas electrical con-
ductivity is given the
different Greek symbol
sigma (a).
O
O
A
o
V
(i)
CH S
In some experiments, we need to enhance the ionic conductivity of a solution,
so we add an additional ionic compound to it. Rather confusingly, we call both the
compound and the resultant solution 'an electrolyte'.
INTRODUCING HALF-CELLS AND ELECTRODE POTENTIALS
303
The preferred electrolytes if the solvent is water are KC1 and NaNC>3. If the solvent
is a non-aqueous organic liquid, then we prefer salts of tetra-alkyl ammonium, such
as tetra-n-butylammonium tetrafluoroborate, "Bu4N + BF4-.
7.2 Introducing half-cells and electrode
potentials
Why are the voltages of watch and car batteries
different?
Relationships between emf and electrode potentials
Being a cell, a battery contains two half-cells separated by an electrolyte. The elec-
trodes are needed to connect the half-cells to an external circuit. Each electrode may
act as part of a redox couple, but neither has to be.
The market for batteries is huge, with new types and applications
being developed all the time. For example, a watch battery is a type
of 'silver oxide' cell: silver in contact with silver oxide forms one
half-cell while the other is zinc metal and dications. Conversely,
a car battery is constructed with the two couples lead(IV)|lead
and lead(IV)|lead(II). The electrolyte is sulphuric acid, hence this
battery's popular name of 'lead-acid' cell (see further discussion
on p. 347).
The first difference between these two batteries is the voltage
they produce: a watch battery produces about 3 V and a lead-acid
cell about 2 V. The obvious cause of the difference in emf are
the different half-cells. The 'electrode potential' E is the energy,
expressed as a voltage, when a redox couple is at equilibrium.
As a cell comprises two half-cells, we can now define the emf
according to
An "electrode poten-
tial' E is the energy
(expressed as a volt-
age) when a redox
couple is at equilibrium.
The value of E cannot
be measured directly
and must be calculated
from an experimental
emf.
Two redox states of the
same material form a
redox couple.
emf
J (positive half-cell)
' (negative half-cell)
(7.22)
This definition is absolutely crucial. It does not matter if the
values of E for both half-cells are negative or both are positive:
is (positive) is defined as being the more positive of the two half-cells,
and ^(negative) is the more negative.
We now consider the emf in more detail, and start by saying
that it represents the separation in potential between the two half-
cell potentials; See Equation (7.22). In order for AG( ce ii) to remain
positive for all fhermodynamically spontaneous cell discharges, the
emf is defined as always being positive.
It is impossible to
determine the potential
of a single electrode:
only its potential rel-
ative to another elec-
trode can be measured.
304
ELECTROCHEMISTRY
Another convention dictates that we write the more positive electrode on the right-
hand side of a cell, so we often see Equation (7.22) written in a slightly different
form:
emf = £ ( rhs) - £(lhs) (7.23)
Being a potential, the electrode potential has the symbol E. We must exercise care
in the way we cite it. E is the energy of a redox couple, since it relates to two redox
species, both an oxidized and reduced form, 'O' and 'R' respectively. We supplement
the symbol E with appropriate subscripts, as £"o,R-
Worked Example 7.9 Consider the electrode potentials for metallic lead within a lead-
acid battery. The lead has three common redox states, Pb 4+ , Pb 2+ and Pb°, so there are
three possible equilibria to consider:
Pb 4
Pb 4
(aq)
+ 2e~
(a q) + 4e-
(aq)
Pb 2+ ran , + 2e
Pb 2
(aq)
s)
Pb° (s
for which E
(equilibrium)
for which E
(equilibrium)
for which E
(equilibrium)
w+.pb 2
^piy+.pb
^p^+.pb
We now see why it is so misleading to say merely E P \, or, worse, 'the electrode potential
of lead'.
We cite the oxidized
form first, as E 0R .
We usue
Ily omit the
superscr
pted
'zero'
on unch
arged
redox
states.
When choosing
bet-
ween two ionic
valen-
ces,
the name
of the
higher (more oxidized)
state
ends with -ic
and
the lower
(less
oxidized) form
ends
with
-ous.
We conventionally cite the oxidized form first within each sym-
bol, which is why the general form is Eo^, so £ Pb 4+ Pb 2+ is correct,
but E Ph 2+ pb 4+ is not. Some people experience difficulty in decid-
ing which redox state is oxidized and which is the reduced. A
simple way to differentiate between them is to write the balanced
redox reaction as a reduction. For example, consider the oxidation
reaction in Equation (7.1). On rewriting this as a reduction, i.e.
Al 3+ ( aq ) + 3e~ = Al( s ), the oxidized redox form will automatically
precede the reduced form as we read the equation from left to
right, i.e. are written in the correct order. For example, £o,r f° r
the couple in Equation (7.1) is E Al i+ A1 .
We usually cite an uncharged participant without a superscript.
Considering the reaction Pb + + 2e~ = Pb, the expression E Pb 2+ Pb
is correct but the '0' in E Pb 2+ Pb o is superfluous.
SAQ 7.10 Consider the cobaltous ion | cobalt redox cou-
ple. Write an expression for its electrode potential.
With more complicated redox reactions, such as 2H + (aq ) + 2e~ =
H2( g ), we would not normally write the stoichiometric number,
so we prefer £h+,h 2 to £ , 2 h + ,h 2 ; tne additional '2' before H + is
superfluous here.
INTRODUCING HALF-CELLS AND ELECTRODE POTENTIALS
305
SAQ 7.11 Write down an expression for the electrode potential of the
bromine | bromide couple. [Hint: it might help to write the balanced redox
reaction first.]
How do 'electrochromic' car mirrors work?
Introducing an orbital approach to dynamic electrochemistry
It's quite common when driving at night to be dazzled by the
lights of the vehicle behind as they reflect from the driver's new-
view or door mirror. We can prevent the dazzle by forming a
layer of coloured material over the reflecting surface within an
electrochromic mirror. Such mirrors are sometimes called 'smart
mirrors' or electronic 'anti-dazzle mirrors'.
These mirrors are electrochromic if they contain a substance that changes colour
according to its redox state. For example, methylene blue, MB + (II), is a chromophore
because it has an intense blue colour. II is a popular choice of electrochromic material
for such mirrors: it is blue when fully oxidized, but it becomes colourless when
reduced according to
Electrochromic mirrors
are now a common
feature in expensive
cars.
MB U -
colourless
MB +
blue
+ e"
(7.24)
(H 3 C) 2 N
N(CH
3)2
(II)
We can now explain how an electrochromic car mirror operates. The mirror is
constructed with II in its colourless form, so the mirror functions in a normal way.
The driver 'activates' the mirror when the 'anti-dazzle' state of the mirror is required,
and the coloured form of methylene blue (MB + ) is generated oxidatively according
to Equation (7.24). Coloured MB + blocks out the dazzling reflection at the mirror by
absorbing about 70 per cent of the light. After our vehicle has been overtaken and
we require the mirror to function normally again, we reduce MB + back to colourless
MB via the reverse of Equation (7.24), and return the mirror to its
colourless state. These two situations are depicted in Figure 7.6.
We discuss 'colour' in Chapter 9, so we restrict ourselves here to
saying the colour of a substance depends on the way its electrons
interact with light; crucially, absorption of a photon causes an elec-
tron to promote between the two frontier orbitals. The separation
in energy between these two orbitals is E, the magnitude of which
relates to the wavelength of the light absorbed X according to the
Planck-Einstein equation, E = hc/X, where h is the Planck constant and c is the
An electron is donated
to an orbital during
reduction. The electron
removed during oxida-
tion is taken from an
orbital.
306
ELECTROCHEMISTRY
Conventional
mirror
Electrochromic
mirror
Incident beam
Emerging beam
Reflective
surface
(a)
Incident beam
Emerging beam
Reflective Electrochromic
surface layers
(b)
Figure 7.6 Mirrors: (a) an ordinary car driver's mirror reflects the lights of a following car, which
can dazzle the driver; (b) in an electrochromic mirror, a layer of optically absorbing chemical is
electro-generated in front of the reflector layer, thereby decreasing the scope for dazzle. The width
of the arrows indicates the relative light intensity
speed of light in a vacuum. The value of E for MB corresponds to an absorption
in the UV region, so MB appears colourless. Oxidation of MB to MB + causes a
previously occupied orbital to become empty, itself changing the energy separation
E between the two frontier orbitals. And if E changes, then the Planck-Einstein
equation tells us the wavelength A of the light absorbed must also change. E for
MB + corresponds to A of about 600 nm, so the ion is blue.
The reasoning above helps explain why MB and MB + have different colours. To
summarize, we say that the colours in an electrochromic mirror change following
oxidation or reduction because different orbitals are occupied before and after the
electrode reaction.
Why does a potential form at an electrode?
Formation of charged electrodes
For convenience, we will discuss here the formation of charges with the example of
copper metal immersed in a solution of copper sulphate (comprising Cu 2+ ions). We
consider first the situation when the positive pole of a cell is, say, bromine in contact
with bromide ions, causing the copper to be the negative electrode.
Let's look at the little strip cartoon in Figure 7.7, which shows the surface of a
copper electrode. For clarity, we have drawn only one of the trillion or so atoms
on its surface. When the cell of which it is a part is permitted to discharge spon-
taneously, the copper electrode acquires a negative charge in consequence of an
oxidative electron-transfer reaction (the reverse of Equation (7.7)). During the oxi-
dation, the surface-bound atom loses the two electrons needed to bond the atom to
the electrode surface, becomes a cation and diffuses into the bulk of the solution.
INTRODUCING HALF-CELLS AND ELECTRODE POTENTIALS 307
m
m
Surface atom
m
Positive ion
(cation)
Electrode Excess surface charge
Figure 7.7 Schematic drawing to illustrate how an electrode acquires its negative charge
m
Positive ion
(cation)
Surface atom
I
Electrode Excess surface charge
Figure 7.8 Schematic drawing to illustrate how an electrode acquires its positive charge
The two electrons previously 'locked' into the bond remain on the electrode surface,
imparting a negative potential.
We now consider a slightly different cell in which the copper half-cell is the positive
pole. Perhaps the negative electrode is zinc metal in contact with Zn 2+ ions. If the cell
discharges spontaneously, then the electron-transfer reaction is the reduction reaction
in Equation (7.7) as depicted in the strip cartoon in Figure 7.8. A bond forms between
the surface of the copper electrode and a Cu + cation in the solution The electrons
needed to reduce the cation come from the electrode, imparting a net positive charge
to its surface.
Finally, we should note that the extent of oxidation or reduction needed to cause a
surface charge of this type need not be large; and the acquisition of charge, whether
positive or negative, is fast and requires no more than a millisecond after immersing
the electrodes in their respective half-cells.
308
ELECTROCHEMISTRY
7.3 Activity
Why does the smell of brandy decrease after
dissolving table salt in it?
We mention the volatile
alcohol here because it
is responsible for the
smell.
Real and 'perceived' concentrations
At the risk of spoiling a good glass of brandy, try adding a little table salt to it and
notice how the intensity of the smell is not so strong after the salt dissolves.
We recall from Chapter 5 how the intensity of a smell we detect
with our nose is proportional to the vapour pressure of the substance
causing it. The vapour pressure of ethanol is Methanol) > its magni-
tude being proportional to the mole fraction of ethanol in the brandy;
brandy typically contains about 40 per cent (by volume) of alcohol.
Although adding table salt does not decrease the proportion of
the alcohol in the brandy, it does decrease the apparent amount. And because the
perceived proportion is lowered, so the vapour pressure drops, and we discern the
intensity of the smell has decreased. We are entering the world of 'perceived' con-
centrations.
Although the actual concentrations of the volatile components
in solution remain unchanged after adding the salt, the system per-
ceives a decrease in the concentration of the volatile components.
This phenomenon - that the perceived concentration differs from
the real concentration - is quite common in the thermodynamics
of solution-phase electrochemistry. We say that the concentration persists, but the
'activity' a has decreased by adding the salt.
As a working definition, the activity may be said to be 'the perceived concentra-
tion' and is therefore somewhat of a 'fudge factor'. More formally, the activity a is
defined by
The 'activity' a is the
thermodynamically per-
ceived concentration.
■y
(7.25)
where c is the real concentration. The concluding term y, termed the activity coeffi-
cient, is best visualized as the ratio of a solute's 'perceived' and 'real' concentrations.
The activity a and the activity coefficient y are both dimension-
less quantities, which explains why we must include the additional
'c & ' term, thereby ensuring that a also has no units. We say the
We only add the term
c e in order to render
the activity dimension-
less.
value of c s is 1 moldm - when c is expressed in the usual units
of mol dm -3 , and 1 molm -3 if c is expressed in the SI units of
mol m~ 3 , and so on.
Why does the smell of gravy become less intense after
adding salt to it?
The effect of composition on activity
Gravy is a complicated mixture of organic chemicals derived from soluble meat
extracts. Its sheer complexity forces us to simplify our arguments, so we will
ACTIVITY
309
approximate and say it contains just one component in a water-based solution. Any
incursions into reality, achieved by extending our thoughts to encompass a multi-
component system, will not change the nature of these arguments at all.
Adding table salt to gravy causes its lovely smell to become less intense. This is a
general result in cooking: adding a solute (particularly if the solute is ionic) decreases
the smell, in just the same way as adding table salt decreased the smell of brandy in
the example directly above.
The ability to smell a solute relies on it having a vapour pres-
sure above the solution. Analysing the vapour above a gravy dish
shows that it contains molecules of both solvent (water) and solute
(gravy), hence its damp aroma. The vapour pressures above the
gravy dish do not alter, provided that we keep the temperature con-
stant and maintain the equilibrium between solution and vapour.
The proportion of the solute in the vapour is always small because
most of it remains in solution, within the heavier liquid phase.
As a good approximation, the vapour pressure of each solute in the vapour above
the dish is dictated by the respective mole fractions in the gravy beneath. As an
example, adding water to the gravy solution dilutes it and, there-
fore, decreases the gravy smell, because the mole fraction of the
gravy has decreased.
Putting ionic NaCl in the gravy increases the number of ions in
solution, each of which can then interact with the water and the solute,
which decreases the 'perceived concentration' of solute. In fact, we
can now go further and say the thermodynamic activity a represents
the concentration of a solute in the presence of interactions.
The pressure above
a solution relates to
the composition of
solution, according
to Henry's law; see
Section 5.6.
An electrochemist asses-
ses the number of ions
and their relative influ-
ence by means of the
'ionic strength' I (as
defined below).
Why add alcohol to eau de Cologne?
Changing the perceived concentration
Fragrant eau de Cologne is a dilute perfume introduced in Col-
ogne (Germany) in 1709 by Jean Marie Farina. It was probably
a modification of a popular formula made before 1700 by Paul
Feminis, an Italian in Cologne, and was based on bergamot and
other citrus oils. The water of Cologne was believed to have the
power to ward off bubonic plague.
Eau de Cologne perfume is made from about 80-85 per cent
water and 12-15 per cent ethanol. Volatile esters make up the
remainder, and provide both the smell and colour.
The vapour pressure of alcohol is higher than that of water, so
adding alcohol to an aqueous perfume increases the pressure of the
gases above the liquid. In this way, the activity a of the organic
components imparting the smell will increase and thereby increase
the perceived concentration of the esters. And increasing fl( es ter)
The word 'perfume'
comes from the Latin
per fumem, meaning
'through smoke'.
These esters are sta-
ble in the dark, but
degrade in strong sun-
light, which explains
why so many perfumes
are sold in bottles of
darkened or frosted
glass.
310
ELECTROCHEMISTRY
has the effect of making the eau de Cologne more pungent. Stated another way, the
product requires less ester because the alcohol increases its perceived concentration.
Incidentally, the manufacturer also saves money this way.
Thermodynamic activity a
Every day, electrochemists perform measurements that require a knowledge of the
activity a. Measurements can be made in terms of straightforward concentrations if
solutions are very dilute, but 'very dilute' in this context implies c « 10~ 4 moldm -3 ,
or less. Since most solutions are far more concentrated than millimoles per litre, from
now on we will write all equations in terms of activities a instead of concentration c.
The values of activity a and concentration c are the same for
very dilute solutions, so the ratio of a and c is one because the
real and perceived concentrations are the same. If a = c, then
Equation (7.25) shows how the activity coefficient y has a value
of unity at low concentration.
By contrast, the perceived concentration is usually less than the
real concentration whenever the solution is more concentrated, so
y < 1. To illustrate this point, Figure 7.9 shows the relationship
between the activity coefficient y (as 'v') and concentration (as 'x') for a few sim-
ple solutes in water. The graph shows clearly how the value of y can drop quite
dramatically as the concentration increases.
The concept of activity
was introduced in the
early 20th century by
one of the giants of
American chemistry, G.
N. Lewis.
0.8-
0.6-
HCI [1 : 1]
KCI [1 : 1]
KOH [1 : 1]
NaBr[1 : 1]
H 2 S0 4 [1 :2]
CuS0 4 [2 : 2]
ln 2 (SQ 4 ) 3 [2 : 3]
1.2
0.4 0.6 0.8
Concentration / mol dm
-3
Figure 7.9 The dependence of the mean ionic activity coefficient y± on concentration for a few
simple solutes
ACTIVITY
311
Unit and unity here
both mean 'one', so
unit activity means
a = 1.
The activity of a solid The activity of a pure solid in its standard
state is unity, so the activity of pure copper or of zinc metal elec-
trodes is one. We write this as a^cu) or a (Zn) = 1-
The activity of an impure solid is more complicated. Such an
'impure' system might be represented by a solid metal with a dirty
surface, or it might represent a mixture of two metals, either as an alloy or an amalgam
with a metal 'dissolved' in mercury.
For example, consider the bi-metallic alloy known as bronze,
which contains tin (30 mol%) and copper (70 mol%). There are
two activities in this alloy system, one each for tin and copper. The
activity of each metal is obtained as its respective mole fraction x,
so x ( sn) = a (Sn ) = 0.3, and a (Cu) = 0.7.
An 'alloy' is a mixture
of metals, and is not a
compound.
Worked Example 7.10 A tooth filling is made of a silver amalgam that comprises
37 mol% silver. What is the activity of the mercury, a
(Hg)?
The activity of the mercury tf(Hg) is the same as its mole fraction,
X(Hg). By definition
*(Hg) + *(Ag) = 1
so
X (Hg)
1 -X
(Ag)
0.63
The sum of the mole
fractions x must always
add up to one because
'the sum of the con-
stituents adds up to
the whole'.
The activity is therefore
*(H g ) = fl(Hg) = 0.63
Reminder: the value of
p & is 10 5 Pa.
The activity of a gas The activity of a pure gas is its pressure
(in multiples of p^), so a(H 2 ) = P(h 2 ) ^ P & - The activity of pure
hydrogen gas a(H 2 ) at p e is therefore unity.
In fact, for safety reasons it is not particularly common to employ
pure gases during electrochemical procedures, so mixtures are pre-
ferred. As an example, the hydrogen gas at the heart of the standard
hydrogen electrode (SHE) is generally mixed with elemental nitro-
gen, with no more than 10 per cent of H2 by pressure. We call the
other gas a base or bath gas. Conversely, we might also say that hydrogen dilutes
the nitrogen, and so is a diluent.
In such cases, we can again approximate the activity to the mole fraction x.
In a mixture of gases,
we call the inert gas a
base or bath gas.
Worked Example 7.11 Hydrogen gas is mixed with a nitrogen 'bath gas'. The overall
pressure is p® . If the mole fraction of the hydrogen is expressed as 10 per cent, what is
its activity?
By definition, x (X ) = partial pressure, Pqq, so
a (H 2 ) — P(H 2 ) + P(total) =0.1
312
ELECTROCHEMISTRY
Amalgams are liquid
when very dilute, but
are solid if the mole
fraction of mercury
drops below about 70
per cent.
The activity of a solution It is unwise to speak in broad terms of 'the activity of
a solution' because so many different situations may be considered. For example,
consider the following two examples.
(1) The activity of a mixture of liquids. It is rarely a good idea to
suggest that the activity of a liquid in a mixture is equal to its mole
fraction x because of complications borne of intermolecular inter-
actions (e.g. see Chapter 2 and Section 5.6 concerning Raoult's
law). Thankfully, it is generally rare that an electrochemist wants
to study liquid mixtures of this sort (except amalgams diluted to a
maximum mole fraction of about 1 per cent metal in Hg), so we
will not consider such a situation any further.
(2) The activity of a solute in a liquid solvent. The activity a and concentration c
may be considered to be wholly identical if the concentration is tiny (to a maximum
of about 10~ 3 moldm -3 ), provided the solution contains no other solutes. Such a
concentration is so tiny, however, as to imply slightly polluted distilled water, and is
not particularly useful.
For all other situations, we employ the Debye-Huckel laws (as below) to calculate
the activity coefficient y. And, knowing the value of y, we then say that a = (c 4-
c & ) x y (Equation (7.25)), remembering to remove the concentration units because
a is dimensionless.
Why does the cell em f alter after adding LiCI?
In fact, a similar result
is obtained when adding
most ionic electrolytes.
Ionic 'screening'
Consider the Daniell cell Zn|Zn 2+ ||Cu 2+ |Cu. The cell emf is about 1.1 V when pre-
pared with clean, pure electrodes and both solutions at unit activity. The emf decreases
to about 1.05 V after adding lithium chloride to the copper half-cell. Adding more
LiCI, but this time to the zinc solution, increases the emf slightly, to about 1.08 V.
No redox chemistry occurs, so no copper ions are reduced to
copper metal nor is zinc metal oxidized to form Zn 2+ . No com-
plexes form in solution, so the changes in emf may be attributed
entirely to changing the composition of the solutions.
Lithium and chloride ions are not wholly passive, but interact
with the ions originally in solution. Let us look at the copper ions,
each of which can associate electrostatically with chloride ions,
causing it to resemble a dandelion 'clock' with the central copper
ion looking as though it radiates chloride ions. All the ions are
solvated with water. These interactions are coulombic in nature,
so negatively charged chloride anions interact attractively with the
positive charges of the copper cations. Copper and lithium cations
repel. Conversely, the additional Li + ions attract the negatively
charged sulphates from the original solution; again, Cl~ and S0 4 ~
anions repel.
We need a slightly
different form of y
when working with
electrolyte solutions:
we call it the mean
ionic activity coefficient
Y±, as below.
ACTIVITY
313
'Associated ions' in
this context means
an association species
held together (albeit
transiently) via electro-
static interactions.
The ionic atmosphere moves continually, so we consider its com-
position statistically. Crystallization of solutions would occur if the
ionic charges were static, but association and subsequent dissocia-
tion occur all the time in a dynamic process, so even the ions in a
dilute solution form a three-dimensional structure similar to that in
a solid's repeat lattice. Thermal vibrations free the ions by shaking
apart the momentary interactions.
The ions surrounding each copper cation are termed the ionic atmosphere. In the
neighbourhood of any positively charged ion (such as a copper cation), there are likely
to be more negative charges than positive (and vice versa). We say the cations are
surrounded with a shell of anions, and each anion is surrounded by a shell of cations.
The ionic atmosphere can, therefore, be thought to look much like an onion, or a
Russian doll, with successive layers of alternate charges, with the result that charges
effectively 'cancel' each other out when viewed from afar.
Having associated with other ions, we say the copper ion is screened from anything
else having a charge (including the electrode), so the full extent of its charge cannot be
'experienced'. In consequence, the magnitude of the electrostatic interactions between
widely separated ions will decrease.
The electrode potential measured at an electrode relates to the
'Coulomb potential energy' V 'seen' by the electrode due to the
ions in solution. V relates to two charges z.\ and z.i ( one being the
electrode here) separated by a distance r, according to
V
z^z
4jre e r r
(7.26)
The 'Coulomb potential
energy' V is equal to
the work that must be
done to bring a charge
z+ from infinity to a
distance of r from the
charge z~.
where e is the permittivity of free space and e r is the relative
permittivity of the solvent. In water at 25 °C, e r has a value of 78.54.
The magnitude of V relates to interactions between the electrode and nearby ions
nestling within the interface separating the electrode and the ionic solution. Since
the 'effective' (visible) charge on the ions decreases, so the electrode perceives there
to be fewer of them. In other words, it perceives the concentration to have dipped
below the actual concentration. This perceived decrease in the number of charges
then causes the voltmeter to read a different, smaller value of E Cu 2+ Cu .
The zinc ions in the other half of the Daniell cell can similarly interact with ions
added to solution, causing the zinc electrode to 'see' fewer Zn 2+ species, and the
voltmeter again reads a different, smaller value of E Zn 2+ Zn . Since the emf represents
the separation between the electrode potentials of the two half-cells, any changes in
the emf illustrate the changes in the constituent electrode potentials.
Background to the Debye-Huckel theory
The interactions between the ions originally in solution and any added LiCl are
best treated within the context of the Debye-Hiickel theory, which derives from a
knowledge of electrostatic considerations.
314 ELECTROCHEMISTRY
Firstly, we assume the ions have an energy distribution as defined by the Boltzmann
distribution law (see p. 35). Secondly, we say that electrostatic forces affect the
behaviour and the mean positions of all ions in solution. It should be intuitively clear
that ions having a larger charge are more likely to associate strongly than ions having
a smaller charge. This explains why copper ions are more likely to associate than
are sodium ions. The magnitude of the force exerted by an ion with a charge z.\ on
another charge zi separated by an inter-ion distance of r in a medium of relative
permittivity e r is the 'electrostatic interaction' 0, as defined by
(7.27)
4jte e r r 2
Positive values of <t>
imply repulsion, and a
negative value attrac-
tion.
Note how this equation states that the force is inversely proportional to the square
of the distance between the two charges r, so the value of decreases rapidly as r
increases.
Since cations and anions have opposite charges, is negative.
The force between two anions will yield a positive value of 0. We
see how a positive value of implies an inter-ionic repulsion and
a negative value implies an inter-ionic attraction.
The Debye-Hiickel theory suggests that the probability of find-
ing ions of the opposite charge within the ionic atmosphere increa-
ses with increasing attractive force.
Why does adding NaCI to a cell alter the emf, but
adding tonic water doesn't?
The effects of ion association and concentration on y
Sodium chloride - table salt - is a 'strong' ionic electrolyte because it dissociates
fully when dissolved in water (see the discussion of weak and strong acids in
Section 6.2). The only electrolytes in tonic water are sugar (which is not ionic) and
sodium carbonate, which is a weak electrolyte, so very few ions are formed by adding
the tonic water to a cell.
The ratio of perceived to real concentrations is called the activity coefficient y
(because, from Equation (7.25), y = a -4- c). Furthermore, from the definition of activ-
ity in Equation (7.20), y will have a value in the range zero to one. The diagram
in Figure 7.9 shows the relationship between y and concentration c for a few ionic
electrolytes.
Adding NaCI to solution causes y to decrease greatly because the number of ions
in solution increases. Adding tonic water does not decrease the activity coefficient
much because the concentration of the ions remains largely unchanged. The change
in y varies more with ionic electrolytes because the interactions are far stronger. And
if the value of y does not change, then the real and perceived concentrations will
remain essentially the same.
ACTIVITY
315
The extent of ionic screening depends on the extent of associa-
tion. The only time that association is absent, and we can treat ions
as though free and visible ('unscreened'), is at infinite dilution.
Why does MgCI 2 cause a greater decrease
in perceived concentration than KCI?
The mean ionic activity coefficient y±
Infinite dilution (extra-
polation to zero con-
centration) means so
small a concentration
that the possibility of
two ions meeting, and
thence associating, is
tiny to non-existent.
The value of y depends
on the solute employed.
The extent of ionic association depends on the ions we add to
the solution. And the extent of association will effect the extent of
screening, itself dictating how extreme the difference is between
perceived and real concentration. For these reasons, the value of
y(= a H- c) depends on the choice of solute as well as its concentration, so we ought
to cite the solute whenever we cite an activity coefficient.
The value of y is even more difficult to predict because solutes
contain both anions and cations. In fact, it is impossible to dif-
ferentiate between the effects of each, so we measure a weighted
average. Consider a simple electrolyte such as KCI, which has
one anion per cation. (We call it a '1:1 electrolyte'.) In KCI, the
activity coefficient of the anions is called Y(cr) an d the activity
coefficient of the cations is 7(k+)- We cannot know either y + or
Y-\ we can only know the value of y±. Accordingly, we modify
Equation (7.25) slightly by writing
We cannot know either
y + or y_; we can only
know the value of their
geometric mean y ± .
-y±
(7.28)
We call KCI a 1:1 elec-
trolyte, since the ratio
of anions to cations is
1:1.
where the only change is the incorporation of the mean ionic activity coefficient y± .
The mean ionic activity coefficient is obtained as a geometric mean via
Y± = VT(K+) x y (cr)
(7.29)
By analogy, the expression for the mean ionic activity coefficient y± for a 2:1
electrolyte such as K2SO4 is given by
Y±
yI x Y-
(7.30)
where the cube root results from the stoichiometry, since K2SO4 contains three ions
(we could have written the root term alternatively as ZJy + x y + x y_, with one y
term per ion). Again, a 1:3 electrolyte such as FeCl3 dissolves to form four ions, so
an expression for its mean ionic activity coefficient y± will include a fourth root, etc.
316
ELECTROCHEMISTRY
SAQ 7.12 Write an expression similar to Equation (7.29) for the 2:3
electrolyte Fe2(S0 4 )3-
Why is calcium better than table salt at stopping soap
lathering?
Ionic strength I and the Debye-Huckel laws
People whose houses are built on chalky ground find that their kettles and boilers
become lined with a hard 'scale'. We say that the water in the area is 'hard', meaning
that minute amounts of chalk are dissolved in it. The hard layer of 'scale' is chalk
that precipitated onto the inside surface of the kettle or boiler during heating.
It is difficult to get a good soapy froth when washing the hands
We look at the actions
of soaps in Chapter 10.
in 'hard water' because the ions from chalk in the water associate
with the long-chain fatty acids in soap, preventing it from ionizing
properly. Conversely, if the water contains table salt - for example,
when washing the dishes after cooking salted meat - there is less of a problem with
forming a good froth. Although the concentrations of sodium and calcium ions may be
similar, the larger charges on the calcium and carbonate ions impart a disproportionate
effect, and strongly inhibit the formation of frothy soap bubbles.
Having discussed ionic screening and its effects on the value
of y±, we now consider the ionic charge z. When assessing the
influence of z, we first define the extent to which a solute pro-
motes association, and thus screening. The preferred parameter is
the 'ionic strength' /, as defined by
In 'dynamic' elec-
trochemistry (when
currents flow) we need
to be careful not to
mistake ionic strength
and current, since both
have the symbol I.
1 '='
(7.31)
i=i
where z, is the charge on the ion i in units of electronic charge, and c, is its concen-
tration. We will consider three simple examples to demonstrate how ionic strengths
/ are calculated.
Worked Example 7.12 Calculate the ionic strength of a simple 1 : 1 electrolyte, such as
NaCl, that has a concentration of c = 0.01 moldm -3 .
Inserting values into Equation (7.31) we obtain
/= \ J]{ |t Na+] x (+ 1 ) 2 ] + ( [C1 " ] x (- 1 ) 2
/
\
terms for the
terms for the
sodium ions
chloride ions
ACTIVITY
317
We next insert concentration terms, noting that one sodium ion and
one chloride are formed per formula unit of sodium chloride (which
is why we call it a 1:1 electrolyte). Accordingly, the concentrations
of the two ions, [Na + ] and [Cl~], are the same as [NaCl], so
/ = i{([NaCl] x 1) + ([NaCl] x 1)}
so we obtain the result for a 1:1 electrolyte that /(NaCl) = [NaCl].
Note that / has the same units as concentration: inserting
the NaCl concentration [NaCl] = 0.01 mol dm -3 , we obtain I —
0.01 mol dm" 3 .
NaCl is called a '1:1
electrolyte' because the
formula unit contains
one anion and one
cation.
We obtain the result
I = c only for 1:1 (uni-
valent) electrolytes.
Worked Example 7.13 Calculate the ionic strength of the 2:2 electrolyte FeSCU at a
concentration c — 0.01 moldm - .
Inserting charges in Equation (7.31):
I = ^{[Fe 2+ ] x (+2) 2 + [SO 2 "] x (-2) 2 }
We next insert concentrations, again noting that one ferrous ion and one sulphate ion are
formed per formula unit:
/ = -{([FeS0 4 ] x 4) + ([FeS0 4 ] x 4)}
so we obtain the result I — 4 x c for this, a 2:2 electrolyte.
Inserting the concentration c of [FeSO/j] = 0.01 moldm -3 , we obtain / = 0.04 mol
dm~ , which explains why hard water containing FeSC»4 has a greater influence than
table salt of the same concentration.
Worked Example 7.14 Calculate the ionic strength of the 1:2 electrolyte CuCl2, again
of concentration 0.01 moldm -3 .
Inserting charges into Equation (7.31):
/ = ^{[Cu 2+ ] x (+2) 2 + [CI"] x (-1) 2 }
We next insert concentrations. In this case, there are two chloride
ions formed per formula unit of salt, so [CI"] — 2 x [CUCI2], but only
one copper, so [Cu 2+ ] = [CuCl 2 ].
/ = -{([CuCl 2 ] x 4) + (2[CuCl 2 ] x 1)}
Note how the calcu-
lation requires the
charge per anion,
rather than the total
anionic charge.
so we obtain the result I = 3 x c for this, a 1:2 electrolyte. And, / = 0.03 moldm
because [CuCl 2 ] = 0.01 moldm" 3 .
-3
318
ELECTROCHEMISTRY
SAQ 7.13 Calculate the relationship between concentration and ionic
strength for the 1:3 electrolyte C0CI3.
Ionic strength I is an
integral multiple of
concentration c, where
integer means whole
number. A calculation
of/ not yielding a whole
number is wrong.
Table 7.5 summarizes all the relationships between concentra-
tion and ionic strength / for salts of the type M x+ X y ~, listed as
a function of electrolyte concentration. Notice that the figures in
the table are all integers. A calculation of / not yielding a whole
number is wrong.
Ions with large charges generally yield weak electrolytes, so the
numbers of ions in solution are often smaller than predicted. For
this reason, values of / calculated for salts represented by the bot-
tom right-hand corner of Table 7.5 might be too high.
Why does the solubility ofAgCI change after adding
MgS0 4 ?
Calculating values of y±
Silver chloride is fairly insoluble (see p. 332), with a solubility
product K sv of 1.74 x 10~ 10 mol 2 dm~ 6 . Its concentration in pure
distilled water will, therefore, be 1.3 x 10~ 5 moldm -3 , but adding
magnesium sulphate to the solution increases it solubility appre-
ciably; see Figure 7.10.
This increase in solubility is not an example of the common ion
effect, because there are no ions in common. Also impossible is
the idea that the equilibrium constant has changed, because it is a constant.
Strictly, we should formulate all equilibrium constants in terms of activities rather
than concentrations, so Equation (7.32) describes K sp for dissolving partially soluble
AgCl in water:
We obtain the
concen-
tration
[Ag + ] =
■- 1.3 x
10- 5 as
the square root
of 1.74
x 10- 10
mol 2
dnrr 6 .
K,
sp
a
(Ag+)«(cn
[Ag+][C1 ] x Y(A g +)Y(cr)
(7.32)
Table 7.5 Summary of the relationship between ionic
strength / and concentration c. As an example, sodium
sulfate (a 1:2 electrolyte) has an ionic strength that is
three times larger than c
X~
x 2 -
x 3 -
x 4 -
M+
1
3
6
10
M 2 +
3
4
15
12
M 3 +
6
15
9
42
M 4 +
10
12
42
16
ACTIVITY
319
0.4 0.6 0.8
[MgSO 4 ]/0.01 mol drrr 3
1.2
Figure 7.10 The solubility s of AgCl (as 'y') in aqueous solutions of MgS0 4 against its concen-
tration, [MgS0 4 ] (as 'x'). T = 298.15 K
The exact structure of the equilibrium constant on the right-hand side of Equation
(7.32) follows from the definition of activity a in Equation (7.25). The product of the
two y terms is y±>
The values of the activity coefficients decrease with increasing ionic strength / (as
below). The only way for ^" sp to remain constant at the same time as the activity
coefficient y± decreasing is for the concentrations c to increase. And this is exactly
what happens: the concentration of AgCl has increased by about 50 per cent when
the concentration of MgS0 4 is 1.2 mol dm -3 .
Changes in solubility product are one means of experimentally determining a value
of activity coefficient, because we can independently determine the concentrations
(e.g. via a titration) and the values of all y± will be 'one' at zero
ionic strength.
Alternatively, we can calculate a value of y± with the Debye-
Huckel laws. There are two such laws: the limiting and the simpli-
fied laws. Calculations with the limiting law are only valid at very
low ionic strengths (i.e. < / < 10 mol dm ), which is very
dilute. The limiting law is given by
An ionic strength of
10 3 mol dm 3 could
imply a concentra-
tion as low as 10~ 4
moldnrr 3 , because
I >c.
logioK± = -Mz + z, |V7
(7.33)
where A is the so-called Debye-Hiickel 'A' constant (or factor),
which has a value of 0.509 mol -1 ^ 2 dm 3 ^ 2 at 25 °C. z + and z.~ are
the charges per cation and per anion respectively. The vertical mod-
ulus lines '|' signify that the charges on the ions have magnitude,
but we need to ignore their signs (in practice, we call them both
positive).
The quantities bet-
ween the two vertical
modulus lines T have
magnitude alone, so
we ignore the signs
on the charges z+
and z~
320
ELECTROCHEMISTRY
From Equation (7.28), we expect a plot of log 10 y± (as 'y') against v7 (as 'x')
to be linear. It generally is linear, although it deviates appreciably at higher ionic
strengths.
Worked Example 7.15 What is the activity coefficient of copper in a solution of copper
sulphate of concentration 10~ 4 moldm - ?
Note how we ignore
the sign of the negative
charge here.
When calculating with
Equation (7.33), be
sure to use 'log' (in
base 10) rather than
'In' (log in base e).
At extremely low ionic
strengths, the simpli-
fied law becomes the
limiting law. This fol-
lows since the denomi-
nator 'I + bv7' tends to
one as ionic strength
I tends to zero, caus-
ing the numerator to
become one.
Copper sulphate is a 2:2 electrolyte so, from Table 7.5, the ionic
strength / is four times its concentration. We say / = 4 x 10 -4
moldm 3 .
Inserting values into Equation (7.33):
log 10 Y ± = -0.509 | + 2 x -2|(4 x 10 -4 ) 1/2
log 10 y ± = -2.04 x (2 x 10 -2 )
logio Y± = -4-07 x 10 -2
Taking the anti-log:
Y± = 10 -00407
Y± = 0.911
We calculate that the perceived concentration is 91 percent of the real
concentration.
For solutions that are more concentrated (i.e. for ionic strengths
in the range 10 -3 < I < 10 -1 ), we need the Debye-Hiickel sim-
plified law:
where all other terms have the same meaning as above, and b
is a constant having an approximate value of one. We include b
because its units are mor~ 1/2 dm 3/2 . It is usual practice to say b =
1 mol - ' dm ' , thereby making the denominator dimensionless.
SAQ 7.14 Prove that the simplified law becomes the limiting law at very
low J.
Worked Example 7.16 What is the activity coefficient of a solution of CUSO4 of con-
centration 10~ 2 moldm - ?
Again, we start by saying that I — 4 x c, so I — 4 x 10 -2 moldm -3 . Inserting values
into Equation (7.34):
log 10 y± =
0.509 |2 x -2|V4 x 10~ 2
1 + V4 x 10~ 2
HALF-CELLS AND THE NERNST EQUATION
321
Table 7.6 Typical activity coefficients y± for ionic electrolytes as a function of concentration c
in water
Electrolyte/
Y±
mol drrT c -
= 10- 3 /mol
dm 3 c =
= 10" 2 /mol
dm
- 3 c~-
= 10" '/mol
dm 3 c =
= 1.0/moldm -3
HC1 (1:1)
0.996
0.904
0.796
0.809
KOH (1:1)
-
0.90
0.80
0.76
CaCl 2 (1:2)
-
0.903
0.741
0.608
CuS0 4 (2:2)
0.74
0.41
0.16
0.047
In 2 (S0 4 ) 3 (2:3)
-
0.142
0.035
-
logio Y± = ~
0.4072
1 + 0.2
logio Y± = -0.3393
Y± = 10
-0.3383
SO
Y± = 0.458
Table 7.6 cites a few sample values of y± as a function of concentration. Note how
multi-valent anions and cations cause y± to vary more greatly than do mono-valent
ions. The implications are vast: if an indium electrode were to be immersed in a
solution of In2(S04)3 of concentration 0.1 mol dm -3 , for example, then a value of
y± = 0.035 means that the activity (the perceived concentration) would be about 30
times smaller!
SAQ 7.15 From Worked Example 7.15, the mean ionic activity coefficient
y± is 0.911 for CUSO4 at a concentration of 10 -4 mol drrT 3 . Show that
adding MgS0 4 (of concentration 0.5 mol dm" 3 ) causes y± of the CUSO4 to
drop to 0.06. [Hint: first calculate the ionic strength. [MgS0 4 ] is high, so
ignore the CUSO4 when calculating the ionic strength /.]
7.4 Half-cells and the Nernst equation
Why does sodium react with water yet copper doesn't?
Standard electrode potentials and the E e scale
Sodium reacts with in water almost explosively to effect the reaction
Na (s) + IT 1
(aq)
Na" 1
(aq) + 2 H 2(g)
(7.35)
322 ELECTROCHEMISTRY
The protons on the left-hand side come from the water. Being spontaneous, the value
of AG r for Equation (7.25) is negative. The value of AG r comprises two components:
(1) AG for the oxidation reaction Na ► Na + + e~; and
(2) AG for the reduction reaction H + + 2e~ ► jH-2-
Since these two equations represent redox reactions, we have effectively separated a
cell into its constituent half-cells, each of which is a single redox couple.
By contrast, copper metal does not react with water to liberate hydrogen in a
reaction like Equation (7.35); on the contrary, black copper(II) oxide reacts with
hydrogen gas to form copper metal:
CuO (s) + H 2(g) ► Cu (s) + 2H+ + O 2- (7.36)
The protons and oxide ions combine to form water. Again, the value of AG r for
Equation (7.36) is negative, because the reaction is spontaneous. AG would be pos-
itive if we wrote Equation (7.36) in reverse. The change in sign follows because the
Gibbs function is a function of state (see p. 83).
The reaction in Equation (7.36) can be split into its two constituent half-cells:
(1) AG for the reaction Cu 2+ + 2e" ► Cu; and
(2) AG for the reaction H 2 ► 2H+ + 2e".
Let us look at Equation (7.36) more closely. The value of AG r comprises two
components, according to Equation (7.36):
AG r = (AG Cu 2 + ^ Cu ) + (AG H2 ^2H+) (7.37)
If we wished to be wholly consistent, we could write both reac-
tions as reduction processes. Reversing the direction of reaction
(2) means that we need to change the sign of its contribution toward
the overall value of AG r , so
This change of sign
follows from the
change in direction of
the second reaction.
AG r = (AG Cu 2 + ^ Cu ) - (AG 2H +^h 2 ) (7.38)
We remember from Equation (7.15) how AG( ce ii) = —nF x emf. We will now
invent a similar equation, Equation (7.39), which relates AG for a half-cell and
its respective electrode potential £o,r, saying:
AGo.r = -nFE ^ (7.39)
Substituting for AGo,r in Equation (7.38) with the invented expression in Equation
(7.39) gives
AG r = (-nFE Cu 2 +Cu ) - (-nFE H+ , H2 ) (7.40)
HALF-CELLS AND THE NERNST EQUATION
323
The 'standard electrode
potential' E^ R is the
value of E ,R obtained
at standard conditions.
This expression does not relate to a true cell because the two electrode potentials
are not measured with electrodes, nor can we relate AG r to the emf, because elec-
trons do not flow from one half-cell via an external circuit to the other. Nevertheless,
Equation (7.40) is a kind of proof that the overall value of AG r relates to the con-
stituent half-cells.
If we write a similar expression to that in Equation (7.40) for the reaction between
sodium metal and water in Equation (7.35), then we would have to write the term
for the hydrogen couple first rather than second, because the direction of change
within the couple is reversed. In fact, any couple that caused hydrogen gas to form
protons would be written with the hydrogen couple first, and any couple that formed
hydrogen gas from protons (the reverse reaction) would be written with the hydrogen
term second.
This observation led the pioneers of electrochemical thermodynamics to construct
a series of cells, each with the H + |H2 couple as one half-cell. The emf of each
was measured. Unfortunately, there were always more couples than measurements,
so they could never determine values for either E Cu 2+ Cu or £h+,h 2
(nor, indeed, for any electrode potential), so they commented on
their relative magnitudes, and compiled a form of ranking order.
These scientists then suggested that the value of £h+,h 2 should
be defined, saying that at a temperature of 298 K, pumping the
hydrogen gas at a pressure of hydrogen of 1 atm through a solution
of protons at unit activity generates a value of £h+,h 2 that is always
zero. They called the half-cell 'H2( g )(/> = 1 atm)|H + (a = 1)' the
standard hydrogen electrode (SHE), and gave it the symbol E^ + H .
The '°' symbol indicates standard conditions.
Then, knowing E^ + H , it was relatively easy to determine values
of electrode potentials for any other couple. With this methodology,
they devised the 'standard electrode potentials' E & scale (often
called the 'E nought scale', or the 'hydrogen scale').
Table 7.7 contains a few such values of E & , each of which was
determined with the same standard conditions as for the hydrogen
couple, i.e. at T = 298 K, all activities being unity and p = 1 atm
(the pressure is not, therefore, p^).
Negative values of E & (such as £ Na + Na = —2.71 V) indicate
that the reduced form of the couple will react with protons to form
hydrogen gas, as in Equation (7.35). The more negative the value
of E & , the more potent the reducing power of the redox state, so
E e for the magnesium couple is —2.36 V, and E K+ K = —2.93.
Zinc is a less powerful reducing agent, so E 2+ = —0.76 V,
and a feeble reducing agent like iron yields a value of £.? 2 + F of
only -0.44 V.
And positive values of E e indicate that the oxidized form of
the redox couple will oxidize hydrogen gas to form protons, again
A pressure of p = 1 atm
is not the same as
p e , but its use is a
permissible deviation
within the SI scheme.
Negative values of
Eq r indicate that the
reduced form of the
couple will react with
protons to form hydro-
gen gas.
Positive values of E^ R
indicate that the oxi-
dized form of the redox
couple will oxidize
hydrogen gas to form
protons.
324
ELECTROCHEMISTRY
Table 7.7 The electrode potential series (against the SHE). The electrode potential series is an
arrangement of reduction systems in ascending order of their standard electrode potential £" e
Couple
i.b.c
E & /V Couple a
£°/V
Sm + + 2e = Sm
Li+ + e~ = Li
K+ + e" = K
Rb+ + e" = Rb
Cs+ + e" = Cs
Ra 2+ + 2e" = Ra
Ba 2+ + 2e" = Ba
Sr 2+ + 2e~ = Sr
Ca 2+ + 2e" = Ca
Na+ + e~ = Na
Ce 3+ + 3e- = Ce
Mg 2+ + 2e" = Mg
Be 2+ + 2e" = Be
U 3+ + 3e" = U
Al 3+ + 3e - = Al
Ti 2+ + 2e" = Ti
V 2+ + 2e" = V
Mn 2+ + 2e" = Mn
Cr 2+ + 2e - = Cr
2H 2 + 2e - = H 2 + 20H~
Cd(OH) 2 + 2e" = Cd + 20H~
Zn 2+ + 2e" = Zn
Cr 3+ + 3e" = Cr
2 + e" = OJ
In 3+ + e~ = In 2+
S + 2e" = S 2 ~
In 3+ + 2e" = In+
Fe 2+ + 2e" = Fe
Cr 3+ + e~ = Cr 2+
Cd 2+ + 2e- = Cd
In 2+ + e" = In+
Ti 3+ + e" = Ti 2+
PbS0 4 + 2e- = Pb + SO 2 "
In 3+ + 3e" = In
Co 2+ + 2e- = Co
Ni 2+ + 2e" = Ni
Agl + e~ = Ag + 1 -
Sn 2+ + 2e _ = Sn
In+ + e" = In
3.12
Pb 2+ + 2e- = Pb
-0.13
3.05
Fe 3+ + 3e _ = Fe
-0.04
2.93
Ti 4 + + e - = Ti 3+
0.00
2.93
2 H + + 2e~ = H2 {by definition)
0.000
2.92
AgBr + e~ = Ag + Br
0.07
2.92
Sn 4+ + 2e - = Sn 2+
0.15
2.91
Cu 2+ + e" = Cu+
0.16
2.89
Bi 3+ + 3e" = Bi
0.20
2.87
AgCl + e" = Ag + Cr
0.2223
2.71
Hg 2 Cl 2 + 2e - = 2Hg + 2C1"
0.27
2.48
Cu 2+ + 2e" = Cu
0.34
2.36
2 + 2H 2 + 4e" = 40H~
0.40
1.85
NiOOH + H 2 + e~ =
: Ni(OH) 2 + OH"
0.49
1.79
Cu + + e~ = Cu
0.52
1.66
I 3 - + 2e - = 31"
0.53
1.63
I 2 + 2e - = 21-
0.54
1.19
MnO^ + 3e" = Mn0 2
0.58
1.18
Hg 2 S0 4 + 2e - = 2Hg
+ so 2 -
0.62
0.91
Fe 3+ + e" = Fe 2+
0.77
0.83
AgF + e" = Ag + F
0.78
0.81
Hg 2+ + 2e" = 2Hg
0.79
0.76
Ag+ + e" = Ag
0.80
0.74
2Hg 2+ + 2e" = Hg 2+
0.92
0.56
Pu 4+ + e" = Pu 3+
0.97
0.49
Br 2 + 2e" = 2Br"
1.09
0.48
Pr 2+ + 2e" = Pr
1.20
0.44
Mn0 2 + 4H+ + 2e" =
Mn 2+ + 2H 2
1.23
0.44
2 + 4H+ + 4e" = 2H 2
1.23
0.41
Cl 2 + 2e- = 2Cr
1.36
0.40
Au 3+ + 3e" = Au
1.50
0.40
Mn 3+ + e" = Mn 2+
1.51
0.37
MnO^ + 8H+ + 5e" =
= Mn 2+ + 4H 2
1.51
0.36
Ce 4+ + e- = Ce 3+
1.61
0.34
Pb 4+ + 2e- = Pb 2+
1.67
0.28
Au + + e _ = Au
1.69
0.23
Co 3+ + e" = Co 2+
1.81
0.15
Ag 2+ + e" = Ag+
1.98
0.14
S 2 2 " + 2e- = 2S0 2 "
2.05
0.14
F 2 + 2e _ = 2F"
2.87
a The more positive the value of E & , the more readily the half- reaction occurs in the direction left to right; the
more negative the value, the more readily the reaction occurs in the direction right to left.
b Elemental fluorine is the strongest oxidizing agent and Sm 2+ is the weakest. Oxidizing power increases from
Sm 2+ to F 2 .
c Samarium is the strongest reducing agent and F~ is the weakest. Reducing power increases from F~ to Sm.
HALF-CELLS AND THE NERNST EQUATION
325
with the magnitude of E e indicating the oxidizing power: E^ 2+
a powerful oxidizing agent such as bromine has a value of E^ _ =
= +0.34 V, but
+ 1.09 V.
In summary, sodium reacts with water and copper does not in consequence of their
relative electrode potentials.
y does a torch battery eventually *go fla
The Nernst equation
A new torch battery has a voltage of about 1.5 V, but the emf decreases with usage
until it becomes too small to operate the torch for which we bought it. We say the
battery has 'gone flat', and throw it away.
We need to realize from Faraday's laws that chemicals within a battery are con-
sumed every time the torch is switched on, and others are generated, causing the
composition within the torch to change with use. Specifically, we alter the relative
amounts of oxidized and reduced forms within each half-cell, causing the electrode
potential to change.
The relationship between composition and electrode potential is
given by the Nernst equation
Eo , R = E^ + ^m( a -p)
nF \a {K) J
(7.41)
Though it is relatively
easy to formulate rela-
tions like the Nernst
equation here for a
cell, Equation (7.41)
properly relates to a
half-cell.
where E Q R is the standard electrode potential determined at s.t.p.
and is a constant, £o,r is the electrode potential determined at non
s.t.p. conditions. R, T, n and F have their usual definitions.
The bracket on the right of Equation (7.41) describes the relative activities of
oxidized and reduced forms of the redox couple within a half-cell. The battery goes
flat because the ratio a(o)/a(R) alters with battery usage, so the value of £o,r changes
until the emf is too low for the battery to be useful.
Worked Example 7.17 A silver electrode is immersed into a dilute solution of silver
nitrate, [AgN0 3 ] = 10~ 3 mol -3 . What is the electrode potential E Ag+Ag at 298 K? Take
"Ag+,Ag
0.799 V.
The Nernst equation, Equation (7.41), for the silver couple is
RT
E Ag \A g = £ Ag +, A g + ~y ln
a (Ag + )
a {Ag)
For simplicity, we assume that the concentration and activity of
silver nitrate are the same, i.e. <3(A g +) = 10 -3 . We also assume that the
silver is pure, so its activity is unity.
We use the approxima-
tion 'concentration
= activity' because the
solution is very dilute.
326
ELECTROCHEMISTRY
The value of RT/F is
0.0257 Vat 298 K.
Inserting values into Equation (7.41):
^Ag+.Ag = 0.799 V + 0.0257 Vln
0.001
1
Note how, as a conse-
quence of the laws of
arithmetic, we multiply
the RT/F term with the
logarithm term before
adding the value of
'"O.R"
SO
and
Ea s \a s = 0.799 V + (0.0257 V x -6.91)
-Ag+,Ag
0.799 V- 0.178 V
Note how the difference
between E and E e is
normally quite small.
£ Ag+ , Ag = 0.621 V
SAQ 7.16 A wire of pure copper is immersed into a solu-
tion of copper nitrate. If E
0.24 V, what is the concentration of Cu'
a (Cu 2 +) is the same as [Cu 2+ ].
= 0.34 V and E r ?
,Cu Cu
,Cu
? Assume that
The Nernst equation cannot adequately describe the relationship between an elec-
trode potential £o,r and the concentration c of the redox couple it represents, unless
we substitute for the activity, saying from Equation (7.28), a = c x y±.
The form of Equation (7.41) will remind us of the equation of a straight line, so a
plot of £o,r as the observed variable (as 'y'), against ln(fl(Q) 4- fl(R)) (as 'x') should
be linear with a gradient of RT -r- nF and with E Q R as the intercept on the y-axis.
Worked Example 7.18 Determine a value for the standard electrode potential Ef + .
with the data below. Assume that y± — 1 throughout.
[AgN0 3 ]/moldm- 3 0.001 0.002 0.005 0.01 0.02 0.05 0.1
£ Ag+, Ag /V 0.563 0.640 0.664 0.682 0.699 0.723 0.741
Figure 7.11 shows a Nernst graph drawn with the data in the table. The intercept of the
graph is clearly 0.8 V.
Why does E AgC \ iAg change after immersing an SSCE in a
solution of salt?
Further calculations with the Nernst equation
Take a rod of silver, and immerse it in a solution of potassium
chloride. A thin layer of silver chloride forms on its surface when
the rod is made positive, generating a redox couple of AgCl|Ag.
We have made a silver-silver chloride electrode (SSCE).
Now take this electrode together with a second redox couple (i.e.
half-cell) of constant composition, and dip them together in a series
Care: In some books,
SSCE is taken to mean
a sodium chloride
saturated calomel elec-
trode.
HALF-CELLS AND THE NERNST EQUATION 327
-6 -4
ln([Ag + ]/mol dm 3 )
Figure 7.11 Nernst graph of the electrode potential E Ag + Ag as 'y' against ln([Ag + ]/moldirT 3 )
as 'x'. A value of E + = 0.8 V is obtained as the intercept on the y-axis
of salt solutions, and measure the emf. The magnitude of the emf will depend on the
concentration of the salt. The silver rod and its outer layer of silver chloride do not
alter, so why does the emf change?
The electrode potential i?AgCi,Ag relates to the following redox reaction:
AgCl (s) + e- ► Ag° (s) + cr
(aq)
(7.42)
This redox couple is more complicated than any we have encountered yet, so the
Nernst equation will appear to be a little more involved than those above:
ExgCU
RT
£ A g ci,Ag + ~y
ln
fl(AgCl)
*(Ag°) a (cn ,
(7.43)
Silver chloride is the oxidized form, so we write it on top of the bracketed fraction,
and silver metal is the reduced form, so we write it beneath. But we must also write a
term for the chloride ion, because Cl~( aq ) appears in the balanced reduction reaction
in Equation (7.42).
If we immerse a silver electrode bearing layer of AgCl in a concentrated solution
of table salt, then the activity fl(cr) W11 l be high; if the solution of salt is dilute, then
^AgCi.Ag will change in the opposite direction, according to Equation (7.43).
SAQ 7.17 An SSCE electrode is immersed in a solution of [NaCI] =
0.1 moldrrr 3 . What is the value of f Ag ci,Ag? Take E* gOAg = 0.222 V. Take
all y± = 1. [Hint: the activities a ( A g ci) and a (Ag0) are both unity because both
are pure solids.]
328
ELECTROCHEMISTRY
Why 'earth' a plug?
Reference electrodes
An electrical plug has three connections (or 'pins'): 'live', 'neutral' and 'earth'. The
earth pin is necessary for safety considerations. The potential of the earth pin is the
same as that of the ground, so there is no potential difference if we stand on the
ground and accidentally touch the earth pin in a plug or electrical appliance. We will
not be electrocuted. Conversely, the potentials of the other pins are different from
that of the earth - in fact, we sometimes cite their potentials with respect to the earth
pin, effectively defining the potential of the ground as being zero.
The incorporation of an earth pin is not only desirable for safety, it also enables
us to know the potential of the other pins, because we cite them with respect to the
earth pin.
A reaction in an electrochemical cell comprises two half-cell reactions. Even when
we want to focus on a single half-cell, we must construct a whole cell and determine
its cell emf, which is defined as '^(positive electrode) - ^(negative electrode)'- Only when we
know both the emf and the value of one of the two electrode potentials can we
calculate the unknown electrode potential.
A device or instrument having a known, predetermined electrode
potential is called a reference electrode. A reference electrode is
always necessary when working with a redox couple of unknown
£o,r- A reference electrode acts in a similar manner to the earth
pin in a plug, allowing us to know the potential of any electrode
with respect to it. And having defined the potential of the refer-
ence electrode - like saying the potential of earth is zero - we then
know the potential of our second electrode.
A reference electrode
is a constant-potential
device. We need such
a reference to deter-
mine an unknown
electrode potential.
Aside
At the heart of any reference electrode lies a redox couple of known composition: any
passage of current through the reference electrode will change its composition (we argue
this in terms of Faraday's laws in Table 7.1). This explains why we must never allow a
current to flow through a reference electrode, because a current will alter its potential.
The standard hydrogen electrode - the primary reference
The internationally accepted primary reference is the standard
hydrogen electrode (SHE). The potential of the SHE half-cell is
defined as 0.000 V at all temperatures. We say the schematic for
the half-cell is
We define the value
of F(she) as zero at al
temperatures.
Pt|H 2 ( fl = l)|H+(a=l,aq)|
HALF-CELLS AND THE NERNST EQUATION
329
The SHE is depicted in Figure 7.12, and shows the electrode
immersed in a solution of hydrogen ions at unit activity (corre-
sponding to 1.228 moldm -3 HC1 at 20 °C). Pure hydrogen gas at
a pressure of 1 atm is passed over the electrode. The electrode
itself consists of platinum covered with a thin layer of 'platinum
black', i.e. finely divided platinum, electrodeposited onto the plat-
inum metal. This additional layer thereby catalyses the electrode
reaction by promoting cleavage of the H-H bonds.
Table 7.8 lists the advantages and disadvantages of the SHE.
We employ hydrochlo-
ric acid of concentra-
tion 1.228 moldm 3
at 20 °C because the
activity of H + is less
than its concentration,
i.e. y± < 1.
Gaseous hydrogen
(at a pressure of 1 atm)
Platinized
platinum
electrode
Aqueous acid
with protons at
unit activity
Figure 7.12 Schematic depiction of the standard hydrogen electrode (SHE). The half-cell sche-
matic is therefore Pt|H 2 (a = l)|H+(a = 1)
Table 7.8 Advantages and disadvantages of using the standard
hydrogen electrode (SHE)
Advantage of the SHE
The SHE is the international standard
Disadvantages of the SHE
Safety The SHE is intrinsically dangerous because H 2
gas is involved
Size The SHE requires cumbersome apparatus,
including a heavy cylinder of hydrogen
Cost The SHE can be expensive because of using
H 2 gas
Accuracy With the SHE it is difficult to ensure that the
activity of the protons is exactly unity
Precision The SHE is prone to systematic errors, e.g.
cyclic fluctuations in the H 2 pressure
330
ELECTROCHEMISTRY
Measurement with the hydrogen electrode The SHE is the primary reference
electrode, so other half-cell potentials are measured relative to its potential. In practice,
if we wish to determine the value of E
M"+,M
then we construct a cell of the type
Remember that the
proton is always sol-
vated to form a hydro-
xonium ion, so a (H3 o+)
= 3(H+).
The words unit activ-
ity here mean that
the activity of silver
ions is one (so the
system perceives the
concentration to be
1 moldrrr 3 ).
Just because a half-
reaction appears in a
table is no guarantee
that it will actually
work; such potentials
are often calculated.
The SHE is sometimes
erroneously called a
normal hydrogen elec-
trode (NHE).
Pt|H 2 (a= l)|H + (a= l)||M"+(a= 1)|M
Worked Example 7.19 We immerse a piece of silver metal into a
solution of silver ions at unit activity and at s.t.p. The potential across
the cell is 0.799 V when the SHE is the negative pole. What is the
standard electrode potential £ & of the Ag + , Ag couple?
By definition
Cmj — lL (positive electrode) -^ (negative electrode)
Inserting values gives
0.799 V = £* + Ag - £ (SHE )
0.799 V= Ef + .
Ag + ,Ag
because £(she) = 0.
The value of i?Ag+,Ag i n this example is the standard electrode
potential because a (Ag +^ — 1, and s.t.p. conditions apply. We say that
£■*„+ a — E . + . = 0.799 V versus the SHE. Most of the values of
E in Table 7.7 were obtained in a similar way, although some were
calculated.
We should be aware from Table 7.8 that the SHE is an ideal device,
and the electrode potential will not be exactly V with non-standard
usage.
Secondary reference electrodes
The SHE is experimentally inconvenient, so potentials are often measured and quoted
with respect to reference electrodes other than the SHE. By far the most common
reference is the saturated calomel electrode (SCE). We will usually make our choice
of reference on the basis of experimental convenience.
Note
that
the 'S
' of
'SCE'
here
does
NOT
mean
'standard',
but
'saturated'
The SCE
By far the most common secondary reference electrode is the SCE:
Hg|Hg 2 Cl 2 |KCl(sat'd)|
HALF-CELLS AND THE NERNST EQUATION
331
■ Lead to voltmeter
■ Glass tube
■ Platinum contact
- Saturated KCI solution
■ Paste of Hg and Hg 2 CI 2
o
- Crystals of KCI
Porous sinter
Figure 7.13 Schematic representation of the saturated calomel electrode (SCE)
The potential of the SCE is 0.242 V at 298 K relative to the SHE.
At the 'heart' of the SCE is a paste of liquid mercury and
mercurous chloride (Hg 2 Cl2), which has the old-fashioned name
'calomel'. Figure 7.13 depicts a simple representation of the SCE.
The half-cell reaction in the SCE is
Hg 2 Cl 2 + 2e"
■* 2CP + 2Hg
(7.44)
So i?(scE) = £Hg 2 ci 2 ,Hg- From this redox reaction, the Nernst equa-
tion for the SCE is
'Hg 2 Cl 2 ,Hg
= E
Hg 2 Cl 2 ,Hg
Calomel is the old-
fashioned name for
mercurous chloride,
Hg 2 CI 2 . Calomel was a
vital commodity in the
Middle Ages because
it yields elemental
mercury ('quick sil-
ver') when roasted; the
mercury was required
by alchemists.
(7.45)
the square terms for mercury and the chloride ion are needed in
response to the stoichiometric numbers in Equation (7.44). Both
mercury and calomel are pure substances, so their activities are
unity. If the activity of the chloride ion is maintained at a constant
level, then £(sce) will have a constant value, which explains why
the couple forms the basis of a reference electrode.
Changing fl(cr) must alter £'Hg 2 ci 2 ,Hg» since these two variables
are interconnected. In practice, we maintain the activity of the chlo-
ride ions by placing surplus KCI crystals at the foot of the tube.
The KCI solution is saturated - hence the 'S' in SCE. For this reason, we should
avoid any SCE not showing a crust of crystals at its bottom, because its potential will
be unknown. Also, currents must never be allowed to pass through an SCE, because
charge will cause a redox change in E^CE)-
Table 7.9 lists the advantages and disadvantages of the SCE reference. Despite
these flaws, the SCE is the favourite secondary reference in most laboratories.
Oxidative currents
reverse the reaction
in Equation (7.44).
cr and Hg are con-
sumed and Hg 2 CI 2
forms. The denomina-
tor of Equation (7.45)
decreases, and F(sce)
increases.
332
ELECTROCHEMISTRY
Table 7.9 The advantages and disadvantages of using the saturated calomel
electrode (SCE)
Advantages of the SCE
Cost
Size
Safety
SCEs are easy to make, and hence they are cheap
SCEs can be made quite small (say, 2 cm long
and 0.5 cm in diameter)
Unlike the SHE, the SCE is non-flammable
Disadvantages of the SCE
Contamination Chloride ions can leach out through the SCE sinter
Temperature effects The value of dE/dT is quite large at 0.7 mVKT 1
Solvent The SCE should not be used with non-aqueous
solutions
The silver-silver chloride electrode
We make the best films
of AgCI by anodizing a
silver wire in aqueous
KCI, not in HCI. The
reasons for the differ-
ences in morphology
are not clear.
Photolytic breakdown
is a big problem when
we light the laboratory
with fluorescent strips.
The silver-silver chloride electrode (SSCE) is another secondary
reference electrode. A schematic of its half-cell is:
Ag|AgCl|KCl(aq, sat'd)
The value of E Agci A = 0.222 V.
The AgCI layer has a pale beige colour immediately it is made,
but soon afterwards it assumes a pale mauve and then a dark purple
aspect. The colour changes reflect chemical changes within the film,
caused as a result of photolytic breakdown:
AgCI + hv + electron donor ► Ag + CI"
(7.46)
exposure to light.
introduce colloids
Table 7.10 lists
The purple colour is caused by colloidal silver, formed in a sim-
ilar manner to the image on a black-and-white photograph after
For this reason, an SSCE should be remade fairly frequently. (We
in Chapter 10.)
the advantages and disadvantages of SSCEs.
Table 7.10 The advantages and disadvantages of using the silver-silver chloride electrode (SSCE)
Advantages of the SSCE
Cost
Stability
Size
Disadvantages of the SSCE
Photochemical stability
Contamination
The SSCE is easy and extremely cheap to make
The SSCE has the smallest temperature voltage coefficient
of any common reference electrode
An SSCE can be as large or small as desired; it can even be
microscopic if the silver is thin enough
The layer of AgCI must be remade often
The solid AgCI loses Cl~ ions during photolytic breakdown
CONCENTRATION CELLS
333
7.5 Concentration cells
Why does steel rust fast while iron is more passive?
Concentration cells
Steel is an impure form of iron, the most common contaminants
being carbon (from the coke that fuels the smelting process) and
sulphur from the iron oxide ore.
Pure iron is relatively reactive, so, given time and suitable con-
ditions of water and oxygen, it forms a layer of red hydrated iron
oxide ('rust'):
Many iron ores also
contain iron sulphide,
which is commonly
called fool's gold.
4Fe (s) + 30 2 + nH 2 ► 2Fe 2 3 • (H 2 0)„ (s
(s)
(7.47)
By contrast, steel is considerably more reactive, and rusts faster and to a greater
extent.
The mole fraction x of Fe in pure iron is unity, so the activity of
the metallic iron is also unity. The mole fraction x of iron in steel
will be less than unity because it is impure. The carbon is evenly
distributed throughout the steel, so its mole fraction x ( q is con-
stant, itself ensuring that the activity is also constant. Conversely,
the sulphur in steel is not evenly distributed, but resides in small
(microscopic) 'pockets'. In consequence, the mole fraction of the
iron host X(p e ) fluctuates, with x being higher where the steel is more pure, and lower
in those pockets having a high sulphur content. To summarize, there are differences
in the activity of the iron, so a concentration cell forms.
The emf of a concentration cell (in this case, on the surface of
the steel where the rusting reaction actually occurs) is given by
We define a concen-
tration cell as a cell in
which the two half-cells
are identical except for
their relative concen-
trations.
emf
RT
nF
In
(7.48)
The electrolyte on the
surface of the iron com-
prises water containing
dissolved oxygen (e.g.
rain water).
Notice how this emf has no standard electrode potential E & terms
(unlike the Nernst equation from which it derives; see Justification
Box 7.2).
A voltage forms between regions of higher irons activity a\ and
regions of lower iron activity a 2 (i.e. between regions of high purity
and low iron purity); see Figure 7.14. We can write a schematic
for a microscopic portion of the iron surface as:
Fe(a 2 ),S,C|0 2 ,H 2 0|Fe(ai),S,C
The commas in this
schematic indicate that
the carbon and sulphur
impurities reside within
the same phase as the
iron.
334
ELECTROCHEMISTRY
Humid air
Sulphur cluster:
area of steel having
a low mole fraction of iron
steel having a
relatively high
mole fraction of iron
Figure 7.14 Concentration cells: a voltage forms between regions of higher iron activity ci\ and
regions of lower iron activity a 2 (i.e. between regions of high purity and low iron purity). The
reaction at the positive 'anode' is 4Fe + 30 2 -*■ 2Fe 2 03, and the reaction at the negative 'cathode'
is S + 2e - -» S 2 ~
There is no salt bridge or any other means of stopping current flow in the micro-
scopic 'circuit' on the iron surface, so electrochemical reduction occurs at the right-
hand side of the cell, and oxidation occurs at the left:
at the LHS, the oxidation reaction is formation of rust (Equation (7.47));
at the RHS, the reduction reaction is usually formation of sulphide, via S +
2e~ -► S 2 ".
We can draw several important conclusions from the example of rusting steel. Firstly,
if the impurities of carbon and sulphur are evenly distributed throughout the steel then,
whatever their concentrations, the extent of rusting will be less than if the impurities
cluster, because the emf of a concentration cell is zero when the ratio of activities is
unity.
Secondly, it is worth emphasizing that while oxide formation would have occurred
on the surface of the iron whether it was pure or not, the steel containing impurities
rusts faster as a consequence of the emf, and also more extensively than pure iron
alone.
If the two half-cells
were shorted then
reduction would occur
at the right-hand half-
cell, Cu 2+ (aq) + 2e- ->
CU( S ), and oxidation
would proceed at the
left-hand side, Cu (s) ->
Cu 2+ (a q) + 2e".
Thermodynamics of concentration cells
A concentration cell contains the same electroactive material in
both half-cells, but in different concentration (strictly, with different
activities). The emf forms in response to differences in chemical
potential [i between the two half-cells. Note that such a concen-
tration cell does not usually involve different electrode reactions
(other than, of course, that shorting causes one half-cell to undergo
reduction while the other undergoes oxidation).
Worked Example 7.20 Consider the simple cell Cu|Cu 2+ (a =
0.002)||Cu 2+ (a = 0.02)|Cu. What is its emf!
CONCENTRATION CELLS 335
Inserting values into Equation (7.48):
0.0257 V / 0.02
emf = In
2 V 0.002
emf = 0.0129 Vln(10)
emf = 0.0129 V x 2.303
emf = 29.7 mV
so the emf for this two-electron concentration cell is about 30 mV. For an analogous
one-electron concentration cell, the emf would be 59 mV.
SAQ 7.18 The concentration cell Zn|[Zn 2+ ](c = 0.0112 moldnrr 3 )|[Zn 2+ ]
(c = 0.2 moldnrr 3 )|Zn is made. Calculate its emf, assuming all activity
coefficients are unity.
Justification Box 7.2
Let the redox couple in the two half-cells be O + «e~ = R. An expression for the emf
of the cell is
emf = £(rhs) £(lhs)
The Nernst equation for the 0,R couple on the RHS of the cell is:
£o,R = < R+ ^ ln(^
nF \a {R)R
and the Nernst equation for the same 0,R couple on the LHS of the cell is:
£o,R = < R + ^ln(^w)
nF W)lhs/
Substituting for the two electrode potentials yields an emf of the cell of
emf = E * R +^L ln (IB**) -E% R -— In ^ (0) -
°' R nF \a mm J °' R nF
It will be seen straightaway that the two £ & terms cancel to leave
nF W) RHS 7 nF V«(R)l
A good example of a concentration cell would be the iron system in the worked example
above, in which a(R) = <Z(Cu) = U accordingly, for simplicity here, we will assume that
the reduced form of the couple is a pure solid.
336
ELECTROCHEMISTRY
The emf of the concentration cell, therefore, becomes
RT RT
emf = — ■ ln[fl (0 )RHs] ln[fl (0 )LHs]
nF nF
which, through the laws of logarithms, simplifies readily to yield Equation (7.48).
If we assume that the activity coefficients in the left- and right-hand half-cells are the
same (which would certainly be a very reasonable assumption if a swamping electrolyte
was also in solution), then the activity coefficients would cancel to yield
emf
RT ln f [Q]RHS
nF \[0]lhs
How do pH electrodes work?
The pH -glass electrode
A pH electrode is sometimes also called a 'membrane' electrode. Figure 7.15 shows
how its structure consists of a glass tube culminating with a bulb of glass. This bulb
is filled with a solution of chloride ions, buffered to about pH 7. A slim silver wire
runs down the tube centre and is immersed in the chloride solution. It bears a thin
layer of silver chloride, so the solution in the bulb is saturated with AgCl.
The bulb is usually fabricated with common soda glass, i.e. glass containing a high
concentration of sodium ions. Finally, a small reference electrode, such as an SCE,
is positioned beside the bulb. For this reason, the pH electrode ought properly to be
called a pH combination electrode, because it is combined with a reference electrode.
If the pH electrode does not have an SCE, it is termed a glass electrode (GE). The
operation of a glass electrode is identical to that of a combina-
tion pH electrode, except that an external reference electrode is
required.
To determine a pH with a pH electrode, the bulb is fully immer-
sed in a solution of unknown acidity. The electrode has fast res-
ponse because a potential develops rapidly across the layer of glass
Empirical means found
from experiment, rather
than from theory.
Buffer solution
(containing chloride ions)
Thin-walled glass bulb
Silver wire
Deposit of silver chloride
Figure 7.15 Schematic representation of a pH electrode (also called a 'glass electrode')
CONCENTRATION CELLS 337
between the inner chloride solution and the outer, unknown acid. Empirically, we find
the best response when the glass is extremely thin: the optimum seems to be 50 (xm or
so (50 |xm = 0.05 mm = 50 x 10 -6 m). Unfortunately, such thin glass is particularly
fragile. The glass is not so thin that it is porous, so we do not need to worry about
junction potentials £j (see Section 7.6). The non-porous nature of the glass does
imply, however, that the cell resistance is extremely large, so the circuitry of a pH
meter has to operate with minute currents.
The magnitude of the potential developing across the glass de-
A pH meter is essen-
tially a precalibrated
voltmeter.
pends on the difference between the concentration of acid inside
the bulb (which we know) and the concentration of the acid outside
the bulb (the analyte, whose pH is to be determined). In fact, the
emf generated across the glass depends in a linear fashion on the
pH of the analyte solution provided that the internal pH does not alter, which is why
we buffer it. This pH dependence shows why a pH meter is really just a pre-calibrated
voltmeter, which converts the measured emf into a pH. It uses the following formula:
2.303 RT
emf = K+ pH (7.49)
F
SAQ 7.19 An emf of 0.2532 V was obtained by immersing a glass elec-
trode in a solution of pH 4 at 25 °C. Taking E (S ce) = 0.242 V, calculate the
'electrode constant' K.
SAQ 7.20 Following from SAQ 7.19, the same electrode
was then immersed in a solution of anilinium hydrochloride
of pH = 2.3. What will be the new emf?
In practice, we do not
know the electrode
constant of a pH elec-
trode.
Electrode 'slope'
We can readily calculate from Equation (7.49) that the emf of a pH electrode should
change by 59 mV per pH unit. It is common to see this stated as 'the electrode
has a slope of 59 mV per decade' . A moment's pause shows how this is a simple
statement of the obvious: a graph of emf (as 'y') against [H + ] (as 'x') will have
a gradient of 59 mV (hence 'slope'). The words 'per decade' point to the way that
each pH unit represents a concentration change of 10 times, so a pH of 3 means
that [H+] = 10" 3 moldm" 3 , a pH of 4 means [H+] = 10" 4 moldm" 3 and a pH of
5 means [H + ] = 10~ 5 moldm -3 , and so on. If the glass electrode does have a slope
of 59 mV, its response is said to be Nernstian, i.e. it obeys the Nernst equation. The
discussion of pH in Chapter 6 makes this same point in terms of Figure 6.1.
Table 7.11 lists the principle advantages and disadvantages encountered with the
pH electrode.
SAQ 7.21 Effectively, it says above: 'From this equation, it can be readily
calculated that the emf changes by 59 mV per pH unit'. Starting with the
Nernst equation (Equation (7.41)), show this statement to be true.
338 ELECTROCHEMISTRY
Table 7.11 Advantages and disadvantages of the pH electrode
Advantages
1 . If recently calibrated, the GE and pH electrodes give an accurate response
2. The response is rapid (possibly millisecond)
3. The electrodes are relatively cheap
4. Junction potentials are absent or minimal, depending on the choice of reference electrode
5. The electrode draws a minimal current
6. The glass is chemically robust, so the GE can be used in oxidizing or reducing
conditions; and the internal acid solution cannot contaminate the analyte
7. The pH electrode has a very high selectivity - perhaps as high as 10 5 : 1 at room
temperature, so only one foreign ion is detected per 100000 protons (although see
disadvantage 6 below). The selectivity does decrease a lot above ca 35 °C
Disadvantages
Both the glass and pH electrodes alike have many disadvantages
1. To some extent, the constant K is a function of the area of glass in contact with the acid
analyte. For this reason, no two glass electrodes will have the same value of K
2. Also, for the same reason, K contains contributions from the strains and stresses
experienced at the glass.
3. (Following from 2): the electrode should be recalibrated often
4. In fact, the value of K may itself be slightly pH dependent, since the strains and stresses
themselves depend on the amount of charge incorporated into the surfaces of the glass
5. The glass is very fragile and, if possible, should not be rested against the hard walls or
floor of a beaker or container
6. Finally, the measured emf contains a response from ions other than the proton. Of these
other ions, the only one that is commonly present is sodium. This error is magnified at
very high pH (>11) when very few protons are in solution, and is known as the
'alkaline error'
Justification Box 7.3
At heart, the pH
electrode operates as a simple
concentration cell. Consider the schematic
H+(a 2 )||H+(o 1 ),
then the Nernst equation can
RT
emf = In
■> p
be written as Equation
(?)
(7.48):
which, if written in terms of logarithms in base 10, becomes
2.303RT
emf =
F
logl " (?)
(7.50)
Subsequent splitting of the logarithm terms gives
2.303RT
emf = log 10 fl2
r
2.303RT
F logio 0i
(7.51)
TRANSPORT PHENOMENA
339
If we say that a\ is the analyte of known concentra-
tion (i.e. on the inside of the bulb), then the last term in
the equation is a constant. If we call the term associated
with 02 'K', then we obtain
2303RT
emf = K -\ log 10 a 2
If ci2 relates to the acidic solution of unknown con-
centration then we can substitute for 'log 10 fl2\ by say-
ing that pH = — log 10 [H + ], so:
2.303RT
emf = K -\ x — pH
which is the same as Equation (7.49)
(7.52)
Care: we have assu-
med here that the
activities and concen-
trations of the solvated
protons are the same.
This derivation is based
on the Nernst equation
written in terms of
ionic activities, but pH
is usually discussed in
terms of concentration.
7.6 Transport phenomena
How do nerve cells work?
Ionic transport across membranes
The brain relays information around the body by means of nerves, allowing us to
register pain, to think, or to instruct the legs to walk and hands to grip. Although the
way nerves operate is far from straightforward, it is nevertheless clear that the nerve
pathways conduct charge around the body, with the charged particles (electrons and
ions) acting as the brain's principal messengers between the brain and body.
The brain does not send a continuous current through the nerve, but short 'spurts'.
We call them impulses, which transfer between nerve fibres within the synapses of
cells (see Figure 7.16). The cell floats within an ionic solution called plasma. The
membrane separating the synapse from the solution with which the nerve fibre is in
contact surrounding the cell is the axon, and is essential to the nerve's operation.
The charge on the inside of a cell is negative with respect to
the surrounding solution. A potential difference of about —70 mV
forms across the axon (cell membrane) when the cell is 'at rest', i.e.
before passing an impulse - we sometimes call it a rest potential,
which is caused ultimately by differences in concentration either
side of the axon (membrane).
Movement of charge across the membrane causes the potential to change. A huge
difference in concentration is seen in composition between the inside of the axon
and the remainder of the nerve structure. For example, consider the compositional
No potential difference
forms along the mem-
brane surface, only
across it.
340 ELECTROCHEMISTRY
Cell exterior
Cell membrane ('axon')
to separate the cell interior and exterior
Cell interior
Ions leaving
cell interior
Ions entering cell
interior from exterior
Figure 7.16 Schematic diagram showing a portion of a cell, the membrane ('axon') and the way
ions diffuse across the axon
Table 7.12
Concentrations of ions
inside and outside nerve
components
[Na + ]/moldirT 3
[K + ]/moldrrr 3
[Cn/moldirT 3
Inside the axon
Outside the axon
0.05
0.46
0.40
0.01
0.04-0.1
0.054
Source: J. Koryta, Ions, Electrodes and Membranes, Wiley, Chichester, 1991, p. 172.
In this context, per-
meable indicates that
ions or molecules can
pass through the mem
brane. The mode of
movement is probably
diffusion or migration.
differences in Table 7.12. The data in Table 7.12 refer to the nerves
of a squid (a member of the cephalopod family) data for other
species show a similar trend, with massive differences in ionic
concentrations between the inside and outside of the axon. These
differences, together with the exact extent to which the axon mem-
brane is selectively permeable to ions, determines the magnitude
of the potential at the cell surface.
The membrane encapsulating the axon is semi-permeable, there-
by allowing the transfer of ionic material into and out from the
axon. Since the cell encapsulates fluid and also floats in a fluid,
we say the membrane represents a 'liquid junction'. A potential
forms across the membrane in response to this movement of ions
across the membrane, which we call a 'junction potential' E y If
left unchecked, ionic movement across the membrane would occur
until mixing was intimate and the two solutions were identical.
For a nerve to transmit a 'message' along a nerve fibre, ions traverse the axons and
transiently changing the sign of the potential across the membrane, as represented
schematically in Figure 7.17. We call this new voltage an action potential, to differ-
entiate it from the rest potential. To effect this change in potential, potassium cations
Some texts give the
name 'diffusion poten
tial'to E } .
TRANSPORT PHENOMENA
341
>
75
o
D.
y^T~~"\
Time f/ms
/ \ ' l '
J Action \
/ potential \ Rest potentia |
Onset of impulse
Figure 7.17 The potential across the axon-cell membrane changes in response to a stimulus,
causing the potential to increase from its rest potential to its action potential
move from inside the axon concurrently with sodium ions moving
in from outside. With a smaller difference in composition either
side of the membrane, the junction potential decreases.
A nerve consists of an immense chain of these axons. Impulses
'conduct' along their length as each in turn registers an action
potential, with the net result that messages transmit to and from
the brain.
To achieve this other-
wise difficult process,
chemical 'triggers' pro-
mote the transfer of
ions.
Liquid junction potentials
A liquid junction potential £j forms when the two half-cells of a cell contain different
electrolyte solutions. The magnitude of £j depends on the concentrations (strictly, the
activities) of the constituent ions in the cell, the charges of each moving ion, and
on the relative rates of ionic movement across the membrane. We record a constant
value of £j because equilibrium forms within a few milliseconds of the two half-cells
adjoining across the membrane.
Liquid junction potentials are rarely large, so a value of £j as
large as 0.1 V should be regarded as exceptional. Nevertheless,
junction potentials of 30 mV are common and a major cause of
experimental error, in part because they are difficult to quantify,
but also because they can be quite irreproducible.
We have already encountered expressions that describe the emf
of a cell in terms of the potentials of its constituent half-cells, e.g.
Equation (7.23). When a junction potential is also involved - and
it usually is - the emf increases according to
emf = .Expositive half-cell) ~ ^negative half-cell) + Ej (7.53)
which explains why we occasionally describe £j as 'an additional source of potential'.
In most texts, the liquid
junction potential is
given the symbol E r In
some books it is written
as Efin or even E,
-dj)
(Up)-
342
ELECTROCHEMISTRY
While it is easy to measure a value of emf we do not know the magnitude of E y
SAQ 7.21 illustrates why we need to minimize E y
SAQ 7.22 The emf of the cell SHE |Ag+|Ag, is 0.621 V. Use the Nernst
equation to show that a (Ag + } = 10~ 3 if fj = V, but only 4.6 x 10 -4 if
fj = 20 mV. E^ g+ Ag = 0.799 V. [Hint: to compensate for Ej in the sec-
ond calculation, say that only 0.601 V of the emf derives from the Ag + |Ag
half-cell, i.e. E Ag+ Ag = 0.601 V.]
What is a 'salt bridge'?
Minimizing junction potentials
In normal electrochemical usage, the best defence against a junction
potential £j is a salt bridge. In practice, the salt bridge is typically a
thin strip of filter paper soaked in electrolyte, or a U-tube containing
an electrolyte. The electrolyte is usually KC1 or KNO3 in relatively
high concentration; the U-tube contains the salt, perhaps dissolved
in a gelling agent such as agar or gelatine.
We connect the two half-cells by dipping either end of the salt
bridge in a half-cell solution. A typical cell might be written in schematic form as:
It's called a bridge
because it connects the
two half-cells, and salt
because we saturate
it with a strong ionic
electrolyte.
Zn (s) |Zn 2+ (aq) |S|Cu 2+ (aq) |Cu
(s)
We write the salt bridge as '|S|\ where the S is the electrolyte within the salt bridge.
But how does the salt bridge minimize Ej? We recognize first how the electrolyte
in the bridge is viscous and gel -like, so ionic motion through the bridge is slow.
Secondly, the ionic diffusional processes of interest involve only the two ends of the
salt bridge. Thirdly, and more importantly, the concentration of the salt in the bridge
should greatly exceed the concentrations of electrolyte within either half-cell (exceed,
if possible, by a factor of between 10-100 times).
The experimental use of a salt bridge is depicted in Figure 7.18. The extent of
diffusion from the bridge, as represented by the large arrows in the diagram, is seen
to be much greater than diffusion into the bridge, as represented by the smaller of
the two arrows. A liquid junction forms at both ends of the bridge, each generating
its own value of E y If the electrolyte in the bridge is concentrated, then the diffusion
of ions moving from the bridge will dominate both of these two E y Furthermore,
these £j will be almost equal and opposite in magnitude, causing them to cancel each
other out.
Table 7.13 shows how the concentration of the salt in the bridge has a large effect
on E-y it is seen that we achieve a lower value of £j when the bridge is constructed
with larger concentrations of salt. A junction potential £j of as little as 1-2 mV can
be achieved with a salt bridge if the electrolyte is concentrated.
BATTERIES 343
Salt bridge
Detail
Figure 7.18 The two half-cells in a cell are joined with a salt bridge. Inset: more ions leave the
bridge ends than enter it; the relative sizes of the arrows indicate the relative extents of diffusion
Table 7.13
Values of junc-
tion potential
in aqueous cells
as a function of the concentra-
tion of inert KC1 within a salt
bridge
[KCl]/c s
£j/mV
0.1
27
1.0
8.4
2.5
3.4
4.2 (sat'd)
<1
Minimizing junction potentials with a swamping electrolyte
The second method of minimizing the junction potential is to employ a 'swamping
electrolyte' S. We saw in Section 4.1 how diffusion occurs in response to entropy
effects, themselves due to differences in activity. Diffusion may be minimized by
decreasing the differences in activity, achieved by adding a high concentration of
ionic electrolyte to both half-cells. Such an addition increases their ionic strengths /,
and decreases all activity coefficients y± to quite a small value.
If all values of y± are small, then the differences between activities also decrease.
Accordingly, after adding a swamping electrolyte, fewer ions diffuse and a smaller
junction potential forms.
7.7 Batteries
How does an electric eel produce a current?
Introduction to batteries
The electric eel (Electrophorus electricus) is a thin fish of length 3-5 feet; see
Figure 7.19. It is capable of delivering an electric shock of about 600 V as a means
344 ELECTROCHEMISTRY
Figure 7.19 The electric eel (Electrophorus electricus) is a long, thin fish (3-5 feet) capable of
delivering an electric shock of about 600 V. (Figure reprinted from Ions, Electrodes and Membranes
by Jiri Koryta. Reproduced by permission of John Wiley and Sons Ltd)
of self-protection or for hunting. The eel either stuns a possible aggressor, or becomes
an aggressor itself by stunning its prey, prior to eating it.
Fundamentally, the eel is simply a living battery. The tips of its head and tail
represent the poles of the eel's 'battery'. As much as 80 per cent of its body is an
electric organ, made up of many thousands of small platelets, which are alternately
super-abundant in potassium or sodium ions, in a similar manner to the potentials
formed across axon membranes in nerve cells (see p. 339). In effect, the voltage
comprises thousands of concentration cells, each cell contributing a potential of about
160 mV. It is probable that the overall eel potential is augmented with junction
potentials between the mini-cells.
The eel produces its electric shock when frightened, hungry or when it encounters
its prey. The shock is formed when the eel causes the ionic charges on the surfaces
of its voltage cells to redistribute (thereby reversing their cell polarities), and has the
effect of summing the emfs of the mini-cells, in just the same way as we sum the
voltages of small batteries incorporated within a series circuit. The ionic strength of
seawater is very high, so conduction of the current from the eel to its prey is both
swift and efficient.
Battery terminology
A battery is defined as a device for converting chemical energy into electrical energy.
A battery is therefore an electrochemical cell that spontaneously produces a current
when the two electrodes are connected externally by a conductor. The conductor will
be the sea in the example of the eel above, or will more typically be a conductive
BATTERIES
345
A battery is sometimes
called a galvanic cell.
metal such as a piece of copper wire, e.g. in a bicycle headlamp. The battery pro-
duces electrons as a by-product of the redox reaction occurring at the cathode. These
electrons pass through the load (a bulb, motor, etc.) and do work, before re-entering
the battery where the anode consumes them. Electrochemical reduction occurs at the
positive pole (the anode) of the battery simultaneously with electrochemical oxidation
at the negative pole (the cathode).
There are several types of battery we can envisage. A majority of
the batteries we meet are classed as primary batteries, i.e. a chem-
ical reaction occurs in both compartments to produce current, but
when all the chemicals have been consumed, the battery becomes
useless, so we throw it away. In other words, the electrochemical reactions inside
the battery are not reversible. The most common primary batteries are the Leclanche
cell, as described below, and the silver-oxide battery, found inside most watches and
slim-line calculators.
By contrast, secondary batteries may be reused after regenerat-
ing their original redox chemicals. This is achieved by passing a
current through the battery in the opposite direction to that during
normal battery usage. The most common examples of secondary
batteries are the lead-acid cell (there is one inside most cars) and
nickel-cadmium batteries (commonly called 'NiCad' batteries).
In the shops, sec-
ondary batteries are
usually called rechar-
geable batteries.
What is the earliest known battery?
Battery types
We have evidence that batteries were not unknown in the ancient world. The Parthians
were a race living in the Mediterranean about 2000 years ago, from ca 300 bc until
ad 224, when they were wiped out by the Romans. They are mentioned in the Bible,
e.g. see The Acts of the Apostles, Chapter 2.
A device was found in 1936 near what is now modern Baghdad, the capital of
Iraq. It consisted of a copper cylinder housing a central iron rod. The identity of the
ionic electrolyte is now wholly unknown. The device was held together with asphalt
as glue.
If the copper was tarnished and the iron was rusty (i.e. each was covered with
a layer of oxide), then an approximate emf for this 2000-year-old battery would
probably be in the range 0.6-0.7 V. We do not know what the battery was used for.
The Daniell cell
One of the first batteries in recent times was the Daniell cell,
Zn|Zn 2+ :Cu 2+ |Cu. This battery comprised two concentric terra-
cotta pots, the outer pot containing a zinc solution and the inner
pot containing a copper solution. Metallic rods of copper and
The vertical dotted
line in this schematic
indicates a porous
membrane.
346
ELECTROCHEMISTRY
zinc were then immersed in their respective solutions. The electrode reaction at the
zinc anode is Zn — »■ Zn 2+ + 2e~ , while reduction occurs at the positive electrode,
Cu 2+ + 2e" -+ Cu.
Although this battery was efficient, it was never popular because it required aqueous
solutions, which can be a danger if they slopped about. Its market share also suffered
when better batteries were introduced to the market.
The Leclanche 'dry-cell' battery
The Leclanche cell was first sold in 1880, and is still probably the most popular
battery in the world today, being needed for everyday applications such as torches,
radios, etc. It delivers an emf of ca 1 .6 V.
Figure 7.20 depicts the Leclanche cell in schematic form. The zinc can is generally
coated with plastic for encapsulation (i.e. to prevent it from splitting) and to stop the
intrusion of moisture. Plastic is an insulator, and so we place a conductive cap of
stainless steel at the base of the cell to conduct away the electrons originating from
the dissolution of the zinc from the inside of the can. A carbon rod then acts as an
inert electrode to conduct electrons away from the reduction of Mn02 at the cathode.
The reaction at the cathode is given by
2Mn0 2(s) + 2H 2 + 2e~
2MnO(OH) (s) +20H
(aq)
(7.54)
and the reaction at the zinc anode is: Zn — > Zn 2+ + 2e~.
We incorporate an ammonium salt to immobilize the Zn 2+ ions: NH4CI is prepared
as a paste, and forms a partially soluble complex with zinc cations produced at the
Positive terminal
Mixture of zinc chloride
ammonium chloride, carbon powder
and manganese dioxide
Carbon rod
Zinc casing (anode)
Negative terminal
(in electrical contact
with the zinc case)
Figure 7.20 Schematic depiction of the Leclanche cell
BATTERIES 347
Table 7.14 Advantages and disadvantages of the Leclan-
che cell
Advantages
It is cheap to make
It has a high energy density
It is not toxic
It contains no liquid electrolytes
Disadvantages
Its emf decreases during use as the material is consumed
It cannot readily deliver a high current
anode. We sometimes add starch to the paste to provide additional stiffness. The
juxtaposition of the zinc ions with the zinc of the casing forms a redox buffer,
thereby decreasing the extent to which the potential of the zinc half-cell wanders
while drawing current.
Table 7.14 lists the advantages and disadvantages of the Leclan-
che cell.
The lead -acid battery
The lead-acid cell was invented by Plante in 1859, and has remai-
ned more-or-less unchanged since Faure updated it in 1881. The
lead-acid cell is the world's most popular choice of secondary
battery, meaning it is rechargeable. It delivers an emf of about
2.0 V. Six lead-acid batteries in series produce an emf of 12 V.
Table 7.15 Advantages and disadvantages of the lead-acid battery
Advantages
It is relatively easy to make, and so can be quite cheap
It has a high energy density, producing much electrical energy per unit mass
It can readily deliver a very high current
Disadvantages
It contains toxic lead
Also, since it contains lead, its power density is low
The acid is corrosive
Furthermore, the acid is a liquid electrolyte
Given time, lead sulphate (which is non-conductive) covers the electrode. Having
'sulphated up', the energy density of the battery is greatly impaired. To avoid
sulphating up, it ought to be recharged often
Alkaline manganese
cells are broadly sim-
ilar in design to the
Leclanche cell, but they
contain concentrated
KOH as the electrolyte
instead of NH 4 CI.
348 ELECTROCHEMISTRY
Plates of lead, each coated with lead dioxide, are immersed in fairly concentrated
sulphuric acid. Lead is oxidized at the lead anode during discharge:
Pb (s) + HS0 4 " (aq) ► PbS0 4(s) + H+ (aq) + 2e" (7.55)
The reaction at the cathode during discharge is
Pb0 2(s) + 2H+ (aq) + H 2 S04(a q) + 2e" ► PbS0 4(s) + 2H 2 (7.56)
Both half-cell reactions are fully electro-reversible. In practice,
Spongy lead has a
higher surface area
than normal lead.
there are two types of lead: the 'collector' electrode is made of lead
alloy 'mesh' in order to give it greater structural strength, and is
made with about 5 per cent antimony. 'Spongy lead' (Pb + Pb0 2 )
is introduced into the holes of the mesh.
Table 7.15 lists the advantages and disadvantages of the lead-acid battery.
8
Chemical kinetics
Introduction
In previous chapters, we considered questions like: 'How much energy does a reaction
liberate or consume?' and 'In which direction will a reaction proceed?' We then asked
questions like: 'To what extent will a reaction proceed in that direction, before it
stops?' and even 'Why do reactions occur at all?' In this chapter, we look at a different
question: 'How fast does a reaction proceed?' Straightaway, we make assumptions.
Firstly, we need to know whether the reaction under study can occur: there is no
point in looking at how fast it is not going if a reaction is not thermodynamically
feasible! So we first assume the reaction can and does occur.
Secondly, we assume that reactions can be treated according to their type, so 'reac-
tion order' is introduced and discussed in terms of the way in which concentrations
vary with time in a manner that characterizes that order.
Finally, the associated energy changes of reaction are discussed in terms of the
thermodynamic laws learnt from previous chapters. Catalysis is discussed briefly
from within this latter context.
8.1 Kinetic definitions
Why does a 'strong' bleach clean faster
than a weaker one does?
Introduction to kinetics: rate laws
We often clean away the grime and dirt in a kitchen with bleach,
the active ingredient of which is the hypochlorite ion C10~ . The
cleaning process we see by eye ('the bleaching reaction') occurs
between an aqueous solution of C10~ ion and coloured species
stuck to the kitchen surfaces, which explains why the dirt or grease,
etc., appears to vanish during the reaction. The reaction proceeds
concurrently with colour loss in this example.
Care: a supermarket
uses the word 'strong'
in a different way from
chemists: remember
from Chapter 6 that the
everyday word 'strong'
has the specific chem-
ical meaning 'a large
equilibrium constant of
dissociation'.
350
CHEMICAL KINETICS
We
ignore
the
com
plication
here
that
solution
-phase
CIO
is in eq
uilibrium
with chlorine
This reaction could be
one of the steps in
a more complicated
series of reactions,
in a so-called multi-
step reaction. If this
reaction is the rate-
determining step of
the overall compli-
cated series, then this
rate law still holds;
see p. 357.
We define the rate of
reaction as the speed at
which a chemical con-
version proceeds from
start to its position
of equilibrium, which
explains why the rate
is sometimes written
as d£/dt, where ? is the
extent of reaction.
We formulate the rate
of a reaction by multi-
plying the rate constant
of the reaction by the
concentration of each
reactant, i.e. by each
species appearing at
the tail end of the
arrow. We can only do
this if the reaction is
elementary (proceeds
in a single step)
We soon discover that a 'strong' bleach cleans the surfaces faster
than a more dilute bleach. The reason is that 'strong' bleaches are
in fact more concentrated, since they contain more C10~ ions per
unit volume than do 'weaker' bleaches.
We will consider the chemical reaction between the hypochlo-
rite ion and coloured grease to form a colourless product P (the
'bleaching' reaction) as having the following stoichiometry:
CIO + grease
(8.1)
We wish to know the rate at which this reaction occurs. The rate
is defined as the number of moles of product formed per unit time.
We define this rate according to
rate =
[product]
At
(8.2)
As far as equations like Equation (8.2) are concerned, we tend to
think of a chemical reaction occurring in a forward direction, so the
product in Equation (8.2) is the chemical at the head of the arrow
in Equation (8.1). Consequently, the concentration of product will
always increase with time until the reaction reaches its position of
equilibrium (when the rate will equal zero). This explains why the
rate of reaction always has a positive value. The rate is generally
cited with the units of mol dm~ s _1 , i.e. concentration change
per second.
The numerical value of the rate of reaction is obtained from a
rate equation, which is obtained by first multiplying together the
concentrations of each reactant involved in the reaction. (Before
we do this, we must be sure of the identities of each reactant - in
a complicated multi-step reaction, the reacting species might differ
from those mentioned in the stoichiometric equation.) The follow-
ing simple equation defines exactly the rate at which the reaction
in Equation (8.1) occurs:
rate = k [CIO ] [grease]
(8.3)
where the constant of proportionality k is termed the rate constant.
The value of k is generally constant provided that the reaction
is performed at constant temperature T. Values of rate constant
are always positive, although they may appear to be negative in
some of the more complicated mathematical expressions. Table 8.1
contains a few representative values of k.
We see from Equation (8.3) that the reaction proceeds faster (has
a faster rate) when performed with a more concentrated ('strong')
KINETIC DEFINITIONS
351
Table 8.1 Selection of rate constants k
Reaction
Phase Temperature/ °C kl units 3
First-order reactions
SO2CI2 -*■ S0 2 + Cl 2
Cyclopropane — >• propene
C2H6 — > 2CH3
CIH2C-CH2CI -► CH 2 =CHC1 + HC1
Gas
Gas
Gas
Gas
Second-order reactions
CIO" + Br" -► BrO + CI" Aqueous
I
CH3COOC2H5 + NaOH -> CH 3 C0 2 Na + C 2 H 5 OH Aqueous
I
H 2 + I 2 -*■ 2HI Gas
2N0 2 -► 2NO + 2 Gas
H 2 + 2NO -► N 2 + 2H 2 Gas
21 -► I 2 Gas
H+ + OH" -*■ H 2 Aqueous
320
500
700
780
25
30
400
300
700
23
25
2 x 10" 5
6.71 x 10" 5
5.36 x 10" 4
4.4 x 10" 3
4.2 x 10" 7
1.07 x 10" 2
2.42 x 10" 2
0.54
145.5
7 x 10 9
1.35 x 10 11
a For first-order reactions, k has the units of s . For second-order reactions, k has units of dm mol s
bleach because the concentration term '[CIO ]' in Equation (8.3)
has increased.
SAQ 8.1 Consider the reaction between ethanoic acid
and ethanol to form the pungent ester ethyl ethanoate
and water:
CH3COOH + CH3CH2OH
CH3COOC2H5 + H2O
Care: A 'rate constant'
is written as a lower
case k in contrast to the
more familiar 'equilib-
rium constant', which
is written as an upper
case K.
Write an expression for the rate of this reaction in a similar form to that in
Equation (8.3), assuming the reaction proceeds in a single step as written.
SAQ 8.2 Write an expression for the rate of the reaction Cu 2+ (aq ) +
4NH 3(a q) ->■ [Cu(NH 3 )4] 2+ (a q), assuming that the reaction proceeds in a
single step as written.
Why does the bleaching reaction eventually stop?
Calculating rates and rate constants
When cleaning in the kitchen with a pool of bleach on tables and surfaces, there
comes a time when the bleaching action seems to stop. We might say that the bleach
is 'exhausted', and so pour out some more bleach from the bottle.
When thinking about reaction kinetics, we need to appreciate that reactions involve
chemical changes, with reactants being consumed during a reaction, and products
352
CHEMICAL KINETICS
being formed. After a time of reacting, the concentration of one or more of the
chemicals will have decreased to zero. We generally say the chemical is 'used up'.
Now look again at Equation (8.3). The rate of reaction depends on the concentrations
of both the grease and the C10~ ion from the bleach. If the concentration [C10~]
has decreased to zero, then the rate will also be zero, whatever the value of k or the
concentration of grease. And if the rate is zero, then the reaction stops.
Although we appreciate from Equation (8.3) that the reaction
will stop when one or both of the concentration terms reaches zero,
we should also appreciate that the concentration terms reach zero
faster if the value of k is large, and the concentrations deplete more
slowly if k is smaller. We see how the value of the rate constant
is important, because it tells us how fast a reaction occurs.
The value of the rate
constant is important
because it tells us how
fast a reaction occurs.
Worked Example 8.1 In solution, the cerium(IV) ion reacts with aqueous hydrogen
peroxide with a 1:1 stoichiometry. The reaction has a rate constant of 1.09 x
10 6 dm mol~ s _1 . How fast is the reaction that occurs between Ce IV and H2O2, if
[Ce IV ] = 1(T 4 moldm" 3 and [H 2 2 ] = 1(T 3 moldnT 3 ?
By 'how fast', we are in effect asking 'What is the value of the rate of this reaction?'
The reaction has a 1:1 stoichiometry; so, following the model in Equation (8.3), the
rate equation of reaction is
(8.4)
lVn
Placing the concen-
tration brackets
together - without
a mathematical
sign between
them - implies that
the concentrations are
to be multiplied.
rate = £[Ce lv ][H 2 2 ]
where k is the rate constant. Inserting values for k and for the two
concentrations:
rate = 1.09 x 10 6 dm 3 mol -1 s _1 x 10" 4 moldm -3
x 10 -3 moldm -3
rate = 0.109 moldm 3 s"
or, stated another way, the concentration of product changes (increases) by the amount
0.109 moldm -3 per second, or just over 1 mol is formed during 10 s. This is quite
a fast reaction.
SAQ 8.3 Show that the rate of reaction in Worked Example 8.1 quadru-
jv
pies if both [Ce lv ] and [H 2 2 ] are doubled.
The rate constant k is
truly constant and only
varies with tempera-
ture and (sometimes)
with ionic strength I.
Worked Example 8.1 shows a calculation of a reaction rate from
a rate constant k of known value, but it is much more common to
know the reaction rate but be ignorant of the rate constant. A rate
equation such as Equation (8.3) allows us to obtain a numerical
value for k. And if we know the value of k, we can calculate
from the rate equation exactly what length of time is required
for the reaction to proceed when performed under specific reac-
tion conditions.
KINETIC DEFINITIONS
353
Worked Example 8.2 Ethyl ethanoate (0.02 moldirT ) hydrolyses during reaction with
aqueous sodium hydroxide (0.1 moldm" ). If the rate of reaction is 3 x 10 2 moldm" s _1 ,
calculate the rate constant k.
The rate equation of reaction is given by
rate = /fc[CH 3 COOCH 2 CH 3 ][NaOH]
Rearranging Equation (8.5) to make k the subject gives
k
rate
[CH 3 COOCH 2 CH 3 ] [NaOH]
Inserting values yields
3 x 10 2 moldm 3 s
3 c-1
and
0.02 moldm x 0.1 moldm
i=1.5x 10 5 mor'dnrV 1
(8.5)
This is quite a large value of k.
Worked Example 8.2 yields a value for the rate constant k, but an alternative and
usually more accurate way of obtaining k is to prepare a series of solutions, and to
measure the rate of each reaction. A graph is then plotted of 'reaction rate' (as 'v')
against 'concentration(s) of reactants' (as 'x') to yield a linear graph of gradient equal
to k.
Worked Example 8.3 We continue with the reaction between Ce IV and H2O2 from
Worked Example 8.1. Consider the following data:
[Ce IV ]/moldnT 3
[H 2 2 ]/moldm- 3
rate/mol dm~ 3 s _1
3 x 10~ 5
8 x 10~ 6
0.0009
5 x 10" 5
2 x 10~ 5
0.0011
8 x 10" 5
3 x 10~ 5
0.0026
1 x 10" 4
5 x 10- 5
0.0055
Equation (8.4) above can be seen to have the form 'y = mx', in which
'rate' is 'y' and the mathematical product '[Ce IV ] x [H 2 2 ]' is 'x'.
The gradient m will be equal to k.
Figure 8.1 shows such a graph. The gradient of the graph is 1.09 x
10 6 dm 3 s" 1 (which is the same value as that cited in Worked Example
8.1). Notice how the intercept is zero, which confirms the obvious
result that the rate of reaction is zero (i.e. no reaction can occur) when
no reactants are present.
We are saying here that
the rate is of the form
y = mx (straight line).
The intercept is zero.
354
CHEMICAL KINETICS
0.006
"i i i i r
0.00E+00 1.00E-09 2.00E-09 3.00E-09 4.00E-09 5.00E-09 6.00E-09
[H 2 2 ]x[Ce lv ]/mol 2 drrr 6
Figure 8.1 Graph of reaction rate against the product '[Ce IV ] x [H2O2]'. The numerical value of
the gradient of the graph is the rate constant k
Why does bleach work faster on some greases than on
others?
Rates expressions, reaction stoichiometry and reaction order
Each reaction has a unique value of rate constant k. For example, the value of k
in Worked Example 8.2 would have been different if we had chosen ethyl formate,
or ethyl propanoate, or ethyl butanoate, etc., rather than ethyl ethanoate. The value
of k depends ultimately on the Gibbs function of forming reaction intermediates, as
discussed below.
Most forms of grease in the kitchen derive from organic materials in the
home - some derive from meat, some from vegetable oils and some from pets in
the home, or even human tissue such as oily fingerprints. (Kitchen grease is, in fact,
a complicated mixture of chemicals, each of which reacts with bleach at a different
rate and, therefore, with a different value of k.)
But the reaction conditions are still more complicated because the stoichiometry
of reaction might alter. Many greases and oils comprise the esters of long-chain
fatty acids. The hydrolysing reaction between NaOH and an ester such as ethyl
ethanoate proceeds with a stoichiometry of 1:1, but a tri-ester, such as most natural
oils (e.g. olive oil or sunflower oil), occurs with a 1:3 stoichiom-
etry, consuming one hydroxide ion per ester bond. Clearly, the
hydroxide will be consumed more quickly when hydrolysing a tri-
ester than a mono-ester. The rate depends on the stoichiometry
of reaction.
The rate of reaction
depends on the stoi-
chiometry.
KINETIC DEFINITIONS 355
Worked Example 8.4 The active ingredient within many weedkillers is methyl violo-
gen, MV+* (I).
H 3 C-N X > ( 7 N-CH 3
\=/ \=/
cr
(I)
Being a weedkiller, MV + " is said to be phytotoxic. As a complication, two radicals of
MV + " will dimerize in solution to form the non-toxic dimer species (MV)2 2+ :
2 MV+' ► (MV) 2 2+ (8.6)
Why does the rate of the reaction in Equation (8.6) quadruple when we double the con-
centration of the radical cation, MV + "?
The rate of reaction is written as the rate constant of reaction multiplied by the concen-
tration of each reacting species on the tail end of the arrow. Accordingly, we write
rate = k[MY + '] 2 (8.7)
Why does the rate quadruple if we double [MV + *]? We start by saying that
[MV + *] (ne w) — 2 x [MV + *] (old) . By doubling the concentration of MV + " and (from
Equation (8.7)) squaring its new value, we see that the new rate = k x {2 x [MV + *]( id)} 2 .
This new rate is the same as 4 x £[MV +, ] 2 old) , which explains why the rate quadruples
when the concentration of MV + * doubles.
The reason why the rate equation includes the square term [MV + *] 2 rather than just
[MV +# ] should not surprise us. Notice that we could have written the equation for the
chemical reaction in a slightly different way, as
MV+' + MV+* ► (MV) 2 2+ (8.8)
which is clearly the same equation as Equation (8.6). If we derive a rate equation for this
alternative way of writing the reaction, with a concentration term for each participating
reactant, then the rate equation of the reaction is
rate = fc[MV + '][MV +# ] (8.9)
and multiplying the two MV + * terms together yields [MV + *] 2 . We see that Equation (8.9)
is the same rate equation as that obtained in Equation (8.7).
SAQ 8.4 Write the rate equations for the following reactions. In each
case, assume the reaction proceeds according to its stoichiometric
equation.
356
CHEMICAL KINETICS
(1) 2N0 2( g) "* N 2 4 (g)
(2) 2S0 2(g)
(3) Ag^
(aq)
-0
CI"
2(g)
(aq)
2S0 3( g)
> AgCI (s)
(4) HCI (aq) + NaOH (aq) -► NaCI (aq)
H 2
Reaction order
The order of a reac-
tion is the same as
the number of concen
tration terms in the
rate expression.
The order of a reaction is the same as the number of concentration
terms in the rate expression. Consider the general rate equation:
rate = fc[A] fl [B]*
(8.10)
The sum of the powers is equal to the order of the reaction.
We see that Equation (8.9) involves two concentration terms, so
it is said to be a second-order reaction. The rate expression for
SAQ 8.4 (1) involves two concentration terms, so it is also a second -order reaction.
Although each of the two concentrations in Equation (8.7) is the same, there are
nevertheless two concentrations, and Equation (8.7) also represents a second-order
reaction. In fact, the majority of reactions are second order.
The rate expression for SAQ 8.4 (2) involves three concentra-
tions, so it is a third-order reaction. Third-order reactions are very
rare, and there are no fourth-order or higher reactions.
Third-order reactions
are very rare.
SAQ 8.5 Analyse the following rate equations, and determine the orders
of reaction:
(1) rate = /c[Cu 2+ ][NH 3 ]
(2) rate = /c[OI-r]
(3) rate = /c[NO] 2 [0 2 ]
It is increasingly common to see the rate constant given a subscripted descriptor
indicating the order. The rate equation in SAQ 8.5 (1) would therefore be written
as k 2 .
Why do copper ions amminate so slowly?
The kinetic treatment of multi-step reactions, and the
rate-determining step
Addition of concentrated ammonia to a solution of copper(II) yields a deep-blue
solution of [Cu(NH 3 )4] 2+ . The balanced reaction is given by
Cu 2+ (aq) + 4NH
3(aq)
-> [Cu(NH 3 ) 4 ] 2+
(aq)
(8.11)
KINETIC DEFINITIONS
357
A quick look at the reaction suggests that the rate of this ammination reaction should
be &[Cu 2+ ( aq )][NH3( aq )] 4 , where the power of '4' derives from the stoichiometry
(provided that the reaction as written was the rate-determining step). It would be
a fifth-order reaction, and we would expect that doubling the concentration of ammo-
nia would cause the rate to increase 16-fold (because 2 4 = 16). But the increase in
rate is not 16-fold; and, as we have just seen, a fifth-order reaction is not likely.
In fact, the ammination reaction forming [Cu(NH 3 )4] 2+ occurs
stepwise, with first one ammonia ligand bonding to the copper ion,
then a second, and so forth until the tetra-amminated complex is
formed. And if there are four separate reaction steps, then there
are four separate kinetic steps - one for each ammination step,
with each reaction having its own rate constant k - we call them
£(i) > k(2) , &(3) and k^. This observation helps explain why the increase in reaction
rate is not 16-fold when we double the concentration of ammonia.
Each step in a multi-
step reaction sequence
proceeds at a different
rate.
Aside
When we write k(i)
with the
subscripted '1' in brackets,
we
mean the rate constant of
the first
step in a multi-step
reaction. When we write k\
with a subscripted '1' but no
brackets
we mean
the rate constant of a first-order reaction.
We adopt this convention
to avoid confusion.
Most organic reactions occur in multi-step reactions, with only
a small minority of organic reactions proceeding with a single
step. We find, experimentally, that it is extremely unlikely for any
two steps to proceed with the same rate constant, which means
that we can only follow one reaction at a time. And the reaction
that can be followed is always the slowest reaction step, which
we call the rate-determining step - a term we often abbreviate
to RDS.
A simple analogy from everyday life illustrates the reasonable-
ness of this assumption. Imagine driving north from Italy to Nor-
way in a journey involving travel along a fast motorway, along
a moderately fast main road, and crawling through a 'contraflow'
system, e.g. caused by road works. No matter how fast we travel
along the main roads or the motorway, the overall time required
for the journey depends crucially on the slowest bit, the tedious
stop-start journey through the 'contraflow'. In a similar way, the
only step in a multi-step reaction that we are able to follow experi-
mentally is the slowest. We call it the rate-determining step because
it 'determines' the rate.
It is not possible to
follow the rates in
a multi-step reac-
tion sequence: only
the slowest step can
be followed.
We call the slow-
est step in a multi-
step process its rate-
determining step, often
abbreviated to 'RDS'.
The overall (observed)
rate of a multi-step
reaction is equal to
the rate of the rate-
determining reaction.
358
CHEMICAL KINETICS
The separate reac-
tion steps will proceed
with the same rates in
the unlikely event that
all steps are diffusion
controlled; see p. 416.
The pair of reactions
in SAQ 8.6 explains
how old wine 'goes
off', i.e. becomes too
acidic to drink when
left too long.
We will discuss multiple-step reactions in much greater detail in
Section 8.4.
SAQ 8.6 Consider the oxidation of aqueous ethanol (e.g.
in wine) to form vinegar:
first reaction
ethanol + 2 > ethanal + H 2 (8.12)
slow
second reaction
2 ethanal + 2 > 2 ethanoic acid (8.13)
fast
Decide which step is rate determining, and then write
an expression for its rate (assuming that each reac-
tion proceeds according to the stoichiometric equation
written).
How fast is the reaction that depletes the ozone layer?
Use of pressures rather than concentrations in a rate equation
Many substances react in the gas phase rather than in solution. An important example
is the process thought to deplete the ozone layer: the reaction between gaseous ozone,
O3, and chlorine radicals, high up in the stratosphere. Ultimately, the chlorine derives
from volatile halocarbon compounds, such as the refrigerant Freon-12 or the methyl
chloroform thinner in correction fluid.
The ozone-depleting reaction involves a rather complicated series of reactions, all
of which occur in the gas phase. Equation (8.14) describes the rate-determining step:
The CI* radical is retrie-
ved quantitatively, and
is therefore a catalyst.
CI* + 3
-* CIO* + 2
(8.14)
The CIO* radical product of Equation (8.14) then reacts further,
yielding the overall reaction:
20 3(el + CI*
(as a catalyst)
30 2( g) +C1*
(retrieved catalyst)
(8.15)
The Earth is constantly irradiated with UV light, much of which is harmful to our
skin. Ozone absorbs the harmful UV frequencies and thereby filters the light before
it reaches the Earth's surface. Normal diatomic oxygen, O2, does not absorb UV in
this way, so any reaction that removes ozone has the effect of allowing more harmful
UV light to reach us. The implications for skin health are outlined in Chapter 9.
The rate equation for Equation (8.14) is expressed in terms of 'pressures' p rather
than concentrations, such as [CI*]. We write the rate as
rate = kp(CV)p(0 3 )
(8.16)
KINETIC DEFINITIONS
359
where k here is a gas-phase rate constant. This rate expression has a similar form to
the rate expressions above, except that the now-familiar concentration terms are each
replaced with a pressure term, causing the units of k to differ.
The kinetics of reactions such as those leading to ozone depletion are treated in
greater depth in subsequent sections.
Why is it more difficult to breathe when up a mountain
than at ground level?
The dependence of rate on reactant pressure
We all breathe oxygen from the air to maintain life. Although we require oxygen,
the air contains other gases. Normal air contains about 21 per cent of oxygen, the
remainder being about 1 per cent argon and 78 per cent nitrogen.
We term the component of the total air pressure due to oxygen its 'partial pressure',
p{02)- From Dalton's law (see Section 5.6), the partial pressure of oxygen p(C>2) is
obtained by multiplying the mole fraction of the oxygen by the overall pressure of
the gases in air.
SAQ 8.7 The air we breathe at sea level has a pres-
sure of 100 kPa. Show that the partial pressure of oxygen
is 21 kPa.
'Standard pressure'
p e equals 10 5 Pa, and
is sometimes called
1 bar.
We start to feel a bit breathless if the partial pressure of oxygen
decreases below about 15 kPa; and we will feel quite ill (light-
headed, breathless and maybe nauseous) if p(02) drops below
10 kPa.
Air is taken into the lungs when we breathe. There, it is trans-
ported through the maze of progressively smaller bronchial tubes
until it reaches the tiny sacs of delicate tissue called alveoli. Each
sac look like bunches of grapes. The alveoli are the sites where oxy-
gen from the air enters the blood, and the carbon dioxide from the
body passes into the air (Figure 8.2). Oxygenated blood then flows
around the body. Each alveolus is tiny, but there are 300000000
in each lung.
We can think of the oxygen transfer from the lung to the blood as a simple chemical
reaction: molecules of gas strike the alveoli. By analogy with simple solution-phase
reactions, the rate equation describing the rate at which oxygen enters the blood is
formulated according to
As the molar masses of
oxygen, nitrogen and
argon are so similar,
we can approximate
the mole fractions of
the gases to their per-
centage compositions.
rate = k x /(O2) x /(alveoli)
(8.17)
where each / simply means 'function of. The thermodynamic function chosen to
represent the oxygen is its partial pressure, p(02)- The alveoli are solid, so we omit
360
CHEMICAL KINETICS
Alveoli
Oxygenated air
in the lung
Oxygen
Capillaries within
the lung, containing
blood
Larynx
Trachea
4t J Principal bronchus
Figure 8.2 Oxygen from the air is taken into the lungs and is transported across the delicate
alveoli tissue covering the inside of the lung and into the blood
them from the rate equation, Equation (8.17). Accordingly, Equation (8.17) becomes
rate = k x p{02)
(8.18)
which is seen to be very similar to the rate equations we formulated earlier for a
reaction between solution-phase chemicals, with partial pressures replacing concen-
trations.
The overall pressure of the air decreases as we journey upwards, away from sea
level, but the proportions of the gases in the air remains constant. At a height of
2 miles above sea level, the air pressure has dropped to about two-thirds of p & .
Worked Example 8.5 If the overall air pressure at a height of about 2 miles has dropped
to 67 kPa, what is the partial pressure of oxygen?
Strategy, we calculate the partial pressure of oxygen p(02) from Dalton's law (see
p. 221), saying /KO2) = total pressure of the air x mole fraction of oxygen in the air.
Substituting values into the above equation:
p(0 2 ) = 67 kPa x 0.21
So the partial pressure of oxygen is 14 kPa.
This simple calculation shows why it is more difficult to breathe when up a mountain
than at ground level: the pressure term in Equation (8.18) decreases, so the rate at which
oxygen enters the blood decreases in proportion to the decrease in the oxygen partial
pressure. And the partial pressure is smaller at high altitudes than at sea level.
QUALITATIVE DISCUSSION OF CONCENTRATION CHANGES 361
By corollary, more oxygen can enter the blood if the oxygen partial pressure is increased.
Two simple methods are available to increase piOx)'-
(1) Breathe air of normal composition, but at a greater overall
pressure. An example of this approach is the diver who
breathes underwater while fitted out with SCUBA gear.
'SCUBA' is an acronym
for 'self-contained
underwater breathing
apparatus'.
(2) In cases where the lungs are damaged, a doctor will place a
patient in an 'oxygen tent'. The overall pressure of gas is the
same as air pressure (otherwise the tent would explode) but the percentage of
oxygen in the air is much greater than in normal air. For example, if the overall
air pressure is the same as p , but the air comprises 63% oxygen rather than
the 21% in normal air, then the partial pressure of oxygen will treble.
SAQ 8.8 Consider the equilibrium reaction between hydrogen and chlo-
rine to form HCI:
forward
H2(g) + Cl2(g) — ~ > 2HCI(g)
back
Write two separate rate laws, one for the forward reaction and one for
back reactions.
8.2 Qualitative discussion of concentration
changes
Why does a full tank of petrol allow a car to travel
over a constant distance?
Qualitative reaction kinetics: molecularity
■
Cars and buses are fuelled by a volatile mixture of hydrocarbons. The mixture is
called 'petrol' in the UK, and 'gas' (short for gasoline) in the USA. One of the
main chemicals in petrol is octane, albeit in several isomeric forms. In the internal
combustion engine, the carburettor first vaporizes the petrol to form an aerosol (see
Section 10.2) comprising tiny droplets of petrol suspended in air (Figure 8.3). This
vaporization process is similar to that which converts liquid perfume into a fine spray.
Liquid
petrol
Spray (aerosol) of petrol
droplets in air
Figure 8.3 A carburettor in a car engine vaporizes the petrol to form an aerosol comprising tiny
droplets of petrol suspended in air
362
CHEMICAL KINETICS
These droplets burn in a controlled manner inside the car engine to release their
chemical energy while the petrol combines chemically with oxygen:
C8Hi 8 ( both g and i) + 12.502(g) ► 8C02(g) + 9H 2
(g)
(8.19)
This equation relates the overall ratios of reactants and products.
It may be termed a fully balanced equation, or is more commonly
termed a stoichiometric equation.
We employ octane as the fuel rather than, say, paper because
burning octane releases such a large amount of heat energy (which
is converted into kinetic energy). Inside the cylinders of the engine,
this energy is released quickly, causing the gaseous products of
combustion to heat up rapidly, which causes the pressure inside
the cylinder to increase greatly. An additional cause of increasing
pressure is the change in the number of moles of gas formed during
combustion, since 13.5 mol of reactant form 17 mol of gaseous
product. The ultimate cause of the car's motion is the large amount
of pressure work performed during burning.
A constant amount of energy is released per mole of petrol. Furthermore, the size
of the car's petrol tank predetermines the car's capacity to contain the alkane fuel;
so we see how the overall amount of energy available to the car between refuelling
stops cannot alter much. As the amount of energy is constant, the amount of work
that the car can perform is also constant.
A well-tuned car will, therefore, travel essentially the same number of miles per
tank of petrol because the amount of energy released for work is simply a function
of the tank's capacity.
The word 'stoichiom-
etry' comes from the
Greek word stoic,
meaning 'indifference'.
A stoichiometric reac-
tion is, therefore, indif-
ferent to all external
conditions, and pro-
ceeds with a predeter-
mined ratio of products
and reactants.
Why do we add a drop of bromine water to a solution
of an a I ken e?
Reaction stoichiometry and molecularity
Bromine readily adds across an alkenic double bond by electrophilic addition
(Figure 8.4). The brominated compound is usually colourless, but bromine in solution
('bromine water') has a red colour. Addition of bromine water to an alkene is
accompanied by a loss of the red colour as reaction proceeds. The stoichiometry of
reaction is almost always 1:1, with one molecule of bromine reacting per double bond.
Elemental analysis can be employed to show that the reaction between Br 2 and
a C=C double bond always occurs with this stoichiometry of 1:1. It is a law of
nature.
Figure 8.4 The red colour of elemental bromine is lost during addi-
tion across an alkenic double bond; the brominated compound is
usually colourless
QUALITATIVE DISCUSSION OF CONCENTRATION CHANGES
363
But sometimes we find that the kinetic data obtained experimentally bear little
resemblance to the fully balanced reaction. In other words, the reaction proceeds by
a mechanism that is different from the fully balanced equation.
The argument (above) concerning petrol centres on the way that chemicals always
react in fixed proportions. The reaction above (Equation (8.19)) is the stoichiometric
reaction because it cites the overall amounts of reaction occurring. But we should
appreciate that the reaction as written will not occur in a single step: it is impos-
sible even to imagine one molecule of octane colliding simultaneously with 12.5
molecules of oxygen. The probability is simply too vast. Even if there is a prob-
ability of getting the molecules together, how do we conceive of one and a half
molecules of oxygen? How can we have half a molecule? Even if we could, is
it possible to arrange these 12.5 molecules of oxygen physically around a single
molecule of octane?
Similarly, we have also seen already how the copper(II) tetrakis
(amine) complex forms in a step manner with four separate stages,
rather than in a single step, forming the mono-ammine complex,
then the bis-ammine, the tris-ammine and finally the tetrakis -ammine,
complex. So we start to appreciate that the actual reaction occur-
ring during the burning of octane is more complicated than it first
appears to be: the ratios in the stoichiometric equation are not
useful in determining the reaction mechanism.
In the case of burning a hydrocarbon, such as octane, the first step of the reac-
tion usually occurs between a peroxide radical 02* generated by the spark of the
sparking plug. A radical inserts into a C-H bond of a hydrocarbon molecule with the
likely mechanism:
~ C-H + 2 ' >~ C-O'-O-H (8.20)
The stoichiometric
ratios in a fully bal-
anced equation are
usually not useful for
determining the reac-
tion mechanism.
The peroxide bond in the product is weak and readily cleaves to form additional radicals.
Because more radicals are formed, any further reaction proceeds by a chain reaction,
termed radical propagation, until all the petrol has been consumed.
The rate-limiting process in Equation (8.20) involves the two
species (peroxide and octane) colliding within the car cylinder
and combining chemically. Because two species react in the rate-
limiting reaction step, we say that the reaction step represents
a bi molecular reaction. In alternative phraseology, we say 'the
molecularity of the reaction is two'.
The overwhelming majority of reactions are bimolecular. Some
reactions are unimolecular and a mere handful of processes proceed
as a trimolecular reactions. No quadrimolecular (or higher order)
reactions are known.
We must appreciate the essential truth that the molecularity of
a reaction and the stoichiometric equation are two separate things,
and do not necessarily coincide. Luckily, we find that reactions are
quite often 'simple' (or 'elementary'), by which we mean that they involve a single
reaction step. The molecularity and the reaction order are the same if the reaction
We describe the mole-
cularity with the famil-
iar Latin-based des-
criptors 'uni'= l/bi' =
2 and Mxi" = 3.
Care: the molecularity
and the order of a
reaction need not be
the same.
364
CHEMICAL KINETICS
Kinetically, a reaction
is simple if the molecu-
larity and the reaction
order are the same,
usually implying that
the reaction proceeds
with a single step.
involves a single step, so we say that many inorganic reactions are
simple because they are both second order and involve a bimolec-
ular reaction mechanism.
In SAQ 8.2, we considered the case of forming [Cu(NH3)4] 2+
from copper(II) and ammonia. We have already seen that a reaction
cannot be quintimolecular - five species colliding simultaneously.
In fact, involves a sequence of a bimolecular reactions.
When magnesium dissolves in aqueous acid, why does
the amount of fizzing decrease with time?
Reaction profiles
Magnesium ribbon reacts with sulphuric acid to cause a vigorous reaction, as demon-
strated by the large volume of hydrogen gas evolved, according to
Mg (s) + H 2 SO
4(aq)
MgS0 4(aq) + H 2(g) f
(8.21)
We know the reaction is complete when no more grey metal remains and the solution
is clear. In fact, we often say the magnesium dissolves, although such dissolution is
in fact a redox reaction (see Chapter 7).
But observant chemists will notice that the rate at which the gas is formed decreases
even before the reaction has stopped. Stated another way, the rate at which H2 is
formed will decrease during the course of the reaction.
We can explain this in terms of a rate expression, as follows. First, consider the
case where 1 mol of magnesium is reacted with 1 mol of sulphuric acid (we will
also say that the overall volume of the solution is 1 dm 3 ). Initially
there is no product, but at the end of reaction there will be 1 mol of
MgS0 4 and mol of magnesium metal or sulphuric acid. There-
fore, the amount of product in creases while the amounts of the two
reactants will both decrease as the reaction proceeds. The amounts
of material change and, as the volume of solution is constant, the
concentrations change.
We have already seen that the concentrations of reactants dictate the rate of reaction.
For the consumption of magnesium in acid, the rate of reaction is given by
The concentrations of
all reactants and prod-
ucts change during the
course of a reaction.
rate = £[Mg][H 2 S0 4 ]
(8.22)
This result was obtained by recalling how the rate of a reaction is equal to the rate
constant of the process, multiplied by the concentration of each species at the tail
end of the arrow in Equation (8.21).
There is 1 mol of magnesium at the start of reaction, and the concentration of
sulphuric acid was 1 mol dm -3 (because there was 1 mol of sulphuric acid in 1 dm 3 ).
QUALITATIVE DISCUSSION OF CONCENTRATION CHANGES
365
We next consider the situation after a period of time has elapsed such that half of the
magnesium has been consumed. By taking proportions, 0.5 mol of the acid has also
reacted, so its new concentration is 0.5 mol dm -3 . Accordingly, the value of 'rate'
from Equation (8.22) is smaller, meaning that the rate has slowed down. And so the
rate at which hydrogen gas is produced will also decrease.
Further reflection on Equation (8.22) shows that the concentrations of the two
reactants will always alter with time, since, by the very nature of a chemical reaction,
reactants are consumed. Accordingly, the rate of reaction will decrease continually
throughout the reaction. The rate will reach zero (i.e. the reaction will stop) when one
or both of the concentrations reaches zero, i.e. when one or all of the reactants have
been consumed completely. The rate at which hydrogen gas is formed will reach zero
when there is no more magnesium to react.
Figure 8.5 is a graph of the amount of sulphuric acid remaining as a function of
time. The trace commences with 1 mol of H2SO4 and none is left at the end of the
reaction, so [H2SO4] = 0. The rate of reaction is zero at all times
after the sulphuric acid is consumed because the rate equation,
Equation (8.22), involves multiplying the k and [Mg] terms by
[H2SO4], which is zero. Note that the abscissa (y-axis) could have
been written as 'concentration', because the volume of the solution
does not alter during the course of reaction, in which case the trace
is called a concentration profile.
Also depicted on the graph in Figure 8.5 is the number of moles of magnesium
sulphate produced. It should be apparent that the two concentration profiles (for
reactant and product) are symmetrical, with one being the mirror image of the other.
This symmetry is a by-product of the reaction stoichiometry, with 1 mol of sulphuric
acid forming 1 mol of magnesium sulphate product.
We call a graph of
concentration of reac-
tant or product (as V)
against time (as V) a
concentration profile.
0.8-
1 •* '
1 /
1 /
1 /
(b)
0.6-
li
0.4-
0.2-
(a)
n
1 1
I
10
Time f/min
15
20
Figure 8.5 Concentration profiles for the reaction between sulphuric acid and magnesium to
form magnesium sulphate, (a) Profile for the consumption of sulphuric acid, and (b) profile for the
formation of magnesium sulphate. The initial concentration of H2SO4 was 1.0 moldm~
366
CHEMICAL KINETICS
Time
Figure 8.6 Concentration profile for the reaction between sulphuric
acid and magnesium to form magnesium sulphate: reaction with
->• three concentrations of [H2S04], = o. The same amount of magnesium
reacted in each case, and cj, > C2 > C\
Figure 8.6 is a similar graph to Figure 8.5, showing the concentration profiles of
sulphuric acid as a function of its initial concentration. The gradients of each trace
are different, with the more concentrated solutions generating the steepest traces.
In fact, we have merely rediscovered the concept of rate, because the gradient of
the concentration profile is reaction rate, being the rate at which a compound or
chemical reacts as a function of time. We say the rate of reaction is the gradient, after
Equation (8.22).
Graphs such as those in Figures 8.5 and 8.6 are an ideal means
of determining the rates of reaction. To obtain the rate, we plot
the concentration of a reactant or product as a function of time,
and measure the slope. (Strictly, since the slopes are negative for
reactants, so the rate is 'slope x — 1'.)
We obtain the rate
of reaction as the
gradient of a concen
tration profile.
Aside
Because product forms at the expense of reactant, the magnitudes of the rates of forming
product and consuming reactant are the same. Although the magnitudes of the rates
are the same, their signs are not: the rate of forming product is positive because the
concentration increases with time.
For example, consider the reaction, a A + bB = cC + dD:
d[Product] d[reactant]
rate
dr
At
For the individual chemical components, we say,
1 d[A]
reactants rate = — 1 x — x
a At
At t [ d[C]
products rate = - x
F c dr
1 d[B]
rate = — 1 x - x
b dr
1 d[D]
rate = — x
d dr
In each case, the minus sign indicates a decreasing concentration with time.
QUALITATIVE DISCUSSION OF CONCENTRATION CHANGES 367
The rate of loss of reactant is negative because the concentration decreases with
time The gradient at the start of the reaction is called the initial rate. Analysing
the initial rates method is an extremely powerful way of determining the order of
a reaction.
Worked Example 8.6 The following kinetic data were obtained for the reaction between
nitric oxide and hydrogen at 700 °C. Determine: (1) the order of the reaction of the reaction
at this temperature; (2) the rate constant of reaction.
Experiment A
Experiment B
Experiment C
Initial concentration of
0.025
0.025
0.0125
NO/mol dm" 3
Initial concentration of
0.01
0.005
0.01
H2/mol dm~ 3
Initial rate/mol dnT 3 s _1
2.4 x 10~ 6
1.2 x 10~ 6
0.6 x 10~ 6
The rates listed in the table were obtained as the gradients of graphs like those in
Figure 8.6.
Answer Strategy
(1) To determine the order of reaction. It is always good research strategy to change
only one variable at a time. That way, the measured response (if any) can be attributed
unambiguously to the change in that variable. And the variable of choice in this example
will be one or other of the two initial concentrations.
The rate equation for the reaction will have the following form:
rate=fc[NOr[H 2 ]- y (8.23)
We determine values for the exponents x and y varying [NO] and [H2]. First, consider
experiments A and B. In going from experiment B to experiment A, the concentration
[NO] remains constant but we double [H2], as a consequence of which the rate dou-
bles. There is, therefore, a linear relationship between [H2] and rate, so the value of y
is '1'.
Second, consider experiments A and C. In going from C to A, we double the con-
centration [NO] and the rate increases by a factor of four. Accordingly, the value of x
cannot be '1' because the increase in rate is not linear. In fact, as 2 2 = 4, we see how
the exponent x has a value of '2'.
The reaction is first order in H2, second order in NO and, therefore, third order overall.
Inserting values into Equation (8.23), we say, rate = A:[NO] 2 [H2].
(2) To determine the rate constant of reaction. We know the rate and concentrations for
several sets of experimental conditions, so we rearrange Equation (8.23) to make k the
subject and insert the concentrations.
rate
k =
[NO] 2 [H 2 ]
368 CHEMICAL KINETICS
The question cites data from three separate experiments, each of which will give the same
answer. We will insert data from 'experiment A':
2.4 x 1(T 6 moldings" 1
[0.25 moldm~'] 2 [0.01 moldm _j ]
so
k = 3.84 x 10 3 dm 6 mor 2 s _1
SAQ 8.9 Iodide reacts with thiosulphate to form elemental iodine. If the
reaction solution contains a tiny amount of starch solution, then this I2 is
seen by eye as a blue complex. The data below were obtained at 298 K.
Determine the order of reaction, and hence its rate constant k.
Experiment A
Experiment B
Experiment C
Initial concentration of
0.1
0.1
0.2
I~/mol dm" 3
Initial concentration of
0.05
0.025
0.05
S 2 8 2 "/mol dm 3
Initial rate/mol
1.5 x 10- 4
7.5 x 10- 5
3.0 x 10- 4
dm~ 3 s *
8.3 Quantitative concentration changes:
integrated rate equations
Why do some photographs develop so slowly?
Extent of reaction and integrated rate equations
A common problem for amateur photographers who develop their own photographs
is gauging the speed necessary for development. When the solution of thiosulphate
is first prepared, the photographs develop very fast, but this speed decreases quite
rapidly as the solution 'ages', i.e. the concentration of S20s 2 ~ decreases because
the thiosulphate is consumed. So most inexperienced photographers have, at some
time, ruined a film by developing with an 'old' solution: they wait what seems like
forever without realizing that the concentration of thiosulphate is simply too low,
meaning that development will never occur. They ruin the partially processed film
by illuminating it after removal from the developing bath. These amateurs need to
answer the question, 'How much thiosulphate remains in solution as a function of
timeT We need a mathematical equation to relate concentrations and time.
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
369
The concentrations of each reactant and product will vary during the course of
a chemical reaction. The so-called integrated rate equation relates the amounts of
reactant remaining in solution during a reaction with the time elapsing since the
reaction started. The integrated rate equation has a different form according to the
order of reaction.
Let us start by considering a first-order reaction. Because the reactant concentration
depends on time t, we write such concentrations with a subscript, as [A] ( . The initial
reactant concentration (i.e. at time t = 0) is then written as [A]o. The constant of
proportionality in these equations will be the now-familiar rate constant k\ (where
the subscripted '1' indicates the order).
The relationship between the two concentrations [A]o, [A] r and t is given by
In
[A]p
[A],
kit
(8.24)
Equation (8.24) is the integrated first-order rate equation. Being
a logarithm, the left-hand side of Equation (8.24) is dimensionless,
so the right-hand side must also be dimensionless. Accordingly,
the rate constant k will have the units of s _1 when the time is
expressed in terms of the SI unit of time, the second.
The first-order rate
constant k will have
the units of s _1 .
Worked Example 8.7 Methyl ethanoate is hydrolysed when dissolved in excess hydro-
chloric acid at 298 K. The ester's concentration was 0.01 moldm -3 at the start of the
reaction, but 8.09 x 10~ 2 after 21 min. What is the value of the first-order rate con-
stant kyl
Answer Strategy. (1) we convert the time into SI units of seconds; (2) we insert values
into Equation (8.24).
(1) Convert the time to SI units: 21 min = 21 min x 60 smin = 1260 s
(2) Next, inserting values into Equation (8.24)
In
0.01 moldm~ J
0.008 09 moldm"
= 1260 s x k\
Notice how the units
of concentration will
cancel here.
so
ln(1.236) = ki x 1260 s
Taking the logarithm yields
0.211 = Jfc, x 1260 s
370
CHEMICAL KINETICS
and rearranging to make k\ the subject gives
0.211
*i
1.68 x 10~ 4 sT l
1260 s
This value of k\ is relatively small, indicating that the reaction is rather slow.
SAQ 8.10 (Continuing with the same chemical example): what is the
concentration of the methyl ethanoate after a time of 30 min? Keep the
same value of ki - it's a constant.
We can calculate a
rate constant k without
knowing an abso-
lute value for [A]
by following the frac-
tional changes in the
time-dependent con-
centration [A] t .
We note, when looking at the form of Equation (8.24), how the
bracket on the left-hand side contains a ratio of concentrations.
This ratio implies that we do not need to know the actual concen-
trations of the reagent [A]o when the reaction starts and [A], after
a time t has elapsed since the reaction commenced; all we need
to know is the fractional decrease in concentration. Incidentally,
this aspect of the equation also explains why we could perform the
calculation in terms of a percentage (i.e. a form of ratio) rather
than a 'proper' concentration.
In fact, because the integrated first-order rate equation (Equation (8.24)) is written
in terms of a ratio of concentrations, we do not need actual concentrations in moles
per litre, but can employ any physicochemical parameter that is proportional to con-
centration. Obvious parameters include conductance, optical absorbance, the angle
through which a beam of plane-polarized light is rotated (polarimetry), titre from a
titration and even mass, e.g. if a gas is evolved.
Worked Example 8.8 Consider the simple reaction 'A — > product' . After 3 min, 20 per
cent of A has been consumed when the reaction occurs at 298 K. What is the rate constant
of reaction k\ ?
We start by inserting values into the integrated Equation (8.24), noting
that if 20 per cent has been consumed then 80 per cent remains, so:
If we know the amount
of reactant consumed,
then we will need to
calculate how much
remains.
Remember that we
should always cite a
rate constant at the
temperature of mea-
surement, because k
itself depends on tem-
perature.
/100% of [A]
In I
V 80%
k x x (3 x 60 s)
of [A]
The logarithmic term on the left-hand side is ln(1.25) = 0.223, so
0.223
k, =
180 s
where the term '180 s' comes from the seconds within the 3 min of
observation time. We see that k\ — 1.24 x 10~ 3 s" 1 at 298 K.
SAQ 8.11 If the molecule A reacts by a first-order mech-
anism such that 15% is consumed after 1276 s, what is
the rate constant k?
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
371
Justification Box 8.1
Integrated rate equations for a first-order reaction
The rate law of a first-order reaction has the form
'rate = &i[A]'. And, by 'rate' we mean the rate of
change of the concentration of reactant A, so
rate =
d[A]
At
= *i[A]
(8.25)
The minus sign in
Equation (8.25) is
essential to show that
the concentration of A
decreases with time.
We will start at t = with a concentration [A]o, with the concentration decreasing with
time t as [A] t . The inclusion of a minus sign is crucial, and shows that species A is a
reactant and thus the amount of it decreases with time.
Separating the variables (i.e. rearranging the equation) and indicating the limits,
we obtain
/ — d[A] = *! /
•/fAln L A Jt Jo
At
Note how we can place the rate constant outside the integral, because it does not change
with time. Integration then yields
-Pn[A]][£=*iM{,
And, after inserting the limits, we obtain
- (ln[A], - ln[A] ) = *if
Using the laws of logarithms, the equation may be tidied further to yield Equation (8.24):
\lMt)
which is the integrated first-order rate equation.
Note that if a multiple-step reaction is occurring, then this equation relates only to the
case where the slowest (i.e. rate-limiting) step is kinetically first order. We will return
to this idea when we consider pseudo reactions in Section 8.4.
Graphical forms of the rate equations
Similar to the integrated first-order rate equation is the linear first-order rate equation :
ln[A],
y
m
t +
x
ln[A]
c
(8.26)
372
CHEMICAL KINETICS
We obtain the first-
order rate constant k
by drawing a graph
with the integrated
first-order rate equa-
tion, and multiplying its
slope by -1.
which we recognize as having the form of the equation for a straight
line: plotting ln[A] ( (as 'y') against time (as 'x') will be linear for
reactions that are first order.
The rate constant k\ is obtained from such a first-order rate graph
as (—1 x gradient) if the time axis is given with units of seconds.
Accordingly, the units of the first-order rate constant are s _1 .
Worked Example 8.9 Consider the following reaction: hydrogen
peroxide decomposes in the presence of excess cerous ion Ce m (which
reacts to form eerie ion Ce IV ) according to a first-order rate law. The
following data were obtained at 298 K:
This reaction appears
to be first order
because the cerium
ions are in excess, so
the concentration does
not really change. We
look at pseudo-order
reactions on p. 387.
Time f/s
[H 2 2 ],/moldm _3
2
6.23
4
4.84
6
3.76
8
3.20
10
2.60
Time f/s
[H 2 2 L/moldm- 3
12
2.16
14
1.85
16
1.49
18
1.27
20
1.01
Figure 8.7 shows the way the concentration of hydrogen perox-
ide decreases with time. The trace is clearly curved, and Figure 8.8
shows a graph constructed with the linear form of the first-order
integrated rate equation, Equation (8.26). This latter graph is clearly
linear.
The rate constant is obtained from the figure as (— 1 x 'gradient'),
so A: = 0.11 s" 1 at 298 K.
E
O
E
O
CM
X
10 15
Time t/ms
Figure 8.7 Plot of [H2O2L against time. Notice the pronounced plot curvature
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
373
■o
o
Figure 8.8 Graph constructed by drawing ln[H 2 02L (as 'y') against time t (as V), i.e. with the
axes of a linear first-order rate law. Notice the linearity of the trace
Justification Box 8.2
The linear form of the integrated first-order rate equation
If we start with the now-familiar integrated rate equation of Equation (8.24):
In
[Ajp
[A],
kit
Using the laws of logarithms, we can split the left-hand side:
ln[A] - ln[A], = kit
Next, we multiply by — 1 :
ln[AT - ln[A] = -M
Then, by adding the [A]o term to the right-hand side we obtain Equation (8.26):
ln[A] t = -kit + ln[A]
which is the linear form of the equation, as desired. We note that the intercept in
Equation (8.26) is ln[A]o-
We could have obtained Equation (8.26) alternatively by integrating without limits
during the derivation in Justification Box 8.1.
374
CHEMICAL KINETICS
The SI unit of time is
the second (see p. 15),
but it is sometimes
more convenient to cite
k in terms of non-SI
units, such as min
even year -1 .
or
Worked Example 8.10 What is the relationship between the values
of rate constant expressed in units of s~ : , and expressed in units of
min -1 and year
-1'?
By Analogy. Let the rate constant be k. Conversion into SI is easy:
k in min -1 = 60 x (k in s _1 ) because there are 60 s per minute,
so 60 times as much reaction can occur during a minute.
Via Dimensional Analysis. Again, let the rate constant be k. In
this example, imagine the value of k is 3.2 year -1 .
The number of seconds in a year is (60 x 60 x 24 x 365.25) = 3.16 x 10 7 s. We
can write this result as 3.16 x 10 7 syr -1 , which (by taking reciprocals) means that
2.37 x 10" 8 yrs" 1 .
To obtain k with units of s _1 , we say
To convert k from SI
units to the time unit
of choice, just mul-
tiply the value of k
by the fraction of the
time interval occurring
during a single second.
^hl
(3.2 yr" 1 ) x (2.37 x 10" s yr s" 1 )
original value of k conversion factor
sok= 1.01 x 10" 7 s _1 .
SAQ 8.12 Show that the rate constants 1.244 x 10 4 yr" 1 and 3.94 x
10" 4 s _1 are the same.
Integrated rate equations: second-order reactions
For a second-order reaction, the form of the integrated rate equation is different:
Notice that th
e units
of the second
-order
rate constant
k 2 are
dm 3 mor 1 s -1
which
are, in effect,
(concen-
tration) -1 s _1 .
We do not need to
know the temperature
in order to answer this
question; but we do
need to know that T
remained constant, i.e.
that the reaction was
thermostatted.
1
1
[A], [A]
= k 2 t
(8.27)
where the subscripted '2' on k reminds us that it represents a
second-order rate constant. The other subscripts and terms retain
their previous meanings.
SAQ 8.13 Show that the units of the second-order rate
constant are dm 3 mor 1 s~ 1 . [Hint: you will need to per-
form a simple dimensional analysis of Equation 8.27.]
Worked Example 8.11 We encountered the dimerization of methyl
viologen radical cation MV + " in Equation (8.6) and Worked
Example 8.4. Calculate the value of the second-order rate constant
£2 if the initial concentration of MV + " was 0.001 moldm -3 and the
concentration dropped to 4 x 10~ 4 mol dirT 3 after 0.02 s. (The tem-
perature was 298 K.)
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
375
Inserting values into Equation (8.27):
1
1
so
and
4 x 10" 4 moldm"
2500 (moldm -3 ) -
10 3 moldm
1000 (moldm -3 )"
= k 2 x 0.02 s
k 7 x 0.02 s
1500 (moldm -3 )" 1
k 7 x 0.02 s
Rearranging, we say
1500 (moldm -3 )"
02~s
so
k 2 = 7.5 x 10 4 dm 3 mol" 1 s" 1 at 298 K
which is relatively fast.
SAQ 8.14 Remaining with the same system from Worked Example 8.11,
having calculated k 2 (i.e. having 'calibrated' the reaction), how much MV + "
remains after 40 ms (0.04 s)?
SAQ 8.15 Consider a second-order reaction which consumes 15 per cent
of the initial material after 12 min and 23 s. If [A]o was 1 x 10 3 mol dm 3 ,
calculate k 2 . [Hint: first calculate how much material remains.]
An alternative form of the integrated rate equation is the so-called linear form
1
[A],
y
= h
t +
X
1
[A]
c
(8.28)
which we recognize as relating to the equation of a straight line,
so plotting a graph of l/[A] r (as 'y') against time (as 'x') will be
linear for a reaction that is second order. The rate constant k 2 is
obtained directly as the gradient of the graph.
We obtain the second-
order rate constant k 2
as the slope of a graph
drawn according to
the integrated second-
order rate equation.
Worked Example 8.12 Consider the data below, which relate to the second-order racem-
ization of a glucose in aqueous hydrochloric acid at 17 °C. The concentrations of glucose
and hydrochloric acid are the same, '[A]'.
Time f/s
[A] /mol dm
-3
0.400
600
0.350
1200
0.311
1800
0.279
2400
0.254
376
CHEMICAL KINETICS
Care: The gradient
is only truly k if the
time axis is given with
the SI units of time
(the second).
A graph of the concentration [A] (as 'y') against time (as 'x') is
clearly not linear; see Figure 8.9(a). Conversely, a different, linear,
graph is obtained by plotting 1/[A] ( (as 'y') against time (as 'x'); see
Figure 8.9(b). This follows the integrated second-order rate equation.
The gradient of Figure 8.9(b) is the second-order rate constant ki, and
has a value of 6.00 x 10 -4 dm 3 mol~' s" 1 .
1000 1500
Time // s
(a)
2500
1000 1500 2000 2500
Time f/s
(b)
Figure 8.9 Kinetics of a second-order reaction: the racemization of glucose in aqueous mineral
acid at 17 °C: (a) graph of concentration (as 'y') against time (as 'jc'); (b) graph drawn according
to the linear form of the integrated second-order rate equation, obtained by plotting 1/[A], (as 'y')
against time (as '*'). The gradient of trace (b) equals the second-order rate constant k 2 , and has a
value of 6.00 x 10~ 4 dnVmor's" 1
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
377
SAQ 8.16 Consider the following data concerning the
reaction between triethylamine and methyl iodide at 20 °C
in an inert solvent of CCI 4 . The initial concentrations of
[CH 3 I] and [N(CH 3 ) 3 ] are the same. Draw a suitable
graph to demonstrate that the reaction is second order,
and hence determine the value of the second-order rate
constant k 2 .
Remem
ber that
when
plotting
kinetic
graphs
the concentration at
t= is
also a
valid
data po
nt.
Time r/s
[CH 3 I]o =
[N(CH 3 ) 3 ] /mol dm
-3
2.112
2400
0.813
5400
0.149
9000
0.122
18000
0.084
Warning: if a chemical process comprises several reaction steps, only the progress
of the slowest step can be followed kinetically. These graphical methods of deter-
mining k are only useful for obtaining the rate-determining step (RDS) of such
reactions. Although the reaction may appear kinetically simple, it is wisest to assume
otherwise.
Justification Box 8.3
Integrated rate equations for a second-order reaction
We will only consider the derivation of the simplest case of a second-order reaction,
where the concentrations of the two reacting species are the same. Being second order,
the rate law has the form rate = ^[A] 2 . The subscript
'2' on k indicates a second-order process. Again, by
'rate' we mean the rate of change of the concentration
of reactant A.
rate ;
d[A]
At
k 2 [A] z
5.29)
The minus sign in
Equation (8.29) is ess-
ential to show that
the concentration of A
decreases with time.
As before, we shall start at t = with a concentration [A]o. The value of [A],
decreases with time t, hence a minus sign is inserted.
Rearranging the equation, and indicating the limits yields
/•[A], J ft
/ —2 d[A] = k 2 /
.'[Alo |AJ, Jo
Integrating gives
-I [A],
.[A],
= k 2 [t]'
-I [Alo
378 CHEMICAL KINETICS
The two minus
signs on
the left will cancel.
Inserting
limits,
and rearranging
slightly
gives Equation
(8.27):
1 1
= k 2 t
This equation is known
as the
integrated second-order rate equation.
Second-order reactions of unequal concentration
We will start with the reactants A and B having the concentrations [A]o and [B]o
respectively. If the rate constant of reaction is k^, and if the concentrations at time t
are [A], and [B] ( respectively, then it is readily shown that
1
[B] - [A],
, , [A]„x[B].
x In = K2I
[B] x [A], ' -
(8.30)
We will need to look further at this equation when thinking about kinetic situations in
which one of the reactants is in great excess (the so-called 'pseudo order' reactions
described in Section 8.4).
SAQ 8.17 A 1:1 reaction occurs between A and B, and is second order.
The initial concentrations of A and B are [A]o = 0.1 moldrrT 3 and [B]n =
0.2 moldrrr 3 . What is k 2 if [A] t = 0.05 moldm -3
to work out a value for [B] f as well.
after 0.5 h? Remember
Why do we often refer to a 'half-life' when speaking
about radioactivity?
Half-lives
A radioactive substance is one in which the atomic nuclei are unstable and sponta-
neously decay to form other elements. Because the nuclei decay, the amount of the
radioactive material decreases with time. Such decreases follow the straightforward
kinetic rate laws we discussed above.
But many people talk emotionally of radioactivity 'because
radioactive materials are so poisonous', and one of the clinching
arguments given to explain why radioactivity is undesirable is that
radioactive materials have long 'half-lives'. What is a half-life?
And why is this facet of their behaviour important? And, for that
matter, is it true that radioactive materials are poisonous?
We shall look at why
radioactive materials
are toxic in Section 8.3,
below.
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
379
Table 8.2 Half-lives of radioactive isotopes (listed in order of increasing atomic number)
Isotope
Half-life
Source of radioactive isotope
l2 B
14 C
40 K
60 Co
129j
238TJ
239p u
0.02
5570
years
1.3 x
10 9
years
10.5
min
1.6 x
10 7
years
4.5 x
10 s
years
2.4 x
10 4
years
Unnatural (manmade)
Natural
Natural: 0.011% of all natural potassium
Unnatural: made for medicinal uses
Unnatural: fallout from nuclear weapons
Natural: 99.27% of all uranium
Unnatural: by-product of nuclear energy
A 'half-life' t 1/2 is the
time required for the
amount of material
to halve.
The half-life of radioactive decay or of a chemical reaction is
the length of time required for exactly half the material under
study to be consumed, e.g. by chemical reaction or radioactive
decay. We often give the half-life the symbol t\/2, and call it
'tee half.
The only difference between a chemical and a radioactive half-life is that the
former reflects the rate of a chemical reaction and the latter reflects the rate of
radioactive (i.e. nuclear) decay. Some values of radioactive half-lives are given in
the Table 8.2 to demonstrate the huge range of values ti/2 can take. The difference
between chemical and radioactive toxicity is mentioned in the Aside box on p. 382.
A chemical half-life is the time required for half the material to have been consumed
chemically, and a radioactive half-life is the time required for half
of a radioactive substance to disappear by nuclear disintegration.
Since most chemicals react while dissolved in a constant volume
of solvent, the half-life of a chemical reaction equates to the time
required for the concentration to halve.
Worked Example 8.13 The half-life of bU Co is 10.5 min. If we start
with 100 g of 60 Co, how much remains after 42 min?
60 Co is a favourite
radionuclide within the
medical profession,
because its half life
is conveniently short.
Answer Strategy
1 . We determine how many of the half-lives have occurred during the
time interval.
2. We then successively halve the amount of 60 Co, once per half-life.
(1) The number of half-lives is obtained by dividing 10.5 min into 42 min; so four
half-lives elapse during 42 min.
(2) If four half-lives have elapsed, then the original amount of 60 Co has halved,
then halved again, then halved once more and then halved a fourth time:
100
1st 2nd 3rd 4th
half-life _ half-life „, half-life ,_ half-life ^ „,
► 50 g ► 25 g ► 12.5 g ► 6.25 g
380 CHEMICAL KINETICS
We see that one-sixteenth of the 60 Co remains after four half-lives, because ( j) 4 = i.
In fact, a general way of looking at the amount remaining after a few half-lives is to
say that
fraction remaining = (|)" (8.31)
where n is the number of half-lives.
SAQ 8.18 The half-life of radioactive 14 C is 5570 years. If we start with
10 g of 14 C, show that the amount of 14 C remaining after 11140 years
is 2.5 g.
Often, though, we don't know the half-life. One of the easier ways to determine a
value of ti/2 is to draw a graph of amount of substance (as ' v') - or, if in solution,
of concentration - against time (as 'x').
Worked Example 8.14 The table below shows the amount of a biological metabolite
T-IDA as a function of time.
1. What is the half-life of T-IDA?
2. Show that the data follow the integrated first-order rate equation.
Time f/min
10
20
30
40
50
60
70
[T-IDA]/|xmol
100
50
25
12.5
6.25
3.13
1.56
0.781
dm" 3
Figure 8.10 shows a plot of the amount of material (as 'v') as a function of time / (as
'x'), which is exponential. This shape should not surprise us, because Equation (8.31) is
also exponential in form.
To obtain the half-life, we first choose a concentration - any concentration will do,
but we will choose [T-IDA] = 50 pmoldm -3 . We then draw a horizontal arrow from this
concentration on the y-axis, note the time where this arrow strikes the curve, and then read
off the time on the x-axis, and call it t\ . Next, we repeat the process, drawing an arrow
from half this original concentration, in this case from [T-IDA] =
25 |xmoldm~ . We note this new time, and call it t2. The half-life is
simply the difference in time between t\ and ti- It should be clear that
the half-life in this example is 10 min.
But then we notice that the time needed to decrease from 60 to
30 |imoldm~ , or from 2 to 1 p,moldm~ will also be 10 min each.
In fact, we deduce the important conclusion that the half-life of a
first-order reaction is independent of the initial concentration of
material.
As long as a reaction
is first order, the dura-
tion of a half-life will
be the same length
of time regardless of
the initial amount of
material present.
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
381
Figure 8.10 Kinetic trace concerning the change in concentration as a function of time: graph
of [T-IDA] (as 'y') against time (as 'x') to show the way half-life is independent of the initial
concentration
Figure 8.11 Graph plotted with data from Figure 8.10, plotted with the axes of the linear form
of the integrated first-order rate equation, with ln[A] as 'y' against time t as 'x'
To show that this reaction is kinetically first order, we take the logarithm of the con-
centration, and plot ln[A]f (as 'y') against time t (as '*'); see Figure 8.11. That the graph
in Figure 8.11 is linear with this set of axes demonstrates its first-order character.
The half-life of second- or third-order reactions is not independent of the initial con-
centration in this way (see p. 387).
382 CHEMICAL KINETICS
Aside
The difference between chemical and radiochemical toxicity
It is good that we should be concerned about the environmental impact of what we,
as chemists, do to our planet. But many environmental campaigners too easily confuse
radioactive toxicity and chemical toxicity. For example, the radon gas emanating from
naturally occurring granite rocks is chemically inert, because it is a rare gas, but it is
toxic to humans because of its radioactivity. Conversely, sodium cyanide contains no
radioactive constituents yet is chemically toxic.
The conceptual problems start when considering materials such as plutonium, which is
a by-product of the nuclear electricity industry. Plutonium is one of the most chemically
toxic materials known to humanity, and it is also radioactive. The half-life of 238 Pu is so
long at 4.5 x 10 8 years (see Table 8.2) that we say with some certainty that effectively
none of it will disappear from the environment by radioactive decay; and if none of it
decays, then it cannot have emitted ionizing a and ft particles, etc. and, therefore, cannot
really be said to be a radioactive hazard. Unfortunately, the long half-life also means that
the 238 Pu remains more-or-less for ever to pollute the environment with its lethal chemistry.
But if we accept that plutonium is chemically toxic,
then we must also recognize that the extent of its toxic-
ity will depend on how the plutonium is bonded chemi-
cally, i.e. in what redox and chemical form it is present.
As an example, note how soldiers were poisoned with
chlorine gas during the First World War (when it was
called Mustard Gas), but chloride in table salt is vital
for life. Some plutonium compounds are more toxic
than others.
At the other extreme are materials with very short
radioactive half-lives, such as 12 B (which has a relatively short t\/2 of 0.02 h). 12 B is
less likely to cause chemical poisoning than 238 Pu simply because its residence time is so
short that it will transmute to become a different element and, therefore, have little time to
interact in a chemical sense with anything in the environment (such as us). On the other
hand, its short half-life means that the speed of its radioactive decay will generate many
subatomic particles (a, fi and y particles) responsible for radioactive poisoning per unit
time, causing a larger dose of radioactive poisoning.
This Aside is not intended to suggest that the threat of radioactivity is to be ignored
or marginalized; but we should always aim to be well informed when confronting an
environmental problem.
If we accept that plu-
tonium is chemically
toxic, then we also
need to recognize that
the extent of its toxicity
will depend on how the
plutonium is bonded
chemically (see p. 59).
How was the Turin Shroud 'carbon dated'?
Quantitative studies with half-lives
The Turin Shroud is a long linen sheet housed in Italy's Turin Cathedral. Many
people believe that the surface of the cloth bears the image of Jesus Christ (see
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
383
Figure 8.12 Many people believe the Turin Shroud bears an image of Jesus Christ, imprinted soon
after his crucifixion. Radiocarbon dating suggests that the flax of the shroud dates from 1345 AD
Figure 8.12), imprinted soon after he died by crucifixion at the hands of the Roman
authorities in about ad 33. There has been constant debate about the Turin Shroud and
its authenticity since it first came to public notice in ad 1345: some devout people
want it to be genuine, perhaps so that they can know what Jesus actually looked like,
while others (many of whom are equally devout) believe it to be a fake dating from
the Middle Ages.
An accurate knowledge of the Turin Shroud's age would allow us to differentiate
between these two simplistic extremes of ad 33 and ad 1345, effectively distinguishing
between a certain fake and a possible relic of enormous value.
The age of the cloth was ascertained in 1988 when the Vatican (which has juris-
diction over Turin Cathedral) allowed a small piece of the cloth to be analysed by
radiocarbon dating. By this means, the shroud was found to date from ad 1320 ± 65.
Even after taking account of the uncertainty of ±65 years, the age of the shroud is
consistent with the idea of a medieval forgery. It cannot be genuine.
But the discussion about the shroud continues, so many people now assert that
the results of the test itself are part of a 'cover up', or that the moment of Jesus' s
resurrection occurred with a burst of high-energy sub-atomic particles, which upset
the delicate ratio of carbon isotopes.
384
CHEMICAL KINETICS
In 1946, Frank Libby of
the Institute of Nuclear
Sciences in Chicago
initiated the dating of
carbon-based artifacts
by analysing the extent
of radioactive decay.
Radiocarbon dating
The physicochemical basis behind the technique of radiocarbon
dating is the isotopic abundances of carbon's three isotopes: 12 C
is the 'normal' form and constitutes 98.9 per cent of all naturally
occurring carbon. 13 C is the other naturally occurring isotope, with
an abundance of about 1 per cent. 14 C does not occur naturally,
but tiny amounts of it are formed when high-energy particles from
space collide with gases in the upper atmosphere, thus causing
radiochemical modification.
All living matter is organic and, therefore, contains carbon; and since all living
material must breathe CO2, all carbon-based life forms ingest 14 C. Additionally, living
matter contains 14 C deriving from the food chain.
But 14 C is radioactive, meaning that atoms of 14 C occasionally
The beta particle emit-
ted during radioactive
decay is an ener-
getic electron.
self-destruct to form a beta particle, /3 , and an atom of 14 N
14 N + /J
14,
(8.32)
Radiocarbon dating is
also called 'radiometric
dating' or 'radiochemi-
cal dating'.
The half-life of the process in Equation (8.32) is 5570 years.
Following death, flora and fauna alike cease to breathe and eat, so the only 14 C in
a dead body will be the 14 C it died with. And because the amounts of 14 C decrease
owing to radioactive decay, the amount of the 14 C in a dead plant or
person decreases whereas the amounts of the 12 C and 13 C isotopes
do not. We see why the proportion of 14 C decreases steadily as a
function of time following the instant of death.
By corollary, if we could measure accurately the ratio of 12 C to
14 C in a once-living sample, we could then determine roughly how
long since it was last breathing. This explains why we can 'date'
a sample by analysing the residual amounts of 14 C.
In the carbon-dating experiment, a sample is burnt in pure oxy-
gen and converted into water and carbon dioxide. Both gases are
fed into a specially designed mass spectrometer, and the relative
abundances of 12 CC>2 and 14 CC>2 determined. The proportion of
14 C02 formed from burning an older sample will be smaller.
Knowing this ratio, it is a simple matter to back calculate to
ascertain the length of time since the sample was last alive. For
example, we know that a time of one half-life tin has elapsed if
a sample contains exactly half the expected amount of 14 C, so the
sample died 5570 years ago.
Great care is needed during the preparation of the sample, since
dirt, adsorbed CO2 and other impurities can all contain additional
sources of carbon. The dirt may come from the sample, or it could
have been adsorbed during sample collection or even contamination
during the dating procedure. The more recent the contamination,
the higher the proportion of carbon that is radioactive 14 C that has
Oil and oil-based prod-
ucts contain no 14 C,
because the crea-
tures from which the
oil was formed died
so many millions of
years ago. Accordingly,
Equation (8.32) has
proceeded to its
completion.
Great care is needed
during the preparation
of the sample before
dating to eliminate the
possibility of contami-
nation with additional
sources of carbon.
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
385
not yet decayed, causing the artifact to appear younger. It has been suggested, for
example, that the Turin Shroud was covered with much 'modern' pollen and dust at
the time of its radiocarbon dating, so the date of ad 1320 refers to the age of the
modern pollen rather than that of the underlying cloth itself.
How old is Otzi the iceman?
Calculations with half-lives
Approximately 5000 years ago, a man set out to climb the Tyrolean Alps on the
Austrian-Italian border. At death, he was between 40 and 50 years old and suffered
from several medical ailments. Some scientists believe he was caught in a heavy
snowfall, fell asleep, and froze to death. Others suppose he was murdered during
his journey. Either way, his body was covered with snow almost immediately and,
due to the freezing weather, rapidly became a mummy - 'The Iceman'. In 1991, his
body was re-exposed and discovered by climbers in the Otzal Alps, explaining why
the 'Iceman', as he was called, was given the nickname 'Otzi' (or,
more commonly, as just Otzi).
His body (see Figure 8.13) was retrieved and taken to the Dep-
artment of Forensic Medicine at the University of Innsbruck. Their
analytical tests - principally radiocarbon dating - suggest that
Otzi died between 3360 and 3100 bc. Additional radiocarbon dat-
ing of wooden artifacts found near his body show how the site of
his death was used as a mountain pass for millennia before and
after his lifetime.
But how, having defined the half-life tin as the time necessary
for half of a substance to decay or disappear, can we quantita-
tively determine the time elapsing since Otzi died? In Justification
Box 8.4, we show how the half-life and the rate constant of decay
k are related according to
Scientific techniques,
such as radiocarbon
dating, applied to
archaeology is some-
times termed archaeo-
metry.
h/2
ln(2)
(8.33)
Equation (8.33) sug-
gests the half-life is
independent of the
amount of material
initially present, so
radioactive decay fol-
lows the mathematics
of first-order kinetics.
Figure 8.13 The body of 'Otzi the Iceman' was preserved in the freezing depths of a glacier.
Radiocarbon dating suggests that Ozti froze to death in about BC 3360-3100
386
CHEMICAL KINETICS
Worked Example 8.15 A small portion of Otzi's clothing was removed and burnt care-
fully in pure oxygen. The amount of 14 C was found to be 50.93 per cent of the amount
expected if the naturally occurring fabric precursors had been freshly picked. How long
is it since the crop of flax was picked, i.e. what is its age?
We start by inserting the known half-life t\/2 into Equation (8.33) to obtain a 'rate constant
of radioactive decay'.
ln(2)
By calculating k, we
are in effect calibrating
the experiment. We
only need to do this
calibration once.
k =
5570 yr
By this means, we calculate the rate constant as k — 1.244 x
10~ 4 yr -1 . (Alternatively, we could have calculated k in terms of
the SI unit of time (the second), in which case k has the value
3.94 x 10
-12
•)
Using our calculators,
we need to type
'ln(0.5093)' as the
numerator rather than
a percentage. The
minus sign comes from
the laws of logarithms.
To calculate t, the length of time since the radioactive decay
commenced (i.e. since the fabric precursor died), we again insert
values into the integrated form of the first-order rate equation,
Equation (8.33). We then insert our previously calculated value of k:
ln(50.93%)
■1.244 x 10" 4 yr"
5420 yr
So the interval of time t since the flax was picked and thence woven
into cloth is 5420 yr, so the cloth dates from about 3420 bc.
SAQ 8.19 To return to the example of the Turin Shroud. Suppose a
sloppy technique caused the precision of the 14 C measurement to decrease
from 92.23 per cent to 90 ±2 percent. Calculate the range of ages for
the shroud. [Hint: perform two calculations, one for either of the extreme
values of percentage.]
Justification Box 8.4
Justification Box 8.1 shows how the integrated first-order rate equation is given by
Equation (8.24):
In
[A]
[A],
kt
After a length of time equal to one half-life ti/2, the concentration of A will be [A] t ,
which, from the definition of half-life, has a value of jfAJo.
Inserting these two concentrations into Equation (8.24) gives
In
[A]
i[A]
= kt
1/2
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS 387
The two [A]o terms cancel, causing the bracket on the left to simplify to just '2'.
Accordingly, Equation (8.24) becomes
ln(2) = kt l/2
So, after rearranging to make t\/2 the subject, the half-life is given by Equation (8.33):
ln(2)
fl/2 = — —
SAQ 8.20 Show from Equation (8.27) that the half-life
of a second-order equation is given by the expression:
1
tl/2 =
[A] /f2
The half-life of a
second-order reaction
is not independent of
the initial concentra-
tions used.
Why does the metabolism of a hormone not cause a
large chemical change in the body?
'Pseudo-order' reactions
A hormone is a chemical that transfers information and instructions between cells in
animals and plants. They are often described as the body's 'chemical messengers', but
they also regulate growth and development, control the function of various tissues,
support reproductive functions, and regulate metabolism (i.e. the process used to break
down food to create energy).
Hormones are generally quite small molecules, and are chemically uncomplicated.
Examples include adrenaline (II) and /2-phenylethylamine (III).
CH(OH)CH 2 NHCH 3
"V
CH 2 CH 2 NH 2
(II)
(HI)
Most hormones are produced naturally in the body (e.g. adrenaline (II) is formed
in the adrenal glands). From there, the hormone enters the bloodstream and is con-
sumed chemically (a physiologist would say 'metabolized') at the relevant sites in
the body - in fact, adrenaline accumulates and is then broken down chemically in
the muscles and lungs. Adrenaline is generated in equal amounts in men and women,
388
CHEMICAL KINETICS
and promotes a stronger, faster heartbeat in times of crisis or panic. The body is
thus made ready for aggression ('fight') or necessary feats of endurance to escape
('flight'). Adrenaline is also administered artificially in medical emergencies, e.g.
immediately following an anaphylactic shock, in order to give the heart a 'kick start'
after a heart attack.
/3-Phenylethylamine (III) is a different type of hormone, and
is metabolized in women's bodies to a far higher extent than in
a man's. The mechanism of metabolism is still a mystery, but it
appears to cause feelings of excitement, alert feelings and in terms
of mood, perhaps a bit of a 'high'. Unfortunately, the bodies of
most men do not metabolize this hormone, so they do not feel the
same 'high'.
Hormones are potent and are produced in tiny concentrations
(generally with a concentration of nanomoles per litre). By con-
trast, the chemicals in the body with which the hormone reacts
have a sizeable concentration. For example, reaction with oxy-
gen in the blood is one of the first processes to occur during the
metabolism of adrenaline. The approximate range of [02](biood) is
0.02-0.05 moldm - , so the change in oxygen concentration is
virtually imperceptible even if all the adrenaline in the blood is
consumed. As a good corollary, then, the only concentration to
change is that of the hormone, because it is consumed.
Although such a reaction is clearly second order, it behaves like a
^zr.st-order reaction because only one of the concentrations changes.
We say it is a pseudo first-order reaction.
We say a reaction is
pseudo first-order if
it is second- or third-
order, but behaves
mathematically as
though it were first
order.
A pseudo-order reac-
tion proceeds with all
but one of the reactants
in excess. This ensures
that the only con-
centration to change
appreciably is that of
the minority reactant.
Why do we not see radicals ft
sunbathing?
orming in
the skin whili
Pseudo-order rate constants
One of the most common causes of skin cancer is excessive sunbathing. Radicals
are generated in the skin during irradiation with high-intensity UV-light, e.g. while
lying on a beach. These radicals react with other compounds in the skin, which may
ultimately cause skin cancer. But we never see these radicals by eye, because their
concentration is so minuscule. And the concentration is so small because the radicals
react so fast. (Photo-ionization is discussed in Chapter 9.)
Reaction intermediates are common in mechanistic studies of
organic reactions. They are called 'intermediates' because they
react as soon as they form. Such intermediates are sometimes
described as 'reactive' because they react so fast they disappear
more or less 'immediately'. Indeed, these intermediates are so reac-
tive, they may react with solvent or even dissolved air in solution,
i.e. with other chemicals in high concentration.
The accumulated con-
centration of such a
fast-reacting inter-
mediate will always
be extremely low:
perhaps as low as
10- 10 moldm" 3 .
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
389
Intermediates react
fast because their
activation energy is
small - see
Section 8.5.
Since the reaction of intermediates is so fast, the concentration
of the radical intermediate changes dramatically, yet the concentra-
tions of the natural compounds in the skin with which the interme-
diate reacts (via second-order processes) do not change perceptibly.
How in practice, then, do we determine kinetic parameters for
pseudo-order reactions such as these?
We will call the intermediate 'A' and the other reagent, which is in excess, will be
called 'B'. In the example here, B will be the skin, but is more generally the solvent
in which the reaction is performed, or an additional chemical in excess.
Because the concentration of B, [B], does not alter, this reaction will obey a first-
order kinetic rate law, because only one of the concentrations changes with time.
Because the reaction is first order (albeit in a pseudo sense), a plot of ln[A] ( (as ' v')
against time (as 'x') will be linear for all but the longest times
(e.g. see Figure 8.14). In an analogous manner to a straightforward
first-order reaction, the gradient of such a plot has a value of 'rate
constant x — 1' (see Worked Example 8.9). We generally call the
rate constant k', where the prime symbol indicates that the rate
constant is not a true rate constant, but is pseudo.
The proper rate law for reaction for the second -order reaction between A and B is:
Pseudo-order rate con-
stants are generally
indicated with a prime,
e.g. k'.
rate = fc 2 [A][B]
(8.34)
as for any normal second-order reaction. By contrast, the perceived
rate law we measure is first order, as given by
rate = k'[A]
where k! is the perceived rate constant.
(8.35)
Writing a pseudo rate
constant k' without an
order implies that it is
pseudo first order.
Time t
Figure 8.14 The reaction of A and B, with B greatly in excess is a second-order reaction, but
it follows a kinetic rate law for a first-order reaction. We say it is pseudo first-order reaction.
The deviation from linearity at longer times occurs because the concentration of B (which we
assume is constant) does actually change during reaction, so the reaction no longer behaves as a
first-order reaction
390
CHEMICAL KINETICS
Comparing Equations (8.34) and (8.35), we obtain
The concentration [B] t
barely changes, so
we can write [B] in
Equations (8.34) and
(8.36).
*' = k 2 [B]
(8.36)
This little relationship shows that a pseudo-order rate constant
k! is not a genuine rate constant, because its value changes in
proportion to the concentration of the reactant in excess (in this
case, with [B]).
Worked Example 8.16 The reaction of the ester ethyl methanoate and sodium hydrox-
ide in water is performed with NaOH in great excess ([NaOH]o = 0.23 moldm~ 3 ). The
reaction has a half-life that is independent of the initial concentration of ester present.
13.2 per cent of the ester remains after 14 min and 12 s. What is the second-order rate
constant of reaction ktl
Strategy. (1) We ascertain the order of reaction, (2) we determine the pseudo rate constant
k', (3) from k', we determine the value of the second-order rate constant k 2 .
(1) Is the reaction a pseudo first-order process? The question says that the half-life t\/2
of reaction is independent of initial concentration of ester, so the reaction must behave as
though it was a. first-order reaction in terms of [ester]. In other words, NaOH is in excess
and its concentration does not vary with time.
(2) What is the value of the pseudo first-order rate constant k'l We calculate the
pseudo first-order rate constant k' by assuming that the reaction obeys first-order kinetics.
Accordingly, we write from Equation (8.24):
/ [ester] o
V [ester].
k't
so
In
100%
13.2%
= k' x [(14 x 60) + 12] s
and
ln(7.57) = k' x 852 s
Because ln(7.57) = 2.02 we say
k'
2.02
852 s
2.37 x 10~ 3 s"
Note how manipulating
the units in the fraction
yields the correct units
for k 2 .
(3) What is the value of the second-order rate constant k 2 l The
value of rate constant k 2 can be determined from Equation (8.36), as
k' 4- [NaOH], so
fe =
2.37 x 10'
0.23 moldm
= 1.03 x 10~ 3 dm 3 mor
QUANTITATIVE CONCENTRATION CHANGES: INTEGRATED RATE EQUATIONS
391
SAQ 8.21 Potassium hexacyanoferrate(III) in excess oxidizes an alco-
hol at a temperature of 298 K. The concentration of l<3[Fe(CN) 6 ] is
0.05 moldrrT 3 . The concentration of the alcohol drops to 45 per cent
of its t — value after 20 min. Calculate first the pseudo first-order rate
constant k', and thence the second-order rate constant /C2-
Justification Box 8.5
When we first looked at the derivation of integrated rate equations, we looked briefly
at the case where two species A and B were reacting but [A]o does not equal [B]o. The
integrated rate equation for such a case is Equation (8.30):
1 /[A] x[B]A
x In = kit
[B] -[A] t V[B] x[A]J
Though we do not need to remember this fearsome-looking equation, we notice a few
things about it. First, we assume that [B]o ^> [A]o, causing the first term on the left-hand
side to behave as l/[B]o.
Also, the major change in the logarithm bracket is
the change in [A], since the difference between [B]o
and [B] f will be negligible in comparison, causing a
cancellation of the [B] terms on top and bottom.
Accordingly, Equation (8.30) simplifies to
1
Wo
x In
[Ajp
[A],
kit
Since ln([A]o/[A],) = k't for a pseudo first-order reac-
tion (by analogy with Equation (8.24)), we say that
1
— — x(kt) = k 2 t
IdJo
and cancelling the two t terms, and rearranging yields
We argue this state-
ment by saying that if
[B] is, say, 100 times
larger than [A], then
a complete consump-
tion of [A] (i.e. a 100
per cent change in its
concentration) will be
associated with only a
1 per cent change in
[B] - which is tiny.
k' = k 2 [B]
so we re-obtain Equation (8.36).
Alternatively, the value of the true second-order rate constant may be obtained by
treating Equation (8.36) as the equation of a straight line, with the form y = mx:
k'
>'
k 2
m
[B]
x
392
CHEMICAL KINETICS
Accordingly, we perform the kinetic experiment with a series of concentrations [B]o,
the reactant in excess, and then plot a graph of k' (as 'y') against [B]o (as 'x'). The
gradient will have a value of £2-
A graphical method such as this is usually superior to a simplistic calculation of
£2 = k! -r- [NaOH] (e.g. in the Worked Example 8.16), because scatter and/or chem-
ical back or side reactions will not be detected by a single calculation. Also, the
involvement of a back reaction (see next section) would be seen most straightfor-
wardly as a non-zero intercept in a plot of k! (as 'v') against [reagent in excess]
(as 'x').
Worked Example 8.17 The following kinetic data were obtained for the second-order
reaction between osmium tetroxide and an alkene, to yield a 1,2-diol. Values of k' are
pseudo-order rate constants because the OSO4 was always in a tiny minority. Determine
the second-order rate constant k^ from the data in the following table:
[alkeneJo/
mol dm -3
/fc'/KT 4 s" 1
0.01 0.02 0.03
3.2 6.4 9.6
0.04 0.05 0.06 0.07 0.08
12.8 16.0 19.2 22.4 25.6
0.003
0.0025
0.002
& 0.0015
0.001
0.0005
0.03 0.04 0.05
[alkene] /mol drrr 3
0.08
Figure 8.15 The rate constant of a pseudo-order reaction varies with the concentration of the
reactant in excess: graph of k! (as 'y') against [alkene]o (as V). The data refer to the formation
of a 1,2-diol by the dihydrolysis of an alkene with osmium tetroxide. The gradient of the graph
yields k 2 , with a value of 3.2 x 10~ 2 dm 3 moP 1 s _1
KINETIC TREATMENT OF COMPLICATED REACTIONS
393
The data clearly show that k' is not a true rate constant, because
its value varies as the concentration of the alkene increases. That k'
increases linearly with increased [alkene] suggests a straightforward
pseudo first-order reaction. Figure 8.15 shows a graph of k! (as 'y')
against [alkene]o (as 'x'). The graph is linear, and has a gradient of
3.2 x 10~ 2 dm moP 1 s _1 , which is also the value of ki.
Graphs such as that
in Figure 8.15 gen-
erally pass through
the origin.
8.4 Kinetic treatment of complicated
reactions
Why is arsenic poisonous?
Care: it is unwise to call
a complicated reaction
such as these a 'com-
plex reaction', since
the word 'complex'
implies an associa-
tion compound.
"Concurrent' or "competing' reactions
Arsenic is one of the oldest and best known of poisons. It is so
well known, in fact, that when the wonderful Frank Capra com-
edy Arsenic and Old Lace was released, everyone knew that it was
going to be a murder mystery in which someone would be poi-
soned. In fact, it has even been rumoured that Napoleon died from
arsenic poisoning, the arsenic coming from the green dye on his
wallpaper. We deduce that even a small amount of arsenic will
cause death, or at least an unpleasant and lingering illness.
Arsenic exists in several different redox states. The characteristic
energy at which one redox state converts to the other depends on
its electrode potential E (see Chapter 7). The nervous system in a human body is
'instructed' by the brain much like a microprocessor, and regulated by electron 'relay
cycles' as the circuitry, which consume or eject electrons. Unfortunately, the electrons
acquired or released by arsenic in the blood interfere with these naturally occurring
electron relay cycles, so, following arsenic poisoning, the numbers of electrons in
these relay cycles is wrong - drastically so, if a large amount of arsenic is ingested.
Improper numbers of electrons in the relay cycles cause them not to work prop-
erly, causing a breakdown of those bodily functions, which require exact amounts
of charge to flow. If the nervous system fails, then the lungs are
not 'instructed' how to work, the heart is not told to beat, etc., at
which point death is not too far away.
But arsenic is more subtle a poison than simply a reducing
or oxidizing agent. Arsenic is a metalloid from Group
V(B) of the periodic table, immediately below the elements
nitrogen and phosphorus, both of which are vital for health.
Unfortunately, arsenic is chemically similar to both nitrogen and
phosphorus, and is readily incorporated into body tissues following
ingestion. Arsenic effectively tricks the body into supposing
that straightforward incorporation of nitrogen or phosphorus has
occurred.
Arsenic and nitrogen
compete for the
electrons participating
in the natural electron-
relay cycles in the
body. The number of
electrons transferred
by nitrogen depends on
the number of arsenic
atoms competing
for them.
394
CHEMICAL KINETICS
Electrons
Product 1
Product 2
Figure 8.16 Arsenic and nitrogen compete for electrons both
for and from the natural relay cycles in the body. The overall
rate at which electrons are transferred by the nitrogen will alter
when arsenic competes for them. The arsenic is poisonous, since
these two pathways yield different products
'Sequester' means to
confiscate, seize or
take control of some-
thing to prevent its
further use. The word
comes from the Latin
sequester, meaning
a 'trustee' or 'agent'
whose job was to
seize property.
The first time the body realizes that arsenic has been incorpo-
rated is when the redox activity (as above) proceeds at potentials
when nitrogen or phosphorus are inert. By the time we detect
the arsenic poisoning (i.e. we feel unwell), it is generally too
late, since atoms of arsenic are covalently bound within body tis-
sue and cannot just be flushed out or treated with an antidote.
The arsenic sequesters electrons that might otherwise be involved
in other relay cycles, which is a concurrent kinetic process; see
Figure 8.16.
So arsenic is toxic because it has 'fooled' the body into thinking
it is something else.
Concurrent (or com-
peting) reactions are
so called because two
reactions occur at
the same time. We
occasionally call them
simultaneous reac-
tions, although this
terminology can be
confusing.
A beam of plane-
polarized light is
caused to rotate by
an angle 9 as it passes
through a solution of
a chiral compound.
The magnitude of
6 depends on the
concentration of the
chiral compound.
Why is the extent of Walden inversion
smaller when a secondary alkyl halide
reacts than with a primary halide?
Reaction profiles for complicated reactions
Alkyl halides react by a substitution reaction with hydroxide ions
to yield an alcohol. A primary halide, such as 1-bromopentane,
reacts by a simple bimolecular Sn2 mechanism, where the 'S'
stands for substitution, the 'N' for nucleophilic (because the
hydroxide ion is a nucleophile) and the '2' reminds us that
the reaction is bimolecular. Being a single-step reaction, the
substitution reaction is necessarily the rate-determining step. The
reaction is accompanied by stereochemical inversion about the
central tetrahedral carbon atom to which the halide is attached - we
call it Walden inversion; see Figure 8.17(a). Each molecule of
primary halide inverts during the Sn2 reaction. For this reason, we
could monitor the rate of the Sn2 reaction by following changes in
the angle of rotation 9 of plane polarized light. This angle 9 changes
as a function of the extent of reaction £, so we know the reaction is complete when
9 remains constant at a new value that we call 0(fi na i).
KINETIC TREATMENT OF COMPLICATED REACTIONS
395
OH-
Br
(a)
HO-
h
,./ Br — "
Ri
k
"(2)
HO-
R R2
H3
(b)
■Br
-H R
OH" I© "OH
. R3 R2 .
R1
R %
-OH
Figure 8.17 Reaction of an alkyl halide with hydroxide ion. (a) A primary halide reacts by an S N 2
mechanism, causing Walden inversion about the central, chiral carbon, (b) A tertiary halide reacts
by an Sn 1 mechanism (the rate-determining step of which is unimolecular dissociation, minimizing
the extent of Walden inversion and maximizing the extent of racemization). Secondary alcohols
often react with both SnI and Sn2 mechanistic pathways proceeding concurrently
Polarimetry is the
technique of following
the rotation of plane-
polarized light.
By contrast, tertiary halides, such as 2-bromo-2,2-dimethylpro-
pane, cannot participate in an Sn2 mechanism because it would be
impossible to fit two methyl groups, one bromine and a hydroxide
around a single carbon. The steric congestion would be too great.
So the tertiary halide reacts by a different mechanism, which we
call SnI- It's still a nucleophilic substitution reaction (hence the
'S' and the 'N') but this time it is a unimolecular reaction, hence
the '1'. The rate-determining step during reaction is the slow uni-
molecular dissociation of the alkyl halide to form a bromide ion
and a carbocation that is planar around the reacting carbon.
Addition of hydroxide occurs as a rapid follow-up reaction. Even
if the alkyl halide was chiral before the carbocation formed, racem-
ization occurs about the central carbon atom because the hydrox-
ide can bond to the planar central carbon from either side (see
Figure 8.17(b)). Statistically, equal numbers of each racemate are
formed, so the angle through which the plane polarized light rotated during reaction
will, therefore, decrease toward 0°, when reaction is complete.
In summary, primary halides react almost wholly by a bimolecular process and
tertiary halides react by a unimolecular process. Secondary halides are structurally
between these two extreme structural examples, since reaction occurs by both Sn2 and
SnI routes. These two mechanisms proceed in competition, and occur concurrently.
When following the (dual-route) reaction of a secondary halide with hydroxide ion,
we find that the angle through which plane polarized light is rotated will decrease,
as for primary and tertiary halides, but will not reach zero at completion. In fact, the
final angle will have a value between 0° and #fi na i because of the mixtures of products,
itself a function of the mixture of SnI and Sn2 reaction pathways.
Genuinely first-order
reactions are unusual.
It is likely that the alkyl
halide collides with
another body (such
as solvent) with suffi-
cient energy to cause
bond cleavage.
396
CHEMICAL KINETICS
We looked briefly at reaction profiles in Section 8.2. Before we look at the reaction
profile for the concurrent reactions of hydrolysing a secondary alkyl halide, we will
look briefly at the simpler reaction of a primary alkyl halide, which proceeds via a
single reaction path. And for additional simplicity, we also assume that the reaction
goes to completion. We will look not only at the rate of change of the reactants'
concentration but also at the rate at which product forms.
Consider the graph in Figure 8.18, which we construct with the
data in Worked Example 8.12. We have seen the upper, lighter line
before: it represents the rate of a second-order decay of molecule A
with time as it reacts with the stoichiometry A + B — > product. The
bold line in Figure 8.18 represents the concentration of the product.
It is a 1 : 1 reaction, so each molecule of A consumed by the reaction
will generate one molecule of product, with the consequence that
the two traces are mirror images. Stated another way, the rate of
forming product is the same as the rate of consuming A:
We include the minus
sign in Equation
(8.37) to show how
the product concentra-
tion INcreases while
the reactant concentra-
tion DEcreases.
rate =
d[A] d [product]
At At
(8.37)
where the minus sign is introduced because one concentration increases while the
other decreases.
Now, to return to the hydrolysis of the secondary alkyl halides, we will call the
reactions (1) and (2), where the '1' relates to the SnI reaction and the '2' relates to
the Sn2 reactions. (And we write the numbers with brackets to avoid any confusion,
i.e. to prevent us from thinking that the '1' and '2' indicate first- and second-order
reactions respectively.) We next say that the rate constants of the two concurrent
reactions are k(\) and k(2) respectively. As the two reactions proceed with the same 1:1
E
O
E
c
o
o
O
1000 1500
Time f/s
2500
Figure 8.18 Concentration profile for a simple reaction of a primary alkyl halide + OH -*
alcohol. The bolder, lower line represents the concentration of product as a function of time,
and the fainter, upper line represents the concentration of reactant
KINETIC TREATMENT OF COMPLICATED REACTIONS
397
0.035
0.03
| 0.025
° 0.015
Reaction (2)
1000 1500
Time f/s
2500
Figure 8.19 Concentration profiles for a concurrent reaction, e.g. of a secondary alkyl
halide + OH~ — y alcohol: reaction (2) is twice as fast as reaction (1) in this example
stoichiometry, the ratio of the products will relate to the respective
rate constants in a very simple way, according to
C(i) moles of product formed via reaction (1)
C(2) moles of product formed via reaction (2)
(8.38)
so we see that the amounts of product depend crucially on the
relative magnitudes of the two rate constants. (We shall return to
this theme when we look at the way the human body generates a
high temperature to cure a fever in Section 8.5).
We now look at the concentration profiles for reactions (1) and
(2). For simplicity, we shall say that reaction (2) is twice as fast
as reaction (1), which is likely - unimolecular reactions are often
quite slow. We should note how, at the end of the reaction at the
far right-hand side of the profile in Figure 8.19, the total sum of
product will be the same as the initial concentration of reactant.
Also note how, at all stages during the course of the reaction, the
ratio of products will be 2:1, since that was the ratio of the two
rate constants.
Care: do not confuse
kx and k (1) . ki is the
rate constant of a first-
order process, and k m
is the rate constant of
the first process in a
multi-step reaction.
We can inter-convert
between 'number of
moles' and 'concen-
tration' here because
the reactions are per-
formed within a con-
stant volume of sol-
vent.
Why does 'standing' a bottle of wine cause it to smell
and taste better?
Consecutive reactions
It is often said that a good wine, particularly if red, should 'stand' before serving. By
'standing', we mean that the bottle should be uncorked some time before consumption,
398
CHEMICAL KINETICS
A wine is also said to
age and breathe, which
means the same thing.
to allow air into the wine. After a period of about an hour or so,
the wine should taste and smell better.
Wines contain a complicated mixture of natural products, many
of which are alcohols. Ethanol is the most abundant alcohol, at a
concentration of 3-11 per cent by volume. The amounts of the
other alcohols generally total no more than about 0.1-1 per cent.
The majority of the smells and flavours found in nature comprise
esters, which are often covalent liquids with low boiling points and
high vapour pressures. For that reason, even a very small amount
of an ester can be readily detected on the palate - after all, think
how much ester is generated within a single rose and yet how
overwhelmingly strong its lovely smell can be!
Esters are the product of reaction between an alcohol and a car-
boxylic acid. Although the reaction can be slow - particularly at
lower temperatures - the equilibrium constant is sufficiently high
for the eventual yield of ester to be significant. We see how even
a small amount of carboxylic acid and alcohol can generate a suf-
ficient amount of ester to be detected by smell or taste.
This helps explain why a wine should be left to stand: some of
the natural alcohols in the wine are oxidized by oxygen dissolved from the air to
form carboxylic acids. These acids then react with the natural alcohols to generate a
wide range of esters, which connoisseurs of wines will recognize as a superior taste
and 'bouquet'.
And the reason why the wine must 'stand' (rather than the
reaction occurring 'immediately' the oxygen enters the bottle on
opening) is that the reaction to form the ester is not a straight-
forward one-step reaction: the first step (Equation (8.39)) is quite
slow and occurs in low yield:
The enzymes in wine
are killed if the percent-
age of alcohol exceeds
about 13 per cent. Fer-
mentation alone cannot
make a stronger wine,
so spirits are prepared
by distilling a wine.
Adding brandy to wine
makes a fortified wine,
such as sherry.
The 'mash' that fer-
ments to form wine
sometimes includes
grape skins; which
are rich in enzymes.
It is likely, there-
fore, that the oxidation
in Equation (8.39) is
mediated (i.e. catal-
ysed) enzymatically.
alcohol (aq ) + O
first reaction
>2(aq)
-> carboxylic acid, , + H2O
'(l)
(8.39)
where the rate constant of Equation (8.39) is that of a second-
order reaction. Once formed, the acid reacts more rapidly to form
the respective ester:
carboxylic acid, ■, + alcohol
second reaction
l(aq)
-> ester
(aq)
(8.40)
where the rate constant for this second step is larger than the respective rate constant
for the first step, implying that the second step is faster.
In summary, we see that esterification is a two-step process. The first
step - production of a carboxylic acid - is relatively slow because its rate is
proportional to the concentration of dissolved oxygen; and the [02]( SO in) is l° w - Only
after the wine bottle has been open for some time (it has had sufficient time to
KINETIC TREATMENT OF COMPLICATED REACTIONS
399
'breathe') will [02]( SO in) be higher, meaning that, after 'standing', the esters that lend
additional flavour and aroma are formed in higher yield.
The case of esterification is an example of a whole class of reac-
tions, in which the product of an initial reaction will itself undergo
a further reaction. We say that there is a sequence of reaction steps,
so the reaction as a whole is sequential or consecutive.
Although this example comprises two reactions in a sequence,
many reactions involve a vast series of steps. Some radioactive
decay routes, for example, have as many as a dozen species
involved in a sequence before terminating with an eventual
product.
For simplicity, we will denote the reaction sequence by
To remember this latter
terminology, we note
that for a consecutive
reaction to occur, the
second step proceeds
as a consequence of
the first.
-> B
-> C
(8.41)
All the other reactants will be ignored here to make the analysis more straightforward,
even if steps (1) and (2) are, in fact, bimolecular. We again write the reaction number
within brackets to avoid confusion: we do not want to mistake the subscripted number
for the order of reaction. We call the rate constant of the first reaction k(\) and the
rate constant of the second will be k(2)-
The material A is the initial reactant or precursor. It is usually
stable and only reacts under the necessary conditions, e.g. when
mixed and/or refluxed with other reactants. We know its concentra-
tion because we made or bought it before the reaction commenced.
Similarly, the material C (the product) is usually easy to handle,
so we can weigh or analyse it when the reaction is complete, and
examine or use it. By contrast, the material B is not stable - if it
was, then it would not react further to form C as a second reaction step. Accordingly,
it is rare that we can isolate B. We call B an intermediate, because it forms during
the consumption of A but before the formation of C.
The concentration profile of a simple 1 : 1 reaction is always easy to draw because
product is formed at the expense of the reactant, so the rate at which reactant is
consumed is the same as the rate of product formation. No such simple relation holds
for a consecutive reaction, because two distinct rate constants are involved. Two
extreme cases need to be considered when dealing with a consecutive reaction: when
The maximum rate of
reaction occurs when
the concentration of
the intermediate B
is highest.
Hi)
C (2)
and when k
(i)
k(2). We shall treat each in turn.
Why fit a catalytic converter to a car exhaust?
Consecutive reactions in which the second reaction is slow
First, we consider the case where the first reaction is very fast compared with the
second, so k(i) > k(2)- This situation is the less common of the two extremes. When
400
CHEMICAL KINETICS
all the A has been consumed (to form B), the second reaction is only just starting to
convert the B into the final product C. A graph of concentration against time will show
a rapid decrease in [A] but a slow increase in the concentration of the eventual product,
C. More specifically, the concentration of intermediate B will initially increase and
only at later times will its concentration decrease once more as the slower reaction
(the one with rate constant k^)) has time to occur to any significant
extent. The concentration profile of B, therefore, has a maximum;
see Figure 8.20.
A good example of a consecutive process in which the second
reaction is much slower than the first is the reaction occurring
in a car exhaust. The engine forms carbon monoxide (CO) as its
initial product, and only at later times will CO( g ) oxidize to form
CC>2(g). In fact, the second reaction (CO( g ) + 502(g) -*■ C02( g )) is
so slow that the concentration of CO( g ) is often high enough to
cause serious damage to health.
Most modern cars are
fitted with a 'catalytic
converter', one pur-
pose of which is to
speed up this second
(slower) process in the
reaction sequence.
Why do some people not burn when sunbathing?
Iff Sf
Consecutive reactions in which the second reaction is fast
If k ( i) < k ( 2), an inter-
mediate is generally
termed a reactive inter
mediate to emphasize
how soon it reacts.
We now look at the second situation, i.e. when k
(i)
C(2).
The
first reaction produces the intermediate B very slowly. B is con-
sumed immediately - as soon as each molecule of intermediate is
formed - by the second reaction, which forms the final product C.
Accordingly, we call it a reactive intermediate, to emphasize how
rapidly it is consumed.
The rate of the first reaction in the sequence is slow because k(i)
is small, so the rate of decrease of [A] is not steep. Conversely, since k@) is fast, we
would at first expect the rate d[C]/df to be quite high. A moment's thought shows
Time
Figure 8.20 Schematic graph of concentration against time ('a concentration profile') for a con-
secutive reaction for which ktn > ken- Note the maximum in the concentration of the intermediate,
Mi)
B. This graph was computed with k@) being five times slower than k
(i)
KINETIC TREATMENT OF COMPLICATED REACTIONS
401
that this is not necessarily the case:
rate of forming the final product, C
d[C]
At
fe(2)[B]
(8.42)
We see from Equation (8.42) that the rate of forming C is quite slow because [B] is
tiny no matter how fast the rate constant k(2)-
We now consider the concentration profile of the reactive intermediate B. Because
the second reaction is so much faster than the first, the concentration of B is never
more than minimal. Its concentration profile is virtually horizontal, although it will
show a very small maximum (it must, because [B] = at the start of the reaction at
t = 0; and all the intermediate B has been consumed at the end of reaction, when
[B] is again zero). At all times between these two extremes, the concentration of B
at time t, [B] t , is not zero. The concentration profile is shown in Figure 8.21.
Sunbathing to obtain a tan, or simply to soak up the heat, is an inadvertent means
of studying photochemical reactions in the skin. It is also a good example of a
consecutive reaction for which k(\) < k(2).
Small amounts of organic radicals are formed continually in the
skin during photolysis (in a process with rate constant &(i)). The
radicals are consumed 'immediately' by natural substances in the
skin, termed antioxidants (in a different process with rate constant
k(2)). Vitamin C (L-(+)-ascorbic acid, IV) is one of the best natu-
rally occurring antioxidants. Red wine and tea also contain efficient
antioxidants.
An antioxidant is a
chemical that prevents
oxidation reactions.
CH 2 OH
Time
Figure 8.21 Schematic graph of concentration against time ('a concentration profile') for a con-
secutive reaction for which k (2 ) > k(i). This graph was computed with k (2 ) being five times faster
than k,
(i)
402
CHEMICAL KINETICS
The vast majority of organic radical reactions involve the radical as a reactive
intermediate, since their values of k(2) are so high, although we need to note that the
second reaction need not be particularly fast: it only has to be fast in relation to the
first reaction. As a good generalization, the intermediate may be treated as a reactive
intermediate if &(2)/&(i) > 10~ 3 .
Integrated rate equations for consecutive reactions
Our consecutive reaction here has the general form A — > B — >• C. Deriving an inte-
grated rate equation for a consecutive reaction is performed in much the same way as
for a simple one-step reaction (see Section 8.2), although its complexity will prevent
us from attempting a full derivation for ourselves here.
For the precursor A, the rate of change of [A] is given by
d[A]
rate = — — = k m [A]
(8.43)
where the minus sign reflects the way that [A] is consumed, meaning its concentration
decreases with time.
The rate of change of C has been given already as Equation
(8.42). Equations (8.42) and (8.43) show why the derivation of
integrated rate equations can be difficult for consecutive reactions:
while we can readily write an expression for the rate of forming
C, the rate expression requires a knowledge of [B], which first
increases, then decreases. The problem is that [B] is itself a function
of time.
The rate of change of [B] has two components. Firstly, we form
B from A with a rate of '+&(i) x [A]'. The second part of the
rate equation concerns the subsequent removal of B, which occurs
with a rate of '—£(2) x [B]'. The minus sign here reminds us that [B] decreases in
consequence of this latter process.
Combining the two rate terms in Equations (8.42) and (8.43)
yields
Writing an integrated
rate expression for
a complicated reac-
tion is difficult because
we don't readily know
the time-dependent
concentration of the
intermediate, [B] t .
The rate expression of
a complicated reaction
comprises one term
for each reaction step.
In this case, species
B is involved in two
reactions, so the rate
equation comprises
two terms.
d[B]
df
k(i)[A] ~ k ( 2)[B]
(8.44)
rate of the process
forming the B
rate of the process
removing the B
Equation (8.44) helps explain why the concentration profile in
Figure 8.20 contains a maximum. Before the peak, and at short
times, the second term on the right-hand side of Equation (8.44) is tiny because [B]
is small. Therefore, the net rate d[B]/df is positive, meaning that the concentration of
B increases. Later - after the peak maximum - much of the [A] has been consumed,
KINETIC TREATMENT OF COMPLICATED REACTIONS
403
meaning more [B] resides in solution. At this stage in the reaction, the first term on
the right-hand side of Equation (8.44) is relatively small, because [A] is substantially
depleted, yet the second term is now quite large in response to a higher value of [B].
The second term in Equation (8.44) is consequently larger than the first, causing the
overall rate to be negative, which means that [B] decreases with time.
Table 8.3 lists the concentrations [A] t , [B] t and [C] ( , which are exact, and will
give the correct concentrations of A, B and C at any stages of a reaction. We will not
derive them here, nor will we need to learn them. But it is a good idea to recognize
the interdependence between [A]^, [B] ( and [C] t .
®
ow do Reactolite sunglasses won
The kinetics of reversible reactions
Reactolite sunglasses are photochromic: they are colourless in the
dark, but become dark grey-black when strongly illuminated, e.g.
on a bright summer morning. The reaction is fully reversible (in the
thermodynamic sense), so when energy is removed from the sys-
tem, e.g. by allowing the lenses to cool in the dark, the photochem-
ical reaction reverses, causing the lenses to become uncoloured and
fully transparent again.
The photochromic lenses contain a thin layer of a silver-
containing glass, the silver being in its +1 oxidation state. Absorp-
tion of a photon supplies the energy for an electron-transfer reaction
in the glass, the product of which is atomic silver:
The word 'photo-
chromic' comes from
the Greek words
photos, meaning
'light' and khromos
meaning 'colour'.
A photochromic
substance acqui-
res colour
when illuminated.
Ag + + e + hv
Ag°
(8.45)
where the electron comes from some other component within the
glass. It is the silver that we perceive as colour. In effect, the colour
Excited states are
defined in Chapter 9.
Table 8.3 Mathematical equations to describe the
concentrations 3 of the three species A, B and C invol-
ved in the consecutive reaction, A — > B — > C
[A], = [A]„ exp(-* ( i)0
[B],
Hi)
H2) - K(l)
-[A] {exp(fc (1) - exp(£ (2) f)}
[C], = [A] 1 -
fc(2) exp(-fc (1) + fc ( i)f
k(2) - fc(i)
a Rate constants are denoted as k, where the subscripts indi-
cate either the first or the second reaction in the sequence.
The subscript '0' indicates the concentration at the com-
mencement of the reaction. Concentrations at other times
are denoted with a subscript t.
404
CHEMICAL KINETICS
indicates a very long-lived excited state, the unusually long time is achieved because
colouration occurs in the solid state, so the reaction medium is extremely viscous.
Atomic silver is constantly being formed during illumination, but, at the same
time, the reverse reaction also occurs during 'cooling' (also termed 'relaxation' - see
Chapter 9) in a process called charge recombination. Such recombination is only
seen when removing the bright light that caused the initial coloration reaction, so the
reaction proceeds in the opposite direction.
Reversible reactions
are also termed equi-
librium or opposing
reactions.
The reaction might
be so slow that the
reaction never actually
reaches the RHS during
a realistic time scale,
but the direction of
the reaction is still
the same.
For a thermodynami-
cally reversible reac-
tion, the rate constants
of the forward and
reverse reactions are
k n and k- n respectively.
Rate laws for reversible reactions
All the reactions considered so far in this chapter have been irre-
versible reactions, i.e. they only go in one direction, from fully
reactants on the left-hand side to fully products on the right-hand
side. (They might stop before the reaction is complete.)
We now consider the case of a reversible reaction:
A
k-.
B
(8.46)
where k\ is the rate constant for the forward reaction and k-\ is the
rate constant for the reverse reaction. The minus sign is inserted to
tell us that the reaction concerned is that of the reverse reaction.
When writing a rate expression for such a reaction, we note that
two arrows involve the species B, so, straightaway, we know that
the rate expression for species B has two terms. This follows since
the rate of change of the concentration of B involves two separate
processes: one reaction forms the B (causing [B] to increase with
time) while the other reaction is consuming the B (causing [B] to
decrease). The rate is given by
rate of change of [B]
d[B]
At
*i[A]-*_i[B]
(8.47)
where the minus and (implicit) plus signs indicate that the concentrations decrease
or increase with time respectively. Note that a new situation has arisen whereby the
expression to describe the rate of change of [B] itself involves [B] - this is a general
feature of rate expressions for simple reversible reactions. Similarly:
rate of change of [A] =
The concentrations
stop changing when
the reaction reaches
equilibrium.
d[A]
At
= -*i[A] + *_i[B]
(8.48)
Note that the rate of change of [A] is equal but opposite to
the rate of change of [B], which is one way of saying that A is
consumed at the expense of B; and B is formed at the expense
of A.
KINETIC TREATMENT OF COMPLICATED REACTIONS
405
When the reaction has reached equilibrium, the rate of change of both species must
be zero, since the concentrations do not alter any more - that is what we mean by
true 'equilibrium'. From Equation (8.48):
= *i[A] (eq) -*_i[B] (eq)
so
fcl[A] (eq ) = *-l[B]
(cq)
which, after a little algebraic rearranging,
ing result:
kl [B](eq)
k-l [A] ( eq)
gives a rather surpris-
(8.49)
We recognize the right-hand side of the equation as the equilibrium
constant K. We give the term microscopic reversibility to the idea
that the ratio of rate constants equals the equilibrium constant K.
The principle of
'microscopic rever-
sibility' demonstrates
how the ratio of rate
constants (forward to
back) for a reversible
reaction equals the
reaction's equilibrium
constant K.
Worked Example 8.18 Consider the reaction between pyridine and heptyl bromide,
to make 1 -heptylpyridinium bromide. It is an equilibrium reaction with an equilibrium
constant K — 40. What is the rate constant of back reaction k_y if the value of the forward
rate constant k\ — 2.4 x 10 3 dm 3 mol _1 s _1 ?
We start with Equation (8.49):
K =
and rearrange it to make k-\ the subject, to yield
K_i = —
K
We then insert values:
so fc_ i = 60 dm 3 mol ' s ' .
2.4 x 10 3 dm 3 moP's" 1
40
SAQ 8.22 A simple first-order reaction has a forward rate constant of
120 s _1 while the rate constant for the back reaction is 0.1 s -1 . Calculate
the equilibrium constant K of this reversible reaction by
invoking the principle of microscopic reversibility.
Integrated rate equations for reversible reactions
In kinetics, we often term the concentrations at equilibrium 'the
(R)
infinity concentration'. The Reactolite glasses do not become
In kinetics, the equi-
librium concentration
[A] ( eq) is often termed
the infinity concentra-
tion, and cited with an
infinity sign as [A]^.
406
CHEMICAL KINETICS
We should never throw
away a reaction solu-
tion without measuring
a value of [A] (eq) .
progressively darker with time because the concentration of
atomic silver reaches its infinity value [Ag]( eq ).
We will consider the case of a first-order reaction, A T± product.
Following integration of an expression similar to Equation (8.48),
we arrive at
In
[A]q - [A] (eg)
[A], - [A]
(cq)
= (ki + k-i)t
(8.50)
which is very similar to the equation we had earlier for a simple reaction (i.e. one
proceeding in a single direction), Equation (8.24). There are two simple differences.
Firstly, within the logarithmic bracket on the left-hand side, each term on top and
bottom has the infinity reading subtracted from it. Secondly, the time t is not multi-
plied by a single rate constant term, but by the sum of both rate constants, forward
and back.
This equation can be rearranged slightly, by separating the logarithm:
ln([A], - [A] (eq) )
y
m
+ ln([A] - [A] (eq) )
(8.51)
Note that the final term on the right-hand side is a constant. Accordingly, a plot of
In ([A]; — [A]( eq )) (as 'y') against time (as 'x') will yield a straight line of gradient
Worked Example 8.19 The data below relate to the first-order isomerization of 2-
hexene at 340 K, a reaction for which the equilibrium constant is known from other
studies to be 10.0. What are the rate constants k\ and &_i?
Time r/min 20 47 80 107 140
([A], - [A] (eq) )/mol dm" 3 0.114 0.103 0.091 0.076 0.066 0.055
Strategy. (1) we need to plot a graph of ln([A], — [A]( eq )) (as 'v') against time t (as 'x');
(2) determine its gradient; (3) then, knowing the equilibrium constant K, we will be able
to determine the two rate constants algebraically.
(1) Figure 8.22 shows a graph for a reversible first-order reaction with the axes for
an integrated rate equation ln([A] t — [A] (eq) ) (as 'y') against time (as 'x'). The
gradient is —5.26 x 10~ 3 min -1 .
The microscopic-
reversibility relation-
ship K = k\ -4-/f_i can-
not be applied unless
we know we have a
simple reversible reac-
tion.
(2) The gradient is -(ki + k-i), so (k x + Jfc_i) = +5.26 x 10~ 3
min - .
(3) Next, we perform a little algebra, and start by saying that
K — ky 4- k-\, i.e. we invoke the principle of microscopic
reversibility . Multiplying the bracket (k t + k-i) by k_\ 4- k_\,
i.e. by 1, yields
gradient = k-i(K + 1) (8.52)
KINETIC TREATMENT OF COMPLICATED REACTIONS 407
-2.2-^
1 1
CT
CD
<
-2.4-
<
-2.6-
-2.8-
-3-
1 1
50 100
Time f/min
150
Figure 8.22 Kinetic graph for a reversible first-order reaction with the axes for an integrated rate
equation ln([A], — [A]( eq )) (as 'y') against time (as 'x'). The gradient is —5.26 x 10~ 3 min" 1
Substituting values into Equation (8.52) gives
5.26 x 10~ 3 min" 1 = fc_i(10.0 + 1)
so
5.26 x lfr 3 min" 1 , ,
k-i = = 4.78 x 10" 4 min" 1
11
Next, since we know both K and &_], we can calculate k\. We know from the gradient
of the graph that
(hi +Jfc_i) = 5.26x 10" 3 min" 1
which, after a little rearranging, gives
fci =5.26 x 10" 3 min -1 - Jfc_,
and, after inserting values
3 ™;„-l
hi = (5.26 x 10"' - 4.78 x 10" 4 ) min -1 = 4.78 x 10" J min
In summary, k\ — 4.78 x 10 3 min ' and fe_] = 4.78 x 10 4
Notice how the ratio
'/fi -e-/f_i' yields the
value of K.
408
CHEMICAL KINETICS
It is wrong (but com-
mon) to see a reversible
reaction written with a
double-headed arrow,
as 'A +* B'. Such an
arrow implies reso-
nance, e.g. between
the two extreme val-
ence-bond structures of
Kekule benzene.
SAQ 8.23 Consider a reversible first-order reaction. Its
integrated rate equation is given by Equation (8.50). People
with poor mathematical skills often say (erroneously!) that
taking away the infinity reading from both top and bottom is
a waste of time because the two infinity concentration terms
will cancel. Show that the infinity terms cannot be cancelled
in this way; take [A] (eq ) = 0.4 moldrrT 3 , [A]o = 1 moldrrT 3
and [A] f = 0.7 moldm -3 .
8.5 Thermodynamic considerations:
activation energy, absolute reaction rates
and catalysis
Why prepare a cup of tea with boiling water?
The temperature dependence of reaction rate
The instructions printed on the side of a tea packet say, 'To make a perfect cup of
tea, add boiling water to the tea bag and leave for a minute'. The stipulation for
'one minute' suggests the criterion for brewing tea in water at 100 °C is a kinetic
requirement. In fact, it reflects the rate at which flavour is extracted from the tea bag
and enters the water.
Pure water boils at 100 °C (273.15 K). If the tea is prepared with
cooler water, then the time taken to achieve a good cup of tea is
longer (and by 'good', here, we mean a solution of tea having a
sufficiently high concentration). If the water is merely tepid, then a
duration as long as 10 min might be required to make a satisfactory
cup of tea; and if the water is cold, then it is possible that the
tea will never brew, and will always remain dilute. In summary,
the rate of flavour extraction depends on the temperature because
the rate constant of flavour extraction increases with increasing
temperature.
Remember that the
temperature of boil-
ing water 7~ (b oii> is
itself a function of
the external pres-
sure, according to the
Clausius-Clapeyron
equation (see
Section 5.3).
Why store food in a fridge?
The temperature dependence of rate constants
Food is stored in a fridge to prevent (or slow down) the rate at which it perishes.
Foods such as milk or butter will remain fresh for longer if stored in a fridge, but they
decompose or otherwise 'go off more quickly if stored in a warmer environment.
The natural processes that cause food to go bad occur because of enzymes and
microbes, which react with the natural constituents of the food, and multiply. When
THERMODYNAMIC CONSIDERATIONS
409
these biological materials have reached a certain concentration, the food smells and
tastes bad, and is also likely to be toxic.
The growth of each microbe and enzyme occurs with its own
unique rate. A fridge acts by cooling the food in order to slow these
rates to a more manageable level. At constant temperature, the rate
of each reaction equals the respective rate constant k multiplied
by the concentrations of all reacting species. For example, the rate
of the reaction causing milk to 'go off occurs between lactic acid
and an enzyme. The rate of the process is written formally as
The rate constant k 2 is
truly a constant at a
fixed temperature, but
can vary significantly.
rate = £2 [lactose] [enzyme]
where, as usual, the subscripted '2' indicates that the reaction is
second order. Neither [lactose] nor [enzyme] will vary with temper-
ature, so any variations in rate caused by cooling must, therefore,
arise from changes in k 2 as the temperature alters.
The rate constant ki is truly a constant at a fixed temperature,
but can vary significantly: increasing as the temperature increases
and decreasing as the temperature decreases. This result explains
why rate of reaction depends so strongly on temperature.
(8.53)
The rate constant
increases as the tem-
perature increases and
decreases as the tem-
perature decreases.
Why do the chemical reactions involved in cooking
require heating?
Activation energy E a and the Arrhenius theory
m
'Naturally occurring'
was the old-fashioned
definition of 'organic
chemistry', and per-
sisted until nearly the
end of the 19th cen-
tury.
Cooking is an applied form of organic chemistry, since the mole-
cules in the food occur naturally. We heat the food because the
reactions occurring in, say, a pie dish require energy; and an oven
is simply an excellent means of supplying large amounts of energy
over extended periods of time.
The natural ingredients in food are all organic chemicals, and
it is rare for organic reactions to proceed without an additional
means of energy, which explains why we usually need to reflux a
reaction mixture.
It is easy to see why an endothermic reaction requires energy to react - the energy
to replace the bonds, etc. must be supplied from the surroundings. But why does an
exothermic reaction require additional energy? Why do we need to add any energy,
since it surely seeks to lose energy?
At the heart of this form of kinetic theory is the activated complex. In this context,
the word 'activated' simply means a species brimming with energy, and which will
react as soon as possible in order to decrease that energy content.
A reaction can be thought of as a multi-step process: first the reactants approach and
then they collide. Only after touching do they react. One of the more useful definitions
410
CHEMICAL KINETICS
The Franck- Condon
principle states that
atomic nuclei are sta-
tionary during a reac-
tion, with only elec-
trons moving -
see p. 451.
of reaction is 'a rearrangement of bonds'. We are saying that, as a good generalization,
the atomic nuclei remain stationary during the reaction while the electrons move. This
idea is important, since it is the electrons that act as the 'glue' between the nuclei.
Such movement occurs in such a way that the bonds between the atoms are different
in the product than in the reactant.
This simple yet profound notion, that atomic nuclei are stationary
during the reaction and that only electrons have time to move, is
called the Franck- Condon principle. We shall see its important
consequences later, in Chapter 9.
We now move on slightly, conceptually. Consider a single pair
of reactant molecules combining to form a product. As electrons
rearrange as the reaction commences, we pass smoothly from a
structure that is purely reactant to one of pure product. The tran-
sition from one to the other is seamless; see Figure 8.23.
There will soon come a point where some bonds are almost broken and others
almost formed. We have neither reactant nor product: it is a hybrid, being a mixture
of both reactant and product. It is extremely unstable, and hence of extremely high
energy (i.e. with respect to initial reactants or the eventual products). We call it the
transition-state complex, and often give it the initials TS. To a first approximation,
the character of the complex is predominantly reactant before the TS is formed, and
predominantly product afterwards.
The transition-state complex TS is only ever formed in minute concentrations and
for a mere fraction of a second, e.g. 10~ 12 s, so we do not expect to 'see' it except
by the most sophisticated of spectroscopic techniques, such as laser flash photolysis.
Forming the TS is like pushing a marble over a large termite hill: most of the
marbles cannot ascend the slope and, however high they rise up the hill's slope, they
do not ascend as far as the summit. Those rare marbles that do reach the summit
appear to stay immobile for a mere moment in time, and are then propelled by their
own momentum (and their own instability, in terms of potential energy) down over
the termite hill and onto the other side. A chemical reaction is energetically similar:
the reaction commences when molecules of reactant collide with 'sufficient energy'.
If sufficient energy is available, then the two or more reactants join to form the
transition-state complex TS, i.e. the electrons rearrange with the net results that, in
effect, atoms or groups of atoms move their positions (the bonds change).
C D
Reactants
C D
Transition-state complex
Products
Figure 8.23 During a reaction, the participating species approach, collide and then interact. A
seamless transition exists between pure reactants and pure products. The rearrangement of electrons
requires large amounts of energy, which is lost as product forms. The highest energy on the
activation energy graph corresponds to the formation of the transition-state complex. The relative
magnitudes of the bond orders are indicated by the heaviness of the lines
THERMODYNAMIC CONSIDERATIONS
411
TS
LU
/ A
t Products
1 A '"'(reaction)
Reactants
►
Reaction coordinate
Figure 8.24 Reaction profile of energy (as 'y') against reaction coordinate (as 'x'). The activation
energy £" a is obtained as the vertical difference between the reactants and the peak of the graph,
at 'TS'
Figure 8.24 shows a graph of energy as a function of reaction progress. The tran-
sition complex is formed at the energy maximum. The figure will remind us of
Figures 3.1 and 3.2, except with the peak on top. It is similar. The enthalpy of reac-
tion is obtained as the vertical difference between 'reactants' and 'products' on
the graph.
The 'sufficient energy' we mentioned as needed to form the
transition-state complex is termed the activation energy, which is
given the symbol Z? a (with the E denoting 'energy' and the sub-
scripted 'a' for 'activation'). The word 'activation' ought to suggest
additional energy is required; in fact, the activation energy is always
positive because the TS is always higher in energy than the reac-
tants. Stated another way, its formation is always endothermic.
The activation energy
E a is always posi-
tive, so the formation
of a transition-state
complex is always
endothermic.
Why does a reaction speed up at higher temperature?
The Arrhenius equation
In Chapters 1 and 2, we met the idea that the simplest way to
increase the energy of a chemical, material or body is to raise its
temperature. So heating a reaction mixture gives more energy to
its molecules. Although only a tiny proportion of these molecules
will ever have sufficient energy to collide successfully and form
an activated complex TS, even a small increase in the amounts of
energy possessed by a molecule will increase the proportion that
The simplest way to
increase the energy of
a chemical, material or
body is to raise its tem-
perature - see p. 34.
412
CHEMICAL KINETICS
have sufficient energy to form the TS. Therefore, heating a reaction
mixture, e.g. by reflux, increases the number of successful colli-
sions between reactant species, increasing the amount of product
formed per unit time.
By increasing the temperature T, we have not changed the mag-
nitude of the activation energy, nor have we changed the value
of AH of reaction. The increased rate is a kinetic result: we
have enhanced the number of successful reaction collisions per
unit time.
The simplest relationship between temperature T and rate constant k is given by
the Arrhenius equation (Equation (8.54)), which relates the rate constant of reaction
k with the thermodynamic temperature T at which the reaction is performed:
Heating a reaction
mixture increases the
number of success-
ful collisions between
the reactant species,
increasing the amount
of product formed per
unit time.
Aexp
RT
(8.54)
The Arrhenius
equation is written in
terms of thermody-
namic temperature.
where R is the gas constant and E a is the activation energy (above),
which is a constant for any particular reaction. T is the thermody-
namic temperature (in kelvin), and A is called usually called the
Arrhenius 'pre-exponential' factor. The value of £ a depends on the
reaction being studied.
The logarithmic form of Equation (8.54)
In k = In A
The activation energy
is obtained as '-1 x R x
gradient', where 'gradi-
ent' refers to the slope
of the Arrhenius plot.
RT
reminds us of the equation of a straight line, ' v = c + mx\ so a
plot of ln(rate constant) (as 'y') against \IT (as 'x') will yield a
straight-line graph of gradient '— E a -r /?'. In practice, we repeat-
edly perform the experiment to determine a value of its rate con-
stant k, each determined at a single value of T .
We should not attempt to memorize the Arrhenius equation until we can 'read'
it, and have satisfied ourselves that it is reasonable. Firstly, we note that R, E a
and the 'constant' term will not vary. We are, therefore, looking at the effect of
T on k. Next, we see that the first term on the right-hand side of the logarithmic
form of Equation (8.54) decreases as the temperature increases and so the loga-
rithmic term on the left must also decrease. However, since there is a minus sign
on the right-hand side of the equation, we are saying that as T increases, so the
right-hand side becomes less negative (more positive). In other words, as the tem-
perature increases, so the logarithm of the rate constant also increases, and hence k
gets larger.
Worked Example 8.20 Consider the following data that relate to the rate of removing
a naturally occurring protein with bleach on a kitchen surface. What is the activation
THERMODYNAMIC CONSIDERATIONS
413
0.003
0.0034
Figure 8.25 An Arrhenius plot of ln(rate constant) (as 'y') against IIT (as 'x'). The data relate
to the rate of removing a naturally occurring protein with bleach on a kitchen surface
energy of reaction?
Temperature 77 °C
Rate constant k/s~ l
20
30
40
50
60
2.20
2.89
3.72
4.72
5.91
Answer Strategy: before we can plot anything, we first convert the
temperatures from Celsius to thermodynamic temperatures in kelvin.
Then we plot ln(k/s~ l ) (as ')>') against l/T (as V).
Such a plot is seen in Figure 8.25. Its gradient is equal to —E a /R.
This graph is seen to be linear, with a gradient of —2400 K. From
the Arrhenius equation, the value of activation energy is obtained as
'—gradient x R' . Therefore:
E. d = -2400 K x 8.314JKT 1 mor 1 = 20000 JmoP 1
The activation energy is 20 kJmol - .
We take the logarithm
of k-
.-1
because
it is mathematically
impossible to take the
logarithm of anything
expect a number.
Aside
Activation in biological systems
Many biological systems show an activated behaviour,
since they appear
to follow the
Arrhenius equation (Equation (8.45)). In fact, all bioloj
;ical
processes are
activated,
but
414
CHEMICAL KINETICS
the complicated array of biological rates means that we only see the rate of the slowest,
rate-limiting step. Nevertheless, look at the following examples:
(1) The rate of development of the water flea (Alona affinis) from egg to adult is
activated, since a plot of ln(development rate) (as 'y') against \IT (as 'x')
is linear.
(2) The walking speed of an ant {Liometopum apiculetum) follows an
Arrhenius plot over a limited temperature range. Over a wider temperature
range, the plot is curved, indicating the involvement of different, finely balanced
processes, each being rate limiting over a different temperature range.
(3) Figure 8.26 shows a plot of ln(heart beat) (as 'y') against \IT (as 'x') for
the diamond-backed terrapin {Malaclemys macro spilota).
(4) The Mediterranean cicadas {Homoptera, Auchenorrhyncha) chirp at a frequency
so closely related to the Arrhenius equation that we can 'hear' the temperature
by measuring its rate.
5.0
4.0
S5 3.0
CD
.Q
t_
CO
S 2.0
1.0
Figure 8.26 An /
diamond-backed ter
peratures, showing
(Figure reproduced
Taylor (eds), Royal
permission)
O ^
c
T\
^^?
^ (
*••*
^
\
3.3 3.4 3.5
1 /(Temperature T/1000 K)
urhenius plot of ln(heart beat) (as 'y') against \IT (as 'x') for the
rapin (Malaclemys macrospilota) is linear over a limited range of tem-
that the rate-limiting process dictating its heart rate is activated,
from Chemical Kinetics and Mechanism, Michael Mortimer and Peter
Society of Chemistry, Cambridge, 2002, p. 77. Reproduced with
THERMODYNAMIC CONSIDERATIONS
415
Why does the body become hotter when ill, and get 'a
temperature' ?
The thermodynamics of competing reactions
■
One of the worst aspects of being ill is the way the body develops a 'temperature'. A
healthy human should have a more-or-less constant body temperature of 37 °C (which,
in the old-fashioned temperature scale of Fahrenheit, is 98.6 °F). A body temperature
of above 100 °F is best avoided, and temperatures above about 104 °F are often lethal.
As a temperature is so harmful to the body, why does the body itself generate the
extra heat?
All reactions proceed via a transition-state complex, and with an
activation energy E a . The values of E a vary tremendously, from
effectively zero (for a so-called diffusion-controlled reaction, as
below) to several hundreds of kilojoules per mole (for reactions
that do not proceed at all at room temperature). The rate constant
of a reaction is relatively insensitive to temperature if £ a is small.
An alternative form of the Arrhenius equation is the integrated
form:
'k at T 2 \ _ £ a / 1 1
k at Ti / R
In
(8.55)
If E a is small, then
the rate constant of
reaction is relatively
insensitive to temper-
ature; if E B is large,
then k is more sen-
sitive to temperature
fluctuations.
Equation (8.55) comes
ultimately from the
Maxwell -Boltzmann
distribution in Equation
(1.16).
which is similar to Equation (8.54), but applies to two temperatures
and two rate constants. If E a is small, then the left-hand side of
the equation will have to be small, implying that the ratio of rate
constants must be close to unity. Conversely, a large value of E a
causes the rate constants k to be more sensitive to changes in
temperature.
Antibodies are naturally occurring substances in the blood that fight an infection or
illness. Like all reactions, the rate at which antibodies fight an illness has an activation
energy, which is quite high. Whenever the illness appears to be 'winning' the battle,
the body raises its temperature to increase the rate at which antibodies react. A higher
temperature means that the body's antibodies operate faster, i.e. with an increased
rate constant. A body, therefore, has 'a temperature' in order to fight an infection
more rapidly.
SAQ 8.24 A person is ill. The rate at which their antibodies react needs
to increase twofold. This rate increase is achieved by raising the body's
temperature from 37 °C to 40 °C. What is the activation energy of the
reaction? [Hint: first convert from Celsius to kelvin.]
Aside
Another reason for the body's temperature rising during illness is an increase in the
rate of metabolism of white blood cells. White blood cells are an essential part of the
416 CHEMICAL KINETICS
body's defence mechanism, and 'attack' any foreign bodies or pathogens in the body,
such as bacteria or viruses. Pathogens are engulfed or consumed by white blood cells
when we are ill by a process known as phagocytosis.
Why are the rates of some reactions insensitive to
temperature?
Diffusion-controlled reactions
■
Some rare reactions occur at a rate that appears to be insensitive to temperature. Such
reactions are extremely rapid, and are termed diffusion-controlled reactions.
If the activation energy is extremely small - of the order of
1 kJ mol~ or so - then all the energy necessary to overcome the
The value of E a is
effectively zero for
diffusion-con trolled
reactions.
activation energy is available from the solvent, etc., so reaction
occurs 'immediately' the reactants collide.
In fact, the only kinetic limitation to such a reaction is the speed
at which they move through solution before the collision that forms
product. This rate is itself dictated by the speed of diffusion (which is not generally
an efficient form of transport). The rate of reactants colliding is, therefore, said to be
'diffusion controlled'. Typically, diffusion-controlled processes in which £ a is tiny
involve radical intermediates.
If a second-order reaction is diffusion controlled, then its rate constant has a mag-
nitude of about 10 10 or even 10 11 dm 3 mol _1 s _1 .
SAQ 8.25 The rate of a reaction is said to be 'diffusion controlled'
because its activation energy is 1.4 kjmol" 1 . The rate constant of reac-
tion is 4.00 x 10 10 dm 3 mor 1 s- 1 at 298 K. Show that the rate constant is
effectively the same at 330 K.
The Eyring approach to kinetic theory
The Arrhenius theory (above) was wholly empirical in terms of it derivation. A more
rigorous, but related, form of the theory is that of Eyring (also called the theory of
absolute reaction rates). The Eyring equation is
( k\ -AH* AS* /k B \
Hrir+T^d) <8 ' 56)
where k% 1S the Boltzmann constant and h is the Planck constant. AH* is the
enthalpy change associated with forming the activated complex and AS* is the change
in entropy.
A plot of \n(k -r T) (as 'y') against IIT (as 'x') should be linear, of gradient
-AH* + R and intercept 'AS* -f- R + ln(fc B H- h)' .
THERMODYNAMIC CONSIDERATIONS
417
Worked Example 8.21 The rate of hydrolysis for a biological molecule was studied
over the temperature range 300-500 K, and the rates found to be as follows. Use a
suitable graphical method to determine A//*, AS* and AG*, for the reaction at 298 K.
77K
fc/dm 3 moP 1 s _1
300 350 400
7.9 x 10 6 3.0 x 10 7 7.9 x 10 7
450
1.7 x 10 8
500
3.2 x 10 8
Figure 8.27 shows a graph of ln(£ -f- T) (as 'y') against '1 4- 7" (as 'x'). Its gradient
of -2386 KT 1 is equal to '-AH* 4- R\ so A//* = 2386 K" 1 x 8.314JKT 1 moP 1 =
19.8kJmol _1 .
The intercept on Figure 8.27 is 18.15, and is equal to 'AS 1 *//? + ln(&B/ h)' . The loga-
rithmic term ln(ks/ h) has a value of 23.76, so
AS*
~R~
18.15-23.76= -5.61
so
AS f = -5.61 x 8.314 JKT 1 mol"
46.6 jr'mor
Finally, the value of AG* is obtained via the equation AG* =
AH f — TAS* (cf. Equation (4.21)). Inserting values and saying
The value of AG* is
positive, implying that
the equilibrium con-
stant of forming the
transition-state com-
plex is minuscule,
as expected.
"i i i i i r
0.0019 0.0021 0.0023 0.0025 0.0027 0.0029 0.0031 0.0033
1/T-1/K
Figure 8.27 An Eyring plot of \n(k/T) (as 'y') against \/T (as 'x'). The data relate to the rate of
hydrolysing a biological molecule in the temperature range 300-500 K
418
CHEMICAL KINETICS
T = 298 K:
AG- = 19800 Jmol" 1 - (298K x -46.6JK" 1 mol -1 )
AG* = 5.9kJmol _1
Aside
Comparing the Arrhenius and Eyring equations
It is extremely common, but wrong, to see the Eyring equation written in a similar
form, with the Arrhenius ordinate of ln(£) (as 'y') and the final logarithmic term written
as ln(kftT 4- h). Although such an equation might be readily achieved from the brief
derivation in Justification Box 8.6, it is seen straightaway to be nonsensical, for the
following three reasons:
(1) If the intercept contains a temperature term T , it is nonsense to have an inter-
cept that contains one of the equation's principal variables.
(2) As an intercept, 1 4- T = 0, so the only sensible temperature to include as T
in the intercept term would be T = oo, which means that AS* = — oo. Again,
this is not realistic.
(3) More importantly, however, is a physicochemical concept behind the equa-
tions: if both equations are written as ln(fe) (as 'y') as a function of 1 -f- T (as
'x'), then it is dishonest to suppose that the gradients of the respective graphs
can be different, one a function of £ a and the other a function of A//*.
As a further implication, A//* cannot be the same as E a . In fact, from the Eyring
theory, we can show readily that
Eyring theory says
that AH* =E a + RT,
explaining why values
of AH* are not con-
stant, but depend on
temperature.
AH
t _
Ea + RT
(8.57)
This equation explains why values of AH * are not
constant, but depend on temperature. Conversely, £ a is
a true constant.
We will employ the form of the Eyring equation writ-
ten as Equation (8.56).
SAQ 8.26 The following table contains the rate constant k for the
demethylation reaction of A/-methyl pyridinium bromide by aqueous sod-
ium hydroxide as a function of temperature:
T/K
k/10 2
dm mol
298
313
333
353
8.39
21.0
77.2
238
THERMODYNAMIC CONSIDERATIONS 419
(1) Calculate the activation energy E a and pre-exponential factor A by
plotting an Arrhenius graph.
(2) Calculate AG*, AH* and AS* for the reaction at 298 K by plotting
an Eyring graph.
(3) What is the relationship between AH* and E a ?
Justification Box 8.6
From Eyring, the rate constant of reaction k depends on a
K % , relating to the formation of a transition- state complex,
be virtually infinitesimal.
The values of k and K % are related as
pseudo equilibrium constant
TS. Clearly, A"* will always
knT
-K
3.58)
where T is the thermodynamic temperature. From an
equation which will remind us of the van't Hoff iso-
therm (cf. Equation (4.59)), we relate the equilibrium
constant with the change in Gibbs function:
Care: there are three
different types of y k'
in Equation (8.58), so
we must be careful
about the choice of big
or small characters,
and subscripts.
AG %
-RT\nK
(8.59)
where AG T here is specifically the change in Gibbs function associated with forming
the transition- state complex. After rearrangement, we obtain
K l = exp
and by substituting for f 1 we obtain
k = exp
h V
-AG*
RT
-AG*
RT
(8.60)
(8.61)
Next, we recall, from the second law of thermodynamics, that AG e = AH® — TAS®.
By direct analogy, AG* = A//* — TAS*, where AH * is the enthalpy of forming the
transition- state complex (akin to the activation energy £ a ). A 5* is the entropy of forming
the transition- state complex.
By inspection alone, we can guess that A 5* will be negative for bimolecular reactions,
since two components associate to form one (the TS). The value of AS* is positive for
unimolecular processes, such as gas-phase dissociation.
Substitution for AG* into Equation (8.61) gives
, k B T
k = exp
h
-A//*
RT
exp (it)
(8.62)
420
CHEMICAL KINETICS
The Arrhenius pre-
exponential factor A
k B T /AS*"
is — — exp
h
R
,.i
where, by dividing both sides by T, we obtain
k k B
■ exp
AH*\ /AS*\
— exp — (8.63)
Taking logarithms of both sides yields the Eyring equa-
tion, Equation (8.56):
-A#* AS* /k B
+ + In —
RT R \ h
This equation can be thought of as a more quantitative form of the Arrhenius equation.
What are catalytic converters?
Catalysis
A catalytic converter
is a part of a car
exhaust. It is some-
times called a 'CAT'
or, worse, given the
wholly non-descriptive
abbreviation 'CC.
A catalytic converter
is a stainless steel
tube located near the
exhaust manifold,
lined with finely divided
metal salts, e.g. of plat-
inum and palladium.
It has been appreciated for many years how the exhaust gases
from a car are hazardous to the health of most mammals. The
most harmful gases are carbon monoxide (CO) and low-valence
of oxides of sulphur (SO2) and nitrogen (NO*). CO is doubly
undesirable in being toxic (it causes asphyxiation) and being a
greenhouse gas. SO2 or NO* are not particularly toxic - in fact,
minute amounts of NO* may even be beneficial to health. But
both gases hydrolyse in water to form acid rain, with attendant
environmental damage. The source of the sulphur and nitrogen,
before combustion, are natural compounds in the petrol or
diesel fuels.
The purpose of the catalytic converter is to oxidize some of
the oxides in the gas phase. NO* is reduced to elemental nitro-
gen. The principal reaction at the catalytic converter is oxidation
of CO:
CO
(g)
+ 2°2(g)
CO
2(g)
(8.64)
This reaction is energetically spontaneous, but it occurs quite slowly if the gases
are just mixed because the activation energy to reaction is too high at 80 kJmol -1 .
The reaction is much faster if the CO is burnt, but a naked flame is considered unsafe
in a car exhaust.
Although the reaction in Equation (8.64) is slow under normal conditions, it can
be accelerated if performed with a catalyst; hence the incorporation of a catalytic
converter in the car exhaust system.
THERMODYNAMIC CONSIDERATIONS
421
Catalysis theory
All reactions occur by molecules surmounting an energetic activa-
tion barrier, as described above.
Consider the diagram of reaction profile in Figure 8.28: the reac-
tion is clearly endothermic, since the energy of the final state
is higher than the energy of the initial state i.e. AH is
positive.
We recall that enthalpy H is a state function (see Section 3.1),
so the overall enthalpy change of the reaction is independent of
the chemical route taken in going from start to finish. It is clear
from Figure 8.28 that the initial and final energies, of the reactants
and products respectively, are wholly unaffected by the presence
or otherwise of a catalyst: we deduce that a catalyst changes the
mechanism of a reaction but does not change the enthalpy change
of reaction.
It is quite common to see reaction profiles that are more com-
plicated than that depicted in Figure 8.28, e.g. with two or three
separate maxima corresponding to two or three activation energies.
The lower reaction profile in Figure 8.29 has two 'peaks' and is
included to show how the second activation energy is obtained.
The second activation energy is usually smaller than the first. (The
words 'first' and 'second' in this context are meant to imply the
sequence during the course of a concurrent reaction from reactant
to product).
TS
a (uncatalysed)
Reactants
'Catalysis' comes from
the Greek cata, mean-
ing 'away from' or
'alongside' and lusis,
meaning 'dissolution'
or 'cleavage'; so a cat-
alyst promotes bond
cleavage away from
the usual mechanis-
tic route.
The principal advan-
tage of catalysis is
to make a reaction
proceed more quickly
because the activation
energy decreases.
A catalyst changes
the mechanism of a
reaction, but not its
enthalpy change.
The energies of the
reactants and products
remain unchanged
following catalysis, so
AH r is unaffected.
Reaction coordinate
Figure 8.28 Reaction profiles for an uncatalysed reaction (upper curve) and
catalysed (lower curve). Note that the reaction energies start and finish at the
same energies, so the magnitude of AH r is not affected by the catalyst
422
CHEMICAL KINETICS
TS
>^ ▲
>>
E?
CD
C
LU
/ / 1
^a (uncatalysed)
Products
Reactants
►
Reaction coordinate
Figure 8.29 Reaction profiles for an uncatalysed reaction (upper single curve) and catalysed
(curve with a double maximum). The two solid arrows represents £ a for the catalysed reactions:
note how T (£ a for the first reaction step) is so much larger than the activation energy for the
second step, '2'
Though a catalyst
alters the mecha-
nism of a reaction,
the reaction's posi-
tion of equilibrium is
unaltered, meaning its
chemical yield remains
the same.
And we see a further point: the equilibrium constant K of a reac-
tion is a direct function of AG r according to the van't Hoff isotherm
(Equation (4.55)). If the overall energy of reaction remains unal-
tered by the catalyst, then the position of equilibrium will also
remain unaltered.
9
Physical chemistry
involving light:
spectroscopy and
photochemistry
Introduction
We look in this chapter at the interactions of light with matter. All photochemical
processes occur following the absorption of a particle of light called a photon, causing
excitation from the ground state to one of several higher energy states. The physico-
chemical properties of the molecule in the excited material will often differ markedly
from that of the ground state.
Spectroscopic techniques look at the way photons of light are absorbed quantum
mechanically. X-ray photons excite inner-shell electrons, ultra-violet and visible-light
photons excite outer-shell (valence) electrons. Infrared photons are less energetic,
and induce bond vibrations. Microwaves are less energetic still, and induce molecular
rotation. Spectroscopic selection rules are analysed from within the context of optical
transitions, including charge-transfer interactions
The absorbed photon may be subsequently emitted through one of several different
pathways, such as fluorescence or phosphorescence. Other photon emission processes,
such as incandescence, are also discussed.
Finally, we look at other spectroscopic phenomena, such as light scattering.
9.1 Introduction to photochemistry
Why is ink coloured?
Chromophores
We know that we have filled our pens with blue ink because the colour looks blue.
Black ink is clearly black because it looks black. We know a colour because we
see it.
424
SPECTROSCOPY AND PHOTOCHEMISTRY
The word 'chromo-
phore' comes from two
Greek words: khroma,
meaning colour, and
phoro, which means
to give or to impart. A
chromophore is there-
fore a species that
imparts colour.
After a further moment's thought, we realize that the only reason
why we see a colour at all is because there is something in the
solution that gives it its colour. There is an innate redness to
chemicals dissolved in red ink, which interact with the light, and
allow us to recognize its colour. There are different chemicals in
blue ink, and altogether different chemicals in green ink. Each of
these chemicals interacts in a characteristic way, which is why we
see different colours.
A chemical that imparts a colour is called a chromophore. We
could see no colour in a solution without a chromophore.
Why do neon streetlights glow?
Emission and absorption
Neon lights are commonly seen on advertising hoardings, outside shops, restaurants,
and cinemas. Their colour is so distinctive that to most people the word 'neon' has
come to mean a dark pink glow, although neon is merely an elemental rare gas
(element number 10).
Neon is wholly colourless if placed in a normal flask, yet the glow of a neon
light is visible even at night, so its colour cannot be due to the way it interacts
with light - there is no ambient light to interact with. Neon lights glow because they
emit light under suitable conditions; see p. 480.
The principal difference between neon lights and the colours seen in daylight is that
a neon light emits light, whereas wine, paint and chromophores, in general, absorb
light. The difference between absorbance and emission is illustrated schematically in
Figure 9.1.
This difference between absorption and emission is crucial, and
underlies many of pivotal aspects of photochemistry. We will look
in depth at the underlying photochemistry of neon in Section 9.3.
At its simplest, photochemical absorbances involve the uptake
of photons, so the number of photons leaving a sample will be less
The word 'light' here
does not just mean
visible light, but all
wavelengths.
■+~*Wr
Absorbing
sample
(a)
light
— vWA>
Emitting
sample
(b)
light
Figure 9.1 The difference between absorption and emission: (a) light enters a sample during
absorption and (b) leaves a sample following emission
INTRODUCTION TO PHOTOCHEMISTRY
425
The light striking the
sample is called the
incident light.
than the number entering it. Some photons are consumed. The
remainder are absorbed. By contrast, a photochemical emission
occurs when there are more photons leaving then entering it, i.e.
more after the interaction than before. It is the phenomenon of
absorbance that we study in most forms of spectroscopy: a sample
is illuminated with photons of various energies, and a detector of some sort analyses
the decrease in the number of photons. Usually, the detector is placed behind the
sample and the light source is in front of it. The light striking the sample is called
the incident light.
In studies involving emission, it is usual for the sample to be
irradiated while the number of photons emitted is measured as a
function of the incident light. To prevent the incident and emitted
beams getting confused, the emitted light is generally analysed
with the detector placed at right angles to the light source. The topic of emission
underlies fluorescence, luminescence and phosphorescence, which are discussed in
Section 9.3.
To irradiate means to
shine light upon.
Why do we get hot when lying in the sun?
Introduction to photochemistry
People are invisible to radio waves. In other words, there should
be no effect when standing between a radio-wave transmitter and
its receiver. Conversely, humans are not invisible to light in the
wavelength range 100-1000 nm - we say they are opaque, which
is most easily proved by asking a person to stand in the path of a
light source such as a torch, and seeing the shadow cast.
We see such a shadow because the light not blocked by a person
travels from the torch and impinges on the ground behind them,
where it interacts, i.e. looks brighter. Light cannot pass through
the person, who is opaque, so no light is available to interact with
those portions of the ground behind them.
A more subtle way of showing that we are opaque (wholly or
partially) to some wavelengths of light is to stand in front of an
oven, or in the sun. If we were invisible, the energy from the
heat source would pass straight through us without our noticing,
just as we do not 'feel' anything when bathed in radio waves.
But we do get hot when lying in the sun because an interaction
occurs between the infrared radiation (the heat) and us, as we are
exposed. We say this infrared light is absorbed, and its energy is
transferred in some way to us; as we saw in Chapter 2, the simplest
way to tell that something has gained energy is that its temperature
goes up.
Opaque is the opposite
extreme to invisible.
The crackle we some-
times hear when walk-
ing round a radio set
is because of metals
and other conductors
in our clothing; human
flesh is 'invisible' to
radio waves.
The science of the way
light interacts with a
species, such as a chro-
mophore in red wine or
a body in the sun, is
called photochemistry.
426
SPECTROSCOPY AND PHOTOCHEMISTRY
The first law of pho-
tochemistry states
that only light that is
absorbed can have any
photochemical effect.
This principle is so simple that it has been given the title the
first law of photochemistry, and was first expressed by Grotthus
and Draper in the early 19th century. They stated it as the (hope-
fully) obvious truth: 'Only light that is absorbed can have any
photochemical effect'.
The primary interaction occurring during a photochemical pro-
cess is between light and an analyte, such as a molecule, ion, atom,
etc. The reaction is often written in the generalized form
M + light
M*
(9.1)
The light contains energy, so Equation (9.1) alerts us to the truth that the body M
has acquired energy during the photochemical reaction. We sometimes say that the
product is a 'hot' atom, ion, etc. for this reason. This energy cannot just be 'tacked
on' to the body M, but is stored within it in some way, so the asterisk against the
product shows that it is formed in an excited state.
These simple concepts underlie the whole of photochemistry.
Why is red wine so red':
Photon absorption: the excited state
The colour of a chro-
mophore is seen only
after light passes
through it, or through
a solution containing it.
The first thing we see when pouring a glass of red wine is its beauti-
ful colour. It seems almost to glow. The colour is more impressive
still if we hold the glass up to the light and see the way light
streams through the wine it contains. Yet we could see no colour if
the room was dark: the wine does not emit light. Indeed, we only
see the colour after the light has passed though the solution. We
see the colour following transmission of the light.
The colours we see around us are each a consequence of light.
Particles of light - we call them photons - enter the wine, pass
through it, and are detected by the eye only after transmission. We
deduce that the action of the chromophore in imparting its colour
is due to the light interacting with the chromophore during the
transmission of light through the wine.
The fundamental feature of photochemistry that separates it from
other branches of chemistry is its emphasis on an excited state. A
coloured chromophore, be it a molecule ion or atom in solution, has
electrons within the bonds which characterize it. These electrons
usually reside in the orbitals of lowest energy. We say the molecule
is in its ground state if it is in its lowest, unexcited state.
The only orbitals we are interested in are the highest occupied molecular orbital
(HOMO) and the lowest unoccupied molecular orbital (LUMO). Collectively, they
We see a colour because
a chromophore inter-
acts with light.
A particle
of electro-
magnetic
radiation
(light) is called a pho-
ton.
INTRODUCTION TO PHOTOCHEMISTRY
427
are often referred to as the frontier orbitals. We can usually ignore
all the other orbitals.
We have already seen that photons of light interact with the
chromophore in solution. In fact, the energy locked up within the
photon is wholly taken up into an electron within the chromophore.
In its place, there exists an electron of higher energy. We say the
photon is absorbed by the chromophore, and its energy enables the
excitation of an electron from a ground-state orbital to an orbital of
higher energy, i.e. from the HOMO to the LUMO, as represented
in the schematic drawing in Figure 9.2.
We say the molecule is in its first excited state. It is possi-
ble to excite into the second excited state, which is of an even
higher energy.
The term ground state
is easy to understand
and then remember:
a ball thrown into the
air will descend, and
come to rest at a posi-
tion of lowest potential
energy. Inevitably, this
usual position of lowest
energy is resting on the
ground.
Why are some paints red, some blue and others black?
Absorption at different wavelengths: spectra
The light we see by eye is termed visible light. Normal, everyday
light is also termed 'white' light, and is a mixture of different
colours. We can readily demonstrate this composite nature of light
by 'splitting' it as it passes through a prism to generate a spectrum.
If white light (e.g. from the sun) passes through a bottle of blue ink,
then some of the photons are absorbed by the chromophore that
makes it appear blue, so the light transmitted (i.e. the remainder that is not absorbed)
only contains the colours that are not absorbed.
Some light is absorbed,
and the remainder is
either transmitted or
scattered; see later.
LUMO
New HOMO
Energy
increment
+ photon
HOMO
Figure 9.2 Schematic representation of a photon being absorbed by a single molecule of chro-
mophore. The photon causes excitation of an electron (depicted by a vertical arrow) from the
HOMO to the LUMO
428
SPECTROSCOPY AND PHOTOCHEMISTRY
'Spectrum' is a Latin
word, so its plural is
spectra, never 'spec-
trums'.
Whatever its acquired
meaning, to a scien-
tist the word spectrum
means 'a range of pho-
ton energies'.
We should be aware from elementary physics that there exists a
continuum of photon energies, with radio waves at one extreme,
which have a tiny energy per photon, through to gamma rays with
a massive energy per photon. We term this continuum of energies a
spectrum. Figure 9.3 is a reminder of the different photon energies
expressed as wavelengths in a spectrum. Because a particle can also behave as a wave
(see p. 431 below), Figure 9.3 also indicates the wavelength A of each photon.
The word 'spectrum' is generally employed in a clumsy way
in everyday life - we might talk about a 'spectrum of opinions',
meaning a range or spread. To a scientist, the word means a spread
of photon energy. As an excellent example, consider the colours
formed when white light passes through a prism: the range of
colours indicates a range of photon energies from 171 kJmol -1
(red light at 700 nm) to 342 kJ mol" ' (violet light of A = 350 nm).
The frequencies on a radio-tuning dial are another example.
The word 'spectrum' also means a graph depicting photon intensity (as 'y') against
photon energy (as 'x'). The discipline of obtaining a spectrum is termed spectroscopy,
and will be discussed in more detail in Sections 9.2 and 9.3.
Figure 9.4 shows a spectrum of methyl viologen in water. The
x-axis of the spectrum covers the wavelength range 220-700 nm.
The range 190-350 nm is the ultraviolet (UV) region (see Fig-
ure 9.3), and 350-700 nm is the region visible to the human eye.
Accordingly, Figure 9.4 is most usually described as a UV-visible
spectrum.
The function on the y-axis of the spectrum in Figure 9.4 is the
absorbance (as defined on p. 441). Absorbance is also called opti-
cal density or optical absorbance in older books; these three terms
each mean the same thing. We can see from the spectrum that more
light is absorbed at 300 nm (in the near infrared) than at 500 nm
Graphs such as
Figure 9.4 are often
called electronic
spectra because the
optical absorption
is caused by the
excitation of electrons
between various
molecular orbitals.
-t-
Visible
10 10 10 8 10 6 10 4
+
"t
■+-
io 1 icr 1 icr 3 icr 5
Radio Micro- Far NIR
waves waves |R
10 2
1 icr 2 io-
■+-
1-7
Xlnm
A./m
10^' icr M icr 11 icr 1
vacuum
uv UV X-rays and y-rays
700
620
580
530 470 420 350 nm
Red Orange Yellow Green Blue Violet
Figure 9.3 A continuum of photon energies exists from radio waves through to y-rays. We call
it a 'spectrum'. The visible region extends from 350 to 700 nm
INTRODUCTION TO PHOTOCHEMISTRY
429
220 270 320 370 420 470
Wavelength i/nm
520
570
Figure 9.4 UV- visible spectrum of methyl viologen in water at a concentration of
10~ 3 moldirT 3 . Methyl viologen compound is the active ingredient in many weedkillers, and has
the IUPAC name l,T-dimethyl-4,4'-bipyridilium dichloride
(in the visible region). The maximum amount of light absorbed is
light of wavelength 260 nm. We often call the peak in this spec-
trum, a spectroscopic band, and the wavelength at which it appears
is termed the wavelength maximum, which we give the symbol
A.( mas ). The value of A.( max ) in a spectrum corresponds to the very
tip of the peak in the spectrum.
The spectrum in Figure 9.4 shows that only light within the range
200-300 nm is absorbed. It follows, therefore, that if light of these
wavelengths is absorbed, then such light cannot also be available
to be 'seen', i.e. to enter the eye. In fact, we only see the remainder
of the light - that which is not absorbed. We say that the colour
observed by the eye is the complementary colour to that absorbed.
For example, removal of red light makes a thing appear blue.
We often call the peak
in this spectrum, a
spectroscopic band.
The word 'spectros-
copy' comes ultimately
from the Greek
skopein, which means
'to watch'. The word
'spectrum' derives from
the Greek metrein,
meaning to measure.
Worked Example 9.1 The permanganate ion absorbs green light (A( max ) = 500 nm).
Why does it appear to the eye to be purple?
Light that is green occurs at the centre of the visible region of the spectrum. If photons of
green light are removed by absorption, then the major colours remaining will be at the two
extremes of the spectrum, at red and blue. The colour we see following removal of green
light will, therefore, be a mixture of blue and red, which the eye perceives as purple.
Why can't we see infrared light with our eyes?
Types' of photon
When sitting on a beach in the sun, or standing in front of an oven, we often refer
to 'waves of heat' hitting us. A thermometer would tell us that our temperature was
increasing. In fact, this heat comprises photons of infrared light, which is why we
430
SPECTROSCOPY AND PHOTOCHEMISTRY
Red light has the
longest wavelengths
in the visible por-
tion of the electro-
magnetic spectrum.
Longer wavelengths
are, therefore, beyond
the red, which, from
Latin is infrared.
We experience pho-
tons of infrared light
as heat.
The visible light having
the shortest wave-
length is violet in
colour. Light hav-
ing an even shorter
wavelength is there-
fore 'more than' or
'beyond' violet which,
from Latin, is ultra-
violet.
sometimes see a heater advertised as an 'infrared lamp'. Although
we cannot see the heat with our eyes, nevertheless the photons of
infrared radiation strike our bodies and cause us to feel a sensation
that we call 'heat'.
The reason why we can see the sun when we look up at the
sky on a clear day is because it emits photons of visible light.
Furthermore, photons of UV light may be 'seen' by the way they
cause our skin to acquire a darker colour following irradiation,
because a tan forms when UV light reacts with melanin in the
skin (see further discussion, later). Again, we notice photons of
radiowaves because the radio beside us on the beach plays music
and relays the latest news to us.
Infrared, visible, UV and radio-wave forms of light each com-
prise photons, yet of different types, which explains why each is
experienced in a different way. Table 9.1 lists the various types of
photon, together with typical wavelengths and applications.
We 'experience' heat, visible light, UV and radio waves by the
way they interact with our thermometers, our eyes, skin and our
radio sets respectively. This is a tremendously important concept.
Photons of infrared light are experienced as heat. The photons
that cause photochemical changes in the retina at the back of the
eye are termed 'visible'. These photochemical reactions in the eye
generate electrical signals which the brain encodes to allow the
reconstruction of the image in our mind: this is why we see a scene
only with visible light - indeed this is why we call it 'visible'.
Just as a golf ball will go down a golf hole and a football cannot,
and cotton can go through the eye of a needle yet string cannot, so
each mode of 'experiencing' these types of light depends on the
Table 9.1 Summary of the different types of photon, and ways in which each is experienced in
everyday life
Name
Source
Typical X/m
Application
y-rays
Nuclear decay
<io- 10
Some forms of medicine
(radiotherapy)
X-rays
High-energy electron
10-8-10-'°
X-rays for detecting broken
collisions
limbs
Ultraviolet
The sun, mercury
vapour lamps
4x l(T 7 -l(r 8
Forming a sun tan
Visible
The sun, incandescence
4 x l(T 7 -7.5 x 1(T 7
All forms and applications of
vision
Infrared
The sun, combustion
7.5 x 10- 7 -10- 4
Warmth; remote-control
handsets
Microwaves
Electron excitation
lfr^-Kr 1
Mobile phone receivers;
microwave ovens
Radio waves
Electron excitation
10 '->10 2
Radar; radio transmission
INTRODUCTION TO PHOTOCHEMISTRY
431
energy of the respective process. Each type of detection has been tailored within the
natural world to respond to only one type of photon.
So we cannot see infrared light even when looking at photons
coming off a heat source because photons of infrared light do
not interact with the chemicals at the back of the eye (see later
example), unlike photons of visible light.
But we must be careful with our words: when we say these are
different 'types' of photon, we do not mean that they are different
in kind, only in magnitude. In kind, all photons are identical: each
comprises a 'packet' of energy, which is termed a quantum, because
the amount of energy in each photon is quantized (or 'fixed').
Whereas the amount of energy per UV photon is fixed, the amount
of energy per type of photon can vary enormously.
The amount of energy
per particle of light
is fixed. We say it is
quantized.
'Quantum' is another
Latin word, so its plu-
ral is quanta, never
'quantums'.
Wave- particle duality
There is a mind-blowing paradox at the heart of all discussions about light. Light is
a form of energy that exhibits both wave-like and particle-like properties. In other
words, a photon is simultaneously both a wave and a particle. We can never fully
understand this paradox, but will merely say that extremely small particles exhibit a
duality of matter.
Figure 9.5 shows a wave of light. It is a form of electromagnetic radiation, with
properties that arise from the electric and magnetic waves oscillating sinusoidally at
right angles to each other. The waves clearly have a 'wavelength' X. Also, when we
think of light as a wave, we must recognize that it travels (propagates) through space
or another medium in a straight line, so it has a direction. The speed of the light is
very fast: all light from y-rays through to radio waves travels at 'the speed of light',
which is usually given the symbol c. Light travelling through a vacuum has a speed
c = 3 x 10 8 ms" 1 .
Light in the visible region of the spectrum is also characterized by its colour, which
is a straightforward function of its wavelength X (see Figure 9.6), so X for green light
is 510 nm, and X for orange light is 590 nm.
Figure 9.5 Light can be thought of as waves of frequency v, speed c and wavelength X. The
frequency is c -h X
432
SPECTROSCOPY AND PHOTOCHEMISTRY
7.5
i i ' i i ~i i i i I i i i r
420 440 460 480 500 520 540 560 580 600 620 640 660 680 700
Wavelength A/nm
Figure 9.6 The colours comprised within 'white' visible light
Worked Example 9.2 How many waves of microwave radiation pass through food per
second in a microwave oven?
In a microwave oven, the microwave 'light' typically has a wavelength of ~12 cm. From
simple mechanics, speed c, wavelength X and frequency v are related as
c — X x v
(9.2)
Frequency and wavelength are inversely proportional, so a longer wavelength propagates
at a lower frequency. If we assume that the speed of light is the same through food as
through a vacuum, then c has the value 3 x 10 8 ms~'.
First, we rearrange Equation (9.2) to make frequency v the subject:
v — c/X.
Second, we convert the wavelength to the SI unit of length, the
metre. The necessary relationship is 1 m = 10 2 cm, so 1 cm = 10~ 2 m.
Accordingly, 12 cm = 12 x 10 -2 m.
Third, inserting values into the rearranged equation yields
Note that the wave-
length is first converted
to metres (which is SI).
One cycle per second
is termed one hertz
(H 3 ), so 2.5 x 10 9 Hz is
2.5 gigahertz (GHz).
3 x 10 8 ms" 1 o ,
v = — = 2.5 x 10 9 s" 1
12 x 10~ 2 m
We see how 2500 million waves of microwave radiation pass through
the food each second.
SAQ 9.1 What is the wavelength of a wave of light having a frequency
of 10 12 Hz?
INTRODUCTION TO PHOTOCHEMISTRY
433
But light is also a particle. Some properties of light cannot be
explained by the wave-like nature of light, such as the photoelectric
effect and blackbody radiation (see Section 9.4), so we also need
to think of light comprising particles, i.e. photons. Each photon has
a direction as it travels. A photon moves in a straight line, just like
a tennis ball would in the absence of gravity, until it interacts in
some way (either it reflects or is absorbed).
And each photon has a fixed energy E. We say the energy is
quantized. The intensity of a beam of light is merely a function of
the number of photons within it per unit time; see below.
Only one photon at a time may interact with matter. This means
that the energy available to each recipient atom or molecule is the
same as the energy possessed by the single photon with which it
interacts. This truth was refined by Stark and Einstein, who called
it the second law of photochemistry: 'If a species absorbs radia-
tion, then one particle (molecule, ion, atom, etc.) is excited for
each quantum of radiation (photon) that is absorbed' .
We will modify and expand this idea when we discuss quantum
yields, in Section 9.2.
The intensity of light is
often expressed as the
number of photons per
second, which is termed
the flux.
The second law of pho-
tochemistry says that if
a species absorbs radi-
ation, then only one
particle is excited for
each photon absorbed.
The photon must have
sufficient energy to
cause the photochemi-
cal changes in a single
molecule, ion, atom,
etc.
How does a dimmer switch work?
Photon flux
The lights in a baby's bedroom, or in a sophisticated restaurant, are dimmed in the
evening, off during the day and bright when visibility needs to be maximized, e.g.
during cleaning. A dimmer switch modifies the brightness of a light bulb, i.e. dictates
the amount of light it emits.
A dimmer switch is merely a device that alters the voltage
applied to the filament in a light bulb. A higher voltage allows
more energy to enter the bulb, to be emitted as a greater num-
ber of photons of visible light. A lower voltage means that less
energy can be emitted, and the light emitted is feeble, with fewer
photons emitted.
A dim light emits fewer photons than does a bright light. So a
dimmer switch is merely a device for varying the flux of photons
emitted by a light bulb. The energy per photon is not significantly
altered in this way.
A laser, with light of wavelength 1000 nm at a power of 100 W (i.e. 10 2 Js _1 ),
emits a flux of 5 x 10 21 photons per second.
In practice, the dimmer
switch incorporates a
small variable resistor
into the switch box,
which allows differing
amounts of voltage to
be 'tapped off'.
434
SPECTROSCOPY AND PHOTOCHEMISTRY
Why does UV-b cause sunburn yet UV-a does not?
Energy per photon
It is easy to get burned by the sun while out sunbathing, because the second law of
photochemistry shows how each UV photon from the sun releases its energy as it
impinges on the skin. This energy is not readily dissipated because skin is an insulator,
so the energy remains in the skin, causes photo-excitation, which is experienced as
damage in the form of sunburn.
The best and simplest way to avoid burning is merely to avoid lying in the sun; but
if we insist on sunbathing, then we must stop the UV light from releasing its energy
into the delicate molecules within the skin. We can reflect the light - which is why
some sunglasses have a mirror finish - or we stop the light from reaching the skin,
by absorbing it first. For this reason, sun creams (see below) are employed to absorb
the UV light before photo-excitation of molecules within the skin can occur.
In recent years, advertisements have tended to sound more tech-
nological, and many now proudly claim protection against 'harmful
UV-b light'. What does this mean?
In practice, the term 'UV light' represents a fairly wide range
of wavelengths from 100 nm through to about 400 nm. Although
there is a continuum of UV energies, it is often convenient to
subdivide it into three 'bands'; see Table 9.2. UV-a comprises
wavelengths in the range 320-400 nm. Such light penetrates the
top, outermost layer of skin, causing photolytic damage to the lay-
ers beneath, as seen by premature aging, sagging, wrinkles and
coloured blotches. UV-a can contribute to the onset of skin cancer
(melanoma) but is otherwise relatively safe provided that suitable
protection is taken; see below.
UV-b radiation has wavelengths in the range 290-320 nm. It is
much more dangerous to the skin than UV-a because each photon
possesses more energy. In consequence, the photolytic processes
caused by UV-b are more extreme than those caused by UV-a. For
example, UV-b causes thermal degradation of the skin (we call
it 'sunburn') but, additionally, it inhibits DNA and RNA replica-
tion, which is why over-exposure to UV-b will ultimately lead to
skin cancer.
UV light in the wavelength range 100-290 nm is called UV-c.
Such UV light can generally be filtered out by ozone (O3) in the
The meaning of 'pho-
tolytic' may be deduced
from its Greek roots:
photo relates to light,
and the ending '-lytic'
comes from lysis, which
means to cleave or
split, so a process is
'photolytic' if splitting
or scission occurs dur-
ing irradiation.
We call the extreme
end of the UV-c range
(of X< 190 nm) the
vacuum UV. It's the
'vacuum' UV because
the 0=0 double bonds
in oxygen absorb UV-
a, -b or -c, causing
air to become opaque.
Although absorbed by
air, such photons read-
ily pass through a
vacuum (or gaseous
atmospheres contain-
ing only monatomic
gases).
Table 9.2 Classification of types of UV light
UV type
Wavelength range/nm
a
b
c
Vacuum
320-400
290-320
100-290
100-190
INTRODUCTION TO PHOTOCHEMISTRY 435
upper atmosphere before it reaches the Earth's surface. Exposure to UV-c is more
hazardous than UV-a or -b because of the greater energy possessed by each photon.
Incidentally, it also explains the current concern about ozone depletion, which allows
more UV-c light to reach the Earth's surface.
So, in summary, the extent of the damage caused by these different types of UV light
depend on the amount of energy per photon of light, itself a function of its wavelength.
The Planck-Einstein relationship
At a quantum-mechanical level, there is a simple relationship that ties together the
twin modes by which we visualize photons: we say that the energy of a photon
particle is E and the frequency of a light wave is v. The Planck-Einstein equation,
Equation (9.3), says
E = hv (9.3)
where h is the Planck constant. Within the SI system, h has the value 6.626 x 10~ 34 Js
if v is expressed as a frequency in cycles per second. The energy E is expressed
in joules.
Equation (9.3) is valid for all photons of all types. We see from Equation (9.3)
how the only difference between different types of photon is the variation in the
energy each comprises, causing the frequency to vary proportionately. Photons of high
energy, such as y-rays and X-rays, are characterized by very high frequencies and high
energy, whereas radio waves are characterized by low frequencies and low energy.
Worked Example 9.3 What is the energy per photon of a y-ray that has a frequency
of3xl0 18 Hz?
We recall that 1 Hz = 1
cycle per second.
Inserting values into Equation (9.3):
E = hxv = 6.626 x 1(T 34 J s x 3 x 10 18 s" 1
E = 1.99 x 1(T 15 J per photon
SAQ 9.2 What is the frequency of a photon having an energy per photon
of 10- 18 J?
Worked Example 9.4 What is the molar energy of the y -particle photons in Worked
Example 9.3?
In Worked Example 9.3, we calculated that each photon has an energy of 1.99 x 10~ 15 J.
To convert from the energy per particle to the energy per mole, we multiply the answer
by the Avogadro number, L — 6.022 x 10 23 mol~ :
Energy per mole = (hv) x L (9.4)
Energy per mole = (1.99 x 1(T 15 J) x 6.022 x 10 23 moP 1 = 1.2 x 10 9 JmoP 1
This energy is so large that we suddenly realize why y-rays are dangerous.
436
SPECTROSCOPY AND PHOTOCHEMISTRY
Furthermore, the magnitude of the energy in this answer helps explains why we some-
times wish to employ light to effect a reaction, because a chemical reagent simply does
not possess enough energy (see later examples).
SAQ 9.3 What is the frequency v of a photon having a molar energy of
l.OMJmor 1 ?
In practice, most chemists prefer to talk in terms of wavelength rather than fre-
quency. For this reason, we often combine Equations (9.2) and (9.3), and so obtain
a modified form of the Planck-Einstein equation:
Equation (9.5) shows
that a spectrum plotted
with wavelength X as 'x'
has its highest energies
on the left-hand side
and the lowest energies
on the right.
E =
he
T
(9.5)
Equation (9.5) is interesting, because it shows that the energy per
photon is inversely proportional to wavelength. The equation also
shows how our usual way of representing a spectrum, with wave-
length X along the x-axis, is illogical: it is more usual, in physical
chemistry, to have numbers increasing from left to right along the
x-axis. We see from Equation (9.5) that if we plot wavelength A
(as 'x'), then in fact the highest energies are on the left-hand side
of the spectrum and the lowest energies are on the right.
Worked Example 9.5 What is the energy per photon of an X-ray of wavelength 5 x
10~ 10 m?
Many textbooks pre-
fer to cite 10 10 m as
1 Angstrom (symbol
A and pronounced as
'ang-strom')- Others
will cite it as 100 pm,
i.e. 100 x 10 12 m.
Inserting values into Equation (9.5):
E =
6.626 x 10~ 34 Js x 3 x 10 8 ms"
5x 10
-10
= 3.98 x 10'
m
This energy equates to a molar energy of 0.24 GJmol -1 , which is
simply massive, and helps explain why X-rays can also cause much
damage to the body. The energy per X-ray photon is so large that
radiographers who work with X-ray machines require complete body
protection.
He
)t
tect
ow does a suntan protect agains
Absorbance as a function of wavelength
Melanin is a com-
plicated mixture of
optically absorbing
materials.
The colour of our skin is caused by the presence of pigments (i.e.
chromophores), of which melanin is the most important. Melanin is
a complicated mixture of optically absorbing materials, formed as
an end product during the photo-assisted metabolism of the amino
acid tyrosine. Both are bound covalently to the surrounding proteins
within the skin, and other pigmented regions of the body.
INTRODUCTION TO PHOTOCHEMISTRY
437
Melanin from natural sources falls into two general classes. The first component
is pheomelanin (I), which has a yellow-to-reddish brown colour, and is found in red
feathers and red hair. The other component is eumelanin (which has two principal
components, II and III). Eumelanin is a dark brown-black compound, and is found
in skin, hair, eyes, and some internal membranes, and in the feathers of birds and
scales of fish. Melanin is particularly conspicuous in the black dermal melanocytes
(pigment cells) of dark-skinned peoples and in dark hair; and is conspicuous in the
freckles, and moles of people with lighter skins.
COOH
(I)
(II)
HOOC N
COOH
Melanin is formed by a photochemical reaction, so the concen-
tration of melanin within the human epidermis (the outer layer of
the skin) increases following exposure to photochemical reactions
with tyrosine in the skin. This increase is seen readily by the for-
mation of a suntan. This increase in the concentration of melanin
following exposure to the sun is more obvious for fair-skinned
people, although darker people usually tan more quickly because
melanin is produced more efficiently in their skins.
The enhanced pigmentation engendered by a suntan is advan-
tageous because the brown melanin pigment absorbs the UV rays
in sunlight, effectively filtering them out, and thereby generates a
barrier between the sensitive inner layers of skin and the harmful
effects of UV. It also explains why the indigenous people who live
at or near the equator need to have more melanin in their skins
than people living nearer the poles.
Unfortunately, melanin does not protect against all UV light, only
that of wavelengths longer than 370 nm. We can readily show this
The word 'epidermis' is
defined on p. 254.
A suntan is merely an
increase in the concen-
tration of melanin in
the skin.
A suntan forms as
nature's way of pro-
tecting the skin against
energetic photons.
438
SPECTROSCOPY AND PHOTOCHEMISTRY
A spectrum is obtained
with a device called a
spectrometer.
to be the case by placing a sample of melanin in the beam of a spec-
trometer, i.e. a device that obtains a spectrum. We have already seen
how a 'spectrum' is defined as a graph of intensity (as 'y') against
photon energy (as 'x'); see Figure 9.4. The spectrometer irradiates
the sample with light of a single frequency. The light beam passes
through the sample of melanin, during which some of the light is absorbed. The
spectrometer determines the exact decrease in intensity that is caused by transmission
through the sample at this wavelength, and then repeats the measurement with as
many other wavelengths as are needed.
The most common form of spectrum required by a chemist is a graph of the
absorbance A (see p. 441), plotted on the y-axis, as a function of wavelength A
(as 'x'). Such a spectrum obtained for melanin is shown in Figure 9.7. It shows
how melanin is not particularly effective for UV-c, thereby allowing some of the
harmful effects of the UV-c light to occur. But the large values of absorbance A
in the wavelength range 200-350 nm clearly show where melanin
absorbs most strongly, and confirms that the more harmful wave-
lengths of UV light are absorbed before they can reach the sensitive
tissues beneath the melanin-containing layer in the skin.
The spectrum of melanin components in Figure 9.7 is different
from that of methyl viologen in Figure 9.4. The latter spectrum
has a clear, symmetrical peak. We call it an optical band. By con-
trast, the spectrum of melanin components increases but never quite
reaches a peak. In fact, the spectrum only shows the side of a huge
optical band. We sometimes say that spectra like Figure 9.7 contain
a band edge.
The spectrum of mela-
nin has minimal optical
absorbances at wave-
lengths greater than
about 750 nm, sug-
gesting that it does not
protect us against the
heat from the sun (i.e.
infrared light).
200
300
400
500 600
Wavelength/nm
700
800
900
Figure 9.7 UV-visible spectrum of the skin pigment melanin. The spectrum contains two traces:
(a) eumelanin and (b) pheomelanin. Both component compounds protect the skin by absorbing
harmful UV light. All pigment concentrations were 1 mgdirT~
INTRODUCTION TO PHOTOCHEMISTRY
439
How does sun cream block sunlight?
Absorbing light
An albino does not have
the ability to generate
melanin, so their hair
is snow white, and
their skin looks pale
pink in colour because
the blood corpuscles
beneath its surface are
the only chromophore
they have.
We have just seen that melanin is quite efficient at stopping UV
light below about 370 nm. If the skin is damaged or the body can-
not generate its own melanin - which is the case for an albino -
then the UV will not be blocked. It is also common for young chil-
dren and babies to have insufficient amounts of melanin in their
skin to block the UV effectively. For this reason, babies should be
protected from direct sunlight.
Furthermore, following concerns over the way the ozone layer is
being damaged by man-made chemicals, some UV-c now reaches
the Earth's surface, particularly in Australasia and some parts of
South America. People living in such places are not adequately pro-
tected against the harmful effects of the sun's rays by the melanin
in their skin.
If insufficient natural melanin is available in the skin, then it
is advisable to protect ourselves artificially; the best form of pro-
tection is to apply chemicals to the skin, such as sun cream (or
'sunscreen'). The best sun creams are emulsions containing chem-
icals that absorb light in a similar manner to naturally occurring
melanin. The organic compounds benzophenones, anthranilates and
dibenzoyl methanes are good at blocking UV-a, and microfine zinc
oxide can also act as a 'physical blocker'. To block UV-b, the
recommended compounds are the salicylate or cinnamate deriva-
tives of camphor or /?-aminobenzoic acid, together with microfine
titanium dioxide (Ti02). Ti02 is the only permitted block against
UV-c. A good-quality sun cream will contain some of each com-
pound to protect against all UV-a, -b and -c.
Figure 9.8 depicts the spectra of a few of the more popular sunblocking compounds
contained within sun creams, and shows just how effective they are. Remember, the
UV region of the spectrum ends at about 400 nm.
Emulsions are dis-
cussed in Section
10.2.
There are now some
health concerns con-
cerning the safety of
these compounds, since
irradiation of TiC>2 or
ZnO causes the forma-
tion of radicals.
Absorbance and transmittance
We say that the beam
of light striking a sam-
ple is the incident light.
The 'transmittance' T of a sample is a quantitative measure of how
much of the light entering a sample is absorbed. (Transmittance is
also called optical transmittance, which means the same thing.)
The amount of light entering a sample is called the incident light.
A simple measure of the number of photons entering the sample
is the intensity /. The intensity of the incident light is conveniently symbolized as
I . Similarly, the number of photons leaving the sample after some is absorbed is
440
SPECTROSCOPY AND PHOTOCHEMISTRY
1.6
1.4
1.2
<D 1 .
O '
C
CO
-e o.8
o
CO
< 0.6
0.4
0.2
0-
v....
--T.-
OMC
Octocrylene
Ti0 2
ZnO
i | i | i | i | i | i | i | i | i | i | i | i
290 300 310 320 330 340 350 360 370 380 390 400
Wavelength/nm
Figure 9.8 UV-visible spectra of some of the active (i.e. sun blocking) components within com-
mercially available sun creams. OMC = ethylhexyl methoxycinnamate
An optical measure-
ment determined at
a single, invariant
wavelength is said
to be monochromatic,
from the Greek roots
mono (meaning one)
and khroma (meaning
colour).
A transmittance of 100
per cent implies no
absorbance at all. A
sample that is wholly
opaque has a trans-
mittance of per cent
because 7 = 0.
best gauged by a new intensity, which we will symbolize as just
I, i.e. without a subscript.
The transmittance T of the sample is given by the ratio of the
intensities of light entering the sample (7 ) and leaving it (/):
I
T
(9.6)
The measurement of an optical transmittance is only accurate if
I and I are both determined at the same, invariant wavelength.
We say the light source is monochromatic.
If no light is absorbed at all, then I = I and the transmittance is
unity. A large number of spectroscopists cite transmittance data as
a percentage. A transmittance of unity represents a transmittance
of 100 per cent, and a value of T of, say, 0.67 is 67 per cent. A
large transmittance indicates that few photons are absorbed, so the
amount of colour will be slight.
Worked Example 9.6 A sample of blackcurrant juice is strongly coloured to the extent
that six photons are absorbed out of every seven in the incident light. What is the trans-
mittance T of the sample?
If six photons in seven are absorbed, then only one photon in seven is transmitted. Inserting
values into Equation (9.6):
1
T = - = 0.14
7
So the transmittance T of the blackcurrent juice sample is 14 per cent.
INTRODUCTION TO PHOTOCHEMISTRY 441
SAQ 9.4 What is the transmittance T when 12 photons are transmitted
for every 19?
The transmittance T is a useful measure of how many photons are absorbed;
nevertheless, most spectroscopists prefer to work in terms of the 'absorbance' A.
(Absorbance is also called optical absorbance and optical density - each of these
three terms means the same thing.)
The absorbance of a sample is defined according to
A = log 10 (j) (9.7)
Values of absorbance obtained from Equation (9.7) are only valid if / and I are
determined at the same, single wavelength, so the absorbance should only be measured
with monochromatic light.
SAQ 9.5 A solution of beetroot juice is strongly coloured. The incident
beam shining through it I emits 4 x 10 10 photons per second, and the
emergent beam emits 10 9 photons per second. Calculate the absorbance
of the solution.
Comparing Equations (9.6) and (9.7), the relationship between absorbance A and
transmittance T is given by
A = log 10 Qr) = -log 10 T (9.8)
If less light remains following transmission through a sample then, from
Equation (9.7), / < I , and A > 1. It should be clear that the absorbance is zero
when the same number of photons enter the sample as leave it. A highly coloured
sample is characterized by a high absorption, so A is large.
On the other hand, the value of A cannot be negative because
it would be bizarre if more light came out of the sample than
entered it!
We should appreciate the consequences of the logarithm term in
Equation (9.7): an absorbance of A = 1 means that one in ten photons is transmitted
and an absorbance of A = 2 indicates a solution in which ten times more are absorbed,
so only one photon in 100 is transmitted. Similarly, an absorbance of A = 3 follows
if a mere one photon per 1000 is transmitted.
At this point we ought to hold our breath: if 999 photons per 1000
are absorbed (with only one being transmitted), then the detector
measuring the intensity of the transmitted light / needs to be excep-
tionally sensitive. Stated another way, an absorbance of A = 3 is
unlikely to be particularly accurate.
At the opposite extreme, if only one photon per 100 is absorbed
(a transmittance of 99 per cent), then the logarithm term in
An absorbance A can-
not be negative.
The best-quality absor-
bance data are obtained
when
the absorbance
A lies
0.75-
in the range
1.
442
SPECTROSCOPY AND PHOTOCHEMISTRY
Care: the words
'strong' and 'weak'
refer to the extent
to which a weak
acid dissociates - see
Section 3.1 - and not
concentrated and dilute
respectively.
Beer's law says
the
absorbance A
of a
chromophore
in
solu-
tion increases
in
direct
proportion to
its
con-
centration.
The relationship
summarized by
Equation (9.9) is only
a law if absorbances
are determined with
monochromatic light
(i.e. a single value
of X), and if the
chromophore does not
associate or dissociate.
Beer's law requires
monochromatic light
because the value of k
depends on X.
Equation (9.7) will magnify any errors. For these reasons, the best-
quality absorbance data are obtained when the absorbance A lies
in the range 0.75-1.
Why does tea have a darker colour if
brewed for longer?
■
Beer's Law
The best way to tell how strong we have made a cup of tea is to
look at its colour. A strong cup of tea has a darker, more intense
colour than does a weak one. In fact, some people, when asked,
'How strong would you like your tea?' will respond by saying
simply, 'Dark!'
Similarly, a glass of fruit cordial or even gravy looks darker if the
coloured compounds in it are more concentrated. This is a general
observation, and has been codified as Beer's law. the absorbance
A of a solution of chromophore increases in direction proportion
to its concentration c, according to
A = kc
(9.9)
where k is merely an empirical constant.
An alternative way of presenting this law is to note that a graph
of absorbance A (as 'y') against concentration c (as 'x') will be
linear. Figure 9.9 shows such a graph for the permanganate ion in
water as a function of concentration.
Equation 9.9 also suggests the (hopefully obvious) corollary that
an absorbing sample has no absorbance if its concentration is zero.
In other words, we have no absorbance if there is no sample in
solution to absorb, which is why the extrapolation of the line in
Figure 9.9 passes through the origin.
Why does a glass of apple juice appear darker when
viewed against a white card?
The Lambert law
Apple juice usually has a pleasant pale orange-yellow colour. But, if the glass is
held against a white card or curtain, the intensity of the colour appears to increase
significantly - it may look brown. It appears darker when held against a white card
because light travels through the glass and juice twice : once before striking the white
INTRODUCTION TO PHOTOCHEMISTRY
443
0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007
Concentration/mol drrr 3
Figure 9.9 An illustration of Beer's law: absorbance of solutions of permanganate ion MnOzt - as
a function of concentration. The optical path length / was 1 cm, and the wavelength of observation
was 523 nm
back card, and a second time after light reflects off the card, as shown schematically
in Figure 9.10. We say that the optical path length has doubled. This length is usually
symbolized as I.
A similar optical effect is seen at the swimming baths. The water at the deep
end appears to have a more intense blue colour than at the shallow end. Again, this
intensification of colour arises because the path length / alters. Again, the photograph
in Figure 9.11 shows the view around the coast of Hawaii. From the varying intensity
of the colour of the sea, it is easy to see which sections of the water are deeper.
Shallow parts of the sea appear as a lighter hue.
The absorbance A has long been known to be a function of path
length /, according to the Lambert law:
A = k'l
(9.10)
The law summarized
by Equation (9.10)
only holds for absor-
bances determined
with monochromatic
light.
where k' is merely another empirical constant (different from that
in Equation (9.9)). We see that the absorbance should increase with
/ while going from the shallow to the deep end of the pool. Also note the (hopefully
obvious) corollary, that an absorbing sample has no absorbance if its path length is
zero. In other words, we can have no absorbance if there is no absorbing sample to
pass through. Stated another way, Equation (9.10) tells us that a graph of absorbance
A (as 'v') against path length / (as 'x') will be linear, and pass through the origin.
444
SPECTROSCOPY AND PHOTOCHEMISTRY
Path length /
A ►
Glass of fruit juice
White,
reflective
card
Figure 9.10 Illustration of Lambert's law: the absorbance A of a glass of juice is proportional to
the optical path length /, so holding the glass against a white card makes its colour appear twice
as intense because the path length has been doubled. The width of the beam here is proportional
to its intensity
Figure 9.11 Part of the Hawaii coastline. The lighter areas indicate areas where the sea is
shallower
Why are some paints darker than others?
The molar absorption coefficient s
To the eye, cobalt blue and woad have a very similar colour. We say they have the
same hue. But woad is considerably paler in intensity, even when the concentration c
INTRODUCTION TO PHOTOCHEMISTRY
445
and path length / are the same. Cobalt blue is simply more intensely blue than woad.
We say that it has a larger 'molar absorption coefficient' e.
At heart, this greater intensity may be explained as follows. The ease with which
an electron may be photo-excited depends on the probability of successful excitation,
which itself depends on the likelihood of photon absorption. If the probability of
excitation in the woad was 20 per cent, then 20 from every 100
incident photons are absorbed (assuming each absorption results in
a successful electron excitation). By contrast, cobalt blue is more
intense because it has a higher probability of photon uptake, so
fewer photons remain to be seen, and the absorbance increases.
A few molar absorption coefficients are listed in Table 9.3. The
higher the value of s, the more intense the colour we see.
Older books often give
the name extinction
coefficient to the molar
absorption coefficient
e, or 'molar extinc-
tion coefficient'.
The Beer- Lambert law
Ink is a fluid or paste having an intense colour. Inks comprise a pigment or dye as
the chromophore dissolved or dispersed in a liquid called the vehicle.
The first inks for writing comprised soot suspended in water.
'India ink' was the first truly waterproof ink and is a dispersion of
carbon black, stabilized to stop sedimentation. India ink is still a
favourite drawing medium with draughtsmen.
The first patent for making coloured inks was issued in Eng-
land in 1772. In the 19th century, ink compositions became more
complicated, as drying agents were added to allow the production
of a wide variety of synthetic pigments for coloured inks. It was
not until the beginning of the early 20th century that ink-making
became a complicated chemical-industrial process.
Modern writing inks usually contain ferrous sulphate, together with a small amount
of mineral organic acid. The resulting solution is light bluish black, and results in
India ink is stabilized to
prevent sedimentation
by adding substances
such as shellac in borax
solution, soap, gela-
tine, glue, gum arabic,
and dextrin.
Table 9.3 Typical values of molar extinction coefficient e
Species
£ a /dnr mol cm
[Mn(H 2 0) 6 ] 2+
[Cr(H 2 0) 6 ] 2 +
[Cr(NCS) 6 ] 3 -
Trytophan
Tyrosine triphosphate
MnO<T
H0.04WO3
Methyl viologen b
0.04 (312) (a spin and Laporte forbidden transition)
10 (610) (a Laporte forbidden d-d band on the chromium)
160 (565) (a Laporte forbidden d-d band on the chromium)
540 (280)
1028 (430)
2334 (523) (a charge-transfer band)
5220 (950) (an intervalence charge-transfer band)
16900 (600) (an intervalence charge-transfer band)
a Wavelength of observation in nanometres indicated in parentheses.
b The radical cation of l,l'-dimethyl-4,4'-bipyridilium.
446
SPECTROSCOPY AND PHOTOCHEMISTRY
writing that is only faint coloured on paper. The molar absorption coefficient s is
small, so a more intense image requires writing over the same part of the paper a few
times, thereby increasing the path length / of chromophore.
To make the written image darker and more legible at the outset, additional dyes
are added to the mixture. Modern coloured inks, and washable inks, contain a sol-
uble synthetic dye as the sole chromophore, ensuring s is large and obviating the
requirement for overwriting.
Modern inks for printing are usually less fluid than writing inks. The composition,
viscosity, density, volatility, and diffusibility of ink may vary, but adding less sol-
vent has the effect of increasing the intensity of the colour, i.e. the absorbance is
proportional to the concentration c of the chromophore.
So a broad overview of ink technology shows how the intensity of colour (its
absorbance) is a function of three variables: molar absorption coefficient e, path length
I and chromophone concentration c. We have already met Beer's and Lambert's laws.
We now combine the two to yield the Beer-Lambert Law:
A = sic
(9.11)
We see how the proportionality factor in Lambert's law (Equation (9.9)) is sc, and
the proportionality factor in Beer's law (Equation (9.10)) is si.
9.2 Photon absorptions and the effect
of wavelength
Most of the previous section concerned UV and visible light. In this section we will
look in greater depth at the other common forms of light. From previous chapters, we
are now familiar with the concept that different physical and chem-
ical processes require differing amounts of energy. More specifi-
cally, it was shown in the previous section how the energies of
photons can also vary. In this section, we see how the energies of
different types of photon are manifested, and how their interactions
may be followed.
A bit of theory introduces this section. We then look first at
infrared light and microwave light. We will also need to look a bit
at the way bonds break following absorption of light, i.e. photolysis.
These ideas are summarized as Table 9.4.
Care: although we
give different names
to these photons, in
reality there exists a
continuum of wave-
lengths from 10 n m
(for y-rays) through to
hundreds of metres
(for radio waves).
The titles we use,
such as 'microwave'
or 'X-ray' are just his-
torical artifacts, and
merely describe the
way scientists first
encountered them.
Why do radical reactions usually require
UV light?
Photo-excitation and bond length
We have seen already how photons of UV light from the sun can
cause burning of the skin. UV photons often cause bonds to cleave,
PHOTON ABSORPTIONS AND THE EFFECT OF WAVELENGTH
447
Table 9.4 Types of photon, and their uses in spectroscopy
Photon
Parameter observed
Spectroscopic technique
y-rays
X-rays
UV
Visible
Infrared
Microwaves
Radio waves
Excitation of the atomic nucleus
Structure determination
Excitation of inner-shell electrons
Excitation of inner and valence
electrons
Excitation of valence and outer-shell
electrons
Excitation between vibrational
quantum states
Excitation between rotational quantum
states
Excitation between electronic spin
states
Excitation between nuclear spin states
Mossbauer
X-ray diffraction
X-ray photoelectron spectroscopy, XPS
UV- visible spectroscopy
UV-visible spectroscopy
Infrared spectroscopy
Rotational spectroscopy
Electron paramagnetic resonance, EPR
Nuclear magnetic resonance, NMR
thereby explaining why cancers of the skin occur if we are over-irradiated with
UV - particularly energetic UV-b or -c.
A majority of radical reactions require irradiation with UV light.
A simple example is the chlorination of methane (CH4), in which
CH4 and elemental chlorine are mixed and irradiated to yield a mix-
ture of chlorohydrocarbons, such as CH3CI and CCI4. The energy
for reaction comes from the UV photons. Diels -Alder and other
pericyclic reactions also require photons of light.
A typical value for bond energy in organic chemistry is
150 kJ mol -1 .
UV photons are ener-
getic enough to break
bonds. Photons of vis-
ible light cannot break
any but the weakest
of bonds.
Worked Example 9.7 What wavelength of light corresponds to an energy of
150 kJ mol -1 ?
We will perform a simple calculation in two parts. First, we will divide by the Avogadro
number to obtain the energy per bond. Second, we convert the energy to a wavelength X
with the Planck-Einstein relation (Equation (9.4)), E = hc/X.
To obtain the energy per bond, we divide 150 kJmol by L:
150000 J mol -1
E =
6.022 x 10 23 mol"
= 2.49 x 10 -19 J
Remember to con-
vert from kilojoules
to joules.
Next, we determine the wavelength, and start by rearranging the Planck-Einstein
relation to make wavelength A the subject:
he
~ ~E
448
SPECTROSCOPY AND PHOTOCHEMISTRY
The units involving J
and s cancel, leaving
a bond length with the
unit of metres.
Finally, we insert values:
6.626 x 10" 34 Js x 3 x 10 8 ms" 1
A
2.49 x 10
-19
800 nm
We rind that photons of wavelength 800 nm are capable (in prin-
ciple) of breaking a bond of energy 150 kJmol -1 . Photons of lower energy (which
have a longer wavelength) will not have sufficient energy. In practice, many chemi-
cal bonds are considerably stronger than 150 kJmol -1 (e.g. see Table 2.4), so bond
cleavage requires light of shorter wavelengths (and high energy).
As experimental chemists, we do not effect a photochemical reaction by irradiating
a reaction with photons having exactly the right amount of energy, but prefer to
supply an excess by irradiating with UV light.
Morse curves
Imagine we have two atoms of the element X - they could be bromine or chlorine.
Each is infinitely far from the other, so there can be no interaction between them.
Now imagine that they slowly approach from infinity, and we monitor their energy
all the time. If we were to draw a graph of their energy as a function of distance
r during their approach, we would generate something like Figure 9.12. We call the
graph a Morse curve.
Although the two atoms of X do not interact at extreme separations r, they do
interact when closer than a minimum distance, which is generally about twice the
normal bond length. At these closer distances, a result of the interaction between the
atoms of X is a decrease in the energy of the two atoms. We recall from Chapter 2
that all matter seeks to decrease its energy, so the lowering of the energy of our
two atoms suggests a willingness to get closer. The interaction causing the decrease
in energy arises from an overlap of the orbitals on the approaching atoms. In other
Electronic ground state
Internuclear distance r
Figure 9.12 Morse curve of the diatomic molecule X 2 in the ground state
PHOTON ABSORPTIONS AND THE EFFECT OF WAVELENGTH
449
The deep energy
mini-
mum in
a Morse
curve
is often
called
an
energy '
well'.
Remember that it is
electrons that hold a
bond together.
words, a bond starts to form. The decrease in energy on overlap is none other than
the first part of the bond energy being released.
The two atoms of X are brought yet closer in our thought exper-
iment, causing the energy to decrease until a minimum is reached.
The energy then rises steeply at short distances r, so a minimum
forms in the graph. The separation at the minimum represents the
preferred separation between the two atoms of X. A molecule has
been formed, which we will call X2.
The rise in energy at very short distances tells us that the two atoms resist any
further approach. They do not want to get closer than the minimum separation.
Another way of looking at the Morse curve in Figure 9.12 is to say it represents
the energy E (as 'y') of the two atoms of X as a function of their bond length r (as
'x'). The two atoms of X form a simple diatomic molecule in its ground state, i.e.
before it absorbs a photon of light.
The preferred bond length is r , because the value on the energy axis is lowest when
the internuclear separation corresponds to this value of bond length. The energy of a
bond having exactly this bond length is E .
Not all molecules of X2 will have the same energy, because
molecules are always swapping energy as a result of inelastic colli-
sions. Accordingly, a graph of energy against number of molecules
of X2 having that energy follows a straightforward Boltzmann type
of distribution, (see Section 1.4), so many molecules will have
more energy than E . Let us concentrate on molecules having a greater energy, which
we will call E\. From the graph it should be clear that a molecule having the energy
E\ will have a bond length between the two extremes of r\ and r[ . In practice, the
two atoms in a molecule of X2 having the energy E\ will vibrate such that the bond
length will vary between these two extremes of r\ and r[, as though connected by
a spring.
And a molecule of X2 having even more energy in its bond will
vibrate more strongly, causing more extreme variations in the bond
length. If the energy in the molecule's bond is enormous, then we
can envisage the (simplistic) situation in which the two atoms of X
are so far apart that the bond has effectively 'snapped', i.e. causing
bond cleavage. The energy of the bond when cleavage occurs will
necessarily correspond to the bond energy AH , so if we shine
light of sufficient energy on the molecule X2, then the molecule
will cleave owing to excessive vibration.
We used the word 'simplistic' in the previous paragraph because
we described the vibrations getting more and more violent, as
though there was no alternative to eventual bond cleavage. In fact,
there is a very straightforward alternative: absorption of a pho-
ton (i.e. energy) to X2 will also cause an electron to photo-excite,
as follows.
Look at Figure 9.13, which now shows two Morse curves. The
lower curve is that of the molecule in the ground state (in fact, it
The V in AH relates to
the bond dissociation
energy of the molecule
in the ground state.
We call it the 'first'
excited state to em-
phasize that an elec-
tron can be excited
further to the second
excited state; if the
photon energies are
vast, then excitation is
to the third state, etc.
450
SPECTROSCOPY AND PHOTOCHEMISTRY
First electronic excited state
AH,
j- Electronic ground state
AH
Internuclear distance r
Figure 9.13 Morse curves of the diatomic molecule X2 in its ground and first excited states
is the same as that in Figure 9.12), and the upper Morse curve shows the relationship
between energy and bond length in the first excited state of the molecule, i.e. in Xj.
Following photon absorption, an electron from the HOMO of X2 is excited from
the ground to the first excited state. The electronic excitation that occurs on photon
absorption is represented on the figure by an arrow from the lower (ground state)
Morse curve to the higher (excited state) curve. The time required for excitation of
the electron is very short, at about 10~ 16 s. By contrast, because the atomic nuclei
are so much more massive than the electron, any movement of the nuclei occurs
only some time after photo-excitation of the electron - a safe estimate is that nuclear
motion occurs only after about 10~ 8 s, which is 10 8 times slower.
The motion of the electrons is so much faster than the motion
of the atomic nuclei that, in practice, we can make an important
approximation: we say the nuclei are stationary during the elec-
tronic excitation. This idea is known as the Born-Oppenheimer
app roximation .
The Born-Oppenheimer approximation has a further serious con-
sequence. At the instant of excitation, the length of the bond in the
excited state species is the same as that within the ground state.
While close inspection readily convinces us that the two Morse
curves are very similar and have the same shape, it is important to
recognize that the equilibrium bond lengths differ.
The Born-Oppenheimer principle says that the atomic nuclei do not move during
the electronic excitation; only later will the excited state structure 'relax' to minimize
its conformational energy. An arrow represents the photo-excitation from the ground
state to the excited state structures. The requirement for the excited-state structure to
The atomic nuclei are
so much heavier than
the electrons that we
approximate and say
the nuclei are sta-
tionary during the
electronic excitation.
We call this idea the
Born - Oppenheimer
approximation.
PHOTON ABSORPTIONS AND THE EFFECT OF WAVELENGTH
451
The Franck- Condon
principle states that
the excited state is
formed with the same
geometry as that of
the ground state from
which it derived. The
transition is from the
ground state to the
excited state lying ver-
tically above it.
move only after the absorption explains why the arrow strikes the
upper curve at a different position than its minimum energy.
This idea may be summarized from within the Franck- Condon
principle. Because the atomic nuclei are relatively massive and
effectively immobile, the transition is from the ground state to
the excited state lying vertically above it. We say that the elec-
tronic excitation is vertical, which explains why the arrow drawn
on Figure 9.13 is vertical.
The molecule in its excited state rearranges after the photon is
absorbed, and rearranges its bond lengths and angles until it has
reached its new minimum energy, i.e. attained a structure corre-
sponding to the minimum in the upper Morse curve.
We see following consideration of Figure 9.13 the common sit-
uation whereby the bond length in the excited state is longer than
that in the ground-state structure. For example, the C=C bond
length in ethene increases from 0.134 nm to 0.169 nm following
photo-excitation from the ground to first excited state.
The second difference caused by photo-excitation is to decrease
the bond dissociation energy (see Figure 9.13). We will call the
new value of bond dissociation energy AH\ ('1' here because we
refer to the, first excited state.) This decrease arises from differences
in the localization of electrons within the molecular orbitals in the
ground and excited states.
Because the value of AH\ is smaller than AH , it becomes
more likely that photo-dissociation of molecular X2 occurs when
the molecule is in the excited state than in its ground state. This
decrease in AH also explains why photo-dissociation can some-
times be achieved following irradiation with light of longer wavelength (lower energy)
than calculations such as that in Worked Example 9.7 first suggest. Nevertheless,
tables of bond dissociation energy generally only contain data for molecules in the
ground state (i.e. values of AH rather than AH\).
So, in summary, photo-initiated chlorination of methane requires UV light to gen-
erate 'hot' atoms of molecular chlorine, which subsequently dissociate:
More subtle change
can occur to a mole-
cule's structure fol-
lowing photo-excita-
tion. For example,
the bond angle in
methanal (formalde-
hyde) increases, so
the molecule is flat in
the ground state but
bent by 30° in the
first excited state; see
Figure 9.14.
Cl 2 + hv
Cl 2 *
2C1*
(9.12)
Figure 9.14 The structure of methanal changes following photo-excitation, from a flat ground-state
molecule to a bent structure in the first photo-excited state
452
SPECTROSCOPY AND PHOTOCHEMISTRY
Why does photolysis require a powerful lamp?
Quantum yield
It is advisable to employ a high-power lamp when performing a photochemical reac-
tion because it produces more photons than a low-power lamp. Its flux is greater.
When we looked at the laws of photochemistry, we saw how the second law stated
the idea that when a species absorbs radiation, one particle is excited for each
quantum of radiation absorbed. This (hopefully) obvious truth now needs to be inves-
tigated further.
The 'quantum yield' is a useful concept for quantifying the number of molecules
of reactant consumed per photon of light. It may be defined mathematically by
=
number of molecules of reactant consumed
number of photons consumed
(9.13)
Its value can lie anywhere in the range 10~ 6 to 10 6 . A value of 10~ 6 implies that
the photon absorption process is very inefficient, with only one molecule absorbed
per million photons. In other words, the energetic requirements for reaction are not
being met.
Conversely, a quantum yield of greater than unity cannot be achieved during
a straightforward photochemical reaction, since the second law of photochemistry
clearly says that one photon is consumed per species excited. In fact, values of > 1
indicate that a secondary reaction(s) has occurred. A value of > 2 implies that the
product of the photochemical reaction is consumed by another molecule of reactant,
e.g. during a chain reaction, with one photon generating a simple molecule of, say,
excited chlorine, which cleaves in the excited state to generate two radicals. Each
radical then reacts in propagation reactions until the reaction mixture is exhausted
of reactant.
To help clarify the situation, we generally define two types of
quantum yield: primary and secondary. The magnitude of the pri-
mary quantum yield refers solely to the photochemical formation
of a product so, from the second law of Photochemistry, the value
of </>(primary) cannot be greater than unity.
The so-called secondary quantum yield refers to the total number
of product molecules formed via secondary (chemical) reactions;
its value is not limited.
The primary quantum yield <fi should always be cited together
with the photon pathway occurring: it is common for several pos-
sible pathways to coexist, with each characterized by a separate
value of 0.
As a natural consequence of the second law of photochemistry,
the sum of the primary quantum yields cannot be greater than unity.
Note that the overall
quantum yield is <t> and
the primary quantum
yield is <t>.
In fact, the term 'sec-
ondary quantum yield'
is a misnomer, since
it refers to chemi-
cal reactions rather
than photochemical
(and hence quantum-
based) processes.
PHOTON ABSORPTIONS AND THE EFFECT OF WAVELENGTH
453
Each electronic state
also has vibrational
sub-states.
Why are spectroscopic bands not sharp?
Vibrational and rotational energy
The energetic separation between the two Morse curves in Figure 9.13 is fairly well
defined. We recall from Chapter 2 how each molecule vibrates somewhat if the tem-
perature is greater than absolute zero, so the actual bond length for the molecule
varies as the two X atoms vibrate about their mean positions, as described above.
The extent of vibration depends on the temperature, and the range of vibrational ener-
gies will follow a straightforward Boltzmann-type distribution law, such as that seen
in Figure 1.9. For these reasons, there will be a small range of energies between the
two Morse curves in Figure 9.13, and we should, therefore, expect
a spectroscopic band to be relatively fine and narrow, reflecting this
small range of internuclear separations. In other words, a spectrum
of X2 should contain a narrow line, with a peak corresponding to
the separations between the two Morse curves (i.e. the transition
X 2 + hv -► X*).
Even a cursory glance at a spectrum will show quite how wide a
typical UV-visible spectroscopic band actually is. For example, the
band for the methyl viologen dication shown in Figure 9.4 is about
100 nm wide at half peak height. There is clearly an additional
variable to consider.
Figure 9.13 is not the whole story, since it only depicts the sim-
plistic situation whereby an electron is excited from the lower
Morse curve to a higher one. In reality, an electron in the ground
state is also described by vibrational energy levels (we sometimes
call them sub-states); see Figure 9.15. This finding applies to all
ground and all excited states, so any photo-effected change in elec-
tronic state also occurs with a change in vibrational energy level:
the electron excites from a vibrational energy level in the lower,
ground-state Morse curve, to a vibrational level in the Morse curve
for the first excited state.
The multiplicity of excitations possible are shown more clearly
in Figure 9.16, in which the Morse curves have been omitted for
clarity. Initially, the electron resides in a (quantized) vibrational
energy level on the ground-state Morse curve. This is the case for
electrons on the far left of Figure 9.16, where the initial vibra-
tional level is v" = 0. When the electron is photo-excited, it is
excited vertically (because of the Franck- Condon principle) and
enters one of the vibrational levels in the first excited state. The
only vibrational level it cannot enter is the one with the same
vibrational quantum number, so the electron cannot photo-excite
from v" = to v' = 0, but must go to v' = 1 or, if the energy
of the photon is sufficient, to 1/ = 1, v' = 2, or an even higher
vibrational state.
An electron excites
from a vibrational
energy level in the
lower, ground-state
Morse curve, to a
vibrational level in the
excited-state Morse
curve.
Instead of saying the
vibrational level v" = 0,
we could say that the
vibrational ^quantum
number' v" = 0.
We describe the vibra-
tional levels in the
lower (ground-state)
Morse curve with the
quantum number v",
the lowermost vibra-
tional level being v" =
0. The vibrational states
in the upper (excited-
state) Morse curve are
described by the quan-
tum number v'.
454
SPECTROSCOPY AND PHOTOCHEMISTRY
i
4 \=
3 \
2 ^
.^ Electronic ground state
4
v
►
Intemuclear distance r
Figure 9.15 Morse curve of a diatomic molecule X2 showing vibrational fine structure
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A .
1 .
1 I
A V
A .
J I
I
A L
i
k A
t
A I
A .
i
k
A i
i
J k
i t
A V
1 L
First excited state
(a)
(b)
(c)
(d)
Ground state
Figure 9.16 Schematic representation of the excitation of an electron from a vibrational level in
a ground-state configuration to vibrational levels in the first excited state. The ground-state electron
can have any vibrational sub-state: (a) v" = 0; (b) v" = 1; (c) v" = 2; (d) v" = 3
Alternatively, the ground-state electron could be in v" = 1. In which case it can
photo-excite to 1/ = 0, 1/ = 2, etc. but not v' = 1, because that would imply no change
in v during the excitation process. Such a situation is also shown in Figure 9.16(b).
The simple quantum-mechanical rule governing transitions between vibrational lev-
els is summarized by
Av = ±l,±2, ±3, etc. (9.14)
so we see how it is possible for molecules of sample in solution to absorb photons
of subtly different energy.
PHOTON ABSORPTIONS AND THE EFFECT OF WAVELENGTH
455
~T
T~
220
230
240 250
Wavelength/nm
260
270
Figure 9.17 UV-visible spectrum of benzene (6 x 10 4 moldm 3 ) in cyclohexane at 298 K,
showing vibrational fine structure
Figure 9.17 shows a spectrum of benzene (IV) in cyclohexane, which clearly shows
small 'peaklets' superimposed on a broader band (or envelope). These peaklets are
called vibrational fine structure. In benzene, they are caused by excitation from v" =
to v' = 1, v' = 2, etc. Consideration of Figures 9.15 and 9.16 suggests that excita-
tion from v" = to v' = 1 requires less energy than from v" = to v' = 2, so the
excitation v" = — > v' = 1 occurs on the right-hand side of the figure, i.e. relates to
processes of lower energy.
(IV)
Close inspection of Figure 9.15 shows how the separations
between the vibrational levels become progressively smaller as we
ascend the figure, going to higher energies. It becomes quite rare
for the energetic separation between the vibrational levels to be
quite as clear as that shown by benzene, so the peaklets are not
often discernible individually, and we only see a broad envelope
incorporating each of them. These spectroscopic peaks are quite
broad, being hundreds of nanometres in width.
It is quite rare for the
energetic separation
between the vibrational
levels to be quite as
clear as that shown
by benzene.
Why does hydrogen look pink in a glow discharge?
Photo-excitation to anti-bonding orbitals
Consider a light bulb containing hydrogen gas at a low concentration of, say, 30 Pa
(which is about 0.03 per cent of atmospheric pressure). A pale pink glow is seen when
456
SPECTROSCOPY AND PHOTOCHEMISTRY
a voltage is applied between two electrodes. The hydrogen molecules are excited to
form Hj, which immediately splits to form two hydrogen atoms. They contain excess
energy, and must emit energy to return to the ground state, which is achieved by the
emission of a photon. We see these photons as pink light.
We saw in Figure 9.15 how photon absorption leads to the excitation of an electron
from the ground state to the first excited state. It is usual for the excited-state structure
to form in a non-equilibrium state, so it must subsequently rearrange to achieve a
lower energy.
The hydrogen molecule comprises only two electrons and may
be described by just two orbitals. The ground-state orbital is a
bonding orbital, and the first excited state is an anti-bonding orbital.
Figure 9.18 shows a Morse curve for the hydrogen molecule. An
electron photo-excited to the first excited state is not in its minimum
energy state, so it will seek to undergo whatever physicochemical
processes are necessary to decrease the energy. The usual process is molecular
rearrangement, such as lengthening of the bond.
The upper Morse curve in Figure 9.18 has no minimum, but the
energy decreases monotonically with increasing bond length. In
other words, the lowest energy occurs at an internuclear separa-
tion of infinity. To decrease the energy of the first-excited-state H^
molecule, the bond length increases until the two hydrogen atoms
are too far apart for there to be any electron density between the
two nuclei: the bond breaks.
The first excited
state in the hydrogen
molecule is an anti-
bonding orbital.
Photo-excitation of the
hydrogen molecule
leads to bond disso-
ciation.
CO
First excited state
Internuclear distance r/10
Figure 9.18 Morse curve of a diatomic H2. Photo-ionization occurs following photo-excitation to
a high-energy state that has no bonding character
PHOTON ABSORPTIONS AND THE EFFECT OF WAVELENGTH
457
Why do surfaces exposed to the sun get so dusty?
Photo -ioniza tion
The top of a bookcase or tabletop exposed to the sun soon shows a thin surface of
dust. It's not just that sunlight makes the dust more visible; rather, it is because light
from the sun actually promotes the collection of the dust.
Look at Figure 9.19, which shows two Morse curves, one each for the ground and
first excited states. It is a fairly typical situation, except that the upper curve is quite
shallow (which is common). Excitation from the ground to the excited state proceeds
following the absorption of a photon.
Because the upper curve is so shallow, the electron enters the Morse curve for
the excited-state structure at an energy that is quite high, causing the bond length to
vibrate over a very wide range of lengths. At its maximum, the bond length r will
correspond to the horizontal plateau on the right of the Morse curve. Being horizontal,
there is essentially no difference in energy as the bond increases in length until there
is no electron density between the two atoms (or groups of atoms), at which point
the bond has broken.
The simplest form of such photo-dissociation is that when one
of the fragments is simply an electron:
Y + hv
-* Y*
-> Y+ + e"
(9.15)
in which case the dissociation process is more correctly termed
photo-ionization.
By convention, the
groups of atoms form-
ed during photo-dis-
sociation are usually
called fragments.
First excited state
Ground state
Intemuclear distance r
Figure 9.19 Morse curve of a diatomic molecule in which the first excited state has a relatively
shallow well
458
SPECTROSCOPY AND PHOTOCHEMISTRY
The table top acquires a
positive charge because
it was originally charge
neutral, and an elec-
tron was lost following
photo-dissociation.
Photo-ionization of
nitrous oxide in the
upper atmosphere only
occurs with light of
wavelengths shor-
ter than 134 nm, to
cause the reaction
NO + hv-> NO + + e-.
UV-photoelectron spe-
ctroscopy (UPS) and
X-ray photoelectron
spectroscopy (XPS)
are powerful analyti-
cal tools: the energy
of the photons required
to eject the electron
is characteristic of an
element and of its oxi-
dation state.
It is usual to employ photons of UV radiation if photo-ionization
is intended, because most ionization energies are relatively large.
Irradiation will cause some ionization. For this reason, a surface
left in the sun - like a tabletop placed in direct sunlight near a
window - will soon acquire a small positive charge. Dust is a
decomposition product from an organic -rich material (such as skin)
and contains lone pairs and other electron-rich moieties. These lone
pairs help the dust particle behave as though it had a slight nega-
tive charge.
We saw in Chapter 2 how Coulomb's law states that charges of
the opposite sign attract each other. We call this an electrostatic
attraction. In consequence, the charges on the dust and tabletop
are mutually attractive, so the dust no longer floats past the table
but forms a weak electrostatic bond with it. The amount of dust on
the table will, therefore, be much greater than would be suggested
by a mere statistical consideration of dust accumulation.
Photo-ionization of this sort is of profound importance in the
upper atmosphere, where photons of vacuum UV light are absorbed
and participate in reactions, e.g. with ozone and nitrogen oxides.
These reactions help explain why the Earth's surface is relatively
free of such harmful UV light, because the photons are absorbed
en route through the Earth's atmosphere.
Experimentally, we often want to ionize a molecule, e.g. for
photo-ionization spectroscopy. In which case, a sample is bom-
barded with energetic photons of UV (i.e. short wavelength) or
low-energy X-rays.
Why is microwave radiation invisible to the eye?
The photochemistry of vision
We have mentioned microwave ovens a few times already. Although we might employ
such an oven regularly, no one has actually seen microwave radiation. The photons are
wholly invisible. The phenomenon of 'sight' can be simplified to the photo-effected
transformation of a pigment related to retinal within the retina at the back of the eye.
Retinal is derived from vitamin A.
The outermost layer of the retina is the pigment epithelium (with supporting cells
for the neural portion of the retina). The epithelium is darkened with melanin (as
described above), thereby decreasing the extent of light scattering within the eye.
Directly beneath this layer is the bacillary layer (also called the rod and cone
layer), which comprises a layer of photoreceptor cells that respond to light. The
rods are highly sensitive, and monitor the intensity of incident light; the cones
are less sensitive, and relay information that is ultimately encoded within the brain
as colour.
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES
459
Figure 9.20 The photo-activated process that accounts for vision: the 11-cis^ 11-trans
photo-isomerism of retinal
The absorption of some
photons does not lead
to bond cleavage, but
to molecular rear-
rangements.
The active compound within the bacillary layer is retinal. To
simplify the photo-physics within the rods and cones hugely, ab-
sorption of a photon initiates a series of conformational changes
that lead ultimately to photo-isomerization of retinal from the li-
tis isomer to the 1 1 -trans isomer; see Figure 9.20. The uncoiling of
the molecule following photo-excitation triggers a neural impulse,
which is detected and deconvoluted by the brain. The photochemical reaction is
breakage and, after rotation, re-formation of the C=C bond.
Photons of microwave light do not possess sufficient energy to break the bond in
retinal, so the photochemical reactions in Figure 9.20 do not occur in the eye when
looking at microwaves. Without the photochemical reaction in the bacillary layer, no
electrical impulse is formed, and the brain does not 'see' the microwave radiation, so
it is invisible light.
9.3 Photochemical and spectroscopic
selection rules
We have now looked at the way photons are absorbed. Photons of UV and visible
light cause electrons to promote between orbitals. Infrared photons have less energy,
and are incapable of exciting electrons between orbitals, but they do allow excitation
between quantized vibrational levels. The absorption of microwaves, which are less
energetic still, effects the excitation between quantized rotational levels.
In this section, we shall look at the way these various absorptions are analysed
by spectroscopists. There are four kinds of quantized energy: translational, rotational,
vibrational and electronic, so we anticipate four corresponding kinds of spectroscopy.
When a photon is absorbed or generated, we must conserve the total angular momen-
tum in the overall process. So we must start by looking at some of the 'rules' that allow
for intense UV-visible bands (caused by electronic motion), then look at infrared
spectroscopy (which follows vibrational motion) and finally microwave spectroscopy
(which looks at rotation).
Why is the permanganate ion so intensely coloured?
Optical charge transfer
The permanganate ion, MnO/t - , has a beautiful, intense purple colour, and is a popular
choice for oxidation reactions. The colour is intense even if the solution is dilute.
460
SPECTROSCOPY AND PHOTOCHEMISTRY
Care: don't confuse
Group VII(a) (man-
ganese, technetium
and rhenium) with
Group VII(b), the halo-
gens.
Care: we write a for-
mal charge with Arabic
numerals, and means
that the full charge
exists as indicated:
Cu 2+ means a cop-
per atom with fully
two electronic charges
missing. We write an
oxidation number with
Roman numerals, and
does not relate to
any physical loss or
gain of electrons: it is
purely a 'book-keeping'
exercise. Mn vn does
not mean that a man-
ganese atom has lost
seven electrons.
The small increment
of charge S might be
as small as 0.1 elec-
tronic charges.
It is simple to determine the molar absorption coefficient s via
Beer's law as 2334 dm 3 mol _1 cm _1 at its A( max) of 523 nm (see
Table 9.3).
The valence of the central manganese atom in the permanganate
anion is +VII. Manganese is in Group VII(a) of the periodic table,
so a valence of VII is achieved by removal of all five d-electrons
and both s-electrons. The manganese in permanganate, therefore,
has an electronic configuration of d . But this fact ought to make
us think: if a colour is seen because electrons are photo-excited,
how can the Mn04~ ion have a colour, since it has no (outer-shell)
electrons? The answer is charge transfer.
A very naive view of the permanganate ion suggests the central
manganese has a +7 charge, and each of the four oxygen atoms is
an oxide ion of charge —2, i.e. the ion is held together with ionic
bonds. This view is wholly false, since the bonding is almost com-
pletely covalent, with overlap of orbitals. But because the central
Mn atom is d°, each of the bonds is dative, with both electrons per
bond coming from the oxygen atoms. In fact, it is probably safest
to say that the formal charge on each oxygen atom is —(2 — S), and
the charge on the manganese is therefore +45. This, then, is the
electronic structure in the ground state, as discussed from within a
valence bond model.
We will discuss the situation in terms of molecular orbitals,
which is a better model. We will, however, simplify it a bit by
talking as though there was a single Mn-0 bond. As usual, absorp-
tion of a photon photo-excites an electron from the ground state
to the first excited state. The molecular orbital for the ground state
has the central manganese with an effectively d° structure, but the
molecular orbital for the first excited state has a different distribu-
tion of electronic charge. This higher energy molecular orbital has
one electron on the central manganese, and so fewer electrons on
the oxygen. This situation is represented by
[Mn +x -0" (2 " ,5) ] + hv
ground state
-> [Mn+'-O -1 ]
first excited state
(9.16)
The system relaxes back to the starting materials (the reverse of Equation (9.16))
within a tiny fraction of a second.
In a normal photo-excitation process, e.g. on a chromophore, the electron does not
move spatially, but stays in the same physical location. Only its energy changes. The
situation for MnO/t - here is different: the electron not only changes its energy when
photo-exited but also its position, from the oxygen in the ground state to the central
manganese in the first excited state. We often say the electronic charge has trans-
ferred or, in short, that there is a charge-transfer process. In summary, although the
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES
461
permanganate ion is d , electrons can still be photo-excited, and hence the ion can
act as chromophore.
The reason why the colour of Mn04~ is so intense follows from the unusual way
in which the electron changes its position. There are no restrictions (on a quantum-
mechanical level) to the photo-excitation of an electron, so the probability of excitation
is high. In other words, a high proportion of the Mn04~ ions undergo this photo-
excitation process. Conversely, if a photo-excited charge does not move spatially,
then there are quantum-mechanic inhibitions, and the probability
is lower.
In fact, the charge-transfer process is wholly allowed, but the
more conventional transitions are only partially allowed. Indeed,
the probability of some photo-excitation processes is so low that
we generally say they are forbidden.
Charge transfer also occurs between ions in solution. The clas-
sical test for the Fe 3+ ion in solution is to mix solutions of Fe 3+
and thiocyanate ion SCN~, to form the [FeSCN] 2+ complex, and a
deep blood-red colour forms. The colour originates from a charge-
transfer transition between Fe 3+ and SCN~. There was no red
colour before mixing, confirming that the optical transition respon-
sible for the colour did not originate from either constituent but
from the new 'compound' formed.
To ascertain if an opti-
cal transition arises
from a charge-transfer
process: (i) prepare
two solutions, each
containing one half
of the charge-transfer
couple; (ii) mix the
two solutions. Charge
transfer is responsi-
ble if a new optical
band (and hence a new
colour) forms.
Why is chlorophyll green?
Metal-to-ligand charge transfer
Chlorophyll is a member of one of the most important classes of pigments involved
in photosynthesis, i.e. the process by which light energy from
the sun is converted into chemical energy through the synthesis
of organic compounds. It is found in virtually all photosynthetic
organisms, including green plants, certain protists and bacteria, and
cyanophytes (algae). The light energy absorbed by chlorophyll is
consumed in order to convert carbon dioxide into carbohydrates.
The chlorophyll molecule comprises a central magnesium ion surrounded by an
organic nitrogen-containing cyclic structure called haem (or 'heme'), (V) which is
based on a porphyrin ring.
The word root 'cyan'
in cyanophyte means
a 'blue-green' colour,
rather than cyanide.
462
SPECTROSCOPY AND PHOTOCHEMISTRY
Naturally occurring
compounds incorpo-
rating haem include
vitamin B 12 , haemo-
globin, and chlorophyll.
The central metal is
iron in haemoglobin,
magnesium in chloro-
phyll and cobalt in Bi 2 .
Both magnesium and porphyrin are essentially colourless, but
mixing solutions of magnesium and haem together causes a colour
to form. The magnesium ion resides at the centre of the haem ring,
which acts as a polydentate ligand.
The colour formed on mixing the two solutions, of magnesium
and haem, arises from a charge-transfer bond. The resultant spec-
trum contains a broad, intense band that we call a metal-to -ligand
charge transfer transition, or MLCT for short.
Why does adding salt remove a blood
stain?
A charge-transfer couple
•
Haemoglobin is an essential component of the blood. The central, active part of the
molecule comprises a haem ring (as above) at the centre of which lies an iron ion.
We call it a charge transfer couple, since it requires two constituents, both the iron
and the haem.
An old-fashioned way of removing the stain caused by blood is to place the stained
item in a fairly concentrated solution of salt, and is usually performed by sprinkling
table salt on the bloodied cloth after dampening it under a tap.
After a while, sodium ions from the salt swap for the iron ion at the centre of the
haem ring. There is no longer a couple (one component is lost), and consequently no
scope for an MLCT transition, so the red colour of the blood fades.
It is likely that the haem remains incorporated in the cloth, but it is not involved
in a charge-transfer type of process and is now invisible.
What
Mixed-valence charge transfer
The 'gold paint'
Other well-known com-
pounds that exist with
a mixed valence are
Prussian blue [Fe 3+ [Fe n
(CN) 6 4 -]], and ferre-
doxin, in which iron has
the valence of +11 and
+III.
with which lamp-posts and other ornamental metalwork are decorated
contains no gold, because of its prohibitive expense. The reflective
gold-coloured substitute is sodium tungsten bronze, Nao.3W03. So
why is Nao 3WO3 reflective?
We make Nao^WC^ by annealing solid WO3 in sodium vapour.
Metallic sodium is a strong reducing agent, and readily reduces
tungsten from W VI to W v , with each sodium atom becoming a
sodium ion. (The sodium ions are necessary to maintain overall
charge neutrality.)
If all the WO3 was reduced with sodium, the product would be
NaWC>3, in which the oxidation state of each the tungsten is +V.
But the stoichiometry of Nao.3 WO3 implies that only 30 per cent of
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES
463
the tungsten atoms have been reduced. A simple calculation to determine the oxidation
number of the tungsten atoms in Nao^WOs soon shows how the compound is not
electronically simple. In fact, the tungsten atoms exist in two separate valences, with
W v and W VI coexisting side by side. Nao^WOs is, therefore, termed a mixed-valence
compound, and shows intervalence behaviour.
Mixed valency of this sort is the cause of the reflective, gold
colour of Nao.3WC>3. In this system, like the MnCv - ion described
above, electrons are excited optically following photon absorp-
tion from a ground-state electronic configuration to a vacant elec-
tronic state on an adjacent ion or atom. The colour is caused by
a photo-effected intervalence transition between adjacent W VI and
W valence sites:
wY 1 + wl + hv
-> w + w
VI
(9.17)
The system relaxes back to the starting materials (the reverse of
Equation (9.17)) within a fraction of a second.
Such intervalence transitions are characterized by broad, intense
and relatively featureless absorption bands in the UV, visible or
near infrared (NIR).
Following the photo-excitation process, the electron moves spa-
tially as well as changing energy, so this is again a charge-transfer
process. Being a wholly allowed transition, the molar absorption
coefficient s is relatively large at 5600 dm 3 mol -1 cm -1 (which is
nearly three times more intense than for permanganate).
The labels 'A' and 'B'
merely serve to identify
two adjacent atoms.
Notice how the atoms
do not move spatially,
but an electron does
move in response to
the photo-excitation,
going from W B to W A .
Many books speak of
near (wavelengths
of 7 x 10~ 7 m to 4 x
10~ 4 m) and far (wave-
lengths of 4 x 10~ 4 m
to about 0.01 m)
infrared.
What causes the blue colour of sapphire?
Intervalence charge transfer
Sapphires are naturally occurring gem stones, and are transparent to translucent. We
know from ancient records that they have always been highly prized. Natural sapphires
are found in many igneous rocks, especially syenites and pegmatites. Alternatively,
synthetic sapphires may be made by doping aluminium oxide AI2O3 with a chro-
mophore.
The principal source of the sapphire's colour is the presence of small amounts of
iron and titanium. The relative ratio of these two contaminants helps explain why the
colour of sapphire ranges from a very pale blue through to deep indigo. The most
highly valued sapphires have a medium-deep cornflower blue colour. Colourless, grey,
yellow, pale pink, orange, green, violet, and brown varieties of the semi-precious
gem corundum are also known as sapphire, although red sapphire is more properly
called ruby.
But why is sapphire blue? Alumina is colourless, yet neither iron nor titanium
commonly form blue compounds. The colour arises from a charge-transfer type of
464
SPECTROSCOPY AND PHOTOCHEMISTRY
interaction. In the ground state, Fe 11 and Ti redox states reside close together. Fol-
lowing photo-excitation, an electron transfers from the Fe 2+ to the Ti IV :
Fe n + Ti rv + /jv
Fe 111 + Ti m
(9.18)
The frequency of the light absorbed to effect the photo-excitation is in the red and
near-infrared parts of the visible region, so the complementary colour seen is blue.
This explains why sapphire is blue. It is again a charge-transfer excitation, but not
of the mixed-valence type. The optical band formed is intense, so a strong colour is
seen even though the concentrations of iron and titanium are minuscule.
The system relaxes back to the starting materials (the reverse of Equation (9.18))
within a fraction of a second (probably 10~ 15 s).
Why do we get hot while lying in the sun?
Vibrational energy
Irradiation with infrared light causes the sensation of warmth, which
is why we sit next to a radiator when cold, and why our temperature
increases when we sunbathe on a sunny beach. Photons of infrared
light are absorbed. Whereas electrons are excited following irra-
diation with visible light, photons of infrared light do not possess
sufficient energy because their wavelengths are longer (remember
that E = hc/X). They cannot photo-excite electrons, but they can
excite the bonds, causing them to vibrate.
Photons of infrared
light are not suffi-
ciently energetic to
excite electrons, but
can excite between
quantum-mechanical
vibrational levels.
We have already seen how, on the microscopic level, the vibrational energies of
bonds are quantized in a similar manner to the way the energies required for electronic
excitation are quantized. For this reason, irradiation with an infrared light from the sun
or a lamp results in a photon absorption, and the bonds vibrate, which we experience
as the sensation of heat.
Mathematically, the movement of vibrating atoms at either end
of a bond can be approximated to simple-harmonic motion (SHM),
like two balls separated by a spring. From classical mechanics, the
force necessary to shift an atom or group away from its equilibrium
position is given by
Equation (9.19) is a
chemical version of
Hooke's law, and only
applies where the
Morse curve is para-
bolic, i.e. near the
bottom of the curve
where molecular vibra-
tions are of low energy
force = — kx
(9.19)
where x is the displacement of the atom from the position of equi-
librium and k is the force constant of the bond. The magnitude of
k reflects the elasticity of the bond, with a large value of k relating
directly to a strong bond.
Force constants for bond stretching are much larger than for bond-angle bending,
which are themselves considerably larger than for torsional vibrations. In consequence,
the energy needed for molecular distortion decreases in going from stretching to
bending to torsion. We can approximate this statement, saying that bonds do not
deform, angles are stiff and torsion angles are flexible.
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES 465
Molecules translate, rotate and vibrate at any temperature (except absolute zero),
jumping between the requisite quantum-mechanically allowed energy levels. We call
the common 'pool' of energy enabling translation, rotation and vibration 'the thermal
energy' . In fact, we can now rephrase the statement on p. 34, and say that temperature
is a macroscopic manifestation of these motions. Energy can be readily distributed
and redistributed at random between these different modes.
We describe as rigid-body rotation any molecular motion that leaves the centre
of mass at rest, leaves the internal coordinates unaltered, but otherwise changes the
positions of the atomic nuclei with respect to a reference frame. Whereas in a simple
molecule, such as carbon monoxide, it is easy to visualize the two atoms vibrating
about a mean position, i.e. with the bond length changing periodically, we may
sometimes find it easier to 'see' the vibration in our mind's eye if we think of one
atom being stationary while the other atom moves relative to it.
It is usually easier, mathematically, not to think in terms of wavelength A (which is
inversely proportional to energy) but to employ variables that are directly proportional
to energy. Most spectroscopists use co, which is the frequency of the vibration nor-
malized to the speed of light c, so co = v -4- c, where v is the frequency. In the context
of infrared spectroscopy, we usually call co the wavenumber of the band vibration.
The wavenumber of a bond vibration is given by
co=—J- (9.20)
In vacuo is Latin for 'in
or through a vacuum'.
where c is the speed of light in vacuo; k is the bond force constant
with units of N m _1 . This way, the wavenumber has the SI units
of m _1 .
The parameter /x in Equation (9.20) is called the reduced mass. For the carbon
monoxide molecule CO, /x is defined by
Mr x Mn , 7
H = — - - x 1.66 x 10" 27 (9.21)
M c + M
where Mr. and Mq are the relative atomic masses of the carbon and oxygen atoms
respectively. We need the additional factor of 1.66 x 10 -27 kg since this is the mass
of one atomic mass unit. With this conversion factor in place, we can express [i in
the SI units of kg molecule -1 .
Worked Example 9.8 What is the reduced mass of carbon and oxygen in the C=0
bond?
The relative atomic masses of oxygen and carbon are 12.00 and 16.01 respectively.
Inserting values into Equation (9.21) yields
12.00 x 16.01 „ ,,
fi = x 1.66 x 10" 27 = 6.858 x 10" 27 kg
12.00+16.01
Notice how /i is smaller than either Mq or Mq-
466
SPECTROSCOPY AND PHOTOCHEMISTRY
Because different bonds connect different atoms or groups of atoms, the energy
needed to excite the bonds varies in a characteristic way, so a C-H bond vibrates
following absorption of a specific energy, which is different from the energy of, say,
an NsN bond. Table 9.5 contains some force constants for a few diatomic molecules.
Clearly, single bonds such as Cl-Cl are weaker than double bonds, e.g. C=0. From
the magnitudes of the numbers in Table 9.5, the triple bond in nitrogen is the strongest,
as expected.
SAQ 9.6 What is the reduced mass of (1) chlorine, (2) hydrogen, and
(3) H-CI in the HCI molecule?
If we know the masses of the atoms involved in a vibration, and the wavenumber
can be determined from a spectrum, then we can readily calculate a value for the
force constant k.
Worked Example 9.9 A spectrum of pipe smoke contains a band at 2180 cm , which
is attributable to carbon monoxide. What is the force constant k for the C=0 bond?
We must first convert the wavenumber co to SI. If there are 2180 vibrations per centimetre,
and there are 100 cm per metre, then there must be 2180 x 100 vibrations per metre, so
<w = 2.18 x 10 5 m" 1 .
Second, we rearrange Equation (9.20) to make k the subject. After squaring both sides:
a? =
1
4jt 2 c 2 /a
Then, multiplying both sides by 4jt 2 c 2 (i yields
k — (4ir 2 c 2 /j,) x co 2
Finally, we insert values. (The value of /x for the C=0 bond was
calculated in Worked Example 9.8 as 6.858 x 10~ 27 kg.)
From Newton's second
law, the unit of kgs 2
equates to N rrr 1 .
k = 4x (3.142) 2 x (3 x 10 8 ms" 1 ) 2 x (6.858 x 10~ 27 kg)
x (2.18 x 10 5 m -1 ) 2
^ = 1158 kgs"
1158 NnT
SAQ 9.7 From Equation (9.20), calculate the wavenumber w of the vibra-
tional band for the fluorine molecule. The force constant k for F 2 is listed
in Table 9.5.
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES
467
Table 9.5 Force constants for some diatomic molecules
Molecule
H-Cl
H-F
Cl-Cl
F-F
0-0
N=0
C=0
N=N
Force constant k/~N m
516
964
320
445
1141
1548
1855
2241
2000 1500
Wavenumber/cm
Figure 9.21 Infrared spectrum of aspirin, i.e. 2-acetoxybenzoic acid. (Solid aspirin was prepared
within a Nujol mull)
What is an infrared spectrum?
Polyatomic molecules
Figure 9.21 depicts the infrared spectrum of aspirin (VI). The presence of several
peaks demonstrates how several infrared photon energies are absorbed while others are
not, so the spectrum shows how aspirin is transparent to photons of energy 2000 cm -1
(corresponding to a molar energy of 26.6 kJmol -1 ), but does absorb very strongly at
2900 cm" 1 .
O OH
468
SPECTROSCOPY AND PHOTOCHEMISTRY
A molecule containing
more than one atom
is termed polyatomic,
since 'poly' is Greek
for 'many'.
A series of vibrations is possible in a complicated molecule,
by which we mean one containing three or more atoms. As a
simple example, let us consider the molecule responsible for the
bleaching action of common household bleach, hypochlorous acid,
H-O-Cl. There are two bonds, H-0 and O-Cl, and each can
vibrate. Although the vibration of one bond is not wholly inde-
pendent of the other, we can usually approximate and treat the
vibrations as if they were independent. A single molecule can, therefore, have a wide
range of frequencies and k values.
In fact, to complicate the situation further, a single bond can have several values of
frequency because there are different kinds of vibration: the simplest bond vibration is
movement along the direction of the bond axis. This is called a stretch mode vibration,
since the bond length changes rhythmically while the bond angle remains unchanged.
A stretch of this sort is depicted in Figure 9.22(a).
Next are the vibrations in which the bond angle varies periodically but the bond
length remains constant. This is called a scissor-mode vibration, because it looks like
the action of a pair of scissors, as depicted in Figure 9.22(b).
O
■o
%/°
(a)
(b)
Figure 9.22 Simple vibration modes for carbon dioxide, 0=C=0: (a) a symmetric stretching
mode and (b) a 'scissor mode' vibration
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES
469
Remember: an infra-
red spectroscopist calls
these frequencies
wavenumbers.
The frequency of these two vibration modes will be different
because the energy required for each is different.
An infrared spectrum is a different way of displaying such infor-
mation, and represents a graph in which the function of energy
chosen for the x-axis is the wavenumber co (frequency). By con-
vention, the x-axis is usually expressed in units of cm -1 , with values decreasing in
going from left to right. Therefore, we say that the vibrations on the left-hand side
(at the highest co) represent vibrations of higher energy. As a good generalization, the
peaks in the left-hand side of the spectrum represent vibrations of the stronger bonds.
For example, a C=C double bond in a benzene ring will absorb photons of energy
20.3 kJmol -1 , which corresponds to a wavenumber of 1700 cm -1 .
Other atoms and groups of atoms have different masses (and hence reduced masses
/x), so they resonantly absorb infrared photons of different energies. Chemists usually
want to know which bonds a molecule is made up of, so they will obtain an infrared
spectrum by scanning over a wide range of energies, and look at which infrared
energies are absorbed. The energy (wavenumber) of each peak is characteristic of a
certain type of bond, so the chemist merely has to compare the peak energies from
the spectrum with wavenumbers in data books.
Why does food get hot in a microwave oven?
Rotational energy
Not everyone cooks with a conventional oven: many people pre-
fer to cook with a microwave oven, because it works faster and
consumes less power (and, therefore, is more efficient) than a con-
ventional oven. A microwave oven heats the food by absorption of
electromagnetic radiation from the microwave region of the spec-
trum. Such microwaves can be viewed as either radio waves of
very short wavelength or infrared light of longer wavelength. In
the case of a typical microwave oven, the frequency is typically
2.45 GHz, and is usually generated by a magnetron. Radio waves
in this frequency range are readily absorbed by water and by small
molecules having the same mass as water.
SAQ 9.8 Show that 2.45 GHz (2450 MHz) corresponds
to a wavelength of 12.2 cm.
Irradiation of food inside the microwave oven causes photon
uptake. The energy liberated each time a photon is absorbed is
not sufficient to cause bond breakage (as was the case with UV
light); nor can these microwave photons cause excitation of elec-
trons (which is why we see a colour during irradiation with visible
light but not with microwaves). Again, the energy is insufficient to
A microwave oven is
more efficient than
a conventional oven
because it only heats
the food without first
heating the oven itself.
Metals tend to reflect
microwaves, hence the
'lightning' seen when
we put a metallic item in
a microwave oven. The
microwaves' energy is
not absorbed by the
food, and is otherwise
absorbed in order to
ionize the (moist) air
inside the oven.
470
SPECTROSCOPY AND PHOTOCHEMISTRY
The food in a micro-
wave oven is effectively
steam -cooked because
the heating occurs
when water molecules
receive the energy of
the microwaves.
The overwhelming
majority of plastics
do not contain water,
nor do glass or ceram-
ics, which is why they
do not get hot when
placed within a micro-
wave oven.
cause molecular vibrations, so there is no vibration of the atoms and groups within
the molecules comprised by the food.
But there is a type of excitation that requires less energy, i.e.
rotation. Irradiation with microwave radiation causes small mole-
cules such as water to absorb energy, and thence rotate at high
speed. These absorptions are 'allowed', quantum mechanically.
Just as the absorption of UV or visible light causes electrons
to excite between different electronic quantum states, so absorp-
tion of infrared photons causes excitation between allowed vibra-
tional states, and absorbing microwave radiation causes excitation
between allowed rotational states in the absorbing molecule. As a
crude physical representation, these quantum states correspond to
different angular velocities of rotation, so absorption of two pho-
tons of microwave radiation by a molecule results in a rotation that
is twice as rapid as following absorption of one photon.
The molecules within the sample (e.g. the food in the microwave
oven) cannot continue to absorb more and more energy indefinitely,
so the energy must be dissipated somehow. In practice, the spinning
water molecules collide with other molecular features within the
topography of the food, resulting in inelastic collisions and the
transfer of energy. We discern this energy transfer as an increase
in heat (see Section 1.4), so the food gets hot.
The US Federal Communications Committee chose a frequency
of 2.45 GHz for microwave ovens to ensure that they do not inter-
fere with other equipment operated by microwaves, such as mobile
phones. Therefore, we see that, contrary to popular belief, a fre-
quency of 2.45 MHz is not a resonant frequency for molecules
of water.
We see how the microwave oven is merely an extremely effi-
cient means of warming anything containing water. Furthermore,
the water is evenly distributed throughout the food, so heating is
usually uniform: heating occurs everywhere all at once because the
water molecules are all excited at the same time. The process of
heating is different from that in a normal oven because we excite
the atoms rather than 'conducting heat'.
Defrosting food in a microwave oven is not very efficient, because
the molecules of water are incapable of rotation, and are 'locked' into
position in the form of ice crystals.
A molecule must have a permanent dipole moment to be micro-
wave active. As it rotates, the changing dipole moment interacts
with the oscillating electric field of the electromagnetic radiation,
resulting in absorption or emission of energy. This requirement
means that homonuclear molecules such as H2 are microwave inac-
tive, but heteronuclear molecules such as SO3, SO2, NO and, of
course, H2O are active.
Although 2.45 MHz
does not coincide with
the maximum micro-
wave absorption for
water, there is a huge
pressure broadening
effect at atmospheric
pressure, so there is
a sufficient overlap for
the absorption process
to be fairly efficient.
A conventional oven
cooks from the out-
side first, explaining
why the crust on the
outside of a loaf of
bread is crispy and
brown yet the inside is
soft and moist. Con-
versely, bread made in
a microwave oven has
no crust.
PHOTOCHEMICAL AND SPECTROSCOPIC SELECTION RULES
471
Rotational quantum numbers
Just as electronic excitations and vibrations are photo-induced, so
rotational energy is also quantized. The energy of a photon of
light is given by the Planck-Einstein equation, E = hv. When a
rotating molecule absorbs a photon of light, it is excited from one
quantum mechanically allowed rotational energy level to another
one of higher energy.
We tend to give the letter / to the rotational quantum states.
The rotational ground state has a rotational quantum number of J'
and the excited rotational quantum number is /. To be allowed
(in the quantum-mechanical sense), the excitation from J' to J
must follow
A J = ±1 (9.22)
Strictly, Equation (9.22) applies to diatomic molecules having a
permanent dipole moment.
We call Equation (9.22) the rotational selection rule. Equation
(9.22) assumes that the bond lengths do not alter during rota-
tion. For this reason, we call the rotating body a rigid rotor (or
'rotator'). In fact, all the bonds will stretch to some slight extent
as the speed of rotation increases, in consequence of centrifu-
gal forces, just as a spring stretches when rotating a weight on
its end.
The energies of rotation are quite small, so we require photons
of relatively low energy to photo-excite between rotational quan-
tum levels. For this reason, the spacings between rotational energy
levels correspond to transitions in the far infrared and microwave
regions of the electromagnetic spectrum.
We say, in the
quantum-mechanical
sense, an excitation
process is allowed if
the probability of it
occurring is very high.
If the probability is low,
we say the process is
forbidden .
Equation (9.22) assu-
mes that the bond
lengths do not alter
during rotation. For
this reason, we call
the rotating body a
rigid rotor.
We follow the rotational
behaviour of molecules
with microwave spec-
troscopy because the
spacings between each
rotational energy level
correspond to transi-
tions in the far infrared
and microwave regions
of the spectrum.
Are mobile phones a risk to health?
Microwaves and human health
Microwave research started in earnest during World War II, when the need to detect
and locate enemy aircraft at long distances, and at night, was crucial to the defence
of Allied forces. The ability of microwaves to cook food was first noticed in a
rather macabre way in 1945: at that time, army personnel operating the radar equip-
ment were experimenting with high-power equipment to enable communication over
longer distances. They noticed how the experiments occurred concurrently with dead
birds dropping to the ground. Closer inspection of the bird carcasses showed that
birds were cooked, or at least partially so. Microwave energy cooks flesh, even if
accidentally!
472 SPECTROSCOPY AND PHOTOCHEMISTRY
The possibility of microwaves being a hazard to health has been investigated exten-
sively, and much care has been spent on optimizing safety features, so it is now clear
that microwave ovens are completely safe.
The new generation of cordless phones transmit their text and verbal messages via
microwaves, prompting persistent rumours that mobile phone can cause a user's ear to
'cook', much like a pie in a microwave oven. Too few studies have been performed
to clarify, so the situation if still inconclusive. Certainly, the energy consumption
of a mobile phone is tiny compared with that in an oven (a microwave oven oper-
ates with a power of about 200-1000 W, whereas a phone operates at a maximum
energy of about 1 W - and the regulatory authorities are decreasing this maximum
quite rapidly).
But there are disturbing signs that mobile phone usage can be quite addictive. Some
researchers think the onset of addiction follows from minute changes deep within the
brain caused by 'cooking' tiny glands embedded there. Others prefer an explanation
from psychology or sociology - such as peer group pressure.
While some studies suggest that these phones operating via microwaves are safe,
others suggest that they are quite dangerous. In conclusion, whereas microwave ovens
are definitively safe (if used correctly), many medical reports now suggest we try not
to communicate via mobile phones too frequently.
9.4 Photophysics: emission and loss
processes
How are X-rays made?
Generation of photons
The transfer or conversion of energy is always associated with the emission of elec-
tromagnetic waves. We met this concept in its simplest form in Chapter 2, when
we looked at the transfer of infrared radiation (i.e. heat). This emission of photons
occurs because all objects contain electrically charged particles; and, whenever an
electrically charged particles accelerates, it emits electromagnetic waves.
All objects at temperatures above absolute zero contain some thermal energy, so
electrically charged particles within them continually undergo thermal motion. If we
could cool matter to K, then the thermal motion would cease and the matter would
not emit any radiation. Again, we saw this idea in Chapter 2.
But absolute zero is unattainable, so all particles move. Furthermore, the particles
never retain an invariant speed because inelastic collisions cause some particles to
decelerate and others to accelerate. As a result, everything emits some electromagnetic
waves, even if merely in the context of a dynamic thermal equilibrium with the object
exchanging energy with its surroundings.
PHOTOPHYSICS: EMISSION AND LOSS PROCESSES
473
The word 'gradient'
here implies the slope
of a graph of par-
ticle velocity (as V)
against time (as V).
The energy of an emitted photon depends on the magnitude of
the change in velocity, so a steep velocity gradient generates (or
requires) a more energetic photon. The intensity of the light emitted
is simply a measure of how many particles change their velocity.
An electron is readily decelerated to a standstill simply by firing
it through a vacuum toward a relatively massive object. The energy
of the photons emitted relates to the difference between the electron's kinetic energy
before and after the collision; so, if the electron is travelling extremely fast before the
collision and stops dead (has a zero velocity), then the energy change is substantial,
and the energy per photon is large.
We often go to hospital to obtain an X-ray photograph if we have a broken bone.
At the heart of the machine generating the X-rays, electrons are accelerated to a very
high energy, which corresponds to a fast velocity. These electrons are then smashed
into a large metal target. Rapid deceleration of the electrons occurs
simultaneously with X-rays emerging at an angle that is normal to
the angle of impact.
The frequency v of the X-rays relates to the energy change E on
deceleration according to the Einstein equation E = hv (Equation
(9.3)). The actual speed at which the electrons decelerate on impact
depends on the identity of metal with which the target is made: most
X-ray generators employ copper as the target. For this reason, we
often see the X-rays employed described as 'Cu Ka', where 'Ka'
merely relates to electronic processes occurring within atoms of
copper in the target.
We simplify the argu-
ment here: in fact, the
energy E of the photon
relates to the frequency
v of the photon and
also the work function
4> of the metal, accord-
ing to the equation
E = hv-d>.
Why does metal glow when hot?
Black body radiation
Warming iron or steel to a temperature of about 500 °C causes it to glow a dull red
colour, as seen on an electric cooker set at 'low'. The oven ring appears bright orange
if the temperature increases further (~1000°C). In these kitchen items, an electric
current inductively heats a coil of wire.
Pure iron melts at 1532 °C, at which temperature the molten iron glows white-
yellow. Further heating to about 2500 °C causes the colour to change again to brilliant
white. In short, all the colours of the visible spectrum are represented. Even before the
iron begins to glow red, we can feel the emission of infrared light as the sensation of
heat on our skin. A white-hot piece of iron also emits ultraviolet radiation, as detected
by a photographic film.
But not all materials emit the same amount of light when heated to the same
temperature: there is a spectral distribution of electromagnetic waves. For example, a
piece of glass and a piece of iron when heated in the same furnace look different: the
glass is nearly colourless yet feels hotter to the skin because it emits more infrared
light; conversely, the iron glows because it emits visible as well as infrared light.
474
SPECTROSCOPY AND PHOTOCHEMISTRY
The correct explanation
of black-body radiation
was an early triumph
of quantum theory.
This observation illustrates the so-called rule of reciprocity: a body radiates strongly
at those frequencies that it is able to absorb, and emits weakly at other frequencies.
The heated body emits light with a spectral composition that
depends on the material's composition. That observation is not the
case for an 'ideal' radiator or absorber: ideal objects will absorb
and thence re-emit radiation of all frequencies equally and fully.
A radiator/absorber of this kind is called a black body, and its
radiation spectrum is referred to as black-body radiation, which
depends on only one parameter, its temperature, so a hotter body absorbs more light
and emits more light.
Strictly, a black body is defined as something that absorbs photons of all energies,
and does not reflect light. Furthermore, a black body is also a perfect emitter of light.
A black body is a theoretical object since, in practice, nothing behaves as a perfect
black body. The best approximations are hot objects such as red- or white-hot metals.
How does a light bulb work?
The emission of light
At heart, a normal light bulb operates via the resistive heating of a metal filament.
It's not particularly efficient, and a bulb only lasts about 1000 h at most. Its poor
efficiency follows because the bulb simultaneously radiates a lot of infrared heat as
well as light; in fact, only about 20 per cent of the energy emitted is light that is
visible to the eye. Since the purpose of a bulb is to generate light, this heat represents
a waste of energy. The bulb does not last long because the tungsten of the filament
evaporates and deposits on the glass. The filament eventually breaks when it becomes
too thin, and we say that it has 'burnt out'.
Why is a quartz- halogen bulb so bright?
Wien's law
Like a conventional light bulb, a quartz-halogen bulb contains a
tungsten filament. The first major difference between the two bulbs
is the encapsulation of the tungsten within a small quartz envelope.
Because the envelope is so small, the distance between the quartz
and the filament is tiny, so the quartz gets very hot. Glass would
melt, so we need the higher melting temperature of quartz.
The second difference is the gas inside the bulb. Inside a normal
light bulb, the gas is usually argon, but the gas inside the quartz
halogen bulb is iodine vapour at low pressure, which has the ability to combine chem-
ically with tungsten vapour. When the temperature is sufficiently high, the halogen
This type of bulb is
called a quartz-halo-
gen lamp because
iodine is a halogen ele-
ment (from Group VII
of the periodic table).
PHOTOPHYSICS: EMISSION AND LOSS PROCESSES
475
gas combines with tungsten atoms as they evaporate from the filament, and redeposits
them on the wire. This process of recycling enhances the durability of the filament.
More importantly, the addition of iodine makes it possible to operate the filament
at a hotter temperature, with a higher proportion of the emitted light being visible
light, although we still form much heat.
We start to discern a dim glow if a surface is heated to about 400 °C in a darkened
room. We see the dull red of a heat lamp after heating to about 500 °C. A conven-
tional light bulb emits light when a coiled tungsten wire is heated to a temperature
of about 2500 °C, and incandesces. The yellow glow of the sun relates to a temper-
ature of 5800 °C. A blue light requires an even hotter temperature. We deduce the
generalization that the light emitted becomes progressively more blue light as the
temperature increases.
Figure 9.23 depicts the distribution of the wavelengths emitted by the bulb. When
heated electrically to about 2500 °C, the coil becomes 'white hot' and emits a great
deal of visible light, although A( max ) is about 1000 nm. The wavelengths emitted
lie in the range 500 to 6000 nm. Quartz -halogen bulbs operate at a temperature of
3000 °C, causing the maximum wavelength to shift to 800 nm. If the temperature
were to increase to 4000 °C, the wavelength corresponding to the maximum intensity
of photons would lie on the upper wavelength limit of the visible, at 700 nm.
Wien's law offers a simple relationship between the wavelength
maximum A( max ) and the thermodynamic temperature T:
*■ (max)
T = constant
(9.23)
Wien's law is also
called Wien's displace-
ment law.
Visible
region 2000 K
2000
4000
A./nm
Figure 9.23 Wien's law and black-body radiation: as the temperature T of the black body is
raised, so the wavelength maximum of the emitted radiation decreases. The area under the curve
indicates the intensity of the energy emitted by the black body, and is proportional to T A
476
SPECTROSCOPY AND PHOTOCHEMISTRY
The filament in a light
bulb is made of tung-
sten because its melt-
ing point is so high.
Operation of the bulb
near its maximum
temperature enables
the value of A. (max) to
shift to shorter wave-
lengths, i.e. closer to
the sensitivity of the
human eye.
The shape of each curve in Figure 9.23 is the same, but the
maximum is higher as the temperature rises, indicating how more
energy is releases at higher T. This shows a second reason why
quartz-halogen bulbs are brighter: more energy is emitted (such a
lamp generally operates at about 300 W, rather than the 100 W in
a more conventional bulb).
What is limelight'?
Incandescence
'Limelight' is a term from the theatre of the 19th century. Thomas
Drummond introduced the first theatrical spotlight in 1816. He was
so successful that limelight soon became a popular term, meaning
'appearing on stage, and being in the public eye'. Drummond' s light consisted of
narrow rods of calcium oxide housed within small burners along the edge of the
stage. Each rod was roasted strongly in jets of burning oxygen and hydrogen.
When something produces light on heating, we say it incandesces. In fact, anything
will glow when heated sufficiently, but the amount of light emitted depends quite
markedly on the material chosen. For example, steel is a fairly efficient producer of
light, but glass is very poor. Heating calcium oxide to a temperature above about
1000 °C causes incandescence and a soft, yet brilliant white light is produced. With a
suitable arrangement of reflectors around the hot lime, this light - the limelight - was
projected onto the actors on the stage, and formed the first recorded type of spotlight.
We choose lime because it has a high melting temperature, so it
can be heated to a white glow without it melting. Pure iron melts
at 1532 °C and lime melts at around 2500 °C. Heating glass causes
it to glow, but it gives off much less light than the same volume
of steel or lime.
Another everyday example of incandescence is the heating of a
metal wire to about 1000 °C in a conventional light bulb. The light
emitted from a candle or other form of fire is a further demonstra-
tion of incandescence.
The light emitted by a candle originates from hot particles of
soot in the flame; these soot particles strongly absorb and thence
re-emit it as visible light. By contrast, the gas flame of a kitchen
oven is paler, despite being hotter than a candle flame, and does not emit much light
owing to an absence of soot.
Lime is the old-fashio-
ned name for calcium
oxide. As a root, it is
also found in limestone
(calcium carbonate),
lime pits (burial sites
for the poor, which
were lined with CaO)
and quick lime (cal-
cium hydroxide).
Why do TV screens emit light?
Luminescence
Luminescence is the emission of light by certain materials when they are relatively
cool, in contrast to light emitted during incandescence, such as when a candle burns,
PHOTOPHYSICS: EMISSION AND LOSS PROCESSES 477
when rock is molten, or when a wire is heated by an electric current within a
conventional light bulb. Such luminescence is most commonly seen in neon and
fluorescent light tubes, and in most television screens (though some are now based
on liquid crystals).
The name 'luminescence' is generally accepted for all the light-emission phenomena
not caused solely by a rise in temperature (as above), but in fact incorporates the
different categories of phosphorescence and fluorescence, although the distinctions
between the terms is still in some dispute.
Electrons are accelerated by a large electron gun positioned behind the screen. Since
they are charged, the electrons are readily accelerated by means of a large voltage.
The electron has a considerable momentum by the time it slams into the inside of the
TV's front screen. In a black and white TV, the inside face is coated with a phosphor,
i.e. a substance that emits light. Electromagnetic coils (or electronically charged metal
plates) direct the beam of electrons from side to side and top to bottom, covering the
entire screen every twentieth of a second or so. Every phosphor that is struck by an
electron emits light; the light emitted by the screen is then perceived by the brain as
an 'image' or 'picture'.
A colour TV screen operates in the same way as a black and
white machine, except that the inside of the screen is coated with
thousands of groups of dots: each group consists of three dots, one
each for red, green and blue.
The kinetic energy of the electron from the electron 'gun' is
absorbed by the phosphor, and re-emitted as the visible light seen
by the viewer.
The groups of dots
are called 'picture
elements' - or pixels
for short.
Aside
Phosphors
The most important phosphors are sulphides and oxides of transition metals. The sul-
phides of zinc and of cadmium are the most important materials of the sulphide type.
An important condition of achieving a highly efficient phosphor is to prepare a salt of
the highest possible chemical purity. The emission of zinc sulphide can be shifted to
longer wavelengths by increasingly replacing the zinc ions with cadmium.
Sulphide-type phosphors are produced from pure zinc or cadmium sulphide (or
mixtures thereof) and heating them together at about 1000°C with small quantities
(0.1-0.001 per cent) of copper, silver, gallium, or other salts, which are termed activa-
tors.
Some oxide-type minerals have been found to luminesce when irradiated. A simple
example is ruby (aluminium oxide with chromium activator), which emits bright-red
light. The phosphors are incorporated into colour television screens to emit the colours
blue (silver-activated zinc sulphide), green (manganese-activated zinc orthosilicate), and
red (europium-activated yttrium vanadate).
478
SPECTROSCOPY AND PHOTOCHEMISTRY
Copper-activated zinc and cadmium sulphides exhibit a rather long afterglow when
their irradiation has ceased, which is favourable for application in radar screens and
self-luminous phosphors.
Why do some rotting fish glow in the dark?
Bioluminescence
In some parts of the world, one of the simplest tests of whether meat or fish has
gone rotten is to look for the emission of light. The test for healthy flesh is to ask
the question, 'Does it glow?' If the flesh does display a ghostly glow then it is
luminescent, and contains biochemical organisms that are hazardous to health.
Bioluminescence results from a chemical reaction, so it is more strictly termed
chemiluminescence. Biochemical energy is converted directly to radiant energy. The
process is virtually 100 per cent efficient, so remarkably little heat is generated during
emission. For this reason the emission is often called 'cold light', or luminescence.
This form of luminescence occurs sporadically in a wide range of natural organisms,
such as protists (bacteria, fungi), animals, marine invertebrates and fish. It even exists
naturally, albeit rarely, in plants or in amphibians, reptiles, birds, or mammals.
In most bioluminescent organisms, the essential light-emitting component is the oxi-
dizable organic molecule luciferin and the enzyme luciferase. The luciferin-luciferase
reaction is an enzyme-substrate reaction (i.e. as a catalyst; see Section 8.5) in which
luciferin is catalytically oxidized by molecular oxygen. Luciferin is oxidized to form
the ketone structure (VII), with emission of light occurring during the enzymic phase
of the reaction.
One of the
names
of the devil
in the
Hebrew Bible
is Lucifer,
and comes
From a
Hebrew wore
meaning
'light bearer'
Luciferin is one of the simplest examples of chemiluminescence,
and it is remarkably efficient. The overall yield is one photon per
molecule of luciferin.
The light emission continues until all of the luciferin has been
oxidized. This type of reaction is found in fireflies, Cypridina,
Latia, and many types of fish, such as lantern fish or hatchet fish.
PHOTOPHYSICS: EMISSION AND LOSS PROCESSES
479
How do 'see in the dark' watch hands work?
Phosphorescence
'See in the dark' watches operate with special material painted onto the watch hands,
which allows them to be seen even in the dark. The paint usually looks pale green
to the eye during daylight, and has a ghostly green-white colour at night. The watch
hands can be seen many hours after sundown. The paint is phosphorescent, even
though we often refer to it as 'fluorescent paint.'
Like fluorescence, phosphorescence is the emission of light by a substance pre-
viously irradiated with electromagnetic radiation. Unlike true fluorescence, however,
the emission persists as an afterglow after the exciting radiation has been removed.
The glow persists from about 10~ 3 s after the irradiation ceases, through to days or
even weeks. In effect, we can say that the light is stored within the phosphorescent
material. The length of time depends on the internal electronic levels involved.
Phosphorescence is different from the more straightforward process of fluores-
cence because an additional excitation occurs, which produces an emission of light
in the visible region. Figure 9.24 highlights the difference between
phosphorescence and fluorescence. Between the ground and excited
electronic levels is a band of intermediate energy, called a meta-
stable level (or electron trap). An electron in an excited level
can demote to the metastable level by emission of radiation or
by energy transfer (as below).
Because transitions between the metastable level and the ground
or excited levels are so slow, we sometimes say they are forbid-
den. Once an electron enters the metastable level, it remains there
until it can make a forbidden transition, in which case a photon is
released. The time of residence for the electron in the metastable
level determines the length of time that phosphorescence persists.
Glow-in-the-dark molecules possess such metastable quantum
states. The reason why the light release persists is that demotion of
an electron from the metastable state to the ground state can only
The quantum-mecha-
nical term forbidden
indicates that the prob-
ability of the event
or process is too tiny
for it to occur to any
significant extent.
+hv
Excited state
Metastable state
Ground state
A metastable state
in physics and chem-
istry is an energetically
excited state in which
an electron resides
for an unusually long
time. A metastable
state, therefore, acts
as a kind of temporary
'energy trap'.
Figure 9.24 Energy-level diagram for a luminescent species, in which a
metastable state slows the rate of emission. The metastable state is also
termed an ion trap
The relative distance of
a line, or level, above
a base line (the ground
level) denotes the
energy of an electron
occupying that level.
480
SPECTROSCOPY AND PHOTOCHEMISTRY
occur if one of the electrons changes its direction of spin. This change of spin in
these molecules is almost totally forbidden by the laws of physics, and proceeds with
a very slow rate constant.
The visible consequence of such a metastable state is phosphorescence: straight-
forward irradiation of such a material readily causes excitation to an excited state
and partial demotion to the metastable state. But the electrons are then trapped in
the metastable states. The faint glow demonstrates that a few electrons demote per
second; the glow persists for as long as it takes for all the electrons to reach the
ground state.
How do neon lights work?
Gas discharge lamps and photon emission
The French scientist Georges Claude was one of the first to experiment with different
types of lamp. In 1910, he first made a glass tube filled with low-pressure neon.
Within less than a decade, signs were being fashioned of glass tubes bent to form
words and designs that glowed red or green or blue when the gases inside them were
subjected to an electric field.
Figure 9.25 shows a typical neon lamp. It consists of a thin glass tube with an
electrode at either end. A neon light requires an extreme voltage, which is provided
by a so-called neon-sign transformer. Neon atoms are neutral, and cannot conduct
electricity, but ionization of the gas-phase neon forms Ne + ions (Equation (9.24)), so
the tube contains a mixture of electrons and ions:
Ne l
( g)
->Ne + (g) +e-
(9.24)
A plasma is a con-
ducting mixture of ions
and electrons.
We say the gas discharges under these conditions, meaning that it
can conduct, i.e. an electric current can flow through it.
Collisions involving the mobile electrons are generally elastic.
They bounce, like a ball off a wall. But a tiny fraction of the elec-
trons undergo an inelastic collision with un-ionized neon atoms, causing a fraction of
the electron's internal energy to transfer to the neon atom. The electron subsequently
moves away after the collision. It has less energy, and so is slower.
Glass tube
~\
Plasma of Ne + * and e
I-
Electrode
Figure 9.25 A neon lamp comprises two electrodes, one at either end of a thin glass tube. An
extreme voltage ionizes the neon to form a plasma of Ne + * and e~ . Inelastic collisions between
Ne* and Ne + allow for the release of energy as visible light
PHOTOPHYSICS: EMISSION AND LOSS PROCESSES
481
The product of the inelastic collisions is a 'hot' atom, Ne* :
Ne + energy from electron ► Ne* (9.25)
The reaction in Equa-
tion (9.25) occurs in
the gas phase, but
the '(g)' subscripts
have been removed
for clarity.
The excited-state neon atom is denoted with an asterisk, as Ne*.
This hot atom must emit energy to return to the ground state, via
the reverse of the reaction in Equation (9.25). The excited-state
structure of Ne* has an electron in a orbital of higher energy (which was previously
the LUMO). Subsequent relaxation of the electron releases a photon of light. It is this
emission that we see. A neon light emits a bright, pink-red glow.
Gas-discharge lamps similarly containing a small amount of the rare gas krypton,
emit a green glow; and helium-based lamps emit a pale blue glow. Lamps containing
argon emit light in the near infrared, so no colour is visible to the eye.
When we look closely at a neon lamp, we should see that it is the gas itself that
emits the light, and not the electrodes.
How does a sodium lamp work?
Initiating a discharge lamp
The first high-intensity sodium lamp was introduced in Europe in
1931. Figure 9.26 shows a schematic view of a sodium lamp; it
comprises a glass shell containing sodium vapour at low pressure,
metal electrodes to generate a current, and neon gas. The pressure
inside the tube is at a relatively low pressure of ~30 Pa, so some
of the sodium evaporates to become a vapour. The inner side of
the lamp is coated with the remainder of the metallic sodium as a
thin film.
An electrode is positioned at either end of the tube, and a large
voltage applied. When current first passes between the electrodes,
the neon is ionized to form a plasma, and starts to glow (as above),
which explains why a sodium lamp first emits a pink shade before
it glows with its characteristic orange colour.
Atmospheric pressure
is 1.01 x 10 5 Pa, so
the pressure inside
the tube is almost
a vacuum.
Sodium lamps glow
pink before orange
because of the neon
they hold, which 'kick
starts' the sodium
emission process.
Glass tube
i
° •
• e
I-
Electrode
Sodium vapour • Sodium cations e Electron
Figure 9.26 A sodium streetlight. The pressure inside the tube is the relatively low pressure of
~30 Pa, so some of the sodium evaporates to become a vapour; the remainder of the sodium lines
the inner side of the lamp as a thin metallic film. Soon after operation commences, 'hot' neon
atoms help generate a plasma of sodium
482 SPECTROSCOPY AND PHOTOCHEMISTRY
When the lamp gets hotter, energy from the neon is transferred to the remain-
der of the sodium, which then vaporizes. Atoms of sodium in the gas phase are
ionized to form a plasma of Na + and e~ . As with the neon lamp (above), some
inelastic collisions between electrons and neon atoms generate excited-state Na*,
which must return to the ground state with the emission of a photon (cf. the reverse
reaction to Equation (9.25)). The energy of this photon corresponds to light of wave-
length 592 nm, which is why the sodium lamp appears orange. The energy release is
extremely efficient.
The lamp above is more properly called a low-pressure sodium lamp. Such lamps are
ideal for street and road illumination, but the monochromatic nature of the emission
makes seeing in colour impossible. An adaptation which emits a range of colours
is the high-pressure sodium-vapour lamp, which is similar to that described above
but contains a mixture of mercury and sodium. Such lamps emit a whiter light and
are useful for extra-bright lighting in places such as road intersections, car parks and
sports stadia.
How do ^fluorescent strip lights' work?
Indirect discharge
Care: the two words
'fluorescence' and
'phosphorescence' are
employed interchange-
ably in everyday life,
but in fact relate to dif-
ferent photon
pathways.
Strip lighting in a classroom, hospital, business hall or kitchen is
often called fluorescent lighting, although in fact it is a phosphores-
cent process, as above. Each bulb consists of a thin, hollow glass
tube that is sealed at both ends. It contains gas such as helium,
argon or krypton, and a drop of liquid mercury (about 0.5 mg
of mercury per kilogram of lamp, or 0.5 parts per million). Like
the neon and sodium lamps above, the pressure inside the tube is
about 30 Pa, so the mercury evaporates to become a vapour. It is
the mercury that yields the light, albeit indirectly.
Again, like the sodium lamp above, application of voltage causes ionization and
a plasma of Hg 2+ ions and electrons soon form. An excited-state ion forms during
inelastic collisions between electrons and mercury atoms (cf. Equation (9.25)): this
excited-state ion must emit light to return to a lower energy state. A mercury lamp
emits this energy in the form of UV light. Because there are only two energy levels
that are energetically accessible (i.e. the ground and first excited state), the frequency
of the UV emitted is almost monochromatic at 254 nm.
Unfortunately, the emission of UV light is not useful for a lamp because our eyes
do not respond to UV. Even if they did, the glass of the tube is not transparent to
light of wavelength 254 nm. The light must, therefore, be converted to visible light.
To this end, the inside glass surface of the tube is coated with a thin layer of phosphor
which is bombarded with photons of UV light. Each time a photon is absorbed it is
re-emitted at a longer wavelength, which we see as visible light.
The reasons why the photon can be re -emitted at a different wavelength were
discussed above when we introduced the topic phosphorescence. The exact
OTHER OPTICAL EFFECTS
483
composition of the phosphor depends on the 'colour type' of the
bulb. Some phosphors emit a light that is almost blue, lending a
cold, almost clinical atmosphere to a room. More modern bulbs
have different phosphors, which emit a more natural light, i.e. with
more red and yellow frequencies. The most common colour types
are 'cool white', 'warm white', and 'deluxe cool white'. In each
case, the phosphors are a mixture comprising varying proportions
of calcium halophosphate, calcium silicate, strontium magnesium
phosphate, calcium strontium phosphate, and magnesium fluoroger-
manate. The crystals are doped with impurities such as antimony,
manganese, tin, and lead.
In fact, most 'neon'
lamps are mercury
lamps in which the
inside of the tube is
coated with a phos-
phor. To see if the
lamp is truly neon-
based, look at the bulb
before it glows: a real
neon lamp needs no
phosphor coating, so
the glass is clear and
without 'frosting'.
9.5 Other optical effects
Opalescence means
the ability to show
different colours, like
the gemstone opal. We
see different colours
when viewing from
different angles.
Why is the mediterranean sea blue?
Light scattering
The Indian physicist Sir Chandrasekhara Venkata Raman was on a
cruise on the Mediterranean Sea in 1921. Some reports suggest it
was his honeymoon. Others say the beauty of its deep blue opales-
cence captivated him. Whatever the reason, he dedicated the rest
of his life to understanding its colour and discovered the so-called
Raman effect.
The Raman effect occurs when a beam of light is deflected. When
a beam of light impinges on a sample of a chemical compound, a
small fraction of the light emerges in directions other than that of the
incident beam. The overwhelming majority of this scattered light is of
unchanged wavelength, but a small proportion possesses wavelengths
different from that of the incident light. This is the Raman effect, and
earned him the Nobel Prize for Physics in 1930.
Light impinging on the surface of the Mediterranean Sea is scat-
tered. Of this light, a small proportion is scattered in such a way
that the frequency changes, causing it to look more blue than was
the incident light. This shift in frequency causes the blue colour of
the Mediterranean Sea.
The Raman effect
The Raman effect relates to scattering of light. Raman found that illuminating a
transparent substance such as water causes a small proportion of the light to emerge
The sky is also blue
because of light scat-
tering.
Raman's studies of the
Mediterranean Sea's
colour led to the phrase
'blue-sky research',
because his work at the
time had no obvious
contemporary applica-
tion.
484
SPECTROSCOPY AND PHOTOCHEMISTRY
Raman spectroscopy
is an inelastic light-
scattering technique,
in which an analyst
directs a monochro-
mated laser beam onto
a sample, and deter-
mines the frequency
and intensity of the
scattered light.
Rayleigh scattering is
named after its dis-
coverer, John William
Strutt (1842-1919),
third Baron Rayleigh,
an English physicist.
He also did important
work on acoustics and
black-body radiation
The Raman effect is
tiny. At most, only
one photon per 10 5
collides in an inelas-
tic manner, the exact
number depending on
the energy (and hence
frequency) of the inci-
dent light.
at right angles to the direction of illumination. Furthermore, some
of this light has a different frequency from that of the incident light.
Consider a beam of incident light, each photon of which has
the same energy E (so it is monochromatic). These photons strike
the sample. Most collisions are elastic, so the energy (and hence
frequency) of the scattered photons does not change. Light that has
the same frequency as the incident beam is said to have undergone
Rayleigh scattering; see Figure 9.27.
But the collision is occasionally inelastic, so a molecule takes
or gives energy to a photon. These photons are scattered, and lose
energy to the sample, so the scattered light has a lower frequency.
The shift in frequency Av exactly matches the vibrational frequen-
cies of molecules in the sample V(vibration) ; so > by measuring the
frequency shifts between the incident and scattered light (v and
y (scattered) respectively), we obtain a graph of the proportion of colli-
sions that are inelastic, i.e. the intensity (as 'y') against vibrational
energy (as 'x'). We obtain a Raman spectrum. Figure 9.28 shows
a Raman spectrum of malachite pigment, the sample coming from
the famous Lucka Bible, which was made in Paris but now resides
in the Czech Republic, at Znojmo.
Figure 9.28 has the appearance of an infrared spectrum, but
the y-axis is absorbance (with peaks pointing upwards), whereas
infrared spectra are drawn with a y-axis of transmittance, with
peaks pointing downwards.
The Raman technique allows us to determine the intensity of
Raman bands, and thereby to quantify the concentration of the
chemical components in a complicated mixture (a Beer's law cal-
ibration graph of intensity against concentration is advisable; see
Section 9.1).
Raman scattered light,
of frequency v s - v - v vib
hv
-► Unchanged (transmitted) light
Rayleigh scattered light,
of frequency v
Figure 9.27 In Raman spectroscopy, light from a laser is shone at a sample. It is monochromated
at a frequency of v . Most of the light is transmitted. Most of the scattered light is scattered
elastically, so its frequency remains at v ; this is Rayleigh scattered light. Raman scattered light
has a frequency Scattered) = v o — v (vibration) • The sample is generally in solution
OTHER OPTICAL EFFECTS
485
1400 1200
Wavenumbers (cm"
Figure 9.28 Raman spectrum of green malachite pigment (basic copper(II) carbonate C11CO3 • Cu
(OH)2 taken from the Lucka Bible, now in the Czech Republic. The sample came from an initial
T from Genesis chapter 1, 'In principio . . .' (Tn the beginning . . .'). Reproduced with permission
by Professor Robin Clarke FRS, University of London
In general, though, Raman spectroscopy is concerned with vibra-
tional transitions (in a manner akin to infrared spectroscopy), since
shifts of these Raman bands can be related to molecular structure
and geometry. Because the energies of Raman frequency shifts are
associated with transitions between different rotational and vibra-
tional quantum states, Raman frequencies are equivalent to infrared
frequencies within the molecule causing the scattering.
Pure rotational shifts are small and generally quite difficult to
observe, unless the molecules are small and analysed in the gas
phase. Rotational motions are hindered for liquid samples, so dis-
crete rotational Raman lines are not observed.
The intensity of the
Raman lines is enhan-
ced significantly if a
light beam of high
intensity (or better,
a laser) is the inci-
dent beam.
Raman spectra may
facilitate qualitative
analysis, 'asking what
is it?' and also quanti-
tative analysis - asking
'how much?'
Do old-master paintings have a 'fingerprint'?
Raman spectra
A new, novel application of Raman spectroscopy has emerged recently. In the Middle
Ages, most artists prepared their own pigments, e.g. by grinding up coloured earth
and binding the powder with linseed oil to make 'burnt umber', or binding soot in
sunflower oil to make 'lamp black'. Since the ingredients to hand differed according
486 SPECTROSCOPY AND PHOTOCHEMISTRY
to geographical region, the chemical composition of the pigments differed from artist
to artist; so, analysing a painting's pigments can function almost like a 'fingerprint'.
Until now, many great paintings have been labelled as being 'by an unknown artist',
but Raman spectroscopy has been shown to be a powerful technique for identifying
the chemical constituents within the pigments of a painting or illuminated manuscript.
Raman spectroscopy is not destructive, so none of the Lucka Bible was removed prior
to analysing for the malachite shown in the spectrum in Figure 9.28, and none of the
page was damaged.
The compositions of pigments is also allowing us to put a precise date to a
manuscript, because inks and paints changed with time, becoming more sophisticated
as the Middle Ages came to an end.
10
Adsorption and surfaces,
colloids and micelles
Introduction
This chapter introduces the topic of adsorption, giving examples of both physical
adsorption and chemical adsorption, and discusses the similarities and differences
between the two. The standard nomenclature of surface science is given from within
this context. The energetics of adsorption are explained in terms of the enthalpies of
bond formation A//( a( j s ). Next, isotherms are discussed.
Colloids are introduced in the second half of the chapter. The various classifi-
cations of colloid types are discussed, together with ways of forming, sustaining
and destroying colloids, i.e. colloid stability. Finally, association colloids ('micelles')
are discussed.
10.1 Adsorption and definitions
Why is steam formed when ironing
a line-dried shirt?
Adsorption
We often generate steam when ironing clothes made of cotton. The steam forms even
if the cloth previously felt dry. But the steam is only formed if the shirt was dried
outside, e.g. on a clothes line. We see no steam if the shirt was
dried on a radiator before ironing.
Anyone who irons a lot will also notice how the texture of
the cotton changes following ironing: beforehand, the cotton felt
coarse and somewhat starched, but afterwards it feels more pliable
and softer.
Water (in the form of vapour) sticks readily to many surfaces.
We say it adheres or adsorbs. The cause of the adsorption is the
Substances partition
preferentially during
adsorption, leaving
the gaseous or liq-
uid phases and forming
a new layer on the
surface of a substrate.
488
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
We call the adsorbing
species the adsorbate.
An adsorbate is ato-
mic, ionic or molecular;
solid, liquid or gas.
The adsorbate adheres
to the substrate, which
is the layer beneath.
The substrate must
be a condensed phase,
either a solid or a liquid.
formation of a bond between molecules of water and a surface, such
as the carbohydrate backbone of the cotton fibre. To be a surface,
it must be a condensed phase, i.e. solid or liquid; clearly, a gas
cannot be a condensed phase. This underlying surface to which
the adsorbate is attached is termed the substrate (or, occasionally,
adsorbent).
The adsorption bond can be quite strong, and is characterized by
an energy known as the 'enthalpy of adsorption' A//( a( js). Since a
hot iron is needed to remove the creases in the cotton, we realize
that the iron must have supplied sufficient heat to the adsorbed
water to overcome and hence break the adsorptive bond joining
it to the underlying cotton substrate. The steam we experience is
adsorbed water released in this way.
A shirt dried previously on a radiator does not form steam while
ironing because the heat from the radiator was itself sufficient
to break the adsorptive bonds between the water and the cotton.
Any adsorbed water was lost before the process of ironing com-
menced.
Steam is again not formed when ironing synthetic fibres, such as
nylon or polyester. Several reasons explain the absence: firstly, the
temperature of the iron is usually much lower with such synthetic
fabrics, to avoid melting them. And we would not necessarily over-
come the enthalpy of adsorption A/f( ac j S ) with a less energetic iron.
But, secondly, the strength of the adsorptive bond formed between
water and artificial fabrics is smaller, so the thermal energy of
room temperature might be sufficient to dislodge most of the water
molecules before ironing commenced.
Also, in being a plastic, the surface of the nylon fibre is smooth,
in contrast to most natural fibres such as wool or cotton, which are
porous. Cotton, therefore, has a larger surface area. Since adsorption only occurs at
the surface of a substrate, there is likely to be more water adsorbed per unit mass of
cotton than on the same mass of nylon.
Figure 10.1 is a schematic representation of an adsorbate during adsorption onto
a substrate.
The word 'substrate'
comes from two roots:
the Latin stratum
means layer, and sub
is the Latin for 'less' or
'lower than'.
The word 'adsorption'
derives from the Latin
word sorbere, which
means 'to suck in'.
Adsorbate
Adsorptive bond of enthalpy Af/ (ads)
Substrate
Figure 10.1 Adsorption of an adsorbate onto a substrate. The charge necessary to form the adsorp-
tive bond comes from the charge centroid of the adsorbate
ADSORPTION AND DEFINITIONS
489
Why does the intensity of a curry stain vary so much?
Adsorption isotherms
An orange-brown stain forms on the inside of a pan while cook-
ing a curry. The colour represents molecules adsorbed from the
constituent spices of the curry.
A strong curry is generally dark red -brown in colour, whereas a
milder curry, such as biryani, is paler in hue. And the difference in
the intensity of the curry stain arises from the varying concentration
of the coloured components in the curry sauce. As an example, we
will discuss the red and spicy tasting compound capsaicin (I),
which is the cause of both the hotness of a chilli and contributes
to its red colour.
Care: note the spellings
here: ADsorption' does
not relate to the better-
known word ABsorp-
tion'. Absorption is the
uptake of something,
like a sponge taking
water into itself, or a
chromophore remov-
ing frequencies from a
spectrometer beam.
OCH,
Care: to a chemist,
the words 'strong' and
'weak' do not mean
'concentrated' and
'dilute' respectively,
but relate to the extent
of ionic dissociation;
see Chapters 4 and 6.
We must first appreciate that a coloured species such as capsaicin
has indeed adsorbed; otherwise, there would be no layer to see. And
a higher concentration of capsaicin yields a more intense colour as a
straightforward manifestation of the Beer-Lambert law. We discern
a relationship between the strength of the curry (by which we mean
the concentration of the spices it contains) and the colour of the
adsorbate, with a strong curry containing more spice and imparting
a more intense colour. Conversely, the amount of dye adsorbed on
the pan will be relatively slight after a mild curry (which is more
dilute in the amounts of 'hot' compounds it contains).
Adsorption occurs on the microscopic level, with molecules of adsorbate sticking
to the atoms on the surface of the substrate (in this case, the pan). In practice, it
is rare for each and every adsorption site to have a dye molecule adsorbed to it.
The coverage is only fractional. We give the name isotherm to the fraction of the
total adsorption sites occupied by adsorbate. An isotherm is denoted with the Greek
symbol 9 (theta).
Next, we realize how intensifying the orange stain of capsaicin confirms these
variations in the magnitude of 9. A small value of 9 tells us that a small proportion
of the possible adsorption sites are occupied by adsorbed molecules of capsaicin, and
490
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
Fraction of all
possible
adsorption sites
occupied, 6
Amount of adsorbate available for adsorption
Figure 10.2 Schematic isotherm for the simplest cases of chemical adsorption from solution onto
a solid substrate. The amount of adsorbate available to adsorb is best gauged by the concentration c
A greater number of
adsorbed molecules
appears as a more
intense stain to the eye
in consequence of the
Beer- Lambert law.
This argument only
holds if each pan-load
of curry is allowed to
reach its thermody-
namic equilibrium.
a higher value of 9 means adsorbate adheres to a larger proportion
of the adsorption sites. And the magnitude of 9 changes because the
metal of the pan (the substrate) is more likely to come into contact
with the capsaicin dye (the adsorbate) when the curry contains a
more concentrated solution of the capsaicin.
Imagine we need to investigate further the intensity of the stain.
We would start by making several curries, each made in an identi-
cal pan and cooked for an identical length of time, although each
containing a different concentration of red capsaicin. We expect
the molecules of capsaicin to occupy a larger proportion of possi-
ble adsorption sites when there is more of it in solution; and we
expect 9 to be smaller when there is less capsaicin in solution. This
is indeed found to be the case.
A graph of the proportion 9 against the (equilibrium) concentration of adsorbate
is also called an isotherm. (This dual usage of the word can cause some confusion.)
A typical adsorption isotherm is shown in the schematic diagram in Figure 10.2,
and shows how the proportion 9 of occupied sites increases quite fast initially as
the concentration [capsaicin] increases. Above a certain concentration of adsorbate,
however, the amount of capsaicin adsorbed does not increase but remains constant. In
this example, the maximum value of 9 is about unity. In other words, all the possible
adsorption sites are bonded to a molecule of capsaicin - but only if the concentration
of the capsaicin is huge.
We might have predicted a value of 9 = 1 at high concentrations
of capsaicin adsorbate. We would say (simplistically as it turns out),
'if each adsorption site is occupied, then no further adsorption
could occur'. We have formed an adsorbed monolayer, with the
substrate bearing a full complement of adsorbate. Notice also from
Figure 10.2 how the value of 9 at the origin of the isotherm is zero.
This ought not to surprise us, because no adsorbate can adsorb if
none of it is in solution.
Similarly, most gases readily adsorb from the gas phase to form
a bound gas, as a direct analogy to the adsorption of capsaicin
Notice how the value
of is zero at the ori-
gin of the isotherm;
this should be no sur-
prise, since there is no
adsorbate in solution to
adsorb!
ADSORPTION AND DEFINITIONS
491
10 15 20
Pressure / kPa
30
Figure 10.3 An isotherm of amount of gaseous cyclohexane adsorbed (as 'y') against pressure p
(as 'x'), depicted as a function of temperature. The substrate is a catalyst comprising a mixture of
metal oxides, called Stirling-FTG. (Figure reproduced by permission of Pergamon from the paper
'Towards a general gas adsorption isotherm', G. M. Martinez and D. Basmadjian, Chem. Eng. Sci.,
1996, 51, 1043)
from solution. (In effect, the molecules of gas changed phase.)
Figure 10.3 shows the isotherm constructed with data for gaseous
cyclohexane adsorbing onto a solid substrate. The graph in Figure
10.3 shows 9 (as 'y') as a function of gas pressure p (as 'x').
The graph only ever goes through the origin at zero pressure.
We have discovered that the only way to have a completely 'clean'
substrate (one with no adsorbate on it, with 9 = 0) is to subject the
surface to an extremely low pressure - in effect, we have subjected
the substrate to a strong vacuum. Effectively, the vacuum 'sucks'
the adsorbate away from the substrate surface. We give the name
desorption to the removal of adsorbate.
The extent of adsorption is a function of temperature T, as
implied by the term 'isotherm', so the construction of an isotherm
graph should be performed within a thermostatted system. When
adsorbing from solution, the value of 9 also depends on the solvent;
generally, if the solvent is polar, such as water or DMF, then the
extent of adsorption is often seen to decrease because molecules of
solvent will occupy sites on the substrate in preference to molecules
of solute.
Finally, note how the isotherm can alter dramatically if adsorp-
tion occurs in the presence of other adsorbates. Curry is actually
a mixture of great complexity, so the example here of capsaicin is
somewhat artificial.
The removal of an
adsorbate is termed
desorption.
The shape of an iso-
therm depends on the
choice of adsorbate,
substrate, temperature
T and, in a solution-
phase system, the
solvent.
The arguments here
are simplified because
all foods contain a
wide assortment of
physically disparate
molecules. In a more
rigorous set of exper-
iments, each solution
should contain a single
solute.
492
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
The two extremes of
'physical adsorption'
and 'chemical adsorp-
tion' are abbreviated
to physisorption and
chemisorption respec-
tively.
Why is it difficult to remove a curry stain?
The strength of adsorption and the magnitude of AH (ads)
Each molecule, ion or atom of solute is bonded to the surface of the adsorbate.
If we wish to remove the adsorbate from its substrate then we must overcome the
enthalpy of adsorption A//( a( j s ). So, removing adsorbate requires the input of energy.
If A//( ac j S ) is large, then more energy is needed to overcome it than if A//( a( j S ) is small.
It is difficult, therefore, to remove a layer of adsorbed curry stain without expending
energy, which we experience physically as a difficulty in removing
the stain; the red colour won't just 'wash off.
There is a wide range of adsorption enthalpies Aff( a d s ), ranging
from effectively zero to as much a 600 kJ per mole of adsorbate.
The adsorptive interaction cannot truly be said to be a 'bond' if
the enthalpy is small; the interaction will probably be more akin
to van der Waals forces, or maybe hydrogen bonds if the substrate
bears a surface layer of oxide. We call this type of adsorption
physical adsorption, which is often abbreviated to physisorption.
At the other extreme are adsorption processes for which A//( a d s )
is so large that real chemical bond(s) form between the substrate and adsorbate. We
call this type of adsorption chemical adsorption, although we might abbreviate this
to chemisorption.
Now, to return to the orange stain, formed on the surface of a
pan by adsorption of capsaicin from a solution (the curry). Such
organic dyes are usually unsaturated (see the structure I above),
and often comprise an aromatic moiety. The capsaicin, therefore,
has a high electron density on its surface. During the formation of
the adsorption bond, it is common for this electron cloud to inter-
act with atoms of metal on the surface of the pan. Electron density
flows from the dye molecule via the surface atoms to the conduc-
tion band of the bulk metal. The arrows on Figure 10.4 represent
the direction of flow as electron density moves from the charge
centroid of the dye, through the surface atoms on the substrate,
and thence into the bulk of the conductive substrate.
As a good generalization, most aromatic and unsaturated species adsorb readily.
Their adsorption is facilitated if the substrate is electronically conductive, being either
A centroid is the loca-
tion of a physicochem-
ical phenomenon or
effect or quantity. A
charge centroid, there-
fore, represents the
part of a molecule or
ion having the highest
charge density.
it clouds _
on adsorbate~
" Molecular plane of adsorbate
, Surface (bound) atoms
OOOOO of substrate
'Bulk substrate
Figure 10.4 Schematic representation of a dye molecule adsorbing on a substrate. The large arrow
indicates the movement of charge from the charge centroid to form an adsorptive bond. Notice the
way charge delocalizes, once it enters the substrate
ADSORPTION AND DEFINITIONS
493
The charge donated
during adsorption con-
ducts away from the
immediate site of ad-
sorption, and delocal-
izes, albeit partially. In
fact, most of donated
charge resides close to
the adsorption site.
metallic or semiconducting, because the charge donated during the
formation of an adsorption bond can readily conduct away from
the precise location of adsorption. We say the donated charge is
delocalized. No such derealization of the charge donated dur-
ing adsorption occurs if the substrate is non-conductive or poorly
conducting. The adsorption of additional adsorbate onto a poor
conductor is, therefore, rendered less likely because electronic re-
pulsions would be induced between the substrate and incoming
molecules of adsorbate. Such repulsions make the donation of more
charge to a surface already bearing an excess surface charge un-
favourable from electrostatic considerations.
A further difficulty arises: when additional molecules of adsorbate approach the
substrate, causing the proportion of sites filled 9 to approach unity, it becomes dif-
ficult for additional adsorbate to find a suitable angle of approach that allows for
successful adsorption. It is like finding a seat on bus: it is easier to see the seats
available in a partially empty bus. It is not only difficult to see the empty seats in a
full bus: it can also be difficult even to squeeze through the crowd to get to them.
The schematic diagram in Figure 10.5 represents such a situation. The molecule of
adsorbate must approach the substrate from within the cone-shaped area above the
vacant adsorption site. Otherwise, steric restriction will prevent successful adsorption.
Any attempts to approach the substrate from an angle outside this cone will cause the
incoming adsorbate to strike a molecule that is adsorbed already, causing the attempt
at adsorption to fail.
The surface of the substrate is not homogeneous, which presents two problems.
First, both electronic repulsions and then steric restrictions promote adsorption at
lower and make it more difficult at higher 0. The influence of these problems
o
Incoming molecule of
adsorbate prior to adsorption
Adsorbate
Substrate
Vacancy on the substrate surface
Figure 10.5 Schematic diagram to show the steric problems encountered when 6 tends to unity.
The incoming molecule of adsorbate must approach from within the confines of the cone shape if
adsorption is to succeed
494
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
Mean enthalpy
of adsorption
AH,
(ads)
Figure 10.6 The enthalpy of adsorption A// (a( js) is a function of 0. The position of the x-axis
represents the mean enthalpy; the magnitude of the vertical deviation from the x-axis is AA// (a( j S )
Care: note the double
'A' symbol in AAH (ads) ,
which represents the
change in AH (adS ) from
its mean value rather
than the enthalpy of
adsorption itself.
is seen in the way the mean enthalpy varies: we tend to talk in terms
of A A // (a( ] s ), where the double A implies we are looking at vari-
ations in AH. Figure 10.6 shows how the value of A// (a( ] s ) varies
as a function of 6. The range of A//( ac j S ) values depends markedly
on the identities of adsorbate and substrate, but the difference in
A //(ads) between 9 = and 1 can be as large as 20 kJmol -1 .
Why is iron the catalyst in the Haber process?
Physisorption and chemisorption
The usual choice of catalyst for the Haber process to produce ammonia is metallic
iron, often mixed with various other additives to alter subtly the magnitude of the
activation energy E a (see Chapter 8). In the Haber process, a mixture of gaseous
nitrogen and hydrogen are heated to 600 °C. The gases are compressed to a pressure
of about 6 atm to facilitate efficient pumping through the plant, and forcing them over
the iron catalyst. The reaction in Equation (10.1) occurs at the catalyst's surface:
N 2(g) + 3H
2(g)
2NH
3(g)
(10.1)
The strength of the N=N
bond decreases after
charge is donated to
the surface of the iron.
The physical conditions to effect a satisfactory extent of reac-
tion are fairly severe, but are needed to overcome the enormous
strength of the nitrogen triple bond. The role of the iron is essen-
tial: chemisorptive adsorption of nitrogen occurs on the surface of
the iron, with charge being donated from the N=N bond to the sur-
face of the iron. As a result, less electron density remains between
ADSORPTION AND DEFINITIONS
495
Molecules of nitrogen
are immobilized by
adsorption, thereby
facilitating collisions
with a molecule of
hydrogen.
the nitrogen atoms, sufficiently weakening the bond. Furthermore,
nitrogen gas is immobilized while adsorbed. While adsorbed, the
probability of a hydrogen molecule striking the nitrogen molecule
is enhanced. In a similar way, it is easier to throw a ball at a
stationary athlete than one who runs fast.
But why do we heat the mixture to effect reaction? We will per-
form a thought experiment: imagine immersing iron into nitrogen
at different temperatures. We will see the effects of the relative
differences between physisorption and chemisorption.
(1) Very cold. Immersing a clean iron surface into liquid nitrogen at 77 K (—196 °C)
yields a weak physisorptive bond. The N=N molecule is probably aligned parallel
to the metal surface, with electron density donating from the centroid of the triple
bond directly to iron atoms on the surface of the metal via a van der Waals type of
interaction. The experimental value of AH^) is small at about 1.5 kJmol -1 .
(2) Room temperature. Now imagine the iron is removed from the liquid nitrogen
and allowed to warm slowly up to room temperature (at, say, 298 K). The energy
from the room is always equal to |/? x T (see p. 33), so the energy of the room
increases with warming. The value of RT at 298 K is 2.2 kJmol -1 . As RT is greater
than A //( a ds) at this temperature, the molecules of nitrogen shake themselves ener-
getically until breaking free. In summary, increases in temperature are accompanied
by desorption of nitrogen from the iron until the iron is bare.
(3) Very hot. We finally consider the case of roasting metallic iron in a nitrogen
atmosphere at about 600 °C. Such roasting occurs, for example, during the Haber
process for making ammonia. When hot, both nitrogen atoms from molecular di-
nitrogen form a different type of bond (a chemical bond) to the surface of the metal,
binding to adjacent iron atoms. The extent of adsorption is thereby increased. The
enthalpy change AZ/( ads ) is greater than 500 kJmol -1 .
Following adsorption, electron density seeps from the inter-nitrogen bond, decreas-
ing its bond order from three to perhaps one and a half. Figure 10.7 depicts schemat-
ically the chemisorption of the nitrogen molecule onto iron. To complete the picture,
Figure 10.8 shows how diatomic hydrogen gas readily adsorbs to the surface-bound
layer of 'complex' on the iron surface - a structure sometimes
called a 'piggy back'.
As a simple proof that a complex-like structure forms on the
surface of metals immersed in a mixture of nitrogen and hydro-
gen gases, try immersing a piece of red-hot bronze in an atmo-
sphere of ammonia. The surface of the metal soon forms a tough,
impervious layer of bronze -ammonia complex, which imparts a
dark -brown colour to the metal. The brown complex reacts read-
ily with moisture if the metal is iron and is impermanent, but the
complex on bronze persists, thereby allowing the colour to remain.
These ammonia complexes explain why so many bronze statues
and medals have a dark-brown colour.
Roasting a bronze arti-
fact in dry NH 3 imparts
to it a deep-brown
appearance, caused by
a chemisorbed adlayer
of ammonia-bronze
complex, which persists
long after removing
the bronze from the
furnace.
496
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
N~N
Surface-bound Fe atoms
Bulk iron
Figure 10.7 During the adsorption of molecular nitrogen onto iron metal, the two nitrogen atoms
donate an increment of charge to adjacent atoms of iron on the metal surface, as depicted by the
vertical arrows. The N=N triple bond cleaves partially, with a resultant bond order of about 1.5
H-H
I I
N N
I i
Chemisorbed hydrogen
Chemisorbed nitrogen
Surface-bound Fe atoms
Bulk iron
Figure 10.8 Since the nitrogen is immobilized, hydrogen gas can adsorb to the surface of the
bound -N-N- structure, thereby effecting a further decrease in the bond order between the two
nitrogen atoms. The bond order between the two hydrogen atoms is also less than one
y is it easier to remove a layer of curry sauce than
to remove a curry stain?
Multiply adsorbed layers
We mean 6 = 1 when
we say 6 is 'unity'.
The idea of having a maximum value of unity is simplistic. Try
this quick experiment: take the pan used to prepare a curry and try
to wash it clean. There will be a stain adjacent to the metal, above
which is a layer of curry sauce. The stain is thin - possibly even
transparent - which we call the adsorbed layer. The layer of stain
has a minimal thickness and it is difficult to remove. Conversely,
we call the overlayers of curry sauce 'bulk material', which can
take any thickness and are relatively easy to remove, even with
a fingernail.
There are layers on layers - we call them multiple layers (or
multilayers). A chemisorbed layer is formed by the creation of
chemical bonds. For this reason, there can only be a single chemi-
sorbed layer on a substrate. Conversely, it is quite likely that a
material can adsorb physically (or physisorb) onto a previously
formed chemisorbed layer, either on more of the same adsorbate or even on a differ-
ent adsorbate.
There can only be one
chemisorbed layer on
a substrate because,
after bonds have for-
med with the substrate,
there is no more sub-
strate with which to
form bonds.
ADSORPTION AND DEFINITIONS 497
This concept is illustrated by the example of curry on a saucepan. A chemisorbed
layer forms on the pan, and physisorbed curry can adhere to the chemisorbed layer.
Similarly, the hydrogen gas in the previous example adsorbs on nitrogen gas chemi-
sorbed on iron.
How does water condense onto glass?
Physisorbed layers adsorbing on chemisorbed layers
We wake up to find it is a bitterly cold day outside. We wish to wash, so we fill
the bathroom sink with hot water; but, as soon as we look up, we see how water
condenses on the cold glass of the mirror.
As soon as steam emanates from the water in the sink, it will rise (owing to
eddy currents; see Chapter 1). A tiny fraction of the airborne water (i.e. steam) will
condense on the mirror, and soon a strongly bound chemisorbed layer forms to cover
its whole surface. The layer is microscopically thin, making it wholly invisible to the
eye. We will call this layer 'layer 1'. Each molecule of water in layer 1 is now
physically distinct from normal water, since charge has been donated to the substrate.
Each water molecule in layer 1 is, therefore, slightly charge deficient, compared with
normal water.
But the water in the sink is still hot, causing more steam to
The layer immediately
adjacent to the sub-
strate need not be
physisorbed.
rise. The air in the bathroom is cold, causing the airborne water
molecules to lose energy as they attain thermal equilibrium. No
more water can form a chemical bond with the glass of the mirror
because there are no adsorption sites available. So they form a
physisorptive bond onto the chemisorbed layer of water, to form
'layer 2'. Each molecule as it forms a physisorbed layer donates charge to water in
layer 1 , which we recall is slightly charge deficient. Physisorption causes the newly
bound water molecules in layer 2 to be slightly charge deficient, though certainly
not as deficient as those in layer 1.
Further layers can physisorb, one above another. The extent of charge deficiency
will decrease until, after about five or six layers have physisorbed, they are ener-
getically indistinguishable. By the time we have about a dozen layers adsorbed
on the glass, we are no longer able to speak about layers, and start to talk about
bulk condensate of water. At this point, more water condenses onto this mass of
water, causing its weight to increase, and eventually it runs down the mirror as warm
condensation.
Another example: consider the adsorption of bromine on silica. We start by placing
elemental bromine and silica at either end of a long sealed vessel. Having a relatively
high vapour pressure, bromine volatilizes and molecules of bromine soon chemisorp
onto the surface of the silica. When the chemisorbed monolayer is complete, succes-
sive layers of bromine form by physisorption. The first physisorbed layer is unique,
498
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
since its under-layer is chemisorbed, but the second physisorbed layer has physisorbed
layers both on its top and bottom. In fact, after several adsorbed layers have been
deposited, such layers become virtually indistinguishable, and may be considered to
be identical in nature to 'normal', i.e. liquid, elemental bromine.
How does bleach remove a dye stain?
Reactivity and properties of adsorbed layers
Before we start, we assume that the dye stains the surface by forming an adsorptive
interaction. The majority of dyes in the kitchen come from vegetables. A particularly
intense dye is ^-carotene (II), which colours carrots, the golden leaves of autumn
and some flowers. The intensity of the /J -carotene colour arises from the extended
conjugation.
Wiping the stained portion of work surface with bleach quickly removes the colour
of the adsorbed dye as a result of the adsorbed material reacting with the bleach.
One of the active components within household bleach is hypochlorous acid, HOC1, in
equilibrium with molecular chlorine. Q2, H + and C10~ add across
the double bonds in the dye (see Figure 10.9) - usually near the
middle of the conjugated portion, because such bonds are weakened
as a result of their conjugation. The colour is seen to vanish because
the conjugation length decreases.
Secondly, by decreasing the extent of conjugation, the electron
density within the adsorbate decreases. Since a large electron den-
sity was possibly a major cause of the adsorptive bond being so
The conjugation
strength is also
decreased because
some of the electron
density has been
donated to the surface.
HOCl +
Figure 10.9 Hypohalite addition across one of the weak, central double bonds of ,6-carotene. The
colour of an adsorbed material is lost when exposed to bleach because of hypohalite addition across
a double bond, thereby decreasing the extent of conjugation, if not removing it altogether
ADSORPTION AND DEFINITIONS 499
strong, and the charge from that double bond is no longer available, the adsorption
strength decreases greatly. We should also expect the solubility of previously insoluble
products of hypohalite to increase, allowing them to wash away.
How much beetroot juice does the stain
on the plate contain?
The Langmuir adsorption isotherm
The dye responsible for the red-mauve colour of beetroot juice is potent, because
a small amount imparts an intense stain on most crockery. It has a large extinction
coefficient e (see Chapter 9). The dye is fairly difficult to remove, so we safely assume
the adsorption bonds are strong and chemisorptive by nature.
We can determine how much of the dye (the 'adsorbate') adsorbs on a plate (the
'substrate') by devising a series of experiments, measuring how much adsorbate
adsorbs as a function of concentration. We then analyse the data with the Langmuir
adsorption isotherm:
c c 1
- = — + -— (10.2)
n n m bn m
where c is the equilibrium concentration of coloured adsorbate in contact with the
substrate, n is the amount of adsorbate adsorbed to the plate (the substrate) and n m
is the amount adsorbed in a complete monolayer. Finally, b is a constant related to
the equilibrium constant of adsorption from solution.
We perform a series of experiments. We purchase a stock of identical plates and
prepare several solutions of beetroot juice of varying concentration. One plate is
soaked per solution. The chemisorptive bond between the dye and plate is strong, and
equilibrium is reached after only a few seconds. The excess juice is decanted off for
analysis, e.g. by means of optical spectroscopy and the Beer-Lambert law, provided
we know the extinction coefficient e for the juice.
We will find a decrease in the amount of dye remaining in
solution because some of it has adsorbed to the plate surface.
The amount adsorbed is readily determined, since the initial con-
centration was known and the equilibrium concentration is read-
ily measured.
From Equation (10.2), we expect a plot of c 4- n (as 'y') against
c (as '%') to yield a straight-line graph of intercept \/{bn m ) and with a gradient l/n m .
Worked Example 10.1 The following data refer to the adsorption of the red-mauve
dye from beetroot juice on porcelain at 25 °C. (1) Show that the data obey the Langmuir
adsorption isotherm. (2) Demonstrate that 1.2 x 10~ 8 mol of dye adsorb to form a mono-
layer. (3) Estimate the area of a single dye molecule if the radius of a plate was 17.8 cm
(we assume the formation of a complete monolayer).
We obtain the amount
of dye n in a mono-
layer as ^(initially) -
^(remaining in solution) ■
500
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
Equilibrium concentration c/mmol dm 3
Amount adsorbed «/nmol
0.012
0.026
0.047
0.101
0.126
2.94
4.98
7.94
9.00
9.59
Answer strategy
Notice the tiny num-
bers here: mmol means
milli-moles (m = 10 3 )
and nmol means nano-
moles (n = 10~ 9 ).
(1) If data obey Equation (10.2), then a plot of c 4- n (as 'y') against
c (as 'x') should yield a straight-line graph of gradient \/n m . Any
non-linear portion(s) of the graph represent concentration ranges not
following Equation (10.2), and hence not following the Langmuir
adsorption isotherm. Figure 10.10 shows the Langmuir plot construc-
ted with the data above. The graph is linear, indicating that the data
obey the Langmuir isotherm.
(2) We then determine the gradient of the line. Its reciprocal has a value of n m , which
is the number of moles of beetroot juice in a monolayer. The gradient of the graph is
0.0796 x 10 6 mol~ . The reciprocal of its gradient equates to n m , so
1 4- (gradient) = n m — 1.256 x 10~ 5 mol.
This is a tiny value, and reveals how a small amount of beetroot dye
will stain a standard aluminosilicate plate; in other words, it demon-
strates how large is the extinction coefficient e of the beetroot juice
chromophore.
(3) We calculate the number of molecules by multiplying n m by the Avogadro number L.
We already know the area over which the monolayer forms via nr 2 , so we calculate the
area per molecule with the following simple calculation:
China or porcelain is
made from clay, itself
made up of aluminosil-
icate.
area per molecule
area of the substrate
number of molecules
(10.3)
0.002
0.14
Concentration at equilibrium cl 10 mol dm
Figure 10.10 Langmuir plot concerning the adsorption of beetroot-juice dye onto a porcelain
substrate at 25 °C: graph of c -h x (as 'y') against c (as 'x')
ADSORPTION AND DEFINITIONS 501
We obtain the number of molecules as n m x L, where L is the Avogadro number:
number of molecules = 1.256 x 10~ 5 mol x 6.023 x 10 molecules mol~
so
number of molecules = 7.56 x 10 18 molecules
We calculate the area A of the plate as nr 2 (in terms of SI units) so A = 3.142 x (17.8 x
10~ 2 m) 2 = 0.093 m 2 . Then, inserting numbers into Equation (10.3), we obtain:
0.093 m 2
area per molecule =
7.56 x 10 18 molecules
A = 1.32 x 10~ 19 m 2 molecule -1
Aside
One of the major components within aluminosilicate china is kaolin clay. Kaolin is
used in the treatment of stomach ache, often being found for example in 'kaolin and
morphine'. Just like the beetroot juice in this example, biological toxins adsorb strongly
to the surface of the kaolin and are immobilized thereafter, and leave the body during
defecation. Many animals in the wild, such as elephants in Africa, spider monkeys in
South America and brightly coloured macaws, eat naturally occurring kaolin clay at the
end of a meal in order to prevent stomach ache or outright poisoning.
Background to the Langmuir model
The Langmuir model first assumes the adsorption sites are energetically identical.
Actually, this assumption is not borne out when adsorption occurs predominantly
by physisorption. The spread of A//( a[ | s ) values between the various sites can be as
high as 2 kJmol -1 , which is often a significant fraction of the overall enthalpy of
adsorption when physisorption is the sole mode of adsorption. By contrast, energetic
discrepancies between sites can be ignored when adsorption occurs by chemisorption.
The second assumption follows from the first: to ensure that the adsorption sites
are energetically identical, we say the adsorption occurs wholly by chemisorption.
Third, we assume the adsorption process is dynamic. Adsorption and desorption
occur to equal and opposite extents at equilibrium, provided the mass of material
adsorbed remains constant.
Justification Box 10.1
For the purposes of the derivation below, we will consider the process of adsorption
from the gas phase. A simple example of a system involving adsorption of gases is the
Haber process, in which N2(g) and H2( g ) adsorb to the surface of metallic iron.
502
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
There are N possible number of adsorption sites on the substrate, the fraction of which
bearing adsorbed material is 9. By corollary, the fraction of empty sites is (1 — 0);
the number of filled sites will thus be NO and the number of empty sites will be
N(l-6).
The rate constant of adsorption is fe a and the rate constant characterizing the way
sites lose adsorbate is k&. From simple kinetics (see Chapter 8), the rate of adsorption
depends on the number of sites available N(\ — 9), the rate constant of the process fe a ,
and the amount of adsorbate wanting to adsorb, which is proportional to pressure p.
Overall, the rate of adsorption is TV x (1 — 9) x fe a x p.
Clearly, desorption can only occur from a site already bearing adsorbed material. The
rate of desorption is proportional to the number of such sites, with a proportionality
constant of fed.
The net rate of adsorption is the difference between the rate of adsorption and the
rate of desorption:
d0
df
N(l-$)pk 3
! t
net rate rate of
adsorption
( NBkj ]
\
rate of
desorption
(10.4)
The fraction 9 stays constant at equilibrium, so the rate of change of 9 will be zero, so
implying
N{\ - 9)pk !i - N9k d =
N(l - 9)pk & = N9k d
(10.5)
The ./V terms cancel, so
(1 - 9)pk & = 9k d
(10.6)
Dividing by k d yields
(1-0)1,-1=,
(10.7)
The ratio of rate con-
stants yields the equi-
librium constant of
adsorption K.
The ratio of rate constants (fe a -r fed) yields the equi-
librium constant of adsorption K; so, multiplying the
bracket gives
pK - pK9 =
(10.8)
Grouping the 9 terms:
P K = 9(pK + l)
(10.9)
ADSORPTION AND DEFINITIONS
503
So the fractional coverage of the total adsorption sites
9 becomes
pK
(10.10)
pK + l
We have not yet expressed Equation (10.10) in its
final, useful form. The fraction 9 can be replaced if we
realize how 9 is the ratio of the volume of gas adsorbed
V( a d S ) divided by the volume of gas adsorbed evenly to
form a complete monolayer V m . Substituting for 9 yields
Care:
the
product of K
and p
cou
Id be
written
as 'K
P' (
i.e. K
xp),
which
cou
Id be
mis-
taken for
a gas-
phase
equilibrium constant
K p .
^(ads)
pK
pK+l
Rearranging Equation (10.11) gives
^(ads)
P_
v m
+
KV m
(10.11)
(10.12)
which is the Langmuir adsorption isotherm.
Accordingly, a plot of p -=- V (as 'y') against p (as 'x') should yield a straight-line
graph of intercept 1 -f- (K x V m ) and gradient l/V m . And knowing V m from the gradient
allows for calculation of K.
We obtain Equation (10.2) when Equation (10. 12) is written in terms of concentration c.
Why do we see a "cloud ' of steam when
ironing a shirt?
Deviations from the Langmuir model
We saw on p. 487 how ironing a shirt with a hot electric iron was usually accompanied
by the release of adsorbed water. But the amount of water adsorbed within a mono-
layer is tiny - no more than about 10~ 8 mol, as shown by Worked Example 10.1.
Even a moment's thought tells us too much steam is released when heating the cloth
beneath a hot iron. In fact, the water does not adsorb according to a simple Lang-
muir model with, for example, multiple layers adsorbing to form a
'sandwich'. We find several layers of water physisorbing, one on
top of another, with the ultimate substrate being an underlayer of
chemisorbed water. This way, the amount of water can be many
times greater than that comprised within a chemisorbed monolayer.
For this reason, we sometimes construct Langmuir plots and
find they are not particularly linear, particularly at the right-hand
side, which represents higher concentrations. This deviation from
The gradient of a Lang-
muir plot is always
positive, since it is
clearly impossible for a
monolayer to contain
a negative amount of
material.
504
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
linearity generally implies layers of physisorbed material have formed on top of the
chemisorbed monolayer at high c. Less frequently, the deviations suggest the system
lies energetically on the borderline between chemi- and physisorption.
SAQ 10.1 The following data refer to the adsorption of butane at 0°C
onto tungsten powder (area 16.7 m 2 g _1 ). Calculate the number of moles
adsorbed in a monolayer, and hence the molecular area for the adsorbed
butane (at monolayer coverage); and compare it with the value of 32 x
10~ 20 m 2 estimated from the density of liquid butane.
Relative pressure p/p°
0.06
0.12
0.17
0.23
0.30
0.37
Volume adsorbed (at s.t.p.) V (ads) /
1.10
1.34
1.48
1.66
1.85
2.05
cm 3 g _1
SAQ 10.2 When nitrogen was adsorbed onto a 213 mg sample of graphi-
tized carbon black, the pressures necessary to maintain a given degree of
adsorption were 15 kPa at 312 K and 17 kPa at 130 K.
(1) Calculate the mean enthalpy of adsorption AH (ads) .
(2) Why 'mean' in this context?
[Hint: the process of adsorption is a condensation reaction, so we can
employ the Clausius-Clapeyron equation (Equation (5.5)) to answer part
(1). Be careful, though, because AH for a condensation reaction will
be negative.]
10.2 Colloids and interfacial science
Why Is milk cloudy?
Particle suspensions and phase dispersal
Milk contains both organic and inorganic components. A majority of the milk is
water based, and contains water-soluble solutes such as calcium compounds. But
milk also contains as much as 15 per cent by mass of water-insoluble, fat-based
compounds.
These two extremes of oil and water do not mix, and remain as separate phases (see
Chapter 5). Mixing these oil- and water-based components causes two distinct layers
to form in a process called phase separation - much as we see when mixing oil and
vinegar when preparing French dressing, or when shaking water and toluene solu-
tions within a separating funnel. We say the water- and fat-based
phases are immiscible. But, unlike the two distinct layers that form
when mixing two immiscible liquids in a separating funnel, the fat
Two immiscible liquids
do not mix.
COLLOIDS AND INTERFACIAL SCIENCE
505
particles in milk retain their microscopic size, and remain sus-
pended or dispersed. We say the water-based phase is the dispersal
medium, and the fat-based phase is the dispersed medium.
We see no particles after dissolving an ionic solute in water,
so aqueous solutions of potassium chloride or copper nitrate are
crystal clear. In fact, a beam of light can pass straight through
such a solution and emerge on the other side of a flask or beaker,
with no incident light scattered by the particles. Scattering would
only occur if the particles were similar in size to, or larger than, the
wavelength A of visible light (350-700 nm). The average diameter
of a solvated ion is 10~ 9 -10~ 8 m, and so is much smaller than the
wavelength X.
But the microscopic fat particles suspended in milk have an aver-
age diameter in the range 10~ 7 to 10~ 5 m, i.e. much larger than A
of visible light. A beam of incident light is scattered rather than
transmitted by a suspension of particles - a phenomenon known as
the Tyndall effect.
In summary, milk appears cloudy because the suspended fat par-
ticles scatter any incident light.
The continuous phase
is said to be the dis-
persAL medium (or
phase), and the sus-
pended particles are
the dispersED medium
(or phase).
The study of the
scattering of light by
colloidal systems has a
long history. The Tyn-
dall effect describes
the scattering of light
by suspended particles.
In fact, the first rigor-
ous theory was that of
Rayleigh in 1871.
Aside
Tyndall scattering also causes the blinding effect of shining a car headlight directly
into a thick bank of fog or mist; it also yields the beautiful iridescent colours on the
wing of a butterfly or peacock tail, and an opal and mother of pearl. A good potter can
reproduce some of these optical effects with so-called iridescent glazes, which comprise
colloidal materials.
The light is scattered in a so-called scattering pattern, which is a function of the
wavelength of the light k and the angle 9 between the incident and scattered light.
Although no precise stipulation relates the size with the light scattered, particles that
cause light scattering generally have diameters that are ten times larger than k, or more.
Incidentally, these different scattering patterns explain why a car headlamp causes
more scattering with a quartz -halogen bulb than with a standard tungsten filament,
because of their differing wavelengths and power.
What is an ^aerosol' spray?
Colloids and aerosols
Squirting a solution of perfume through a fine tube, near the end of which is a slight
constriction, generates a fine spray of perfume comprising tiny particles of liquid.
The spray remains airborne for a short time, and then settles under the influence of
506
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
A colloid is one phase
suspended within
another.
gravity. We call this suspension an aerosol, which is one form of
a colloid.
A colloid is a broad category of mixtures, and is defined as
one phase suspended in another. A perfume spray is made up of
a liquid (the perfume) dispersed in a gas (the air). The principle underlying the
perfume atomizer is the same as the nozzle on a can of polish, and the jets within
the carburettor in the internal combustion engine. In each case, the colloid formed is
an aerosol.
Aside
In 1860, the Scottish chemist Thomas Graham noticed
how substances such as glue, gelatin, or starch could
be separated from other substances, like sugar or salt,
by dialysis across a semi-permeable membrane, such
as cellulose or parchment (made from treated animal
skin). He gave the name colloid to substances unable
to diffuse through the membrane - because the particles were too large. In fact, colloidal
particles are larger than molecules, but are too small to observe directly with a standard
optical microscope.
The word 'colloid'
comes from the Greek
root coll, meaning
'glue'.
What is 'emulsion paint'?
Emulsions and colloid classification
Emulsion paint com-
prises pigment bound in
a synthetic resin such as
urethane, which forms
an emulsion with water.
Emulsion paint is easy to apply, aesthetically good looking, and
forms a hard wearing and waterproof coating. Like milk in the
previous example, the paint comprises one phase dispersed within
another. It is a colloid, but this time a liquid finely dispersed in
another liquid, which we call an emulsion.
In emulsion paint, the dispersion phase is the liquid of the paint
and is generally water-based for emulsion paints. The water is
buffered (p. 269) with ammonia or simple amines to yield a slightly alkaline solution.
The dispersed phase in most emulsion paints comprises small particles of immisci-
ble urethane.
Water evaporates from the paint during drying, with two consequences. Firstly,
molecular oxygen initiates a polymerization reaction. And, secondly, as water evap-
orates from the layer of paint, so the volume of paint decreases, causing the urethane
particles to come together more closely, thereby encouraging propagation of the
polymerization reaction, causing juxtaposed urethane particles to join and hence
form a continuous layer. The eventual paint product is a solid, polymeric layer of
poly(urethane).
COLLOIDS AND INTERFACIAL SCIENCE 507
Classification of colloids
A colloid is defined as a non-crystalline substance consisting of ultra-microscopic
particles, often of large single molecules, such as proteins, usually dispersed through
a second substance, as in gels, sols and emulsion.
There are eight different types of colloid, each of which has a different name
according to the identity of the dispersed phase and the phase acting as the dispersion
medium.
A gas suspended in a liquid is called a foam. Obvious examples include shaving
foam (the gas being butane) and the foam layer generated on the surface of a warm
bath after adding a surfactant, such as 'bubble bath'. The gas in this last example will
be air, i.e. mainly nitrogen and oxygen.
A gas suspended in a solid is also called a foam. This form of colloid is relatively
rare in nature, unless we stretch our definition of 'solid' to include rock, in which case
pumice stone is a colloidal foam. Synthetic foams are essential for making cushions
and pillows. There is also presently much research into forming metal foams, which
have an amazingly low density.
A liquid dispersed in a different liquid is called an emulsion, as
The word 'emulsion'
comes from emulsio,
Latin for the infinitive,
'to milk'.
above. In addition to emulsion paint, other simple examples include
butter, which consists of fat droplets suspended in a water-based
dispersion medium, and margarine, in which water particles are
dispersed within an oil-based phase.
A liquid dispersed in a solid is called a solid emulsion. Few
examples occur in nature, other than pearl and opal, the solid phase of which is based
on chalk.
A solid or liquid suspended in a gas is called an aerosol. A good example of a
solid-in-gas aerosol is smoke, either from a fire or cigarette. An example of a liquid-
in-gas aerosol is the liquid coming from a can of polish or paint, or the perfume
emerging from an atomizer.
A solid suspended in a liquid is called a sol, although a high concentration of solid is
also called a paste. Some paints are sols, particularly those containing particles of zinc
to yield weatherproof coatings. Toothpaste is also a sol. Many simple precipitation
and crystallization processes in the laboratory generate a sol, albeit usually a short-
lived one, since the solid settles under the influence of gravity to leave a clear solution
above a layer of finely divided solid. Try mixing solutions of silver nitrate and copper
chloride; this generates a cloudy sol of white silver chloride, which soon settles to
form a white powder and a clear, blue solution. Simple sols are rarely stable unless the
liquid is viscous, or if chemical interactions bind the dispersed phase to the dispersion
phase.
A solid suspended in a solid is called a solid suspension. Pig-
mented plastic - particles of dye suspended in a solid polymer - is
a simple example. Freezing a liquid-liquid emulsion, such as milk,
also yields a solid suspension. A layer of partially set paint, in
which the urethane monomer has been consumed but the polymer
has yet to form long chains, can also be thought of as representing
a solid suspension.
Short, imperfectly for-
med polymer chains
are properly called
oligomers, from the
Greek oligo, meaning
'small'.
508
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
Table 10.1
Types of colloid
dispersion
Dispersed phase
Dispersion medium
Name
Example
Gas
Gas
a
-
Liquid
Gas
Liquid aerosol
Fog, perfume spray
Solid
Gas
Solid aerosol
Smoke, dust
Gas
Liquid
Foam
Fire extinguisher foam, shaving foam
Liquid
Liquid
Emulsion
Milk, mayonnaise
Solid
Liquid
Sol, paste
Toothpaste, crystallization
Gas
Solid
Solid foam
Expanded polystyrene, cushion foam
Liquid
Solid
Solid emulsion
Opal, pearl
Solid
Solid
Solid suspension
Pigmented plastics
a A gas-in-gas system is not a colloid, it is a mixture.
Finally, there are no gas-in-gas colloids: we cannot suspend a gas within another
gas, since it is not possible to have gas 'particles' of colloidal dimensions. Introducing
one gas to another generates a simple mixture, which follows the thermodynamics of
mixtures, e.g. Dalton's law (p. 221).
We summarize these colloid classifications in Table 10.1.
Aside
Ice cream is simultaneously both an emulsion and a partially solidified foam, so it
comprises three phases at once. The ice cream would be too solid to eat without the
air, and too cold to eat without discomfort. The air helps impart a smooth, creamy
consistency. The solid structure is held together with a network of globules of emulsified
fat and small ice crystals (where 'small' in this context means about 50 (xm diameter).
Interfaces and inter-phases
The plural of 'medium'
is media, not 'medi-
ums'.
Not all colloid systems are stable. The most stable involve solid
dispersion media, since movement through a solid host will be
slow. Emulsions also tend to be stable; think, for example, about
a glass of milk, which is more likely to decompose than undergo
the destructive process of phase separation. Aerosols are not very
stable: although a water-based polish generates a liquid-in-air colloid, the particles of
liquid soon descend through the air to form a pool of liquid on the table top. Smoke
and other solid-in-gas aerosols are never permanent owing to differences in density
between air and the dispersed phase.
COLLOID STABILITY
509
In fact, we generate the most unstable colloids immediately before we need them.
A good example is oil and water, which explains why we shake the bottle before
serving French dressing to a salad.
The problems causing the observed instability of colloids stems from the interface
separating the two phases. In French dressing, for example, each oil particle is sur-
rounded with water. The instability of the water- air interface causes the system to
minimize the overall area of contact between the two phases, which is most readily
achieved by the colloid particles aggregating and thereby forming two distinct phases,
i.e. oil floating on water.
Aside
'Milk fat' comprises lipids, which are solid at room temperature. If they were liquid,
we could correctly call them 'oils'.
The melting temperature of milk fat is 37 °C, which is significant because 37 °C is the
body temperature of a cow, and milk needs to be a liquid at this temperature. As well
as the obvious dietary properties imparted by the fat, milk fat also imparts lubrication,
which is why cream has a 'creamy' feel.
10.3 Colloid stability
How are cream and butter made?
Such separation occurs
naturally, since cream
is less dense than milk,
but such separation is
too slow for commercial
cream production.
Colloid stability
Ordinary milk displays an amazingly diverse range of colloid chem-
istry. It owes its white colour to multiple light scattering from
globules of fat suspended in an aqueous phase (see p. 504); pro-
tein aggregates supplement the scattering efficiency. The colloidal
milk coming straight from a cow is stable except from slight grav-
itational effects, so centrifuging the milk separates the cream from
the milk, thereby forming two stable colloidal systems, both of
which comprise fat and water phases.
Mechanical agitation of the cream - a process called whipping - creates a meta-
stable foam (i.e. it contains much air). Further whipping causes this foam to collapse:
some water separates out, and the major product is yellow butter. Incidentally, butter
is a different form of colloid from milk, since its dispersed medium is water droplets
and its dispersal phase is oil (milk is an oil-in-water colloid). Forming butter from
milk is a simple example of emulsion inversion.
Similarly, the protein in milk is very rich in colloidal chemistry. Most of the
protein is bound within aggregates called casein micelles (see p. 512). The colloids
in milk are essentially stable even at elevated temperatures, so a cup of milky tea, for
510
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
In the dairy indus-
try, milk is warmed to
kill potentially dan-
gerous bacteria such
as E. coli, lysteria, and
salmonella - a process
known as 'pasteuriza-
tion' after Louis Pas-
teur (1822-1895) who
discovered the effect
in 1871.
example, will not separate, although boiled milk sometimes forms
a surface 'skin' of protein. This skin chemisorbs to many metals
and ceramics, thereby explaining why it is generally so difficult
to clean it off a saucepan used to boil the milk. The formation of
such skin is termed fouling in the dairy industry, and represents a
potential health risk.
But most colloids are intrinsically unstable, and can be 'broken'
by applying heat, changing the pH or adding extra chemicals, such
as salt, addition of which causes a large change in ionic strength.
All colloids are thermodynamically unstable with respect to the
bulk material, but the kinetics of destroying the colloid are often
quite slow. Colloidal milk will persist almost indefinitely before
other biological processes intervene (we say the milk 'goes off).
How is chicken soup 'clarified' by adding eggshells?
The 'breaking' of colloids
An expert cook starts the process of making chicken soup by boiling the carcass of a
chicken in water. The resultant broth is cloudy. The broth is allowed to cool, and the
chicken bones are removed. The chef then adds pieces of pre-washed eggshell to the
soup. The shells of a dozen or so eggs will be sufficient to remove the cloudiness of
one pint of soup: we say the soup has been clarified. We can finish making the soup
by adding vegetables and stock, as necessary, after removing the eggshells.
Chicken broth is cloudy because it is colloidal, containing micro-
scopic particles of chicken fat suspended in the water-based soup.
Like milk, cream or emulsion paint, the cloudy aspect of the soup
is a manifestation of the Tyndall effect. Adding the eggshells to the
colloidal solution removes these particles of fat, thereby removing
the dispersed medium. And without the dispersed medium, the col-
loid is lost, and the soup no longer shows its cloudy appearance.
We say we have broken the colloid.
In this example, a simple mechanism for breaking a colloid was chosen. The
eggshells are made of porous calcium carbonate, their surface covered with innu-
merable tiny pores. The particles of fat in the broth accumulate in these small pores.
Removing the eggshells from the broth (each with oil particles adsorbed on their
surfaces) removes the dispersed medium from the broth. One of the two components
of the colloid is removed, preventing the colloid from persisting.
We say the colloid is
broken when the dis-
persion medium no
longer suspends the
dispersed medium.
How is 'clarified butter' made?
The thermal breaking of colloids
Clarified butter was once a popular delicacy, though it is no longer in vogue to the
same extent. It is ideal for making dishes that benefit from a buttery flavour but need
COLLOID STABILITY
511
The smoking point of
butter is the temper-
ature at which par-
tial combustion starts,
yielding smoke. An oil or
fat is only safe to cook
with below its smoke
point.
to be cooked over a strong flame, such as omelets, since it does
not burn as easily as ordinary butter. We clarify normal butter by
removing its milk solids, which incidentally increases its smoking
point temperature.
To prepare it, small nuggets of unsalted butter are melted slowly
in a deep saucepan. The water in the butter evaporates, causing the
milk solids to sink to the bottom of the pan, and a foam rises to the
surface and is skimmed off. The clear, yellow melted butter is then
poured off to leave the milk solids at the bottom of the saucepan,
to be discarded.
On cooling, the butter has essentially the same taste as conventional butter, but
the appearance has changed to an almost waxy consistency. And the butter is clear,
because the colloidal system of water in oil is broken, hence Tyndall light scattering
is no longer possible. For this reason, the process is often called clarification. Perhaps
the weight-conscious need to know its calorific value per unit mass is greater than
normal butter as a consequence of losing most of its water and milk solids.
We break the colloid by heating. In fact, the only temperature -independent colloidal
systems are those involving a solid dispersion medium, such as lava. As with all
thermodynamic quantities, the equilibrium constant associated with forming a colloid
depends strongly on temperature. In general, warming a colloid results in two possible
outcomes: either the dispersed medium dissolves into the dispersion medium (they
become miscible), or the dispersion and dispersed media separate to form two separate
layers. The crystallization of small particles of solute from solution is a good example
of the former situation, when crystals form as a solution cools; the solution is clear
and homogeneous at higher temperatures.
Why does hand cream lose its milky appearance
during hand rubbing?
The mechanical breaking of colloids
Most hand creams are colloidal, and generally have a thick, creamy
consistency. The majority of hand creams are formulated as a
liquid-in-liquid colloid (an emulsion), in which the dispersion
medium is water based, and the dispersed phase is an oil such
as palm oil or 'cocoa butter' . These oils are needed to replenish in
the skin those natural oils lost through excessive heat and work.
The hand cream is opaque as a consequence of the Tyndall effect.
The milky aspect is lost soon after rubbing the cream into the skin
to yield a transparent layer on the skin. Furthermore, the hands feel
quite damp, and cool. We notice with pleasure how our hands feel softer after the
water evaporates.
The mechanical work of rubbing and kneading the hand cream breaks the colloid.
The oil enters the skin - as desired - while the water remains on the skin surface
before evaporating (hence the cooling effect mentioned above).
Most oil-in-water
emulsions (like cream
or hand cream) feel
creamy to the touch,
and most water-in-
oil emulsions (like
margarine) feel greasy.
512
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
The mechanical breaking of colloids is also essential when making butter from milk:
the solid from soured cream is churned extensively until phase separation occurs. The
water-based liquid is drained away to yield a fat-rich solid, the butter.
Why does orange juice cause milk to curdle':
Colloid stabilization and emulsifiers
The word 'vitamin' is
an abbreviation of the
two words 'vital min-
eral'. Vitamins were
once considered to be
those minerals that
were vital for a healthy
life. The modern mean-
ing is somewhat more
comprehensive.
We have already seen how milk is an emulsion comprising oil as a
dispersion medium in a water-based dispersion medium. Milk fats
also form colloids. The aqueous component of milk contains many
vitamins, especially the salts of calcium, which baby mammals
need to produce strong teeth and bones.
Colloidal proteins in milk are chemically unstable, so adding
an acidic solution such as orange juice to a glass of milk will
'break' it, causing the milk to curdle. The milk rapidly separates
into two separate layers, termed curds and whey. The orange juice
supplements the aqueous phase, so the bottom layer in the glass
will look orange. The upper layer will remain somewhat cloudy,
and comprises the fats and organic components within the milk.
Drinking the curdled mixture is likely to taste revolting.
Milk is an unusual colloid in comprising oil particles suspended
in water. Adding, say, olive or sunflower oil to water will not
produce a stable colloid. Two layers will re-form rapidly even after
vigorous shaking, with the oil floating above the water. Milk is
stable because it contains an emulsifier, i.e. a compound to promote
the formation of a colloidal emulsion.
The difference in stability between milk and other colloidal oil-
in-water colloids, such as French dressing, is the naturally occurring
protein casein (see p. 509), which milk contains in tiny amounts.
Casein is a phospholipid also found in egg yolk (from whence
it gets its name), animals and some higher plants. Molecules of
casein stabilize the colloid by adsorbing to the interface separating
the minute particles of oil dispersed in the water-based dispersion
medium. The usual repulsions experienced between oil and water
are overcome by surrounding the oil particle in this way, thereby promoting the
persistence of oil particles as a suspension.
We now look more closely at the structure of casein. It is a long
molecule with different ends: one end is polar and the other is non-
polar. In milk, the polar group (ending with a phosphate group) is
positioned to face the polar water, and the non-polar end faces the
oil. In effect, each particle of oil has a double coating: the inner
layer is the non-polar end of the casein emulsifier, and the outer layer is a sheath of
polar phosphate groups.
The word 'curdling'
comes from curd, the
coagulated solid formed
by adding acid or ren-
net to milk. Curds are
the precursors of most
cheeses.
The emulsifier casein
adsorbs to the inter-
face between the oil
and water.
Each colloid particle
is surrounded with an
electric double layer.
COLLOID STABILITY
513
Table 10.2 Vocabulary concerning the breaking of colloids
Term
Definition
Example
Aggregation The reversible coming together of
small particles to form larger
particles
Coagulation The irreversible formation of the
thermodynamically stable phase in
bulk quantities
Flocculation Adding fibrous or polymeric materials
to entrap solid particles of colloid
Coacervation Separation of a liquid-in-liquid
colloid into its two separate phases
A suspension of micro-crystals
forming soon after nucleation
The precipitation of crystals
Adding silicate to turbid drain water
causes the solution to clear
French dressing forms an upper layer
of oil and a lower layer of aqueous
vinegar
But we must appreciate how the phosphate group acts much like the anion from a
weak acid (see Chapter 6), so its exact composition will depend on the pH of solution.
The pH of cow's or human milk is about 7 (see Table 6.4). If the pH decreases much
below about 6 (e.g. by adding an acid in the form of orange juice), the phosphates
become protonated. The emulsifying properties of casein cease as soon as its structure
changes, causing the milk to separate.
Table 10.2 contains additional vocabulary relating to the thermodynamics of break-
ing colloids.
How are colloidal particles removed
from waste water?
Aggregation, coagulation and flocculation
The adjective 'parti-
culate' means 'in the
form of particles'.
Waste water from a drain or rain-overflow usually contains sediment,
including sand, dust and solid particles such as grit. But smaller,
colloidal particles also pollute the water. Water purification requires
the removal of such particulate matter, generally before disinfecting
the water and subsequent removal of any water-soluble effluent.
Larger particles of grit and dust settle relatively fast, but colloidal solids can require
weeks for complete sedimentation (i.e. colloid breaking) to occur completely. Such
sedimentation occurs when microscopic colloid particles approach, touch and stay
together because of an attractive interaction, and thereby form larger particles, and
sink under the influence of gravity. We call this process aggregation.
But time is money. The waste industry, therefore, breaks the colloid artificially
to remove the particulate solid from the water. They employ one of two methods.
Firstly, they add to the water an inorganic polymer such as silicate. The colloid's
thermodynamic stability depends on the surface of its particles, each of which has
a slight excess charge. As like charges repel (in consequence of Coulomb's law;
514
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
The word 'flocculate'
comes from the Latin
floccus, meaning an
aggregate (originally of
sheep, hence their col-
lective noun of 'flock').
see p. 313), the colloid particles rarely come close enough for
any attractive interactions to develop. But the colloid particles
are attracted strongly by ionic charges along the backbone of
the silicate (principally, existing as pendent groups of -0~). The
Coulombic attraction between the microscopic colloid particles and
the silicate polymer increases the weight of the chains, promoting
its settling under the influence of gravity. We call this process
flocculation.
Secondly, the water-treatment companies add aluminium compounds, such as alu-
minium sulphate, AI2 (804)3, to break the colloid. The aluminium ion Al 3+ ( aq ) has
a large surface charge, and adsorbs strongly to the surface of any colloidal particle,
particularly those possessing a negative surface charge. The net surface charge of such
a 'modified' colloid particle is neutralized, thereby obviating the repulsions between
colloid particles, and enhancing the rate of sedimentation. This second method of
breaking a colloid is termed coagulation. Changing the solution pH will also change
the surface charge of colloid particles, again breaking the colloid.
Coagulation is thermodynamically irreversible, but flocculation is reversible.
Aside
It is fairly common to employ the words 'flocculation' and 'coagulation' instead of
'aggregation'. Strictly, aggregation means particles of, for example, colloid coming
together without external assistance.
Conversely, flocculation implies those aggregation processes effected by the inter-
twining of fibrous particles, for example in the wool trade, or the entrapment of silt
particles in foul water, as above.
10.4 Association colloids: micelles
Why does soapy water sometimes look milky?
Association colloids
Soapy water often looks milky. This milky appearance indicates that a colloid has
formed, with one phase suspended in another. But soapy water introduces another
complexity: whereas water containing a lot of soap does indeed have a cloudy
aspect, dilute solutions of soap are not cloudy, but clear. We see this behaviour
when washing our face in the sink (yielding a concentrated and, therefore, milky
soap solution) or washing in a larger volume of water, such as a bath, when the
water can remain clear. Whether or not a soap solution forms colloid depends on its
concentration.
ASSOCIATION COLLOIDS: MICELLES
515
The word 'micelle'
comes from the dimi-
nutive of the Latin word
mica, meaning 'crumb'.
We are dealing here with a new type of colloid: the micelle.
A micelle forms when molecules aggregate to form particles sus-
pended in solution (i.e. a colloid). A micelle is often called an
association colloid, because it forms colloid by the association of
a discrete number of components. Such colloids form by the self-
organizing (or 'self-assembly) system. The most common micelles
form from detergents and surfactants, but alcoholic drinks contain-
ing absinth (Pernod, Ricard, Ouzo, etc.) also form micelles, thereby
explaining why Ouzo becomes cloudy after adding water.
Although the micelle particle is an aggregate, it behaves like a
liquid; indeed, it is often convenient to regard these micelle aggre-
gates as a separate phase. For this reason, we usually class micelles
as a liquid-in-liquid colloid.
The micelle is an aggregate containing several molecules: we will call them 'mono-
mers', M. In this example, each monomer is a molecule of soap. In general, the micelle
'particle' forms via the step-wise addition of n monomer molecules:
The colloid is held
together with inter-
actions of various types,
as described on p. 517.
M + M (n
(«-D
M„
(10.13)
It is impossible to specify, let alone quantify, all the equilibrium steps, so we formulate
our ideas in terms of approximate models.
One of the models best able to describe the properties of micellar
colloid solutions is the closed-association model. In it, we start
by assuming the colloid comprises n molecules of monomer. We
approximate by saying the colloid forms during a single step:
«M
M„
(10.14)
The long-forgotten
English chemist Cooper
(1759-1839) was the
first to give a sat-
isfactory explanation
of colloid formation
(including the CMC,
as below).
The equilibrium constant of colloid formation (Equation (10.14))
is given by
[M„]
^(micellation) = (10.15)
As n increases, so the value of A'(miceiiation) changes quite dramatically with concen-
tration. Being an equilibrium constant, the value of ^(miceiiation) is fixed (at constant
temperature), so there is effectively no micelle at low concentration but, at a sharply
defined concentration limit, just about all of the monomer M becomes bound up
within the micelle.
This concept is well illustrated in Figure 10.11, which shows the way n dictates
the proportion of monomer units incorporated into the micelle colloid. As the value
of n increases, the steeper the graph in Figure 10.1 1 becomes. For example, a micelle
formed from 30 monomer units has a very steep concentration dependence, i.e. almost
a 'step', but below a certain concentration - which is characteristic of the temperature
and the monomer - about 10 per cent of the colloid is micellar. Above this limit,
however, more than 90 per cent is bound up within the micelle.
The concentration range over which the transition 'monomer — »■ micelle' is effected
decreases significantly as the value of n increases.
516
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
2
o
03
t=
3
c
"D
0.5
CMC
MC
[surfactant] tota / dm -3
Figure 10.11 As the aggregate number n increases, so the fraction of the added surfactant that
goes into the micelle (as 'y') varies more steeply with total concentration of surfactant monomer
(as 'x'). The critical micelle concentration (CMC) is the midpoint of the region over which the
concentration of the micelle changes (Reproduced by permission of Wiley Interscience, from The
Colloidal Domain by D. Fennell Evans and Hakan Wennerstrom)
The CMC relates to
the total concentration
of monomer.
The graph in Figure 10.11 shows well how a micelle would forms at a sin-
gle concentration, rather than over a range. Although a single concentration is a
theoretical construct, the concentration range can be remarkably
narrow.
We call the centre of the concentration range the critical micelle
concentration (CMC). As an over-simplification, we say the solu-
tion has no colloidal micelles below the CMC, but effectively all
the monomer exists as micelles above the CMC. As no micelles
exist below the CMC, a solution of monomer is clear - like the
solution of dilute soap in the bath. But above the CMC, micelles
form in solution and impart a turbid aspect owing to Tyndall light
scattering. This latter situation corresponds to washing the face in
a sink.
The number of monomers in a typical micelle lies in the range
30-100, which is large enough for a reasonably well defined CMC
to show. The value of n depends on the choice of monomer,
with large monomer molecules generally forming larger micelles,
according to Table 10.3.
The properties of the micelle are very well defined; so, for
example, the maximum micelle radius is a simple function of the
hydrocarbon chain in the monomer.
In fact, a truly micellar
solution will comprise
micelles having a sta-
tistical spread of n
values and hence a
range of molar masses.
For example, a sul-
phonic acid with a
chain of 14 CH 2 units
produces a micelle of
n = 80, albeit with
a standard deviation
a = 16.5
ASSOCIATION COLLOIDS: MICELLES
517
Table 10.3 A micelle comprises n monomer units. The value of n
depends on the length of the monomer chain
Surfactant
CMC/mol dm"
Temperature/ C
C 6 H 13 S0 4 Na
0.42
25
17
C 7 Hi 5 S0 4 Na
0.22
25
22
C 12 H25S0 4 Na
8.2 x 10- 3
25
64
C 14 H 29 S0 4 Na
2.05 x 10~ 3
40
80
What is soap?
The physical nature of colloids
One of the principal causes of the English Civil War was the sudden imposition
of steep taxes on soap, in 1637. The people could not afford to clean themselves,
and rioted.
But how does soap work; why did the people want soap rather
than water alone? To explain the mode by which soap works, we
start by describing the structure of soap molecules in water. A
good example of such a soap is the dual-nature molecule, sodium
dodecyl sulphate (SDS, III), with its long, snake-like structure. Its
head is an ionized sulphonic acid group -SOJNa + , ion paired with a sodium cation
to yield a salt. The remainder of the molecule is a wholly non-ionic straight-chain
alkyl group. We give such molecules the general name of surfactant.
The word surfactant
is an abbreviation of
'surface-active agent'.
(Ill)
O
II
^O-S-0 Na +
II
O
The sulphonic acid 'head' is capable of electrostatic interactions
just like any other charged species; the 'tail' can only interact
weakly via induced dipoles of the London dispersion force type (see
p. 47). For this reason, the head and tail of SDS behave differently,
and often in a contradictory manner. In Chapter 5 we saw how
polar and non-polar species tend not to associate. As an example,
mixing benzene and water forms separate layers, i.e. forming sep-
arate phases, with the organic layer floating above the aqueous. In
a similar way, the hydrocarbon ends of the SDS molecules seek
to avoid contact with water. We say they are hydrophobic. The
sulphonic acid groups differ and seek solvation by water: they are
hydrophilic. In common language, we might say they are 'water
hating' and 'water loving' respectively.
The words 'hydro-
phobic' and 'hydro-
philic' derive from
Greek roots: 'hydro'
comes from hudoor
meaning water and
'phobic' comes from the
Greek phobos meaning
fear or hate (hence
the English word 'pho-
bia'). The Greek philos
means love or friend-
ship.
518
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
O^A/WW
QWvWW
f SDS *
Polar 'head' of
sulphonic acid
Non-polar 'tail'
of hydrocarbon chain
Figure 10.12 Micelles of sodium dodecyl sulphate (SDS) comprise as many as 80 monomer units.
The micelle interior comprises the hydrocarbon chains, and is oil like. The periphery presented to
the water of solution is made up of hydrated hydrophilic sulphonic acid groups
The interactions between SDS and water represent a compromise between the
extremes of complete phase separation (as happens when benzene and water mix) and
molecular dispersion (SDS in dilute aqueous solution). A micelle forms. To minimize
the energetically unfavourable interactions with water, SDS molecules aggregate to
form a variety of microscopic structures, such as the 'dandelion head' in Figure 10.12.
We can summarize the principal properties of these aggregates, saying: they form
spontaneously at a well-defined concentration, the CMC (see p. 516); and adding
more monomer to the solution yields more micelles, each colloidal particle having
the same size, and ensuring the concentration of free monomer does not change.
We now consider the two competing forces affecting the forma-
tion of these micelles. On the one hand, the hydrophobic hydrocar-
bon end of the molecule is taken away from the polar environment
of water into the oil-like interior of the micelle - a process which
provides the driving energy of the micellation process. On the
other hand, repulsions are minimized between the charged heads
of each monomer as they come together during micellation, so
the hydrophilic sulphonic acid groups are spatially well separated.
These two processes compete: the first has the effect of increasing
the micelle size, and the second has the effect of decreasing it.
The aggregation process stops when the micelle reaches its opti-
mum size (e.g. see value of n in Table 10.3). This competition
helps explain why adding more monomer does not augment the existing micelles, but
merely generates further micellar colloid in solution.
The properties of SDS in dilute solutions generally remind us of a simple ionic
solute much like NaCl or KNO3. But above a concentration of 8 x 10~ 3 moldm -
the tensions inherent in the two distinct natures of SDS become apparent, and the
SDS becomes a soap.
The association forces
between juxtaposed
surfactant monomers
is physical, not chem-
ical, so the motion of
the hydrocarbon tails
within a micelle is sim-
ilar to the local motion
in a sample of pure
hydrocarbon.
Why do soaps dissolve grease?
Detergency
We are now able to describe the way soaps clean the skin. Soaps were first men-
tioned on p. 239, when we introduced the action of aqueous alkali on the skin,
ASSOCIATION COLLOIDS: MICELLES
519
reacting chemically with oils, fats and greases in a process we
call 'saponification'. But the best soaps are not inorganic alkalis,
they are organic salts of long-chain carboxylic acids. They are also
milder on the skin.
We also need soaps for cleaning the crockery after a meal of
chips, pizza or greasy sausages. Such cleaning can be difficult and
time-consuming unless we first add to the water an effective soap
or detergent such as 'washing-up liquid'.
Each micelle has a polar periphery and an oil-like core. When
molecules of monomer collide with the solid surface of, say, a dirty
plate, the non-polar ('hydrophobic') end adsorbs to the non-polar
grease. Conversely, the polar ('hydrophilic') end readily solvates
with water. Soon, each particle of oil or grease is surrounded
with a protective coating of surfactant monomer, according to
Figure 10.13.
Having 'disguised' each particle of oil or grease, it can readily
enter solution while sheathed in its water- attracting 'overcoat' of
surfactant. And if the oil particles enter the solution, then the oil
is removed from the plate, and is cleaned.
After leaving the plate, the grease particle remains encapsulated
within the micelle, surrounded with the oil-like hydrocarbon chains
of the soap monomers. The soap cleans the plate by allowing the
grease to enter solution.
A compound able to
'dissolve' grease by
forming micelles is
called a detergent.
Naturally occurring
detergents are also
called soaps.
We often describe the
structure of this coat-
ing as a bi-layer, with
the inner (oil-facing)
part made up of water-
repelling hydrocarbon
chains, and the outer
(water-facing) layer
comprising the sul-
phonic acid groups.
A detergent forms a
micelle from an other-
wise insoluble phase.
Why is old washing-up water oily when cold but
not when hot?
The Krafft point temperature
Washing up the dishes after a meal demonstrates how the temperature affects the
colloid's stability. The dirty water is cloudy after washing up because it contains
Polar 'head' of
sulphonic acid
S SDS V
Non-polar 'tail'
of hydrocarbon chain
Adsorbed layer of SDS
Substrate of
non-polar oil or grease
Figure 10.13 The non-polar tail of an SDS molecule readily adsorbs to the surface of non-polar
oil or grease on a plate or hand. The polar sulphonic acid heads point toward the solution, and
are hydrated
520
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
micelles of surfactant; each micelle particle contains a tiny particle of grease. But, after
cooling the washing water below about 35 °C, an oily scum forms on its surface and
the water beneath becomes clear. Re-warming the water causes the oily compounds
that make up the scum to re-enter the water, to re-form the colloid.
Micelles only form above a crucial temperature known as the Krafft point temper-
ature (also called the Krafft boundary or just Krafft temperature). Below the Krafft
temperature, the solubility of the surfactant is too low to form micelles. As the tem-
perature rises, the solubility increases slowly until, at the Krafft temperature 7k, the
solubility of the surfactant is the same as the CMC. A relatively large amount of sur-
factant is then dispersed into solution in the form of micelles, causing a large increase
in the solubility. For this reason, IUPAC defines the Krafft point as the temperature
(or, more accurately, the narrow temperature range) above which the solubility of a
surfactant rises sharply.
In reverse, the surfactant precipitates from solution as a hydrated crystal at temper-
atures below 7k, rather than forming micelles. For this reason, below about 20 °C,
the micelles precipitate from solution and (being less dense than water) accumulate
on the surface of the washing bowl. We say the water and micelle
phases are immiscible. The oils re-enter solution when the water
is re-heated above the Krafft point, causing the oily scum to pep-
tize. The way the micelle's solubility depends on temperature is
depicted in Figure 10.14, which shows a graph of [sodium decyl
sulphate] in water (as 'y') against temperature (as 'x').
To peptize means a
bulk phase enters solu
tion as a colloid (here,
as a micelle).
0.3
0.2
a 0.1
o
c
o
O
Solubility curve
Hydrated crystals
Micelles
t ^CMC curve
Monomers in H 2
J I I
10
20 'k
30
40
Temperature/°C
Figure 10.14 Graph of [surfactant] (as 'y') against T (as V) for sodium decyl sulphate in water.
The Krafft temperature is determined as the intersection between the solubility and CMC curves,
yielding a 7k of about 22 °C. At lower temperatures, the micelles convert to form hydrated crystals,
which we might call 'scum' (Reproduced by permission of Wiley Interscience, from The Colloidal
Domain by D. Fennell Evans and Hakan Wennerstrom)
ASSOCIATION COLLOIDS: MICELLES 521
Table 10.4 Krafft point temperatures 3 for sodium alkyl sulphates in water
Number of carbon atoms 10 12 14 16 18
Krafft temperature/ C 8 16 30 45 56
a Data reproduced with permission from J. K. Weil, F. S. Smith, A. J. Stirton and
R. G. Bristline, J. Am. Oil. Chem. Soc, 1963, 40, 538.
The value of 7k is best determined by warming a dilute solution of surfactant, and
noting the temperature at which it becomes clear. Table 10.4 lists the Krafft points
for a series of colloidal systems based on aqueous solutions of sodium alkyl sulphate
(cf. structure III).
Aside
Tap water in some parts of the country contains large amounts of calcium and magnesium
ions. The calcium usually enters the water as it seeps through limestone and chalk, since
these rocks are sparingly soluble.
One of the simplest ways to tell at a glance if our tap
water contains much Ca 2+ - we say it is 'hard' - is to
look at the bath after letting out the water. Surfactants
form micelles with the calcium di-cation at tempera-
tures above 7k, i.e. in a hot bath. But after cooling to
a temperature of about 20 °C, the micelles precipitate
to yield hydrated crystals - which we observe as a ring
of 'scum' along the waterline.
A quick glance at Figure 10.14 suggests a simple way of saving time when cleaning
away the bath ring is to scrub with hot water since the solid crystals convert back
to water-soluble micelles above 7k. The micelles can flow away from the bath after
removing the plug.
The brown colour of
this bath ring generally
derives from occlusion
of dirt and particles of
skin within the crystals.
Why does soap generate bubbles?
Surfactants and surface tension
A surfactant yields bubbles by decreasing the surface tension y of the solution. Most
detergents are surfactants. A detergent such as a long-chain alkyl sulphonic acid
dramatically decreases the surface tension y of water by adsorbing at the air- water
interface. Stated another way, the sulphonyl groups attach to the meniscus of the
water via strong hydrogen bonds. The polar sulphonyl group points toward the water,
while the long alkyl chain points away, into the air above the water. Figure 10.15
shows a schematic representation of the structure, which looks much like a long-pile
carpet in miniature.
522
ADSORPTION AND SURFACES, COLLOIDS AND MICELLES
Air
Meniscus
■ Water
OWAW
/ \
Polar 'head' of Non-polar tail'
sulphonic acid of hydrocarbon chain
Figure 10.15 Molecules of surfactant adsorb at the air - water interface, thereby decreasing the
surface tension y . The polar sulphonic acid heads point toward the solution, thereby ensuring some
extent of hydration
The water forms beads,
rather than any other
shape, because a sp-
here hasthe lowest pos-
sible surface area per
unit volume, thereby
minimizing the surface
area presented to the
atmosphere.
The first effect of decreasing the surface tension is to make it
easier for the water to adhere to a surface, be it a crockery bowl or
waste food on a plate; we say the 'wetting properties' of the water
are enhanced. Of course, the phrase 'adsorbed water' is an over-
simplification. As soon as a dish enters a detergent solution, the
molecules of detergent residing on the surface of the water realign
to form a layer between the water and the dish. This multi-layer
structure helps explain detergency: water forms a thin, continuous
layer on a surface if detergent is added; otherwise, the water forms
small, spherical beads.
Why does detergent form bubbles?
Surfactants and bubbles
A surfactant is a soap.
The air in water colloid formed during the turbulent flow of water,
such as water passing over a dam or running water into a bath,
is a particularly unstable colloidal system. The bubbles rise to the
surface within a few seconds to leave a clear, homogeneous liquid. But adding a
soap or detergent completely changes the properties of the water, allowing the ready
formation of an air-in-water colloid. Why is there a difference?
Detergents aid the removal of dirt. Commercial synthetic detergents were first devel-
oped in the 1950s. Detergents act mainly on the oil-based films that trap dirt particles.
Most detergents have an oil-soluble portion (usually a hydrocarbon chain), and a
water-soluble portion, which is generally ionic.
Detergents act as emulsifiers, breaking the oil into tiny droplets, each suspended in
water. The disruption of the oil film allows the dirt particles to become solubilized
(they peptize). Soap, the sodium or potassium salt of long-chain fatty acids, is a good
detergent, although it often forms insoluble compounds with certain salts found in
hard water, thus diminishing its effectiveness.
ASSOCIATION COLLOIDS: MICELLES
523
Detergents are classified into four groups:
(1) Anionic, or negatively charged, e.g. soaps.
(2) Cationic, or positively charged, e.g. tetra-alkyl ammonium chloride (used as
fabric softeners).
(3) Non-ionic, e.g. certain esters made from oil (used as degreasing agents in
industry).
(4) Zwitterionic, containing both positive and negative ions on the same
molecule.
Surface tension is the tendency of liquids to reduce their exposed
surface to the smallest possible area. A single drop of water - such
as a rain drop - tries to take on the shape of a sphere. We attribute
this phenomenon to the attractive forces acting between the mole-
cules of the liquid. The molecules within the liquid bulk are attrac-
ted equally from all directions, but those near the outer surface
of the droplet experience unequal attractions, which cause them to
draw in toward the centre of the droplet - a phenomenon experi-
enced as a tension.
Adding a surfactant such as decadodecylsulphonic acid to the solution changes the
magnitude of the surface tension.
The high surface ten-
sion of a water menis-
cus explains why var-
ious small insects are
able to skate across the
surface of a pond.
Answers to SAQs
1 Introduction to physical chemistry
1.1 If the resistance per degree is 6 x 10~ 6 £2°C _1 and R = 3.0 x 10~ 4 £2 at
0°C, then T = 140 °C.
1.2 'length' is the variable, '3.2' is the number, 'k' is the factor and 'm' is the
unit.
1.3 T = (273.16 + 30) K = 303.16 K.
1.4 (287.2 - 273.16) K = 14.04 °C.
1.5 Draw a graph of volume V (as 'y') against temperature T (as 'x').
1.6 From Equation (1.8), V 2 = 1.17 dm 3 .
1.7 From Equation (1.11), V 2 = 0.4 dm 3 .
1.8 We rearrange Equation (1.6) to yield — = — , so a tenfold rise in
V2 T 2
temperature results in a tenfold increase in volume.
1.9 From Equation (1.13) and remembering that 1 dm 3 = 10~ 3 m 3 , p 2 =
13.7 x 10 6 Pa.
1.10 From Equation (1.14), T 2 = 508 K.
1.11 The area inside the cylinder is 297 cm 2 (i.e. 297 x 10~ 4 m 2 ). Then, using
Equation (1.15), F = 1.68 x 10 7 Pa.
1.12 The 'room temperature energy' is | x R x T, so energy = 3.3 kJmol - .
526 ANSWERS TO SAQs
2 Introduction to interactions and bonds
2.1 V = 1.5 x 10" 2 m 3 = 15 200 cm 3 .
2.2 In each case, we assume the bond will be polar if the separation between
the two electronegativities is large, and non-polar if the separation is
small, (a) HBr is polar; (b) SiC is non-polar; (c) SO2 is non-polar; (d) Nal
is very polar, indeed, fully ionized.
2.3 302 K.
2.4 p = 2.18 x 10 6 Pa; p is about 0.5 per cent lower when calculated with the
ideal-gas equation, Equation (1.13).
2.5 The molar mass of water is 18 gmol -1 , so 21 g represents 1.167 mol. The
energy liberated is then 1.167 mol x 40.7 kJmol -1 = 47.5 kJ.
3 Energy and the first law of thermodynamics
3.1 The molar mass of water is 18 gmol - . CV in J K _1 g _1 x 18 gmol - =
75.24 JK _1 mol~ . (Note how the units of g and g _1 cancel here.)
3.2 The rise in temperature is 80 °C. 1.35 x 10 3 g of water represents 75 mol.
The energy necessary is, therefore, 80 °C x 75 mol x 4.18 JK _1 mol -1 =
25.08 kJ.
3.3 Inserting values into energy is V x / x t (Equation (3.10)) yields 1920 J.
3.4 The energy produced by the electrical heater (Equation (3.10)) is 108 kJ;
we call this value C. The molar mass of anthracene is 178 gmol -1 , so
0.40 g represents 2.25 x 10~ 3 mol. The molar enthalpy liberated is,
therefore, 4806 kJmol" 1 .
3.5 From Equation (1.13), 0.031 m 3 is the same as 31 dm 3 .
3.6 The change in volume AV is 1.99 dm = 1.99 x 10~ 3 m 3 . Using Equation
(3.12), w = 199 J.
3.7 The fully balanced equation is 1C 2 H 6 + 3.50 2 -> 2C0 2 + 3H 2 0. From
Equation (3.22), the value of AC p is 144.9 JK _1 mol -1 . (By 'per mole'
here we mean per mole of reaction.) Then, from Equation (3.21), the value
of A// C e at 80 °C is -1550.8 kJmol" 1 .
3.8 A//- = fl£ (NO2) - {2H° (NO) + H* (0 2 )}
3.9 The fully balanced equation is C 6 H 12 6 + 90 2 -► 6C0 2 + 6H 2 0. Using
Equation (3.29), the value of A// C e is -2804.8 kJmol" 1 .
3.10 Having drawn a suitable Hess-law cycle, AH° = —0.21 kJmol - (or
-210 J mol -1 ).
3.11 A H^ce) = -H85 kJ mol" 1 .
ANSWERS TO SAQs 527
Reaction spontaneity and the direction
of thermodynamic change
4.1 The liquid water will have a lower entropy than the gaseous water, so the
process will not be thermodynamically spontaneous.
4.2 The number of moles decreases from 1.5 mol of gas to 1 mol. Assuming
equivalent entropies per mole, the entropy decreases. AS will be negative,
so we do not expect a spontaneous process.
(273 K\
I =2.2 J K" 1
258 K/
mol -1 .
/350 K\ , ,
4.4 AS = 25.8 In + 1.2 x 10" 2 (350 K - 300 K) = 4.58 JKT 1
V300 KJ
mol -1 .
4.5 AH, - 230.6 = kJ mol -1 , AS r = -423.2 JKT 1 mol -1 , and hence
AG r = -104.5 kJmol _1 .
4.6 From Equation (4.21), AG r - 4.09 = kJ mol
-l
4.7 (1) Taking straight ratios yields ^(vacuum) = 3682 Pa. (2) From Equation
(4.39), AG = -8.2 kJmor 1 .
4.8 The left-hand side is a logarithm and, therefore, has no units. AH r has
units of J mol -1 and R has units of J K _1 mol -1 , so AH Y -r R has units of
K. The units of the reciprocal temperature in the right-hand bracket is
K _1 , which cancels with the K from AH T -f- R; therefore, the right-hand
side is also dimensionless.
4.9 Remembering to convert from kJ to J, K = 127.
4.10 (1) AS = 560 JK _1 mol~ . Its positive value implies a thermodynamically
spontaneous process. (2) From Equation (4.62), AG r at 332 K is —220.8
kJmol" 1 .
4.1 1 Using Equation (4.74), AG e is negative only above 985 K. Maintaining this
high temperature explains why synthesis gas is not economically viable.
4.12 Using Equation (4.78), AH = 14.7 kJmol" 1 .
4.13 Draw a van't Hoff graph of In K (as 'y') against \IT (as 'x'), then
multiply the gradient by —R to obtain AH = 36.7 kJmol -1 .
5 Phase equilibria
5.1 About 100 mmHg, which equates to 1.31 x 104 Pa (0.13 x p & ).
5.2 A fivefold increase in pressure means p 2 — p\ = 5p e — p & = 4/? e . Using
Ap® as the 'dp' term in Equation (5.1) yields A V m = 75.6 x 10~ 6 m 3 mol -1 .
528 ANSWERS TO SAQs
5.3 The pressure p = force 4- area (Equation (1.15)), so p = 1000 Nvlm 2
(he has two snow shoes) = 10 3 Pa. Inserting values again into
Equation (5.1) yields AT = —7.3 x 10~ 5 K, i.e. unnoticeable.
5.4 From Equation (5.5), and remembering to convert from kJ to J,
T 2 = 352.9 K (=79.8 °C).
5.5 We start by writing the equilibrium constant K as ' [sucrose] ( water ) -r
[sucrose](CHCi 3 )\ which equals 5.3. So for every 6.3 portions of sucrose,
one portion dissolves in the chloroform and 5.3 in the water. One portion
out of 6.3 is 15.8 per cent.
5.6 Remember to convert the volume of water into a mass of 900 g. From
Equation (5.17), AT = 0.305 K and r (freeze) = 273.45 K.
5.7 10 g of N 2 is 0.357 mol and 15 g of CH 4 is 0.938 mol.
p {CHi) = 0.357 H- (0.357 + 0.938) = 0.275 or 27.5 per cent.
5.8 From Equation (5.22), p( to tai) = 0.286 x p e .
5.9 From p(t tai) = 0.286 x p & , the vapour comprises 69 per cent toluene.
6 Acids and bases
6.1 (a) base = carbonate and acid = bicarbonate; (b) base = H 2 EDTA 2 ~ and
acid = H 3 EDTA"; (c) acid = HN0 2 and base = N0 2 ".
6.2 pH = 0.7.
6.3 [HC1] = 5 x 10" 7 mol dm -3 when pH = 6.31 and 4.79 x 10" 7 mol dm -3
when pH = 6.32.
6.4 [HN0 3 ] = 6.3 x 10" 2 mol dm -3 .
6.5 The exponent is —5, so the pH is +5.
6.6 pH=11.8.
6.7 [H 3 + ] = 10 -9 mol dm" 3 , so [OH~] = 10 -5 (because K w = 10 -14 ).
6.8 pH = 12.
6.9 K a = K w + K b = 5.75 x 10" 10 .
[H 3 EDTA-][H+] [H 2 EDTA 2 -][H+]
[H4EDTA] (> [H 3 EDTA _ ]
[H!EDTA 3 -][H+] [EDTA 4 -][H+]
[H 2 EDTA 2 "] [HiEDTA 3- ]
6.11 Using Equation (6.43), V( a ikaii) = 8.6 cm 3 .
6.12 With s = j and using Equation (6.46), V( a ikaii) = 12.5 cm 3 .
ANSWERS TO SAQs 529
6.13 Using Equation (6.50), pH = 9.04.
6.14 The pH is 9.22 before adding any alkali. 8 cm 3 of this alkali contains
8 x 10~ 4 mol of NaOH, so 8 x 10~ 4 mol of NH 3 is formed and
8 x 10~ 4 mol of NH4" 1 " is consumed. Accordingly, there are 0.0508 mol of
NH 3 and 0.0492 mol of NH 4 + . Then, from Equation (6.50), pH = 9.23.
7 Electrochemistry
7.1 1 mol of electrons = 1 F, so 10~ 9 F are passed, which represent
6.022 x 10 14 electrons.
7.2 If n = 1, then 1 F generates 1 mol, so 10~ 10 mol are formed.
7.3 | mol.
7.4 The iron couple is the more positive, so © Co( S )|Co 2+ ( aq )||Fe 3+ (aq )|Fe( s )
7.5 Using Equation (7.15), AG (ceU ) = -289 kJmol" 1 .
d(emf) . ,
7.6 = 2.79 x 10" 4 VK" 1 using Equation (7.20), so
6.T
AS(ceii) = 26.8 JK _1 mol -1 using Equation (7.18).
7.7 emf= 0.0648 V at 610 K.
7.8 emf= 0.0271 V at 660 K, so AS (ceU) = -146 JK" 1 mol -1 with n = 2.
7.9 AG (cii) = -212 kJmol" 1 , AS (ceU) = 67.5 JK" 1 mol -1 and
A# (ce ii) = -192 kJmol" 1 .
7.10 E Co 2 +Co .
7 -ll E BllBl -.
7.12 y± = 4/y + x y + x y_ x y_ x y_
7.13 / = 6xc.
7.14 At low ionic strengths, the -Jl term in the denominator becomes
negligible, so (1 + b-Jl) tends to unity, yielding the limiting
Debye-Huckel law.
7.15 / = 4 x c. y± = 0.064, i.e. decreases by 93 per cent.
7.16 Using Equation (7.41), a(Cu 2+ ) = 4.17 x 10 -4 .
7.17 Using Equation (7.43), £ A gCi,Ag = 0.251 V.
7.18 Using Equation (7.48), emf= 0.037 V.
7.19 Using Equation (7.49), K = 0.017 V.
7.20 emf = 0.153 V.
530 ANSWERS TO SAQs
2.303 RT
7.21 The numerical value of at 298 K is 0.059 V or 59 mV.
F
7.22 Use Equation (7.53). Subtract £j from both sides of the equation, then
manipulate the value of Slight-hand side), i-e. ^Ag+.Ag, to obtain the activity
as normal.
8 Chemical kinetics
8.1 rate = £2[CH 3 COOH][CH 3 CH 2 OH]
8.2 rate = £[Cu 2 a + q) ][NH 3(aq )] 4
8.3 rate = k[Ce ][H 2 2 ]. If each concentration doubles, then
[Ce IV ] (new) = 2 x [Ce IV ] (o id) and [H 2 2 ]( n ew) = 2 x [H 2 2 ] (o id). Inserting
the new concentrations into the rate expression will increase the new rate
fourfold.
8.4 (1) rate = k x p(N0 2 ) x p(N 2 4 )
(2) rate = k x p(S0 2 ) 2 x p(0 2 )
(3) rate = k x [Ag + ][CT]
(4) rate = jfc x [HC1] + [NaOH]
8.5 (1) Second order; (2) first order; (3) third order.
8.6 Equation (8.12) is rate limiting because it is the slow step. Its rate is
k x [ethanal] x [0 2 ] 2 .
8.7 From Equation (5.19), partial pressure = mole fraction, x x p & , so
p(0 2 ) = 0.21 x 10 5 Pa, so p(0 2 ) = 2.1 x 10 4 Pa.
8.8 rate (forward) = k x p(H 2 ) x p(Cl 2 ) and rate (backwards) = fc x p(HCl) 2 .
8.9 rate = it x [I - ] x [S 2 Og~], so rate = k 2 = 0.03 dm 3 mol" 1 s _1 .
8.10 Using Equation (8.24), and converting to seconds,
[A], = 7.39 x 10" 3 mol dm -3 .
8.11 jfei = 1.27 x 10" 4 s _1 .
8.12 Calculate the number of seconds in a year
(= 365.25 x 24 x 60 x 60 = 3.156 x 10 7 syear" 1 ), then divide
1.244 x 10 4 year" 1 by it.
8.14 Using Equation (8.27), [MV + '] ( = 0.25 x 10" 3 mol dm -3 .
8.15 If 15 per cent is consumed then 85 per cent remains, so
[A], = 0.85 x 10" 3 mol dm" 3 , yielding k 2 = 0.238 dm 3 mol" 1 s" 1 .
8.16 Draw a graph of 1/[A], (as 'y') against time t (as 'jc'). The gradient
equals k 2 , so k 2 = 6.36 x 10~ 4 dm 3 mol -1 s _1 .
ANSWERS TO SAQs 531
8.17 Because it is a 1:1 reaction, if [A] t = 0.05 after 0.5 h then
[B] f = 0.15 moldm" 3 . t = 1800 s. Using Equation (8.30),
k 2 = 2.25 x 10" 3 dm 3 mol -1 s _1 .
8.18 The length of time represents two half-lives. 10 g halves to 5 g after one
half-life; this mass halves again after a second half-life.
8.19 If 14 C remaining is 88 per cent, then the age is 1031 years. If 14 C
remaining is 92 per cent, then the age is 670 years. Quite a difference!
8.20 From Equation (8.27), 1/[A], = 1/0.5[A] . Accordingly, algebraic
addition of the two concentration terms as 'l/[A] r — l/0.5[A]o = kti/2'
1
yields 'l/[A]o = kt\/ 2 , which, after dividing by '&' yields t\/ 2
[A] k 2
8.21 t = 600 s and k' = 1.33 x 10" 3 s" 1 . k 2 = k' -=- [K 3 [Fe(CN) 6 ]]
= 2.266 dm 3 mol" 1 s" 1 .
.22 Equating Equation (8.49) to K, K = 1200.
(1 -0.4) moldm" 3
8.23 Equation (8.50) includes In
The units cancel. If
(0.7 -0.4) moldm -3
the mathematics is performed erroneously, the log becomes
ln(l 4- 0.7) = 0.357. If, however, we perform the mathematics in the two
brackets first, we obtain In (0.6 4- 0.3) = 1.20.
.24 Using Equation (8.55), and saying the left-hand side is merely ln(2), and
converting each temperature to kelvin, we calculate E a = 41.6 kJmol -1 .
.25 Using Equation (8.55), k 2 = 4.23 x 10 10 dm 3 mol" 1 s" 1 at 330 K, i.e.
about 5 per cent faster.
.26 (1) Using Equation (8.55), £ a = 53.6 kJmol" 1 and A = 1.98 x 10 12 dm 3
mol~ s _1 .
(2) Using Equation (8.56), A//* = 50.9 kJmol" 1 ; using Equation (8.61),
AG* = 56.3 kJmol -1 , and AS* = -18.1 JK^mor 1 .
(3) E a — A//* = 2.7 kJmol - , which is very close to the theoretical
value of 2.4 kJmol -1 .
9 Physical chemistry involving light:
spectroscopy and photochemistry
9.1 From Equation (9.2), A = 3 x 10~ 4 m, which is a microwave photon.
9.2 From Equation (9.3), v = 1.51 x 10 15 Hz.
9.3 From Equation (9.4), 1.0 MJmol -1 equates to an energy per photon of
1.66 x 10~ 18 J, and describes a frequency v = 2.5 x 10 15 Hz.
532 ANSWERS TO SAQs
9.4 From Equation (9.6), T = 0.63.
9.5 From Equation (9.7), A = 1.39.
9.6 From Equation (9.21). (1) jx = 1.614 x 10" 27 kg; (2) 2.947 x 10" 27 kg.
9.7 From Equation (9.21), /x = 1.577 x 10~ 27 kg. From Equation (9.20),
co = 89 118 m _1 =891 cm -1 , in the infrared region of the spectrum.
9.8 From Equation (9.2), c = A x v, A = 0.1224 m = 12.2 cm.
10 Adsorption and surfaces, colloids
and micelles
10.1 (1) We obtain the number of moles adsorbed in a monolayer as the
reciprocal of the Langmuir plot's gradient: V m = 8.49 cm 3 g _1 . From
the ideal-gas equation (Equation (1.13)), this volume equates to
3.74 x 10" 4 molg" 1 .
(2) This number of moles adsorbs to form a monolayer of area 16.7 m 2 ,
so 1 mol has the almost incredible area of 44 650 m 2 . 1 mol
comprises 6.022 x 10 23 molecules, so one molecule has an area of
44650 m 2 -=- 6.022 x 10 23 = 7.42 x 10" 20 m 2 .
10.2 (1) A//( a d S ) = —232 Jmol~ (or —0.23 kJmol - ). The mass of sample is
irrelevant. (2) This enthalpy is a mean because all enthalpies depend on
temperature (see Equation (3.19)), and the temperature was varied over
143 K.
Bibliography
The books and comments contained in this bibliography are each cited at the recom-
mendation of the author alone.
Any web address (URL) is prone to alteration without warning. All the URLs here
were correct when the bibliography was completed in Easter 2003. Please be careful
with upper and lower cases when typing in the URL on your browser.
General bibliography
The best selling textbook of physical chemistry in the world is undoubtedly Atkins 's
Physical Chemistry. The latest edition is the seventh by P. W. Atkins and Julio de
Paula, Oxford University Press, Oxford, 2002. Many students will find it rather math-
ematical, and its treatment is certainly high brow. Its 'little brother' is Elements of
Physical Chemistry (third edition), P. W. Atkins, Oxford University Press, Oxford,
2001, and is intended to overcome these perceived difficulties by limiting the scope
and level of its parent text. Both are thorough and authoritative.
Walter Moore's two books Physical Chemistry, Longmans, London, 1962, and its
abridgement Basic Physical Chemistry, Prentice Hall, London, 1983, both deserve
their popularity. Although Moore is rigorous, he never loses sight of his stated aim
to teach his subject, perhaps explaining why his prose is just as informative as his
mathematics.
Physical Chemistry (third edition) by G. W. Castellan, Addison Wesley, Reading,
MA, 1983, is not much in vogue these days, in part because many of his symbols do
not conform to the IUPAC system. Castellan's strength is his explanations, which are
always excellent. His mathematical rigour is also notable.
Worked examples
Several texts approach the topic by means of worked examples. Physical Chemistry
(2nd edition), C. R. Metz, McGraw Hill, New York, 1989, is a member of the Schaum
534
BIBLIOGRAPHY
'out-line' series of texts, and Physical Chemistry, H. E. Avery and D. J. Shaw,
Macmillan, Basingstoke, 1989, is part of the 'College Work-out Series'. Both books
are crammed with worked examples, self-assessment questions, and hints at how to
approach typical questions. Avery and Shaw is one of the few general textbooks on
physical chemistry that a non-mathematician can read with ease.
Molecular chemistry texts
Many books approach physical chemistry from the molecular level, looking first at
atoms, ions and molecules in isolation, and only then analysing the way such species
interact one with another. In this approach, macroscopic properties are shown to relate
to the underlying microscopic phenomena. Such books are strong on quantum mechan-
ical and statistical mechanical theory, implying a far more conceptual approach. They
are always far more mathematical than texts majoring on phenomenological chemistry.
Their approach is, therefore, the exact opposite to that taken in this book. Some (possi-
bly most) students will struggle with them, but those whose appetites have been whet-
ted will find ultimately the 'microscopic to macroscopic' route is a far more powerful
way to understand the world and its physical chemistry. Physical Chemistry: A Molec-
ular Approach, Donald A. McQuarrie and John D. Simon, University Science Books,
Sausalito, CA, 1997, is a good example of this kind of approach, and comes highly rec-
ommended. Principles of Chemistry, Michael Munowitz, W. W. Norton, New York,
2000, comes from the same conceptual stable, and is slightly less mathematical. Phys-
ical Chemistry, John S. Winn, HarperCollins, New York, 1995, is also quite good.
As an adjunct, Theoretical and Physical Principles of Organic Reactivity, Addy
Pross, Wiley, New York, 1995, presents a penetrating analysis of the way reaction
profiles help the understanding of the physical chemist.
Experimental chemistry
Accurate and reliable determination of physicochemical data lie at the heart of
our topic, but good books addressing this aspect are surprisingly difficult to find.
Experimental Physical Chemistry: A Laboratory Textbook (second edition), Arthur
M. Halpern, Prentice Hall, Upper Saddle River, NJ, 1988, is one of the better books.
It does presuppose a thorough understanding of fundamental methods of analysis, but
it includes all the necessary theory for each of its 38 experiments.
Also worth a glance is Practical Skills in Chemistry, by John R. Dean, Alan.
M. Jones, David Holmes, Rob Read, Jonathan Weyers and Allan Jones, Prentice
Hall, Harlow, 2002. Being a general text, the techniques required by the experimental
physical chemist occupy a relatively small proportion of the book.
Popular science
Many 'fun' science books are now available. For example, try Len Fisher's How to
Dunk a Doughnut: The Science of Everyday Life, Weidenfield and Nicholson, London,
BIBLIOGRAPHY 535
2002. Gary Snyder's The Extraordinary Chemistry of Ordinary Things (third edition),
Wiley, New York, 1998, has the associated Website http://www.wiley.com/college/
extraordinary. Louis A. Bloomfield's book How Things Work (second edition), Wiley,
New York, 2001, aims more at an understanding of physics; it can be investigated
electronically at the Website http://www.wiley.com/college/howthingswork. This latter
site has a message board, so you can ask him your own questions. For a related
physics-based Website, try http://www. howstujfworks.com.
The history of chemistry
The late J. R. Partington's four-volume work, A History of Chemistry, Macmillan,
London, completed in 1970, must be counted as the work on the subject. Volume IV
recounts the history up to about 1960. He is enormously learned and erudite. He is
also authoritative; the book is well supported with footnotes.
For another masterly introduction, try The World of Physical Chemistry by Keith
Laidler, Oxford University Press, Oxford, 1995. Laidler is himself a proficient physi-
cal chemist. It represents a complete survey of the scientific development of physical
chemistry from the 17th century to the present day. Excellent.
Laidler' s next book To Light Such a Candle: Chapters in the History of Science
and Technology, Oxford University Press, Oxford, 1997, explores the links between
science and technology. Laidler follows the Victorian historian Carlyle's advice that
'history is the biography of great men', so we find chapters on Maxwell, Faraday,
Planck and Einstein, among others. It is a well-presented book, and highly informative.
Most recently, Laidler published Energy and the Unexpected, Oxford University
Press, Oxford, 2003. This book falls more readily into the popular science genre, but
it does have a historical emphasis.
The Website http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians is hosted
by St Andrew's University in Scotland, and cites biographies of several hundred
scientists, including most of the 'greats' mentioned in this text.
Finally, The Royal Society of Chemistry's magazine Chemistry in Britain has an
article on the history of chemistry in most issues.
Introduction
Julian of Norwich
One of the best versions of Julian of Norwich is Revelations of Divine Love, Pen-
guin, Harmondsworth, 1966, with Clifton Wolters's masterly introduction. Julian's
Way by Rita Bradley, HarperCollins, London, 1992, brings out the allusions to the
interconnectedness of things.
Definition*
Several sources were used regularly as sources of the definitions found in this
text. The IUPAC Website http://www.iupac.org/publications/compendium/index.html
536 BIBLIOGRAPHY
defines several hundred terms and concepts. Peter Atkins's little book Concepts in
Physical Chemistry, Oxford University Press, Oxford, 1995, is presented in the form
of a pocket dictionary, and is invaluable not just for definitions. The Oxford English
Reference Dictionary, Oxford University Press, Oxford, 1995, is surprisingly rich in
scientific information. Finally, try the Oxford English Dictionary, available free at
http://www. oed. com.
Linguistic introduction
Minerva, publishes Umberto Eco's masterpiece The Name of the Rose (1994) in a
superb English translation by William Weaver. A 'must read'.
Anyone wanting a gentle, humorous, introduction to the topic should read Bill
Bryson's Mother Tongue: Our Language, Penguin, Harmondsworth, 1990. More
detailed is the gem Our Language by Simeon Potter, Pelican, Harmondsworth, 1958,
which glistens with detail and is always a delight to read.
For the budding etymologist, all good dictionaries include etymological detail.
English from Latin and Greek Elements by Donald M. Ayers, University of Arizona
Press, Tuscon, AZ, 1985, is a must for any budding etymologist, and The Oxford
English Reference Dictionary and the Oxford English Dictionary (available as above)
are both invaluable.
1 Introduction to physical chemistry
General reading about variables and relationships
The conceptual frameworks underpinning physical chemistry are discussed well in
The Foundations of Physical Chemistry, in the Oxford Primer Series, Oxford Uni-
versity Press, Oxford, 1996. Each of its authors, Charles Lawrence, Alison Rodger
and Richard Compton, is an experienced teacher of physical chemistry. Its sister
volume, Foundations of Physical Chemistry: Worked Examples, by Nathan Lawrence,
Jay Wadhawan and Richard Compton, Oxford Primer Series, Oxford University Press,
Oxford, 1999, is a treasure trove of worked examples.
The zeroth law of thermodynamics
The zeroth law is described at http://www.sellipi.com/science/chemistry/physical/
thermodynamicsZzero.html, although not in any great depth.
Ink-jet printers
Peter Gregory's short article, 'Colouring the Jet Set' in Chemistry in Britain, August
2000, p. 39, concerns the fabrication of inks for PC printers. He also discusses the
BIBLIOGRAPHY
537
underlying mechanical features allowing a printer head to function. Alternatively, the
'How Stuff Works' Website at http://www.howstujfworks.com features the relevant
pages at, ~/questionl63.htm, ~/inkjet-printer.htm and ~/inkjet-printer2.htm.
King Edgar
Details concerning the reign of this remarkable and wise man are mentioned at the
informative site http://www.chrisbutterworth.com/hist/edgar.htm, although it does not
mention the legend of how the first foot rule was made. For more detailed information
about King Edgar, try The Saxon and Norman Kings, Christopher Brooke, Fontana,
London, 1963, chapter 8, or Sir Frank Stenton's magisterial volume in the Oxford
History of England, Anglo-Saxon England (third edition), Oxford University Press,
Oxford, 1971.
easuring temperature
For a superior introduction to this difficult topic, try Peter Rock's now classic book,
Chemical Thermodynamics, Oxford University Press, Oxford, 1983. The treatment in
Temperature Measurement (second edition), by Ludwik Michalski, Joseph McGhee,
Krystyna Eckersdorf and Jacek Kucharski, Wiley, New York, 2001, is aimed at engi-
neers manufacturing temperature-measuring machines, such as electrical and optical
sensors, but some of its introductory material might help.
Thunder and lightning
The sites http://www.usatoday.com/weather/thunder/wlightning.htm and http://www.
nssl.noaa.gov/edu/ltg both discuss the fascinating topic of thunder and lightning. Nei-
ther extrapolates in order to explore the validity of the gas laws. The site http://bcn.net/
~lti/j '_setsAinks.html has more scientific content, but is more difficult to navigate.
System e Internationale
One of the best scientific sites on the web is IUPAC's own site, http://www.iupac.org/
publications/compendium/index.html, which defines several hundred terms and
concepts.
The Maxwell- Boltzmann law
For biographies of these great men, try http://www-groups.dcs.st-and.ac.uk/~history/
Mathematicians/Boltzmann. html and http: //www -groups, dcs.st-and. ac. uk/~history/
Mathematicians/Maxwell. html.
538 BIBLIOGRAPHY
2 Introducing interactions and bonds
General reading
The Chemical Bond, by J. N. Murrell, S. F. A. Kettel and J. M. Tedder, Wiley, Chich-
ester, 1978, is an excellent text, and describes the interactions inherent in all bond
formation. It is a very mathematical read, and tackles the topic in terms of valence
bond and molecular orbitals: not an easy read, but well worth a try.
The content of The Forces Between Molecules, by Maurice Rigby, E. Brian Smith,
William A. Wakeham and Geoffrey C. Maitland, Oxford University Press, Oxford,
1986, is more explicitly about interactions than formal bonds. Again, it will be a
fairly austere and mathematical read. In the Oxford 'Primer' series, try Energy Levels
in Atoms and Molecules by W. G. Richards and P. R. Scott, Oxford University Press,
Oxford, 1994. It's easier than the two books above, and again helps provide some of
the background material to the subject. It is still mathematically based.
One of the better books - precisely because it is less mathematical - is Jack Bar-
rett's Structure and Bonding, RSC, Cambridge, 2001, in the new Royal Society
of Chemistry 'tutorial chemistry texts' series. Although the layout and style were
designed to make the topic accessible, be warned that even this book can be very
rigorous and looks a bit daunting.
Finally, for organic chemists wanting to look at bonds and interactions, try Mech-
anism and Theory in Organic Chemistry (third edition) by Thomas H. Lowry and
Kathleen Schueller Richardson, Harper & Row, New York, 1987, which looks at
the physical chemistry underlying so much of organic chemistry - a topic sometimes
known as 'physical-organic chemistry'. A similar approach is apparent in Orbital
Interactions Theory of Organic Chemistry by Arvi Rauk, Wiley, New York, 1994,
although all his discussions are cast in terms of orbitals.
Liquid crystals
The literature on liquid crystals and LCDs is simply vast. A good start is Thermotropic
Liquid Crystals, by George Gray, Wiley, Chichester, 1987. Gray was one of the
principal pioneers in the early days of liquid crystals.
Alternatively, those wanting more material could consult Dietrich Demus, who,
with his colleagues, has published numerous texts on the subject. The introduc-
tory chapters in volume I, Handbook of Liquid Crystals: Fundamentals, by Diet-
rich Demus, John W. Goodby, George W. Gray, Hans W. Spiess, Volkmar Vill,
Wiley-VCH, Weinheim, 1998, are all high-brow but encompass the topic with much
useful descriptive prose. The site http://www.eng.ox.ac.uk/lc/research/introf.html gives
a beautifully clear good lay-man's introduction, and http://www.eio.com/lcdhist.htm
contains dozens of super links.
For those wishing to follow up the topic and are interested in coloured LCDs,
try Chapter 5 of Peter Bamfield's Chromic Phenomena, Royal Society of Chemistry,
Cambridge, 2001.
BIBLIOGRAPHY
539
Nucleation
Nucleation is a difficult topic. The physical chemistry of small particles attracting
and aggregating is described in Chapters 9-11 of D. H. Everett's authoritative text,
Basic Principles of Colloid Science, Royal Society of Chemistry, Cambridge, 1988.
Hydrogen bonds and the structure of water
Martin Chaplin of South Bank University, London, has produced an extensive series of
Web pages concerning water, its structure and properties. The home page is located at
http://www.sbu.ac.uk/water. The structure of water is discussed at http://www.sbu.ac.
uk/water/clusters. html.
The site http://www.nyu.edu/pages/mathmol/modules/water/info_water.html hosts
a nice discussion of water, including two short video clips: (1) the quantum-
mechanically computed movement of two water molecules united by means of a single
hydrogen bond, at http://www.nyu.edu/pages/mathmol/modules/water/dimer.mpg;
(2) a short film of several hundred water molecules dancing within a cube at
http://www. nyu. edu/pages/mathmol/modules/water/water_dynamics. mpg.
Cappuccino coffee
The story about the Capuchin monk Marco d'Aviano can be found at http://www.
mirabilis.ca/archives/000710.html or the sometimes tongue-in-cheek site http://www.
sspx.ca/Angelus/2000_January/A_Politically _Incorrect_Monk.htm.
DNA and heredity
Organic Chemistry: A Brief Introduction by Robert J. Ouellette, Prentice Hall, New
Jersey, 1998, contains a super introduction to the history of DNA and heredity.
Stephen Rose's now classic book The Chemistry of Life, Penguin, Harmondsworth,
1972, goes into more depth, and includes a good discussion of H-bonds in nature
and DNA. The sites http://www.dna50.org.uk/index.asp and http://www.nature.com/
nature/dna50/ have good pictures and links.
The site http://www.netspace.org/MendelWeb/Mendel.html reproduces the full text
of Mendel's original research papers.
London dispersion forces
A good start is the long and well-referenced review article 'Origins and applica-
tions of London dispersion forces and Hameker constants in ceramics' by Roger J.
540
BIBLIOGRAPHY
French in Journal of the American Ceramics Society, 2000, 83, 2117. Its introductory
sections are excellent; its application sections are not particularly germane to us
but are interesting. It's worth downloading an electronic copy of this paper from
http://www. Irsm. upenn. edu/~frenchrh/download/0009jacersdispersionf eature.pdf.
For a brief biography of London, go to http://onsager.bd.psu.edu/~iircitano/London.
html, or read Fritz London: A Scientific Biography by Kostas Gavroglu, Cambridge
University Press, Cambridge, 1995.
Critical and supercritical fluids
Steve Howdle's short article 'Supercritical solutions' in Chemistry in Britain, August
2000, p. 23, is a good introduction to the topic.
The Web page http://www-chem.ucdavis.edu/groups/jessopAinks.html contains
many good links to other sites. Anyone wanting a more in-depth look at the topic
should consult the bi-monthly Journal of Supercritical Fluids.
Etymology of the word 'Molecule'
The etymological detail concerning Faraday and Cannizzaro come from Everett's
Basic Principles of Colloid Science, Royal Society of Chemistry, Cambridge, 1988.
The other citations in the chapter come from the Oxford English Dictionary: G.
Adams, Nat. & Phil., 1794, I, hi, 79; Kirwen, Geol. Ess., 1799, 478; W. Wilkinson,
Oult. Physiol, 1851, 9; and Tyndall, Longm. Mag., 1882, I, 30.
Nitrogen fixation
Most biochemistry texts discuss the so-called 'nitrogen cycle' in some depth. Rose
(above), for example, touches on the topic a few times. Additional information may be
found at the http://academic.reed.edu/biology/Nitrogen, http://www.infoplease.com/
ce6/ scUA0860009.html and http://helios.bto.ed.ac.uk/bto/microbes/nitrogen.htm Web-
sites.
The van der Waals equation
The page http://www.hull.ac.uk/php/chsajb/general/vanderwaals.html, at Hull Uni-
versity's Website, includes an interactive page - the 'van der Waals calculator' - to
determine values for real and ideal gases, with the van der Waals equation. The site
http://antoine.frostburg.edu/chem/senese/javascript/realgas.shtml includes a different
calculator with more variables, but is not quite so easy to use.
The page http://www. chuckiii. com/Reports/Science/Johannes _van_der_waals. shtml
contains a brief biography of van der Waals. The page http://www.hesston.edu/
BIBLIOGRAPHY 541
academic/FACULTY/NELSONK/PhysicsResearch/Waals mentions other detail, and cites
a few interesting links.
Linus Pauling
For a brief life of this brilliant man, try the Web page of the Swedish Academy of
Science, who awarded him two Nobel prizes: go to http://www.nobel.se/chemistry/
laureates/1954/pauling-bio.html. Alternatively, his authorized biography, Linus Paul-
ing, is by Anthony Serafini, Paragon House, 1991.
Etymology of the word covalent
Langmuir defined 'covalent' in Proc. Nat. Acad. Sci., 1919, V, 255. His quote (repro-
duced on p. 68) relates to sodium chloride.
Electron affinity and ionization energy
It is surprisingly difficult to find reliable values of / and £( ea )- Probably the most
extensive collection of data is Bond Energies, Ionization Potentials and Electron Affini-
ties by V. I. Vedeneyev, V. L. Gurvich, V. N. Kondrat'yev, Y. A. Medvedev and
Ye. L. Frankevich, Edward Arnold, London, 1966. The Chem Guide Website has sev-
eral good pages, e.g. look at http://www.chemguide.co.uk/atoms/properties/eas.html.
3 Energy and the first law of thermodynamics
General reading
Gareth Price's book Thermodynamics of Chemical Processes in the affordable Oxford
Primer series, Oxford University Press, Oxford, 1998, describes much of the back-
ground material discussed here, and goes into some depth.
Chapter 2 of E. Brian Smith's book Basic Chemical Thermodynamics, Clarendon
Press, Oxford, 1990, is a superb introduction to the topic. His Chapter 1 discusses
concepts such as reversibility and the broader question, 'Why do we need thermody-
namics?' His Chapter 5 covers the measurement of thermodynamic parameters.
In the new RSC 'Tutorial Chemistry Texts' series, Jack Barrett's Structure and
Bonding, RSC, Cambridge, 2001, is another superb resource because it covers many
of the thermodynamic aspects featured in this chapter. Be warned, though: the overall
treatment is a rigorous and somewhat mathematical tour of the topic.
542
BIBLIOGRAPHY
Sweat and sweating
The Website http://www.sweating.net contains generous chunks of relevant infor-
mation. Alternatively, try the page http://www.howstuffworks.com/sweat.htm at the
'How Stuff Works' site. The page http://www.howstuffworks.com/sweat3.htm intro-
duces the necessary thermodynamics, and the page http://www.howstuffworks.com/
sweat2.htm describes how sweat is made.
Lord Kelvii
The site http://scienceworld.wolfram.com/biography/Kelvin.html has a relatively short
but informative biography of Kelvin, the man. Little science is mentioned. Kelvin also
appears often in all the biographies of Joule (e.g. see below).
James Joule
For a brief but lively biography of Joule the man, try 'Heated exchanges' by Leo
Lue and Les Woodcock, in Chemistry in Britain, August 2001, p. 38. The article
well describes the confusion surrounding the formulation of the laws of thermody-
namics, explaining why the first law was defined before the zeroth, etc. The site
http://scienceworld.wolfram.com/biography/Joule.html also has a photo of the great
man. D. S. L. Cardwell has written a more detailed biography of this remarkable
man: see James Joule: A Biography, Manchester University Press, Manchester, 1989.
Cardwell's account contains sufficient scientific detail to place Joule's work in its
proper context.
Kirchhoff's Law
The Website http: //www -groups, dcs. st-and. ac. uk/^history/Mathematicians/Kirchhoff
html contains a substantial biography and several photographs of this great but
long forgotten scientist. The shorter biography at http://scienceworld.wolfram.com/
biography ZKirchhoff.html concentrates more on his discoveries in physics than in
thermochemistry.
Be warned that a Web search for 'Kirchhoff will yield dozens of pages on Kirch-
hoff's rules, which relate to electronic circuits.
Conversion of diamond into graphite
There is a wealth of information on this thermodynamic transformation, e.g. in most
textbooks of physical chemistry. The site http://chemistry.about.com/library/weekly/
aa071601a.htm has copious links, while the short Web page http: //members. tripod.
com/graphiteboy/Graphite _Diamond.htm cites a few nice details
BIBLIOGRAPHY
543
Enthalpies of reaction
The short, fun Website http://schools.matter.org.uk/Content/Reactions/BE_enthalpy
NO.html has a few examples that might amuse.
Enthalpies of combustion
Any edition of the Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL,
will contain an authoritative array of data, and all standard texts on physical chemistry
publish tables of A// C e data. The Website http://www.innovatia.com/Design_Center/
rktprop2.htm is relatively mathematical, but of good quality. Alternatively, try the site
http://www.geocities.com/CapeCanaveral/Launchpad/5226/thermo.html, which has a
fairly good treatment.
Bond enthalpies
Any edition of the Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL,
will cite some data, and all standard texts on physical chemistry will publish tables of
AH— data. Also, try the Web page http://www.webchem.net/notes/how_far/enthalpy/
BE
bondentha Ipy. h tm .
Hess's law
The site http://www.chemistry.co.nz/hess_law.htm from New Zealand has a short
biography of Hess, including photographs. The page http://dbhs.wvusd.kl2.ca.us/
Chem-HistoryZHess-1840.html has an .html copy (in English translation) of Hess's
original paper, dating from 1840.
Peppermint and cooling agents
It is relatively difficult to find details of cooling agents, but try T. P. Coultate's,
Food: The Chemistry of its Components (fourth edition), Royal Society of Chemistry,
Cambridge, 2002, p. 241.
Reaction spontaneity and the direction
of thermodynamic change
One of the better books on the topic is Peter Atkins's The Second Law: Energy, Chaos,
and Form, P. W. Atkins, Scientific American, New York, 1994, which explains the
544
BIBLIOGRAPHY
complicated and interpenetrating truths underlying this topic. Its 'popular science
book' style may irritate some readers, but will undoubtedly help many, many more.
Clausius
The Website http://www-groups.dcs.st-and.ac. uk/^history/Mathematicians/Clausius.
html contains a short biography of Clausius.
Entropy as *the arrow of time'
The idea of entropy being 'the arrow of time' has attracted a huge following from mysti-
cal poets through to Marxists (see http://www.marxist.com/science/arrowoftime.html).
The outline of the science in the Marxist.com site is actually very good.
The site http://www-groups. dcs. st-and. ac. uk/^history/Mathematicians/Eddington.
html contains a brief biography of Sir Arthur Eddington, who first proposed the
phrase' arrow of time' .
Most people seem to think T.S. Eliot's poem The Wasteland (1922) contains the
lines 'not with a bang . . .' but in fact they conclude the poem The Hollow Men
(1925). Incidentally, the 'whimper' is an obscure reference to Dante's description
of a newborn baby's cry upon leaving one world to enter another. The text of The
Hollow Men is available at http://www.aduni.org/~heather/occs/honors/Poem.htm.
Gibbs function
A short biography of Gibbs may be found in Physical Chemistry: A Molecular
Approach, p. 924. A longer version is available at http://www-groups.dcs.st-and.ac.
uk/~history/Mathematicians/Gibbs.html. To read some of his delightful quotations,
try http: //www -groups, dcs. st-and. ac. uk/~history/Quotations/Gibbs. html.
Le Chatelier's principle
Le Chatelier's original work was published in the journal of the French Academy of
Sciences, as H. L. Le Chatelier, Comptes rendus, 1884, 99, 786. It's in French, and
even good translations are hard to follow.
The best biography is available at the French Website http://www.annales.com/
archives/xAc.html. The short article 'Man of principle' by Michael Sutton in Chem-
istry in Britain, June 2000, 43, also includes a nice introduction to the man and the
background to his science. Websites bearing the same title as Sutton's article prolifer-
ate: two of the better ones (which include photographs) are http://www.woodrow.org/
teachers/ci/1992/LeChatelier.html and http://www.stormpages.com/aboutchemists/
lechatelier. html.
BIBLIOGRAPHY
545
The example of chicken breath was inspired by the article, 'From chicken breath
to the killer lakes of Cameroon: uniting seven interesting phenomena with a single
chemical underpinning' by Ron DeLorenzo, Journal of Chemical Education, 2001,
78(2), 191. The article also discusses boiler scale, the way that carbon dioxide parti-
tions between fizzy drink and the supernatant gases (see p. 165), and stalactites and
stalagmites.
5 Phase equilibria
General reading
In the affordable Oxford Primer series, Thermodynamics of Chemical Processes by
Gareth Price, Oxford University Press, Oxford, 1998, describes some of the back-
ground material discussed here, albeit in modest depth.
Tin and Napoleon
Whether true or not, the story about Napoleon is quite well known, and is mentioned
on the Websites http://www.tclayton.demon.co.uk/metal.htmWSn and http://www.
corrosion-club, com/tinplague. htm.
Thermodynamic data for tin may be found in Tin and its Alloys and Compounds,
B. T. K. Barry and C. J. Thwaits, Ellis Horwood, Chichester, 1983. The International
Tin Research Institute has an informative Website at http://www.tintechnology.com,
the 'library' page of which is particularly good.
Supercritical fluids
For the underlying science of supercritical fluids, try Steve How die's short article
'Supercritical solutions' in Chemistry in Britain, August 2000, p. 23, which represents
a useful introduction to the topic. For more applications of such fluids, try the short
review article 'Some applications of supercritical fluid extraction', by D. P. Ndiomu
and C. F. Simpson in Analytica Chimica Acta, 1988, 213, 237. The article is somewhat
dated now but readable. A look at the contents list of The Journal of Supercritical
Fluids will be more up-to-date: go to http.V/www.umecheme. maine.edu/jsf
Decaffeinated coffee
The article 'Caffeine in coffee: its removal: why and how?' by K. Ramalakshmi
and B. Raghavan in Critical Reviews in Food Science and Nutrition, 1999, 39, 441
provides an in-depth survey of the physicochemical factors underlying decaffeination
of coffee with supercritical CO2.
546
BIBLIOGRAPHY
D. J. Adam, J. Mainwaring and Michael N. Quigley have described a simple exper-
iment to remove caffeine from coffee with a Soxhlet apparatus: see Journal of Chem-
ical Education, 1996, 73, 1171. Their solvent was a chlorinated organic liquid rather
than supercritical CO2. The abstract is available at http://jchemed.chem.wisc.edu/
journal/issues/1 996/Dec/absll 71.html.
Chemical potential
Chemical potential can be a very difficult topic to grasp, but any standard textbook
of physical chemistry will supply a more complete treatment than that afforded here.
A particularly useful introduction to the thermodynamics of solutions and mixtures is
Chapter 6 of Basic Chemical Thermodynamics, by E. Brian Smith, Oxford University
Press, Oxford, 1990.
Be careful when searching the Web, though, because typing the phrase 'chemical
potential' will locate a great many pages that relate to solid-state and atomic physics:
both disciplines use the term to mean different concepts.
Cryoscopy and colligative properties
Perhaps the best short description of cryoscopy and the uses of colligative properties
is Michael Sutton's delightful article 'One cool chemist', Chemistry in Britain, 2001,
June, p. 66, concerning Francois-Marie Raoult. Sutton sketches a brief portrait of
Raoult before widening the scope to include the physicochemical factors underlying
the depression of freezing point. He gently guides the reader through the early history
of how our Victorian forebears made ice cream at a time predating refrigerators, before
explaining the physicochemical principles underlying the action of salt on a frozen
path. Highly recommended.
A slightly different approach will be found in Mark Kurlansky's fascinating book
Salt: A World History, Walker Publishing Company, New York, 2002, which is more
than a 'popular science' book, and will be enjoyed as well as being informative.
The site http://www. northland. cc. mn. us/Chemistry/colligative_properties_constants.
htm cites many ^(cryoscopic) and ^(ebuiiioscopic) constants.
Anaesthesia
Michael Gross has written a beautifully clear article 'The molecules of pain' describ-
ing how the sensation of pain originates. He also discusses a few recently discovered
molecules that act as chemical 'messengers' to the brain: see Chemistry in Britain,
June 2001, p. 27.
The site http://www.oyston.com/history has a fascinating history of the topic, mention-
ing such early anaesthetics as ether, chloroform and nitrous oxide. Local anaesthetics
are often injected in the form of liquids or solutions, see the article 'pharmacology
of local anaesthetic agents' by a British anaesthetist, Dr J. M. Tuckley, may be found
BIBLIOGRAPHY
547
at http://www.nda.ox.ac.uk/wfsa/html/u04/-u04_014.htm. General anesthetics are more
usually administered as a gas: see the article, 'General anaesthetic agents' at http://www.
mds.qmw.ac.uk/biomed/kb/pharmgloss/genanaesth.htm, again from a British hospital.
The anaesthetic properties of halothane are described in some depth at the French
Website http://www.biam2.org/www/Spe2391.html, which includes health and safety
data.
__^_^_^_^_^_^_^_^_^_^_^_^_^_^_^_^_^_m
Solid-state gas sensors
The complicated topic of solid-state electrical conductivity is well described in Solid
State Chemistry and its Applications, A. R. West, Wiley, Chichester, 1984, although
it does not explicitly discuss sensors. Those wanting more depth should look at
Transition Metal Oxides, P. A. Cox, Clarendon Press, Oxford, 1992, which provides
a readable account of the conduction of ions and electrons through solids.
For a background introduction to the doping of semiconductors, try the fun Website
Britney Spears' guide to semiconductor physics at http://britneyspears.ac/lasers.htm,
which is actually quite good in parts.
The 'press centre' on the Website of British Petroleum pic (BP) is a good source
of information, at http://www .bpamoco.com/centres/press/index.asp . For example, the
page http://www.bpevo.com/bpevo_main/asp/evo_glo_0003.asp lists the terms used by
most petrol companies, and http://www.bp.com/location_rep/uk/bus_operating/
manu_ops.asp cites the amounts of the known carcinogen, benzene, found naturally
in petrol.
The page http://www.radj vrd.edu/~wkovarik/lead cites some fascinating sidelights
on the history of petrol and its additives, and its links are carefully chosen. For example,
the excellent page http://www.chemcases.com/tel summarizes the history of tetraethyl
lead and the attendant controversies over its addition to petrol. Though quite small, the
site http://www. exxon. com/exxon _productdata/lube _encyclopedia/knock. html contains
useful information on the anti-knock properties of petrol and its additives.
Incidentally, as our discussion initially focused on the way we smell petrol, Natalie
Dudareva's article 'The joy of scent' is a good introduction. She concentrates on
flowers, but her theory and references are germane: see Chemistry in Britain, February
2001, p. 28.
Finally, one of the better books to analyse the environmental impact of petrol is
Green Chemistry: An Introductory Text by M. Lancaster, Royal Society of Chemistry,
Cambridge, 2002.
Steam distillation
The best book describing essential oils and their extraction is The Essential Oils:
Individual Essential Oils of the Plant Families (in six volumes) by Ernest Guenther,
1948 (reprinted 1972-1998).
548 BIBLIOGRAPHY
For discussions concerning the extraction of essential oils, e.g. for aromatherapy, see
the sites http://www.origanumoil.com/steam_distillation.htm, http: //www. distillation.
co.uk and http://www.lifeblends.com/howitworks5.html. For a more in-depth descrip-
tion of oil extraction, see Guenther (above) or http://www.fatboyfresh.com/essentialoils/
extraction.htm. The site http://www.chamomile.co.uk/distframe.htm has some super
graphics of steam distillation.
The Chemistry of Fragrances, D.H. Pybus and C. S. Sell, Royal Society of Chem-
istry, Cambridge, 1999, gives an extremely balanced assessment of aromatherapy as
a scientific discipline. See also 'The joy of scent', above.
6 Acids and bases
General reading
Aqueous Acid-Base Equilibria and Titrations, Robert de Levie, Oxford University
Press, Oxford, 1999, is a good resource, whose scope extends far beyond this book.
Its particular emphasis is speciation analyses, which are discussed in an overtly math-
ematical way. The maths should be readily followed by anyone acquainted with
elementary algebra. Water Chemistry, by Mark M. Benjamin, McGraw Hill Interna-
tional Edition, New York, 2002, is a longer book and covers the same material as de
Levie but in substantially greater depth.
The dissociation of water
One of the best resources for looking at the way equilibrium constants K vary with
ionic strength is the Web-based resource Joint Expert Speciation System (JESS) avail-
able at http://jess.murdoch.edu.au/jess/jess_home.htm. Notice the way that values of
any equilibrium constant (K a , K w , etc.) changes markedly with ionic strength /.
Chlorine gas as a poison gas
The information concerning the Second Battle of Ypres is embedded within a disturb-
ing account of man's scope for evil, World War One: A Narrative by Philip Warner,
Cassell Military Classics, 1998. The ghastly effects of poisoning with chlorine gas
are recounted at http://www.emedicine.com/EMERG/topic851.htm.
Sulphuric acid formed in the eyes
The paper 'The design of the tears' by Jerry Bergman in The Technical Journal
2002, 16(1), 86, is one of the best short introductions to the operation of the tear
BIBLIOGRAPHY
549
duct, and mentions SO3 in the vapours emanating from onions; or go to http://www.
answersingenesis.org/home/area/magaz.ines/tj/TJ_issue_index.asp. The book Clinical
Anatomy of the Eye, R. Snell and M. Lemp (second edition), Blackwell, Boston, 1998,
discusses tears and onions on p. 110.
The literature on acid rain is simply vast. One of the better introductory texts is Acid
Rain: Its Causes and its Effects on Inland Waters, by B. J. Mason, Clarendon Press,
Oxford, 1992. Dozens of Websites supplement and update Mason's book. For a gen-
eral but more widely ranging survey of pollution and its legacy, try Pollution: Causes,
Effects and Control (fourth edition), edited by Roy M. Harrison, Royal Society of
Chemistry, Cambridge, 2001.
The Websites http://lihrary.thinkquest.org/CR021547I/acid_rain.htm and http://
www.geocities.com/CapeCanaveral/Hall/91 1 1/ACIDRAIN. HTML simplify the issues
(some might add 'over-simplify'). The Website of Greenpeace at http://www.
greenpeace.org can be sensational at times, but its information is usually reliable.
For more interesting sidelights on the topic, try the Canadian Government's site at
http: //www. atl. ec.gc. ca/msc/as/as _acid.html.
Bases in nature
The Websites http://www.consciouschoice.com/herbs/herbsl308.html and http://www.
homemademedicine.com/insectbiteandbeesting.html cite the many naturally occurring
plants having interesting medicinal properties.
Neither of these sites contains much scientific theory. A more in-depth survey of the
phenomenon of pain may be found at http://www.chic.org.uk/press/releases/pain2.htm,
produced by the Consumer Health Information Centre (CHIC); and Michael Gross'
short article 'The molecules of pain' describes the sensation of pain and the mode(s)
by which pain killers operate: see Chemistry in Britain, June 2001, p. 27.
The Website http://www.ediblewild.com/nettle.html lists many more properties of
stinging nettles - particularly impressive being the number of ways to consume nettles
in the form of edible foodstuffs, drinks and beers.
Carbolic acid and Lord Lister
A good introduction may be found at the Websites http://web.ukonline.co.Uk/b.gardner/
Lister.html and http://limiting.tripod.com/list.htm. The former site also contains some
fascinating links.
Indigestion tablets and neutralization reactions
iiges
The Internet site http://www.picotech.com/experiments/antiacid/results.html investi-
gates the chemistry behind several well-known tablets.
550
BIBLIOGRAPHY
Concerning the effects of aluminium and the onset of Alzheimer's disease, the
site http://www.alzscot.org/info/aluminium.html gives good information - but it can
be a bit scary! The site http ://www .hollandandbarrett.com/healthnotes/Drug/Sodium_
Bicarbonate.htm describes the action of NaHC03 (an aluminium-free anti-acid prod-
uct) in indigestion formulations.
Finally, the site http://www.naplesnews.com/today/neapolitan/a84826n.htm gives
good advice on diet, aimed at avoiding the problem before we need to ingest an
anti-acid tablet.
The colours of flowers and indicators
The fascinating book Sensational Chemistry, in the Open University's 'Our Chemical
Environment' series, OU Press, Milton Keynes, 1995, is an excellent introduction to
the topic. The Web has several hundred relevant sites, most of which are simple to
follow: just go to a reputable search engine, and type 'acid-base indicator'.
Finally, Food Flavours: Biology and Chemistry (fourth edition) by C. Fisher and
T. Scott, Royal Society of Chemistry, Cambridge, 2002, supplements many of the
titles above.
7 Electrochemistry
General reading
There are a large number of books on electrochemistry. The electrochemistry text
is undoubtedly Electrochemical Methods: Fundamentals and Applications (second
edition) by A. J. Bard and L. R. Faulkner, Wiley, New York, 2001. This book is
unfortunately a mathematical read, but it contains absolutely everything we need at
our level; and the prose is generally a model of clarity.
In the Oxford Primer series, the book Electrode Potentials by Richard G. Compton
and Giles H. W. Sanders, Oxford University Press, Oxford, 1996, is an introduction.
It is intended for the absolute novice, but develops themes to a satisfactory level.
Its treatment of the Nernst equation is both thorough and straightforward. It contains
copious examples and self-assessment questions.
Fundamentals of Electroanalytical Chemistry, by Paul Monk, Wiley, Chichester,
2002, is intended to be an easy read. It was written for those learning at a distance.
Its style is non-mathematical, and involves a series of 'discussion questions'. It's also
packed full of worked examples and self-assessment questions.
Dynamic electrochemistry
One of the better articles is 'Electrochemistry for the non-electrochemist' by Peter
T. Kissinger and Adrian W. Bott, Current Separations, 2002, 20(2), 51. This brief but
informative article outlines 'nine of the most fundamental concepts of electrochemistry
BIBLIOGRAPHY
551
which must be mastered before more advanced topics can be understood' . The article
can be downloaded from http://www.currentseparations.com/issues/20-2/20-2d.pdf.
Also, try Electrode Dynamics, by A. C. Fisher, Oxford University Press, Oxford,
1996, which is another title in the Oxford primer series. Its early chapters discuss
the transport of analyte through solution and the various rates inherent in a dynamic
electrochemistry measurement. It is a readily affordable and readable introduction and
highly recommended.
Electrolysis and pain
Perhaps the best site is that of Amnesty International at http://www.amnesty.org. Its
page http://web.amnesty.org/library/eng-313/index cites many dozens of case studies,
a significant fraction of which involve electricity. Or type 'electricity' into their search
engine (almost hidden at the foot of the page), and be shocked at the number of so-
called 'friendly' and 'civilized' countries that employ torture: Israel, Indonesia, the
USA, and many of the 'new' countries of Europe. Be warned: some of the details are
horrific - electricity is a very efficient way of generating pain.
Hair removal Celectrology')
The commercial sites http://www.electrology.com/info.html (The American Electrol-
ogy Association) and http://www.betterhairremoval.com/electrolysis.htm contain copi-
ous information about the effect of electrolysis on follicles.
Fuel cells
By far the best book is Fuel Cell Systems Explained by James Larmanie and Andrew
Dicks, Wiley, Chichester, 2000. It's an expensive gold mine of a book.
A simpler 'first glimpse' at fuel cells comes from the brief layman's article in The
Economist, 'The fuel cell's bumpy ride', published in the 24 March 2001 edition.
Its scientific content is good, although its general approach centres on economics.
A similarly easy read, but for chemists, is Rob Kingston's 'Powering ahead', in
Chemistry in Britain, June 2000, p. 24.
Scientific American published a special issue on fuel cells in July 1999. Alter-
natively, A. John Appleby's introductory article 'The electrochemical engine for
vehicles' is a good read; the Internet sites http://www.chemcases.com/cells/index.htm,
http://www.fuelcells.org and http://www.howstuffworks.com/fuel-cell.htm and ~fuel-
cell2.htm are also worth a try.
The Campaign for a Hydrogen Economy (formerly The Hydrogen Association of
UK and Ireland, HEAUKI) is a treasure-trove of information and advice and may be
contacted at 22a Beswick Street, Manchester M4 7HR, UK. It is a non-profit orga-
nization that 'works for the world-wide adoption of renewably generated hydrogen
552
BIBLIOGRAPHY
as humankind's universal fuel'. CREC can be contacted at co-ordinator@hydrogen-
heauki.org.
Standard cells
A brief discussion of standard cells, together with a short biography of Edward
Weston, may be found at http://www.humboldt.edu/~scimus/HSC.54-70/Descriptions/
WesStdCell. htm. Alternatively, http://www. humboldt. edu/~scimus/Instruments/Elec-
Duff/StdCellDuff.htm is also useful, although its symbolism does not always adhere
to SI or IUPAC.
Corrosion
The 'Corrosion Doctors' site at http://www.corrosion-doctors.org contains lots of
interesting case studies.
Electrochromisi
The best introductory text on electrochromism is Electrochromism: Fundamentals
and Applications by P. M. S. Monk, R. J. Mortimer and D. R. Rosseinsky, VCH,
Weinheim, 1995, although the introductory sections of Handbook of Inorganic Elec-
trochromic Oxides, by C. G. Granqvist, Elsevier, Amsterdam, 1995, are also invalu-
able. For a shorter but nevertheless thorough introduction, read Chromic Phenomena
by P. Bamfield, Royal Society of Chemistry, Cambridge, 2001. The popular article
'Through a glass darkly' by Paul Monk, Roger Mortimer and David Rosseinsky in
Chemistry in Britain, 1995, 31, 380 is very short, but might represent a less threatening
introduction.
The Website http://jchemed. chem. wise. edu/J ournal/Issues/1 997/Aug/abs962. html
explores simple laboratory demonstrations of electrochromism, whereas the commer-
cial sites http://www.refr-spd.com/hgi-window.html, http://www.ntera.com/nano.pdf
and http://www.saint-gobain-recherche.com/pages/angl/materia/mtintell.htm are more
product oriented. Some contain pictures and video sequences.
Eau de Cologne and perfumes
The Chemistry of Fragrances, by D. H. Pybus and C. S. Sell, Royal Society of Chem-
istry, Cambridge, 1999, contains all the amateur needs to know on the subject. Addi-
tionally, the site http://www.eau-de-cologne.com (in German) describes the history of
the perfume, and the site http://www.farinal709.com contains additional information
for the interested novice.
BIBLIOGRAPHY
553
Activity and ion association
For a short biography of G. N. Lewis, the early giant of 20th century American
chemistry, visit the site http://www.woodrow.org/teachers/ci/1992/Lewis.html.
Ionic Solution Theory, H. L. Friedman, Wiley, New York, 1953, is a standard work
in the field, but is a bit mathematical and can be difficult to follow. An easier book
to follow is Ions, Electrodes and Membranes (second edition) by Jifi Koryta, Wiley,
Chichester, 1992, and is an altogether more readable introduction to the topic. It can
also be trusted with details of pH electrodes and cells. Its examples are well chosen,
many being biological, such as nerves, synapses, and cell membranes. It is probably
the only book of its kind to contain cartoons.
Easier still is John Burgess' book, Ions in Solution, Ellis Horwood, Chichester,
1999. Though it does not go into great detail about activity coefficients y, its treatment
of ionic interactions and solvation is excellent.
coefficient and solubility products
The example on p. 319 comes from Equilibrium Electrochemistry, The Open Uni-
versity, The Open University Press, Milton Keynes, 1985. The book contains many
other excellent examples.
Reference electrodes
The classic book on the topic remains Reference Electrodes by D. I. G. Janz and
G. J. Ives, Academic Press, New York, 1961. This book is still worth consulting
despite its age. One of the best articles is 'Reference electrodes for voltammetry'
by Adrian W. Bott, Current Separations, 1995, 14(2), 33. The article can be down-
loaded from http://www.currentseparations.com/issues/14-2/csl4-2d.pdf. Also, Elec-
trochemistry by Carl H. Hamann, Andrew Hamnett and Wolf Vielstich, Wiley-VCH,
Weinheim, 1998, has extensive discussions on reference electrodes.
*H electrod
concentrations cells
Try looking at The Elements of Analytical Chemistry, by Seamus P. J. Higson, Oxford
University Press, Oxford, 2003. It is an affordable addition to an analyst's library,
since it covers all the basics.
Batteries
Understanding Batteries by R. M. Dell and D. A. Rand, Royal Society of Chemistry,
Cambridge, 2001, represents a good introduction to the topic, as does Electrochemistry
554 BIBLIOGRAPHY
(above). Both books discuss battery design, construction and operation. One of the bet-
ter books is J. O. Besenhard, Handbook of Battery Materials, Wiley, Chichester, 1998.
For an easier read, try the feature article in the occasional 'Chemist's Toolkit'
series, entitled 'Batteries today' by Ron Dell, in Chemistry in Britain, March 2000, p.
34. It goes into greater depth than here, and is informative and relevant. The recent
paper 'The lead acid battery' by A. A. K. Vervaet and D. H. J. Baett, Electrimica
Acta, 2002, 47, 3297, discusses the recent developments of this, the world's best-
selling battery.
The site http://antoine.frostburg.edu/chem/senese/101/redox/faq/lemon-battery.shtml
cites details of how to make a battery from a lemon.
Ancient batteries
The 'battery' found near Baghdad is mentioned in a footnote in Modern Electrochem-
istry, J . O'M. Bockris and A. K. N. Reddy, Macdonald, London, 1970, p. 1265.
8 Chemical kinetics
General reading
Probably the best book on the topic is Chemical Kinetics and Mechanism, edited by
Michael Mortimer and Peter Taylor, Royal Society of Chemistry, Cambridge, 2002.
This book is really excellent, and comes complete with a superb interactive CD. The
other good book on kinetics currently in print is Fundamentals of Chemical Change,
by S. R. Logan, Longmans, London, 1996. Its mathematical treatment may appear
daunting at times, but its prose is good. Its coverage also includes the kinetics of
electrode reactions, photochemical reactions, and discusses catalysis to a relatively
deep level. Modern Liquid Phase Kinetics, by B. G. Cox, Oxford University Press,
Oxford, 1994, in the Oxford Primers series, is a relatively cheap text. Its illustrations
are not particularly good, but the book as a whole is readable.
Although long out of print, a super book for the complete novice is Basic Reaction
Kinetics and Mechanisms by H. E. Avery, Macmillan, London, 1974. Its author is a
superb teacher, whose insights and style will benefit everyone. Although quite old
now, the general approach and the theory sections in Fast reactions by J. N. Bradley,
Oxford University Press, Oxford, 1975, is still of a high standard.
Ozone depletion
A vast library waits to be read on ozone depletion. The best book by far is 'Topic
study 1: the threat to stratospheric ozone' in the Physical Chemistry: Principles of
Chemical Change series, published in the UK by the Open University, Milton Keynes,
1996. From the UK's Royal Society of Chemistry come Climate Change, Royal Soci-
ety of Chemistry, Cambridge, 2001, Green Chemistry, M. Lancaster, Royal Society
BIBLIOGRAPHY
555
of Chemistry, Cambridge, 2002, and Pollution, Causes, Effects and Control (fourth
edition), edited by Roy M. Harrison, Royal Society of Chemistry, Cambridge, 2001.
Each gives a clear exposition.
The magazine Website http://www.antarcticconnection.com has much up-to-date
information, as does http://www.ciesin.org/TG/OZ/oz-home.html and its links.
The full text of the 1985 Vienna Convention for the 'Protection of the Ozone
Layer' is available at http://www.unep.org/ozone/vienna_t.shtml, and contains much
of the primary data that alerted the scientific community to the threats of CFCs, etc. It
is somewhat dated now, but the Web page of the United Nations 'Ozone Secretariat'
is more reliable: http://www.unep.org/oz.one/index-en.shtml.
The 'popular science' pages of 'Beyond Discovery' are compiled by the American
National Academy of Sciences and are generally highly readable, and written for the
intelligent novice. They may be accessed at http://www.beyonddiscovery.com. Do a
search using 'ozone' as a key word. On a lighter note, the fun site at 'Fact Monster'
http://www.factmonster.com/ipka/A0800624.html displays much 'pre-digested' data.
Viologens
The viologens are mentioned a couple of times in this chapter, and are formally
salts of 4,4'-bipyridine. The only book dedicated to them is The Viologens: Physic-
ochemical Properties, Synthesis and Applications of the Salts of 4 , 4' -Bipyridine
by P. M. S. Monk, Wiley, Chichester, 1998, and is up-to-date and comprehensive.
Although badly out of date, a shorter work is 'Electrochemistry of the viologens',
C. L. Bird and A. T. Kuhn, Chem. Soc. Rev., 1981, 10, 49.
Radiocarbon dating
A good layman's guide to radioactive dating may be found in Searching for Real Time,
by Richard Corfield, Chemistry in Britain, January 2002, p. 22. It presupposes only
slight background knowledge of radiochemistry. For its worked examples, it considers
the age of the Turin Shroud and the stone of which sedimentary rocks are made, i.e.
dating the movements of tectonic plates. The excellent book Archeological Chemistry
by A. Mark Pollard and Carl Heron, Royal Society of Chemistry, Cambridge, 1996,
will reward a thorough reading.
The Website http://www.cl4dating.com comprises a good treatment of the topic and
hosts a huge amount of information. The academic journal Radiocarbon always con-
tains examples of radiocarbon dating, as well as dating using the isotopic abundances
of other elements (abstracts of its papers may be accessed online, at http://www.
radiocarbon.org) .
The Turin Shrc
The site http://www.shroudstory.com/cl4.htm is less informative, but does discuss
all the evidence underpinning the claims of a botched radiocarbon dating of the
556
BIBLIOGRAPHY
Turin Shroud, e.g. via contamination with other sources of carbon. Its treatment is
informative and generally applicable.
Otzi the Ice Mai
For the non-scientist, the fascinating book The Man in the Ice, by K. Spindler, Wei-
denfeld and Nicholson, London, 1993, is well worth a look.
The BBC Website has a page dedicated to Otzi the Iceman, with some nice pictures:
go to http://www.bbc.co.uk/science/horizon/2001/iceman.shtml, which also reproduces
the full transcript of an hour-long documentary.
The journal Radiocarbon regularly publishes fascinating technical papers and case
studies of radiocarbon dating. For example, the original data concerning Otzi derive
from the paper, 'AMS- 14 C dating of equipment from the Iceman and of spruce logs
from the prehistoric salt mines of Hallstatt', W. Rom, R. Golser, W. Kutschera,
A. Priller, P. Steier and E. M. Wild, Radiocarbon, 1999, 41(2), 183.
For a briefer scientific analysis of Otzi, Archeological Chemistry (above) devotes
a small case study to Otzi, see its p. 251 ff. And several Websites are devoted to
him, including http://info.uibk.ac.at/c/c5/c552/Forschung/Iceman/iceman-en.html and
the French site http://www.archeobase.com/v_texte/otzi/corp/cor.htm.
W a Id en inversion reactions
Walden first published his observations on inversion in Berichte 1893, 26, 210; 1896,
29, 133; and 1899, 32, 1855, long before the inversion mechanism was proposed by
Ingold in J. Chem. Soc, 1937, 1252. The idea that the addition of one group could
occur simultaneously with the removal of another was first suggested by Lewis in
1923, in Valence and Structure of Atoms and Molecules, Chemical Catalog Company,
New York, 1923, p. 113. Olsen was the first to propose that a one-step substitution
reaction leads to inversion, in /. Chem. Phys., 1933, 1, 418.
Catalytic converters
The hugely complicated chemistry of the catalytic converter is described in some depth
in the Open University's excellent book Physical Chemistry: Principles of Chemical
Change, 'Topic study 2'. Its part 1 is entitled 'the three-way catalytic converter', Open
University, Milton Keynes, 1996. It covers the composition of the catalytic surface
and the role of each dopant, the actual chemical reactions occurring, and details of
the current legal situation regarding atmospheric pollution.
The 'How Things Work' Website is worth a glance: look at pages http://www.
howstuffworks. com/question66. htm and http://www. how stuff works. com/question482.
htm.
Finally, Green Chemistry, by Lancaster (above), discusses these converters to some
small extent in Chapter 4, especially pp. 107-109.
BIBLIOGRAPHY
557
Photochromic lenses
A layman's guide to the action of photochromic lenses may be found at the 'How Stuff
Works' Website, at http://www.howstujfworks.com/question412.htm. The Website of
Britglass, http://www.britglass.co.uk/publications/mglass/making2.html, also cites a
few interesting facts. A more thorough treatment is available in the excellent text
Chromic Phenomena, Peter Bamfield, Royal Society of Cambridge, 2002, section 1.2
of his Chapter 1 .
Hormones
The topic of hormones is horribly complicated, so read widely and with care. Most
Websites that advertise information about hormones are too simplistic and they should
be avoided - particularly those sites that sell hormones, and start with a paragraph
or two of 'pseudo science'. Most textbooks of physiology and medicinal chemistry
contain sufficient detail. For example, Medicinal Chemistry, F. D. King (ed.), Royal
Society of Chemistry, Cambridge, 2002, has a good introduction.
Chocolate
The Science of Chocolate by Stephen Beckett, Royal Society of Chemistry, Cam-
bridge, 2000, is probably the best general introduction to the subject, and contains
sufficient scientific content to stimulate, while avoiding being overtly technical. Highly
recommended.
Activation energy
Addy Pross's book, Theoretical and Physical Principles of Organic Reactivity, Wiley,
New York, 1995, is an invaluable tool for understanding the way constructing a
reaction profile can help the physical chemist to predict the outcome of a chemi-
cal reaction. Lowry and Richardson's Mechanism and Theory in Organic Chemistry
(above) is also germane.
9 Physical chemistry involving light:
spectroscopy and photochemistry
General reading
Several books describe the background to this topic. Perhaps the best general introduc-
tions come from the Royal Society of Chemistry: Colour Chemistry by R. M. Christie,
RSC, Cambridge, 2001, is written for the beginner, but does extend to some depth. It
558 BIBLIOGRAPHY
is particularly good at describing chromophores of every describable type. Chromic
Phenomena by P. Bamrield, RSC, Cambridge, 2001, is written more for the spe-
cialist, and concentrates on colour changes for device applications. Nevertheless, its
introductory material is superb.
Useful books from the Oxford University Press 'primer' series include: Foundations
of Spectroscopy, by Simon Duckett and Bruce Gilbert, OUP, Oxford, 2000; Introduction
to Organic Spectroscopy, by Laurence M. Harwood and Timothy D. W. Claridge, OUP,
Oxford, 1997; and Energy Levels in Atoms and Molecules, by W. G. Richards OUP,
Oxford, 1996. Each, particularly the last, represents a clear and lucid introduction.
More in-depth treatments include Ultraviolet and Visible Spectroscopy (second edi-
tion) by Michael Thomas, Wiley, Chichester, 1997. This book was written as part of a
distance-learning course within the Analytical Chemistry by Open Learning (ACOL)
series, so it contains a good number of examples and sample questions. Its typical
ACOL format will probably annoy some readers.
The two books Modern Spectroscopy (second edition), by J. Michael Hollas, Wiley,
Chichester, 1992, and Fundamentals of Molecular Spectroscopy, by C. N. Banwell
(third edition), McGraw Hill, Maidenhead, 1983, are each highly recommended. Each
contains instrumental detail, in addition to the theory and applications of the spectro-
scopic techniques discussed. Both will look too mathematical to many readers, but
they are both authoritative and clear.
A simpler version of Hollas 's book is now available in the Royal Society of
Chemistry's new Tutorial Chemistry Texts series: Basic Atomic and Molecular Spec-
troscopy, J. Michael Hollas, RSC, Cambridge, 2002. It gives a super introduction, and
its academic level is well gauged, although it does require a knowledge of molecular
orbitals and maths.
Ink-jets
Peter Gregory's article, 'Colouring the jet set' in Chemistry in Britain, August 2000, p.
39, introduces the chemistry of colour, and the colour science of inks for PC printers.
He places the subject in context, and also discusses the chemical composition of the
many of the most common inks. Essays in Ink (for Paints and Coatings too) by J.
Kunjappa, Nova Science Publishers, New York, 2002, provides a good introduction
for the non-specialist who may be curious about the world of ink.
Pigments, paints and dyes
Philip Bell's text Bright Earth: the Invention of Colour, Viking, 2001, is a begin-
ner's guide intended for those with absolutely no background in science. The book
commences with a history of the subject, detailing the prehistoric painting found in
caves such as Lascoux, before moving on to the Aztecs, Incas and other races peo-
pling the history of colour. The sections on theory should be readily followed by
absolutely everyone.
BIBLIOGRAPHY
559
A slightly more in-depth study of colour is afforded by The Physics and Chemistry
of Colour by Kurt Nassau, Wiley, Chichester, 2001. The author describes many
everyday examples of colour, from peacock tails through to the Northern Lights,
Aurora Borealis. Its Chapter 1 is an overview, and is probably a little highbrow at
times, but overall is a fascinating read.
Ch
romophores in nature
There are many case studies and examples from everyday life in Light, Chemical
Change and Life: A Source Book for Photochemistry , edited by J. D. Coyle, R. R. Hill
and D. R. Roberts, Open University Press, Milton Keynes, 1988, and Our Chemical
Environment, Book 4: Sensational Chemistry, Open University Press, Milton Keynes,
1995. Both are outstanding and well worth a read.
The sixth chapter of Food: The Chemistry of its Components (fourth edition), by
T. P. Coultate, Royal Society of Chemistry, Cambridge, 2002, revels in the colours
of nature, as well as synthetic chromophores.
Melanin in the skin
There is an embarrassing wealth of material concerning melanin, and the interac-
tions of light on the skin. The Website http://omlc.ogi.edu/spectra/melanin by Steven
Jacques of the Oregon Laser Centre introduces the topic well, with a readable and
highly informative discussion.
Although badly dated now, The Pigmentary System, edited by J. J. Norlund, Oxford
University Press, Oxford, 1988, is still worth reading, particularly its chapter 'The
chemistry of melanins and related metabolites', by G. Prota, M. D'Ischia and
A. Napolitano.
Several companies now produce sunglasses with melanin pigment embedded within
the lenses; the intention is for the commercial lens to mimic the way the eye blocks
some wavelengths, i.e. forming 'natural sunglasses'. The informative Website http://
www.fcgmelanin.com has both 'history' and 'science' pages.
A more in-depth study of the photochemistry of melanin - natural or synthetic - is
available at http://www.findarticles.com. Type in 'melanin' and look for the article by
Corinna Wu of 18 September 1999.
Finally, as an adjunct, try Peter Bamfield's book Chromic Phenomena (above) for
a brief discussion of hair dyes, on its pp. 1 10 ff.
Franck-Condon
The recent book Conformational Analysis of Molecules in Excited States, by Jacek
Waluk, Wiley, New York, 2000, describes the way we can experimentally determine
the shapes of molecules in the ground and excited states. It can be a little high brow at
560
BIBLIOGRAPHY
times, and is clearly written for physicists (perhaps explaining its overtly mathematical
treatment), but it will be useful for those wanting to study a little further.
Incandescent lights
Louis A. Bloomfield's entertaining book How Things Work: The Physics of Everyday
Life (second edition), Wiley, New York, 2001, discusses neon bulbs and fluorescent
strip lighting, see pp. 395-399. For a more scientific look at fluorescent dyes, try
Chapter 3 of Peter Bamfield's Chromic Phenomena (above), especially pp. 182-184.
Bioluminescence
This intriguing subject is introduced by Tony Campbell in his article 'Rainbow mak-
ers' in Chemistry in Britain, June 2003, p. 30.
R
aman spectroscopy
The Raman effect' by Neil Everett, Bert King and Ian Clegg in Chemistry in Britain,
July 2000, p. 40, is a good general introduction, written for scientists with no prior
experience of Raman spectroscopy. Each of the books cited above under 'general
reading' discuss Raman spectroscopy, but in greater depth.
Raman spectroscopy and pigments
This new topic is well described in the article 'A Bible laid open', by Stephen Best,
Robin Clark, Marcus Daniels and Robert Withnall, Chemistry in Britain, February
1993, 118. The article shows the Raman spectra of several pigments. A more technical
description may be found at S. P. Best, R. J. H. Clark and R. Withnall, Endeavour,
1992, 16, 66.
Robin Clark's group operates from University College in London; their Website
http://www.chem.ucl.ac.uk/resources/raman/speclib.html is informative, and contains
links to many spectra. The Raman spectrum of malachite is interpreted in M. Schmidt
and H. D. Lutz, Physics and Chemistry of Minerals, 1993, 20, 27.
For more information about the pigments themselves, try A History of Lettering,
N. Gray, Phaidon, Oxford, 1986, or The Icon: Image of the Invisible, Egon Sendler,
Oakwood, 1988. Chapters 12-15 of the latter book are a mine of detail concerning
the manufacture of pigments.
The colour of water
A simple yet useful answer to the question 'Why is water blue?' may be found in
the article by Charles L. Braun and Sergei N. Smirnov, J. Chem. Educ, 1993, 70,
BIBLIOGRAPHY 561
612. Alternatively, try 'The colours of water and ice', by Terence Quickenden and
Andrew Hanlon, Chemistry in Britain, December 2000, p. 37.
10 Adsorption and surfaces, colloids
and micelles
General reading
Perhaps the best introduction for the novice is the recent article 'The impact of colloid
science' by Mike Garvey in the 'Chemists' Toolkit' series in Chemistry in Britain,
February 2003, p. 28.
Also for the novice, and also very good, are Introduction to Colloid Science,
D. J. Shaw, Butterworths, London, 1980, and Basic Principles of Colloid Science,
by D. H. Everett, Royal Society of Chemistry, Cambridge, 1988. Both are rigorous:
the former is more pedagogical in style, and an easier read; the latter is very thorough
but somewhat mathematical.
Robert J. Hunter has written two good books on colloid science: the magisterial
Foundations of Colloid Science (second edition), Oxford University Press, Oxford,
2000, is surely the benchmark text on this topic, but it is not cheap. Its smaller
offspring is Introduction to Modern Colloid Science, Oxford University Press, Oxford,
1993. This latter text loses none of the rigour but is much shorter and cheaper.
The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet
(second edition) by D. Fennell Evans and Hakan Wennerstrom, Wiley, New York,
1999, is a superb book which satisfactorily demonstrates the interdisciplinary nature
of the topic. Its biological examples are particularly good. They also present a nice
discussion on pp. 602-603 of how colloid science contributed to the growth of several,
disparate strands of science.
In the American Chemical Society 'ACS Professional Reference Book' series
comes, The Language of Colloid and Interface Science - A Dictionary of Terms, Lau-
rier L. Schramm, American Chemical Society, Washington, 1998; and a related text
is Dictionary of Colloid and Interface Science, Laurier L. Schramm, Wiley, New
York, 2001. Both were compiled to be read as dictionaries, but are thorough and
well presented.
The book Principles of Adsorption and Reaction on Solid Surfaces by Richard I.
Masel, Wiley, New York, 1996, concentrates on mechanistic detail and adsorption at
the solid-gas interface, but, within its self-imposed limitations, it is a superb book.
Any in-depth study of adsorption requires us to know something of the surfaces
involved. One of the better books describing how we characterize a surface is Surface
Chemistry, by Elaine M. McCash, Oxford University Press, Oxford, 2001. It is both
affordable and well paced, and consistently concentrates on concepts rather than
diverting the reader's attention to experimental details. Recommended.
Finally, the journal Langmuir usually contains relevant papers. Go to its homepage
at http://pubs. acs. org/journals/langd5 .
562
BIBLIOGRAPHY
Adsorption of biological toxins on kaolin
The nature section of the BBC Website describes several examples of animals eating
kaolin clay to immobilize toxins by adsorption on the clay's surface. For example,
see the first entry on the page http://www.bbc.co.uk/nature/weird/az/mo.shtml. Alter-
natively, read The Life of Mammals by Sir David Attenborough, BBC Books, 2002,
p. 170.
Food chemistry
A great many of the examples in this chapter involve food. The best chemistry text
on the subject is probably An Introduction to Food Colloids by E. Dickinson, Oxford
University Press, Oxford, 1992, and is an absolute gold-mine of a book. For fur-
ther reading, though, try Food: The Chemistry of its Components (fourth edition) by
T. P. Coultate, Royal Society of Chemistry, Cambridge, 2002, or Food Microbiology
(second edition) by M. R. Adams and M. O. Moss, Royal Society of Chemistry, Cam-
bridge, 2002, both of which are filled with suitable examples. Food Flavours: Biology
and Chemistry by C. Fisher and T. Scott, Royal Society of Chemistry, Cambridge,
1997, supplements these titles.
Emulsions
The short Web page at http://www.surfactants.net/emulsion.htm has a few interest-
ing links. Food: The Chemistry of its Components by Coultate (as above) discusses
emulsifiers on p. 114 ff.
Pasteur iza tion
The French site http://www.calixo.net/braun/conserve/pasteurisation.htm and the Cana-
dian site http://www.foodsci.uoguelph.ca/dairyedu/pasteuriz.ation.html both cite perti-
nent details.
Many Websites willingly divulge the secrets of butter making, such as http://www.
foodfunandfacts.com/milk.htm. The site http://www.culinarycafe.com/Eggs-Dairy/
Clarified_Butter.html describes how (and why) to make clarified butter.
Absinthe
The paper 'Absinthe: enjoying a new popularity among young people?' by C. Gam-
belunge and P. Melai, Forensic Sci. Int., 2002, 130, 183 gives a brief history of
BIBLIOGRAPHY
563
absinthe-based drinks and their effect on 'the artistic muse'. See also, 'Absinthe:
behind the emerald mask', D. D. Vogt and M. Montagne, Int. J. Addiction, 1982, 17,
1015 for a little more social background; its description of the effects of absinthe
addiction are salutary.
Milk
There is a colossal amount of information on the Web concerning milk. For example, see
the page http://www.sciencebyjones.com/MILK_NOTES.HTM or the 'Dairy Chemistry
and Physics page' at http://www.foodsci.uoguelph.ca/dairyedu/chem.html, hosted by
the Canadian University of Guelph.
Ice cream
Most of the information about ice cream comes from Dickinson (above), but Food:
The Chemistry of its Components by Coultate (above) is also worth a glance.
Soap and Civil War
The soap tax was one of the 'trigger points' leading inexorably to the Great English
Civil War. This tax is but one example of the foolishness of the British King Charles
I, although it is wise to appreciate how many people aggressively capitalized on his
foolishness. The best account of the causes of the Civil War is The King's Peace
1637-1641, by C. V. Wedgewood, Penguin, Harmondsworth, 1955. The soap tax is
mentioned on p. 160; the introductory background to the King's ill-advised policy
commences on p. 157.
The Tyndall effect
'Blue skies and the Tyndall effect' by M. Kerker in /. Chem. Educ, 1971, 48, 389 is a
nice introduction. Alternatively, try Chapter 7 'Some important properties of colloids:
II scattering of radiation' in Everett (above), which is extremely thorough.
The history of how the science of light scattering grew is well delineated in the
fascinating book, The Scattering of Light, M. Kerker, Academic Press, 1967, in which
Kerker outlines how the work of various scientists has intertwined.
The Krafft temperature
One of the most cited papers is J. K. Weil, F. S. Smith, A. J. Stirton and R. G. Bristline,
/. Am. Oil Chem. Soc, 1963, 40, 538. One of the better sources of data is Shaw, Intro-
duction to Colloid Science (see above) p. 87, and Evans and Wennerstrom, The Colloidal
564 BIBLIOGRAPHY
Domain: Where Physics, Chemistry, Biology, and Technology Meet (see above) p. 1 1 .
The paper, 'Use of quantitative structure- property relationships in predicting the Krafft
point of anionic surfactants' by M. Jalali-Heravi and E. Konouz, Internet Electronic
Journal of Molecular Design, 2002, 1, 410, has a nice introduction and useful refer-
ences. It can be downloaded at http://www.biochempress.com/av01_0410.html.
The journal Colloid and Polymer Science occasionally has relevant papers: go to
its homepage at http://link.springer.de/link/service/journals/00396.
Water treatment
One of the best books on the topic is Green Chemistry: An Introductory Text by M.
Lancaster, Royal Society of Chemistry, Cambridge, 2002. Its treatment is thorough,
but not always in great depth. It is clearly aimed at the complete novice. Chapter 5 of
Pollution, Causes, Effects and Control (fourth edition), edited by Roy M. Harrison,
Royal Society of Chemistry, 2001, describes the material in a little more depth; do
not be confused by the careless title of the book Basic Water Treatment by C. Binnie,
M. Kimber and G. Smethurst, Royal Society of Chemistry, Cambridge, 2002, which
is neither an elementary text nor a treatise on treating alkaline water.
Index
Natural materials cited in the text: flora, fauna, food, and natural chemicals
Absinthe, 562
Adenine, 45
Adrenal glands, 387
Adrenaline, 387, 388
Agar, 342
Albumen, 95, 166, 170, 252
Algae, 461
Alona affinis, 414
Alveoli, 359
Ant, 414
Ant, red, 253
Antibodies, 415
Apple, 244
juice, 442
Apricots, 252
L-(+)-ascorbic acid, 244, 401, 512
Axon, 339, 340, 341
Bacillary layer, 458
Bacteria, 63, 255, 461
Barley, 63
Beans, 63
Beetroot juice, 441, 499, 500
Bergamot, 309
Bile, 252
Birds, 471
Biryani, 489
Blackcurrant juice, 440
Blood, 165, 222, 251, 262, 267, 359,
388, 414, 439, 462
plasma, 251
white cells, 415
Bones, 473
Brain, 339, 393, 458
Brandy, 308
Bread, 470
Bronchus, 360
Butter, 68, 408, 509, 510, 511, 512,
562
clarified, 510
Butterfly, 505
Caffeine, 189, 192, 545
Capsaicin, 489, 490, 491
Carbohydrates, 94, 111
Carrots, 498
Casein, 509, 512, 513
Castor oil, 263
Centaurea cyanus, 275
Cepahalopod, 340
Cheese, 60, 61
Cherries, 275
Chickens, 165
Chilli, 489
Chips, 519
Chlorophyll, 461, 462
Chocolate, 557
Citrus oil, 309
Coffee, 61, 62, 185, 188, 189, 190,
192, 229, 539, 545
Cappuccino, 61
Corn, 63, 211
Cornflower, 275
Cotton, 488
Coulomb's law, 513
Cream, 216, 217, 509, 511, 563
Cream, whipped, 509
Crisps, 241
Curds, 512
Curry, 489, 490, 491, 492, 496
Cyanophytes, 461
Cypridina latia, 478
Cytosine, 45
Dandelion, 314
Delphinidin, 274
Delphiniums, 275
Dextrin, 445
Diamond-backed terrapin, 414
DNA, 44, 45 434, 539
Dock leaf, 234, 245, 261
Dragons, 148
E. coli, 510
Eel, electric, 343, 344
Eggs, hen, 94, 165, 166, 170, 203
shell of, 165
Eggs, human, 47
Electrophorus electricus, 343, 344
Elephants, 501
Enzymes, 267, 397, 408, 409, 478
Epidermis, 253, 437
Epithelium, 458
Eugenia caryophyllata, 230
Eugenol, 229, 230
Eumelanin, 437, 438
Eye, human, 239, 428, 429, 444,
458
Faeces, 252
Fat, 505, 508
566
INDEX
Fatty acids, 354
Ferredoxin, 462
Fireflies, 478
Fish, 208, 223, 478
Flowers, 230, 498
colour of, 550
Follicle, 283, 285
Food (general), 77, 104, 148, 205,
216, 217, 241, 244, 270, 272,
280, 308, 408, 409, 469, 470,
550, 562
French dressing, 509, 513
Fruit, 216, 217
Gastric 'juice', 252
Gelatine, 342, 445, 506
Geraniums, 275
/3-D-glucose, 96, 97, 113, 174, 375,
376
Grape, 398
Grape, skins, 398
Gravy, 308
Grease, 349, 354, 519
Guanine, 45
Gum Arabic, 445
Haemoglobin, 252, 462
Hair, 283, 285
Hatchet fish, 478
Hazelnuts, xv
Histamine, 262
Homoptera Auchenorrhyncha, 414
Hormones, 387, 388, 557
Hydrangea, 274
Ice cream, 77, 104, 216, 217, 508
Insects, 523
Jam, 139, 140, 141
Jasmine, 230
Lactose, 409
Lantern fish, 478
Larynx, 360
Leaves, 498
Lemons, 133
Limes, 252
(+)-limonene, 133
Linseed oil, 485
Liometopum apiculetum, 414
Litmus, 273
Luciferase, 478
Luciferin, 478
Lungs, 120, 359
Lysteria, 510
Macaw, 501
Maize, 63
Malaclemys macrospilota, 414
Malic acid, 244
Maple syrup, 252
Margarine, 511
Mayonnaise, 508
Meat, 308, 316
Mediterranean cicadas, 414
Melanin, 430, 431,438, 559
Mentha arvensia, 125
Mentha piperita, 1 25
(-)-menthol, 125
Milk, 61, 252, 408, 409, 504, 505,
506, 507, 509, 510, 511, 512,
513, 563
milk fat, 509
Mother of pearl, 505
Nerves, 339, 340, 341, 344
Nettle, 245, 253, 264, 549
sting of, 264
Oil of cloves, 230
Oils, natural, in human skin, 81
Olive oil, 354, 512
Onions, 239, 549
Orange, 133
juice, 512, 513
Ouzo, 515
Papaver rheas, 275
Peacock, 505
Pearls, 505, 508
Peas, 63
Peppermint oil, 125, 543
Pernod®, 515
/J-phenylamine, 387
Pheomelanin, 437, 438
Phospholipids, 512
Pizza, 519
Pollen, 385
Popcorn, 31
Poppy, 275
Potato, 241
Potato crisps, 241
Protein, 267, 412, 413, 436
denaturing of, 95
Protists, 461
Red ant, 253
Remex crispus, 26 1
Rhizobium, 63
Rhubarb, 252
Ricard®, 515
RNA, 434
Roses, 275
Rubber, 89
Rumex obtusifolia, 261
Saliva, 244, 252, 279
Salmon, 252
Sausage, 87, 519
Shellac, 445
Skin, 35, 79, 240, 253, 358, 388,
434, 458, 473, 518, 559
oils, in, 81
Soap, 240, 245, 315, 445, 563, 517
Soil, 274
Sperm, 47
Spices, 489
Spider monkeys, 501
Spinach, 252
Spinal fluid, 252
Squid, 340
Starch, 346, 368, 506
Stomach, 262
Strawberries, 275
Sucrose, 139, 140, 141, 206
Sugar, 130
Sunflower oil, 354,485,512
Sweat, 81
Synapse, 339, 340
Tea, 201, 254, 408, 442, 509
Tears, 239, 548
Teeth, 279, 311
Terrapin, diamond-backed, 414
Thermodynamics, of colloids, 513
Thymine, 45
Tobacco, 120
Tongue, 9, 140
Tonic water, 314
Trachea, 360
Tryptophan, 445
Tuberose, 230
Tyrosine triphosphate, 445
Urine, 252
Vegetables, 217, 498
Vinegar, 233, 241, 244, 282
Vitamin B 12 , 462
Vitamin C, see ascorbic acid
Water Ilea, 414
Whey, 512
Wine, 358, 397, 426
Woad, 94, 98, 300, 444, 445
Wood, 94, 98
Wool, 488, 514
Yellow dock leaf, 261
Yolk, egg, 95, 166
INDEX
567
Chemicals cited in the text: elements, compounds, alloys and materials
This list does not include all compounds and materials cited in tables.
Acetic acid, see ethanoic acid
Acetonitrile, 302
2-Acetoxybenzoic acid, 467
Acetylene, 53
Adenine, 45
Aldehydes, 172
Alkene, 392
Alkyl halides, 394
Alkyl sulphonic acids, 517
Alumina, see aluminium oxide
Aluminium, 280, 282, 285, 301,
304
Aluminium chloride, 75, 262
Aluminium hydroxide, 262
Aluminium oxide, 285, 286, 463,
477
Aluminium sulphate, 514
Aluminosilicate, 499, 500, 501
Amines, 506
p-aminobenzoic acid, 439
Ammonia, 38, 52, 53, 108, 115,
135, 168, 240, 242, 258, 271,
272, 351, 356, 364, 494, 501,
506
Ammonium chloride, 242, 271, 272,
346, 347
Ammonium hydroxide, 241, 253
Ammonium ion, 242, 258
Ammonium nitrate, 63
Anilinium hydrochloride, 327
Anthracene, 97, 274
Anthrocyanin, 274, 275
Antimony, 348, 483
Aragonite, 122
Argon, 30, 58, 74, 481, 482
Arsenic, 393, 394
L-(+)-ascorbic acid, 244, 401, 512
Asphalt, 345
Aspirin, see acetoxybenzoic acid
Benzene, 55, 225, 226, 228, 454,
469
Benzoic acid, 96, 104
Bicarbonate ion, 242, 259, 267
Borax, 445
Boron- 12, 382
Brass, 183
Bromide ion, 292, 301, 325,
Bromine, 49, 108, 292, 325, 362,
497
2-bromo-2,2-dimethylpropane, 395
Bromobenzene, 226, 228
1-bromopentane, 394
Bronze, 311, 495
Buckminster fullerene, see fullerene
Butane, 49, 1 14, 504
1-Butanol, 114
1-Butene, 174
Butyl ethanoate, 115
Cadmium, 477
Cadmium sulphate, 296
Cadmium sulphide, 477
Cadmium, 295, 296
Caesium-133, 16
Caffeine, 189, 192, 545
Calcite, 122
Calcium, 301
Calcium carbonate, 122, 139, 165,
211, 245, 262, 269, 315
Calcium carboxylate, 245
Calcium chloride, 69, 321
Calcium fluoride, 124
Calcium halophosphate, 483
Calcium hydroxide, 266, 268
Calcium ions, 315, 521
Calcium oxide, 274, 476
Calcium silicate, 483
Calcium strontium phosphate, 483
Calomel, see mercury(I) chloride
Camphor cinnamate, 439
Camphor salicylate, 439
Carbohydrates, 94, 111
Carbolic acid, see phenol
Carbon, 18, 171, 238, 333, 346, 445,
504
Carbon-12, 18
Carbon- 13, 384
Carbon- 14, 380, 384, 386
Carbon dioxide, 37, 50 ff., 105, 111,
112, 114, 121, 165, 178, 184,
185, 189, 190, 191, 205, 206,
237, 238, 244, 245, 262, 267,
359, 384, 400, 420, 468, 545,
546
Carbon monoxide, 121, 147, 171,
224, 400, 420, 465
Carbonate ions, 316
Carbonic acid, 238, 242, 259, 268
Carboxylate anions, 260
Carboxylic acid, 397
/6-carotene, 498
Caustic soda, see sodium hydroxide
Ceric ion, see cerium( IV)
Cerium(IV), 352, 353, 354, 372
CFC, 33, 60, 61, 555
Chalk, see calcium carbonate
Chloride ion, 312, 327, 356
Chlorine, 49, 59, 69, 124, 135, 136,
243, 350, 361, 466, 498, 548
Chlorine radical, 358
Chloroform, 143, 206, 546
Chloromethane, 447
Chromium, 463
Citric acid, 265
[C1(H 2 0) 6 ]-, 137
Cobalt, 292, 304, 462
Cobalt-60, 379
Cobalt(II) chloride, 319
Cobalt(II) ion, 292, 304
Copper, 183, 285, 291, 292, 306,
310, 311, 312, 313, 322, 323,
325, 326, 334, 342, 345, 345,
346, 473, 477
Copper carbonate, 485
Copper chloride, 317, 507
Copper nitrate, 326
Copper sulphate, 127, 306, 310,
320, 321
Copper ion, 291, 292, 312, 313,
322, 323, 325, 326, 334, 342,
346,351, 356,357
[Cu(NH 3 ) 4 ] 2 +, 351, 356, 357, 363,
363
Cyclohexane, 491
Cytosine, 45
Decadodecylsulphinic acid, 523
Delphinidin, 274, 275
Diamond, 68, 109, 183, 542
1 ,2-dihydroxy ethane, 118
Dimethylformamide, 302, 491
Dimethylsulphoxide, 302
Dinitrogen tetroxide, 356
1,2-diol, 392
EDTA, 242, 250, 260, 265
Ethanal, 172, 358
Ethane, 105
Ethanoate anion, 162, 234, 271
Ethanoic acid, 114, 156, 158, 162,
164, 174, 233, 234, 241, 254,
267, 271, 351, 358
Ethanol, 68, 118, 164, 276, 308,
309, 351, 358, 391, 397
Ethanol, industrial, 118
Ethene, 53, 118, 362
Ethyl butanoate, 354
Ethyl ethanoate, 164, 351, 353, 354
Ethyl formate, 354
Ethyl methanoate, 390
568
INDEX
Ethyl propanoate, 354
Ethylene glycol, 118, 172, 220
Ethylhexyl methoxycinnamate, 440
Ethyne, 53
FC1, 42, 135
Ferric, see iron(III)
Ferrous, see iron(II)
Fluorine, 49, 74, 135, 466
Fool's gold, see iron sulphide
Formamide, see methanol
Formic acid, see methanoic acid
Fuller's earth, 239
Fullerene, 67, 109
Gallium, 477
Glycerol, 240
Gold, 14, 65, 301, 462
Granite, 382
Graphite, 109, 160, 183, 542
Guanine, 45
[H(H 2 0) 4 ]+, 235
Haem, 461
Halothane, 222
Helium, 55, 74, 481, 482
Heme, see haem
Heptyl bromide, 405
Hexacyanoferrate(II), 391
Hexacyanoferrate(III), 76
2-hexane, 406
Hydrides, 64
Hydrobromic acid, 297, 298
Hydrocarbons, 49, 361, 363
Hydrochloric acid, 247, 248, 271,
310, 321, 329, 332, 356, 375,
369, 370, 376
Hydrogen, 44, 46, 53, 62, 61, 65,
67, 108, 135, 144, 147, 167,
171, 289, 290, 292, 297, 298,
304, 311, 321, 322, 323, 329,
361, 364, 365, 367, 455, 470,
476, 494, 501, 539
Hydrogen bromide, 43, 297, 298
Hydrogen chloride, 42, 46, 242,
243, 466
Hydrogen iodide, 75
Hydrogen peroxide, 352, 353, 354,
372, 373
Hydrogen sulphide, 148
Hydroquinone, 148
Hydroxide ion, solvated, 235, 236,
242, 243, 244, 246, 249, 252,
253, 346, 396
Hydroxybenezene, see phenol
Hypochlorate ion, 244
Hypochlorite, 349, 350, 351
Hypochlorous acid, 243, 244, 468,
498
Indium sulphate, 310, 321
Iodide ion, 368
Iodine, 49, 74, 134, 202
Iridium-platinum alloy, 15
Iron, 287, 292, 323, 333, 345, 463,
473, 494, 495, 496, 501
Iron(II) ion, 287, 323
Iron(II) oxide, 333, 334, 345
Iron(II) sulphate, 445
Iron(II) sulphide, 333
Iron(III) chloride, 315
Iron(III) ion, 292, 461, 462
Iron(III) sulphate, 316, 317
Kaolin (clay), 501, 562
Krypton, 15, 74, 481, 482
Krypton-86, 15
Lead, 301, 303, 304, 348, 483
Lead sulphate, 348
Lead tetraethyl, 225, 547
Lead(II) ion, 304
Lead(IV) ion, 303, 304
Lead(IV) oxide, 348
Lead, spongy, 348
Lithium, 160, 294, 297, 301
Lithium chloride, 311, 313
Lithium ion, 294, 312
Lithium perchlorate, 160
Magnesium, 74, 301, 364, 365
Magnesium fluorogermanate, 483
Magnesium hydroxide, 262
Magnesium ion, 461
Magnesium sulphate, 123, 319,
321
Malachite, 485, 486, 560
Malic acid, 244
Manganese, 460, 477, 483
Manganese dioxide, 294, 346
Manganese oxyhydroxide, 346
Menthol, 125
Mercury, 5, 296, 299, 301, 311, 312,
330, 331,482
Mercury sulphate, 296, 299
Mercury(I) chloride, 297, 330, 331
Mercury(II) ions, 301, 482
Methanal, 451
Methane, 49, 52, 53, 55, 56, 67, 75,
112, 114, 117, 221,447
Methanoate anion, 253
Methanoic acid, 253, 261, 264
Methanol, 68, 147
methylated spirit, 68
Methyl ethanoate, 369, 370
Methyl iodide, 377
Methyl viologen, 148, 355, 374,
375, 429, 438, 445
Methylchloride, 447
Methylene blue, 305
Methyl-ethyl ether, 154
N-methylpyridine bromide, 418
Mustard Gas, 382
[Na(H 2 0) 6 ]+, 137
Nation®, 290
Neon, 31, 55, 69, 74, 424, 480, 481,
482
Nitric acid, 63, 238, 247, 250, 251,
265, 266
Nitric oxide, 108, 112, 267, 268,
356
Nitrogen, 47, 52, 63, 67, 68, 72,
108, 135, 186, 221, 223, 238,
311, 359,393, 394,466,494,
495, 496, 501, 504, 507
Nitrogen- 14, 384
Nitrous acid, 63
Nitrous oxide, 63, 108, 168, 237,
267, 268, 270, 367, 546
NO^ gas, 267, 268, 420, 458, 470
Nujol, 467
Nylon, 488
Octadecane, 49, 67
Octane, 362, 364
Octocrylene, 440
Opal, 483, 507
Osmium tetroxide, 392
Oxide ion, 290, 322
Oxygen, gas, 29, 57, 58, 72, 94,
105, 135, 143, 144, 207, 223,
237, 290, 333, 356, 358, 359,
360, 361, 362, 364, 388, 397,
398, 400, 476, 507
Ozone, 358, 359, 434, 439, 458,
554
depletion of, 358, 359
Palmitic acid, 240
Paraffin wax, 49, 67, 195
Pegmatite, 463
Pentadecane, 67
Peppermint oil, 125, 543
Permanganate ion, 76, 429, 443,
445, 459, 460, 461
Petrol, 49, 221, 224, 226, 361
Petroleum gas, 52
INDEX
569
Petroleum gel, 67, 68
Phenol, 254, 255, 257, 258, 549
Phenolphthalein, 276, 277
/S-Phenylamine, 387
Phosphate buffer, 269
Phosphoric acid, 259
Phosphorous, 111, 393, 394
Platinum, 289, 290, 297, 298, 301,
329
'Platinum black', 329
Platinum-iridium alloy, 15
Plutonium-238, 382
Polyethylene glycol (PEG), 160
Polystyrene, 508
Polyurethane, 68, 506
Porphyrin, 461
Potassium chloride, 220, 303, 310,
315, 330, 331, 332, 342,343
Potassium hexacyanoferrate(III),
391
Potassium hydroxide, 249, 310, 321,
347
Potassium nitrate, 64, 123, 171, 210,
342,518
Potassium palmitate, 245
Potassium stearate, 245
Potassium sulphate, 315
Propylene carbonate, 302
Proton, solvated, 156, 163, 234, 235,
236, 241, 242, 246-252, 255,
261, 289, 290, 292, 304, 321,
322, 323, 330, 337, 348, 498
'abstraction' of, 239, 241
Prussian blue, 462
Purple of Cassius, 65
Pyridine, 405
Quartz, 293, 474
Radon, 74, 382
Retinal, 459
Rhenium, 460
Rock salt, 216, 220
Ruby, 463, 477
Rust, see iron(II) oxide
Sapphire, 463
Silica, 68, 497
Silicate, 513, 514
Silicon carbide, 43
Silver, 280, 282, 285, 297, 298, 303,
311, 325, 326, 327, 330, 322,
336, 343, 403, 477
Silver bromide, 75, 297
Silver chloride, 75, 297, 318, 319,
326, 327, 322, 332, 336, 356,
507
Silver ethanoate, 283
Silver iodide, 75
Silver ion, 325, 326, 330, 343, 356,
403
Silver nitrate, 507
Silver oxide, 234, 280, 282, 303
Soda, see sodium carbonate
Sodamide, 242
Sodium, 70, 124, 234, 301, 321,
322, 323, 481, 482
Sodium bromide, 310
Sodium carbonate, 240, 244
Sodium chloride, 59, 68, 69, 70, 81,
123, 124, 126, 217, 218, 219,
314,316,356,518
Sodium cyanide, 382
Sodium decyl sulphate, 520
Sodium dodecyl sulphate, 517, 518
Sodium ethanoate, 241
Sodium hydrogen carbonate, 550
Sodium hydroxide, 239, 249, 245,
253, 264, 265, 353, 354, 390,
392, 418
Sodium iodide, 43, 75
Sodium nitrate, 303
Sodium tungsten bronze, 462, 463
Steel, 333, 473
Steel, stainless, 420
Stirling-FTG, 491
Strontium magnesium phosphate,
483
Sulphur, 67, 111, 134, 148, 238,
333, 334
Sulphate anion, 239, 250, 312
Sulphide, 334
Sulphur dioxide, 237, 267, 268, 356,
420, 470
Sulphur trioxide, 75, 112, 135, 135,
148, 470, 549
Sulphuric acid, 239, 250, 251, 259,
303, 310, 347, 348, 364, 365,
548
Sulphurous acid, 238
Syenites, 463
Table salt, 69, 123, 217, 308, 309,
316, 382
Technetium, 460
Tetraalkyl ammonium chloride,
523
Tetrabutylammonium
tetrafluoroborate, 303
TEA, see trifluoroethanoic acid
Thiocyanate anion, 461
Thionyl chloride, 135
Thiosulphate ion, 368
Thymine, 45
Tin, 182, 301, 311 ff, 483,
545
Tin(IV), 301
Titanium, 463
Titanium dioxide, 439, 440
Toluene, 228
Trichloroacetic acid, see
trichloroethanoic acid
Trichloroethanoic acid, 260,
261
Trifluoroethanoic acid, 261
Trirthylamine, 377
Tungsten, 504, 505
Tungsten bronze, 445
Tungsten trioxide, 75
Urea, 243
Urethane, 506
Urethane, 507
Viologens, 555
Water, 41 ff., 61, 62, 66, 72, 79,
92-94, 97, 102, 105, 106,
111, 112, 114, 123, 127, 132,
134, 135, 140, 141, 148, 162,
168, 171, 172, 175, 178-182,
184, 185, 190, 192, 194, 198,
199, 200, 202-204, 206, 207,
216, 218-220, 234-239,
241-244, 250, 289, 290, 292,
302, 309, 333,346, 351, 358,
408, 470, 483, 488, 497, 503,
504, 505, 506, 508, 513, 548
waste water, 513
Xylitol, 126
Yttrium vanadate, 477
Zinc, 183, 292, 292, 296, 299, 301,
303, 307, 311, 312, 313, 323,
335, 345, 346, 477, 507
Zinc orthosilicate, 477
Zinc oxide, 263, 439, 440
Zinc sulphate, 296, 299
Zinc sulphide, 477
Zinc ion, 307, 291, 292, 312, 313,
323, 335, 346
570
INDEX
Etymologies, definitions and the meanings of special words
The majority of these entries relate to etymologies. The remainder relate to precise scientific usage.
Abscissa, 4
Abstract, 241
Anaesthetic, 222
Anode, 280
Antiseptic, 255
Argentina, 5
Autoprotolysis, 236
Autoprotolysis, 236
Basic, 243
Blue-sky research, 483
Books on, 535
Calibrate, 11
Calomel, 331
Calorie, 61
Calorific, 112
Calorimetry, 61
Cappuccino, 62
Carbolic, 255
Cascade, 280
Catalysis, 421
Cathode, 280
Cell, 288
Centigrade, 11
Chlorine, xix, 59
Chlorophyll, xix
Chromophore, 274, 424
Complementary, 3
Compound variable, 5
Conjugate, 241
Corollary, 10
Criterion, 236
Cryoscopy, 217
Cyan, 461
Ductile, 15
Dust, 139
Dynamic, 7
Dynamo, 7
Ebullient, 127
Ebullioscopy, 217
Electrochemical, 283
Electrolysis, 283
Empirical, 281, 336
Emulsion, 507
Endogenic, 163
Endothermic, 82
Energy, 86
Engine, 132
Entropy, 131
Epidermis, 253
Epidural, 254
Etymology, xix
Exogenic, 164
Exothermic, 80
Experiental, 26
Factor, 19
Flocculate, 514
Fugacity, 155
Fugitive, 155
Giant, 19
Gigantic, 19
Gradient, 472
Homonuclear, 67
Hydraengea, 274
Hydrolysis, 238
Hydrophilic, 517
Hydrophobic, 517
Ideal, 10
In vacuo, 15, 465
Infinitesimal, 90
Infrared, 430
Integer, 247
Intermolecular, 38
Ion, 302
Irradiate, 425
Isochore, 171
Isotherm, 23, 162
Isothermal, 87
kilo, 101
Labile, 234
Latent heat, 82
Law, 7
Lime, 476
Litmus, 273
Medium, 508
Meta, 180
Micelle, 515
Microscope, 139
Miscibility, 207
Miscible, 207
Molecule, 65, 540
Monochromatic, 440
Narcosis, 222
Narcotic, 222
Normal, 178
Oligomer, 507
Opalescence, 483
Opaque, 425
Ordinate, 4
Ossified, 244
Oxygen, xix
Perfume, 309
Permeability, 340
pH, 246
Phase, 178
PhD, 14
Philosophical, 14
Photochemistry, 425
Photochromic, 403
Photolytic, 434
Photon, 426
Physicochemical, 2
Plural, 5
Principal and principle, 63
Product, 252
Pseudo, 167
Quanta, 430
Quotient, 52
Reciprocal, 23
Redox couple, 279
s.t.p., definition of, 35
Saponify, 240
Sequester, 394
Sigmoid, 266
Sour, 233
Spectroscopy, 429
Spectrum, 428
Stoichiometry, 362
Stoichiometry, 135
Strategy, 93
Sublime, 37
Substrate, 488
Surface tension, 139
Surfactant, 517
Surroundings, thermodynamic, 139
System, 139
System, thermodynamic, 137
Tangent, 92
Thermochemistry, 10
Thermometer, 8
Thermos flask, 7
Thermostat, 35
Trimer, 75
Ultra-violet, 430
Unit, 311
Unity, 311
Universe, thermodynamic, 137
Valency, 75
Virial, 57
Vitamin, 512
INDEX
571
Names cited in the text
Adams, G., 65
Andrews, Thomas, 50
Avogadro, Amedeo, 65
Boltzmann, Ludwig, 35
Bonaparte, Napoleon, 182, 393,
545
Bond, James, 109
Born, Max, 123
Boyle, Robert, 24, 273
Br0nsted, 242
Cannizarro, Stanislao, 65
Capra, Frank, 393
Celsius, 11
Charles I, King of England, 563
Charles, J. A. C, 21
Christ, Jesus, 382, 383
Clark, Latimer, 296
Clausius, Rudolf, 85, 132, 544
Cooper, Thomas, 515
Crick, Francis, 45
d'Aviano, Marco, 62
Dante, Alighieri, 544
Devil, the, 182, 478; see also lucifer
Donald Watson, 45
Draper, 426
Drummond, Thomas, 476
Eddington, Sir Arthur, 136
Edgar, King of England, 14, 537
Einstein, Albert, 136, 433
Eliot, T. S., 136, 544
Epictetus, 18
Faraday, Sir Michael, 65, 281
Farina, Jean Marie, 309
Faure, 347
Feminis, Paul, 309
Franklin, 242
Gay-Lussac, 21
George III, King of England, xix
Gibbs, Josiah Willard, 145, 544
Graham, Thomas, 506
Grotthus, 426
Guldberg, Cato Maximilian, 158
Haber, Fritz, 108
Hemlholtz, Hermann von, 85
Henry, William, 222
Hess, G. H., 98, 123
Ingold, Sir Christopher, 556
Jean, Sir James, 136
Jeremiah, Hebrew prophet, 240
Jesus, see Christ
Job, in Bible, 240
Joule, James Prescott 85, 86, 542
Julian of Norwich, Mother, xv, 535
Kelvin, Lord, 22, 54, 85, 542
Kirchhoff, Gustav, 542
Kirwen, 65
Langmuir, Irving, 68
Le Chatelier, Henri, 166, 544
Leclanche, Georges, 346
Lewis, G N., 64, 66, 242, 310,
556
Libby, Frank, 384
Lister, Lord Joseph, 254, 549
London, Fritz, 47, 48
Lucifer (the devil), 478
Maxwell, James Clark, 35
Meischer, 44
Mendel, Gregor, 46
Napoleon, see Bonaparte
Newton, Sir Isaac, 24
Pasteur, Louis, 510
Pauling, Linus, 66, 541
Plante, 347
Raman, Sir Chandrasekhara, 483
Rankine, William, 85
Raoult, Jean-Maries, 216, 546
Rayleigh, Baron, 484, 505
Renaldi, 11
Rey, Jean, 10
S0renson, 248
Stark, 433
Strutt, John William, see Rayleigh,
Baron
Thompson, William, see Kelvin,
Lord
Tyndall, John, 65, 505
van der Vries, Hugo, 46
van der Waals, Johannes Diderik,
43,48
van't Hoff, Jacobus, 162
Waage, Peter, 158
Walden, 556
Weston, Edward, 296
Wilkinson, William, 65
Places mentioned in the text
Arctic circle, 34
Argentina, 284
Athens, 84
Baghdad, Iraq, 345, 554
Briinn Monastery, Czech Republic,
46
Cologne, 309
Equator, 35
Everest, Mount, 201
Glasgow Royal Infirmary, Scotland,
254
Grand Canyon, 5
Hawaii, 434
Innsbruck, 385
London, xix, 84
Lucka, Paris, 484, 485
Manchester, UK, St Anne's Square,
86
Mediterranean, 345
Mediterranean, Sea, 483
Moscow, Russia, 182
572
INDEX
Mount Everest, 201
New York, 84
North Sea, Britain, 286
Otzal Alps, 385
Pashawa, Pakistan, 64
Scandanavia, 286
Sevres, Paris, 14, 18
Switzerland, 85
Turin Cathedral, Italy, 382, 383, 555
Tyrolean Alps, 385
Ypres, France, 242, 548
Znojmo, Czech Republic, 484
General index
Tables are indicated as (T)
Abscissa, 4
Absolute rates, theory of, see Eyring
Absolute temperature scale, see
Kelvin
Absolute zero, 21, 22, 30, 55, 472
Absorbance, 370, 428, 436, 441, 442
sign of, 441
Absorption, optical, 424
coefficient, 444
Acceleration, 4
units of, 18
Acid, see acid-base
conjugate, 241 ff.
constant, see acidity constant
dissociation constant, see acidity
constant
Lowry-Br0nsted theory of, 156,
233 ff., 251, 256, 268, 274
pH and, 251
properties of, 234(T)
weak and strong, 156, 253 ff.,
268, 274
weak acids, as indicators, 274
Acid rain, 237, 267 ff., 549
Acid-base indicators, 273 ff., 550
as weak acids, 274
colours of, 276(T), 277
end-point pH, 276, 277
Acid-base neutralisation, 262 ff.
Acidity constant K a , 255, 257(T),
259 ff., 268, 270, 271
from titration curve, 268
variations in, 259 ff.,
Acoustics, 484
Action potential, 341
Actions and reactions, Newton's
laws, 32
Activation, energy, 57, 203, 409,
411-415, 419, 420-422, 477
absolute rates, see Eyring
activated complex, 409
biological systems and, 413
catalysis and, 421, 422
activators, 477
Activity, 171, 210, 308 ff., 311, 319,
320, 321, 325-327, 329, 330,
331, 334, 339, 343, 553
dimensionless, 308
electrode potential and, 325 ff.
function of state:
gas mixtures, 311; impure solids,
311; liquid mixtures, 312;
pure gases, 311; pure
solids, 171, 311
mole fraction and, 311
solutions and, 312 ff.
value of unity, 311, 325, 327
Activity, coefficient, 308 ff., 314,
315, 319, 320, 321(T), 343,
553
calculation of, 318 ff.
effect of ionic strength, 314
mean ionic, 315
Adhesion, 487 ff.
Adiabatic, 88, 109, 131,150, 177
Adsorption, 384, 487 ff., 490-496,
499,501,519,521
adsorption, immobilising,
494-496
adsorbate, 490, 493, 501, 519,
521
enthalpy of, 492, 501
interfaces and, 521
isotherms of, 493, 499
materials:
detergent, 519; surfactant,
521; toxins, 501
multi-layers, 496, 497
rate constant of, 502
steric effects and, 492
substrate sites, 490, 501
temperature and, 491
Advertising, 32
Aeroplane, 152, 153
Aerosol, 361, 505 ff., 507
spray, 505
Afterglow, of phosphors, 478
Aggregation, 513
'Aging' of wine, 397
Air conditioning, 61
Air freshener, 25, 32, 33, 49
propellant in, 49
Air pressure, see pressure, air
Albino effect, 439
Alkali, reactions of, 244, 245 (T)
alkaline battery, see battery,
alkaline
Allergy, 262
Allotropes, 109, 111, 182
'Allowed' transitions, 461, 471
Alloy, 311
Alzheimer's disease, 263, 550
aluminium and, 263
Amalgam, 296, 301 ff., 311,
notation for, 301
Amnesty International, 551
Amount of substance, SI unit of, 16
Anaesthetics, 222, 546
Anaphylactic shock, 388
Andrews' plot, 52, 53
Angle of rotation, 394
Anglo Saxon England, 14
Angstrom, non-SI unit of, 17, 436
Anion stabilization, 260
Anions, 72, 284 ff.
Anode, 295 ff.
Anodized aluminium, 285
Antacids tables, see indigestion
stomach ache, 501
Anti-dazzle mirror, see mirror
Anti-freeze, 194, 220, 254
Antihistamine, 262
Anti-oxidants, 401
Anti-smoking pipe, 121
Apostle, 345
Arabic numerals, 75, 460
Arable farming, see farming
Archaeology, 555
Archeometry, see radiocarbon
dating
Area, molecular, 500, 501
Army, 93
Aromatherapy, 229
INDEX
573
Arrhenius equation, 203, 409, 411,
414
Eyring theory different, 418
graph of, 412
pre-exponential factor, 412, 420
rate of heartbeat and, 414
Arrow of time (entropy), 136, 544
Arsenic and Old Lace, 393
Asymmetric stretch vibration, 468
Athletes, 113
Atmosphere, non SI unit of, 17, 54,
186
Atmospheric pressure, see pressure,
atmospheric
Atomic nuclei, 450
Atomizer, 361
Attractions, 38, 314
between ions, 314
between molecules, 38
'Audit of energy', 145
Autoprotolysis constant A" w , 236 ff,
237(T), 249, 251, 252, 253,
258
Average bond enthalpies, see
enthalpy of bonds
Avogadro constant, 16, 284, 500
Ball, 427
Balloon, 20
hot-air, 20
party, 86
Bands (spectroscopic), 429, 434,
453, 464
band edge, 438
band vibration, 465, 466
Bar, non-SI unit of, 17, 34, 54
Barometer, 5
Base gas, 311
Base, conjugate, 241 ff.
Bases, in Lowry-Br0nsted theory,
233 ff., 239 ff
Basicity constants A"b, 257 ff.
'Bath' gas, 311
Batteries, 159, 160, 245, 246, 283,
284, 288, 294, 296, 303, 325,
343 ff., 553, 554
acid, 245, 246
discharge of, 160(F)
oldest, 345, 554
polarization of, 283
torch, 325
storage of electrical energy, 288
types of, 283
primary, 294, 297, 303, 345:
manganese dioxide, 294;
lithium, 294, 297; silver
oxide, 303
rechargeable, see battery,
secondary
secondary, 296, 284, 303,
344-348, 554:
'alkaline', 296, 346,
347; lead-acid, 284,
303, 347(T), 348, 554;
NiCad, 345
variation in potential, 294
voltage of, 303
Battle of Ypres, 242, 548
Beach, 83, 429
Beer's law, 442, 443, 484
Beer-Lambert law, 4, 445, 446,
489, 490, 499
Bernoilli effect, 152
Beta particles, 384
Bible, 240, 244, 345, 484, 485
Lucka, 484, 485, 486
Bicycle, 7, 59, 88, 345
tyre, deflation and inflation of, 88
ff.
tyre, inflation of, 59
Bioluminescence, 478
Bioluminescence, 560
Black ice, 193
Blackbody radiation, 433, 473, 484
Blagden's law, 216
Blast furnace, see furnace
Bleach, 243, 349, 351, 413, 468, 498
Blood, 165, 217, 222, 251, 262, 267,
462
pressure of, 217
stains of, 462
Boiling point, 11, 68, 69
elevation of, see ebullioscopy
saturated vapour pressure and,
180
variables effect on:
pressure, 106, 200;
temperature, 49, 106,
200, 408
volume change during, 98 ff.
Boltzmann constant, 416,
Boltzmann, energy distribution, see
Maxwell-Boltztnann
Bomb calorimeter, 94, 95, 121
Bonds, 38, 44, 60, 61, 107 ff., 114,
115,447,451,458,459,464,
465, 467, 469, 487, 494, 496
adsorption and, 487
angle of, 44, 451, 464
in water, 44
bond order, 494, 496
cleavage during reaction, 1 14
change during reaction, 107 ff.
definition of 'formal', 38
dissociation energy, 115, 447,
469
energetics of forming, 60, 61,
114; of breaking, 61
enthalpy of, see enthalpy of
bonds
length, 448, 450, 451
in water, 44
Born-Haber cycles, 123
Born-Oppenheimer approximation,
450
Boyle's law, 23, 27, 28, 50, 197
Breaking colloids, see colloid,
breaking
Breathing, 223, 397
fish, 223
wine and, 397
Bricks, 18
Brownian motion, 139
Bubbles, 26, 27, 316, 507, 523
Bubble bath, 507
Bubble-jet printer, 26
gas in liquid, 27
surfactants and, 523
soapy, 316
Bubonic Plague, 309
Buffer, 267 ff., 336
acetate, 271
phosphate, 269
Burial pits, 476
Burning, 111, 139, 141, 237, 362,
400
Burns, chemical, 245
Burnt umber (pigment), 485
Buttons, 182
Calibration, 11, 96, 337, 338, 386
Calorie, non-SI unit of, 17, 54
Calorimetry, 61, 94 ff., 167
introduction to, 61
Calorimeter, 94, 126
bomb, 94, 95, 121
Camping gas, 23
Cancer, 447
Cappuccino coffee, 61, 539
Car, 89, 101, 193, 194, 198, 220,
283, 305, 361, 362, 399, 400,
420, 505
components within: battery, 283;
exhaust, 420; headlamp,
505; radiator, 194; tyre, 89,
101; windscreen, 220
Carbocations, 395
Carbon dating, see radiocarbon
dating
Carburettor, 224, 361, 506
574
INDEX
Catalysis, 289, 397 ff., 420-422,
494, 495, 556
activation energy and, 421, 422
enzymes and, 397 ff.
catalytic converter, 399, 400,
420, 556
Cathedral, xix, 182
St Paul's in London, xix
Cathode, 290, 295
Cations, 284 ff.
Cattle prod, 281
Cause and effect, 1
Caustic soda, see sodium hydroxide
CD player, 284
Cells, electrochemical, 190, 280 ff.,
292, 295 ff., 302 ff., 345, 551,
552
concentration, see concentration
cell
emf, see emf
fuel, 289 ff., 301, 551
galvanic, 345
Gibbs function of, 293
number of electrons, 293 ff.
right and left hand sides, 292
schematic, see schematic
standard, 295 ff., 552
thermodynamics of, 295 ff.
Celsius, see centigrade
Centigrade, 11
relationship with kelvin, 22
Centroid, of charge, 492
Ceramic, 470
CFC (chlorofluorocarbon), 33
Chaos, 130
Charge accumulation, 307
Charges, 45, 59, 260, 306, 493
centroid of, 492
charge transfer, 148, 294, 459,
461, 462, 463, 464
metal-to-ligand, 461, 462
mixed valency, 462 ff.
delocalisation of, 260
donation of, 493
excess, 45
flow of, 235, 280
formation of, 306
Charles's law, 21, 26, 27
Chemical adsorption, see
chemisorption
Chemical energy, 78
Chemical inertia, Le Chatelier's
principle, 166
Chemical inertness, see inertness
Chemical messengers, see hormone
Chemical potential, 212 ff., 334, 546
definition, 215
effect of temperature, 214
elevation of boiling temperature
and, 217
mixtures, 213
mole fraction and, 213
standard, 213
Chemical warfare, 242
Chemiluminescence, 478 ff.
Chemisorption, 492 ff.
China clay, 499, 500, 501
Chromophore, 274, 424-426, 442,
446, 463
in nature, 559
Civil war (English), 563
Clapeyron equation, 193-198,
196-198,202, 204
approximations to, 197
Clapeyron graph, 202
Clark cell, 295, 297, 299
Clausius equality, 136, 137, 146
Clausius- Clapeyron equation, 197,
198 ff., 204, 504
'linear' and 'graphical' form of,
202
Clothes, 2, 14, 134, 182, 385, 386,
462
Coacervation, 513
Coagulation, 513, 514
Coal, 98, 121, 237, 255
Coalescence, 39
Cobalt blue, 444, 445
Coefficient of friction, 192
Coke, 121
Cold light, see luminescence
Colligative properties, 212 ff., 546
Collisions, 32, 33, 39, 55, 131, 133,
411,412,472,480,481
between gas particles, 32, 33, 39,
55, 131, 133, 411,412,
472, 480, 481
between reactants, 411, 412
coalescence following, 39
elastic, 39
gas molecules with container
walls, 32, 33
inelastic, 39, 131, 133, 480, 481
Colloids, 504 ff.
breaking, 510 ff.: chemically,
511; mechanically, 511;
thermally, 510
classification, 507
physical nature of, 517 ff.
stability of, 509
Colour, 2, 75, 129, 211, 275, 303
ff., 322, 349, 356, 362, 403,
421, 426, 429, 432, 455, 458,
459, 461, 463, 477, 481, 483,
489, 498, 560,
complementary, 429
inorganic compounds:
silver chloride, 322; silver
iodide, 75
organic compounds:
cherries, 275; cornflowers,
275; delphiniums, 275;
eggshells, 211;
geraniums, 275;
hydrangeas, 274;
raspberries, 275;
strawberries, 275
redox state and, 305, 306
white light and, 432
Combination pH electrode, see
electrode, pH
Combustion, 94, 112, 120
incomplete, 120
enthalpy of, see enthalpy of
combustion
Competing reactions, 395 ff.
Complementary colours, 429
Complexation, enthalpy, see
enthalpy, complexation
Complicated reactions, 393 ff.
Compound units, 18
Compound variables, 5, 103
Compression, 25, 54, 153, 197
compressibility, 54
compressibility factor, 54
Computer, printer, 26
Concentration, 247, 264, 190, 323,
308, 314, 333-329, 333 ff.,
364 ff., 366, 396, 401, 403,
442, 443, 336, 484, 553
cells, 323, 333-329, 333 ff., 553
changes during reaction, 364 ff.
consecutive reactions, 403(T)
definition of, 264
gradient of, 290
infinity, see infinity concentration
profile, 365, 366, 396, 401
real and perceived, 308, 314;
see also activity
SI unit of, 247
Concurrent reactions, 394 ff.
Condensation, 50, 51, 79, 178(T)
of C0 2 , 51
of water, 50, 79
Condensed phases, definition, 38
Condenser, reflux, see reflux
condenser
Conduction, 370
of electrons, 69, 72
INDEX
575
of ions, 301
of heat, 89
Conductivity, ionic, 302 ff., 235
Conjugate acid and base, 141, 270
Conjugation, of dye, 498
Consecutive reactions, 397 ff.
Constant pH solution, 271
Consumption of reactant, see
kinetics
Convection, 20
Cooking, 95, 87, 166, 170, 199, 203,
217, 409, 470, 471
rate of, 203, 217
'Cooling agents', 125, 543
Corrosion, 243, 286, 552, 347
Cotton, 487
Coulomb, 284
Coulomb's law, 123, 312, 458
Coulomb potential energy, 313,
314
Couple, redox, see redox couple
Covalency, 50
Covalent bonds, 64, 68
definition, 68
introduction, 64
Covalent compounds, properties,
68(T)
Covalent solids, giant, 68
Critical fluids, 51, 190 ff., 540
critical point, 179, 190, 191
determination of, 190
critical pressure, 51
critical state, 50
critical temperature, 50
Critical micelle concentration, 515
ff., 520
Crucifixion, 383
Crying, see tears
Cryoscopy, 212 ff., 216, 546
cryoscopic constant, 218
rate of, 220
Cryostat, 31
Crystallization, 137, 513
Crystals, hydrated, 521
'Curly d' 3, 149
Current, 16, 26, 186, 224, 280, 282,
287, 293, 295, 301, 301, 339,
347, 473, 477
anodic and cathodic, 186
electric, 16, 26, 301, 473, 477
SI Unit of, 16
emf and, 293, 295
forward and backward, 287
sensors and, 224
Curtains, 442
Cushions, 507, 508
Cylinder, volume of, 32
Dalton's law, 221 ff., 360
'Dandelion clock', 314
Daniell cell, 312, 313, 314, 345 ff.
Dating, see radiocarbon dating
Debye-Huckel, 312, 313, 318, 319,
320, 321
'A factor', 319
limiting law, 319
simplified law, 320
Decaffeinated coffee, 189 ff., 545
Deflation, of tyre, 88
Deflation, temperature of air
expelled during, 88
Defrosting food, 470
Degradation, 322
De-icer, 220
Delocalisation, of charge, 260
Delta, definition of A operator, 79
Denaturing, of proteins, 95, 166,
1170
Density, 18, 20 ff., 37, 191, 508,
509
effect of temperature, 20, 37
units of, 18, 21
Depression of freezing point, 212,
214
Depression of melting point, see
cryoscopy
Depression of melting temperature,
see cryoscopy
Desorption, 491, 501
rate constant of, 502
Detector, 425
Detergency, 518, 520, 521, 522
Dew, 42, 178
Dewar flask, 7, 79, 94
Diamond, instability of, 109, 183,
542
Diamonds are Forever, 109
Diaper, see nappy
Diatomic molecules, photochemistry
of, 449
Dielectric constant, 302
Diels-Alder reaction, 447
Diet, 267
Diet, healthy, 217
Diffusion controlled reaction, 415,
416 ff.
Diffusion, 151, 342, 358
Digital watches, see watch
Diluent, 311
Dimensional analysis, 13, 162, 228
Dimerization, 354, 374
Dimmer switch, 433
Dipole, 41, 43, 47, 59, 67, 234,
470
dipole-dipole interactions, 43
Diprotic acids, 250
Direction of change, 130 ff.
Direction of thermodynamics
change, 83
Dirt, 349, 521, 522
Dirty water, 513 ff.
Discharge lamps, 480 ff.
Discharge, electric, 72
Disorder, 129, 130, 131, 137
spatial, 129, 130, 137
Dispersal medium, 505 ff., 507
Dispersed medium, 505 ff., 507
Dispersion forces - see London
Displays
liquid crystal, 40 ff., 538
picture element, 41
Dissociation, of solutes, 234
Dissociation constant, see acidity
constant
Dissociation energy, see bond
enthalpy
Distillation, 102, 229, 397, 547
steam, 229
Distribution, of energy, 35, 131
Dominant, genetic trait, 47
Doped oxide, 224
Doped semiconductors, 209
Doping, 209, 224
Dot cross diagrams, 65
Double layer, see electric double
layer
Dry ice, 37, 178, 184
Drying, of clothes, 134
Duality, wave-particle, 431 ff.
Dust, 457, 458, 508
Dye, 129, 273, 285, 286, 393, 558,
489, 492, 498, 499
Dynamic electrochemistry, see
electrochemistry
Dynamo, 7
Dyspepsia, 262
Eau de Cologne, 309 ff., 552
Ebullioscopy, 217 ff.
ebullioscopic constants, 219(T)
Eddy currents, 20
Effervescence, 244
Eggshells, and partition, 211
Einstein equation, see
Planck-Einstein
Elastic collisions, of gas molecules,
39 ff.
Electric discharge, 72
Electric double layer, 512
Electric iron, 503
Electrical conductivity, 69, 302
576
INDEX
Electrical energy, 2, 288
Electrical resistance, 12
Electrochemistry, 279 ff., 305 ff.
dynamic, 285, 287 ff., 316, 550
Electrochromism, 305, 552
Electrocution, 235
Electrode, 41, 261, 286, 297, 301
ff., 336-338, 553
amalgam, see amalgam
composition of, 286
formation of charge at, 306
passivating layer on, 301
pH, 261, 336 ff., 553
'slope' of, 337
reference, see reference electrode
saturated calomel, see reference
electrode, saturated
calomel
silver- silver chloride, see
reference electrode,
silver-silver chloride
standard hydrogen (SHE), see
standard hydrogen
electrode
types of:
glass, 336, 337, 338(T); glassy
carbon, 301; inert, 301;
metallic, 301; redox,
301 ff.; silver- silver
bromide, 297; wood,
300
Electrode potential, 303, 304, 314,
321, 322 ff., 330, 333, 342
activity and, 325 ff.
current passage and, 328
definition of, 303
em/ and, 303
half cell and, 325 ff.
sign of, 322 ff.
standard, 321, 323, 324(T), 326,
327, 330, 333, 342
Electrology, 283, 285, 551
Electrolysis, 280, 283, 302 ff., 314,
316 ff., 342, 551
cells, 280, 282
notation i.e. '1:1', '1:2', 316 ff.
strong and weak, 314
swamping, 342
Electromagnetic waves, of light, 41
Electromotive force, 288
Electronegativity, 42
Electronic state, 453
Electron, 64, 84, 286, 290, 301, 305
ff., 410, 427, 450-457, 460
ff., 477, 479, 473, 498
accumulation, 64
attachment energy, see electron
affinity
charge on, 284
density, 498
excitation of, 453, 454
gun (in TV), 477
movement during electrode
reaction, 286, 305 ff., 410
photo-excitation of, 460 ff.
Electron affinity, £( ea ), 72, 73(T),
123, 124, 541
Electron-pair theory, 66
Electrophilic addition, 362
Electropositivity, 42
Electrostatic interaction, 43, 313 ff.,
458
Elevation of boiling temperature,
217
Emergency heat stick, 127
emf 160, 288 ff., 290-298, 303,
312-314, 322-325, 328, 337,
342, 345-346
cells and, 288 ff.
concentration cells, 334 ff.
current, effect of drawing, 288,
293, 295
electrode potentials and, 303
Gibbs function and, 160, 322
non-equilibrium measurements
of, 294
reversible measurement of, 300
sign of, 288, 294
temperature effect of, 296, 293 ff.
virial series for, 297
Emission, of light, 424, 425, 472,
479, 481
Emulsions, 439, 562, 506 ff, 509.
inversion, 509
paint, 506
solid, 507
Emulsifiers, 508 ff., 512 ff.
Endogenic, 163
Endothermic, 81 ff., 126, 411
Energetic disorder, 135
Energy, 2, 7, 10, 30, 33, 34, 35, 39,
46, 60, 62, 78, 80, 83, 85, 86,
131,140, 141, 145, 157, 159,
289, 409, 433, 456, 457, 467,
469, 479
activation and, see activation
energy
'audit' of, 145
bond rearrangement, 60, 157, 289
chemical, 159
conventions for sign of, 80, 83
cycles with, 85; see also Hess's
law
disorder of, 131
distribution of, 35, 131
electrical, 2, 159, 288
equivalence to raising/ lowering
weights, 86
exchange following collision, 10,
39
gain and loss, 80
hydrogen bond, 46
kinetic, see kinetic energy
light and, 433
molar, 62
photons and, see photon, energy
potential, 60, 78, 91, 427
room temperature and, 33
quantised, 30, 78
rotational, 78; translational,
30, 78
temperature and, 7, 60, 87
traps of, 479
Engine, 132, 224
Enthalpy of activation, and
activation energy, 417
Enthalpy, 102 ff., 104, 107, 108,
109, 110-112, lll(T), 114,
115, 116(T), 118, 120, 121,
123 ff., 126, 193, 194, 195,
200, 543
activation, 416, 417, 419
adsorption, 492
cells, 298 ff.
definition of, 109
formation, HO-lll(T), 118
indirect measurement of, 126
introduction to, 102 ff.
lattice, 123 ff.
processes:
activation, 416 ff.; adsorption,
492, 494, 495, 496, 501,
504; boiling, 200; bonds,
114 ff., 115, 116(T),
123, 124, 449, 450 ff.,
458, 543; combustion,
112, 112(T), 121;
complexation, 127, 337,
543; melting, 194, 195;
reaction, 108, 411, 543;
solution, 125, 210;
sublimation, 123, 124;
vaporization, 108
sign of, 104
standard, 107
state function, 120, 421
via van't Hoff isochore, 174, 175
Entropy, 60, 131 ff., 136, 139, 144,
168, 293, 544
arrow of time, 136, 544
INDEX
577
cells and, 296 ff.
Gibbs function and, 144
heat capacity and, 139 ff.
phase and, 133
processes:
activation, 416, 417, 419;
mixing, 136
temperature and, 139
Envelope, spectral, 454
Epidural, 254
EPR, 447
Equilibrium, 8, 10, 156, 166, 181,
241, 287, 294, 295, 350, 490
Equilibrium, adsorption and, 490
current and, 287
dynamic, 10, 241
emf and, 294
Gibbs function and, 157
electrochemical, 287
phase change and, 181
resistance to change, 166
speed attaining, 156
thermal, 8
Equilibrium constant K, 158 f., 162
ff., 167, 171, 205, 218 ff.
Gibbs function and, 162, 164
processes:
acid dissociation, see acidity
constant; adsorption,
502; micellation, 515;
partition, 205
pseudo, 167
reaction quotient and, 159
solids and, 171
types of:
cryoscopic, 218; ebullioscopic,
219; solubility product
and, see solubility
product
units of, 159
Equilibrium reactions, kinetics, see
reversible reactions, kinetics
Equivalence point, in titration, 263
Essential oils, 190, 229, 230, 547
Esterification, enthalpy of, 1 14
Esters, 115
Evaporation, 81, 133, 134
Excess charges, 45
Excited state, 403, 426, 449, 450 ff.,
453, 454, 456, 457, 460, 479,
481, 482
long-lived, 403
Exhaust, of car, 399, 400
Exogenic, 164
Exothermic, 79 ff., 112, 114
definition of, 79
Experiment, recording of, 34
Experimental chemistry, books on,
534
Explosion, 144, 329
Exposure, 127, 182
Extensive quantities, definition, 93
Extent of reaction f , 156 ff., 254,
256, 350, 394
dissociation of weak acids and,
156, 157
equilibrium constants and, 165
Extinction coefficient, see
absorption coefficient
Eyring, 416-420
absolute rates and, 416 ff.
Eyring equation, 416
Eyring plot, 417.
Eyring theory, 416 ff., 419
differences from Arrhenius, 418
Factors, standard, xxviii ff., 19, 101,
102
Fahrenheit, 86, 415
Far infrared, 463
Faraday, 283 ff., 287, 295, 300, 325,
charge, 284, 285
constant, 284
laws of electrolysis, 283, 284(T),
287, 295, 300, 325
Farmers, 63, 165
Feasibility, thermodynamic, see
spontaneity
Fermentation, 65
Fertiliser, 63
Filament, in bulb, 474 ff., 476
Fingerprints, 354, 485
Fire, 35, 94, 111, 148, 507, 508
fire extinguisher, 508
First law, see thermodynamics
First-order reaction, 351, 356, 369
ff.
graphical methods, 371
units of rate constant, 369
and intermediates, 389
Fixation, nitrogen, 63, 540
Fizzy, drink, 205
Fizzy, lolly, 244
Flames, temperature of, 114
Flash Photolysis, 410
'Flat' drink, 206
Flavour, 241, 408, 550
Flocculation, 513, 514 ff.
Fluorescence, 479 ff. 482
Fluorescent 'strip lights,' 482
Fluorescent paint, 479
Flux, 433, 452
of photons, 433
Foam, 507
Fog, 185
Fool's gold, 333
Football, 430
Forbidden, quantum mechanically,
479
Force, 195, 196, 198, 464, 468
force constant, 464, 468
force exerted, 195, 196, 198
Formal bond, 38
Formal charge on ions, 59, 260, 360
Formation of bonds - see bond
'Fouling', 510
Fragments, following
photo-dissociation, 457
Fragrance, 133, 185, 190, 548, 552
France, 14
French Academy of Sciences, 14
French Revolution, 14
Franck-Condon principle, 410, 451,
453, 559
Free energy, see Gibbs function
Freezing, 178(T)
freeze drying, 185
freezing point, 11, 178, 212
depression of, 212
Frequency, 293, 432, 435, 469, 484
shift in (Raman), 484
Friction, 86
friction factor, see coefficient of
friction
Fridge, 60, 408
Frontier orbitals, 59, 60, 305
Frost, 178
Fuel, 111, 289, 551,237
fuel cell, 289, 551
hydrogen, 289 ff., 301
Fugacity, 155
fugacity, coefficient, 155
Fuller's earth, 239
Function of state, see state function
Functions, notation for, 2, 6
Furnace, 495
blast, 69
Fusion, 178(T); see also phase
Gales, 286
Galvanic cell, see cell, galvanic
Gamma rays, 428, 431, 435, 446
Gas constant, 28, 54(T)
effect of units when citing, 54
Gas, 20 ff., 25, 52, 55, 222; see also
phase
camping and, 23
molar volume of, 55
non-ideality of, 52
578
INDEX
Gas (continued)
propellant in cans of air
freshener, 25
volume, effect of temperature,
20,25
solution of, 222
Gasoline, see petrol
Gay-Lussac law, see Charles's Law
Gel, 67
Gelling agents, 342
Geometric mean, 315
Giants, 19
Gibbs energy, see Gibbs function
Gibbs function of activation,
417-419, 422
Gibbs function, 144-149, 153,
160-163, 166 ff., 168, 181
ff, 196, 293, 300, 322,
354, 417, 418, 544
cells and, 160, 293, 300, 322
definition of, 145
emf and, 322
equilibrium, 160
equilibrium constant and, 162,
164
partial molar, see chemical
potential
phase change and, 181 ff.
relationship between AG and
AG G , 160
system and, 144, 147
sign of, 146, 147, 163, 293
standard, 160 ff.
state function, 147
variables:
pressure, 148, 149, 153;
temperature, 149, 166 ff.
Gibbs', sense of humour, 145
Gibbs-Duhem equation, 148, 154,
196
Gibbs-Helmholtz equation, 166 ff.
Glands, 81
Glass, 309, 337, 470, 474, 475, 480,
481, 482, 497
glass electrode, see electrode,
glass
Glassy carbon electrode, see
electrode, glassy carbon
Glucose tablets, 113
Glue, 506
Gold paint, 462
Gold, fool's, 333
Golf, 430
Gravity, 6, 15, 86, 97, 505, 509
acceleration due to, 6, 86, 97
Greek, language, 536
Greenpeace, 549
Ground state, 427, 449, 450 ff., 453,
454, 457, 460, 482
Haber process, 108, 135, 494
Hail, 178
Hair dryer, 69
Hair removal, see electrology
Half cell, 290, 302, 300, 303, 312,
313, 321 ff., 326, 334, 342
activity and, 325 ff.
electrode potential and, 325 ff.
Half lives t l/2 , 378 ff., 390
Halons, 60
Hand cream, 511
Hard and soft, 67, 69
Hasselbalch, see
Henderson-Hasselbach
Headlamp, car, 505
Health, 217, 241, 420, 471
diet for, 217
Heartbeat, activated, 414
Heat, 7, 35, 87, 89, 430, 464, 472
change, 87
conduction, 35, 89
exchanger, 60
movement of, 7
solution, of, see enthalpy of
solution
vaporization, of, 230
Heat capacity, 91-93, 104, 105,
107, 139-142
entropy and, 139 ff.
temperature and, 141, 142
isobaric, definition of, 93
isochoric, 92
molar, definition of, 92
specific, definition, 92
'Heat death', 136
Heat sticks, 127
Heater, 91, 93, 96, 98
electrical, 93, 96
Heating, change in volume during,
98 ff.
Helix, of DNA, 44
of liquid crystals, 41
Henderson-Hasselbach equation,
271 ff.,
Henry's law, 222-225, 309
Henry's constant, 223(H)
Heredity, 44, 539
Hess's law, 98 ff., 108, 118 ff., 123,
148, 178, 180, 194, 543
Born-Haber cycle, 123
cycles with, 118 ff., 123,
cycles with, using ions, 123
High-performance liquid
chromatography (HPLC), 206
History of chemistry, books on, 535
Hollow Men, The, 136, 544
HOMO, 426, 427
Homonuclear, molecules, 47
Honeymoon, 483
James Joule's, 85
Hooke's law, 464
Hormones, 387, 388, 557
Horror films, 184
Hot-air balloon, see balloon
HPLC, 206
Hydration, energy of, 127
Hydrogen bond, 44, 46, 62, 61, 167,
539
energy of, 46, 62
fuel cell with, see fuel cell,
hydrogen
test for, 144
Hydrolysis, 237 ff.
reactions involving, 118
Hydronium ion, see proton, solvated
Hydrophilic, 517
Hydrophobic, 517
Hydroxide, different from hydroxyl,
239
Hydroxonium ion, see proton,
solvated
Hydroxyl, different from hydroxide,
239
Hypohalite addition, 498
Ice, see water; also
cubes, 175
misty centre of, 208
dry, see dry ice
iceberg, 194
Ice Man, the, see Otzi
shoes, 192
skater, 192, 198
slippery, 192
Icons, 560
Ideality,
absence of, 52
gases, effect of temperature on,
53
Ideal mixture (Raoult's law), 228
Ideal-gas equation, 28, 38, 52, 53,
55, 57, 107, 154, 203, 212,
221
Illness, temperature induced, 415
Immiscibility, 504, 520
Immobilizing, during adsorption,
494-496
Incandescence, 473, 476, 560
Incident light, 425, 439, 484
Incineration, of waste, 32
INDEX
579
India ink, 445
Indicators, see acid-base indicators
Indigestion tables, 262, 549
stomach ache, 501
Induced dipole, 47
Industrial alcohol, 118
Inelastic collision, of gas molecules,
39 ff.
Inert electrode, see electrode, inert
Inertness, chemical, 74
Infection, temperature increase, 415
Infinite dilution, 315
Infinity concentration, in kinetics,
405 ff.
Inflation of tyre, 59, 89
Inflation, temperature of air inside
tyre, 89
Infrared light, 427, 428-430, 463,
464, 467, 469, 484
lamp, 430, 464
near and far, 463
spectra, see spectrum
Initial rates method, 366, 367
Ink, 421, 427, 445, 446
for bubble-jet printer, 26
India Ink, 445
Ink jet, printers, 536, 558
Inner orbitals, 60
Insulator, of heat, 80
Integrated rate equations, 369 ff.
consecutive reactions, 402 ff.
first order, 369-374
linear form, 372, 373
reversible reactions, 405 ff.
second order, 374-378
Intensity of light, 433 ff.
Intensive properties, 191
Intensive quantities, definition, 93
Interactions,
between gas particles, 59
dipole-dipole, 43
electrostatic, 43
electrostatic, see electrostatic
interactions
ionic, see ionic interactions
Interface, and adsorption, 509
Intermediates, 388 ff., 398, 399, 400
chemical, 388 ff.
reactive, 400 ff.
Intermolecular, definition, 38
Internal energy, 77 ff., 87, 98, 102
ff.,
energy cycles with, 98 ff.
sign of, 87
work and, 102 ff.
International Bureau of Weights and
Measures, 14
Inter-particle interactions, 55
Ions, 26, 59, 126, 137, 254, 302,
309, 314 ff., 312, 489, 553
association of, 314 ff., 553
atmosphere of, 312
charge of, 314, 316
dissociation to form, 254, 489
interactions between, 312
movement of, 340
screening of, see screening
solutions of, 126, 137, 254, 302,
312,489
conductivity of, 302
solvated, 126, 137, 302
Ionic compounds, properties, 69(T)
Ionic strength, 309, 315 ff., 319,
320, 321, 352, 510
Ionization, 59, 481
Ionization energy, 7 e , 70, 71(T),
123, 124, 541
Ionization, photolytic, 458
IQ, 5
Iridescence, 505
Ironing, 487 ff., 503
Irradiation, 425, 469
Irreversibility, 167, 514
chemical, 167
flocculation, 514
Isobaric heat capacity, see heat
capacity
Isochore, the, see van 't Hoff
isochore
isochore plot, 174, 210
Isochoric heat capacity, see heat
capacity
Isomantle, 102
Isomerism, 458
Isotherm, 50
adsorption, 489 ff., 491, 496,
499
isotherm, van't Hoff, see van't
Hoff isotherm
Isothermal, definition of, 87
Isotopes, 379(T), 383
IUPAC, 46, 68, 76, 191, 510, 535
IVF, 47
Joules, 447
Junction potential, see potential,
liquid junction
Kaolin and toxicity, 562
Kelvin temperature, 412
relationship with centigrade, 22
temperature scale, 22, 172, 174
Kettle, 2, 79, 91, 99, 178
element of, 91
whistling, 99
kilo, 101, 103
Kilogram, anomaly of in SI, 17
Kinetics, 156, 183, 349, 403 ff.; see
also first order, integrated rate
equations, rates, and second
order
introduction, 349
of phase change, 183
of reversible reactions, 403 ff.
integrated rate law for, 405 ff.
Kinetic energy, 30, 33, 39, 130, 224,
362, 409
collisions, inelastic, and, 39
gases and, 30
relationship with temperature, 30,
33
theory, 409
Kirchhoff s law, 104, 106, 172, 197
'Knocking', of petrol, 225, 547
Krafft point temperature, 519 ff,
563
Lakes, 267, 268
Lambert's law, 442, 443
Lamp, 345, 452, 476
bicycle, 345
Langmuir adsorption isotherm, 499
ff., 503
Language, books on, 536
Laser, 410, 484
Latent heat, 82
Latin descriptors, 363
language, 536
Lattice enthalpy, see enthalpy
lattice energy, incorrect term, 123
Lava, 511
Laws of motion, see Newton's laws
Le Chatelier's principle, 166 ff., 544
Lead-acid battery, see battery
Leclanche cell, 346, 347(T)
Length,
archaic unit of foot and yard, 1
Lewis structures, 64, 65 ff,. 70
hydrogen, 65; sodium chloride,
70; water, 66
Liebig condenser, see reflux
condenser
Light, 15, 69, 306, 403, 425, 476
bulb, 433, 474 ff.
interaction with, 425, 426
polarization of, 132
refraction of, 20
scattering of, 427483 ff. 505 ff,
563
580
INDEX
Light (continued)
speed of, 15, 431, 432
storage of, see phosphorescence
UV, see UV light
Lightning, 15, 25, 63, 72, 537, 469
'Lime', see calcium oxide
Limelight, 476
Limestone, see calcium oxide
Liquid crystals, 40, 538
Liquid junction potential, see
potential, liquid junction
Liquification, 178(T)
Litmus test, the, 273
Litre, non SI unit of, 17
Load, in battery or cell, 294
Locomotives, 132
Logarithm, 320, 339, 413
London dispersion forces, 47, 49,
60, 74, 517, 539
Lowry-Br0nsted acids, see acids,
Lowry-Br0nsted
Lowry-Br0nsted bases, see bases,
Lowry-Br0nsted
Luminescence, 476 ff.
Luminous intensity, 15
LUMO, 426, 427
Macroscopic, 32
Magnet, 41, 43
Magnetron, 469
Marbles, 91, 410
Marriage, 241
Mars, planet, 17
Mass action, law of, 158
Mass spectrometer, 384
Mass, see weight
Matches, 94
Mathematician, 149
Maxwell relations, 143, 151
Maxwell -Boltzmann distribution,
35, 314, 415, 449, 453, 537
Measurement, scientific, 9
Medicine, 430
Melanoma, 434
Melting, 68, 69, 177 ff., 178(T),
192, 212, 214, 218 ff.,
ice, 177 ff., 178(T)
salt effect on, 218 ff.
temperature, 68, 69, 177 ff.,
178(T), 192, 212, 214
chemical potential and, 214
depression of, 212, 214, 218 ff.,
impurities, effect on, 212
pressure and, 192
Melting point, mixed, 212
Membrane, 339, 340, 345, 346, 506
biological, 339, 340
permeable, 340
porous, 345, 346
semi-permeable, 340
Meniscus, 207
Metabolism, 415
Metastability, 180, 479
phases, 180
states, 479
Micelle, formation, 515
Micelles, 514 ff.
Microscopic reversibility, 405 ff.
Microscopic, 32
Microwaves, 432, 428, 446, 458,
459, 469 ff.
microwave oven, 469 ff., 430
Middle Ages, 383, 485, 486
Minus-oneth law, see
thermodynamics
Mirror, 39, 305, 306
anti-dazzle, 305, 306
electrochromic, 305, 306
Mist, 50
Mixed melting point, 212
Mixed valency, 463 ff.
Mobile phone, 284, 430, 471
Modulus, 319, 218
Molality, different from molarity,
218
Molar absorptivity, 499
Molar gas constant, see gas constant
Molar heat capacity, see heat
capacity
Molar volume, see volume, molar
Mole fraction, 213, 214, 215, 221,
223, 226, 309, 310
chemical potential and, 213
Molecular area, 500, 501
Molecular orbitals, see orbital,
molecular
Molecularity, 362 ff.
Momentum, 477
Monatomic gases, 70
Monk, xix, 62
Monochromatic, 440, 442
Monolayer, 490, 499, 504
Monoprotic acids, 250
Morse curves, 448, 449, 450, 453,
456
Mossbauer spectroscopy, 447
Motion, of gas particles, 30
'Mountain sickness', 359
Mountaineer, 201
Mouth, 77, 125, 126
sensations in, 77, 125, 126
mouthwash solution, 126
Multi-layer, adsorption, 496, 497
Multi-step reactions, 350, 357
rate-determining step in, 357
Murder, 385, 393
Name of the Rose, The, xix, 536
Nappy, baby's, 263
NASA, 17
Near infrared, 463, 481
Nematic structure, of liquid crystal,
41m
Neon lamp, 69, 480, 481, 482
Nemst equation, 321 ff., 325, 330
concentration cell and, 333, 335,
338, 339
graph with, 326, 327
redox systems:
saturated calomel electrode,
331; silver ion, silver,
326; silver- silver
chloride, 327
Nernstian response, 337
Neutral, solutions, 236
neutralization, 242, 262, 265
stoichiometric ratio and, 265
Newton's laws, of motion, 32, 466
NiCad battery, 345
Nitrogen fixation, 63, 540
NMR, 447
Nobel prize, 483
Noble gases, compounds of, 74
Non-ideality, of gases, 52
Non-SI units, see SI
Normagraph, 186, 187
Normal hydrogen electrode (NHE),
see standard hydrogen
electrode
Normal, meaning at p°, 104, 178,
188, 195
Nose, 152
Nuclear, 78, 378, 379
decay, 378, 379
energy, 78
Nucleation, 42, 513, 539
water droplets, 42
Nucleophiles, 394
Occlusion, 286
Octet rule, 66, 74
Ohm's law, 281, 295
Oilrigs, 286, 287, 288
Opal, 507
Open-circuit potential, 294
Opposing reactions, kinetics, see
reversible reactions, kinetics
Optical absorbance, see absorbance
Optical density, see absorbance
INDEX
581
Optical path length, 443
Orbital, 59, 60, 75, 305, 426, 427,
448, 455 ff.
anti-bonding, 455-457
electrons in, 305
frontier, 59, 60
inner, 60
molecular, 64
HOMO, 426, 427
LUMO, 426, 427
outer shell, 59, 460
overlap of, 75
Order of reaction, 363, 356 ff.
Ordinate, 4
Organ pipes, 182
Organometallics, 225
Osmotic pressure, 212
Otzi, The Ice Man, 385, 386, 556
Outer space, 64, 199
Oven, 87, 90, 239, 429
microwave, 469 ff.
rate of warming, 90
Oxidation number, 75 ff, 460
rules for assigning, 76
Oxidation, 111, 283, 290, 292, 305,
306, 307, 312, 323, 325, 331,
420
photo-effected, 457 ff.
Pain, 79, 254, 261, 279, 281, 339,
546, 551
Paint, 2, 129, 427, 444, 462, 479,
507, 558
watercolour, 129
oil, 485
Parabola, 51
Parchment, 506
Parthians, 345
Partial differential, 92, 104
Partial molar Gibbs function, see
chemical potential
Partial pressure, 221, 223, 308, 311,
358, 359, 360
Partially soluble salts, see solubility
product
Particle accelerator, 472
Particles of gas, 30
trajectory of, 30
Particle- wave duality, 431 ff.
Partition, 209 ff, 487
constant, 205
eggshells and, 211
equilibrium constants and, 205
recrystallization and, 209
solubility product and, 209
rate of, 208
temperature effect of, 208
Partition function, 205
Pasteurization, 562
Pastry, 139
Patent, 445
Path, salted when icy, 218 ff.
Pathogens, 416
PEM, see polymer exchange
membrane
Peptization, 520, 522
Percolator, for coffee, 189
Perfume, 361, 505, 507, 508, 552
Pericyclic reactions, 447
Periodic table, 66, 474
Permitivity, 313, 314
free space, 313
relative, 313, 314
Perturbation, of electrons, 38
Petrol, 221, 224, 225, 361, 547
'green', 224, 225
Petroleum gas, 52
pH, 245 ff, 148, 251, 179,
336-338, 510,513, 514
buffers, see buffers
neutrality and, 251
pH combination electrode, see
electrode, pH
pH electrode, see electrode, pH
meter, 337, 337
units of, 248
Phagocytosis, 416
Phase, 61, 79, 178 ff, 183-185,
188, 192, 196, 197, 200, 212,
292, 293, 508
boundary, 179, 192, 196, 197,
292, 293
defined by Clapeyron
equation, 196, 197
change, 61, 79, 178(T), 178 ff,
181 ff.
direction of, 194
Gibbs function of, 181 ff.
kinetics of, 183
thermodynamics of, 188
diagram, 179, 183-185, 192, 200
diamond-graphite, 184; tin,
183; water, 185
dispersal, 504
equilibria, 178, 185 ff, 212
pressure effect of, 185 ff
purity effect of, 212
terminology of, 178
separation, 508
transition, see phase change
Philosopher, 136
Phosphors, 477, 482
phosphorescence, 479 ff, 482
Photochemistry, 426 ff, 433 ff.
first law of, 426
photochemical yield, see
quantum yield
second law of, 433
Photochromism, 557
photochromic lenses, 557
Photo-dissociation, 456, 457
Photo-electric effect, 433
Photo-excitation, see photon
excitation
Photograph, 368
print development, 368
X-ray, 473
Photo-ionization, 388, 457
Photo-isomerism, 458
Photolysis, 322
Photons, 426, 429, 439, 441, 445,
482
absorption, 403, 426, 434, 456,
457-459
emission, 472
excitation, 449, 450 ff, 460
energy, 428, 435, 438, 449, 467
flux of, 433
frequency, 473
generation of, see photon
emission
Photo-stability, 309
Photo-synthesis, 461
Physical adsorption, see
physisorption
Physical hardness, see hardness
Physical softness, see softness
Physisorption, 492 ff, 496, 501
Phytotoxicity, 354
Pi (11) product, 159
Pickle, 267
Picture element, of display, 41, 477
Piezoelectric effect, 293
Pigments, 461, 485, 558, 560
Pillows, 507
'Pinking', 225
Pixel, see picture element
pKa, 269, 271
pAT w , 249, 251
Plague, Bubonic, 309
Planck constant, 416, 435 ff.
Planck function, 169
Planck-Einstein equation, 78, 305,
306, 435, 436, 447, 464, 473
Plasma, 480, 481
Plastics, 470
Playground games, 39
Plimsoll sign, o, 108
Plug, electric, 328
pOH, 249
582
INDEX
'Poised' reactions, 170
Poison, see toxicity
Polarimetry, 370, 394, 395
Polarity, of solvents, 69
Polarizability, 47
Polarization, of light, 132, 394
polarizer, 41
Pollution, 549, 563
Polyatomic molecules, 467
Polymer exchange membrane
(PEM), 289
Polymerization, 506
Popular science, books on, 534
Position of equilibrium, see extent of
reaction
Potential,
action, 341
liquid junction E s , 337, 340 ff.,
liquid junction, magnitude of,
341, 342, 343(T)
potentiometer, 288
rest, 339, 340, 341
Potential energy, 60, 78, 91, 427
'well,' 449, 450
Potential difference, 288, and see
emf
Pottery, glaze on, 505
Power, electrical, 159
Precipitation, 513
Prehistoric people, 111, 112, 114
Pressure, 5, 23, 25, 27, 33, 34, 86,
103, 106, 148 ff., 153-155,
166, 192, 212, 358, 359,
362, 408, 455, 480, 502,
503
atmospheric, 5, 34, 103, 154
colligative, 212
reaction rate expressions, 358,
359
differences causing vacuum,
153
effect of on
temperature, 33;
effect on:
boiling temperature, 106, 408;
on Gibbs function, 148
ff.; on melting
temperature, 192; on
phase equilibria, 186 ff.;
on volume, 23, 25, 27
phenomenon of, 30
resistance to change, 166
standard, 86, 227, 311, 323
use of fugacity, 155
within tyre, 86
Pressure broadening, 470
Pressure cooker, 106, 199, 203
Pressure-volume, work, see work
Printer, 26, 563
bubble-jet, 26
Probability, of finding electrons, 64
Propagation, of polymer growth, 506
Propellant, in air freshener can, 25
Prussian blue, 462
Pseudo-order reactions, 372,
387-393
and second-order reactions,
391 ff.
calculations with, 389 ff.
notation of, 389
Psychology, 472
Pumping, 86, 132
engine, 132
car tyre, 86
Purification, recrystallization, 209,
210
Purity, 212
effect on phase equilibria, 212
mixed melting points, 212
Purple of Cassius, 65
Quantity calculus, 13, 175, 264
Quantum mechanics, 48, 454, 474,
479
quanta, 431
quantal states, 91
quantum numbers, 453, 455, 470
electronic, 453, 455
rotational, 470
quantum states, 470
quantum yield, 452
quantum yield, 452 ff.,
primary, 452 ff.
secondary, 452 ff.
Quartz
watch, see watch
quartz-halogen bulb, 474 ff., 505
Quick lime, see calcium hydroxide,
Racemization, 375
Radar, 430, 471
Radiator, 1, 7, 194, 487, 488
Radicals, 72, 117, 355, 358, 363,
388, 446
Radio, 3, 64, 425, 430
volume control, 3
Radio waves, 425, 428, 431, 469
Radioactivity, 378, see also nuclear
Radiocarbon dating, 383 ff., 555
Railway, 132
Rainwater, 235
Raman effect, 483, 484 ff., 560
spectra, 484, 485
Raoult's law, 225 ff.
Rates, 90, 134, 183, 203, 208, 217,
220, 290, 350-355, 366-374,
377, 398, 399, 403, 408 ff.
411, 416, 502
adsorption and, 502
cryoscopy, 220
definition of, 350
diffusion controlled, 415, 416
fuel-cell operation, 290
initial rates method, 366, 367
oven, warming of, 90
partitioning, 208
rate constant, 203, 350, 351(T),
352, 355, 366, 367, 369,
370
fluorescence and, 480
temperature effect of, 408 ff.
411
rate determining step, 357, 377,
398, 399
rate law, 349 ff., 354, 364, 366,
367
pseudo order, 389 ff.
consecutive reactions, 403 (T)
units of, 369, 372, 374
variables, effect of:
pressure, 358; temperature,
408 ff. 411, see also
Arrhenius, diffusion
control / and Eyring
Rayleigh scattering, 484, 354, 355
Reaction, 162, 354
affinity, 162
enthalpy, see enthalpy
free energy, 162
intermediates, 354
profiles, see concentration
profiles
stoichiometry, 354, 355
Reaction quotient Q, 159, 163
relationship equilibrium constant,
159, 162
variation during reaction, 159
Reactive intermediates, see
intermediate, reactive
Reactolite® sunglasses, 403, 405
Real gas, 52
Recessive, genetic trait, 47
Rechargeable battery, see battery
secondary
Recrystallization, 171, 209
Redox, 279, 282 ff., 291, 301-304
couple, 279, 291, 303, 304
energy of, 301
reactions, 282 ff.
Reduced mass /x, 465, 466, 469
INDEX
583
Reduction, 283, 285, 292, 305, 307,
323, 325, 331
Reference electrode, 297, 324-337,
553
primary, 328 ff.
saturated calomel electrode SCE,
297, 330, 331, 332(T), 337,
342
secondary, 330 ff.
silver-silver chloride SSCE, 326,
327, 332(T)
standard hydrogen electrode SHE,
297, 324, 328, 329(T), 330
Reflux condenser, 144
Refraction of light, 20
Relationships, 2, 4
graphical, 4
mathematical, 2
Relativity, 136
Reproducibility, 5
Repulsion, between ions, 314
Resistor, electrical, 3, 12, 281, 294,
301, 433
Resonance, 261, 470
Rest potential, see potential, rest
Revelations of Divine Love, xv
Reversibility, 89, 90, 142, 160, 167,
274
acid-base indicators and, 274
cells and 300
chemical, 167
coagulation and, 514
infinitesimal changes, 90
thermodynamics of, 89, 404
reactions, kinetics of, 403 ff.
work maximised by, 90
Revolution, French, 14
Rigid rotor, 471
Road, salted with ice, 218 ff.
Rock concert, 184
Roman numerals, 76, 460
Romans, 345
Room temperature, 34 ff.
energy of, 34, 495
Root mean square, speed of gas
particles, 31
Rotary evaporator, 188
Rotational motion, 485
Rotational spectroscopy, 447
Rule of reciprocity, 474
Rust, 333 ff.
s.t.p., 104, 108, 109, 111, 185,
212, 325, 330
Salary, 103
Salt bridge, 291, 292, 293, 342, ff.
Salted path, 218 ff.
Saponification, 240, 245
Satellites, 64
Saturated solution, 209
saturated calomel electrode, see
reference electrode,
saturated calomel
saturated vapour pressure, 180,
188, 224, 309
Scattered light, see light scattering
Schematic, cell, 291
rules for constructing, 291
Scientific notation, 19
Screening, ionic, 312 ff., 315, 316
SCUBA, 361
Sculpture, 495
Scum, on bath, 521
Sea level, 195, 360
Second law, see thermodynamics
Second-order reaction, 351, 356,
374 ff.
units of rate constant, 374
and intermediates, 389
Selection rules, 454
rotation, 471
Selectivity, of glass electrode, 338
Semiconductors, 209
Sensors, 224, 547
Separating funnel, 207, 208, 504
Separation, distance between gas
molecules, 31, 38
Shaving foam, 507
Sherbet, 234
Shirt, 487
Shock, electric, 344
Shorting, 334
SI, 14-17, 186, 247, 293, 465, 537
base units, 15
deviations from, 17(T), 186
units of:
amount of substance, 16;
concentration, 247;
current, 16; frequency,
293; length, 14; mass,
17; temperature, 16;
time, 15
Simple harmonic motion, 464
Size effects, in light dispersion, 505
Sky, colour of, 132, 563
Slippery ice, 192
Smell, 221, 225, 309, 309
Smoke, 120, 184, 466, 508, 511
smoking point, butter, 511
smoking, anti-smoking pipe, 121
SnI reactions, 395
S N 2 reactions, 394
Sneeze, 152
Snow, 33, 240
Soap operas, 245
Soapy water, 514
Sociology, 472
Sodium lamp, 481
low pressure, 482
Softness (physical), of Agl, 75
Sol, 507
Solid, 507
emulsion, 507
suspension, 507
Solidification, see fusion
Solid-state sensor, see sensor
Solubility, 210
solubility product, 209, 210, 318,
553
partition and, 209
recrystallization and, 209
Solution, of gas, 222
enthalpy of, see enthalpy of
solution
Solvated ions, see ions, solvated
Solvent, 207, 230, 242
bound and free molecules of, 137
extraction, 207, 230
polarity, 69
Soxhlet apparatus, 189, 190
'Sparging', 223
Speciation, 548
Specific heat capacity, see heat
capacity
Spectrum, 427, 436, 438, 440, 466
infrared, see 467, 469, 484
Raman, 484, 485
spectroscopy, 465, 558
UV- visible, 428 ff.,
Spectrometer, 438, 489
Speed, 30, 31, 39
gas particles, 30, 31(T)
molecules following collisions,
39
reaction, see rate
root mean square, 30
Spin, of electrons, 480
Splint, 144
Splitting nuclei, 78
Spontaneity, 130, 131, 134, 135,
144, 181
'by inspection,' 134
enthalpy and, 130
phase changes, 181
reactions, 130, 131, 134, 135,
144, 181
Spotlight, 476
Spray cans, 506
Spring, 464
584
INDEX
SSCE, see reference electrode,
silver-silver chloride
St Paul's Cathedral, London, xix
Stability
anions, 260
colloids, see colloid
in light, see photostability
Stain, 489, 492
Standard cells, 295 ff., 552,
Clark, 295; Weston, 295
Standard factors, xxviii ff., 19, 101,
102, 447
Standard hydrogen electrode (SHE),
297, 324
Standard calomel electrode
(incorrect term), see saturated
calomel electrode
Standard states, 34, 108, 248, 358
standard conditions, 34
concentration, 248; pressure,
358
standard electrode potential, see
electrode potential
standard enthalpy, see enthalpy,
standard
State functions, 83, 112, 120, 147,
421
Steam, 39, 52, 79, 98, 99, 106, 470,
487, 497
adsorbed, 487
condensation of, 39, 52
volume changes, 98, 99, 106
Steam distillation, 229, 548
Steam engine, 85
Steric obstruction, during
adsorption, 493
Stoichiometry, 105, 115, 265, 354,
355, 362, 363 ff.
reaction, of, 354, 355, 362, 363
ff.
stoichiometric ratio s, 265
stoichiometric numbers, 105, 115
Stone Age, 1 1 1
Strain, mechanical, 293
Stratosphere, 358, 554
String, 14
Strip lights, see fluorescent 'strip
lights '
Strong acids, see acids, weak and
strong
Structure elucidation, X-ray, 45
Sublimation, 37, 123, 124, 134,
178(T), 184, 202
Subscripts, typographical usage, 23
Substrate, 487 ff., 490, 493, 499
Suck, of vacuum, 151
Sun cream, 439
Sun, the, 388 ff., 400, 403, 425,
430, 434, 457, 464, 557, 559,
sunbathing, 388 ff., 400, 425, 464
sunburn, 434
sunglasses, 403, 557, 559
photochromic, see
photochromism
sunlight, 436 ff.
sun-rise, 132
sun-tan, 430, 434, 437
Supercritical fluids, 189, 190 ff.,
540, 545
and chromatography, 191
Superman, 15
Surface tension, 81, 521, 522, 523
Surfactants, 517 ff., 521, 522
types of, 523
Surgery, 222, 254
Surroundings, thermodynamic, 144
Swamping electrolyte, 342
Sweating, 81, 83
Sweets, 148, 280
Symmetric stretch vibration, 468
System,
Gibbs function and, 144
thermodynamic, 137
Systeme Internationale (SI), see SI
Table salt, 308, 314, 315, 382
Tables, guide to writing, 175
Tangents, 92
Tarnish, 282
Tart, jam, 139, 140, 141
Tautomerism, 255
Tax, on soap, 563
Teeth, 126, 280, 281, 311, 507, 508
amalgam filling of, 280, 281, 311
toothpaste, 126, 507, 508
Television, 64, 476
remote control, 430
Temperature, 7, 9, 16, 20, 25, 27,
30, 32, 33, 60, 93, 81, 87, 88,
106, 107, 140, 146, 293 ff,
296, 309, 350, 352 370, 374,
408 ff., 453, 475 509, 537
effect on:
adsorption, 491; on density,
20; on emf, 293 ff., 296;
on gas volume, 20, 25;
on kinetic energy, 30,
32; on pressure, 33, 106;
on rate constant, 408 ff.;
effect on volume of gas,
25, 27
absolute scale of, see Kelvin
boiling, varies with pressure, 106
critical, 167, 170
calculating the rise during
heating, 93
colloids and, 511
during reaction, 107
energy and, 429
following deflation of tyre, 88
gauge of energy, 60
increase in body during illness,
81
inside oven, 91
measurement of, 537
of human, body, 7
philosophical discussion about,
14
relationship with energy, 7, 60,
87, 464
resistance to change, 166
SI Unit of, 16
thermodynamic, see Kelvin
Temperature coefficient of voltage,
see temperature voltage
coefficient
Temperature coefficient, see
temperature voltage coefficient
Temperature voltage coefficient,
293, 296 ff.
Terpenes, 133
Terracottta, 345
The Iceman, see Otzi
Theatre, 476
Thermochemistry, 95
Thermodynamics, 7, 9, 60, 67, 78
ff., 85, 87-89, 130 ff., 136,
137, 151, 163, 180, 211, 295
ff., 300, 322, 334, 536
cells and, 295 ff., 322
concentration cells, 334
reversible measurement of,
300
direction of change, 130 ff.
first law of, 85 ff.
introduction, 7
minus-oneth law of, 7, 78, 87, 89
phase change, 188
reversibility, see reversibility
second law of, 131 ff., 136, 151,
419
spontaneity and, see spontaneity
system, 137
temperature, 163, 211
universe, 137
work, see work
zeroth law of, 7, 9, 60, 78, 87,
88, 89, 180, 536
Thermometer, 7 ff., 11 ff., 85, 132,
429
INDEX
585
types: gas, 12; ideal, 11 ff.;
mercury in glass, 8;
platinum resistance, 12;
water in glass, 10
Thermos flask, see Dewar flask
Thermostat, 35
Thinking, 339
Third law, see thermodynamics
Third-order reactions, 356
Throat, effect of sneeze on, 152
Thunder, 25, 63, 537
Tig, playground game of, 39
Time, SI unit of, 15
Titrations, 261 ff., 266-269, 548
endpoint and, 269
titration analysis, 261 ff.,
titration curve, 266, 267, 268
Torch, 159, 284
Torture, 281, 551
Total differential, 149, 154, 215
Toxicity and kaolin, 562
Toxicity, 64, 118, 121, 220, 225,
261, 301, 347, 378, 379, 393,
409, 501, 562
adsorption and, 501
chemical and radiochemical, 382
poison gas, 242
redox state and, 393
Traffic jam, 357
Trajectory, of gas particles, 30
Transformer, 480
Transition state complex, 410
Translational energy, 30, 39
Transmittance, 426, 439, 440, 441
Transport, of ions, 339 ff.
Triple point, 179 ff.
Triprotic acids, 250
Tubidity, 513
Turbulent flow, 153
Turin Shroud, 382, 383, 385, 386,
555
TV, see television
Tyndall effect, 505 ff., 510, 511, 563
Typography, 23, 24
differences of k and K, 23
italics, use of, 24
subscripts, use of, 23
Tyre, 86, 89, 151, 153, 198
inflation of, 86
United Nations (UN), 281
Charter of Human Rights, 281
Units,
cancelling of, 62, 264
compound, 18
effect when citing on gas
constant, 54
inter-conversion between, 374
parameters:
acceleration, 18; amount of
substance, 16;
concentration, 247;
current, 16; density, 18,
21; frequency, 293;
length, 14; mass, 17;
pressure, 18;
temperature, 16; time,
15 velocity, 18; volume,
18, 29
Universal gas constant, see gas
constant
Universe, 136-138
thermodynamic, 137
US Federal Communications
Committee, 470
UV light, 358, 388, 389, 428, 430,
431,434, 437,438, 446, 447,
448, 451, 458, 469, 482
types of, 434
Vacuum, 89, 151, 152, 154, 188,
481,491
outer space, 199
vacuum distillation, 188
vacuum flask, 89
Valency, 75, 304
valence bond theory, 66
van der Waals, 42, 54, 540
equation, 54, 540
forces, 42
van't Hoff, 162, 167, 171 ff., 174
ff., 210
isochore,174 ff., 210
isochore plot, 174, 210
'linear'/'graphical' form, 174
partition and, 210
isotherm, 162, 167, 171 ff.
Vaporization, 70
Vapour pressure, 154, 224
colligative, 212
Variables, 2 ff., 19, 536
citations, 536
controlled, 3
introduction to us of, 2
notation of, 19
observed, 3
Vatican, The, 383
Velocity, 4, 473
Vibration, 314, 449, 464 ff., 468,
470
asymmetric stretch, 468
bands, 465 ff
levels, 453, 454
scissor mode, 468
stretch modes, 468
spectrum, 485
symmetric stretch, 468
Victorian, 216, 546
Virial, 57, 58, 155, 297
coefficients, 57, 58, 155
virial equation, 57 ff.
virial series, 297
Vision, 430, 458 ff.
chemistry of, 458 ff.
Vitamins, 512
Voltage coefficient of temperature,
see temperature voltage
coefficient
Voltage coefficient, see temperature
voltage coefficient
Voltage, 433, 480
under load, 294
Voltammetry, 553
Voltmeter, 288
Volume, 20, 25, 23, 25, 27, 55, 98
ff., 154, 194
change during heating, 98 ff.
change during phase change, 194
gas, effect of temperature, 20, 25
molar, 55, 154, 194
units of, 29
variables:
pressure, 23, 25, 27;
temperature, 25, 27
Volume-pressure, work, see work
Vortex, 153
Walden inversion, 556
Walkman®, 294
Wallpaper, 393
War, 182, 548, 242, 382, 471
Wasteland, The, 544
Watch, 293, 294, 479
Water, 91, 102, 235-237, 251, 252,
514 ff., 560, 563
boiling of, 91, 102
bottled, 236
colour of, 560
dirty, 514 ff.
dissociation of, 548
expansion while freezing, 194
soapy, 514 ff.
super pure, 235, 237, 251, 252
treatment of waste, 563
Watercolour paint, 129
Waterfall, 85
Water pump, 153
Watt, unit of power, 91
Wavelength, 305, 306, 428, 431,
436, 475, 505
586
INDEX
Wavelength (continued)
wavelength maximum, 429, 460,
475
Wavenumber, 465 ff.„ 469
Weak acids, see acids, weak and
strong
Weedkiller, 354, 429
Weeping, see tears
Weight, 86, 194 ff.
equivalence with energy, 86
Welding apparatus, 23
Weston standard cell, 295, 297
'Wetting' of a plate, 522
Wien's law, 474 ff.
Windows, 18
Windscreen, car, 220
Words, study of, xix
Work function, 472
Work, 59, 86 ff., 90, 99-100, 102
ff., 150, 153, 294
gas and, 88
heating and, 99
inflation of tyre and, 101
internal energy and, 102 ff.
maximised for reversible change,
90
pressure-volume, 99 ff.
reversible measurement and,
90
sign of, 100
thermodynamics of,. 88, 89
Worked examples, books on, 533
Xi (£), extent of reaction, 156 ff.
XPS, 447
X-rays, 428, 435, 436, 446, 458,
472 ff.
diffraction with, 45, 447
Ypres, Battle of, 242, 548
Zeroth law, see thermodynamics
Zinc and castor oil cream, 263
Zone refining, 209
Zwitterions, 523
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