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Wiley
This book provides a basis for the study of matter by
describing its atomic and molecular nature and by
showing how the macroscopic behaviour of matter
can be related to its fundamental structure. These
relationships are examined without recourse to
quantum mechanics or thermodynamics, although the
results of these theories are described where neces
sary.
Many interesting problems, together with their solu
tions, are included to test a student's understanding
and extend his interest.
Contents
1 The Study of The Properties of Matter
2 Atoms, Molecules and the States of Matter
3 Interatomic Potential Energies
Energy, Temperature and The Boltzmann
Distribution
5 The Maxwell Speed Distribution and the
Equipartition of Energy
6 Transport Properties of Gases
7 Liquids and Imperfect Gases
8 Thermal Properties of Solids
9 Defects in Solids; Liquids as Disordered Solids
Solutions to Problems
Reading List
Index
Properties of Matter
The Manchester Physics Series
General Editors
F. MANDL : R. J..EJJ1IS0N : D. J. SANDIFORD
Physics Department, Faculty of Science,
University of Manchester
This series is planned to include the following volumes:
Properties of Matter: B. H. Flowers and E. Mendoza
Electromagnetism: I. S. Grant and W. R. Phillips
Atomic Physics: J. C. Willmott
Optics: F. G. Smith and. J. H. Thomson
Statistical Physics: F. Mandl
Solid State Physics: H E Hall
PROPERTIES
OF
MATTER
B. H. Flowers, F.R.S.
Langworthy Professor of Physics,
University of Manchester
E. Mendoza
Professor of Physics,
University College of North Wales, Bangor
John Wiley & Sons Ltd.
LONDON NEW YORK SYDNEY TORONTO
Copyright © 1970 John Wiley & Sons Ltd. All
Rights Reserved. No part of the publication may
be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, elec
tronic, mechanical, photocopying, recording or
otherwise, without the prior written permission
of the Copyright owner.
Library of Congress Catalog Card No.70118151
ISBN 471 26497 Cloth bound
ISBN 471 26498 9 Paper bound
Set on Monophoto Filmsetter and printed by
J. W. Arrowsmith Ltd., Bristol, England
Editors' Preface to the
Manchester Physics Series
In devising physics syllabuses for undergraduate courses, the staff of
Manchester University Physics Department have experienced great diffi
culty in finding suitable textbooks to recommend to students; many
teachers at other universities apparently share this experience. Most books
contain much more material than a student has time to assimilate and are
so arranged that it is only rarely possible to select sections or chapters to
define a selfcontained, balanced syllabus. From this situation grew the
idea of the Manchester Physics Series.
The books of the Manchester Physics Series correspond to our lecture
courses with about fifty per cent additional material. To achieve this we
have been very selective in the choice of topics to be included. The emphasis
is on the basic physics together with some instructive, stimulating and
useful applications. Since the treatment of particular topics varies greatly
between different universities, we have tried to organize the material so
that it is possible to select courses of different length and difficulty and to
emphasize different applications. For this purpose we have encouraged
authors to use flow diagrams showing the logical connection of different
chapters and to put some topics into starred sections or subsections.
These cover more advanced and alternative material, and are not required
for the understanding of later parts of each volume.
vi Editors' preface to the Manchester Physics Series
Since the books of the Manchester Physics Series were planned as an
integrated course, the series gives a balanced account of those parts of
physics which it treats. The level of sophistication varies : 'Properties of
Matter' is for the first year, 'Solid State Physics' for the third. The other
volumes are intermediate, allowing considerable flexibility in use. 'Elec
tricity and Magnetism' , 'Optics' and 'Atomic Physics' start from first year
level and progress to material suitable for second or even third year
courses. 'Statistical Physics' is suitable for second or third year. The books
have been written in such a way that each volume is selfcontained and
can be used independently of the others.
Although the series has been written for undergraduates at an English
university, it is equally suitable for American university courses beyond
the Freshman year. Each author's preface gives detailed information
about the prerequisite material for his volume.
In producing a series such as this, a policy decision must be made about
units. After the widest possible consultations we decided, jointly with the
authors and the publishers, to adopt SI units interpreted liberally, largely
following the recommendation of the International Union of Pure and
Applied Physics. Electric and magnetic qualities are expressed in SI units.
(Other systems are explained in the volume on electricity and magnetism.)
We did not outlaw physical units such as the electronvolt. Nor were we
pedantic about factors of 10 (is 0.012 kg preferable to 12 g?), about
abbreviations (while s or sec may not be equally acceptable to a computer,
they should be to a scientist), and about similarly trivial matters.
Preliminary editions of these books have been tried out at Manchester
University (and in the case of 'Properties of Matter' also at Bangor
University) and circulated widely to teachers at other universities, so that
much feedback has been provided. We are extremely grateful to the many
students and colleagues, at Manchester and elsewhere, who through
criticisms, suggestions and stimulating discussions helped to improve the
presentation and approach of the final version of these books. Our partic
ular thanks go to the authors, for all the work they have done, for the
many new ideas they have contributed, and for discussing patiently, and
frequently accepting, our many suggestions and requests. We would also
like to thank the publishers, John Wiley and Sons, who have been most
helpful in every way, including the financing of the preliminary editions.
Physics Department F. Mandl
Faculty of Science R. J. Ellison
Manchester University D. J. Sandiford
Preface
A radical revision of the undergraduate physics syllabus of the Uni
versity of Manchester was undertaken in the year 1959. This exercise
involved the participation of many members of the academic staff. It was
eventually decided to base the whole of the syllabus upon two introductory
firstyear courses, one concentrating on the general properties of wave
motions, the other based upon the statistical properties of matter con
sidered as a collection of interacting atoms and molecules. The latter
course, consisting of about 34 50minute lectures, was first given in the
1959/60 academic session under the title 'Properties of Matter'; over the
years it has developed into the present book.
Our aim has been to show how the macroscopic quantities describing
matter in bulk can be related to each other in terms of the microscopic
properties of molecules and their interactions. This of course is the subject
matter of statistical thermodynamics. To the purist this subject can only
be tackled after a thorough grounding in advanced mechanics, thermo
dynamics and the quantum theory. But the spirit of inquiry amongst
undergraduates, and the incentive to devote their time and energies to
these rigorous pursuits can more readily be generated, it seemed to us, if
they can first be made aware of what much of physics is about in a more
rough and ready fashion. It is perhaps contrary to the present fashion,
but we have omitted all quantum considerations from the foundations of
this work, confining ourselves to a few passages here and there which are
in the nature of 'see the next exciting instalment' when, indeed, quantum
viii Preface
theory is necessary rather than merely desirable in order to understand
some macroscopic phenomenon. Similarly, we have excluded any discus
sion of the second law of thermodynamics and its consequences — at the
risk, here and there, of doing violence to the distinction between internal
energy (which we calculate) and free energy (which we do not). We hope
that we have at least identified the points at which the distinction matters.
However, we consequently have not always been able to avoid the phrase
'It can be shown that . . .', although we have tried to avoid any implication
of it except in peripheral matters. We comfort ourselves by suggesting that
physics would be very dull unless there were always some things left
outstanding in this way.
More importantly, however, we have been forced to restrict severely
the number of kinds of matter we were prepared to discuss. We have
excluded all discussion of ionized plasmas, of polymers and of biological
materials. Each of these, it seemed to us, requires a book to itself. We
have touched, although briefly, on the engineering properties of materials
limiting ourselves to a discussion of the strength of real solids — for our
concern has been rather to show that these properties can in principle be
related to the microscopic properties. Argon, gaseous, liquid and solid,
figures ubiquitously. It is perhaps the simplest element from our point of
view, the ideal element, about which much experimental information is
available to us for our simpleminded analysis. Apart from that, we have
mostly confined ourselves to gases, liquids and solids consisting of small
molecules, and to simple ionic substances and metals.
This is the way in which much of the study of the properties of matter
developed historically. We hope that we have succeeded in bringing back
some of the excitement of the original discoveries ; certainly we found it
exciting to rediscover some of these ourselves.
The course has been given in modified form in Manchester since 1959
and in Bangor since 1965, by others as well as by ourselves. We are
indebted to several of our colleagues who, as lecturers or tutors, have
contributed much to its gradual development. We are particularly in
debted to Dr David Caroline, as well as to the editors of the Manchester
Physics Series for their friendly but penetrating criticisms and suggestions.
Most of all we are indebted to more than a thousand of our students whose
own efforts to understand what we were trying to do has been our main
encouragement and incentive. They are not, of course, responsible for the
remaining imperfections in our book.
B. H. Flowers
Eric Mendoza
List of Symbols
A,B
constants
A
area
A
activation energy
CI) CIq
atomic or molecular diameter
a,b
constants in van der Waals' equation
a
linear expansion coefficient ; Madelung constant
P
volume expansion coefficient
y
ratio of specific heats ; surface tension
y G
Griineisen constant
c
specific heat, usually with suffix: C p , C v
c
speed of molecule ; speed of light
D
diffusion coefficient
d
distance
E
energy
e
charge on electron
s
depth of interatomic potential well
F
force
/()
function
&
faraday
r\
viscosity coefficient
J
flux of particles
K
bulk modulus
°K
degrees absolute
k
Boltzmann's constant
K
velocity coefficient of chemical reaction
K
thermal conductivity
JT
kinetic energy
u
latent heat at low temperatures
$£
Lorentz number
£
length
A
wavelength
X
mean free path
M
molecular weight
m
mass of atom
x List of Symbols
N Avogadro's number
n number, number density
rt coordination number
Jf number per unit area
v frequency
v E Einstein frequency
P pressure
P[ ] probability function
Px'PyPz momentum components
p, q indices of interatomic potential energy
r radial distance
p density
s strain
a collision crosssection ; conductivity
T temperature
t time
t characteristic time
U x drift velocity
V large volume
V molar volume
v small volume
v x ,v y , v z velocity components
V potential energy
(D angular velocity
tAt sections or subsections marked with a star may be omitted,
if the reader so wishes, as they are not required later in the book
Contents
1 THE STUDY OF THE PROPERTIES OF MATTER
1 . 1 The Study of the Properties of Matter .... 1
1.2 Orders of Magnitude ....... 2
1.3 Units and Systems of Units ...... 5
2 ATOMS, MOLECULES AND THE STATES OF MATTER
2.1 Atoms, Ions and Molecules ...... 8
2.2 Gases, Liquids and Solids . . . .15
3 INTERATOMIC POTENTIAL ENERGIES
3.1 Molecular Dimensions . ...... 22
3.2 Interactions between Electrically Neutral Atoms and
Molecules ........ 23
3.3 Binding Energy and Latent Heat . . . .31
3.4 Surface Energy ........ 33
3.5 Elastic Moduli 40
3.6 Vibrations in Crystals : Simple Harmonic Motion . 44
3.7 Metals 47
3.8 Ionic Crystals ........ 53
Problems ......... 57
Contents
ENERGY, TEMPERATURE AND THE BOLTZMANN
DISTRIBUTION
I. Probability Functions
4. 1 Heat and Energy
4.2 Concepts of Probability Theory
4.3 Thermal Equilibrium ......
4.4 Boltzmann Distributions — I. A Gas of Independent
Particles under Gravity .....
Appendix A
A. 1 Dependence of the Probability Function on Energy and
Temperature
A. 2 Form of Probability Function ....
A. 3 Extension to Macroscopic Systems
Problems ........
62
67
74
82
90
92
94
96
THE MAXWELL SPEED DISTRIBUTION AND THE
EQUIPARTITION OF ENERGY
5 . 1 VelocityComponent Distribution P[u J
5.2 Speed Distribution P[c]
5.3 The Equipartition of Energy
5.4 Specific Heats C p and C v
5.5 Activation Energies
Problems .
97
106
114
117
127
131
6 TRANSPORT PROPERTIES OF GASES
6.1 Transport Processes .....
6.2 Solutions of the Diffusion Equation : the yjt Law
6.3 Diffusion and the Random Walk Problem
6.4 Distribution of Free Paths ....
6.5 Calculation of Transport Coefficients
6.6 Knudsen Gases ......
Appendix B
B.l Diffusion Coefficient in Gases
Problems .......
136
145
150
153
159
169
171
174
7 LIQUIDS AND IMPERFECT GASES
7.1 Relations between Solid, Liquid and Gas
7.2 The Approach to the Liquid State .
7.3 Van der Waals' Equation
7.4 Application to Gases ....
"Ar7.5 Refinements to Van der Waals' Equation
"Ar Starred sections or subsections may be omitted, if the reader so wishes, as they are not
required later in the book.
176
185
188
193
203
Contents
xin
7.6 Critical Constants 208
7.7 Fluctuation Phenomena . • .211
7.8 Properties of Liquids on Estimated Van der Waals' Equa
tion 217
Problems 222
8 THERMAL PROPERTIES OF SOLIDS
8.1 The External Forms of Crystals
8.2 XRay Structure Analysis
8.3 Amplitude of Atomic Vibrations in Solids
8.4 Thermal Expansion and Anharmonicity .
8.5 Thermal Conduction in Solids
8.6 Electrons in Metals .
Problems ..... •
225
233
236
238
244
254
261
9 DEFECTS IN SOLIDS: LIQUIDS AS DISORDERED SOLIDS
9.1 Deformation of Solids .
9.2 Brittle Materials .
9.3 Deformation of Ductile Metals
*9.4 Growth of Crystals
*9.5 Point Defects
^r 9.6 Diffusion in Solids
*9.7 Diffusion in Liquids
Problems ....
Solutions to Problems
Reading List
Index ....
Physical Constants and Conversion Factors
263
269
274
281
286
290
297
304
306
311
313
inside back cover
CHAPTER
The study of the properties of
matter
1.1 THE STUDY OF THE PROPERTIES OF MATTER
Throughout the whole of the nineteenth century, one of the open ques
tions of science was whether matter was composed of atoms or not.
Nobody had yet been able to perform experiments with single atoms,
certainly noOne had ever seen one, and for a long time noone knew even
the order of magnitude of the sizes of atoms — whether their diameters
were typically of order 10" 5 cm or 10" 50 cm. One method of attack was to
try and correlate as many different properties of solids, liquids and gases
as possible on the basis of simple postulates about the forces which atoms
exerted on one another. The earliest attempt at describing these forces was
made by Boscovitch in 1745. Sixty years later, a triumph was scored when
Laplace, arguing from the fact that the rise of a liquid in a capillary tube
was observed to be independent of the thickness of the wall of the tube,
deduced that atomic forces must act only over short distances. He was
able to deduce theoretically the form of the surfacetension law for
liquids, that the force exerted by surface tension should be proportional
to the length of a cut in the surface — and this was verified experimentally.
Much later, in the 1860's and 70's, the transformation of gas into liquid
was demonstrated for many substances when it became technically
possible to produce high pressures and low temperatures. The similarities
2 The study of the properties of matter Chap. 1
and regularities in behaviour of several substances, predicted on the basis
of crude atomic models, added plausibility to those models. Above all,
the rough agreement between estimates of the sizes of atoms based on
widely differing kinds of experiments (about eight completely different
methods all gave atomic diameters of the order of 10~ 7 10~ 8 cm) made
the atomic hypothesis fairly secure by 1900.
Thus the subject called Properties of Matter' or 'Heat' was at one time
an exciting one. Physicists measured surface tensions and latent heats and
elasticities and tried to correlate them under an allembracing atomic
theory. But with the discovery of subatomic particles and the invention of
counting devices which could detect single atoms or ions, the subject lost
its urgency. By the early years of the twentieth century, noone anywhere
doubted that matter was atomic in structure and that atoms were of the
order of 10 8 cm in diameter. Experimenters still measured surface ten
sions, latent heats and elasticities, but these had now become respectable, if
routine, activities in their own right. Books came to be written entitled
'Properties of Matter' which described highly sophisticated apparatus for
measuring quantities of this kind, and gave elaborate calculations on the
twisting of laminas and the bending of beams, but never mentioned the
word 'atom'.
It is the purpose of this book to try and recapture some of the spirit of
the old approach. We will in fact start from statements about the shapes
and sizes of atoms and the forces holding them together, and then show
how the properties of solids, liquids and gases can be deduced. It is our
purpose to show that, given the potential energy between two atoms of
known atomic weight, it is possible to estimate the density of the solid
and its specific heat, its thermal expansion and elasticity, the surface
tension and latent heat and viscosity of the liquid, the diffusion constant
and thermal conductivity and specific heat of the gas and the velocity of
sound through it : they are all related properties of matter.
1.2 ORDERS OF MAGNITUDE
The estimates we shall make will rarely be exact ones. Since the object
of our discussions will be mainly to show that we can identify the forces
or mechanisms underlying certain phenomena, it will serve our purpose
if we can show that using approximate methods we can get roughly the
right answer. To improve on rough estimates usually demands a great
increase in mathematical complexity, and it would achieve little if we
risked obscuring the line of the argument by getting involved in com
plicated manipulations merely to add a few percent to the accuracy of the
result. In many operations, it is by contrast extremely important to know
1 .2 Orders of magnitude
Hydrogen gas
Fig. 1.1. A diagram drawn by John
Dalton, in his epochmaking book A
New System of Chemical Philosophy
published in 1810. The gas is shown as
a regular arrangement of atoms, which
is quite wrong. Not till this idea was
supplanted could any real progress be
made.
Fig. 1.2. An early idea of the effect of one atom on
another — from a book published by Boscovitch in 1 745.
Compare this diagram with Fig. 3.4. The oscillations
in this graph were postulated to account for the
structure of a gas as pictured above, but this remains
an astonishingly penetrating attempt to explain the
properties of matter in fundamental terms.
4 The study of the properties of matter Chap. 1
the exact values of certain parameters. Chemical engineers, for example,
need to know thermal constants to fivefigure accuracy in order to predict
whether they can manufacture a certain product economically or not.
Similarly, a bridge might collapse if a designer made a mistake by a factor
of two in some of his data. But for our present purposes, we will be content
if we can estimate that a thermal or chemical change may take place at
some temperature of the order of a. few hundred degrees absolute; and we
will regard it as satisfactory if, given the atomic constitution of both, we
can predict from first principles that steel is a good deal stronger than
butter.
oo°
<oooo
Qto OO
oo
Fig. 1.3. Possibly the very first 'modern' drawing of the molecules in the
solid, liquid and gas phases. A 'doodle' from one of Joule's notebooks, done
while he was working out the implications of the conservation of energy and
realizing that the old static picture of gases was wrong (1847).
This emphasis on the importance of orders of magnitude must not be
taken to disparage the crucial role of accuracy in experimental measure
ments nor to suggest that exact theories need not be pursued. Indeed the
existence of new and unexpected phenomena is sometimes shown up
when the discrepancy between observation and theory is quite small.
Measurements of the specific heats of gases, for example, give results
which can be predicted in order of magnitude by simple theories based on
the laws of classical physics, but the persistent disagreement between
precise measurements and exact classical theories was the first evidence
that the laws of classical mechanics themselves were not applicable in all
circumstances. Again, unexpected discrepancies between existing theories
and measurements of the specific heats of solids at lower temperatures
could similarly only be explained by using a quantum approach ; measure
ments on metals showed that electrons deviated sharply from the classical
behaviour that had been expected. These important phenomena (which
will all be discussed later in this book) would not have been discovered had
physicists been content merely to make rough estimates. Nevertheless, if
a rough estimate does give a result which agrees in order of magnitude
1 .3 Units and systems of units 5
with observation, this can usually be taken to mean that the correct
mechanisms have at least been identified. This will be the main theme of
this book.
Fig. 1.4. A very early picture, one of the first to be drawn to
scale, of a gas — the molecules of air in a volume 10~ 4 cm
square by 10~ 8 cm thick— not long after Avogadro's
number had been reliably estimated. Kelvin 1883.
1.3 UNITS AND SYSTEMS OF UNITS
When we make a statement about a physical quantity, like 'the mass of a
proton is 1.66 x 10 ~ 24 g', the datum consists of three parts : the number of
order unity, the power of 10 and the unit (1.66, 10~ 24 and the gram,
respectively). Many students have a habit of remembering the first figures
to high accuracy, but of forgetting the power of 10 or ignoring the units.
In our collective memory we can recall students who have insisted that the
sun is 3,000 miles from the Earth, that the diameter of an atom is
109,737 cm, that Planck's constant is 6.6 x 10 23 unknown units, that
gravitation is the cohesive force that holds solids together. Wildly wrong
statements like these are sillier than saying that butter is a good material
for building bridges. In quoting physical quantities, it is usually more
6 The study of the properties of matter Chap. 1
important to get the power of 10 and the units correct — to be sure of the
order of magnitude — than it is to quote the digits at the beginning. But
having got the figures correct, the units must be quoted; data without
units are devoid of meaning.
To be of any use, a system of units should be selfconsistent and prefer
ably it should deal with numbers which are comprehensible to the human
imagination. All wellknown systems satisfy the first condition (S.I., c.g.s.,
even British units) but it is the second that it is difficult to fulfil. In almost
any physical problem, one encounters numbers which are extraordinarily
large or small by everyday standards; numbers like 10 23 or 10" 16 which
occur often in physics, cannot easily be visualized. Further, any unit
quantity which is suitable for one problem is often quite unsuited for
another. A coulomb (C), for example, is a tiny thing compared with the
charge on a gram ion (10 5 C) but enormous compared with the charge
on a single ion or electron (10" 19 C). To avoid very large or small factors,
it is natural to choose units which are large when we measure large things
and small when we measure small ones. It is natural to measure atomic
weights in grams, and the mass of single atoms in atomic mass units ;
to insist on measuring all masses in kilograms, say, is merely perverse.
Similarly the centimetre or the metre are suitable units of length for many
common objects but single atoms are best measured in Angstrom units :
lA = 10" 8 cm. The reader must, therefore, be prepared to change his
units with the problem.
It is a sad fact that different authors of research articles use different
systems of units, where not only do the symbols stand for different mag
nitudes and dimensions, but different numerical constants also appear.
In electromagnetic equations, statements of the same equation may or
may not contain factors like c, 4n, e /i which arise from the units, and make
them difficult to understand. In fact, few difficulties of this kind appear in
this book : but the competent physicist must be prepared to be able to
read papers written by authors who may be working in any system, and
the student must be facile in all of them.
1.3.1 Energy units
The joule (J) is the common unit of energy. For some problems it is a
suitable unit since the gas constant R (defined in section 4.4.2) is 8.31 J
per degree and this is not a large number. For measuring molar binding
energies or latent heats or heats of reaction, the kilojoule (10 3 J) is more
appropriate. For measuring the corresponding energies of a single atom
or molecule, however, which are roughly 10 24 times smaller, the joule is
not appropriate. Nor is the common 'small' unit of energy, the erg,
because it is only a factor of 10 7 times smaller than the joule and one
1.3 Units and systems of units 7
finds oneself dealing with awkwardsounding amounts of energy such
as 10" n or 10" 14 erg. There does in fact exist an energy unit which is
suitable for these purposes. It is the electron volt. This is the amount of
energy acquired by an electronic charge when it falls through a potential
difference of 1 volt: 1 electron volt (1 eV) = 1.60 x 10~ 12 erg = 1.60 x
10" 19 J. (This relation can be calculated from the knowledge (a) that
when 1 coulomb falls through 1 volt, 1 joule of energy is released, and (b)
that the charge on an electron (section 2.1.2.) is 1.60 x 10~ 19 C). In these
units molecular binding energies lie between 10 and 0.01 eV, and these
numbers are easy to visualize and handle.
CHAPTER
Atoms, molecules and the
states of matter
2.1 ATOMS, IONS AND MOLECULES
It is convenient for us to start with the statement that matter is com
posed of atoms. Atoms are not the fundamental units of nature because
they themselves can be broken up into a few smaller, and in a sense more
fundamental, units or elementary particles (electrons, neutrons and
protons). But the conditions needed to make atoms disintegrate are
rather extreme and are not normally met with (at any rate if we except
radioactive substances whose nuclei disintegrate but which do not concern
us particularly), so that from the present point of view we need go no further
than to say that matter is built up of atoms.
Under ordinary conditions, matter seems to be continuous. Given a
small piece of any solid, for example, it is possible to cut it up into smaller
fragments and to go on repeating this process ; there seems to be no limit
to the fineness of subdivision, other than that set by the instruments
available. But in fact (if we carry out the process by any method, under
conditions of temperature and pressure which are not too extreme) there
is a limit when we reach atomic dimensions. The illusion that matter is
continuous is due to the extreme smallness of even the largest atoms, and
to the very large numbers of them which are present even in a microscopic
speck.
2.1 Atoms, ions and molecules 9
We shall be concerned mostly with the forces between atoms, and it is
therefore necessary to describe their structure and the formation of ions
and molecules, so that the origin of the forces which they exert on one
another can be understood.
2.1.1 Atomic number
If a sample of a substance can be shown to consist of atoms all of one
kind, that substance is said to be a chemical element. Three examples of
elements are hydrogen (H), under normal conditions a gas which easily
takes part in a number of chemical reactions with other elements ; helium
(He), a gas which is chemically inert and hardly reacts at all with other
elements ; and lithium (Li), a highly reactive metal.
It is possible to find similarities and regularities among the physical
and chemical properties of elements. Of the many characteristics which
it is possible to select and use to arrange the elements in some sort of
order, one has been found to have a special significance. When samples
of the elements are bombarded with energetic electrons, Xrays are emitted,
any element giving a spectrum containing many characteristic wave
lengths. Each of them, however, includes a recognizable group of four
lines, called the /Clines, whose wavelengths vary from element to element.
The lines from hydrogen have the longest wavelength, those from helium
the next longest, then lithium, and in this way it is possible to arrange the
elements in order. On this basis, hydrogen is said to have atomic number
1, helium 2, and so on.
2.1.2 Structure of atoms
An atom consists of a nucleus, consisting of neutrons and protons, which
is extremely small (about 10 5 times smaller in diameter than the atom
as a whole) but which contains almost all the mass. Around this nucleus
is a cloud of electrons. This cloud is easy to visualize when there are many
electrons ; but it is a fact, made comprehensible by quantum mechanics,
that a single electron in an atom also behaves somewhat as if it were
spread tenuously throughout a certain volume. Even though the electron
is a point charge, it appears to an outside observer as if it were continuously
spread out. The term 'electron cloud' is therefore appropriate even to an
atom containing only a single electron. Each electron carries a negative
electric charge, whose magnitude is a fundamental constant :
e= 1.602 x 10" 19 C.
It is found that the number of electrons inside any atom of an element is
equal to the atomic number of that element. To maintain the electrical
10 Atoms, molecules and the states of matter Chap. 2
neutrality of each atom, the nucleus is positively charged, with a mag
nitude equal to that of all the electrons in the cloud outside it. Thus a
hydrogen atom has one negatively charged electron surrounding a
nucleus which has one unit of positive charge. Each helium atom has two
electrons outside a nucleus carrying two units of charge, and so on.
Between each electron and the nucleus there exist forces of attraction
(Coulomb forces) which bind electrons to the nucleus — that is, they make
it difficult for the electrons to escape. However, some electrons may be
more tightly bound than others — they require greater energy to separate
them from the nucleus — and at the same time the clouds are usually
smaller in size. In helium and the other 'inert' or 'rare' gases such as
neon and argon which resemble it, all the electrons are tightly bound
and no further electrons can be added to the system if it is to remain
stable. In lithium and the other alkali metals like sodium and potassium
and the 'noble' metals copper, silver and gold, most of the electrons are
tightly bound but there is one which is rather loosely bound, so that it
does not take much energy to detach it from its atom. In hydrogen also,
the single electron is only loosely bound. These elements are said to be
monovalent and to contain one valence electron. Other elements have
more than one loosely bound electron in each atom ; the alkalineearth
metals such as beryllium and magnesium each have two. In another group
of elements, the halogens, which include the gases fluorine and chlorine,
all the electrons are tightly bound but it is possible for another single
electron to enter the existing cloud and become tightly bound too. Other
atoms can accept more than one extra electron in this way ; oxygen for
example can accept two, nitrogen three. Only the loosely bound electrons
can enter into combinations in this way. The tightly bound ones remain
undisturbed by ordinary chemical changes.
2.1.3 Ions
These are atoms which have lost or gained one or more electrons (while
keeping their nuclei unchanged) so that they are no longer electrically
neutral. An atom of hydrogen can lose its electron (which is of course
negatively charged) so that it has an excess of one unit of positive charge.
Its mass is very little different from that of a hydrogen atom since most of
the mass resides in the nucleus which is unchanged. A chlorine ion is
formed by adding one electron to the atom, oxygen can form two kinds
of ions according as one or two electrons are added to the atom. The
properties of ions are quite different from those of atoms not only because
ions are charged but also because in ions all the electrons are tightly
bound. The relative sizes of atoms of mercury, of a free ion and of the ions
in metallic mercury are shown in Fig. 2.1.
2.1 Atoms, ions and molecules 1 1
[a) {b)
Fig. 2. 1 . Scale drawings of (a) a mercury atom in the vapour at room temperature
(4.4 A in diameter), and (b) a mercury ion (Hg ++ ) formed from an atom which has
lost two loosely bound electrons (2.24 A in diameter), (c) The ions in the solid or
liquid metal are a little bigger than this.
Free ions can be formed in two ways. They can be produced by chemical
action, when the initially neutral atom is very close to other atoms which
contribute or take away electrons. They can also be produced when neutral
atoms are bombarded by beams of particles such as energetic electrons,
when the electrons inside the atoms are knocked out by what can be
regarded as direct collisions.
Any mass of matter in equilibrium must be electrically neutral, or very
nearly so ; if any ions are present it is exceedingly probable that oppositely
charged ions are to be found not far away.
2.1.4 Molecules
It is possible for the loosely bound electrons to be shared in various
ways between different atoms, so as to bind those atoms together to
form molecules. Some substances do so in the gaseous state but the
molecules do not preserve their identity in the solid state; other sub
stances form molecules which exist in all three states, solid, liquid and gas.
Hydrogen atoms find it favourable at ordinary temperatures to share their
electrons so as to form molecules each containing two atoms. They can
be pictured as two nuclei embedded in a cloud of electrons enveloping
them both. The molecule forms a separate, stable entity, and at ordinary
temperatures hydrogen gas consists almost entirely of such molecules.
When cooled to very low temperatures so that the gas liquefies or solidifies,
the molecules become quite tightly packed together but they still more or
less retain their identity, though a given hydrogen atom may occasionally
wander from molecule to molecule. Halogen gases also form diatomic
molecules. Helium and the other rare gases are composed of molecules
each containing a single atom (so that it is immaterial whether one calls
these atoms or molecules).
12 Atoms, molecules and the states of matter Chap. 2
Metallic elements such as lithium exist as molecules only in the gas
phase produced by heating the metal, though many of these molecules
are ionized particularly if the temperature is high. Solid lithium metal
however consists entirely of ions, each of which has lost one electron ;
these valence electrons may be regarded as a gas which fills the space
between the ions. Other metals have similar structures. The electron gas is
mobile inside metals and this confers the electrical conductivity which is
their characteristic.
Similar processes of electron sharing can occur between atoms of
different elements, to form molecules of compounds. For example, the
water molecule is formed from one oxygen atom and two hydrogen
atoms. It can be pictured as a rather large electron cloud (formed from
those from the hydrogen atoms and the least tightly bound electrons
from the oxygen atom) which envelops the oxygen nucleus and its most
tightly bound electrons, somewhere deep inside near its centre, with the
two hydrogen nuclei a little distance away. Steam at high temperatures
consists predominantly of such molecules. In ice, the same molecules can
also be distinguished, though the hydrogen atoms spend a proportion of
their time wandering from molecule to molecule. Water also contains a
proportion of molecules which have split up into ions, consisting of
positively charged hydrogen ions and negatively charged hydroxyl ions
(oxygen and hydrogen atoms with one extra electron).
CI
I Li
#
(a) ^^ ^^pWm?
id)
Fig. 2.2. (a) A molecule of lithium chloride which can exist in the gas. (b) In the solid
each Li ion is surrounded by six CI ions — there are no distinguishable molecules.
The electron clouds have been drawn with well defined surfaces.
Another simple molecule is that of lithium chloride consisting of a
lithium ion bonded to a chlorine ion. These molecules exist only in the
gas phase, produced when the substance is vaporized, though even then
a number of molecules break up into the constituent atoms or ions. In the
liquid and solid, and when dissolved in water, any one ion cannot be
2.1 Atoms, ions and molecules 13
said to be definitely associated with any one other ion, so that no molecules
exist any more. In the solid, any one lithium ion is surrounded by six
chlorine ions, all symmetrically disposed around it, and each chlorine ion
is similarly surrounded by six lithium ions. The electrons cannot be
accelerated so the substance is ah insulator. In dilute solutions, every ion
is separately covered by a layer of water molecules. In neither case can it
be said that the compound exists in the form of molecules.
Carbon can enter into the composition of a vast number of molecules of
many varied shapes, sizes and properties and these constitute organic
compounds. They include biological substances whose molecules are
often of an astonishing complexity. Such molecules remain almost un
changed in solid, liquid and gas. The benzene molecule, for example, is a
fiat ring of six carbon atoms each with a hydrogen atom close to it with
electron clouds between, and more tenuous clouds above and below the
ring. These molecules are present in benzene whether solid, liquid or
vapour.
2.1.5 Nuclear and chemical reactions
Atoms and molecules are not indestructible, since the forces holding
them together are not infinitely strong. Atoms can be converted into other
atoms by nuclear reactions which alter the nuclei, molecules can be
changed into other molecules by chemical reactions which affect the
looselybound electrons and hence alter the groupings of the atoms.
However, the forces (or amounts of energy) needed to make nuclei break
up or combine are much greater than those required to promote chemical
changes. This is made manifest by the temperature which must be attained
before reactions of different kinds can be initiated — we shall see later that
these temperatures on the absolute scale are a rough measure of the
energies involved. Nuclear reactions, such as the combination of hydrogen
nuclei to form helium, can only be made to take place at very high tem
peratures of the order of those found in the interior of stars, but many
chemical reactions will proceed at ordinary temperatures, 10 3 or 10 4 times
lower on the absolute scale. This topic is discussed again in section 4.4.3.
2.1.6 Atomic masses and Avogadro's number
Atomic and molecular masses can be determined to good accuracy by
studying the masses of substances taking part in chemical reactions, and
making suitable assumptions as to the nature of the reactions. Nowadays,
atomic masses are measured to a high degree of precision by mass spectro
meters. The vapour is ionized by bombardment with a beam of electrons
and the charged ions are deflected by electric and magnetic fields in such
a way that the ratio of charge to mass can be determined. Since the charge
14 Atoms, molecules and the states of matter Chap. 2
must be equal to that of an electron or a multiple of it, the mass can be
found.
Observations with mass spectrometers showed that any element, as
found naturally, contains atoms of different masses but having the same
chemical properties. These are called isotopes of the element. It is the
nuclei which differ from one another in mass although they all carry the
same charge and the electron clouds round them are identical.
To construct a scale of atomic masses, we take the mass of a given
isotope as a standard and compare others with it. The mass of the most
abundant isotope of carbon is conventionally taken as 12 atomic mass
units, written 12a.m.u. Natural carbon is a mixture of isotopes with
masses close to 12 and 13 a.m.u. and the average is 12.011 a.m.u. The mass
of the average hydrogen molecule is 2.016 a.m.u ; it is a mixture of atoms
of masses near 1 and 2 a.m.u. Oxygen has isotopes of masses close to
16, 17 and 18 a.m.u. Thus the water molecule can have masses anywhere
between 18 and 24 a.m.u. the value 18 being overwhelmingly the common
est. For many purposes, however, it is sufficient to quote average masses,
rounded off to the nearest whole number. If the mass of a single molecule
is M a.m.u. we define one gram molecule, often called one mole, of that
substance to be M gm. The molecular weight of the substance however is
usually denoted by M, without any units.
It follows from this definition that the number of molecules contained
in one mole of any substance is independent of its chemical composition
or physical form. This number is called Avogadro's number and is given by
N = (6.02257 + 0.00009) x 10 23
which for most practical purposes can be taken as
N = 6xl0 23 .
This is the number of hydrogen molecules in 2.016 g of hydrogen, of
carbon atoms in 12.011 g of carbon, of water molecules in 18.015 g of
water or ice or steam, and so on. The value of the atomic mass unit in
grams (the mass of an imaginary molecule of molecular weight equal to 1)
is the reciprocal of N :
1 a.m.u. = (1/N)g = 1.66xl0" 24 g.
It is the enormous magnitude of Avogadro's number, which implies
that the individual atoms are so extremely small, that gives the illusion
that matter is continuous on the ordinary scale.
The gram ionic weight can be defined in a similar way to the gram
molecule. The total charge on a gram ion of any substance in which each
ion carries a single electronic charge e is Ne units of charge. It is called
2.2 Gases, liquids and solids 15
the faraday and is equal to
1#" = 1.60 x 10" 19 x 6.023 x 10 23 = 0.965 x 10 5 C.
For most practical purposes 1 faraday can be taken as 10 5 C.
So far, we have made several dogmatic statements about the atomic
nature of matter, the masses of single atoms and the value of Avogadro's
number. Of course, these facts are not self evident and historically they
took a long time to prove and were bitterly disputed till surprisingly
recent times. In the nineteenth century the proof that the atomic hypothesis
was correct hinged on the observation that Avogadro's number could be
measured in a large number of completely independent ways and in spite
of the crudeness of the measurements, the result was always roughly the
same. The earliest method involved a study of the rate of diffusion of gases
combined with crude estimates of the volumes that the same gases would
have occupied if they could have been liquefied or solidified. Other early
methods depended on studies of the heat of formation of brass ; of the
relation between the surface tension and latent heat of liquids ; of the colour
of the sky and the absorption of starlight in the atmosphere ; of the charge
on the electron and its relation to electrochemical changes ; other methods
were also used. All gave values for N which at any rate had the same
number of zeros after the first figure and in view of the extraordinary
range of phenomena covered, this constituted convincing proof of the
atomic hypothesis — though even as late as 1907 it was not universally
accepted. Nowadays, it is possible to detect the effect of single atoms and
by indirect methods to render them visible, and Avogadro's number is
known to high accuracy.
2.2 GASES, LIQUIDS AND SOLIDS
As a useful, though not complete, classification it can be said that matter
exists in three states, as gas, liquid or solid. This statement is justified by
the fact that there exist many substances which can undergo sharp, easily
identifiable, reproducible, and reversible transitions from one state to the
other. Water is the classical example : its freezing and melting, boiling and
condensation have been contemplated since the time of the ancient Greek
scientists. There are obvious contrasts between the properties of ice,
water and steam or water vapour which make their description as solid,
liquid and gas quite unambiguous. Similarly, most metals are solid, they
melt under well defined conditions of temperature and pressure to form
liquids and boil at higher temperatures to produce gases.
If all substances possessed such clear demarcations, it would be easy
to define the different states of matter. But there are very many substances
16 Atoms, molecules and the states of matter Chap. 2
like glasses or glues which one normally thinks of as being solid but which
do not melt at sharply denned temperatures ; when heated they gradually
become plastic, till they become recognizably liquid. Other solids such as
wood or stone are inhomogeneous and it is difficult to describe their
structure in detail.
We will therefore not attempt to present definitions of solids, liquids or
gases ; there would be too many exceptions. Instead we shall describe the
principal characteristics of these three states of matter in bulk, and relate
them to their structure on the molecular scale. Later we shall describe
some of the conditions under which any one state can take on some of the
characteristics of the others. In the course of these rather brief summaries,
we shall use terms like 'small' or 'large', 'quickly' or 'slowly' which are as
yet a little vague. In later chapters, it will be shown how they can be
precisely defined.
2.2.1 Compressibility, rigidity, viscosity
Among other properties, it will be useful to compare the compressibility
and rigidity or viscosity of the different states of matter. If a given pressure
acting on a substance produces a large relative change of volume, that
substance is highly compressible. Rigidity is the ability to oppose or
withstand forces directed towards changing the shape of a body, while
keeping its volume constant ; this property refers to a purely static situ
ation. A related quantity is the viscosity, a measure of the resistance to
changes of shape taking place at finite speeds. For example, a body moving
through a medium has to keep pushing it aside to keep moving ; if the
forces required to be exerted on the body are large, the medium has a
high viscosity.
2.2.2 Properties and structures of gases
Gases have low densities, they are highly compressible over wide ranges
of volume, they have no rigidity and low viscosities. The instantaneous
structure of a small volume of gas is illustrated in Fig. 2.3. The molecules
are usually a large distance apart compared with their diameter and there
is no regularity in their arrangement in space. Given the positions of two
or three molecules, it is not possible to predict where a further one will
be found with any precision — the molecules are distributed at random
throughout the whole volume. They are moving randomly with a mean
velocity comparable with that of sound, of the order of 10 4 cm/s. Occasion
ally two or three of them may be found very close to one another so that
their electron clouds overlap and they bind together. Such clusters are
common at high pressures but they are usually shortlived.
2.2 Gases, liquids and solids
17
The low density can be readily understood in terms of the compara
tively small number of molecules per unit volume, and the high compressi
bility follows from the fact that the average distance between molecules
can be altered over wide limits. The lack of rigidity can be explained by the
molecules being able to take up any configuration with equal ease. Further,
the molecules can move long distances without encountering one another,
so that there is little resistance to motion of any kind, which is the basis
of the explanation of the low viscosity.
/ifra
wmm
Fig. 2.3. Molecules in a volume of gas at room temperature, a cube of about 20A at
a pressure of about 20 atmospheres (the molecules are pictured as simple spheres).
2.2.3 Properties and structure of liquids
Liquids have much higher densities than gases — comparing liquids with
common gases under ordinary conditions the factor is of the order of 10 3 .
Their compressibility is low. They have no rigidity but their viscosity is of
18 Atoms, molecules and the states of matter Chap. 2
the order of 10 2 times greater that that of ordinary gases. It is difficult to
give a detailed picture of the structure of a liquid ; an attempt has been
made in Fig. 2.4. The molecules are packed quite closely together so that
each one is bonded to a number of neighbours ; in the illustration, that
number is between 4 and 5. Given the position of one molecule, it is now
possible to state how many molecules should be found in contact with it,
which is a good deal more information than it is possible to give about the
arrangement of the molecules in a gas. But still the pattern as a whole is a
disordered one. The molecules are moving with just the same order of
velocity as in a gas at the same temperature, though the motion is now
partly in the form of rapid vibrations and partly translational. The
configuration is therefore continually changing.
This picture can be correlated with the macroscopic properties — the high
density from the large number of molecules per unit volume, the lack of
rigidity from the lack of order and the continual alteration of the arrange
ment. The comparatively close packing explains the low compressibility.
Fig. 2.4. Molecules in a volume of liquid about 20A x 20A x 3A.
2.2 Gases, liquids and solids
19
The fairly high viscosity arises from the fact that the molecules have to
wriggle past one another in this irregular but closely packed arrangement,
rather like people moving past one another in a dense crowd, where slow
relative movements are easy but rapid ones are difficult.
2.2.4 Properties and structure of solids
Solids have practically the same densities and compressibilities as
liquids. In addition they are rigid ; under the action of small forces they
do not easily change their shape.
An important property of those solids which have a welldefined
meltingpoint is that if they are formed very slowly from the liquid state
they are crystalline — that is, they form shapes bounded by plane faces with
Fig. 2.5. Molecules in a volume of solid about 20A x 20A x 3A. They are close
packed, and the arrangement is highly regular. There is however a fault (dislocation)
in the arrangement towards the left, which can be seen by holding the page horizontal
and looking upwards along the rows. (Dislocations are discussed in detail in
section 9.3.1).
20 Atoms, molecules and the states of matter Chap. 2
characteristic angles between them. Sometimes however the crystalline
form is not obvious, especially if the solid is not produced under suitably
controlled conditions ; the crystals may be too small to be seen so that the
solid as a whole does not show the expected facets. Substances which do
not melt sharply but show a gradual transition to the liquid when heated
are said to be amorphous and show no trace of regularity of external shape.
In crystalline solids, the molecules are arranged in regular three
dimensional patterns or lattices, of which Fig. 2.5. is an example. If the
crystal has been carefully prepared, the regular arrangement persists over
distances of several thousand molecules in any direction before there is an
irregularity, but if it has been subjected to strains or distortions the regular
arrangement may be perfect and uninterrupted only over much shorter
average distances. In metals the ions are closely packed together, so that
the distance between the centre of an ion and that of one of its nearest
neighbours is equal to the diameter of one ion, or something close to it. In
other crystals, the packing together of the molecules may be relatively
open, but even in a light solid such as ice the distance between the centres
of any molecule and its near neighbours is only twice the diameter of a
molecule. In solids, the molecules are again moving with the same order of
magnitude of velocity as in gases or liquids, but the motion is confined to
vibrations about their mean positions.
Amorphous solids can be described as liquids of extremely high vis
cosity, which over long periods of time have been 'frozen' into one par
ticular configuration. Figure 2.4 could be taken to illustrate the arrange
ment of molecules in such a solid, with the great difference that the con
figuration hardly alters with time, the atoms hardly ever changing their
relative positions, although they continue to vibrate.
In terms of these structures, it is not surprising that solids have com
pressibilities and densities like those of liquids. The high rigidities can
also be understood, for the molecules can only move with respect to one
another with difficulty. In crystalline solids, changes of shape can only take
place through molecules slipping into holes which exist at irregularities
in the lattice and this is not easy. In amorphous solids, there are plenty of
holes for the molecules to slip into in order to initiate a change of shape,
but the bonds between neighbours cannot be easily broken by external
forces acting on the solid as a whole. All solids therefore resist the action
of external forces and this is just what we mean when we say they are
rigid.
2.2.5 Characteristic times
There are some substances which are undeniably solids, yet which over
a long period of time alter their shape under the action of only small
2.2 Gases, liquids and solids 21
forces. Glaciers, for example, although made of ice which is undoubtedly
solid and crystalline, are found to be flowing slowly downhill if they are
observed over a period of years. Lead is another wellknown example, a
metal of high density but comparative softness. Sheets of lead, sometimes
used for covering the roofs of large buildings, will alter their shape over a
period of decades, slowly creeping downwards under the action of their
own weight. Under large pressures, a number of metals flow quite quickly
and their ability to fill small cracks makes them suitable for use as washers
or gaskets in situations where more fluid sealing compounds cannot be
used. Indium and pure gold, as well as lead and other soft metals, can be
used in this way. Thus, if we observe their behaviour for short times under
the action of small forces, we class these metals as solids ; but if we study
them under high pressures, or for times of the order of decades (10 8 10 9
seconds) if only small forces act on them, we say that they behave to some
extent like liquids.
By contrast, liquids and gases show resistance to bulk motion whose
speed is comparable with the speed with which the molecules are moving.
Water is well known to feel like a solid if one dives on to it instead of
through it. If a gas is made to move with a speed comparable to the velocity
of sound it can sustain very sharp changes of density ; instead of being
uniform, it is divided into distinct regions of different temperatures and
densities. The boundaries between them are called shock waves. Thus
liquids or gases subjected to large forces for short periods of time exhibit
some of the characteristics of solids.
Descriptions of the properties of solids, liquids and gases should there
fore include estimates of the times over which it is necessary to extend the
observations in order to decide whether a substance has rigidity or not,
whether local variations of density can be sustained or not. Under ordinary
conditions of temperature and pressure, lead is a solid if we are concerned
with events taking place in times which are less than, say 10 7 seconds, but
for experiments lasting more than, say 10 9 seconds, lead is a liquid.
Similar but much shorter characteristic times can be defined for substances
which are gases or liquids. In a similar way, when we say that a substance
softens or melts over a certain range of temperature, we mean that below
that range the characteristic time for flow under small pressures is very
long, inside the range it decreases with rising temperature, and at higher
temperatures it is small.
CHAPTER
Interatomic potential energies
3.1 MOLECULAR DIMENSIONS
A rough calculation of molecular dimensions can be made if the molar
volume of a solid or liquid is known. This is given by
V = M/p
where M is the gram molecular weight and p the density in g/cm 3 .*
For water, M = 18 g and p is about 1 g/cm 3 for both liquid and solid.
V is therefore about 18 cm 3 . Now this is the volume occupied by
N = 6 x 10 23 molecules. Thus the average volume occupied by a single
molecule is 3 x 10" 23 cm 3 . If we regard this volume as a cube, its side
must be about 3 x 10" 8 cm = 3 A. If it is a sphere or any other simple
shape its linear dimensions will not differ much from this. The distance
between molecules in water or ice must therefore be about 3 A. It has
already been mentioned (section 2.2.4) that the lattice in ice and the
packing in water are relatively open, so that this figure is greater than the
diameter of a water molecule.
In metals, the packing is usually very close, so that the diameter of an
ion cannot differ much from (VJN) 1 ' 3 . For potassium, a light metal with
large ions, M = 39 g, p = 0.86 g/cm 3 , so that V = 45.4 cm 3 and the
* Many students firmly believe that the molar volume of all substances is 22.4 litres. This
is indeed roughly true for gases under 'standard' conditions of temperature and pressure,
°C and 1 atmosphere. It is NOT true for solids or liquids.
3.2 Interactions between electrically neutral atoms and molecules 23
diameter of an ion is about 4.2 A. For gold, one of the densest metals,
M = 197 g, p = 19.3 g/cm 3 ; V = 10.2 cm 3 and the ionic diameter =
2.6 A. Indeed the diameters of all monatomic ions or molecules and the
mean diameters of the smaller polyatomic molecules are all between
1.5 and about 5 A. From the fact that the densities of common gases are
of the order of 10~ 3 g/cm 3 (of the order of grams per litre) it follows that
the mean distance between molecules is of the order of a few times 10 A,
which is much greater than the diameter of a molecule. This agrees with
the situation pictured in Fig. 2.3.
3.2 INTERACTIONS BETWEEN ELECTRICALLY NEUTRAL
ATOMS AND MOLECULES
In the previous chapter we described the different spatial arrangements
which the molecules or ions could take up in solids, liquids and gases.
These can be related in terms of the forces which the molecules or ions
exert on one another as a function of their distance apart. In this section
we will concentrate on electrically neutral atoms and molecules : metals
will be dealt with in section 3.7, ions in section 3.8.
We have to reconcile two apparently contradictory statements. First :
liquids and solids are highly incompressible. From this it follows that
when two molecules are squashed together so that they approach one
another closely, they repel one another. Second: solids and liquids
cohere — that is, their molecules tend to pull themselves close to one
another. It takes force to stretch a solid, therefore molecules attract one
another. We will denote the repulsive forces by F R and the attractive
forces by F w .
F R must be dominant when the distance between molecules is about
12 A or less. F w must be dominant when the separation is about 23 A
or greater. At some intermediate distance, say roughly 2 A, the two forces
must be equal and opposite — repulsion balances attraction. (Obviously
these figures must not be taken too literally. They are quoted as crudely
typical.) We seek an algebraic representation for this.
First we must have a sign convention for the direction of the force
exerted by a molecule. Imagine the origin of coordinates to be taken at
the centre of the molecule and a line drawn outwards towards another
molecule on which it exerts a force. Call this the raxis. A force which
acts in the direction of r increasing — a force of repulsion, tending to
separate the molecules — is reckoned positive (Fig. 3.1). A force of attrac
tion is, on the same convention, reckoned negative in sign. We will use this
convention consistently.
24 Interatomic potential energies Chap. 3
Force exerted by molecule
&
Molecule
r  axis
Fig. 3.1. Convention for direc
tions of axes and forces. A
repulsive force exerted by a
molecule, in the direction of r
increasing, is positive.
Interatomic distance
Interatomic distance
Fig. 3.2. (a) Short range and long range forces ; both are repulsive, (b) Expo
nential falloff. Again, a repulsion is shown. (Plotted in arbitrary units).
All intermolecular forces, attractions and repulsions, become smaller
as the separation increases. If we choose to represent this variation by a
simple power law, this law might be
Force = + (const.)
choosing the + sign for a repulsion and the — sign for an attraction, a is
some standard length and the index n is positive. Now if n is large, r~ n
becomes rapidly smaller when r is increased and rapidly bigger when the
distance is decreased. If n is small, however, then the force falls off com
paratively slowly at large distances though it also increases comparatively
slowly at small distances, To take a specific example, Fig. 3.2(a), consider
two forces proportional to (a/r) 1 and (a/r) 2 and called F x and F 2 respec
tively. The index 7 is for present purposes a large number and the 2 is a
small number. At r — a, both forces are numerically equal. However at a
small separation r = a/10, F x is 10 5 times bigger than F 2 ; but at a large
3.2 Interactions between electrically neutral atoms and molecules 25
distance r = 10a, F 2 is 10 5 times bigger than F : . A force like F x , dominant
at short distances but negligible at large, is called a short range force.
F 2 is a long range force. Besides simple power laws like (a/rf other forces
are found in nature which vary like exp( — r/a) where a is some characteris
tic distance. Every time r is increased by a, this kind of force decreases by a
factor e = 2.728, so that it falls to 1/20 of its value for every 3a (Fig. 3.2(b)).
Thus exponentially varying forces are certainly short range.
In this terminology, the repulsive forces between atoms are short range,
the attractive contributions are rather longer range. The best expression
for the total force has been shown theoretically to be of the type
F = F R + F W
= A.e~ rja B(a/r)\
where the variable r is the distance between the centres of the atoms, a is
some measure of the 'diameter' of an atom — not, of course, that it has a
sharply defined surface like a billiard ball.
The repulsive force is caused by the overlapping of the two electron
clouds — this gives the exponential variation. The attractive force is
called the van der Waals force. It arises from the distortion of the electron
cloud of one molecule by the presence of the other. It exists even though
the atoms are electrically neutral.*
For many purposes it is an advantage to have an expression for the
interatomic force which fits the true curve adequately but which has a
simple analytical form. In fact, the exponential term can be replaced by
one of the type (a/rf where n is 10 or 13 or some number like that, so that
This is good enough for our purposes and we will adopt it.
3.2.1 Potential energy
Rather than deal with forces, it is more convenient to deal with the
potential energy of two molecules with respect to each other. This is a
* Many students firmly believe that the only force of attraction which can exist between
electrically neutral atoms is the Newtonian gravitational attraction due to the masses. They
assume that the attractive forces between atoms are gravitational. This assumption was
made in the early nineteenth century by Dalton and other pioneers. Apart from the fact
that the index n is wrong (2 for gravity, 7 for interatomic forces), the magnitude of the binding
energy (see section 3.3) is a factor of about 10 3 ° times too small:
1 ,000,000,000,000,000,000,000,000,000,000
times too small. Indeed, gravitational forces can safely be forgotten in all problems which
do not involve the Earth or bodies of comparable mass.
26 Interatomic potential energies Chap. 3
scalar quantity and therefore simpler to discuss than forces, which are
vectors.
With the sign convention of the last section — F positive if it acts in the
direction of r increasing — the potential energy can be defined by the
equation*
Fir) = t^X (31)
ar
where F(r) means 'the force which depends on r' and i^{f) means 'the
potential energy which depends on r\ The force is the gradient of the
potential energy. Alternatively,
T^W = ^(r ) [ F{r).dr, (3.2a)
where r is a standard point and i^(r ) is the potential energy there.
It is usually most convenient to take this constant ^(r ) as zero; if
this is done then we have
nr) =  f
F(r) . dr. (3.2b)
We are at liberty to choose the standard point r where we please. For
many problems it is convenient to take r = oo. Then the potential
energy at r is given by
TT(r)=  f F(r).dr. (3.2c)
J oo
In some problems however it is convenient to define 'V to be zero at the
origin, that is to take r = 0. The latitude in the absolute value of the
potential energy causes no difficulty because we are usually concerned
with measuring only changes of potential energy.
As an example, consider the potential energy of two atoms derived from
the force of repulsion
A
F = ?'
where n > 1. Then
f r a>_ A r 1
1 r " ^T
TT = A
J r
* Really the equations should be written in vector notation
F(r) =  grad jr(r) ; V{t) = ! F . dr
but as we are dealing with central forces the simpler formulation is adequate.
3.2 Interactions between electrically neutral atoms and molecules 27
It is obviously convenient to take r = oo and to write
nr)= A l
n1 r" _1
A graph of this function resembles Fig. 3.3(a). The physical situation
being described is that initially two atoms are an infinite distance apart
and they are pushed together infinitely slowly so that finally they are
separated by r. The potential energy is then increased by this amount.
Whether one says that the energy of the second atom is increased with
respect to the first, or vice versa, is irrelevant ; in fact it is better to talk
about the potential energy of the whole system. The energy in fact resides
in the field of force between the atoms.
The position of stable static equilibrium of a system occurs when the
forces are zero, and this is where the potential energy is a minimum. Left
to itself, a system will always move so as to reduce its potential energy.
It follows straightforwardly from the definitions that a repulsive force
which decreases with distance always has a i^(r) curve of the type shown
in Fig. 3.3(a) — or Fig. 3.3(b) which differs from it merely in the addition
of an arbitrary constant — so that when the system moves to reduce 'V,
it does so by increasing the separation r. Conversely, attractive forces
have rising curves like Fig. 3.3(c).
A system having a potential energy which is a function of displacement
like Fig. 3.3(d) will tend to move into the position of minimum energy,
where
^ = °
This is the position marked r in the diagram. The system is then said to
be in a potential 'well'. It is called a well because it looks like a hole in the
ground. Work must be done, energy must be supplied, to get the system
out of the well. The amount of energy needed to take the system far to the
left would be e t ; to move it far to the right, e 2 .
3.2.2 Interatomic potential energy
The interactions between atoms can be represented by a graph of their
potential energy as a function of the distance between their centres.
Following the argument of section 3.2, it can be expected to be of the form
where p is approximately 9 or 12, q is a smaller number and X and \i are
constants.
28 Interatomic potential energies Chap. 3
Fig. 3.3. (a) i^(r) for a repulsive force, (b) The same situation with a shifted
arbitrary zero for y. (c) ^"(r) for an attractive force, (d) A potential well.
A very convenient form of this type of equation is
r(r) =
(3.3)
This looks a good deal more complicated than the equation just above it
but in essence it is the same, namely the sum of a l/r p term and a — 1/V
term. A graph of this kind of function (for the special case of p = 12,
q = 6) is shown in Fig. 3.4 and another example (p = 11, q = 1) is shown
in Fig. 3.15(a).
The reader should check that Eq. (3.3) has the following properties.
The potential energy at r = oo, when the molecules are infinitely far apart,
is zero. By putting dl^/dr = 0, it can be readily verified that the minimum
3.2 Interactions between electrically neutral atoms and molecules 29
value of the energy is — e, so that we can say that the depth of the well is e.
This minimum occurs at r = a . This is the position of static equilibrium.
In order to pull the atoms apart to infinity, the attractive forces would
have to be overcome and this would require the expenditure of an amount
of energy e.
A/flL
Fig. 3.4. Interatomic potential energy, Eq. (3.3), plotted
for the important case of p — 12, q = 6 (the Lennard
Jones 612 potential). The potential energy has been
plotted in units of s, the separation between centres in
units of a . The curve crosses the axis at r = a (Eq. 3.5).
One case which is specially useful for simple molecules is the Lennard
Jones 612 potential when p = 12, q = 6: it reduces to
nr)
(3.4)
This is the function plotted in Fig. 3.4. There is a more symmetrical form
of this equation, which introduces another parameter a :
V(r) = 4e
(3.5)
This is exactly the same as Eq. (3.4), if we put a = Xjl ■ a — 1.12a. When
the separation r between the centres is equal to a, the potential energy is
zero ; thereafter, if the two atoms are squashed together a little more the
potential energy rises steeply ; in other words, the force of repulsion in
creases greatly. If we regarded the atom as a kind of ball with a hard
30 Interatomic potential energies Chap. 3
surface, we would identify the diameter of the ball with the separation
between centres at which the repulsion rises steeply. Fig. 3.4 would then
be interpreted to mean that the diameter of one atom is a and that the
position of static equilibrium occurs when the separation is a = 1.12a,
so that atoms are 'nearly touching'. Since a and a are nearly equal to one
another, both are good measures of the diameter of one atom.
Two qualifications must be made about the use of this curve. The first
is that it refers to the interaction between two molecules only. If a third
molecule is in the vicinity, the force between the first two may be modified
by the movement of the charges in the electron cloud. But we will assume
that this effect is negligible. As a result, the total energy of an assembly of
molecules will be taken to be the sum of the energies of every pair as given by
this curve. The second qualification is that we have tacitly assumed that the
molecules are spherical so that the energy is uniquely defined by their
distance apart. If this is not so, then their relative orientations may be
important. In general, however, we will limit discussion to simple molecules
which do not depart too much from sphericity, where their separation r
is easily defined and i^{r) does not depend on orientation.
3.2.3 Nearest neighbour interactions
When two atoms are in equilibrium and 'nearly touching', their
potential energy is — £ (Fig. 3.4). If their separation is approximately
doubled, their potential energy decreases to about — e/30, that is, by a
large factor. Whereas two atoms which are nearest neighbours are bound
together by an amount of energy e ('bound' in the sense that energy is
required to separate them), two atoms which are next nearest neighbours
are only very loosely bound to each other (Fig. 3.5(a)). This is another
way of saying that the van der Waals forces are short range forces —
though of course the repulsive forces are of even shorter range.
Fig. 3.5. (a) Nearest and nextnearest neighbours. The
potential energy of A and B is e; that of AC is about
30 times smaller, (b) An aid to counting nearestneighbour
interactions. Energy e is needed to 'cut' each pair apart.
3.3 Binding energy and latent heat 31
Provided the picture is not taken literally, it can be considered as if each
atom were held to its nearest neighbours by some kind of bond which
requires an amount of energy £ to cut it (Fig. 3.5(b)). Bonds between atoms
which are further apart however are so weak that we can forget them.
The purpose of this picture is to indicate the amount of energy that must be
supplied in order to break up the structure. It must NOT be thought to imply
that the electrons are more concentrated in certain regions.
3.2.4 Potential energy dominant at low temperatures
In the rest of this chapter we will show that it is possible to relate some
of the macroscopic or large scale properties of solids and liquids to the
potential energy between the atoms or molecules.
We will make an important assumption, namely that the kinetic
energy of the atoms or molecules is small compared with their potential
energy. This is equivalent to saying that we will assume that the tempera
ture is low. At this stage, before we have described in detail what we mean
by temperature, it is not possible to say precisely what is meant by a 'low'
temperature ; in fact, a temperature which is low enough for our approxi
mation to hold for one substance may not be low enough for another. In
quoting any data however, we will always take the precaution of referring
to temperatures which are low enough for the substance concerned. The
effect of this procedure will be to simplify our calculations. In order to
estimate the energy of an assembly of a large number of atoms or molecules
in a mass of liquid or solid at low temperatures we need only take into
account the potential energy due to their interactions and we can neglect
their kinetic energy.
The result of the discussion of the last section, where we saw that the
potential energy of such an assembly is dominated by the potential
energy between nearest neighbours simplifies our calculations even further.
3.3 BINDING ENERGY AND LATENT HEAT
The energy required to change one mole of solid or liquid into gas at
low pressure is called the binding energy. It is closely allied to quantities
which can easily be measured experimentally, the latent heats of evapora
tion (liquid to gas) or sublimation (solid to gas).
For precise calculations, there are difficulties when these quantities are
compared, however. When we calculate binding energies, we usually take
the pressure of the gas to be zero so that the separation between atoms is
infinite. Experimentally, we usually take the pressure to be the vapour
pressure at the temperature concerned. (For example, one might measure
the latent heat to evaporate water at 100 °C to produce steam at 1 atmo
sphere.) The difference has to be allowed for, although it is small at low
32 Interatomic potential energies Chap. 3
temperatures. A more serious source of error is that latent heats of
evaporation of liquids are functions of temperature — they decrease with
increasing temperature and become zero at a high temperature above
which the liquid cannot exist. (See for example Fig. 3.13(b) for the latent
heat of liquid argon.) It is therefore meaningless to quote 'the latent heat
of vaporization' of water or any other liquid as if it were a constant.
However, we are restricting our discussion to low temperatures, and in
that region latent heats tend towards limiting values : it is these which we
consider.
For rough estimates, we note that latent heats of melting, to convert
solid to liquid, are small compared with latent heats of evaporation to
convert liquid to vapour. For example, to convert ice to water at °C
requires about 6x 10 10 erg/mol (that is, 340 J/g), to convert water to
steam at a comparable temperature requires 45 x 10 10 erg/mol (2,500 J/g)
— almost eight times as large. So we can approximate even further and,
if no better data are available or only rough estimates are needed, we
can say that the binding energy is not very different from the latent heat
of evaporation.
3.3.1 Estimation of £ from latent heat data
On our approximation, the binding energy at low temperatures is equal to
L — ex (number of pairs of nearest neighbours).
It is useful to define the coordination number n, the number of nearest
neighbours which surround a given atom or molecule. It can never
exceed 12. For closepacked solids it can reach 12 ; for more open arrange
ments it is smaller, 6 or 10. In dense liquids, the coordination number is
about 10. (In the twodimensional pictures, Figs. 2.4 and 2.5, n is about
4 or 5 and about 6 respectively.)
For coordination number n, an assembly of N atoms has ?nN pairs of
nearest neighbours ; the factor \ arises from the fact that each bond pictured
in Fig. 3.5(b) links two atoms but it must only be counted once. Thus :
L = ±nNe. (3.6)
If we know L , this allows us to estimate e. At the same time, the smallness
of the latent heat of melting can be understood, since the change of co
ordination number between solid and liquid is quite small (12 to 10 say),
whereas between liquid and gas it is large (10 to zero). We can get consist
ent results for £ by using the latent heat of evaporation together with
n = 10.*
* Strictly our calculations refer to single atoms but they can be applied to molecules which
do not depart too far from spherical shape. These include diatomic molecules like N 2 or
molecules like CC1 4 which are roughly tetrahedral. Long chain molecules are ruled out.
3.4 Surface energy 33
For liquid nitrogen, consisting of diatomic molecules N 2 , the latent
heat at low temperatures is about 210 J/g, the molecular weight is 28,
so the molar latent heat of evaporation is 6 x 10 10 erg/mol. Then e is
2 x 10~ 14 erg ~ 0.01 eV. This is the energy needed to separate two nitrogen
molecules from one another. For carbon tetrachloride, CC1 4 , the latent
heat is about 210 J/g, the molecular weight is 153 so that e is about
10~ 13 erg or 0.05 eV.
These are typical values. Most molecules have e of the order of 0.01
to 0.1 eV. In Fig. 3.4, each division of the vertical (energy) axis is therefore
of this order of magnitude ; each horizontal division represents a distance
of a few Angstrom units. Compared with ionization energies, the energies
required to remove an electron from the cloud surrounding typical atoms
to convert them into ions, these are small amounts of energy. Ionization
energies are commonly of the order of 1 to 10 eV, which is 100 to 1,000
times as big.
3.4 SURFACE ENERGY
The surface energy of a solid or liquid is the amount of energy that is
needed to create 1 cm 2 of new surface. The process can be pictured as
follows. Imagine a column of solid or liquid of 1 cm 2 crosssection to be
broken apart by some means, Fig. 3.6. Energy must be used in order to
overcome the interactions between molecules on either side of the break.
We will calculate this energy. Let there be JT molecules per cm 2 of cross
section ; if the diameter of one molecule is a cm, then Jf is something
like 1/a.o per cm 2 .
Fig. 3.6. Creating new surfaces by cutting a solid or
liquid column in two.
34
Interatomic potential energies Chap. 3
After the break, each molecule in the surface is no longer surrounded
by the full number n of nearest neighbours. Instead it has, on the average,
only j\x neighbours, in the one hemisphere; therefore \nJf nearest
neighbour interactions must be broken. This requires an amount of
energy \ nJ^e : but it produces 2 cm 2 of new surface, 1 cm 2 each for the
top and bottom halves of the column. The surface energy is therefore
^wjVe erg/cm 2 .
3.4.1 Surface tension
Surface tension, a quantity which is easily measured experimentally
and is allied to surface energy, is usually defined as the force exerted on a
cut 1 cm long in the surface of a solid or liquid, a force which tends to
close the cut. It will be denoted by y dyn/cm. Imagine a thin film of liquid,
with upper and lower surfaces like a soap film, to be stretched across a
wire frame, Fig. 3.7. One side of the frame is moveable. It is assumed that
the film is many molecules thick. The force on the slider is 2yl dyn, where
/ is its length. The factor 2 appears because there are two surfaces. If the
slider is moved back a distance d, the work done is 2yld erg. We imagine
this process to be done so slowly that heat can flow into the film so that
any tendency to cool is counteracted and the temperature and y remain
constant. Since the total area of surface is now increased by 2ld, the amount
of energy supplied is y erg per unit area. The surface tension is therefore
clearly related to the surface energy. They are not identical however,
because of the heat energy flowing in during the process to keep the tem
perature constant. This is the same kind of difference as that between a
binding energy and a latent heat. Again, however, if measurements are
extrapolated to low temperatures there is little difference between the
two, although for the roughest estimates it is not necessary to make even
this correction. ((Fig. 3.13(d) shows the variation of surface tension with
temperature for liquid argon and this is typical.) We will therefore write
y = irvVe
(3.7)
Fig. 3.7. Stretching a liquid film to create new surface.
3.4 Surface energy 35
3.4.2 Estimation of £ from surface tension data
It is clear that, with the considerable oversimplifications in our model
of a liquid, we can relate the surface tension to the latent heat. Although
the phenomena that we usually associate with these quantities are quite
different, both are simply measures of the depth e of the potential well and
of the sizes of molecules. In 1870 Kelvin used a similar analysis to estimate
the size of water molecules from the known molar volume V , the heat of
vaporization and the surface tension, effectively writing our equations in
the form a = 2yV /L . This was one of the first methods for estimating
Avogadro's number.
Here we will use surface tensions to work out e for the same liquids as
in section 3.3.1 and show that the results are comparable with the values
deduced from latent heats. At its normal boiling point (77°K), liquid
nitrogen has a density of 0.81 g/cm 3 and its surface tension is 8.7 dyn/cm.
Its molecular weight is 28. The molar volume is therefore 35 cm 3 , and
following the argument of section 3.1.2 the diameter of the molecule is
3.9 A; hence Jf is 6.7 x 10 14 per cm 2 . If we take n = 10, £ must be
0.5 x 10" 14 erg or 0.003 eV. This must be compared with 2 x 10" 14 erg
(0.01 eV) which we deduced from latent heats. For carbon tetrachloride
at room temperature the data are 1.6 g/cm 3 , 26 dyn/cm and molecular
weight 153. The molar volume is 96 cm 3 , the diameter of a molecule 5.4 A,
JT is 3.5 x 10 14 per cm 2 and if n = 10, e is 3 x 10" 14 erg or 0.02 eV. This
must be compared with 10" 13 erg (0.05 eV) from the latent heat data.
Both liquids therefore give figures which are consistent within a factor 4,
and this must be considered good agreement in view of the crude handling
of the data.
* 3.4.3 The rise of liquids in capillary tubes
One of the commonest methods of measuring the surface tension of a
liquid is to measure its rise in a capillary tube of known radius. In this
section we will discuss, in terms of interatomic potential energies, why
many common liquids rise in a glass tube but mercury falls.
When a glass tube is exposed to an atmosphere containing the vapour
of a liquid, its surface is bombarded with molecules and some of these
stick to the glass. The process is called adsorption. The whole surface
quickly becomes covered with a layer one or two molecules thick. The
molecules next to the glass may be attached very firmly: this can be
deduced from the amount of energy (the heat of adsorption) which is
observed to be given out when the surface is exposed in this way.
* Starred sections or subsections may be omitted, if the reader so wishes, as they are not
required later in the book.
36 Interatomic potential energies Chap. 3
But this tight binding usually does not extend very far because inter
molecular forces are of short range and the forces acting on a molecule
outside the solid are determined by the nature of the outermost layers.
After the surface of the glass has been covered with the first one or two
layers of vapour molecules, further vapour molecules approaching the
surface experience an attraction which is almost the same as if the entire
tube were composed of these molecules. Its surface energy per unit area
is practically equal to that of the liquid from which the vapour was
produced.
When a tube is first dipped into a liquid it has not risen in the tube
and the surface is flat. Let us find the height to which it rises by calculating
the change of potential energy when the tube is filled to an arbitrary height
h and then let us write down the condition that the potential energy
should be a minimum ; this determines the equilibrium value of h. Let the
liquid have density p and surface tension y and let the radius of the tube
be r. We can consider the tube to be filled in the following stages. We
imagine a volume of liquid to be removed from the flat surface, just enough
to fill the tube to the height h. It must have volume nr 2 h and surface
area 2nrh (Fig. 3.8). To remove it from the rest of the liquid an amount of
energy equal to (2nrh)y must be expended. Then we imagine this liquid
to be changed to a cylindrical shape (which requires no change of surface
area and hence no expenditure of energy). When it is raised vertically, its
potential energy due to its weight is increased to {nr 2 hp)gh/2 since its
mass is nr 2 hp and the height of its centre of mass is h/2. Finally the liquid
can be imagined to be put inside the tube. A surface area of the inside of
the tube equal to (2nrh) is covered and an equal area of the surface of the
cylinder of liquid is also covered. Thus, since both surfaces have surface
energy y per unit area, the surface energy is reduced by 4nrhy.
surface of tube
/"not covered
correct amount
of liquid removed
surface of tube
/not covered
liquid raised into
/'cylindrical shape
surface of liquid
'not covered
surfaces
/covered
(a) it>) (c)
Fig. 3.8. Energy changes when liquid rises in a capillary tube.
3.4 Surface energy 37
Thus the total increase of potential energy when the liquid rises to
height h is
U = 2nrhy + nr 2 h 2 pg — 4nrhy
= nr 2 h 2 pg — 2nrhy.
This equation shows that the reason the liquid rises in the tube is that
the surface energy of the interior of the tube is reduced.
The condition that U should be a minimum is that
= 0,
dU
~dh
that is
7ir 2 hpg — 2nry =
2y
h = — . (3.8)
rpg
For carbon tetrachloride in a tube of 1 mm bore, h is equal to 6.5 mm
since y = 26 dyn/cm and p — 1.6 g/cm 3 .
This discussion should be valid for any liquid whose molecules are
adsorbed on to the surface of the tube. For mercury in glass, however,
conditions are very different. Under normal conditions mercury does not
adhere to glass. Droplets of this liquid simply run off a glass surface.
It should be noted however that under very special conditions mercury
can be made to stick to glass, but that even then the adhesion is very weak.
The effect is sometimes observed in McLeod gauges used to measure
pressures in high vacuum systems. McLeod gauges have two limbs con
taining mercury, one open and the other closed. When the pressure is
being measured in a system at extremely low pressure the mercury is
pushed right up to the closed end of the tube and fills it completely.
As part of the measuring procedure, the mercury in the other limb is
then lowered and under normal conditions the mercury in the closed
limb also falls, so that the two menisci keep at practically the same height
as one another. But when conditions are exceptionally clean (the glass
surfaces have been heated and the system has been evacuated for a con
siderable time), it is occasionally observed that the mercury in the closed
limb does not fall but remains stuck to the glass. If the difference of
heights is h, the mercury at the top of the closed limb is under a tension
of pgh dyn/cm 2 , where p is the density of the mercury, and this tension
must be resisted by the adhesion to the glass. As the level is further lowered,
the mercury in the closed limb suddenly falls when the adhesion is broken.
38 Interatomic potential energies Chap. 3
Level differences up to about 10 cm are sometimes seen, corresponding
to tensions of about 10 5 dyn/cm 2 . We can use this fact to estimate the
energy required to separate 1 cm 2 of mercury from glass. Let us assume
that once we have separated them by a distance of one or two atomic
diameters, the force becomes very small. The work required to do this
is equal to the tension times the distance, which is 10 5 dyn/cm 2 multiplied
by 10" 8 cm, that is 10" 3 erg/cm 2 for a range of 1 A; for a range of 10 A
it is 10" 2 erg/cm 2 . Interpreting this in terms of interatomic potential
energies as we did in section 3.4, the depth of the well must be of the order
of 10" 5 or 10" 6 eV (using Jf of the order of 10 15 per cm 2 and n about 10).
This is a very small figure compared with quite weak van der Waals
energies. Of course, the mercury glass bond always breaks at its weakest
place and the average energy of adhesion must be rather larger than we
have estimated. But allowing for this, it seems safe to assume that even
under the most favourable conditions mercury adheres only weakly to
glass.
This means that when a glass surface is covered with mercury, the
surface energy decreases at best by very little. Following the previous
analysis, the condition for minimum total potential energy is that the
liquid should be depressed inside a tube, which is what is observed.
The important point about capillarity experiments is that they measure
the interactions between molecules of the liquid and those of the surface
of the tube. In order to measure the surface tension of the liquid alone,
other methods have to be used.
3.4.4 Speed of ripples over a liquid surface
One interesting method of measuring the surface tension of a liquid is
to find the speed of propagation of ripples across its surface.
When a wave is travelling across a surface and the wave profile is
sinusoidal, the area is greater than when the surface is plane. The surface
energy is therefore increased (Fig. 3.9). This effect— which leads to a
finite speed of propagation of the waves — is the one we are interested in.
But at the same time, the weights of the parts of the wave which are
displaced upwards and downwards also increase the potential energy of
the system and this affects the speed of propagation too. It can be shown
that for waves of wavelength A and frequency/, the speed defined by
c = fA
is given by
3.4 Surface energy
39
where y is the surface tension of the liquid whose density is p, and g is the
acceleration due to gravity.
Fig. 3.9. Sinusoidal wave on a liquid. The perimeter ABCDE is longer than
the undisturbed distance ACE (equal to the wavelength A) so that the surface
energy is increased.
It follows from the dependence of the two terms on A and A~ 1 respec
tively that when the wavelength is very great the second term under the
square root sign is small ; the speed is then controlled by gravity alone
and is not affected by surface tension. But when the wavelength is small,
the gravity term becomes small and the speed is dominated by the surface
tension term. For these short wavelength ripples,
2ny
so that they travel faster the shorter their wavelength.
The method consists of generating ripples usually at audio frequencies,
on the liquid in a tank. The frequency must be known and the wavelength
is usually measured either by stroboscopic photography or by setting up
a stationary wave pattern and measuring the distance between nodes. The
liquid must be of sufficient depth for the bottom of the tank to have no
influence on the waves. The disturbance does not in fact penetrate very
deeply into the liquid ; the effective mass which takes part in the motion
is only about onetenth of a wavelength deep, to be precise A/4n. The
amplitude of the disturbance at a depth of one wavelength is negligible
and this gives a criterion for the depth of liquid to use. The analysis of
the motion is set as a problem at the end of the chapter. It depends on the
fact that if a system is disturbed and its potential energy is proportional
40
Interatomic potential energies Chap. 3
to the square of the displacement, the motion is periodic in time. This
topic is discussed in section 3.6.
3.5 ELASTIC MODULI
Elastic moduli are all defined by equations of the type
(change of pressure) = (modulus) (fractional change of dimensions).
For small changes of dimensions, usually less than 1 % or 0.1 %, the body
regains its original shape and size when the forces are removed; the
behaviour is said to be reversible (or elastic), and we will concentrate on
this type of change. Furthermore, when the fractional changes of dimen
sions are extremely small — a factor 10 or so smaller than the limit where
elastic behaviour ceases — the changes of dimensions are quite accurately
proportional to the pressures. Thus over these very limited ranges, the
elastic modulus is a constant for the material. (We will discuss the break
down of proportionality in section 3.7.1. and nonelastic behaviour in
section 9.1.1.)
Stretching and twisting by different geometrical arrangements of
forces, and compression by uniform 'hydrostatic' pressures are different
ways of producing deformations and correspondingly we define Young's
modulus, the rigidity or shear modulus and the bulk modulus as in Fig.
3.10.
r
l
~"~r
i_
1
1
1
1
1
1
1
1
(a)
ib)
(c)
Fig. 3.10. Systems of forces and deformations denning elastic moduli, (a)
linear tension producing extension, related by Young's modulus, (b) tan
gential forces producing an angle of shear, related by the rigidity and (c)
hydrostatic pressure producing a change of volume, related by the bulk
modulus.
For many practical purposes, it is necessary to emphasize the differences
between the three moduli but here we will concentrate on their similarities.
For any one substance they are of the same order of magnitude. Usually
the bulk modulus and Young's modulus are almost equal and the rigidity
3.5 Elastic moduli 41
modulus is a factor 2 or 3 smaller. This can be seen in the Table. But it
must always be remembered that for liquids, the rigidity is zero (section
2.2.3).
Material Young's mod. Rigidity Bulk mod.
dyn/cm 2 dyn/cm 2 dyn/cm 2
Solid argon 7.0 xlO 9 3.0 xlO 9 6.0 xlO 9
Sodium chloride 4.0 x 10 1 1 1.3 x 10 1 1 2.5 x 10 1 i
Steel 2.0 xlO 12 0.8 xlO 12 1.8 xlO 12
We will take the bulk modulus as the typical elastic parameter, because
it is the easiest to calculate.
The bulk modulus K of a material, the reciprocal of the compressibility,
is defined by
 (S)
(3.10)
where V is the volume, which is decreased when a pressure P is exerted
uniformly in all directions. Usually, it is assumed that the temperature is
kept constant during the compression. K can be measured directly by
exerting a known pressure and measuring the change of volume — a whole
technology has grown up for producing enormous pressures without the
substance leaking past the piston which compresses it. Usually the main
source of error is due to the nonuniformity of the forces acting in different
directions. Alternatively, the speed of propagation of sound waves through
a material can be found. This depends on the compressibility (as mentioned
in section 3.6.1.), though a number of corrections have to be applied if exact
values of K are required.
3.5.1 Bulk modulus and the 'Vif) curve
A compressed body can do work if the pressure is released; thus a
compressed body has potential energy. This energy is given by a term of
the type (force) x (distance) or (pressure) x (volume) — this is shown
explicitly for a gas in Fig. 4.2 but it holds for any body. Notice that the
energy E increases when the volume decreases so that,
d£ = PdV.
42 Interatomic potential energies Chap. 3
Hence we can write
This is an important relation. Pressure can usually be interpreted as an
energy per unit volume, an energy density. However, one must be careful
about this expression ; it assumes that no heat flows in during the process
of compression, or in thermodynamic language the compression must be
adiabatic. This conflicts with the usual definition of bulk modulus given
above. But thermal effects become small at low temperatures, so once
again our estimates will become better the lower the temperature.
Substituting (3.11) in (3.10):
ld 2 E
K = y W.
This expression is in macroscopic terms — that is, the variables are E, the
energy of the whole block of material, and V its volume. We wish to
express these in terms of 'Vif) the potential energy of a pair of molecules
and r the separation between two molecules. We do this as follows.
If we can express V in terms of r, we can express d 2 E/dV 2 in terms of r.
In general,
d£ _ dr d£
dK~cfF'~a>'
d 2 £ d IdE dr\ d 2 E /dr\ 2 d£ dV
dV 2 dV\dr dVj dr 2 \dV) dr dV
This holds whatever the relation between Fand r. In particular, if we regard
the molecules as little cubes.
V = Nr 3 ,
where N is the number of molecules in the block and r the distance between
two neighbours. Therefore
dr
Further, if deformations are small and we take only nearest neighbour
interactions into account, we can say that to a good approximation r = a ,
where a is the separation for static equilibrium where the potential energy
is a minimum, i.e. d£/dr = 0. Thus the second term on the right hand side
3.5 Elastic moduli
43
of (3.12) is zero, and
d 2 E
dV
K =
9N 2 a*,
'9Na .
(3.13)
Finally we must relate E, the energy of the block of material, to the energy
of the individual molecules. As already emphasized, we will assume that
their kinetic energy is small compared with their potential energy due to
the intermolecular forces, which is equivalent to saying that we consider
only low temperatures. We therefore relate E to *T(r). Taking only nearest
neighbour interactions into account, we consider ^Nn nearest neighbour
pairs; then
E = iATnlT(r),
and
K = n
d 2 TT(r)
dr 2
18a .
(3.14)
This expression holds whatever the form of f~(r). Let us assume the 612
potential
TT(r) = e
?r>r
Then
whence
d 2 T(r)
dr 2
72s
a 2
(3.15)
K =
4ns
al
4iVne 8L f
Nat
V n
(3.16)
where V is the molar volume and L the binding energy per mole. This is
a rather extraordinary relation. It predicts that, for solids which are bound
together by van der Waals forces so that the 612 potential is a good
description of the interatomic potential, the bulk modulus is 8 times the
binding energy per unit volume. Implicitly, we are referring to low
temperatures. This relation allows us to predict the order of magnitude of
44 Interatomic potential energies Chap. 3
any elastic constant, if we know the latent heat of evaporation or the
surface tension. The factor 8 depends on the assumption of the 612
potential but one expects a similar sort of factor for any molecular solid
or liquid.
3.5.2 Comparison of bulk modulus and latent heat data
As in sections 3.3.1 and 3.4.2, we will use liquid nitrogen and carbon
tetrachloride as typical molecular liquids. For liquid nitrogen, SL /V =
(8 x 6 x 10 10 /35) erg/cm 3 = 1.4 x 10 10 erg/cm 3 . The measured bulk modu
lus of the solid at about the same temperature is 1.26 x 10 10 dyn/cm 2 .
The agreement is excellent. For CC1 4 , at room temperature 8L /% =
8 x 3.2 xlO 1 796 = 2.5 x 10 10 erg/cm 3 . The measured bulk modulus is
1.1 x 10 10 dyn/cm 2 . The agreement within a factor of 2 must be considered
good.
3.6 VIBRATIONS IN CRYSTALS: SIMPLE HARMONIC
MOTION
We have already seen that molecules in a solid or liquid are vibrating
about their mean positions (sections 2.2.3, 2.2.4). The purpose of the present
discussion is to show how the frequency of vibration can be estimated,
knowing the T^*(r) curve.
Firstly we must establish some relations about simple harmonic
motion. Consider a system subject to a restoring force directed towards
an origin and proportional to the displacement x. With the sign convention
of section 3.2,
F = —ax.
Such a system executes simple harmonic motion of frequency
1 Ax
2n\j m
where m is the mass in motion. This well known result can be recast in
terms of potential energies. We are at liberty to take the zero of potential,
energy anywhere we wish. In this case we will choose it at x = 0, and
V = ax 2 . (3.17)
The system is said to be in a parabolic potential well (Fig. 3.11(a)).
We can define the curvature at any point of a curve as the reciprocal
of the radius of curvature. It is shown in mathematics texts that for the
curve y = f(x),
d 2 y /dx 2 mR .
curvature = r. — . , , , , 2 >3/2 • ^.io)
{l+(dy/dx) 2 } 3/2
3.6 Vibration in crystals : simple harmonic motion
45
Fig. 3.11. (a) Parabolic potential well, (b) A curved potential well which is
roughly parabolic at the bottom.
If the curve passes through a minimum then dy/dx is zero there and the
curvature at the minimum is simply d 2 y/dx 2 . Thus for a parabolic potential
well V = jtxx 2 , a is the curvature at the bottom of the well.
Finally we can say that any reasonable curve which goes through a
minimum is not very different from a parabola in the vicinity ^bf the
minimum, Fig. 3.11(b).
Gathering these results together, we can say that a system which has
a i^(r) curve of the usual interatomic type will come to rest in static
equilibrium at a separation a at the minimum. But if it is displaced
slightly, it will undergo simple harmonic oscillations of frequency
v =
2n
\d 2 y/dr\ =t
m
1/2
(3.19)
3.6.1 Einstein frequency
Imagine now a solid in which all the molecules are fixed at their equi
librium positions in the perfect crystal lattice, except one which is free to
vibrate. As a first approximation, dissect this one out of the lattice together
with two neighbours, one on either side, and assume that the vibration
takes place along the line joining them, Fig. 3.12. When the molecule
moves to the left it goes nearer to one neighbour and further away from
the other, To the approximation that the potential well near the minimum
is a symmetrical parabola, the change of potential energy is twice that due
to one neighbour. So, using the result of Eq. (3.15),
d 2 1T
dr 2
144e
a 2
(2 neighbours in line).
46 Interatomic potential energies Chap. 3
Fig. 3.12. Linear vibrations of a molecule with
two nearest neighbours.
To a better approximation, imagine that we dissect out the one molecule
surrounded by n nearest neighbours distributed uniformly over the
surface of a sphere : these neighbours are now at all angles to the direction
of vibration. The potential well is of depth ne and is a parabolic function of
radial distance r. But to calculate the change of potential energy with
displacement along a certain direction we have to average a factor cos 2 6
over all directions. It emerges that
—I = — ^ (n neighbours spherically disposed)
2 1 "1 si* 1
dr 2 J r=ao 3a 2 ,
like the previous expression but with the factor 3 coming from the aver
aging. Thus the frequency
1 /24ne
Ve 2ti\I mal
(3.20)
where m is the mass of one molecule. It is called the Einstein frequency.
This is a very rough estimate of the frequency of vibration, because of
course all the molecules are vibrating at once and the potential energy
of one molecule depends not only on its own position but on its neighbours'.
Whereas this analysis implies that all molecules are vibrating at a single
frequency, in a real solid many frequencies are present. But the order of
magnitude is significant. It is no coincidence that the Einstein frequency
and the bulk modulus both depend on (d 2 lT/dr 2 ) r=ao . The connection is
that the speed of sound is given by
//bulk modulus \ .. __
speed = / l>^l)
Y \ density /
and using Eqs. (3.16) and (3.20) it can be verified that the Einstein frequency
is the frequency of sound waves whose wavelength is about twice the
intermolecular spacing. A full understanding of this result depends how
ever on a study of the propagation of waves through lattices of points
rather than continuous media.
3.6.2 Estimation of Einstein frequency
From the data of 3.3. 1 and 3.4.2, the frequency of molecular vibrations in
both liquid nitrogen and carbon tetrachloride is of the order of 10 12 c/s.
3.7 Metals 47
This has no immediate interest for us though later we shall see that it has
important consequences for the thermal properties, notably the specific
heat, of these substances (5.4.4).
3.6.3 Experimental data for argon
Argon is a rare gas which liquefies at about the same temperature at
which air liquefies, 80° K. The molecule is a single spherical atom. Inter
atomic attractions are purely van der Waals forces. The solid has a close
packed structure. It is therefore an 'ideal' molecular crystal and has been
extensively studied down to very low temperatures. Experimental data
are given in Fig. 3.13; suggestions for analysing them are given in a
problem at the end of the chapter.
One interesting use to which these data can be put is to verify experi
mentally that the 612 potential is a good representation. If we use the
generalized pq form of the interatomic potential energy, Eq. (3.3), then
it can be verified that
d 2 T\ pqs
~Tir\ = ~\ (322)
dr lr=ao "o
Thus comparing the bulk modulus with the binding energy L per unit
volume, as in equation (3.16), the product pq can be measured. It will be
found to be about 64, which is surprisingly close to 72.
The sublimation energy has only been measured down to 70° K and
this makes exact extrapolation to 0°K difficult. However, specific heats
have been measured down to very low temperatures and we can then use
the following energy cycle to find L . This uses quantities which have not
yet been denned but is given here for completeness, (i) Start with the solid
at T = 0°K and vaporize it; the energy required is L . (ii) Heat the gas
to 83°K ; to present accuracy it is a perfect gas of specific heat f R, where
R is the gas constant = 8.31 J/deg. (iii) Condense the gas to solid at 83°K—
the energy released can be read off the graph, (iv) Cool to near 0°K— the
energy extracted, deduced from specific heat measurements at these low
temperatures, is 165 J/mol. The argon is now back in its initial condition
and from the energy balance L can be calculated.
3.7 METALS
Metals consist of arrays of positive ions permeated by an atmosphere
of free electrons (section 2.1.4). Each ion carries a positive charge and the
electrostatic coulomb repulsions between the array of like charges would
be enormous ; the electrons, all negatively charged, would also repel one
another equally strongly. Thus at first sight we would not expect metal
48
Interatomic potential energies Chap. 3
to cohere but to fly apart. But in fact any small region in the metal tends
to be electrically neutral and the electrons tend to concentrate between
the ions so that their negative charges cancel out or screen the effect of
the positive ions on one another.
10
'o
h.
E
o
<>0
50
100 0< 150
(a)
50 OK 80
(c)
Fig. 3.13. Data for Argon. Atomic weight = 40. Sources of data : (a) Densities
— Dobbs and Jones, Rept. Prog. Phys. 20, 516 (1957); Mathias, Onnes and
Crommelin, Leiden Comm. 131a (1912). (b) Latent heats— computed from den
sities and vapour pressure measurements of Clark, Din and Robb, Michels,
Wassenaar and Zwietering, Physica 17, 876 (1951). (c) Compressibility — Dobbs
and Jones, as above, (d) Surface tension — Stansfield, Proc. Phys. Soc. 72, 854
(1958).
3.7 Metals 49
The problem of calculating the potential energy of an ion in the lattice
is an extremely difficult one because the electrons are mobile and can
redistribute themselves if the mean distance between ions is changed.
Nevertheless we can make some general statements about the shape of
the interionic potential energy curve.
First, there must be a minimum in the curve because the metal coheres
and energy equal in magnitude to a binding energy is needed to evaporate
it. We also observe that a metal resists great compression and we interpret
this to mean that when the ions themselves begin to overlap the potential
energy increases very rapidly, in much the same way as in a molecular
solid. Thus the i^(r) curve must resemble Fig. 3.4 near the minimum and
at small values of r the curve must rise very steeply. As we have seen
(section 3.2) we can use almost any rapidly increasing algebraic function
of r to represent this.
Next we must discuss the i^{r) curve on the other side of the minimum.
A metal also resists being stretched or expanded so that the curve must
rise in a similar sort of way. Now when we dealt with molecular solids we
represented the potential energy of the van der Waals attractions between
neutral molecules by an r 6 law; there are sound theoretical reasons for
doing this. By contrast, there is no simple expression of this type to
represent the subtle interplay of attractions and repulsions between the
ions and the mobile electrons. Nevertheless the screening effect causes the
interionic forces to be of short range. Quite arbitrarily we will therefore
adopt an r~ 6 law for the potential energy at large r in metals also. The
justification for this procedure is that it allows us to reach results which
have the right order of magnitude so that, arguing in reverse, we can say
that there must be a fairly strong resemblance between the real i^(r)
curve in metals, and Fig. 3.4. We will therefore use the LennardJones
612 potential energy for metals also. But it must be clearly understood
that for metals, in contrast to molecular crystals, it is only a crude approxi
mation having no theoretical justification.
Some data for potassium and mercury are collected in the Table below.
The coordination number in metals is always high, about 10 or 12. From
the surface tension of the liquid (potassium at high temperature, mercury
at room temperature) and also from the latent heat the depth e of the
potential well can be calculated. The two estimates agree tolerably well.
The value of e is comparable with that for molecular solids.
The bulk modulus does not agree very well with the ratio SL /V
(eight times the binding energy divided by the molar volume ; see section
3.5.1). For potassium there is a discrepancy by a factor of 5 and for other
metals it can be 10. The failure of this rather sensitive test shows that a
612 potential is not a good representation of the potential for some
50 Interatomic potential energies Chap. 3
Potassium Mercury
Atomic weight 39 200
Density (g/cm 3 ) 0.86 14
Binding energy L (erg/g atom) 1 1 x 10 1 1 7.8 x 10 1 l
Surface tension of liquid (dyn/cm) 364 465
Bulk modulus (dyn/cm 2 ) 0.4 x 10 1 1 2.7 x 10 1 '
Atomic volume (cm 3 ) 45 14
Diameter of ion (A) 4.2 2.8
efromL (eV) 0.19 0.16
e from surface tension (eV) 0.14 0.09
8L /K, Eq. (3.16) (erg/cm 3 ) 2 x 10 1 * 4.5 x 10 1 1
Einstein frequency (c/s) ~ 10 ~ 10
metals. If instead of 6 and 12 for the indices we use p and q, Eq. (3.3), then
following through the calculation suggested in section 3.6.3 it is not
difficult to show that the compressibility should be given by pqL /9V
(SL /V is a particular case when pq = 72). Presumably the data for some
metals mean that the product pq is sometimes a good deal smaller than
72 ; the repulsions might vary more slowly than r" 12 or the attractive part
of the interatomic potential might be of longer range than r~ b .
3.7.1 Departures from Hooke's law
In section 3.5.1 we limited the consideration of the elastic moduli of a
solid to small departures from the position of minimum potential energy.
The changes of dimensions and the pressures acting on the solid were
both assumed to be small; under these conditions the deformation is
proportional to the pressure and the substance is said to obey Hooke's
law. Conditions like these are the ones usually encountered.
Now it must be emphasized that a solid may be perfectly elastic (in
the sense that the body regains its original shape and size when all the
forces are removed) and yet it may not obey Hooke's law. Indeed strictly
speaking for any finite deformation, Hooke's law cannot hold. If from
experiment the deformation is proportional to the pressure, this merely
means that the deformation has not been measured accurately enough.
We can calculate the relation between pressure and deformation
without imposing the condition that we are always near the minimum of
the potential energy curve. Recapitulating some of the equations of
section 3.5.1 : pressure P = (dE/dV) where E = \N\iV and TT is the
pair potential. Therefore P = ^Nn(di^/dV). Though it is not a very
good approximation, we will use the 612 potential for metals. It is
3.7 Metals 51
convenient to write it in the generalized form of Eq. (3.3) :
^HUt)"W
and then to rewrite it in terms of volumes. Putting V = Na% for the
initial volume and V = Nr 3 for the volume under pressure :
H(3H*)"
Substituting this in the equation for the pressure
'HSlHfflfiS)
We can write this more elegantly by using the fact that the compressibility
at small pressures is given by K = 4Nne/V . Therefore
Further, it is common practice to use the fractional change of volume
(V— v o)/K as a measure of the deformation; it is called the strain and is
denoted by s. Thus
v = TTS ^
Substituting this and expanding by the binomial theorem for small s,
P = K(s%s 2 +f s 3 ). (3.25)
This relation between pressure and strain should be obeyed by any solid
with a 612 potential — molecular solids like argon, as well as metals. But
metals have been extensively studied and can be prepared as specimens
capable of undergoing large deformations, and more data exist for them
than for any other class of material, so we will concentrate on them.
Our expression suggests plotting the ratio P/K as a function of strain s.
This is reasonable : the definition of bulk modulus, extrapolated naively,
implies that a pressure equal to K would reduce the volume by a factor e,
so we can take K as a unit of pressure. The curve is plotted in Fig. 3.14.
Positive pressures, which cause compression, are plotted downwards and
negative pressures (that is, tensions) are plotted upwards. Hooke's law
then appears as a straight line at 45°. As we shall see shortly, the small range
of strains (between ±4 %) covered by the graph encompasses an extremely
wide range of conditions, far outside any encountered in ordinary
engineering.
52
Interatomic potential energies Chap. 3
Fig. 3.14. Stress/strain curve predicted for a solid with a 612 potential (full
line). Hooke's law is the dotted line at 45°. Positive strains, to the right, are
extensions ; negative strains are compressions. Pressures plotted downwards ;
tensions upwards ; both measured in units of K. Crosses are measurements of
the elongation of an iron whisker (Brenner, J. Appl. Phys. 27, 1484 (1956)).
Compression measurements on iron by Bridgman {Proc. Am. Acad. Arts. Sci.
11, 187 (1949)) agree at s = and s = 0.017, marked by an open circle; at
intermediate stresses Bridgman gives the coefficient of s 2 as 6.1 where we have
4.5. Explosion waves show that for extremely large strains the points lie
above our curve ; this is suggested by the arrow. (Al'tschuler et al, Soviet Phys.
J.E.T.P.l, 606 {195S)).
The trend of the curve is reasonable. When a solid is compressed, the
repulsive forces come into play and a given pressure produces less strain
than predicted by Hooke's law. The opposite holds for stretching.
Results for iron are shown at a number of points. These represent a
range of techniques which is probably as wide as can be imagined. On
3.8 Ionic crystals 53
the left the compression measurements are taken from experiments in
which enormous pressures, of the order of 30,000 atmospheres, were
generated hydraulically using thickwalled vessels with ingeniously
designed pistons. These experiments showed that for small strains the
coefficient of s 2 is 6.1, where our simple theory gives 4.5, so that the points
lie below our curve ; but then they move upwards and at about s — — 0.017
the points lie on the predicted curve. Other experiments have also been
done at far higher pressures, right off the graph — experiments in which
high explosive was detonated on the face of a thick iron plate and the speed
of the shock wave determined. Measurements at s = — 0.2 and beyond
show that the points lie above our predicted curve ; the arrow attempts
to indicate this. This result may mean that the real repulsive force cannot
be represented by a simple power law like r~ 12 .
In contrast to these massive techniques, the other half of the curve
represents experiments performed under a microscope. It is a fact that if
iron or any other ordinary metal is strained beyond about 0.1%, it breaks.
But it is possible to prepare 'whiskers', that is thin threads of the metal
which — for reasons which will be described in section 9.4.2 — are by
comparison immensely strong. The one used in these experiments was
1.6 x 10 ~ 4 cm in diameter and a few millimetres long. It was stretched by
applying a force equal to the weight of about 10 g and the extension
measured. Of course, such an experiment measures Young's modulus, not
the bulk modulus, but as we have stated in section 3.5, these are almost
the same.
Our simple theory gives deviations from Hooke's law of the correct
sign, and correct magnitude within a factor of 2, which is satisfactory.
3.8 IONIC CRYSTALS
Lattices like those of sodium chloride and lithium chloride consist of
arrays of ions, each positively or negatively charged (Fig. 2.2). The forces
and potentials between two ions therefore consist now of three compo
nents : the repulsive part, the van der Waals part (exactly as for neutral
atoms) and in addition the electrostatic attraction or repulsion. This is
given by Coulomb's law. If each ion carries a charge of magnitude e the
potential energy = + e 2 /4ne r (like charges) or — e 2 /4ns r (unlike
charges). The energy is measured in joules if e is expressed in Coulombs,
r in metres and e , the permittivity of free space is given by 4ne = 10~ 9 /9
farads/metre. The signs follow the convention of Fig. 3.1, to express the
direction of the attraction or repulsion. We summarize this as
,^ e 2 f Like charges +
' (^Coulomb = i'
4ns r ) Unlike charges
54 Interatomic potential energies Chap. 3
Thus the total potential energy between two ions is
nr) = ~5±
r p r b 4ns r
where p is about 10 and k and n should be of the same order of magnitude
as for neutral atoms.
We will now show that the r 6 term, the van der Waals attraction, is
negligible compared with the coulomb potential and can be discarded.
If the charge cloud were spherically symmetric, the Coulomb force and
potential at an external point would be the same as if its charge were
concentrated at the centre. This is a consequence of the inverse square law.
Thus the potential energy between two ions cannot be very different from
that between an ion and an electron the same distance away. But we know
that to pull an electron out of an atom to make an ion, we have to do an
amount of work (equal to the ionization energy) of the order of 10 eV,
that is 10 11 erg. Therefore, since the diameter of an atom is of the same
order as the interionic spacing in crystals, the potential energy of two
ions must be of this order of magnitude. The van der Waals energies are of
the order of 0.1 eV, one hundred times smaller.
In Fig. 3.15 we have attempted to plot the potential energy of two ions
of like sign and unlike sign as a function of their separation. Also shown is
Xb)
Fig. 3.15. (a) The potential energy of two ions of unlike charge as a function of the
distance r between their centres. The dashed curve is the potential energy of two
neutral atoms, Fig. 3.4, plotted on the same scale, (b) The potential energy of two
ions of like charge, which is repulsive at all distances.
3.8 Ionic crystals 55
the potential energy of two neutral atoms, to the same scale. This em
phasizes in an obvious way that van der Waals energies can be neglected
for ions. We can therefore write for two ions :
y/ ~, A e 2 f Like charges +
^■^ Unlike charges . <326)
3.8.1 The binding energy of sodium chloride. The Madelung sum.
Figure 3.15 also emphasizes another point : that the Coulomb potential
is long range (section 3.2). It is therefore no longer justifiable to deal
only with nearest neighbour interactions. The Coulomb potential due to
an ion can be felt far into the lattice, and another method of calculating
binding energies is needed.
To start with, consider the Coulomb energy only of a line of ions, each of
charge e alternately positive and negative as in Fig. 3.16, which has been
dissected out of a lattice like that in Fig. 2.2.
Let the interionic spacing be r. Then the energy of one ion due to its
two neighbours (which necessarily have the opposite sign to it) is
— 2e 2 /4ne r. The next nearest neighbours, distance 2r away and necessarily
having the same sign as the ion considered, give potential energy
4 2e 2 /4ns 2r ; and so on.
Fig. 3.16. A line of ions dissected out of an ionic lattice
Thus the potential energy of the single ion in the infinitely long line is
2e 2
4ns r
(li+W+"0.
Note that
so that
log e (l + x) = xy+y
li+ii = log c 2 = 0.69.
Therefore the potential energy of the single ion in the line is  l.3$e 2 /4ne r.
This holds for any ion, positive or negative, anywhere in the line.
56 Interatomic potential energies Chap. 3
The constant which we have just worked out is called the Madelung
constant a for a line of ions. The Madelung constant for the three
dimensional sodium chloride lattice has been calculated: it is 1.75, which
is not very different from our 1.38. Indeed, all Madelung sums for simple
lattices of ions of alternate sign are of the order of unity. They must be
so, since the effect of a positive ion is to some extent cancelled out by the
next nearest negative ion and so on. Thus although the Coulomb potential
is long range, our calculations are greatly simplified by being able to say
that the potential energy of an ion in a lattice is given by the energy of a
nearest neighbour pair times the Madelung constant which is of the order
of unity.
So far we have dealt with the Coulomb part of the potential. When it
comes to the short range repulsions, the k/r p term in which p is about 10,
we may guess that only nearest neighbours need be counted. The repulsive
potential energy of a single ion is therefore nX/r p , where n is the coordina
tion number.
The potential energy of a pair of ions in the crystal can therefore be
conveniently written
ir =
4ne o a
!0 _ «0
p\rj \r
(3.27)
This is of the pq type of Eq. (3.3) with q = 1. By comparison, the depth
of the well
\ pj4ns a
In one gram ion of sodium chloride, there are N Na + ions and N Cl~
ions, that is there are N pairs of ions. Thus when the interatomic spacing
is r, the energy is NV per gram ion; when it is a , the energy is —Ne.
This gives the binding energy of the substance.
Now the index p of the repulsive potential is about 10. So to about 10%
accuracy, we may say that the binding energy of an ionic crystal is equal
to e 2 /4ne a the potential energy of one pair of adjacent ions times the
Madelung constant times Avogadro's number. The binding energy is
dominated by the Coulomb energy.
The binding energy for sodium chloride has been determined as
763 kJ/mol. The interionic spacing has been determined by Xray analysis
as 2.8 A that is 2.8 x 10 ~ 10 m. The charge e, on each ion, is one electron
charge, 1.6 x 10" 19 C. Therefore
Note 2
= 860 k J.
4ne a
Problems 57
If we diminish this by 10% to allow for the factor (1  1/p), the binding
energy agrees extremely well.
3.8.2 Elasticity of ionic crystals
We can calculate the bulk modulus of an ionic crystal using the same
method as for molecular crystals. The volume occupied by N pairs of
positive and negative ions is
V = 2Nal
and equation (3.13) becomes
{d 2 E/dr 2 ) r=ao = N(d 2 r/dr 2 ) r = ao
lSNa 18Na
where 'V is the pair potential energy. The easiest way to evaluate the
second differential is to quote the result of Eq. (3.22), that it is equal to
pqe/al, with q = 1. This leads straightforwardly to the result
PL
K =
9V
where L is the binding energy and V the molar volume. This is a special
case of the relation pointed out in 3.6.3, namely that if the indices in the
interatomic potential are p and q, the bulk modulus depends on the
product (pq); here q — 1, so we can measure p directly.
For sodium chloride, the molecular weight is 58.5 and the density
2.18 g/cm 3 so that the molar volume V is 27 cm 3 . The binding energy
is 7.6 x 10 12 erg/mol, and the measured bulk modulus at low temperature
is 3.0 x 10 1 1 dyn/cm 2 . Substituting, p is found to be 9.4, which is reasonable.
The speed of sound, calculated from the bulk modulus and the density,
is about 4 x 10 5 cm/s. The Einstein frequency v E , corresponding to a wave
length of about twice the interionic spacing, is therefore about 5 x 10 13 c/s.
Since the propagation of such a wave means that ions (charges) are moving
it can lead to the absorption of electromagnetic waves of this frequency.
This frequency lies in the infrared.
PROBLEMS
3.1. For gravitational forces, which obey an inverse square law, a sphere of large
radius behaves as if it were a point located at the centre. In this question, the
object is to link up the definition of potential energy V with that of gravitational
potential energy as usually defined in elementary treatments.
Write Newton's constant as G, the mass of the Earth M, the radius of the
Earth a.
(a) Calculate the force on a mass m at a distance r from the centre of the Earth
(r > a). Take r radially outward, note that r can only be positive. Get the
sign of this force correct.
58 Interatomic potential energies Chap. 3
(b) Calculate the potential energy, taking the value at r = oo to be zero.
(c) Draw a graph of this function between r = a and r = oo.
(d) Calculate the force on m at the surface of the Earth and by equating it to the
weight mg, get an expression for g at the surface.
(e) Calculate the potential energy at r = (a + h) where h is small compared with
a, so that the square of h/a can be neglected. Show that the potential energy
at (a+h) is greater than that at a by an amount mgh. Mark this increase
clearly on your graph.
3.2. Two small magnets are arranged as shown. The lower one is fixed, the upper
one is restrained from moving horizontally, but is free to move vertically.
^
77777777777777777
The force between them is a repulsion of magnitude (2/x M 2 /h 4 ) where M is
the magnetic moment of each, p, the permeability of vacuum and h is the
distance apart of the two magnets.
(a) Write down the potential energy due to the repulsion.
(b) Draw a graph of this as a function of h.
(c) Draw a graph of the potential energy of the upper magnet as a function of h,
due to the Earth's gravity.
(d) Draw a graph of the total potential energy as a function of h.
(e) Calculate h for static equilibrium and indicate this point on the graph.
(/) Using the methods of section 3.6, calculate the period of oscillation after
the upper magnet is given a small displacement downwards and then
released.
3.3. A sinusoidal wave y = a sin 2%x/A, of wavelength A and amplitude a disturbs
the surface of a liquid of density p and surface tension y. The problem is to
calculate the frequency and speed of propagation; see section 3.4.4 and Fig. 3.9.
(a) Calculate the potential energy of a whole wavelength, due to the up and
down displacements of the weights of the two halves, as follows. Show that
the displaced mass of the half wavelength between x = and x = A/2 is
palA/n, where / is the width of the wavefront. Prove that its centre of gravity
is at height na/S and hence that the potential energy of a whole wavelength
is \pa 2 lAg. Note : J sin 2 a da — ^{asin a cos a).
(b) Write down an integral expressing the length of perimeter of a whole
wavelength of the sinewave. Assume a/A is small, expand the integrand by
the binomial theorem and integrate it. Show that the increase of surface
energy is n 2 a 2 ly/A. The total potential energy is the sum of this and (a);
note the proportionality to a 2 .
(c) Assume (from comparison with a complete analysis) that the penetration
depth is effectively A/47i so that the mass in motion is effectively A 2 pl/4n.
Problems 59
Hence show that the frequency is
v = 1 / A g 2n y
A\J In pA '
(d) The (phase) velocity is vA. Sketch it as a function of A.
(e) For waves with A = 1 mm on liquid argon at 100°K, show that surface
tension contributes 30 times as much to the energy as the weight. Calculate
the frequency and velocity. Design an experimental setup to measure the
surface tension.
3.4. In a rough demonstration experiment, liquid nitrogen was contained in a small
thinwalled, spherical glass dewar with a narrow neck. This was connected to
a gasmeter so as to measure the volume of nitrogen boiled off. The meter and
the gas passing through it were at room temperature, 20°C. While 25 g of liquid
evaporated, the meter registered 0.76 cu ft of gas (1 cu ft = 28.3 litres). What is
the molecular weight of nitrogen? The dewar was known to hold 122 cc up to
a mark on the narrow neck. It was estimated that about 1 cc was always occupied
by bubbles. The flask was weighed empty and full up to the mark ; the liquid
weighed 98 g. What is the density? the molar volume? the diameter of a molecule?
A capillary tube of internal diameter 0.55 mm was dipped into the liquid. The
capillary rise was 8mm. What is the surface tension, and depth e of potential well?
3.5. Calculate some of the atomic constants of argon using the data of Fig. 3.13;
see section 3.6.3.
(a) From the density, Fig. 3.13(a), calculate the molar volume V and hence the
diameter a of an argon atom, assuming each atom to be a little cube.
(b) If the details of the argon crystal lattice are taken into account, see section
8.1.2, it can be shown that the molar volume is not Na% but Nal/y/l.
Calculate a better value of a Q than in (a) above.
(c) Estimate the depth e of the potential well from the latent heat data of
Fig. 3.13(6), taking the heat of sublimation at 70°K to be a sufficiently good
measure of L (Eq. (3.6)).
(d) Extrapolate the heat of sublimation to T = 0°K using the energy cycle
described in section 3.6.3 and calculate a better value of e than in (c) above.
(e) Estimate e from the surface tension data of Fig. 3.13(d), extrapolating y to
T = 0°K.
(/) Estimate the bulk modulus K at T = 0°K from the compressibility graph,
Fig. 3.13(c). Compare this value with 8L /% (Eq. (3.16)).
(g) Check Eq. (3.22). (Refer if necessary to section 8.4.1.) Hence show that if the
indices of the interatomic potential energy are p and q instead of 6 and 12,
the bulk modulus is given by
Use your values of K, L and V to estimate the value of pq.
(h) Estimate the Einstein frequency v E (Eq. (3.20)).
3.6. The molecules of a complicated organic molecule can be considered to be
discshaped, of radius rem and thickness r/lOcm. When two molecules are
close together face to face (like two pennies one on top of the other) an energy
E is required to separate them; when they are placed end to end (like two
pennies touching one another, edge to edge) an energy JE/30 is required. For
these molecules, which are far from spherically symmetrical, it is no use quoting
formulae for L or y (Eqs. (3.6), (3.7)) for spherical molecules ; it is necessary to
imagine how these molecules can be packed together and to tackle the problem
ab initio.
60 Interatomic potential energies Chap. 3
(a) Estimate the molar heat of evaporation.
(b) Estimate the surface tension.
(c) Near the surface of a solid or liquid, will the molecules tend to be oriented
(0 randomly, {if} with their planes parallel to the surface, (Hi) with their planes
normal to the surface? Give reasons.
3.7. The crystal of sodium fluoride is a cubic structure in which alternate sites are
occupied by Na + and F" ions, each ion having 6 nearest neighbours. The
Madelung constant is 1.75 when referred to the smallest Na + F~ separation,
which will be denoted by a . The heat of formation of the crystal from its
constituent ions is 900 kJ per mole of sodium fluoride. The density of crystalline
sodium fluoride is 2.9 g cm 3 , and the atomic weights of sodium and fluorine
are 23 and 19. Estimate a and Avogadro's number from the data. State clearly
any approximations you are using in your calculations.
3.8. The interaction between ions in sodium chloride can be described by their
Coulomb interaction, plus a repulsive potential energy Aexp(rfp) acting
between nearest neighbours only. (This exponential term is used in place of the
r" 12 term; A and p are constants). Obtain an expression for the lattice energy
in terms of the nearestneighbour separation a and the Madelung constant
a for sodium chloride. Given a = 2.8 A, a = 1.75, and that the lattice energy
is 763 kJ/mol, find p.
3.9. In a medium of dielectric constant K, the potential energy of two charges e x
and e 2 separated by distance r is e 1 e 2 /4n£ Kr.
Water H 2 0, ethyl alcohol C 2 H 5 OH and ammonia NH 3 have dielectric
constants of 80, 25 and 18 respectively at room temperature. These high values
are due to the fact that the molecules are electric dipoles, with regions of + and
— charge, which can be easily aligned by external fields ; they can crowd round
a charged particle with their oppositelycharged ends all pointing towards it
and so screen the particle from its neighbours. H 2 and NH 3 are compact mole
cules, C 2 H 5 OH is relatively large.
(a) Use the data for the binding energy of sodium chloride, section 3.8.1, to
calculate the binding energy of a single sodium chloride molecule.
(b) Explain qualitatively the fact that the solubility of sodium chloride in 100 g
of solvent is 37 g in water, 0.07 g in alcohol, 3 g in liquid ammonia ; the
dissolved sodium chloride exists as Na + and Cl~~ ions.
3.10. Calculate the Madelung constant for a line of dipoles, (a) all aligned in the same
direction so that they repel one another with anr" 4 force, and (b) alternately
parallel and antiparallel so that nearest neighbours attract, next nearest neigh
bours repel and so on. Evaluate the series numerically or by looking up Riemann
zeta functions in Dwight's 'Mathematical Tables' or 'Tables of Integrals'.
3.11. The molecular weight M, latent heat of evaporation L, and surface tension
y at 20°C, of certain liquids are given in the table. Make a crude estimate of the
molecular diameters in each case stating what approximations are made.
M
LJ/gm
y dyn/cm
Alcohol
46
856
22
Benzene
78
389
29
Mercury
200
272
475
Water
18
2,250
73
Problems 61
3.12. (i) Prove that for a substance whose interatomic potential energy is given by
Eq. (3.3), the relation between pressure P (applied hydrostatically) and
volume v is
where K is the bulk modulus at very small strains, v is the initial volume,
and a and /? are equal to (p+3)/3 and (q + 3)/3 respectively. Hence show
that for small strains s less than about 0.1
a ,E+p±* + ...\.
(ii) For an ionic crystal, p is about 11 and q = 1. Compare this simple theory
with measurements by Bridgman on the compression of sodium chloride.
p
Kg/cm 2
s
P
Kg/cm 2
s
lxlO 4
2
3
0.038
0.068
0.093
4xl0 4
6
8
10
0.115
0.152
0.183
0.210
One method of deducing the bulk modulus for very small strains is to note
that on almost any theory P = —K(s — ys 2 ) where y is a constant, so that
a graph of P/s against s should be a straight line from which K can be
deduced. Use the measurements in the first table for this.
(Hi) Convert the data of the next table into (P/K) against (v /v) raised to the
appropriate powers, compare calculated and measured pressures and
comment on the results.
3.13. What conclusions can you draw about the natural vibrational frequencies of
diamond, iron and lead from the following data (Y, M and p are respectively:
Young's modulus in dyn/cm 2 , the atomic weight, and the density in g/cm 3 ) :
M
Diamond 8.4 xlO 12 12 3.5
Iron 2.0 xlO 12 56 7.9
Lead 0.18 xlO 12 208 11.4
CHAPTER
Energy, temperature and the
Boltzmann distribution
4.1 HEAT AND ENERGY
We will take it for granted that heat is a form of energy. This statement
is based on the experiments of Joule, the majority of which were of the
same basic pattern. Weights held on strings could descend and so provide
mechanical energy to drive a mechanism whose motion was resisted by
friction of some kind and which grew hotter as it was driven. The change
in mechanical energy was measured by the loss in potential energy of the
weights, and the quantity of heat produced was measured in terms of the
rise in temperature of the apparatus and its heat capacity. The mechanisms
were very varied : a dynamo dissipating its energy in a resistance, a per
forated piston moving through viscous liquids, a conical bearing with
friction between the rubbing surfaces, a system of paddles churning viscous
liquids. In another investigation, air was compressed by a pump into a
cylinder and the temperature rise was measured — here no frictional force
was encountered during the compression of the gas, but work had to be
done against the pressure it exerted. In all these experiments, the conver
sion factor relating the energy absorbed by the mechanism and the heat
produced in it was the same within rough limits, ± 15 %. Since the mech
anisms were so diverse in type, it was unreasonable to suggest that this
rough constancy could be a property of the substances or devices employed ;
it could only be explained if heat and energy were physically identical.
4.1 Heat and energy 63
4.1.1 Ordered and random movements of molecules
It has already been mentioned that the molecules of any substance are
in ceaseless, rapid motion. The molecules of a gas move in straight
trajectories till they collide, the molecules of a solid are in vibration about
their mean positions, those in a liquid vibrate and also slip through the
holes in the structure. These motions are random, in the sense that
movement in one direction is just as likely as movement in any other, and
also in the sense that any molecule changes its speed many times per
second so that if we were to follow all the details of the motion we would
find that the kinetic energy went irregularly through all possible values,
from zero up to some large value.
Consider a body which is big enough to contain a large number of
molecules (though it might be of dimensions which are small on the
ordinary scale) and let it be at rest. Then the total momentum of the
molecules must be zero. If, however, we examined the momentum of one
single molecule at any instant, we would almost certainly find it to be large.
It is only by finding the vector sum of all the momenta or finding the
average momentum of a very large number of molecules at any instant, or
alternatively by finding the average momentum of one molecule over a
long period of time, that we can come to any conclusion about the move
ment of the body as a whole. In the same way, we can consider a region of
a body which is moving with a certain velocity. If we were to determine the
instantaneous velocity of any one molecule we might well find that it was
moving very fast in the opposite direction to the bulk motion, and we could
deduce nothing about that bulk motion. But if we averaged the momentum
of a large number of molecules in the region at one particular instant, or
averaged the momentum of one molecule over a long time, we would detect
a nett momentum, corresponding to the drifting of the body as a whole.
It is important to point out that the statements just made are in fact a
little too dogmatic. If we found the vector sum of the momenta of all the
molecules in a body, it would be unlikely to be exactly zero. It would fluctu
ate about the value zero. For the averaging to have any physical signifi
cance, these fluctuations must be relatively small. We will see later (in section
7.7.2) that this condition is satisfied if the number of particles in the assem
bly is large, and it will be assumed in the rest of this chapter that this is so.
We are thus led to distinguish between the random movements of the
molecules of a body (which add up vectorially, and hence average out
to zero) and the movement of the body as a whole. The random move
ments are superimposed on the bulk or ordered movement, and the two
can only be separated by averaging. An example of ordered motion is the
macroscopic movement of any body such as a ball. A less obvious example
is the flow of a liquid, either streamlined or turbulent, where a bulk velocity
64 Energy, temperature and the Boltzmann distribution Chap. 4
can be defined over each small region of the fluid. Another example is
provided by waves of compression or rarefaction passing through any
medium, or torsional waves through a solid, when the velocity and dis
placement due to the wave can be defined. The passage of an electro
magnetic wave through a transparent medium may also cause the atoms
to vibrate in an ordered way. But in all these examples, the ordered move
ments have the random movements of the molecules superimposed on them.
4.1.2 Temperature and random motion
We can now extend the statement that heat is a form of energy and
make it more precise, as follows : Temperature is a measure of the energy
of the random motion of the molecules of a substance. (Temperature is
not the only measure of this energy, but we will not pursue this fact here.)
This statement is ultimately based on Joule's experiments. Later in this
chapter we will show how the form of the relation can be deduced.
A molecule in general possesses both kinetic and potential energy.
Sometimes the kinetic energy dominates the situation, as in a gas of low
density where the potential energy arising from close collisions can often
be neglected. On the other hand, a solid owes its regularity of structure
to the potential energy of interaction of an ion with its neighbours and as
the kinetic energy merely leads to vibrations of the ions about their mean
positions, the random variation of the potential energy is important.
It must be emphasized that it is only the randomly varying contribution
to the energy which is related to the temperature. Any ordered movements
must be transformed away by a suitable choice of coordinate system.
However, it is possible for ordered molecular movements to be converted
into random ones. In fact, this is just what happens whenever mechanical
energy is converted into heat. In Joule's experiments where liquids were
stirred, the movements of the weights were converted by the mechanisms
into bulk movements of the liquids — the surface waves, the eddies. But
all the time, part of the energy of the ordered motion of the liquids was
being converted or degraded by molecular collisions into random move
ments of the molecules and this caused the rise in temperature. In his
experiments where gases were compressed by a piston, the ordered motion
of the layer of gas being pushed back by the movement of the piston
was transferred to a random movement of molecules throughout the gas
so that the energy of the piston caused a rise in temperature. At the same
time, the molecules of the piston were also heated.
4.1.3 Degradation of ordered into random motion
To see how this conversion or degradation is carried out, we consider
first a simple example. Let a single molecule be projected into a layer of
4.1 Heat and energy
65
gas (Fig. 4.1), with a velocity much higher than the random velocities of
the molecules of the gas. In principle, we can imagine that this energy
could be measured by some simple mechanical device, since only one
particle is involved. The molecule must soon undergo a collision in which
the laws of conservation of energy and momentum are obeyed. A propor
tion of its energy is transferred to the other molecule, both are deflected
and the first continues at a slightly lower velocity. It undergoes a large
number of such collisions, each time transferring some of its energy and
deflecting the molecules of the gas till finally it emerges with a smaller
kinetic energy than it started with ; in principle this loss of energy can be
measured by mechanical means. Each molecule struck undergoes further
collisions, at different glancing angles, with changes of direction each time,
so that after quite a short time the excess of energy is carried to distant
parts of the gas and is shared between all the molecules as extra kinetic
energy of motion in all directions. The energy of the molecules is increased
by an amount equal to the difference between E and £', but in macro
scopic terms we merely say that the temperature has increased.
O
O
o
o
o
o
o
o
o
o
o
\o
o
o
o
o
o
o
Fig. 4.1. Passage of a single fast molecule of initial energy E through a layer
of gas. It emerges with a lower energy E'.
We can now consider a gas contained inside a cylinder fitted with a
piston (Fig. 4.2). Let the area of cross section be A and the pressure of the
gas P, and imagine the piston to be moved inwards a distance dx. This
requires the expenditure of an amount of work PA • dx. But each molecule
66 Energy, temperature and the Boltzmann distribution Chap. 4
near the piston acquires an excess velocity and now plays the role of the
single incident molecule in Fig. 4.1. Once again, the excess energy supplied
by the moving piston is, after a large number of collisions, shared between
all the molecules as a purely random movement, as a rise in temperature.
Area/4
Fig. 4.2. Gas contained in a cylinder with
piston. When the piston moves the ordered
motion of the molecules near it is ran
domized by collisions.
If a solid is struck, the energy of deformation will at first probably be in
the form of a compression wave travelling through it, so that it rings with
a frequency of a few hundred or thousand cycles per second. This highly
ordered motion can also be degraded into random motion, though the
process is more complicated than in gases, but the final result is that the
energy of the original disturbance is degraded into vibrations in all direc
tions and with all frequencies. Each vibrating molecule has both kinetic
and potential energy at any instant (the potential energy between any pair
of molecules being given by Fig. 3.4). Each of them separately varies
randomly with time, and so does the sum of the two, but it is this total
energy which we use to measure the temperature. If liquids are stirred, the
molecules can'vibrate and also change their positions by slipping past one
another. Again some of the energy of the ordered movement can be de
graded into increases of random molecular movement constituting rises
in temperature.
In all these examples, where work is done on a system and energy is
apparently lost but is in fact divided up among a large number of particles,
a thermometer would indicate a rise in temperature. Temperature is one
of the measures of this randomized energy.
4.1.4 Macroscopic variables and statistical specifications
In practice, measuring instruments which we use for ordinary kinds of
physical measurements are relatively massive, slowlyresponding devices
which are incapable of detecting the effect of individual molecules. Thus
when we measure the pressure of a gas using an ordinary sort of pressure
gauge, a mercury manometer or a diaphragm actuating a lever, the gauge
4.2 Concepts of probability theory— I. Probability functions 67
measures a timeaverage of the pressure. It does not register the individual
impacts of the molecules, which occur with extremely high frequency.
Similarly, when we measure the density of a gas we normally use a volume
whose dimensions are very large compared with the distances between
molecules. Figure 2.3, which pictures the molecules in a gas unevenly
distributed throughout a volume, implies that the density fluctuates from
place to place, but these fluctuations are never detected by a measurement
using a large volume. Thermometers, similarly, measure timeaverages of
molecular energies. It is in fact possible to detect fluctuations in pressure, or
density or energy under the right conditions using appropriate detectors,
but for the moment we will restrict our considerations to ordinary instru
ments which measure average values of molecular quantities.
The contrast between the complexity of the situation from the atomic
point of view, and the simplicity of the largescale measurements is a
profound one. The one requires a knowledge of say 10 23 positions and
velocities, the other a single dial reading. It is our task to relate these points
of view and it is suggested at once that the elaborate specification of all
the positions and velocities of the constituent molecules is not only
impossibly complicated, but is actually irrelevant.
We need only concern ourselves with those features of the assembly
of molecules which allow us to calculate averages of molecular quantities.
It turns out that the most we ever need to specify is the fraction of the total
number of molecules to be found within certain limits of position or having
speeds within a certain range. We need to know, for example, that 1 %
of the molecules of a liquid of molecular weight 30 at 300°K have speeds
between 10,000 and 10,320 cm/s ; it is not necessary to enumerate which
molecules they are at any instant nor exactly where they are to be found.
Specifications of this kind are all that is necessary to relate the macroscopic
properties to the motions of the molecules. One often meets rather, similar
statements about, say, the economic state of a nation. For many purposes,
it is sufficient to enumerate what percentage of the population has earning
power within certain limits; the specification of the individuals does not
matter. Statements of this kind are statistical, in the sense that they deal
with percentages or probabilities but do not specify individuals. We
must therefore study some of the concepts and theorems of probability
theory.
4.2 CONCEPTS OF PROBABILITY THEORY— I. PROBABILITY
FUNCTIONS
We will study a statistical problem which has nothing to do with physics.
Consider a large population of people, several million in number, out of
68 Energy, temperature and the Boltzmann distribution Chap. 4
which we select a small sample of 100. Imagine that the height of each of
these individuals has been measured. One good way of displaying the infor
mation is to plot a histogram which we construct as follows. We divide the
total possible range of heights into convenient intervals or ranges (say
2 cm wide), choosing them so that there is no ambiguity about classifying
an individual whose height is at the end of a range. (For example, if the
accuracy of measurement were .01 cm, we might choose ranges 160
161.99 cm, 162163.99 cm and so on.) Then we count up the number of
individuals in each range and plot them as a graph. The result might look
like Fig. 4.3(a). The number in each range is drawn as a horizontal line
covering the interval. From this diagram we can at once see that the
population contains many members around 175 cm tall, and that very
short and very tall members are rare.
The histogram is not a smooth curve. Of course it consists of a series of
steps because we are dealing with intervals of finite width — but sometimes
the steps go up when they might be expected to go down. Intuitively one
expects that the distribution of heights should be a smoothly varying
function, but it is more or less obvious that the irregularities are present
because the number of individuals in any range is small. For example, if
the 100 people had by chance included only one more member in the
range 166168 cm and one less in the next lower range, then one of the
dips would have disappeared. It will be shown in section 7.7.3 that in
many circumstances, if the number of individuals in any class is most
probably n, then quite probably the number might lie in the range between
(n + Jn) and (nJn). For example, if the number of individuals might be
expected to be 10, then (using different samples from the same population)
we would probably find counts anywhere between about 7 and 13. If we
expected 100 in any range, then counts between 90 and 110 would be
common, and so on. The numbers are said to fluctuate between certain
limits. Now it is obvious that as n becomes larger, Jn becomes larger too,
but more slowly ; so the fluctuations become proportionally smaller, the
larger the numbers are. For example, with 10 members in a range, the
expected fluctuation is about ±30%; with 100 members it is ±10%;
with 10 6 members it would be only ±0.1 % and so on.
We will in future assume that the sample of the population which we
take is very large— so large that the number in any range is itself so large
that we can neglect fluctuations. We will then get a smoothly stepped
histogram like Fig. 43(b), which refers to a sample of 10 6 people.
The number in any range depends of course on the total number in the
sample, being proportional to it. For many purposes (for example, com
paring samples of different size) this is inconvenient and it is better to refer
to some standard number for the total sample. The best choice is 1.
4.2 Concepts of probability theory — I. Probability functions
69
To convert Fig. 43(b) to this standard, all the ordinates of Fig. 4.3(b)
have simply to be divided by 10 6 . If we take the ordinates of all the steps
in Fig. 43(b) they add up to 10 6 ; in Fig. 4.3(c) they add up to 1. The histo
gram is said to have been normalized to a total population of 1.
15
 10
0)
f 5
'Ha)
A
■J* k
160 170 180 190
Height h
0.15
0.10
0.05
160 170 180 190 h
160 170 180 1'
160 170 180 190 h
Fig. 4.3. (a) Typical histogram showing distribution of heights of 100 people.
(b) Using scale at left : Expected distribution with 1 million people, (c) Using
scale at right : Probability of a person's height being within 2 cm range of h.
(d) Using scale at left : Probability of a person's height being within \ cm
range of h. (e) Using scale at right : Same data, but referring to 1 cm ranges oih.
(f) Probability function P[h], identical with (e) but using infinitesimally small
steps.
These ordinates are now called probabilities. From Fig. 43(b) we can
read that if we take 10 6 individuals, there would probably be 80,000 whose
height lay between 168 and 170 cm. But we can extend this statement by
referring to Fig. 4.3(c) and saying that if out of the population we selected
one single individual at random, the probability that his height lay between
these limits would be 0.08. These two statements are equivalent to each
other.
This notion of probability is replete with philosophical difficulties
which must be faced if a full understanding of many branches of physics is
70 Energy, temperature and the Boltzmann distribution Chap. 4
to be reached. But from our present point of view we can regard the
probabilities of Fig. 4.3(c) as being merely a convenient shorthand for
deriving the numbers of Fig. 4.3(b); the number in any range is the proba
bility of being within that range multiplied by the total number in the
sample.
We can now ask : what happens to the probability histograms if we
reduce the size of the intervals — if for example we choose ranges of width
\ cm instead of 2 cm. First, the steps now get narrower so that the histo
gram approximates better to a smooth curve. Secondly, the probabilities
of lying within each range must now decrease, being proportional to the
width of the ranges. (We must emphasize that the size of sample from which
the histogram is derived must be so large that the numbers in any range
still remain large or else the large fluctuations will reappear.) The curve
for \ cm intervals is given in Fig. 4.3(d). But we can go further. It is possible
to plot just the same results in a more useful way, so that the ordinates
are independent of the width of the ranges. We can still do the counting
and work out the data at \ cm intervals, but we work out the probability
of finding an individual within ranges of unit width— conventionally,
of width 1 cm. Here, one has to multiply the probabilities of Fig. 43(d)
by 2, because the standard ranges are twice as wide as the one chosen in the
counting. We then get Fig. 4.3(e). Of course we could get practically the
same figures by taking Fig. 4.3(c) and dividing those ordinates by 2, because
the standard range is only half as wide as the 2 cm ranges used there ; this
would merely give a more crudely stepped histogram. Finally, we can
take the limiting case of imagining the intervals to be infinitesimally wide
but still calculating the probabilities for unit range of heights, Fig. 4.3(/).
The only difference between this and Fig. 4.3(e) is that it is a smooth curve.
It is called 'the probability function of h\ a phrase which is written P[h].*
The equation for the curve of Fig. 4.3(/) is
P[h] = 0.067905 exp[(/i 175) 2 /69.031]. (4.1)
The value of this function at h = 185 is 0.01597 ; thus the probability that a
member of the population has a height between 184.9 and 185.1 cm is
P[h] dh = 0.003194. If there were 10,000 people in a sample, the number
of people whose heights lay between these limits would most likely be 32.
The properties of the probability function P[h] may be summarized
as follows. The probability that an individual has a height within the range
dh about h is P[h] dh. In a population of n people, the number with
* This symbol must not be thought to imply the same analytical function of the variable
every time. For example, we will later meet two probability functions P[u] and P[c] where
u and c are respectively a velocity component and the total speed of a molecule ; the two are
different in analytical form from one another.
4.2 Concepts of probability theory — I. Probability functions 71
heights within these limits is nP[h] dh. Finally, the fact that the graph has
been normalized to a total population of 1 means that by convention
f
Jo
P[h] dh = 1 (4.2)
that is, the area under the graph is unity. In ordinary language, a proba
bility of unity is called certainty, and this equation says that it is certain
that the height of any individual lies between zero and infinity, which is
correct.
One further point must be mentioned. In many situations where we
consider the statistical distributions of the properties of molecules, we
can take two quite different points of view. The first is similar to the one
we have just considered. We can look at the assembly of molecules at
any one instant and find the distribution of the relevant quantities (position
or momentum coordinates) over them, in just the same way as we did for
the heights in a population. But the second procedure has no analogue
in the counting of people. It is to follow a single molecule for a long time ;
its coordinates change continually, and we will find that they take on all
possible values. For example, the speed of a molecule may be high or low,
in any direction, at one time or another. We can then, in principle, find
the fraction of the total time that the molecule spends in a given state.
It is a fundamental assumption that these two kinds of distribution are
identical. Thus if 1 % of the molecules of a certain liquid have speeds
between 10,000 and 10,320 cm/sec (at any one instant), then one single
molecule will, for 1 % of its time, be travelling with a speed between the
same limits. When we come to take averages, we will in other words
assume that sample averages are identical with time averages.
4.2.1 Mean values
The mean value of the height of an individual in a population is defined
by
, . , , total height of all members
mean height h = —
total number of members
_ sum of terms : (height h) x (number with he ight h, dh)
total number of members
that is :
C hP[K\ dh r
J n = J [/l] dH (43a)
72 Energy, temperature and the Boltzmann distribution Chap. 4
where the integration must be carried out over all possible values of h.
For the population of Fig. 4.3, the mean height is
/•oo
h = 0.067905 h exp[  (h  1 75) 2 /69.03 1] dh,
J — 00
where the range of h has to be taken from  oo to oo even though that is
unrealistic in practice, the integrand being practically zero over most of
the range.
For finding mean values, both in this example and for the functions
which will occur later in physical problems, the following integrals will
be found useful :
j V " 2d * = ^ J"
V a
» oo j
xe~ a * 2 dx = —
Jo 2a
/» 00
xe~ a * 2 dx =
v — oo
jVe—d*^
j; b a d,i^
/•OO 1
x 3 e" ax2 dx = x i 2
Jo 2a 2
/•oo
x 3 e _ax2 dx =
•/ — oo
With these integrals it is not difficult to prove that the mean height in the
above example is 175 cm.
In general, the mean value of a quantity x is given by
fxP[x]dx. (4.3b)
In the same way we can also find the mean value of any power of x.
Later it will be seen that mean square values of certain quantities are
significant :
= fx 2 P[x]dx. (4.3c)
This is not in general identical with x 2 ; there is no reason why Jx 2 P[x] dx
should be equal to (j"xP[x] dx) 2 .
4.2 Concepts of probability theory — I. Probability functions 73
4.2.2 Independent probabilities
Consider now a second characteristic of each member of the same
population which is quite independent of the first. As an example which
we will assume to be independent of height, consider the marks m gained
in any particular examination (and let us assume that m is a continuous
variable). If h and m were not independent, we might find that the examina
tion score was proportional to the height of the examinee or some such
relation which we will assume does not hold. We can again measure the
probability that any individual scores between m and (m + dm). Let us
write this
P[m] dm
where P[m] is the probability function of m, perhaps a different function
from P[h\
Given these two characteristics h and m, we can ask what is the probabili
ty that a given individual has a height between h and (h + dh) and also
scores between m and (m + dm) — a probability which we can write
P[h, m] dh dm,
where the notation P[h, m] means a function of the two independent
variables h and m. We could find this from the data about the population
by drawing h and m axes (Fig. 4.4) and representing each individual by a
point with his coordinates. Then we could divide up the area into rec
tangular cells of size dh by dm, and we could count the number of points
inside the appropriate cell ; finally we would divide this number by the
total population n. In Fig. 4.4, only a few points have been drawn in the
cell, but we must assume that the number is really so large that P can be
considered as a continuous function of h and m.
We can also relate P[h, m] to P[h] and P[m\ For if we choose any person
at random from the population the probability that his height is within
the desired range is P[h] dh. At the same time, the probability that his
score is within the desired range is P[m] dm. Thus
P[h, m] dh dm = P[h]P[m] dh dm. (4.4)
This important equation expresses the fact that for any two character
istics which are entirely independent of one another, the probability of their
happening together is the product of their separate probabilities.
Familiar examples of this occur in games of chance. For example, a
die has six faces which are equal except for the markings and the probability
of throwing a given number, say a one, is therefore £. If two dice are
74
Energy, temperature and the Boltzmann distribution Chap. 4
thrown, the numbers which turn up are independent of one another, so
that the probability of throwing two ones is (£) 2 .
Further aspects of probability theory, notably fluctuations, are con
sidered in section 7.7.
d/7
height h
Fig. 4.4. Counting the number of individuals whose height is between h
and (h+dh) and score between m and (m + dm) in an examination.
4.3 THERMAL EQUILIBRIUM
We will in what follows consider systems which are in thermal equilib
rium ; we must try to describe what we mean by this.
From the macroscopic point of view, or from the practical point of
view, equilibrium is not too difficult to define even though it is seldom,
if ever, strictly attained. The essential conditions are that the temperature
must not change with time, however long the system is left, and that the
temperature should be uniform throughout the system.
These conditions imply other restrictions. It follows from our definition
that there can be no flux or current of heat through the system — for this
would produce nonuniformities of temperature. Thus a bar with heat
fed into one end and extracted at the other may reach a steady state when
4.3 Thermal equilibrium 75
temperatures do not change with time, but this is not equilibrium because
temperatures are not the same everywhere. It follows also from our
definition that there cannot be any current or bulk movements of particles
through the system, because these would give rise to differences of tem
perature. Thus a closed vessel partly filled with a liquid is not initially in
thermal equilibrium, because the liquid must partly evaporate so that
there must be a bulk movement of molecules upwards, driven by a minute
temperature difference between liquid and vapour. Only when the space
is saturated with vapour and the whole system has settled down to exactly
uniform temperature will it be in equilibrium. It follows also from our
definition that no system can be in equilibrium when it undergoes variable
acceleration, or when dissipative forces are acting within it.
The simple definition of equilibrium is therefore a restrictive one if
strictly applied. The atmosphere of the Earth, for example, is not in
equilibrium because molecules are escaping from the top of it and because
it is absorbing radiation from the sun. Indeed our whole universe is
able to function because it is not in equilibrium. But we might take a
measurement on some property of a small part of the Earth's atmosphere,
and if the changes which took place in it during the time occupied by the
measurement were small enough, we could forget that the air was not in
strict equilibrium. In a similar way, we can look again at the liquid
evaporating into a closed space — strictly speaking, such a system only
approaches equilibrium asymptotically and never actually attains it.
But in practice we need often only wait a short time — minutes, say, or
days — for the greater part of the change to have taken place. Subsequent
changes are likely to be so slow that during the course of any experiment
they are negligible. In such cases, equilibrium has for all practical purposes
been reached.
From the practical point of view, then, equilibrium is not too difficult
an idea to describe. From the molecular point of view, however, all the
simplicity of definition evaporates and the situation becomes one of the
most elusive to describe. Because of fluctuations, the density in a small
volume of gas inside a 'uniform' atmosphere continually changes so that
on the atomic scale there are still random fluxes of particles. At the inter
face of a liquid and a vapour, seemingly quiescent and unchanging on the
macroscopic scale, molecules are jumping out and returning all the time.
A solid rod is continually expanding and contracting in length, imper
fections in the lattice are continually moving about. Let us for the moment
then adopt a practical definition of thermal equilibrium and hope that
by the time he has reached the end of this book the reader will have a
deeper understanding of the meaning of thermal equilibrium at the
molecular level.
76 Energy, temperature and the Boltzmann distribution Chap. 4
4.3.1 The Boltzmann law
Let us now return to the problem of specifying the state of an assembly
of a large number of molecules. We will assume that the assembly as a
whole is at rest, but the molecules have random thermal motion.
In classical mechanics, the state of a molecule must be specified by its
coordinates of position, x, y and z, and its components of momentum
p x , p y , p z parallel to the three axes. In the ordinary cartesian system, we
might as well use velocities instead of momenta (they are related by simple
equations of the type p x = mv x ). But since position coordinates and
momenta are the conjugate variables used in the Hamiltonian dynamical
equations which arc valid in all systems of coordinates, we will use
momenta here.
Let us now concentrate on one single molecule in an assembly in a
container maintained at a temperature T. It does not matter whai:
molecules are being examined, nor what the container is, as long as it
functions as a thermostat which is set to maintain the temperature T
constant. In fact, we can take an assembly of the same molecules — a
portion of a solid embedded in the middle of a larger lump of the solid,
for example — considering the whole mass acting as the thermostat for
the smaller portion. It is one molecule in this assembly that we concentrate
on.
When a system is in thermal equilibrium, the coordinates and momenta
of this one molecule are independent of one another. This means that
wherever a molecule is located, its momentum may be of any magnitude
in any direction. Any interdependence between the coordinates, such as
the condition that fast molecules can only be found near the origin, is
ruled out. Thus, six numbers or parameters are required to specify the
state of one molecule.
For simplicity, let us deal first with p x , the component of the momentum
parallel to the xaxis, of the one molecule whose random thermal motion
we are considering.
The energy of this molecule is in general the sum of contributions from
all the position coordinates and components of momentum, but the part
of the energy contributed by p x is
^(p x ) = ~, (45)
2m
where Jf stands for kinetic energy and the notation Jf(p x ) emphasizes the
fact that Jf depends upon p x ; m is the mass of the particle and pl/2m
might have been written \mv 2 x but we have chosen to use the more general
formulation.
4.3 Thermal equilibrium 77
Then Boltzmann's law states that the probability that, in thermal
equilibrium, the component of the momentum lies between p x and
(Px + dp x )isgivenby
Ac jriPx)lkT dp x (4.6a)
or in other words the probability function for this one component of
momentum is
Ac *ip x )ikT t ( 46b )
This law is stated at this stage without derivation. The exponential
factor itself is called the Boltzmann factor.
In these expressions, A is a quantity which has to be chosen so that the
integral of the probability over the whole possible range of p x is 1 — the
probabilities have to be normalized.
The quantity T is called the absolute temperature, and these equations
define what we mean by absolute temperature. Later we will see that it is
identical with the 'perfect gas scale' temperature (section 4.4. 1).
The quantity k is a constant, called 'Boltzmann's constant', having the
units J/degree, whose magnitude we will work out in section 4.4.1.
This statement of Boltzmann's law is fairly general, but we will now
write down an explicit example of a Boltzmann factor and use it to
point out a number of features of Boltzmann's law. We choose cartesian
coordinates, when the Boltzmann factor for the momentum component
p x is
Q pH2mkT
Let us see what the form of this factor implies — in particular we will show
that it is consistent with the intuitive ideas of temperature already des
cribed. Firstly, it implies that the most probable value of p x (the xcom
ponent of momentum of the molecule in the assembly in equilibrium)
is zero, and that large (positive and negative) values of p x are less probable.
(The distribution follows the same kind of law as the distribution of heights
of a population about their mean, the example of section 4.2.) Although
for the molecule the mean value of p x is zero — this is obvious because the
distribution is symmetrical about p x = 0, so that positive and negative
values of p x are equally probable — the mean value of pi is not zero ; this
in turn means that the contribution to the mean kinetic energy arising
from p x (the mean value of Jt(p x ) or pl/lm) is not zero. In the next chapter,
we work out this mean value and show that it is equal to jkT, where k
is Boltzmann's constant. This is a very important result. It confirms the
statement made in section 4.1.2 of this chapter that temperature is a
measure of the mean kinetic energy of any molecule in an assembly in
78 Energy, temperature and the Boltzmann distribution Chap. 4
equilibrium ; we can say therefore that Boltzmann's law is consistent with
our intuitive ideas. At the same time it is worth pointing out an elementary
mathematical feature of the Boltzmann factor, namely that whereas we
usually measure temperature in degrees, (kT) is a measure of the tem
perature in energy units, ergs or electron volts. The ratio Jf/ZcTis dimen
sionless — and since one can only take the exponential of a dimensionless
number, it is not surprising that the kinetic energy and the temperature
are associated in this particular way. Indeed, in Appendix A we demon
strate how the form of the Boltzmann factor can plausibly be derived from
simple considerations based on Joule's experiments; we start with the
fact that the probability function must be a function of (energy)/(tem
perature) where the temperature is measured in energy units — this is the
only mathematically acceptable form it could possibly have.
Having pointed out these aspects of Boltzmann's law, using a simple
explicit form of the kinetic energy term, let us return to the more general
formulation and extend its use.
The Boltzmann law holds for each of the components of momentum
p x , p y and p z independently. For example, the probability that the com
ponent of momentum parallel to y lies between the limits p y and
(p y + dp y )is
AQ* (p * )lkT dp y
Combining these two probabilities, we can say that the probability
that the x and y components of momentum lie simultaneously between
the limits p x , (p x + dp x ) and p y , (p y + dp y ) is
A 2 e X( Px )/kT dpx Q xr( Py )/kT dp ^
Let us rearrange this expression. When exponentials are multiplied to
gether, we add the indices. So we can write the probability
A 2 e^^ + x ^ )] i kT dp x dp y .
By extension, we can calculate the probability that the three components
of momentum lie simultaneously within limits p x , (p x + dp x ) ; p y , (p y + dp y ) ;
p 2 ,(p z + dp z ). This is
A 3 e* lkT dp x dp y dp z , (4.7)
where Jf is the total kinetic energy of the molecule :
X = X(p x ) + JT{p y ) + Jf(p z ) = ^(pl + P 2 y +Pl) = im(v 2 x + v 2 y + v 2 ). (4.8)
We might have written Jf(p x ,p y ,p z ) to emphasize that the total kinetic
energy depends on p x , p y and p z .
The Boltzmann law also holds for the contributions to the energy
which come from the position coordinates — these are always potential
4.3 Thermal equilibrium 79
energies. For example, the potential energy of a molecule undergoing
simple harmonic motion is of the form 'Viz) = ^<xz 2 , where z is a dis
placement and the symbol 'Viz) emphasizes the fact that V depends on z ;
the potential energy of a molecule in the Earth's gravitational field is of
the form Viz) = mgz where g is the acceleration due to gravity and z is a
vertical coordinate. These are two examples of the fact that energies which
depend on position coordinates are potential energies.
The Boltzmann law states that the probability that a molecule has a
coordinate between z and (z + dz) is
(constant)e r(2)/kr dz,
where 'Viz) is the contribution to the potential energy which depends
on z. The constant is found by normalizing. For a harmonic oscillator,
the probability is of the form (const)exp(az 2 /2/cT)dz; for a particle in
the Earth's gravitational field, it is (const)exp(mgz//cT)dz.
Extending the argument, the probability that the molecule is to be
found between x, (x + dx), y, (v + dv), z, (z + dz) is
(constant)e ~ ir/kT dx d v dz, (49)
where V is the total potential energy — it might have been written
V(x, v, z) meaning that it depends on x, y and z. Finally, the probability
of finding the molecule in the state specified simultaneously by momenta
and positions p x , ip x + dp x ) • • • z, (z + dz) is
(constant) e~ E/kT dp x dp y dp z dx dy dz, (4.10a)
where E is the sum of the kinetic and potential energies
E = jf+ ir
and the constant has again to be found by normalizing. Note that in all
these expressions, the probability function has always been of the form
e £/*r
where E is the energy which depends on the coordinates and momenta
considered.
So far we have dealt with the probability of finding a single molecule
in a certain state specified by momenta p x , ip x + dp x ), ... and coordinates
x, (x + dx), .... It is not difficult to extend the application of Boltzmann's
law to two molecules and thence to a large number N.
To do this, we first need a notation. Let us use p xl to denote the x
component of momentum of molecule 1, p x2 for the corresponding
quantity for molecule 2, and so on. Then we can write down the Boltzmann
factors for all the momentum and position coordinates, combine all the
80 Energy, temperature and the Boltzmann distribution Chap. 4
exponentials together by adding their indices. The result is that the
probability of finding one molecule between limits p xl , (p xl + dp xl ), ...,
z 1} (z 1 +dz 1 ) and simultaneously the second molecule between limits
P X 2 , (Pxi + dp x2 ) ■■■z 2 ,(z 2 + dz 2 ) is
(constant) Q' mT dp xl dz x dp x2 • ■ ■ dz 2 , (4.10b)
where E is now the sum of the kinetic and potential energies of the two
molecules. In other words, the probability function for this state is
(constant) Q~ mT .
Similarly, the probability function for N molecules to be found with
momentum near p xl ,...,p zN and coordinates near x 1 ...z N (6N para
meters in all) is
(constant) e~ E/kT ,
where £ is the total energy of the whole assembly. This energy depends on
the position and momentum coordinates of all the molecules, and it is
(usually) an exceedingly complicated expression. Nevertheless, the form
of the probability function is so simple that it is consistent with the need to
describe a system in equilibrium by only a small number of parameters.
4.3.2 Validity of Boltzmann factors
The problem confronting physicists around the beginning of the cen
tury was whether matter really was composed of atoms and molecules,
and the statistical methods whose main results we have just presented were
evolved in order to tackle this question. To begin with, calculations
were performed on exactly specified systems (such as gases, perfect and
imperfect) but it was noted that terms of the type exp(£//cT) kept
appearing. Eventually it was realized that this kind of expression was not
a characteristic of any one model of the way a particular lump of matter
was constituted but was of the most general validity.
In Appendix A an outline of an approach is given which emphasizes
this generality of application. In it, one discusses the thermal equilibrium
of a 'subsystem' within a 'system'. The result, that the Boltzmann factor
gives the probability function for the subsystem to possess a given energy,
is quite independent of what it is made of— provided only that it is in
thermal equilibrium. It could be a collection of molecules of the same
kind as those of the system, or different from them ; it could be a collection
of large particles or it could be a single particle while the rest of the system
could be a liquid. The subsystem could even be a large object (a metal bar,
for example) 'immersed' in a 'gas' of similar but purely imaginary copies
of the same object, exchanging energy with one another by some
4.3 Thermal equilibrium 81
unspecified mechanism whose only function was to ensure that the whole
system was in thermal equilibrium. In order to relate the behaviour of the
real object in these imaginary surroundings to its behaviour inside a real
thermostat we need only assume that the behaviour of any body in thermal
equilibrium under given conditions is independent of the mechanism used
to bring it to that equilibrium.
The final result of these discussions is that the Boltzmann factor is
valid not only for a molecule (which we discuss in section 4.4 and later
throughout this book) but for a large object such as a particle undergoing
Brownian motion (which we discuss in section 4.4.2).
We will therefore assume that Eq. (4.7) and Eq. (4.9) are both applicable
to any object which is in thermal equilibrium and is subject to the laws
of classical mechanics. For example, the probability that a particle of
microscopic dimensions (large on the molecular scale), suspended in
a liquid will be found within the range of coordinates z to (z + dz) is
AQ\p{Y'(z)/kT).dz where i^(z) is the contribution to the potential
energy which depends on z.
It is ironic that the intense scrutiny that the laws of statistical mechanics
came under resulted in the establishment of their general validity. For
this scrutiny was forced on physicists because certain results, notably
predictions about specific heats at low temperatures, were in conflict
with experiment. But the error lay not in the statistical methods but in the
assumption that atoms were subject to classical mechanics.
4.3.3 Kinetic and potential terms
So far we have defined what we mean by a probability function and have
quoted the form of the Boltzmann factor. This is the probability function
for any system in thermal equilibrium at temperature T to have energy
E. In turn, this will allow us to calculate mean values of molecular velocities
and other parameters depending on them, and so to calculate the thermo
dynamic behaviour of many systems
We always begin by writing down an expression for the energy E. For
a collection of molecules this is immensely complicated in general, and to
reduce it to manageable proportions we have to introduce simplifying
assumptions. But in systems obeying the laws of classical mechanics, the
Boltzmann factor can always be separated into a product of kinetic and
potential energy terms for the molecules. The kinetic energy always con
sists only of quadratic terms of the type pl/2m (or jmv*). Whenever a
molecule moves, whether it is moving in a straight line or in an orbit or
oscillating about a fixed point, its kinetic energy can always be expressed
by terms of this form. Therefore, when we write down a Boltzmann factor
for a single molecule or for an assembly, the evaluation of the kinetic energy
82 Energy, temperature and the Boltzmann distribution Chap. 4
term, giving the distribution of velocities, does not depend on the physical
nature of the assembly, on whether we are dealing with a solid, liquid or gas.
When we come to calculate velocity distributions or the mean speeds or
energies of molecules, this fact leads to some remarkable generalizations.
The potential energy of the whole system is similarly the sum of energies
of all the individual molecules. The contribution to this energy arising
from fields of force such as gravitational, electromagnetic, electrostatic
or magnetic fields, is in general a simple function of the coordinates and
the Boltzmann factor breaks up into a product of simple terms. There is
a further contribution due to interactions with other molecules, but
since the distance apart of each molecule from every other molecule
enters into the expression for this potential energy, it is usually an ex
tremely complicated function of the coordinates. It must be emphasized
that in contrast to the kinetic energy term, the evaluation of the potential
energy term depends very much on the physical nature of the assembly,
on whether we are dealing with a solid, liquid or gas.
The plan of the rest of this chapter is as follows. Since the Boltzmann
factor incorporates the temperature, and we have stated that this is
called the absolute temperature, we must relate it to other scales. We do
this by considering (section 4.4) the equilibrium of an assembly of non
interacting particles which is an idealized model of a perfect gas and which
bears some resemblance to many real gases. As a result we will be able to
identify T and also calculate Boltzmann 's constant k. Underlying this
discussion will be the need to show that Boltzmann's law gives self
consistent results. Then we will deal with another similar system, a
suspension of tiny particles in a liquid (section 4.4.2).
4.4 BOLTZMANN DISTRIBUTIONS— I. A GAS OF
INDEPENDENT PARTICLES UNDER GRAVITY
Consider a mass of gas, maintained by some means at a uniform tem
perature T. It is imagined to be in a very tall vessel — it will emerge later that
the results are more interesting when the height is several kilometres
and it is in the gravitational field of the Earth. The problem is to calculate
the distribution of the gas molecules with height.
We will not calculate the distribution of kinetic energies (that will be
done in the next chapter) but we will deal only with potential energies ;
the problem can be simplified so that we deal with a single potential
energy term and the corresponding Boltzmann factor.
Consider a single molecule. If we assume that the acceleration due to
gravity g is constant with height (an approximation which is sufficient
for our considerations) then the potential energy of each molecule of mass
4.4 Boltzmann distributions — I. A gas of independent particles 83
m due to its height z above an arbitrary zero (usually at the Earth's surface)
is mgz.
Each molecule also has potential energy due to the presence of neigh
bouring molecules. Most of the time, of course, the molecules are far
apart and their interatomic potential energy is negligible. But when they
collide with one another, their potential energy is, for a short time, by no
means negligible. Now it is essential that there should be collisions
between gas molecules because this is the mechanism whereby a gas
reaches thermal equilibrium ; if it were not for these collisions the gas
could never reach uniform temperature. But we will now make a further
assumption — namely that the collisions are relatively rare so that
averaged over a long time the interatomic potential energy is negligible.
This can be achieved by ensuring that the gas is at low pressure. It is
true that under these conditions, we might have to wait a very long time
for the gas to reach equilibrium, but that does not concern us. We call
such an assembly a gas of independent particles.
Thus we reach a compromise : we choose an assembly of molecules
where collisions do take place, even though the potential energy is large
whenever they occur, because collisions are essential to enable thermal
equilibrium to be reached. On the other hand, we choose an assembly
where the collisions are relatively rare so that on the average the inter
atomic potential energy is negligible. Under these conditions, we can
say that the only significant contribution to the potential energy of a
molecule is that due to the Earth's gravity, namely mgz, where z is its
height above an arbitrary zero.
The probability of finding a molecule at a height between z and (z + dz)
is given by the Boltzmann law :
P[z] dz = Be n ' )lkT dz = Be' mgz,kT dz. (4.11)
We can evaluate the constant B by applying the condition that the
molecule must be found somewhere between the zero of height and the
top of the column — which for simplicity we will assume is at z = oo :
f
Jo
P[z] dz = 1 (4.2)
whence B = mg/kT.
Hence the probability of finding the molecule between z and (z + dz) is
P[z]dz = ^e^ z/fcT dz. (4.12)
84 Energy, temperature and the Boltzmann distribution Chap. 4
It is worth remarking that had we chosen any other arbitrary zero for z,
all that would have happened is that a constant factor would have ap
peared in our expression which would later have disappeared in the
normalizing.
(Though it does not concern us and the answer is obvious by common
sense, we will digress to find the dependence on one of the horizontal
coordinates, x. The potential energy does not depend on x : i^ix) = 0.
Hence the probability of finding the molecule between x and (x + dx)
contains the Boltzmann factor exp( — 0/kT) which is unity; and this
probability is simply proportional to dx — in other words this implies
that the molecule may be found anywhere in the xy plane with equal
probability.)
Returning to the zvariation, let us now consider a large number n
of molecules. The number which we will find between z and (z + dz) is
^e—^dz.
kT
Now the density of any substance is equal to the number of molecules
per unit volume, multiplied by the mass of one molecule. Hence if p is
the density at zero height (strictly, the limiting density between z =
and dz where dz is small) then this last equation gives
p(z) = p exp(mgz/kT). (4.13)
We can plot the variation of density with height, at any given tem
perature. The density falls off exponentially with height. A graph of
exp( — z/z ) as a function of z is given in Fig. 4.5; it is identical in form
with Fig. 3.2(b). When z is equal to zero it has the value unity. If we
compare the function at two heights differing by z , say at z = h and
(h + z ), we find that the value falls by a factor e = 2.717 ... ; z is called
the scale height. At a height of 2z , the function falls by a factor of about 7 ;
at 4z by a factor of about 50.
Here, the scale height
z = — ' (4.14)
mg
but we cannot yet evaluate this because we do not know the value of
Boltzmann 's constant k. We will now find this. From this relation between
density p and temperature T in a gravitational field, we can derive another
between p and T as a function of pressure P. We do this by changing our
approach completely — taking a macroscopic view of the same phenom
enon, regarding the gas now as a fluid having weight and capable of exert
ing a pressure without bothering about its molecular constitution and not
4.4 Boltzmann distributions— I. A gas of independent particles 85
Fig. 4.5. The exponential function exp(z/z ). For an increase of z by an
amount z , the function decreases by a factor e.
concerning ourselves with how the gas exerts its pressure. We consider
a tall vessel filled with the gas and deal with the equilibrium of a slice of
it between the heights z, and (z + dz), Fig. 4.6. The pressure at z is greater
than that higher up because of the weight of the gas above it ; this is true
for a column of any material. Thus
dP = — pg dz.
I
dz
pressure (P + dP)
pressure P
Fig 4.6. Equilibrium of a gas in the Earth's
gravitational field.
86 Energy, temperature and the Boltzmann distribution Chap. 4
The total pressure at height z is the integrated weight of all the gas above :
/»00 /«00
P = J dP= g\ pdz,
and using the relation (4.13) above,
f 00 kT kTp
P = gPo exp(mgz/kT)dz = — p [exp(mgz/kT)]? = .
J z m m
It is convenient here to refer to a mole of the gas instead of a single
molecule, by multiplying top and bottom by Avogadro's number N :
NkTp
M '
where M = Nm is the molecular weight. If V = M/p is the volume
occupied by a mole of the gas, it follows that :
PV= NkT. (4.15)
This is a relation between pressure, volume and temperature for the gas
of noninteracting particles which we have deduced from Eq. (4.12). It
must be remembered that T is still defined by the Boltzmann law (4.11).
4.4.1 Perfect gases
It is well known that real gases obey an equation of just this type, under
the correct conditions — namely that the pressure is sufficiently low. Air,
for example, obeys this relation with accuracy at pressures of the order of
a few atmospheres at ordinary temperature; at low temperatures the
range of pressure over which the equation holds becomes very much
smaller — but at all temperatures air, like any other gas, obeys this kind of
relation in the limiting case of zero pressure.
The temperature used in practice was originally the centigrade scale
defined in terms of certain properties of water, and the equation was
written
PV= const(273 + T°C).
It is convenient to shift the zero to  273°C and to write the equation
PV= RT (4.16)
where T is known as the temperature on the perfect gas scale.
We can identify (4.15) with (4.16) and draw the following conclusions.
Firstly, a perfect gas behaves like a gas of noninteracting particles.
Secondly, the temperature defined by the Boltzmann equation is indeed
4.4 Boltzmann distributions — I. A gas of independent particles 87
identical with that on the perfect gas scale ; it was this statement that we
set out to prove. Temperature on the absolute scale is written T °K.
Finally, Boltzmann 's constant k can now be found. As a result of
measurements on a number of gases at low pressures, extrapolated back
to what are called standard conditions of temperature and pressure,
namely 1 atmosphere or 1.013 x 10 6 dyn/cm 2 , and 0°C or 273°K, it
is found that V is then 2.24 x 10 4 cm 3 for all gases and that therefore R is
8.31 J/mol.deg. for all substances.
Since Nk = R, and N = 6x 10 23 , we have k = 1.38 x 10 23 Jdeg 1 =
0.86xlO _4 eVdeg 1 .
With these data we can now find the scale height of the Earth's atmos
phere, making a gross assumption which is quite untrue — that it is all
at the same temperature. We use Eq. (4.14). For air, we can take
m = 30 a.m.u. = 5 x 10" 23 gm. Let us take T = 300°K and g = 10 3 cm/s 2 .
Then z = /cT/mg is about 8 kilometres. (We might equally well write
z — RT/Mg, and take M — 30.) Thus if atmospheric pressure is taken
to be 76 cm of mercury at the Earth's surface, it is 28 cm at 8 Km a height
which is roughly the height of the world's highest mountains. At twice
this height the pressure is about 10 cm. (Note that these heights are small
compared with the Earth's radius which justifies our approximation that
g is constant with height.)
4.4.2 Examples of Boltzmann distributions— II. Brownian movement
Consider a tiny solid particle immersed in a liquid. It exhibits Brownian
motion, which means that it can be observed with a microscope to be in
ceaseless, rapid motion. It moves randomly about in short discontinuous
jumps, though if a higher magnification were used together with a higher
speed of observation, each of those jumps would be seen to consist of a
number of shorter jumps. These random movements are the result of the
impacts of molecules of the liquid against the particle, and this is the
mechanism for keeping the temperature of the particle constant on the
average.
We can write down the Boltzmann factor for one single particle, and
hence calculate a quantity which is almost directly observable : the proba
bility of finding the particle between a height z and (z + dz) from the
bottom of the vessel. To do this, we need to know its potential energy
at a height z. It is
V(z) = m*gz
where m* may be called the effective mass of the particle which takes into
account the mass of liquid displaced :
m* = v(pp')
88 Energy, temperature and the Boltzmann distribution Chap. 4
where v is the volume of the particle, p and p' are the densities of solid and
liquid respectively. (This result can be derived from the fact that the force
on the particle in the direction of z increasing is — m*g.)
We can therefore write: the probability of the particle being found
between the limits of height z and (z + dz), at temperature T, is given by
P[z] dz = P Q m * gzlkT dz,
where P is a normalizing factor which we will not evaluate. If we observed
the particle wandering through the liquid over a long time, this expression
would give the fraction of the total time it spent within these limits of
height. Alternatively, if we had a large number of particles present (large
enough for us to apply statistical methods but small enough for us to
neglect any mutual attractions between particles), this expression would
determine the average number to be found at any time between the given
limits :
ndz = n Q m * gz/kT dz, (4.17)
where n dz is the number between z = and dz.
This is just the same form of variation as for molecules of a perfect gas,
but now the scale height kT/m*g is in practice very small instead of being
many kilometres because of the magnitude of m*. Typical numbers are
given in a problem at the end of the chapter. The historical significance of
this experiment is that it is possible to prepare particles (of a resin, gam
boge) whose sizes are great enough for their masses to be found, and yet
small enough to undergo Brownian motion with a measurable scale
height. Knowing m* (or more precisely, measuring v, p and p'\ the accelera
tion due to gravity and the temperature on the perfect gas scale, it was
possible firstly to get a direct, almost visual, proof of the validity of the
Boltzmann law and secondly (by measuring the scale height) to determine
Boltzmann's constant. This was done by Perrin in the early years of this
century. Fig. 4.7 is an adaptation of a photograph taken by focusing a
microscope at different levels and then making a montage of them ; z
was of the order of 10~ 3 cm.
4.4.3 Characteristic temperatures
By raising the temperature sufficiently, it is possible to break up any
kind of bonding between particles. If at a low temperature the bonding
together of the given particles lowers the energy of the assembly by an
amount e per particle, then when there is a reasonably large probability
that the energy of thermal agitation is also e, the interaction will be over
come and the bond broken. The nucleons forming a nucleus, the electrons
and nucleus forming an atom, atoms combined together to form a
4.4 Boltzmann distributions — I. A gas of independent particles 89
molecule and molecules condensed to form a liquid or solid will all be
knocked apart at sufficiently high temperatures.
This will happen above a temperature T given approximately by
kT
(4.18)
because the Boltzmann factor exp( — e/kT) is then of order unity (we justify
this statement a little more precisely in section 5.5). If the temperature
were 10 times lower the Boltzmann factor would have the value e 10
which is very small, about 10 ~ 4 , and it would be improbable that sufficient
thermal energy would be available.
•
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z = *L
o rng
Scale Height
Fig. 4.7. Distribution of resin particles in water as
a function of height. Adapted from a photograph
by Perrin (1910), actually a montage of several
sections at different heights.
This is a rough rule but it is remarkably powerful. For example, the
binding energy of two molecules of many substances is of the order of
10~ 14 to 10~ 13 erg. Correspondingly, each substance has a critical
temperature above which the liquid phase can never form, usually of the
order of 10" 14 /Jt to 10 _13 /fc degrees— that is, 10 2 to 10 3o K. Ordinary
boiling and melting points are usually not very different in order of
magnitude. In a different region of energy, atoms can be ionized at room
temperature when they are bombarded with electrons whose energy is of
90
Energy, temperature and the Boltzmann distribution Chap. 4
the order of 10 eV (that is, of the order of 10 11 erg). We can therefore
estimate that the atoms in a gas can be ionized by collisions with other
gas atoms at temperatures of the order 10 _11 //c degrees, or roughly
10 5o K. Indeed, gases do form ionized plasmas at such temperatures.
Finally, in another and much higher energy range, nuclei can be dis
rupted by nuclear particles of about 1 MeV energy (around 10 " 6 erg).
This means that one can expect nuclear reactions to take place in gases
heated to about 10 9 or 10 10o K, which corresponds to conditions in the
interiors of stars.
APPENDIX A
A.l Dependence of the probability function on energy and temperature
In this Appendix we will attempt to make plausible the form of the
probability function, quoted without proof in section 4.3.1.
Consider a lump of matter, composed of molecules. Its temperature
is maintained at a constant value T by some mechanism, a thermostat,
which we will not consider. We call this lump of matter the system. We
select a small part of the interior of the system which we call the 'sub
system', Fig. A.l. For example, the system might be a block of metal with
a cavity containing gas : and we might select this gas to be the subsystem —
or we might select part of the metal to be the subsystem.
subsystem
system
Fig. A.l. A lump of matter (the system) maintained at
temperature T; we select part of it, called the subsystem.
We ask what is the probability that the molecules of the subsystem
have particular positions and momenta. Since the system is in equilibrium,
Appendix 91
this probability will not change with time, even though the molecules are
moving rapidly and randomly.
If we had just one molecule in the subsystem, we could agree to write
P[x, y, z ; p x ,p y ,p z ] dx dy dz dp x dp y dp z
for the probability that the molecule was at a point whose coordinates
lay in the range x to (x + dx), y to (y + dy), and z to (z + dz), while the
components of momentum lay between p x and (p x + dp x ), p y and (p y + dp y ),
p z and{p z + dp z ).
If we had more molecules — and we usually have large numbers — then
we have to write something more complicated. If there were n molecules,
we would write
P[x t . yi > 2 i ; Px, , P yi , P Zl ;x 2 ,y 2 ,z 2 ; p X2 ,p y2 ,p Z2 ;...x„,y„,z„; p Xn , p yn , p z J
x dx! dy t dzj dp xl . . . dp yn dp Zn
for the probability that molecule 1 is within a range dx! and dy x and dz x
of the point (x l , y t , z t ) and moving with a momentum whose components
are between limits p Xl and {p xi + dp Xi ) and so on ; while at the same time
molecule 2 is at a point near (x 2 , y 2 , z 2 ) with momentum near (p X2 , p y2 , p Z2 ) ;
and so on for all the molecules.
This function P, which is denoted above to be a function of all these
variables, could in general be very complicated. The fact that we are
dealing with a system in equilibrium however leads to something much
simpler.
Since the subsystem is continually interchanging energy with the rest
of the system by molecular collisions, it is natural to suppose that P might
be a function simply of the energy E of the subsystem. (E itself depends
on all the coordinates and momenta of all the molecules.) It also depends
on the temperature T which characterizes the system as a whole. No
other simple mechanical quantities can be thought of which could be
relevant. (The total linear momentum and angular momentum of the
system might be included but we rule them out by considering the system
as a whole to be at rest.)
Now the probability function is essentially dimensionless because of
the meaning of probability. It must therefore be a function of the ratio
between E and some other quantity which has the dimensions of energy.
The only quantity we have at our disposal is T — which is a measure of
energy, but whose dimensions are largely arbitrary since we can measure
temperature in an arbitrary way. The simplest possibility is to say that
P is a function of (E/kT) where k is a universal constant whose actual
92 Energy, temperature and the Boltzmann distribution Chap. 4
magnitude depends upon the units used to measure the temperature :
P = f(E/kT). (A.1)
This already looks a good deal simpler than our last expression for P, but
of course E itself still depends on all the coordinates and momenta and is
usually a very complicated expression.
A.2 Form of probability function
We can determine the form of the function/by considering the following
special situation. Consider two independent systems both maintained at
temperature T; for example, two cavities each containing gas, inside the
same block of metal maintained at T. Let the energies of the systems be
E x and E 2 respectively. The probability of finding system 1 in a given
configuration is f(EJkT) and the probability of finding system 2 in
another given configuration is f(E 2 /kT). But if now we consider the two
systems together, the probability of finding both systems 1 and 2 in their
given configurations at the same time must be
f / £i + £ 2
J \ kT
since the energy of the two systems together is the sum of their separate
energies. But since the two systems are independent, the probability must
be the product of two separate probabilities :
We will now show that this serves to determine the form of the function
/. This can be seen intuitively by noting that since P (or/) is multiplicative
whereas E is additive, the function must be of the kind
log f(E/k T) = BE /k T+C (A.3)
where B and C are constants not involving E. Alternatively, this result
can be derived explicitly from (A.2) as follows. For simplicity write x 1 and
x 2 for EJkT and E 2 /kT, and write /'(x) for df/dx where x is x t or x 2 or
(x 1 +x 2 ). Carry out a partial differentiation of Eq. (A.2) all through
with respect to x t keeping x 2 constant; then
/'(*!+ x 2 ) = f\x x )f{x 2 ).
Differentiating (A.2) with respect to x 2 keeping x t constant gives
f'(x 1 +x 2 ) = f(x l )f'(x 2 ).
Appendix 93
Therefore
f'(x 1 )f{x 2 ) = f{x,)f\x 2 )
or
/'(*i) f'(x 2 )
fix,) f(x 2 )
= B
where B is a constant, or more precisely B does not contain E t or E 2
although it might depend on T. The solution of this equation is (A. 3)
above.
Therefore
f(E/kT) = (constant) e BE/kT
where the constant depends on temperature. Note that we lose no generality
by writing B equal to + 1 or — 1 since we can absorb any other numerical
factors in the constant k. Thus we have reduced the original complicated
form of the expression for the probability of finding the assembly in the
specified state to
(constant) e BE/kT dx x dvi . . . dp zn
where B is + 1 or — 1.
Consider now the special case of one molecule in the assembly, which
happens to be moving in the xdirection. The energy is simply pl/2m,
independent of x. The molecule may therefore be found with equal prob
ability to have any x coordinate, but the probability of having momentum
between p x and (p x + dp x ) is proportional to
exp(Bp 2 x /2mkT)dp x .
The top graph in Fig. A.2 shows this function for B = + 1. It would imply
that it is almost certain that the molecule would have infinitely large
momentum: this is not acceptable. The lower graph is for B = — 1, and
implies that small momenta are more probable: this is reasonable.
Therefore B = — 1.
Thus we conclude that the probability of finding the assembly in the
condition specified is
(constant) e~ E/kT dx, dy x . . . dp z „.
The value of the constant of proportionality is determined by the fact that
the molecules must be found somewhere inside the accessible range of
coordinates and momenta. Thus the integral over all variables must be
equal to unity.
94 Energy, temperature and the Boltzmann distribution Chap. 4
Fig. A.2. Form of the functions exp(+p 2 /a 2 ) and exp(— p 2 /<x 2 ).
By expressing the energy where possible as the sum of terms each
depending on one coordinate or one component of momentum, the
probability separates into simple terms each of the form of Eq. (4.6a).
The above expression satisfies all our conditions. It allows the energy of
independent systems to be additive but probabilities to be multiplicative.
It includes the temperature in a way which agrees with intuitive ideas but
at the same time it can serve as the definition of what we mean by tem
perature and in the text we call it the absolute temperature.
A.3 Extension to macroscopic systems
The language which we used in this discussion of the form of the
probability function was a very general one : we spoke of systems and
subsystems. Though in the text we concentrated on large assemblies of
molecules, and though in deriving the form of the probability function we
gave an example of a system consisting of gas molecules inside a cavity,
Appendix 95
there is no need to limit the discussion in this way. The Boltzmann law
can be applied to any system in thermal equilibrium.
For example, consider particles (each containing many molecules)
suspended in a liquid and undergoing Brownian motion. The problem
(which is discussed in section 4.4.2) is to calculate the distribution of the
particles with height, taking into account the potential energy due to the
Earth's gravity. We are not now interested in how the individual molecules
behave inside each particle but rather in how the particle behaves as a
whole ; we are interested in how the centre of mass of the whole particle
moves, we do not enquire how the molecules move with respect to this
centre of mass.
We could begin by accepting the validity of the Boltzmann law for
molecules and the distribution of their energies. With this as a starting
point, we could then write down the energy of each molecule inside the
particle in terms of its own position coordinates and momenta with
respect to fixed axes. This expression would be an immensely complicated
one. Then we could change the axes of reference to a set through the centre
of mass of the particle and moving around with it : this would be a simple
linear transformation in which for example the zcoordinate of the nth
molecule would be written as z H = z'„ + h where z' n is the zcoordinate
with respect to the new axes and h the height of the centre of gravity above
the fixed origin. The expression for the energy would still be complicated.
However, we would be able to group together a number of terms and
separate out a term exp( — m*gh/kT) from the probability function, where
m* is the effective mass of the whole particle. This is the sort of term we are
interested in, since both h and m* refer to the whole particle, or in other
words this term does not contain any molecular coordinates or momenta.
But there is no need to go through this complicated procedure. There
is no need to assume that the Boltzmann law is valid for molecules only,
molecules which are members of large assemblies ; hence there is no need
to begin each calculation at the molecular level. Our derivation of this
law was valid for any system in thermal equilibrium. At one stage for
example we considered a system of a block of metal with two cavities each
filled with gas ; we might as well have considered a liquid containing two
particles. A single particle can act as the subsystem and the Boltzmann
law applies to it. Thus, we can select coordinates (such as h) which refer
to one particle, calculate the corresponding contribution to the energy
and hence write down the Boltzmann factor for this particle. This gives
exactly the same answer as before, of course, but far more directly.
Two remarks can be made here. First, this discussion was forced upon
us because ordinary objects contain vast numbers of molecules and only
statistical statements can be made about them. We have however ended
96 Energy, temperature and the Boltzmann distribution Chap. 4
by applying the results of this discussion to single objects and this might
seem inconsistent. In fact it is not, because the mechanism for keeping its
temperature constant is one of continuous bombardment — for example,
collisions of a particle with molecules of the liquid in which it is suspended.
This situation is so complex that again it cannot be followed in detail.
Secondly, let us go back and examine a little more closely what is
involved in accepting that the Boltzmann law is applicable to molecules.
After all, each molecule itself has a structure and may contain many, even
several hundred particles (electrons, nucleons). Thus if we accepted a
strictly classical point of view it would not be sufficient to write down 3
coordinates and 3 momenta for each molecule: many more would be
needed. But our discussion has shown that, just as we need not write down
the separate Boltzmann factors for every molecule inside a macroscopic
particle but can write down the factor for the particle as a whole, so we
need not begin by considering the constituent subatomic particles but can
deal with the molecule as a whole. At least, then, our approach is self
consistent. (In addition, quantum mechanical considerations show that
many types of motion, of the atoms inside a molecule or of the nucleons
inside a nucleus which would be expected to occur if classical mechanics
were universally valid, do not in fact take place at normal temperatures.
The energy needed to excite them is much greater than kT where T is
a normally accessible temperature; see section 4.4.3. This merely rein
forces the result for completely different reasons.)
PROBLEMS
4.1. In one of his experiments with resin particles suspended in water, Perrin used
a microscope with a short depth of focus to count the number of particles
in horizontal layers 6 microns apart (1 micron, jx = 10 6 m = 10~ 4 cm). At
17°C the numbers which he observed in the field of view were 305, 530, 940 and
1880. He found the particles to have a radius of 0.52 /z and a density of 1.063
gmcm 3 ; the density of water at 17°C is 0.999 gm cm 3 . Use Perrin's results
to calculate Avogadro's number, given that the gas constant R is 8.314 J
deg  * mol" * and g = 980 cm sec 2 .
What would be the distribution with height for particles of density 0.935
gmcm
■39
CHAPTER
The Maxwell speed distribution
and the equipartition of energy
5.1 VELOCITYCOMPONENT DISTRIBUTION P[v x ]
In Chapter 4, we saw that the probability of an assembly of n molecules
possessing coordinates p xi ■ ■ • z n is given by
(const)e £/kT dp xl • • • dz„ (5.1a)
where the x, y and z and the p x , p y and p z coordinates refer respectively to
the positions and momenta of the individual molecules. Since the energy
splits up into independent contributions, the potential energy ^depending
only on the positions and the kinetic energy Jf only on the momenta, the
probability can be expressed as the product of two entirely independent
groups of terms :
(const) e*' kT dp xl ■ ■ • dp xn x e~ w dx t • • • dz„. (5.1b)
We then considered an assembly of independent particles in which the
potential energy due to the interactions between the molecules was taken
to be negligible on the average even though collisions between molecules
must occur in order to preserve thermal equilibrium. We did not consider
the kinetic energy of the molecules, but by considering a special case of the
assembly in the Earth's gravitational field, we showed that the quantity T
appearing in the Boltzmann factor is identical with the thermodynamic
temperature.
98 Maxwell speed distribution and the equipartition of energy Chap. 5
In this chapter we will concentrate on the kinetic energy rather than on
the potential and deduce the probability that a molecule is in an assembly
in thermal equilibrium at a given temperature and has a momentum
component or total momentum or speed, within a certain range.
The results which we will deduce are widely applicable. They hold for
any state of matter, solid, liquid or gas, provided the laws of classical
physics apply.
To emphasize this point, we will first consider how the total potential
energy of an assembly of N interacting molecules could be written down —
interacting in the sense that each pair of atoms has a mutual potential
energy. Considering the interaction of the ith molecule with the y'th for
example, the potential energy depends on the distance r tj between them
raised to the inverse 6th or 12th power (if they are neutral molecules) and
to express these quantities in terms of the coordinates of the individual
molecules is complicated. To extend this to all possible pairs of molecules
gives a vast array of terms which includes many product terms and which
cannot be arranged as a simple sum of terms each depending on a single
coordinate. This is why in the previous chapter we limited the discussion
to one of the few physical cases where there are on the average only
negligible interactions — a perfect gas where the molecules are for most of
the time at a great distance from one another. Without introducing gross
simplifying assumptions, we cannot extend the discussion of the potential
energy term to any interacting assembly such as a liquid or a solid or a
dense gas.
Even in strongly interacting assemblies, however, the total kinetic
energy can always be written down as a sum of individual kinetic energies
each depending on only one velocity component. For example, in a solid
where the molecules are close together and mostly oscillate about their
mean positions, the displacement of one molecule from its equilibrium
position certainly causes its neighbours to move, either setting them
oscillating at a single frequency or with a complicated frequency spectrum,
or perhaps changing their places in the lattice. Nevertheless, if the instanta
neous velocity of one molecule is (v xl , v yl , v zl ) and that of a neighbour is
( v x2> v yi> v z2) tne energy of each is still the sum of terms of the type jmv xl or
\mvl 2 , the kinetic energy therefore always separates out into the sum of
simple terms and the corresponding Boltzmann factor always separates
out into a product of single terms.
Thus the momentum or velocity distribution which we will deduce is
applicable to all physical systems. A solid and a liquid and a gas at high
pressure and a gas at low pressure, all of the same molecular weight and
at the same temperature have the same velocity distribution, even though
the types of movements the molecules perform are quite different. In a
5.1 Velocitycomponent distribution P[v x ] 99
gas at low pressure the molecule moves in straight lines between relatively
rare collisions, and their average interaction potential energy is negligible ;
in a solid, the molecules are oscillating about their lattice points and have
high potential energy. Nevertheless, if we considered a substance of given
molecular weight in the gaseous and in the solid states at the same tempera
ture, the proportion of molecules whose velocity lay within a given range
would be found to be just the same in the two states. The potential energies
would be quite different, the distribution of kinetic energies would be
identical. The discussion that follows is in no way limited to a gas — though
we shall in fact apply the results to a gas and thereby gain insight into the
way the pressure of a gas is produced. Later we shall apply the same
results to solids and liquids.
Consider an assembly of molecules each of mass m. It is convenient
now to deal with velocities instead of momenta; let v x , v y and v z be the
components of velocity parallel to x, y and z, so that p x = mv x , p y = mv y ,
p z = mv z . Since m is constant, the probability that the xcomponent of
velocity lies between v x and (v x + dv x ) is
PM dv x = A exp(  mv 2 x /2kT) dv x (5.2)
where A is determined by the fact that the probability that v x must have
some value between — oo and + oo is unity :
/•oo
J,
exp(mvl/2kT)dv x = 1.
We use one of the integrals listed in the Table on p. 72, namely
exp(ax 2 )dx = /;
Joo V a
whence A = (m/2nkT) 1/2 so that the probability distribution is
m
1/2
P[Vx]dv * = [^f] 1 e ~ mv2x ' 2kTdv *> (53)
v x is the xcomponent of the velocity. A molecule travelling in any direc
tion has a component v x unless it happens to be moving exactly at right
angles to the xaxis, when v x is zero. P[v y ] dv y and P[v z ] dv z are identical
in form.
A graph of P[v x ] as a function of v x is given in Fig. 5.1 expressed in
dimensionless form. It is a Gaussian distribution centred about the velocity
component v x = 0. The symmetry of the curve means that for every
molecule travelling with a certain velocity in the + x direction, it is equally
probable that another is travelling in the x direction. Thus the average
value of the total component of momentum is zero, as we would expect in
a system at rest.
100 Maxwell speed distribution and the equipartition of energy Chap. 5
Fig. 5.1. Probability distribution P[>] of a component of
molecular velocity in any direction called the xaxis. v =
(mv 2 J2kTY 12 , a dimensionless measure of velocity.
5.1.1 Experimental verification of the P[v x ] distribution for gases and solids
It is possible to determine the velocity in any direction of a molecule in
thermal equilibrium inside a piece of matter by observing the Doppler
shift of a spectral line emitted by the molecule. When stationary, an atom
emits a spectral line of a certain wavelength, A say, with the same in
tensity in all directions. An observer can measure this wavelength, the
direction of observation being called the xaxis, taken as positive away
from the observer. When the atom moves, in any direction, the observed
wavelength is altered to a value A given by
AA (
(5.4)
where v x is the velocitycomponent of the atom along the xaxis away from
the observer and c is the speed of light. If the atom is moving at any angle to
the line of observation, it is always the xcomponent of the velocity which is
measured as a change of wavelength. Radiation can be received from all
the atoms in the assembly, whether they are travelling towards the observer
and give a shorter wavelength, or receding from him and give a longer
wavelength, or moving at rightangles when their wavelength is unaltered.
Thus the whole assembly of molecules containing these atoms produces a
spread of wavelengths whose intensity at any wavelength is given by the
number of molecules with the appropriate value of v x , that is, by the
5.1 Velocitycomponent distribution P[v x ] 101
probability distribution of v x :
1(A) dA = J(A ) exp[  mc 2 (A  A ) 2 /2A 2 kT] dA. (5.5)
A spectral line which would, in the absence of thermal motion, be sharp is
therefore broadened into one of gaussian shape whose width increases with
temperature (Fig. 5.2(a)).
(a)
0.53 • 0.53
6374.51 A
Fig. 5.2. (a) Variation of intensity with wavelength of a spectral line broadened
by the Doppler effect due to thermal motion, (b) Broadening of a line emitted
by ionized iron atoms (M = 56) in the Sun's corona. Data from Dollfus, Compt.
Rend. Acad. Sci. 236, 996 (1953).
For observing this effect in gases, and for checking the validity of the
P[v x ] distribution curve, optical emission lines in the visible region of
wavelengths can be used. Even in gases however, the thermal motion of the
molecules is not the only cause of the broadening of spectral lines. Every
line has a 'natural width', a quantum effect, determined by the finite
lifetime of the excited state of the atom which emits it ; many lines however
have small natural widths. When molecules collide, the wavelength of the
emitted line may be altered by the presence of the nearby molecule, but this
effect can be reduced by working at very low pressures where collisions are
infrequent. Finally, every optical instrument will broaden a monochro
matic line because of its finite resolving power, but this effect can easily
be allowed for. Under favourable conditions, therefore, the Doppler
broadening can be detected. The agreement with theory is always good.
102 Maxwell speed distribution and the equipartition of energy Chap. 5
Interesting applications of this effect have been made in astronomy,
to determine the surface temperatures of stars. The Sun's corona, for
example, is known to be a very tenuous gas of uniform high temperature.
Among the spectral lines it emits is a red line (A = 6374 A), from highly
ionized iron atoms which have lost nine electrons. This line has small
natural width, but in the Sun's corona it has been observed to be a com
paratively broad line of Gaussian profile, 0.53 A from the centre of which
the intensity falls to 1/e of its maximum value Fig. 5.2(b). The molecular
weight of iron is 56 and with these data the temperature of the corona is
2.1 x 10 6o K. This agrees with other independent estimates.
Solids do not emit sharp visible spectral lines, so that this method of
measurement cannot be extended to study the motion of atoms in solids.
But nuclear radiations, gamma ray spectra, can under the right circum
stances be sharp enough. The complication here is that the radiation is so
energetic that the nucleus recoils when it emits a gamma ray (an effect which
is negligible with the lowerenergy optical radiation) and there is a Doppler
broadening due to recoil. However, radiation can be emitted with the
recoil momentum being transmitted to the whole crystal instead of being
taken up by the single nucleus — a quantum phenomenon called the
Mossbauer effect — so that the recoil velocity is negligible, and the emission
is sharply monochromatic. In principle, we could then study the Doppler
broadening of such a line emitted from a crystal at a finite temperature ;
it should resemble Fig. 5.2.(a).
Cross
section
=fr^~
(a)
(6)
Energy (meV)
Fig. 5.3. {a) Schematic layout of apparatus for measuring the distribution of
the velocitycomponent in solids using the Mossbauer effect. A — device for
moving the source rapidly backwards and forwards. B — cooled crystal
emitting gamma rays. C — absorbing crystal at room temperature. D — counter
which measures gamma rays passing through the absorbing crystal, (b)
Absorption cross section per nucleus using iridium emitter and absorber, as a
function of energy difference between emitted and absorbed radiation,
allowing for recoil energy. The cross section falls by a factor e for 69 meV
change of energy. From Visscher, Ann. Phys. 9, 194 (1960).
5.1 Velocitycomponent distribution P[v x ] 103
But there is a further complication because we cannot make a gamma
ray spectrometer to disperse different wavelengths and measure the profile
of a line directly to the accuracy required. Instead we make use of the fact
that if a given nucleus can decay from an excited state and emit radiation
of energy E, then the same nucleus can absorb radiation of energy (E + E K ),
where E R is the known energy of recoil, and thereby become excited.
(Recoilless absorptions also occur but these can be disregarded.) We utilize
this effect as follows (Fig. 5.3(a)). We have two crystals, one a source kept
at low temperature so that the emitted gamma rays are monochromatic.
It is mounted on a device which oscillates backwards and forwards at high
speeds, comparable with thermal speeds of 10 4 or 10 5 cm/s. This motion
creates a controllable Doppler shift of frequency of the radiation which
then falls on a second crystal. This is the crystal the velocity distribution of
whose atoms we wish to explore ; it is kept at a high temperature (say,
room temperature). We measure the energy absorbed per second as a
function of the frequency of the incident gamma rays which we can
calculate from the known velocity of the source ; we must also allow for
the effect of the recoil energy of the absorbing nuclei.
The prediction is that the absorption should show a Gaussian variation
with gamma ray energy, of width 2yJ(E R kT) where T is the temperature
of the fixed crystal C and the factor J(kT) comes from the Boltzmann
distribution of thermal velocities. For iridium crystals with the absorber
at 300°K, the curve is shown in Fig. 53(b). (The energy axis has been
shifted to allow for the recoil energy and a spike due to recoilless absorp
tions has been deleted.) For iridium the gamma ray energy is 129 keV, so
that E R = 0.046 meV. The essential point is the resemblance of the absorp
tion curve to the Boltzmann curve for velocitycomponent distribution,
Fig. 5.1.
5.1.2 The pressure of a perfect gas
We have repeatedly emphasized that the P[v x ] distribution given by
Eq. (5.3) is obeyed by assemblies of molecules in all states of rarefaction
or condensation. The first application of this distribution law will, how
ever, be to a perfect gas and we will derive the PV = RT law again.
Consider a gas contained in a vessel, one wall of which is plane, the vessel
and gas being in thermal equilibrium. Let the y and z axes be drawn in
the plane of the wall, the x axis normal to it. Molecules travel towards the
wall, hit it and rebound. Thus each molecule suffers a change of momentum,
the force on the wall being the rate of change of momentum. The impacts
occur so frequently that this appears as a steady pressure (although
refined measurements would show that it does fluctuate about its mean
value) which we will now calculate.
104 Maxwell speed distribution and the equipartition of energy Chap. 5
We first make the rather unrealistic assumption that each molecule is
reflected elastically at impact. In other words, if its initial velocity before
impact is (v x ,v y ,v z ) then after impact it is ( — v x , v y , v z ); the normal
component is reversed but the others are unchanged (Fig. 5.4(a)). This
implies that the wall must be smooth on the atomic scale, which is of
course impossible. At the end of this section we will show that this assump
tion is unnecessary. Since only molecules travelling towards the wall hit it,
v x may have any value between and oo whereas v y and v z may have
values from — oo to + oo.
\a)
(t)
Fig. 5.4. (a) The impact of a molecule on a wall, assuming an elastic impact
and specular reflection from a smooth surface, (b) More realistically, a molecule
probably sticks to the wall for a finite time and is reemitted at random.
The nett changes of momentum in the y and z directions from one impact
are zero, but in the x direction there is a change of 2mv x per impact.
Let us select out of the whole assembly those molecules with a velocity
component between v x and (v x + dv x ). If there are a total of n molecules per
cm 3 , there are nP[v x ] dv x molecules in this class.
Any molecule at a distance equal to, or less than v x from the wall must
hit it during one second. Therefore the number hitting an area A of wall in
one second is equal to the number contained in a volume of area A and
length v x .
There are therefore nAv x P[v x ] dv x impacts per second on area A, each
bringing a change of momentum 2mv x ; so the force on this area is
2mnAvlP[v x ] dv x due to these molecules, since the force is the rate of
change of momentum. Thus the contribution to the pressure is
2mnvlP[v x ]dv x .
5.1 Velocitycomponent distribution P[v x ] 105
The total pressure from molecules of all velocities is found by integrating
over all relevant values of v x :
* go / m \ 1/2 f °°
P = 2mn J^ v 2 x P[v x ] dv x = 2mnU^ J i£ e"*'» r dt; x .
(There should be no confusion between pressure P and probability func
tion P[ ].) This integral can be evaluated by writing
mvl/lkT = a 2 ; , dv x = \ da,
\ m I
and by using one of the integrals on page 72 :
I
e* 2 da = ^.
o 4
Then P = nkT, where n is the number of molecules per cm 3 . Let V be
the volume of M grams of gas, containing N molecules ; then n = N/V.
Thus
PV= NkT= RT.
This is the perfect gas law which we have already deduced by considering
the potential energy (section 4.4).*
We based this calculation on the assumption that the molecules are
reflected specularly on impact, which implies among other things that the
walls are smooth compared with molecular dimensions. It is much more
realistic to assume that the individual impacts are as shown in Fig. 5.4(b) —
a molecule sticks to the wall for a finite time and is reemitted later. In sec
tion 9.4.1 we show that 10" 8 s is a reasonable estimate of the 'sticking time'
under certain conditions. While this is a short time in ordinary terms, it is
long on the molecular scale; the atoms of the wall perform 1,000 or
10,000 vibrations in that interval. When a molecule jumps off again there
fore, it does so at an angle and at a speed unrelated to the incident angle
and speed. Let us recast the argument about the momentum change.
Whatever the details of the impacts, we will nevertheless assume that the
gas is in equilibrium. Then the number of molecules within any range of
velocity must remain constant. Therefore if nAv x P[v x ] dv x molecules with
velocity between v x and (v x + dv x ) are removed from the gas in each second
by sticking to the area A, the same area must reemit the same number per
second to preserve the equilibrium. Therefore, the overall momentum
change is the same as we calculated before, even though any one individual
molecule may be emitted with quite a different velocity from that at impact.
This demonstrates that in order to derive the PV = RT law, we do not
* This cannot pretend to be a new result — it is merely a confirmation of the perfect gas law
by a different method.
106 Maxwell speed distribution and the equipartition of energy Chap. 5
have to assume any special form of impact at the walls, but merely that the
assembly is in thermal equilibrium.
5.2 SPEED DISTRIBUTION P[c]
The previous discussion leads to the statement that the probability
that any one molecule in any assembly (solid, liquid or gas) in thermal
equilibrium has velocity components between v x and (v x + dv x ), v y and
(v y + dv y ), v z and (v z + dv z ) is equal to the product of three factors:
P[v x , v y , v z ] dv x dv y dv z = P[v x ] dv x P[v y ] dv y P[v z ] dv z
l m \ 3/2
\2nkTj
e M»i + v$ + vl)l2kT dVxdv ^ dVt
= /_m_\ 3 e «c*/2*r d d d (56)
\2nkTj
where we have written the total speed
c = (v 2 x + v 2 + v 2 ) 1 ' 2 . (5.7)
We will now find the probability that the total speed of a molecule
lies between limits c and (c + dc). We will no longer be concerned with
velocity components v x , v y and v z , or in other words all molecules moving
with the same speed will be classed together, no matter which direction
they are moving in. We have, in fact, to integrate over all angles. Thus we
require an expression of the type P[c] dc.
5.2.1 Transformation of coordinates
When we were concerned with v x , v y and v z , the velocity of any particle
was represented by a point within a framework of (v x , v v , v z ) axes, and the
number of points within a parallelepipedal element of volume dv x dv y dv z
was counted in a manner analogous to Fig. 4.4. In this new system of
counting, the state of the assembly is represented by exactly the same set
of points, but the speed c of a molecule is represented by the length of a
radius vector from the origin and we no longer have a simply shaped
element of volume.
We can transform from dv x dv y dv z to dc using the following intuitive
method. The range between c and (c + dc) is represented by a spherical
shell bounded by spheres of radius c and (c + dc), Fig. 5.5. The required
number of points, integrated over all directions, is therefore proportional
5.2 Speed distribution P[c] 107
to the volume of the shell which is 4nc 2 dc. We can therefore write
' m \
3/2
P[c]dc = 4n\
2nkT]
c 2 e mc2/2kT dc
(5.8)
for the probability of finding a molecule with speed between c and (c + dc)
irrespective of the direction in which it is travelling. This is the Maxwell
speed distribution, the result we set out to find.
Fig. 5.5. Counting of representative points in a shell
bounded by spheres of radius c, (c + dc).
A more rigorous method of transforming from dv x dv y dv z to dc is as follows.
The velocity of a molecule is uniquely specified by its speed c, the angle 9 it
makes with an axis (which we may take without loss of generality to be coinci
dent with the z or v z axis) and another angle <f> which it makes with a plane
through this axis (which we may take to be the xz or v x v z plane). This is a system
of spherical polar coordinates, Fig. 5.6(a). 9 can vary from to n, cf> from to
2n. When c is varied by dc, 9 by d9, <f> by d^>, a volume
(dc)(c d9)(c sin 9 d</>) = c 2 sin 9 dc d9 d<f>
is generated, Fig 5.6(6). Thus the probability that the velocity of the molecule
lies between c and (c + dc), at angles between 9 and {9+d9\ (f) and ((f>+d(f)) is
/ m \ 3 > 2
P[c, 9, 0] dc d9 dtf> = — ~\ c 2 e  mc2/2fcr sin 9 dc d9 dd>
\2nkTJ
which is exactly equivalent to Eq. (5.6). P[c] does not contain 9 or <£, which
merely expresses the fact that all directions are equally probable. If this
108
Maxwell speed distribution and the equipartition of energy Chap. 5
expression is integrated over all possible directions, the little volume element
becomes the spherical shell of Fig. 5.5. We then recover the Maxwell distribution
(Eq. (5.8)) since
f f sin0d0<ty*=4w.
Jo Jo
(b)
Fig. 5.6. (a) Spherical polar coordinates, (b) Generation of volume element by
variations dc, d6, d(f>.
5.2 Speed distribution P[c] 109
5.2.2 The Maxwell distribution
The form of the Maxwell distribution is shown in Figs. 5.7(a) and 5.7(b).
The curves are unsymmetrical (in contrast to the curve of Fig. 5.1 for the
velocitycomponent) and pass through the origin and only positive values
of speed have any meaning. Writing the ratio
$mc 2 /kT = a 2
so that a is a dimensionless quantity proportional to the speed, the
distribution law becomes
PO]dff = ^<7 2 e~ ff2 d<7
and this function is plotted in Fig. 5.7(a). All masses and temperatures are
represented by this one graph. The area under the curve is unity — this
follows directly from the normalization of the individual components
and expresses the fact that the speed of a molecule is certain to lie between
zero and infinity. We can graph P[c] directly however, if we select any
given mass of molecule and any given temperature. Two such curves, for
M = 28 and T = 100°K and 1,000°K respectively are shown in Fig. 5.7(b).
The areas under these graphs are also unity. When the temperature is
raised, the maximum of the curve moves to higher values of speed, as is
to be expected. At the same time, the spread of speeds increases, the curves
becoming broader.
Three characteristic values of the speed can be usefully defined — the
most probable speed where P[c] goes through a maximum, the mean
speed and the rootmeansquare speed. They do not differ very greatly
from one another.
The most probable speed c m can be found by setting dP/dc = which
gives
c m = (— ) • (5.9)
The mean speed c is found (following section 4.2.1) from
f 00 2 l2kT\ 1/2
Jo cp[c]dc= >M  U28c  l5l0)
The mean value of c 2 , called the mean square speed c 2 (which is useful in
finding the mean kinetic energy and is not equal to the square of the mean
speed), is given by
3kT
c 2 P[c]dc = . (5.11a)
m
c =
f
Jo
110 Maxwell speed distribution and the equipartition of energy Chap. 5
P[&\
3x10 
2x10
1x10
cm/s
Fig. 5.7. (a) The Maxwell distribution of speeds expressed in terms of the
dimensionless measure of speed a = (mc 2 /2kT) 112 . (b) The Maxwell
distribution for M = 28, T= 100°K and T = 1,000°K. The unit of
speed is 1 cm/s. The probability that the speed of a molecule lies
between 20,000 and 20,001 cm/s is 3 x 10~ 5 at 100°K; in 28 g of this
substance, there would be nearly 2 x 10 19 such molecules.
5.2 Speed distribution P[c] 111
The square root of this quantity is called the rootmeansquare speed
c
— /3/cT\ 1/2
c rms = (c 2 ) 1/2 =p^ = 1.225 c m . (5.11b)
The value of the mean kinetic energy of all the molecules follows from the
meansquare speed :
mean kinetic energy = \mc 2 = \kT. (5.12)
This result, Eq. (5.12), is a special case of a very general theorem, the
equipartition of energy, which will be dealt with at length later in this
chapter. In view of its importance, it will be derived again in another way.
Instead of finding the mean value of c 2 directly, we may proceed by finding
the mean values of v 2 and v 2 and v 2 . Since the components are independent
variables, the mean values are additive :
c 2 = vl + v^+v^. (5.7)
In outline, the calculation is as follows. The mean value of v 2 is given by
_ r °° / m \ 1/2 f 00
K = J v 2 x P[v J dv x = \—A J v 2 x exp(  mvl/lkT) dv x .
The same integral has been encountered in section 5.1.2 in the calculation
of the pressure of a perfect gas, although the limits of integration are now
— oo to oo, instead of to oo. The result is that
uj = kT/m. (5.13)
The same expression holds for the other two contributions, so that once
again
c I =3kT/m (5.11a)
and the mean kinetic energy is \kT.
5.2.3 Magnitude of the characteristic speeds
The magnitude of these average speeds can be calculated once (kT/m) 1 ' 2
is known, or (RT/M) 1/2 . Putting R = 8.31 Jrnol" 1 deg 1 , and referring
to nitrogen (M = 28) and T = 0°C = 273°K, this factor is 2.8 x 10 4 cm/s,
so that the r.m.s. speed is nearly 5 x 10 4 cm/s. Since M varies between 2
and 200 from the lightest to the heaviest element, these speeds all lie in
the range 10 4 10 5 cm/s for the elements, whether solids, liquids or gases,
at room temperature.
Sound waves consist of ordered movements of molecules, in which
energy and momentum are propagated through the medium from
112 Maxwell speed distribution and the equipartition of energy Chap. 5
molecule to molecule superimposed on the random movements of the
molecules. The ordered motion in a solid, for example, might consist of
sinusoidal vibrations of the molecules in the direction of propagation or
transverse to it, and being a collective mode of motion it can be separated
from the random thermal motion on which it is superposed. It is shown in
standard texts on wave motion that in all substances the speed of propaga
tion of the sound is comparable with the characteristic speeds of the
Maxwell distribution. The calculation for gases is given in section 5.4.2.
In air, for example, at 300° K the mean speed is 470 m/s, the speed of
sound which consists of longitudinal vibrations only, is 350 m/s. In copper
at the same temperature, the mean speed is 316 m/s, while the speeds of
longitudinal and transverse sound vibrations through an unbounded
volume of the metal are 456 and 225 m/s respectively.
5.2.4 Experimental verification of the P[c] distribution
The Maxwell distribution has been verified experimentally for gases.
The most direct methods depend on two techniques — the production of
molecular beams and the measurement of their speed distribution using a
timeofflight or chopper method, analogous to Fizeau's method for
measuring the speed of light. The experiments of Lammert (1929) are
typical. Mercury was heated to 100°C in an oven which had a small hole
in one side through which the vapour could escape as a molecular beam.
The entire apparatus was under high vacuum. Inside the oven the mole
cules were practically in equilibrium at 100°C since the rate of loss of
molecules was small, so the speeds were distributed according to the
Maxwell law. Once a molecule escaped through the hole its speed was not
likely to change since it probably never collided with another molecule.
Thus though the beam travelled through a space which was not maintained
at the same temperature, it was a sample of those molecules inside the
oven whose direction of travel happened to lie in the direction of the beam.
Let us call this the xaxis — then the method produced molecules whose
total velocity vector (of magnitude c) was parallel to x. It was the c
distribution of the beam which was measured ; it was not possible to
measure the velocity component v x of molecules not travelling parallel to
x, for such molecules were simply not in the beam. Inside the oven, any
molecule travelling towards the hole with speed c could escape in time dt
if it were within a distance c dt of the hole. Hence the number of such
molecules escaping per second is proportional to cP[c], which means that
the distribution function of the speeds of the molecules in the beam was
of the type c 3 exp(mc 2 /2kT), not the Maxwell distribution though
clearly derived from it.
The speeds were found by passing the beam through a velocity selector
5.2 Speed distribution P[c]
113
consisting of two discs (Fig. 5.8(a)), each having 50 narrow radial slits in
it, rotating rapidly on a common axis parallel to the beam. The disc
further from the oven was turned through a small angle 5 with respect to
the first. A molecule passing through a slit in the first disc and travelling
with a speed c took a time l/c to travel the distance / between discs. If
the speed of rotation of the discs on their axis was co rad/s, the molecule
met a slit in the second disc if 8 = col/c ; or if the angular width of each
slit was 2y, molecules with speeds in the range col/(S + y) to col/(S — y) could
get through both slits; molecules with speeds outside this range were
stopped, co could be varied so as to select different ranges of speeds. In the
edge of the discs were wider slits of total angular width equal to that of all
the narrow slits together, but these were so wide that they passed molecules
W
w
+ 2y
= =•.=•= =
=■ = =r ^= =
V
J
l/C
/
it
&
\a)
r^'H
Fig. 5.8. (a) Slotted wheel with slits acting as a velocity selector. Schematic
layout: W — wheels (seen edge on), rotating on common axis; O — oven;
C — cold surface. All located inside a high vacuum.
20%
10% 
I
 1 !
1
 ,—l L^
1 1 1 1 1 1— L 1 fc
4 x10 4 cm/s
(b)
Fig. 5.8. (b) The percentage of the total intensity of a molecular
beam within given limits of speed. Mercury vapour at 100°C.
calculated from Maxwell distribution. ob
served. Data from Lammert, Z. Physik 56, 244 (1929).
114 Maxwell Speed distribution and the equipartition of energy Chap. 5
of all speeds. The two emergent beams, selected and unselected, fell on a
surface cooled with liquid air which trapped the molecules, and their in
tensities were compared by finding the times needed to produce deposits
which were just visible. In one experiment, / = 6 cm, <5 = 4.18°, 2y =
Tiro rad and the discs rotated at 70 rev/s so that the range between 340 and
390 m/s was selected. The beam through the wide slits was just visible
after 4' 40", the other after 51' 45" ; thus 9.0 % of all the molecules in the
beam had speeds in this range. The complete plot is shown in Fig. 5.8(b),
which gives the observed intensities and that predicted for a beam with
M = 200, T = 373° K. Other methods of measuring the intensity of the
beam have been used.
It has not been possible to measure the speed distribution of molecules
in solids directly. Xray and neutron diffraction measurements can give
information about the distribution of the amplitudes of vibration of
molecules about their lattice points, but it would be necessary to know the
frequencies of the vibrations to convert these measurements into speeds.
Evidence of a different kind comes from the passage through solids of slow
neutrons which suffer large changes of momentum whenever they collide
with atoms. After several such collisions the neutrons come into thermal
equilibrium with the solid. The speed distribution of an emergent beam
(which can be determined by a timeofflight technique) is always found
to be of the Maxwell type appropriate to the temperature of the solid.
Only if the atoms in the solid had the same distribution (though corre
sponding to their heavier mass) would this result be found.
We will see later that specific heat measurements give an insight into
the velocity and speed distributions in solids and that these show that the
Maxwell distribution only holds at sufficiently high temperatures. At low
temperatures, the motion of the atoms is not described accurately enough
by classical mechanics. Quantum mechanics has to be used instead and
this leads to different results.
5.3 THE EQUIPARTITION OF ENERGY
When we write down the Boltzmann factor for a particle in an assembly
in thermal equilibrium, we have to know its total energy E. So far, we have
made use of the fact that E can be split up into kinetic and potential energies
which depend on different and independent variables, and by separating
those variables we were able to deduce some useful results. One of these
was (Eq. (5.12)) that when the kinetic energy of one molecule in an assembly
is \mv 2 x +\mv 2 y +\myl, then the total kinetic energy of all N molecules in
thermal equilibrium at temperature T is f JVfeT, and the mean energy of
one molecule is f/cT. Thus the mean energy does not depend on v x or v y
5.3 The equipartition of energy 115
or v z in any way but only on the temperature T. It is the purpose of this
section to show that this result is a special case of a much more general
theorem, which is one of the most important results of classical physics.
It is useful to begin by noting that so far we have not written down expli
citly all the forms of energy which a molecule can possess. (We will assume
that external fields of force, such as the Earth's gravity, or electric or
magnetic fields are absent ; this restricts the discussion but the results are
nevertheless of significance). The only form of energy which has been
explicitly used in calculations has been the translational kinetic energy
of the centre of mass, consisting of the three terms of the type \mv 2
mentioned above. We will now consider the energy of angular rotation
which every molecule of finite size must possess, and the energy of vibration
due to % internal oscillations inside a molecule containing more than one
atom, or due to the oscillations of a molecule about its equilibrium position
inside a solid lattice. Our procedure should really be to consider what
conjugate momenta and position coordinates are needed to write down
the Hamiltonian expression for the energy. Here we will quote some
results without proof and we will use velocities rather than momenta.
First let us consider a rotator, that is a rigid body of arbitrary shape,
rotating without any constraint. Then its energy of rotation can be written
E r = \l^\ +$I 2 oj 2 2 +i/ 3 e»5 (5.14)
where the J's are the principal moments of inertia and the co's are the
angular velocities about the three mutually perpendicular principal axes
of the body. This is the kinetic energy term. We will not consider any
potential energy which depends on the angular orientation of the body.
Thus, if a molecule of a perfect gas can be considered as a rigid rotator,
the energy must have six terms in it, three of the type \mv 2 x and three of
the type jlco 2 . Classically, every co can vary between — oo and +00.
Now let us consider a linear simpleharmonic oscillator, that is a point
particle which oscillates about an equilibrium position along a line or,
for example, a pair of particles whose oscillation about their centre of
mass is in one dimension only. Then two coordinates, a displacement x
and velocity v x are needed to specify the instantaneous state of the
oscillator. The energy is partly kinetic, partly potential. For the single
particle :
E = jmv x +j(xx 2
where a is the restoring force per unit displacement. It follows that a
particle capable of simple harmonic oscillations in three dimensions, such
as a molecule in a solid lattice, has six terms in the expression for its energy
of oscillation. It is, in principle, possible for all the velocities and all the
116 Maxwell speed distribution and the equipartition of energy Chap. 5
displacements to take any values between + oo and — go. (Notice that E
for the onedimensional oscillator could be written as ?ooco where x is
the amplitude ; the sum of the two terms is a constant. But this expression
for the energy is not appropriate for the present purpose, since it is
required to write it in terms of those coordinates which are needed to
specify the instantaneous state of the body completely ; for an oscillator
in thermal equilibrium, the amplitude is not constant. In just the same way,
the translational kinetic energy must be written in terms of v x , v y and v z
and not merely of c.)
Typical terms in these expressions are
\mv 2 x , ?Ico 2 ,
i<xx 2
and all of them are of the same type, a constant times the square of a
coordinate. Such terms are called degrees of freedom*
Because of the similarity in the form of these terms we can say at once
that angular velocities, for example, are distributed in just the same way
as translational velocities. The probability that co 1? a single component
of the angular velocity of a molecule in thermal equilibrium, has a value
between co x and (a> x + da> 1 ) is
/ / \ 1/2
P\ca l '\dxo 1 = U^T exvilMfi kT ) da >i
an expression completely analogous to P[v x ] dv x (Eq. 5.3). Further it was
proved that the mean value of \mv 2 x was \kT (which follows from Eq. (5.13)).
Exactly the same result must hold for the mean value of any of the other
terms in the energy. The mean value of \l<£>\ for an assembly in thermal
equilibrium must be jkT. Exactly the same result must hold for the mean
value of \l(o\ and \l(X)\. The mean value of \a.x 2 for an assembly of oscil
lators in thermal equilibrium must also be \kT.
This is the theorem of the classical equipartition of energy. Every degree
of freedom, that is, every quadratic term in the energy — translational,
oscillatory or rotational — contributes y/cTto the mean energy. An assembly
of N particles in thermal equilibrium, each with/ degrees of freedom has a
mean energy per particle of \fkT, a total energy oijfNkT.
The implications may be stated in another way. Consider an assembly
of molecules, each one capable of several sorts of motion (oscillation about
* It has become customary in statistical mechanics to use the phrase 'number of degrees of
freedom' in this way, namely to mean the number of quadratic terms in the energy. The
reader should be warned that this is in conflict with the more usual definition in ordinary
mechanics which restricts the number of degrees of freedom to the number of kinetic energy
terms.
5.4 Specific heats C p and C v 117
a lattice point together with rotation about one or more axes, for example :
or translation throughout a volume together with one or more modes of
internal oscillation of each molecule). Then according to the equipartition
theorem, in thermal equilibrium each possible mode of motion will be
excited and the amount of energy in each mode is predictable if the tem
perature is known.
5.4 SPECIFIC HEATS C p AND C v
One purpose of the next section is to describe experiments which provide
the most direct tests of the validity of the theorem of the equipartition of
energy and of the classical mechanics on which it is firmly based. The
mean energy of each molecule in a mass of material cannot be measured
in any direct experimental way, but specific heats can be measured and
these are closely related to it. Our first task will be to define specific heats
and to develop some relations between them, then to describe the experi
mental results. It will emerge that the theorem of equipartition of energy
is not of universal validity, and that this is because classical mechanics
cannot adequately describe atomic vibrations and oscillations under all
conditions and must be replaced by quantum mechanics. Historically the
failure of the equipartition theorem to predict the correct specific heats of
gases was one of the first symptoms of the inadequacy of classical mech
anics ever to be observed.
If dQ is the quantity of heat energy supplied to a standard mass of a
substance under certain conditions and the temperature is thereby raised
by dT, then the specific heat is defined as dQ/dT under those conditions.
The most convenient standard mass is the mole and the appropriate units
are J/mol deg. Grams can be used instead of gram mols, and it is occasion
ally useful to deal with the specific heat per unit volume of a substance.
Practically all bodies when heated under conditions of constant
pressure will expand. This expansion absorbs energy in two possible ways.
First the external forces acting on the body are pushed back. At the same
time, the mean distance between the atoms of the body increases and the
potential energy of interaction between atoms is changed — under normal
conditions it is increased. It is therefore useful to define two limiting sets of
conditions under which specific heats can be measured — constant volume
when the body is constrained by external forces not to expand and all the
heat energy goes into raising the temperature, and constant pressure where
some of the energy is absorbed by the process of expansion. The specific
heat at constant volume is denoted by C v , the specific heat at constant
pressure by C p .
118 Maxwell speed distribution and the equipartition of energy Chap. 5
5.4.1 The difference (C p  C v )
C v is the quantity which is more readily calculated theoretically for any
assembly of atoms or molecules since potential energies of interaction
between atoms are constant under conditions of constant volume. In fact
if E is the mean energy of a standard mass of material, equal to the mean
energy of one molecule multiplied by the number of molecules, then
C p is the quantity most easily measured experimentally since laboratory
work is usually carried out under conditions of constant pressure. The
difference between the two quantities, (C p C v ), is therefore of significance,
to compare experimental measurements with theoretical predictions.
For a perfect gas, (C p — C v ) is easy to calculate because the potential
energy of interaction between the atoms is always zero. The energy E,
which in thermodynamics is conventionally called the internal energy,
therefore does not change when the gas expands at constant temperature.
The extra energy absorbed by the expansion at constant pressure must be
equivalent only to the work done in overcoming the external pressure.
The energy required to produce a given temperature rise under conditions
of constant pressure is equal to that needed to produce the same tem
perature rise when the volume is kept constant, plus the energy absorbed
as work done against the external pressure. It has already been shown
(4.1.3) that if a body expands in volume by dV against pressure p, the
work done is p dV (writing the A dx of 4.1.3 as dV). Thus
C p dT= C v dT+PdV
or
p
C n C=P—\ (5.16)
It must be stressed that this holds only when the internal energy of the
substance does not change with volume. For a perfect gas, PV = NkT,
whence it follows that
C p C v = Nk = R. (5.17)
Different gases can have appreciably different values of the molar specific
heats C p and C„, but the difference (C p C v ) for any one gas must always
be equal to 8.31 J/deg, at all temperatures.
For a substance whose internal energy varies with volume, the difference
(C p  C v ) must also vary with volume, and must involve the compressibility
5.4 Specific heats C p and C v 1 19
and expansion coefficient. It is shown in standard thermodynamic texts
that
~ IdE \
dV
c — c =
P+\t\
—
V
_ \dV) T _
\dT
and using the second law of thermodynamics it may be shown that this
can be written
C p C v = TP 2 KV (5.18)
where ft is the volume coefficient of thermal expansion, K the isothermal
bulk modulus :
o 1 l dV \
'vM, (519)
K=  V (%) T < 310 >
and V is the molar volume if C p and C v are molar specific heats. Inserting
typical figures, (C p  C v ) for most liquids and solids at room temperature
are of the order of R/3 and R/10 respectively. Though the volume expansion
is very small, the pressures which would be needed to counteract it are
very large so that their product is by no means negligible. For very hard
substances such as diamond, (C p  C v ) is however very small, of the order
offl/1,000.
Experimental methods for measuring C p for solids and liquids are
straightforward in principle. A known quantity of heat is introduced into
the specimen, usually from an electric heater in good thermal contact, and
the temperature rise measured, usually with a thermocouple or resistance
thermometer. It is essential to reduce heat losses from the specimen as
much as possible. The specimen is isolated in a high vacuum and the
surroundings are heated separately so as to follow the temperature of the
specimen as closely as possible. It is also essential to make sure that the
temperature is uniform throughout the specimen before any readings are
taken. From the measurements of C p and a knowledge of the expansion
coefficient and compressibility, C v can be calculated.
5.4.2 Ratio of specific heats C p /C v
The methods just described are not very suitable for taking measure
ments on gases, particularly at the low densities required for them to
behave like perfect gases. Their specific heat per unit volume is small, and
the heat absorbed by the containing vessel may be comparatively large.
120 Maxwell speed distribution and the equipartition of energy Chap. 5
While it is therefore difficult to measure either C p or C v directly, it is
however simple to measure their ratio
7 = C p /C v . (5.20)
A knowledge of y, together with the fact that (C p — C v ) = R, allows both
C p and C v to be found :
R y R
c — c = —
y — 1 v y—\
y is measured by studying adiabatic changes in a gas.
Adiabatic changes are processes (such as changes of volume) which
occur so slowly that the assembly of molecules never departs far from
thermal equilibrium even though a finite time is required to reestablish
the Boltzmann distribution disturbed by the change ; and at the same time
no heat must flow into the body, d<2 = 0. The adiabatic compression of a
gas must be a slow process in which heat cannot escape or enter and since
work is done on the gas its temperature must rise. In practice, a compromise
has to be struck between the need for slowness of change and the require
ment of thermal isolation, and it is usually necessary to accomplish any
changes quite quickly.
Sound waves through a gas are adiabatic. They consist of local alterna
tions of compression and rarefaction which produce small local alterations
of temperature and these are not dissipated. It is shown in standard texts
on wave motion that the speed of a sound wave is given by
//bulk modulus \
Speed = 7( density ) < 3 ' 21)
where the bulk modulus is — V(dP/dV) under adiabatic conditions. We
now calculate this quantity for a perfect gas.
Consider 1 mole of a gas which does work by changing its volume and
into which heat also flows, so that the pressure, volume and temperature
all change infinitesimally. Then
dQ = C v dT+PdV (5.21)
Now the relation PV = RT also holds, so that
PdV+VdP = RdT
and we can use this to eliminate dT:
dQ = ^(PdV+VdP) + PdV.
R
5.4 Specific heats C p and C v 121
Using the relation C p C v = R, this becomes
RdQ = C p P dV+C v VdP.
If dQ = 0, the adiabatic bulk modulus
~ v Itt] =< ^P = yP> (522)
so that the speed of sound is given by
Measurements of the speed of sound therefore give y. At the same time,
the reason that the speed of sound is comparable with but not equal to
the mean speed of the molecules, Eq. (5.10), can be appreciated, since they
differ only by a factor y/{ny/S) which lies between 0.75 and 0.8 for most gases.
The speed of sound can be measured from the transit time of a pulse of
sound (such as a gunshot) between two points a known distance apart, or
from simultaneous measurements of frequency and wavelength of a
sinusoidal note. Ruchhardt's method is an interesting alternative way of
finding the adiabatic bulk modulus yP directly. A resonator consisting of
a vessel of a few litres' capacity and filled with the gas under examination
is fitted with a vertical glass tube into which a steel ball fits closely, Fig. 5.9.
The ball must be accurately spherical, the tube accurately cylindrical and
the clearance between the two must be of the order of 0.001 cm so that if
the ball is at the top of the tube it sinks only very slowly due to leakage of
gas past it. If the ball is given a sudden displacement from its position of
near equilibrium it oscillates, typically with a period comparable with one
second. The restoring force is the adiabatic elasticity of the gas in the vessel.
The system may be considered as a Helmholtz resonator with the ball
acting as a heavy driving piston.
If the displacement is x, the change of volume is Ax, where A is the
cross sectional area of the tube. Then from the definition of elasticity, the
change in pressure is
dV yP
dP = yP = ^A X
where Kis the volume of the gas. The force on the ball is the pressure times
the area
yPA 2
F = — — x
122
Maxwell speed distribution and the equipartition of energy Chap. 5
which shows that the motion is simple harmonic, of period
,. Vm
x = 2n
yPA'
where m is the mass of the ball. Observation of the period therefore gives y.
OT—
ft
direction of
displacement x
and of positive
forced
— ' rest position
(a)
[b)
Fig. 5.9. (a) Simplified version of Ruchhardt's apparatus, {b) Ball dis
placed from rest position.
5.4.3 Results of specific heat measurements
The previous sections have described the principles behind the methods
of measuring specific heats and how C v can be deduced from them. For
solids and liquids the most easily measured quantity is C p and then
{C p — C v ) can be calculated knowing the expansion coefficient and com
pressibility.
For gases at low pressures the most easily measured quantity is C p /C v
and since (C p — C v ) always has the value R, C v can be calculated. Thus,
since C v = (dE/dT) v we have a searching method for testing the truth of
the law of equipartition of energy.
We will deal with the results for solid elements first. The predicted molar
specific heat for a solid having one atom (or ion or rigid molecule) at each
5.4 Specific heats C p and C v 123
lattice point, capable of oscillating harmonically in three dimensions is
C * = N [£ffr T ) = 3R (524)
which has a value near 25 J deg" 1 mol~ * at all temperatures. This rule is
in fact obeyed by a large number of solid elements at room temperature.
It was discovered empirically in the early nineteenth century by Dulong
and Petit and was even used as a guide in determining atomic weights in
the days when there was some ambiguity about them. But at low tempera
tures, the specific heat always falls below this predicted value. The graphs
for all elements have the same form and can be superposed by merely
altering the horizontal scales ; Fig. 5.10 shows the curves for three elements.
At high temperatures, copper and lead have their expected specific heats
of 3R. At 300° K, however, the value for copper begins to fall off, at 150°K
it is decreasing rapidly with temperature, and at 50° K it is very small
indeed. Argon does not begin to show a decrease until quite low tempera
tures are reached. Diamond follows the same sort of curve except that it
would only approach its full value of 3R if the graph were extrapolated to
almost 2,000° K. These results for low temperatures are in sharp conflict
with the equipartition law.
The same kind of discrepancy occurs for gases as well. Atoms have finite
sizes so that they must each have three principal moments of inertia. An
atom of a monatomic gas should therefore have three quadratic terms for
its energy of rotation of the type ^hcoi as well as three terms of the type
\mv\ for its translational energy. Thus there should be 6 degrees of free
dom per atom each contributing \kT to the energy, so that C v should
be equal to %R and y = f = 1.33. In fact helium and other rare gases
including argon, as well as mercury, all of which are monatomic, are found
to have y quite close to 1.67 so that C v = f.R. Evidently three of the
possible degrees of freedom are not excited.
5.4.4 Quantum theory and the breakdown of equipartition
These evident failures of the equipartition theorem mean that some of
the ideas of classical mechanics itself are at fault. For solids, the explanation
of the small specific heats at low temperatures hinges on the fact that the
frequency of vibration of the atoms or molecules or ions about their lattice
positions is very high, of the order of 10 12 or 10 13 cycles per second (see
sections 3.6.2, 3.7 and 3.8.2 for estimates of the Einstein frequency). Now it
is a result of quantum theory that the energy of an oscillator of frequency v
can only take discrete values, jhv, § hv, f hv and so on, where h = 6.6 x 10" 27
erg. s or 4 x 10~ 15 eV. s is Planck's constant. We no longer separate the
energy into potential and kinetic contributions, but talk only of the total
energy and this can only change by discrete amounts hv.
124
Maxwell speed distribution and the equipartition of energy Chap. 5
3/?
2/?
Equipartition holds
Copper
/Lead
/ /
'/
100 200
Temperature
(a)
<3*
50 °K
Temperature
(6)
300 K
3/?
Equipartition holds
2/?
R
n
£
2/h/
A
k
Temperature
(c)
Fig. 5.10. (a) The specific heat C„ of lead and copper as a function of tempera
ture, (b) Specific heat data for solid argon. C v was derived from the measured
C p using Eq. (5.18); this can be checked using K (the reciprocal of the com
pressibility) and V from Fig. 3.13 and /? derived from the density data :
T°K
20
40
60
80
jSdeg"
0.4
1.2
1.5 1.8 xlO 3
(c) Curves for C v for all substances can be superposed by altering the horizontal
scale as shown; the Einstein frequency for copper and lead can then be
estimated by comparing the two curves.
5.4 Specific heats C p and C v 125
In solid organic crystals or solidified rare gases or other soft substances,
where v E is of the order of 10 12 cycles per second, the energy of a vibrating
molecule can only increase in steps of about 7 x 10 ~ 15 erg or 0.004 eV. At
very low temperatures (when kT is small compared with this quantity, so
that the factor exp(hv E /kT) representing the probability of a molecule
being in the next higher state is small ; to be precise, when T is small com
pared with 40°K), the oscillators can vibrate only in their lowest state. The
energy of oscillation is therefore hardly changed by changing the tem
perature, so the specific heat is very small. At high temperatures (say
400° K in this case), the discreteness of the vibrational energy levels makes
little difference to the mean amount that the oscillators can take up, and
the classical law holds. In hard substances like ionic crystals or diamond,
where the Einstein frequency is 10 or 50 times greater than in soft crystals,
the specific heat is low even at room temperature.
To explain the unexpectedly low specific heats of gases at ordinary
temperatures we have to invoke the fact that the energy of a rotating body
is quantized. It can only possess discrete values and can only change by
discrete amounts. For monatomic helium molecules, the steps are of the
order of 10 eV in size ; thus the minimum energy of rotation that a helium
atom may possess is of this order. If its total average energy is very much
less than this amount, it is highly improbable that it should be rotating
at all. Since kT at room temperature is of the order of ^j eV, it is evident
that rotation of the atoms in helium at room temperature does not take
place. For monatomic mercury, the steps are of the order of 100 times
smaller, but still the thermal motion is not sufficient to excite a significant
number of mercury atoms into rotation. Thus they only possess three
translational degrees of freedom and the ratio of specific heats is 1.67.
We have just discussed monatomic gases ; there is disagreement with
the classical equipartition theorem for diatomic molecules also. Many
such molecules — like hydrogen, oxygen, chlorine — are dumbell shaped, as
in Fig. 5.11. We would expect each molecule to be capable of translation
(three energy terms of the type \mv\ where m is the mass of the molecule
OO db OOe
(a) [b) (c)
Fig. 5.11. Modes of rotation of a dumbellshaped diatomic
molecule. Specific heat measurements indicate that (a) and
(b) occur, (c) does not.
126 Maxwell speed distribution and the equipartition of energy Chap. 5
and v x is the velocity component of its centre of mass), as well as rotation
(three terms of the type jliCoj) and also vibration in and out along the
line joining the two atoms as if they were connected by a spring (two terms,
for potential and kinetic energies as for any harmonic oscillator). This
gives a total of 3 + 3 + 2 = 8 possible degrees of freedom corresponding
to y = ^ = 1.25. In fact, it is observed that many diatomic gases have y
close to 1.4 (that is, j) at room temperature which seems to imply that
only 5 degrees of freedom are excited at room temperature. Thus, the
value for H 2 is 1.408; for N 2 it is 1.405, for NO 1.400 and for 2 1.396.
Again, in order to explain these data we invoke the quantization of energy.
The rotation (c) resembles that of a single atom and is again eliminated
because the minimum energy of rotation is much larger than kT at
ordinary temperatures. Of the 8 degrees of freedom, 7 remain ; evidently
2 more must be eliminated. It is not obvious whether these are the two
terms in the springlike internal vibration, or the two rotations (a) and
(b) in Fig. 5.11. An exact analysis in fact shows that the vibration is of
very high frequency and it is the energy of that motion which is not equi
partitioned ; the two rotations (a) and {b) are excited.
The quantization of energy is of course not invoked only to explain the
specific heat data ; other phenomena are also explained at the same time.
For gases, the most direct confirmation comes from their spectra, from
the interpretation of which it can be confirmed that certain rotations and
internal vibrations do not take place at ordinary temperatures.
We have seen that at temperatures where kT is small compared with
the energy steps, the value of C v is lower than predicted on classical equi
partition theory. We would however expect that at sufficiently high tem
peratures, more modes of rotation or vibration will be excited. An increase
of C v and a decrease of y at high temperatures is therefore to be anticipated.
This is indeed found experimentally — notably with hydrogen whose C v
changes from jR below 50° K to %R at room temperature. This must mean
that at 300° K, 5 degrees of freedom are excited but at 50° K only 3. Further,
it must mean that two of the modes of rotation or internal vibration must
have energy steps whose magnitude is comparable with kT where T is
100°K or 200°K — say 0.01 eV. To take another example, chlorine is ob
served to have y equal to 1.355 at room temperature and this corresponds
neither to 5 degrees of freedom (y = 1.4) nor to 6 degrees (y = 1.33).
Presumably at room temperature 5 of its degrees are fully excited and
another is partially excited and the value of y ought to decrease to 1.33
at temperatures not too far above room temperature.
Among polyatomic molecules, H 2 S has y = 1.340 which is close to the
value f , corresponding to 6 degrees of freedom; this is taken to mean
that the internal vibrations are not excited but all 3 of the rotational
5.5 Activation energies 127
modes occur. Larger polyatomic molecules have low values of y showing
that some internal vibrations are excited, presumably because their
frequencies are sufficiently low.
Our detailed study of specific heats demonstrates the power as well as
the limitations of the equipartition of energy theorem. We may restate this
theorem as follows. In an assembly in thermal equilibrium at temperature
T, each degree of freedom is excited and contributes jkT to the total
energy provided that kT is large compared with any energy level spacings
which an exact quantummechanical analysis shows must exist. If the
equipartition theorem gives a result which is in conflict with experiment,
then this gives some insight into the energy spacings of the modes of
motion which are not excited.
5.5 ACTIVATION ENERGIES
We will now use the Maxwell speed distribution to derive a result of
general applicability in chemistry as well as physics. It concerns activation
energies.
It frequently happens that a system can lose energy provided it can first
overcome a barrier. For example, a molecule sitting on the surface of a
solid in site X (Fig. 5.12(a)) may be able to find a site Y where its energy
is lower. But in order to get there it has to jump or roll over an intervening
molecule. Its interatomic potential energy as a function of position there
fore resembles that in Fig. 5.12(b). A , the height of the barrier which
must be jumped, is called the activation energy.
Chemical reactions are often characterized by activation energies. It
often happens that two substances can be mixed together without reacting
but if they are heated a reaction starts. This reaction may itself give out
heat. A wellknown example is provided by iron and sulphur which can
exist as a mixture at room temperature without undergoing any change,
but which when heated react together to form iron sulphide — the reaction,
once started, giving out much heat.
The interatomic potential energy as a function of the distance between
the centres of two molecules must be of the type in Fig. 5.12(c). As they
approach they repel one another (compare Fig. 3.3(a) and (b)) but if they
can overcome the barrier of height A then they can stick together. This
complex molecule is presumed to initiate or to be the product of the
chemical reaction.
We wish to calculate the probability that this takes place. Let us for
simplicity assume that the reaction takes place in the gas phase — that is,
the two reacting gases F and G are mixed together. Whenever a molecule
of F approaches a molecule of G with a kinetic energy greater than A ,
128 Maxwell speed distribution and the equipartition of energy Chap. 5
the two can react. Having fallen into the well, the complex molecule must
then lose its excess energy but we will assume that this process can take
place easily. We call this a reactive collision.
X Y Position
(b)
'X
Distance between centres
(c)
Fig. 5.12. (a) and (b) A molecule can lose energy by transferring from site X to site Y
but needs activation energy A in order to do so. (c) Potential energy of two molecules
which can undergo a chemical reaction.
The problem of calculating the relative speeds of two molecules (and
hence the kinetic energy which one imparts to the other on collision) is a
complicated one — some of its aspects are dealt with in the next chapter.
Let us therefore make two gross simplifying assumptions, firstly that all
the molecules are stationary except for one F molecule which is moving
with some arbitrary speed c, and further that whenever it meets a G
molecule it hits it head on. The effect of these assumptions may (as usual)
be expected to make the result incorrect only by a factor of order unity
but to leave the form of the result correct — and it is the form which is
important here.
Let there be n G molecules per cm 3 of the species G. In one second, the
one moving F molecule travels a distance c. If its area of crosssection is a
(a quantity which is defined more exactly in the next chapter) it sweeps
out a volume ac in one second. In this volume there are n G ac molecules
of the other gas ; this is therefore the number of collisions it makes in one
second with molecules of G (and this result remains true even if the path
is not a straight one). The probability that one molecule is indeed moving
5.5 Activation energies 129
with speed between c and (c + dc) is given by P[c] dc, Eq. (5.8), so that if
the one F molecule follows the Maxwell distribution it will probably
encounter
n G (TcP[c] dc
molecules of G in one second at a speed between c and (c + dc).
If now we imagine n F molecules of F per cm 3 , all of which we can treat
in the same way, the number of such encounters is
n F n G acP[c] dc
in each cm 3 per second.
Now let us put in the condition that the kinetic energy \mc 2 must be
equal to A or exceed it, that is c ^ (2A /m) 1/2 . The number of reactive
collisions per unit volume per second is
/•OO
n R = n F n G acP[c\ dc
J /2A
V m
= 4™ F n G <x(^) ' J
c 3 Q mc2/2kT dc
/ 2A
V m
after writing P[c] in full, and rearranging.
When realistic modifications are made to allow for the facts that all F
and G molecules are moving so that we must calculate the relative kinetic
energies, and that not all collisions are headon, the result happens to be
identical with this expression except that a reduced mass m F m G /(m F + m G )
must be used in place of the m.
We can substitute x = (mc 2 /2kT) as the variable. This gives
A mC A
dx = — — ■ dc
kT
IkTV 12 f°°
n R = 2 3l2 n F n G <r\ — xe x dx.
R \TimJ J Ao/kT
This can be integrated straightforwardly (it is left for the student as an
exercise in integration by parts) and the result is a sum of two terms of the
form {kTy l/2 exp(A /kT) and (/cT) 1/2 exp(A /kT). The important
point is that the result contains the factor exp( A /kT). Because of the
rapid variation of exp(X) with X, the variation of the exponential factor
dominates the variation of the number of reactive collisions ; over the
fairly narrow temperature interval usually encountered, the other factors,
130 Maxwell speed distribution and the equipartition of energy Chap. 5
T~ 1/2 and T 1/2 , vary only slowly by comparison and can be regarded as
roughly constant. The presence of the factor exp( AJkT) in the expres
sion for the number of molecules which can jump a barrier of height A
is an important result. We have deduced it for a simplified model of a
chemical reaction in the gas phase but it is valid for any system where
the Maxwell distribution holds. In the system shown in Fig. 5.12(a) and (b)
for example, the number of molecules which can jump over the barrier of
height A is proportional to exp( — A /kT).
If we can observe experimentally the number of activated molecules (or
some macroscopic property which is directly proportional to this number)
at different temperatures, then we can deduce the activation energy. The
easiest way is to use a graphical method, plotting the number on a logarith
mic scale against 1/T. This gives a straight line whose gradient is — A /k.
When measuring the gradient it is important to remember that an increase
of a number by a factor 10 adds 2.303 to its natural logarithm.
The rates of very many chemical reactions have been measured and
analysed in this way to find the activation energy. Our simple theory
corresponds, in chemists' language, to a bimolecular reaction ; they express
the rate of chemical change in terms of a 'velocity coefficient' K, which is
proportional to the constant connecting n R with the product n F n G in our
notation. Curves of log K against 1/T and of log(n R /n F n G ) against 1/T
therefore have the same gradient.
To take one example, the reaction
CH 3 I + C 2 H 5 ONa=CH 3 OC 2 H 3 + Nal
in alcohol solution was one of the earliest to be measured accurately
enough to give consistent results. The concentration of the reactants was
determined to begin with, and by measuring the concentration of one of
them after the lapse of several minutes the speed of reaction was measured
and hence the velocity coefficient obtained. A graph of K on a logarithmic
scale against 1/Tis shown in Fig. 5.13. Note that the reaction rate increases
by a factor 40 while the temperature increases only by 10 % from 273°K
to 303°K. The slope indicates that the activation energy is 8.1 x 10 4 J/mol
or about 0.85 eV/molecule, an amount comparable with the ionization
energy of sodium or iodine.
If we allow ourselves to take a highly simplified view, this fact tells us a
good deal about the molecules themselves and the complex which must
be formed when the two reacting molecules are in the act of colliding. If
the Na in the C 2 H 5 ONa were in the form of an ion Na + and the I were
present as I" in the other reacting molecule, then we would expect an
attraction, not a repulsion, between the two molecules. The bonds which
link the Na and I atoms inside their respective molecules cannot therefore
5.5 Activation energies 131
be ionic but are in fact covalent. During the collision the electrons must
be redistributed inside the molecules so as to ionize these atoms — which
requires about 1 eV of energy — and sodium iodide can be formed.
200
100 
S 1
3.7
x10" 3 VT
Fig. 5.13. Plot of velocity coefficient K (on logarithmic scale) against 1/Tfor
chemical reaction in solution. The temperature varies from 303°K at the
left to 273°K at the right. An increase of K by a factor 10 increases log e .K by
2.303. Data from Hecht and Conrad, Z. Physik. Chem. 3, 450 (1889) reworked
in MoelwynHughes Kinetics of Reactions in Solution, Oxford 1947.
PROBLEMS
5.1 The length of the metre is defined in terms of the wavelength of an orange line
in the spectrum of the krypton isotope of mass 86 a.m.u., a line which under
certain conditions is very narrow. The wavelength is denned to be 6057.8022 A.
With a lamp immersed in liquid nitrogen at 63°K, the width of the line for the
intensity to fall by a factor e is 0.0037 A. (a) Estimate how much of this width
is due to Doppler broadening, (b) What would be the width if the lamp were
run at 1,000°K?
5.2 (a) A spherical planet of mass M has some gas molecules near it. Write down
an exact expression for the force on a gas molecule at a large distance r from
the centre of the planet, and hence for the potential energy. (Use G for New
ton's universal constant of gravitation.)
132 Maxwell speed distribution and the equipartition of energy Chap. 5
(b) The planet and the molecules are in thermal equilibrium at temperature T.
Write down the Boltzmann factor for a molecule at distance r from the
centre of the planet. Get the sign of the exponent correct.
(c) Write down the volume of a spherical shell bounded by radii r, (r+dr), and
hence calculate the probability of finding the molecule between these limits.
(d) What is the value of this expression when r is infinite?
(e) This means that one of the following is true :
(i) the density of the atmosphere decreases exponentially with r
(ii) the density of the atmosphere cannot reach equilibrium
(Hi) the density of the atmosphere is zero outside a certain radius
(iv) the density of the atmosphere decreases as 1/r.
5.3 Prove that the height of the centre of gravity of an atmosphere is equal to its
scale height. (Assume a plane Earth, g = constant.) Write down an expression
for the total energy (kinetic energy of the molecules plus potential energy due
to the Earth's gravity) for one mole of an atmosphere of a monatomic perfect
gas. Hence prove that its specific heat is f R.
5.4 At ordinary temperatures, nitrogen tetroxide (N 2 4 ) is partially dissociated
into nitrogen dioxide (N0 2 ) as follows :
N 2 4 ^2N0 2
0.90 g of liquid N 2 4 at 0°C are poured into an evacuated flask, of 250 cm 3 .
When the temperature in the flask has risen to 270°C, the liquid has all vaporized
and the pressure is 960 mm Hg. What percentage of the nitrogen tetroxide has
dissociated?
5.5 Calculate the escape velocity from the Earth's gravitational field. Calculate the
r.m.s. velocity of helium atoms at room temperature. According to some
theories, the Earth's atmosphere once contained a large percentage of helium.
Explain the fact that helium is now a rare gas. (Assume that the Earth's tempera
ture has remained constant at all times at 300° K, molecular weight of helium =
4.)
5.6 This problem gives an interesting insight into the energy relations in adiabatic
expansions. A piston moves with constant velocity and each molecule under
goes a kind of 'Doppler' change of velocity after reflection from it.
(a) A molecule of mass m approaches a wall with velocity u x and is specularly
reflected. The wall moves with velocity £, as shown. Which of the following
expressions is correct for the velocity after reflection?
(u x 2£) (u,£) (U. + S) (u x +2£).
Write down the kinetic energy after reflection, assuming that £ is very
small compared with u x , so that £ 2 can be neglected.
^ U x £.
O * — ► £■
I
Problems 133
(b) Consider a molecule travelling in an arbitrary direction towards the wall,
with C still in the xdirection. Following an argument like that in section
5.1.2, write down the number of impacts on area A of the wall in a small
time dt, undergone by molecules whose xcomponent of velocity is between
u x , (u x +du); hence the change of kinetic energy of this class of molecule in
this time. Hence find the change of kinetic energy of all molecules. Express
the increase of volume dV in terms of the distance moved by the wall, etc.
(c) Assume the gas is perfect and monatomic. During the small time, the volume
is practically constant Write down the heat capacity of the gas in terms of
the specific heat per molecule and the total number of molecules ; do not
confuse number of molecules/cc with total number of molecules. Hence
calculate the change of temperature dT.
(d) Show that dT/T + 2 dV/lV = 0, and that this is consistent with the PV y law
of adiabatic expansions.
(e) Write down the analogous equation for a polyatomic perfect gas.
(/) What modifications to the discussion are needed if the molecules are not
assumed to be reflected specularly?
(g) Describe briefly how the ordered motion of the molecules after reflection is
degraded into thermal motion (i) in microscopic terms (ii) in macroscopic
terms.
5.7 Consider a magnetic dipole in a magnetic field. Using the coordinate system
of Fig. 5.6(a) let the vertical axis represent the direction of the field, and the
radius vector represent the dipole. From Fig. 5.6(b), it can be seen that for a
sphere of unit radius constructed about the origin, an element of area generated
by dd, d<j> is sin dd d(j>.
The potential energy of a dipole of moment m at an angle 6 to a field H is
— m fx H cos 6.
For an assembly of N independent dipoles at temperature T :
(i) Write down the probability that a dipole is oriented between 6, (6 + dd), </>,
(ii) Normalize this expression by integrating over all possible values of 6 and </>
and equating this probability to 1.
(Hi) Write down the probability of a dipole being oriented between 6, (6+dd),
irrespective of the angle $.
(iv) Each dipole has a moment m cos 6 parallel to the field. Write down the
contribution from those of AT dipoles which are oriented between d, (9+ dd).
(v) Hence find the total magnetic moment from all N dipoles. Show that it
tends to Nmln H/3kT when (m n H/kT) is small.
(Note : (e x + e " x ) = cosh x ; ^(e*  e " x ) = sinh x ; the limit of (coth x  1/x)
when x is small is x/3.)
5.8 Crystals of sodium chloride show strong absorption of electromagnetic radia
tion at wavelengths of about 6xl0 3 cm. Assuming this to be due to the
vibrations of individual atoms, calculate (a) the frequency of the vibrations ;
(b) the potential energy of a sodium atom as a function of its distance d from its
equilibrium position, assuming the vibration to be simple harmonic; (c) the
probability distribution of 5 at T = 400°K; (d) the r.m.s. value of S at 400° K.
The atomic weight of sodium is 23.
5.9 A galvanometer mirror is suspended on a thread, inside a box containing air.
Its moment of inertia for torsional swinging is /. The torsion constant of the
fibre is n. 6 is the angle of torsion, co the corresponding angular velocity. While
swinging, the total energy (given by jla> 2 +jnd 2 ) is constant.
134 Maxwell speed distribution and the equipartition of energy Chap. 5
(a) How many degrees of freedom does the system have as regards this motion?
(b) What is the total energy of random swinging of the mirror?
(c) If /i = 10" 6 dyn cm, what is the r.m.s. value of the angle of deflection in
radians?
(d) Which of the following mechanisms produces these movements?
(i) expansion and contraction of the mirror, varying J
(ii) radial expansion and contraction of the fibre, varying n
(Hi) thermal motion of screw dislocations in the fibre
(iv) random collisions of air molecules with the fibre, interchanging spin
(v) random collisions of air molecules with the mirror.
(e) The box is evacuated so that there are very few molecules in it. The box
remains at temperature T. Which of the following values does the r.m.s.
deflection now have?
(i) zero
(ii) reduced in the ratio of the pressures
(Hi) same as before
(iv) increased in ratio of the pressures
(v) infinite.
(/) What is the mechanism which produces the movement now?
5.10 The molecules of a substance are known to consist each of two atoms, rigidly
fixed to one another, like a dumbell. The mass of each atom is 10 a.m.u. so that
the molecular weight is 20 a.m.u. Each atom consists of a heavy but extremely
small nucleus containing practically all the mass, surrounded by a larger
spherical 'cloud' of electrons whose total mass is only about l/1000th that of the
nucleus.
(a) Estimate the order of magnitude of the moment of intertia of the molecule
for rotation about the axis shown.
(b) What is the mean kinetic energy in this mode of rotation of one molecule in
the solid phase at room temperature (assuming it is not prevented from
rotating in any way)?
(c) What is the mean frequency of this rotation at room temperature?
(d) What value of C v , the specific heat at constant volume, would be expected
if the substance were solid at room temperature if the laws of classical
physics were applicable?
(e) What value of C v would be expected if the substance were a gas at room
temperature if the laws of classical physics were applicable?
(/) What value of y = C p /C v would you then expect to find?
(g) What value of y would you expect to find in practice?
Problems 135
5.11 Hydrogen gas is contained in a thermally insulated cylinder with a moveable
piston. If the pressure on the piston is suddenly reduced to 0.38 of its original
value as a result the volume of the gas is immediately doubled, estimate the
specific heat at constant volume per mole for hydrogen.
5.12 The electrical conductivity a of a certain class of solid is predicted to vary
according to the law
r
^ A jkT
T
where C is a constant, k = 0.86 x 10 ~ 4 eV/deg and T is the temperature. A is
an activation energy, the energy required for an elementary charge to be moved
from its atomic site.
Measurements of a (in arbitrary units) for ice as a function of Tare as follows :
a 31 135 230 630
T 200°K 220°K 230°K 250°K
Plot a suitable straightline graph and deduce A .
Decide whether conduction in ice is dominated by (a) electron conduction
with an energy gap A = 0.1 eV, {b) proton transport involving the breaking of
a hydrogen bond, A = 0.25 eV, (c) transport of complex ions requiring
simultaneous breaking of 4 hydrogen bonds, A = 1 eV.
CHAPTER
Transport properties of gases
6.1 TRANSPORT PROCESSES
So far we have concentrated on the properties of solids, liquids and gases
which are in equilibrium. In this chapter we will deal with systems which
are nearly but not quite in equilibrium — in which the density (or the tem
perature or the average momentum) of the molecules varies from place to
place. Under these circumstances there is a tendency for the nonuniformi
ties to die away through the movement — the transport — of molecules down
the gradient of concentration (or of their mean energy down the tem
perature gradient or their mean momentum down the velocity gradient).
We will define certain transport coefficients and show how they can be
estimated for gases.
Although the systems we consider are nonuniform in some way and
cannot therefore be in thermal equilibrium nor obey the Maxwell speed
distribution exactly, we will always make the assumption that the depar
ture from equilibrium is only small. We will therefore assume that no
error will be introduced if we take the speed distribution inside any region
of the substance to be Maxwellian.
6.1.1 Diffusion as a transport process
Diffusion is the movement of molecules from a region where the con
centration is high to one where it is lower, so as to reduce concentration
gradients. This process can take place in solids, liquids and gases (though
6. 1 Transport processes 137
this chapter will be mostly concerned with gases). Diffusion is quite
independent of any bulk movements such as winds or convection currents
or other kinds of disturbance brought about by differences of density or
pressure or temperature (although in practice these often mask effects due
to diffusion).
One gas can diffuse through another when both densities are equal. For
example, carbon monoxide and nitrogen both have the same molecular
weight, 28, so that there is no tendency for one or other gas to rise or fall
because of density differences; yet they diffuse through each other.
Diffusion can also take place when a layer of the denser of two fluids is
initially below a layer of the lighter so that the diffusion has to take place
against gravity. Thus, if a layer of nitrogen is below a layer of hydrogen, a
heavy stratum below a light one, then after a time it is possible to detect
some hydrogen at the bottom and some nitrogen at the top, and after a
very long time both layers will be practically uniform in concentration.
Diffusion coefficients of gas a in gas /? can be measured with a suitable
geometrical arrangement of two vessels with different initial concentra
tions together with some method of measuring those concentrations — a
chemical method or mass spectroscopy for example. If the rates of change
of concentration with time are plotted, the diffusion coefficient can be
deduced ; the equations describing the process are given in section 6.2.
It is also possible to measure coefficients of selfdiffusion, of a gas a in
gas a for example. This can be done by using two isotopes having the same
shape and size of molecules, the same interaction potential and almost the
same mass, but which are nevertheless detectably different — one isotope
might be radioactive, the other not. One method is described in section
6.2.1. Mass spectrometer methods can also be used. Applied to a solid, one
is described in section 9.6.1.
6.1.2 The diffusion equation
We will begin by taking a macroscopic view of the phenomenon, that is,
we will write down equations which involve such variables as concentra
tions or fluxes but will not specifically mention individual molecules. We
define the concentration of a as the number of molecules n per unit volume,
and we consider the simple case where n varies with one coordinate only
which we call the xaxis. In Fig. 6.1, the concentration at all points in the
plane x is n, at (x + dx) it is (n + dn). Then diffusion takes place down the
concentration gradient, from high to low concentrations ; we are assuming
that bulk disturbances are absent. We next define the flux J of particles
as the number of particles on average crossing unit area per second in the
direction of increasing x. Notice that both concentration and flux can be
138
Transport properties of gases Chap. 6
measured in moles instead of numbers of molecules : this is equivalent to
dividing all through our equations by Avogadro's number N.
In general, the flux J may change with position x and may also change
with time t . In other words, J may be a function of x and t so we write it
as J(x, t). Of course, there are circumstances where J may be the same
for all x, or where it is constant with time, but the most general situation
is that J does depend on both.
n n + d/7
x x + dx
Fig. 6.1. Coordinates used in the definition of
diffusion.
It is an experimental fact that, at any instant, the flux at any position x
is proportional to the concentration gradient there :
J(x, t) oc
dn
dx
or
J(x,t) = D
dn
dx
(6.1)
where D is called the diffusion coefficient. This is known as Fick's law.
By itself, Eq. (6.1) is adequate to describe 'steadystate' conditions
where currents and concentrations do not change with time so that the
flux can be written J(x). For example, if a tube of length / cm and constant
crosssection A cm 2 has molecules continually introduced at one end and
extracted at the other at the same rate, the concentration gradient becomes
— An//, where An is the difference of concentration between the two ends.
The number of particles crossing any plane in the tube per second is then
— DA An// and this does not change with time.
Consider, however, the much more general situation where initially a
certain distribution of concentration is set up and then subsequently the
molecules diffuse so as to try to reach a uniform concentration. Concentra
tions are, therefore, changing with time and particles must be accumulating
in the region between x and (x + dx) or moving from it. Therefore, the
6.1 Transport processes 139
number crossing area A of the plane x is not equal to that crossing the
same area at (x + dx). The flux entering this volume is
The flux leaving the slice can be written J xn + dx where
J xo +dx = J xo +\ — \dx+ •
xo + dx
dJ]
dxJT'
and we can neglect higher terms. The rate of movement of molecules
from the slice is equal to the difference between the two values of A J, and
also equal to the volume of the slice, A dx, times the rate of decrease of n :
dJ * , dn ,
——Adx = — ^dx
ox dt
that is
dJ dn
dx dt'
Combining this with equation (6.1) and eliminating J :
dn 8 ( dn\ d 2 n
(6.2)
if we assume that D is a constant independent of the concentration. This
is called the diffusion equation, and since n depends on x and t it could be
written n(x, t)*
Thus we have a system of three equations. (6.1) is an experimental law
linking the flux at any point with the concentration gradient there. (6.2) is
the continuity equation expressing the fact that molecules cannot dis
appear, and (6.3) combines these two equations. Eq. (6.1) is adequate for
steadystate conditions, where conditions do not vary with time ; but for
the general case (6.3) may be used.
* If the process takes place in 3 dimensions, J is a vector whose components are (J x , J y , J z )
and Eqs. (6.1) and (6.2) become
t • , . , . . ~/.<3" .d n dn\
J = iJ x +]J y + kJ 2 = £> i— +j_ + k— = Dgradn
\ ox dy dzj
dn dJ x dJ v dJ,
— = ^+— + — = div J
dt ox dy oz
where i, j, and k are unit vectors parallel to x, y and z. Eliminating J :
8n *■ . ~ * ^ ^2 (d 2 n d 2 n d 2 n
= _div £>gradn )=EF7 2 n = D — +_+_
dt \dx 2 dy 2 dz 2
140 Transport properties of gases Chap. 6
These are typical of transport equations — with the proviso that for
energy and momentum diffusion, the coefficients in the three equations
are not all identical as they are here.
6.1.3 Heat conduction
The conduction of heat is also a process of diffusion in which random
thermal energy is transferred from a hotter region to a colder one without
bulk movement of the molecules themselves. In a hot region of a solid body,
the molecules have large amplitudes of vibration ; in a hot region of a gas
they have extra kinetic energy. By a collision process, this energy is shared
with and transferred to neighbouring molecules, so that the heat diffuses
through the body though the molecules themselves do not migrate. The
macroscopic equations describing conduction in one dimension x are,
firstly, the experimental law for the heat flux
Q =  K _ (6.4)
(where Q is the heat flux across unit area, measured in W cm 2 , k is the
thermal conductivity and T is the temperature) and, secondly, the con
tinuity equation
dQ dT
which expresses the conservation of energy in the form that the heat which
is absorbed by a slice of a body goes into raising its temperature. C is the
specific heat per unit mass, p the density so that Cp is the specific heat per
unit volume. Combining these two equations to eliminate Q :
dT l K \ d2T l(k *
where (ic/Cp) is called the thermal diffusivity by analogy with Eq. (6.3).
Eq. (6.4) by itself is adequate for steadystate conditions, as when for
example heat is fed into one end of a bar and extracted at the other and all
temperatures are constant with time, and Tcan be calculated as a function
of x alone. But when conditions are not steady, and T varies with time as
well as position, Eq. (6.6) describes the situation.*
6.1.4 Measurement of thermal conductivity of gases
To measure the thermal diffusivity, one has to arrange for temperatures
to vary with time and to measure the speed of propagation of these
* Many students are familiar with (6.4) and with the concept of thermal conductivity
but have never met (6.6) and thermal diffusivity. In fact, transient heat flows are of great
technical importance.
6.1 Transport processes 141
temperature changes. This is difficult with gases (which we are concen
trating on in this chapter), where convection currents may be set up. It is
most convenient, therefore, to measure the thermal conductivity. Usually
this is done by applying Eq. (6.4) directly, using the simplest geometrical
arrangement. Commonly, the gas is enclosed between two concentric
heavy, metal cylinders. Power is supplied electrically to the inner one ; the
temperature difference is measured directly with a thermocouple. It is
important to correct for the conduction of heat through the electrical
leads which can be done by pumping all the gas out and measuring how
much heat is still conducted across. It is also important to make sure that
the gas does not set up a pattern of convection currents, which it can do
rather unexpectedly at certain values of the gap width and pressure. This
effect can be detected by using a different size of apparatus and also by
checking that the power varies inversely as the gap.
6.1.5 Viscosity
For completeness, a third simple transport process — the diffusion of
momentum by viscous forces — will be mentioned here, though only
briefly. Viscous motion of fluids can be far more complicated than diffusion
or heat conduction and we will be forced to consider only the steadystate
equation.
Consider a gas or liquid confined between two parallel plates (Fig. 6.2).
Let the lower plate be stationary and the upper plate be moving in the
direction shown, which we will calJL the xdirection. Molecules of fluid
very near the plate will be dragged along with it and have a drift velocity,
U x parallel to x, superposed on their thermal velocity. We will assume
that U x is much less than the mean thermal speed or the speed of sound.
Molecules of fluid near the stationary plate will, however, remain more or
less with zero drift velocity.
moving plate
z
~ UL
'y
stationary plate
Fig. 6.2. Coordinates used in the definition of viscosity.
Eventually a regime will be set up in which there is a continuous velocity
gradient across the fluid from bottom to top. In this state, molecules will
be continuously diffusing across the space between the plates and taking
142 Transport properties of gases Chap. 6
their drift momentum with them. Considering an area of a plane parallel
to the xy plane in the fluid, molecules which diffuse across from above
to below will carry more drift momentum than those which diffuse from
underneath to on top. In other words, the more rapidly moving layer
tends to drag a more slowly moving layer with it, because of this diffusion
of momentum.
In macroscopic terms, a shearing stress (force per unit area) is necessary
to maintain this state of motion. The experimental law is
dU x
P xz = r\~ (6.7)
oz
where P xz is the force per unit area in the x direction due to a gradient of
U x in the zdirection and rj is called the coefficient of viscosity. Provided the
direction of the force is clearly understood, it is not necessary to include a
minus sign, as this depends on the convention for the choice of axes.
We started by considering a fluid in Fig. 6.2, but Eq. (6.7) can be applied
to solids because the righthand side can be written dd/dt, where 6 is an
angle of shear. It is difficult to imagine a solid subjected to a shear which
goes on increasing with time, but it is quite common for solids to be sheared
to and fro in an oscillatory fashion. Forces are then required to provide the
accelerations, but in any case the viscosity gives rise to the dissipation of
energy and the production of heat. It is usual to refer to this as due to the
internal friction of solids.
It is implied in Fig. 6.2 that dUJdz is a constant and that U x increases
proportionally to z. This is so if the coefficient rj is a constant. For many
liquids this holds, but there are notable exceptions where n varies with
the velocity gradient or rate of shear so that the velocity profile is not linear.
Blood, for example, flows with a much lower viscosity through narrow
capillaries than measurements of the flow through wide tubes would
indicate — which is fortunate because otherwise one's heart would have to
generate several horsepower to maintain circulation. Other suspensions
such as cement also have low viscosities when agitated. Oil paint is fluid
when worked rapidly with a brush, but when laid on a vertical surface
and sheared only by a small force due to its weight, it does not fall off.
Such liquids are called thixotropic. Other liquids have opposite behaviour.
Whenever we apply Eq. (6.7) to a fluid, therefore, it will be assumed that we
are dealing with a gas or a 'Newtonian' liquid for which n is independent
of the rate of shear.
When we come to write down equations representing the motion of a
fluid while it is not in a steady state but accelerating, we meet a situation
which is much more complicated than the diffusion or heat conduction
cases. For one thing, there are always massacceleration terms which have
6.1 Transport processes 143
no analogue in the other phenomena. For another, a kind of regime may
be set up where the flow is not streamline as illustrated in Fig. 6.2 but
turbulent, and vortices or eddies are present which add an element of
randomness to the flow pattern. Whether or not it is set up depends on
the ratio of the inertial to the viscous terms. We can, however, usefully
adopt a mathematical representation of the simple situation of Fig. 6.2.
We can imagine the liquid divided into layers, each one sliding over the
one underneath it on imaginary rollers like long axle rods parallel to the
yaxis. These rollers are not there in any real sense, but they can lead one
to define a quantity called the vorticity which is always present in a flowing
fluid even when no macroscopic vortices are present. (In a simple case
like Fig. 6.2 the vorticity degenerates into the velocity gradient.) Now in
the general case of an accelerating fluid with nonuniform velocity it is
the vorticity which diffuses throughout the fluid, though the equation it
obeys is not of a simple form. For obvious reasons we will not pursue this
topic but will be content with the steadystate Eq. (6.7).
6.1.6 Measurement of the viscosity of gases
In his classic experiments to measure the viscosity of gases at low pres
sures, Maxwell used a torsion apparatus in which a number of circular
glass discs were arranged to swing in between fixed ones (Fig. 6.3). He
found the damping coefficient of the oscillations. If we neglect the energy
loss in the torsion wire itself and assume that the discs would go on
swinging for a very long time if all the gas were removed, we can calculate
the damping as follows.
Consider one surface of one plate, and select an annulus between radii
r and (r + dr). Then (assuming streamline flow) the force on this annulus,
whose area is 2nr dr, is
dF = — — (2nr dr)
a
where the linear velocity is rco, co being the angular velocity, and d is the
spacing between adjacent moving and stationary surfaces. The contribu
tion to the couple is the radius times the force :
dG = 2^ r 3 dr
and the total couple is
f
Jo
27tna> C a , , nnoj .
d J 2d
where a is the radius of the disc. If there are n discs, each with two surfaces,
there are 2n such contributions. The equation of motion of the system
144
Transport properties of gases Chap. 6
when swinging freely is
d 2 {nna*ri)dO n
dr d dt
where co — d9/dt, I is the moment of inertia, [i the torsion constant of
the suspension. This is the equation of a damped oscillation. The time
required for the amplitude to decrease by a factor e is 21 /B, where B is the
coefficient of the second term in the equation. Thus rj can be determined.
*■
Fixed surfaces
Moving surfaces
Fig. 6.3. Principle of the apparatus for the measurement of viscosity by the
damping of torsional oscillations, (a) assembly of discs, (b) section of apparatus.
In Maxwell's final apparatus, there were 3 swinging discs (n = 3) with
d = 0.469 cm. / was determined as 7.33 x 10 4 g.cm 2 ; the radius a was
effectively 13.1 cm, after allowing for the width of the suspension arrange
ment in the centre ; the period was 72.5 s. In one experiment with air at
21°C, the (natural) logarithmic decrement was determined as 0.073, which
meant that 13.7 swings were needed to damp the amplitude by a factor e.
From these data, r\ = 2.47 x 10 ~ 4 g/cm s. A number of corrections were
needed to allow for edge effects and for torsional damping in the suspending
6.2 Solutions of the diffusion equation : the yjt law 145
wire. This method has been used for measuring the viscosity of liquids as
well as gases.
6.2 SOLUTIONS OF THE DIFFUSION EQUATION : THE jt LAW
It is worthwhile studying two solutions of the diffusion equation. The
first corresponds to the following initial conditions : A semiinfinite prism
of material has area of cross section A ; the length is along the xaxis and
the ends are at x = and x = oo. On the face x = 0, N molecules are
initially all concentrated in a thin layer and are subsequently allowed
to diffuse into the material. We will denote the number at time t which are
within a slice between x and (x + dx) by n(x, t)A dx. Then the appropriate
solution of Eq. (6.3) shows that the concentration
The function is shown in Fig. 6.4 for a number of values of the time. The
following statements should be verified : (a) that the function does indeed
satisfy the diffusion equation, which can be shown by direct substitution,
and (b) that the total number of molecules remains constant and equals
AT at any time t, which can be shown by integrating n(x, t)A dx from
/ = V4D
Fig. 6.4. Concentration as a function of x for different
values of time.
146 Transport properties of gases Chap. 6
to oo, using an integral from the Table on p. 72. It is obvious, from the
diagram, that the concentration always remains greatest near the starting
place and falls off with increasing distance, and that the spread increases
with time, which is all very reasonable.
One very interesting aspect of the diffusion process can be deduced
from this solution. On the microscopic scale, diffusion is of course a
random process, and it is impossible to predict exactly how far one
particular molecule will travel. But if we were to scale down the curves
of Fig. 6.4 so as to refer to one molecule instead of N , these curves would
then be the probability function P[x] for the nett distance travelled by a
single molecule at any time (see section 4.2).
We can, therefore, use the curves of Fig. 6.4 to calculate the mean nett
distance travelled by a molecule at any time t. This is
A f 00
x(t) — — xn(x, t) dx.
Using an integral from the Table on p. 72, we find
x = ^{Dtf 12 . (6.9)
V 71
Thus the mean nett distance travelled is proportional to the square root
of the time. This is perhaps an unexpected result : one is used to travelling
twice as far in twice the time, but for the random process of diffusion this
is not so. Of course, some molecules go much further than this, others
less far, and it is the mean which we have calculated. Stated differently, our
result shows that to diffuse a mean distance x, the time required is propor
tional to x 2 . This is an important characteristic of the diffusion process.
Before leaving this problem, note that substituting T for n and taking
D to signify the thermal diffusivity, we have the solution to the problem of
a semiinfinite slab with a finite amount of heat generated on the surface
and subsequently allowed to be conducted away.
Another solution to the diffusion equation refers to the problem of a vessel
of crosssection A with two layers (say of liquid or gas, so long as convec
tion is avoided) each of depth 1/2 and of initial concentration n molecules/
cm 3 and zero respectively, Fig. 6.5(a). Diffusion starts at zero time. It is
obvious that after an infinite time the concentration throughout the vessel
must be uniform and equal to n /2 molecules/cm 3 , Fig. 6.5(b). After a
time t , the concentration as a function of distance along the vessel is given
by Fig. 6.5(c). (The solution is a Fourier series.)
Initially the concentration difference is n , but the mean concentration
decreases in one half and increases in the other. The time interval required
6.2 Solutions of the diffusion equation : the yjt law 147
for the difference of concentration to decrease by a factor e is called the
relaxation time x for the diffusion ; it is a natural unit of time to use for
describing the process. It emerges from the analysis that
t = l 2 /n 2 D. (6.10)
Note that once again, a time is porportional to the square of a length.
Area A
1/2
1/2
05/7.
o
c
o
o
o
/ // = r
//a>
1
1 *
(b)
1 l
(c)
Position
Container
' Q
Narrow 8
' tube g
Container
P
id)
(e)
Fig. 6.5. (a) Molecules of gas initially occupying lower half of vessel, upper
half being filled with another gas (not shown) to avoid convection, (b) after
an infinite time, (c) concentration as a function of position at different times,
(d) 'lumped' volumes and a tube, (e) concentrations in P and Q as function of
time.
Without doing an exact analysis, some insight into this result can be
gained from a crude model of the process. Imagine both halves of the vessel
replaced by containers P and Q (Fig. 6.5(d)) one filled with the same total
number of molecules as before and the other empty. Let these two be
connected by a narrow tube which allows diffusion to take place. In the
148 Transport properties of gases Chap. 6
language of the electrical engineer we have replaced the distributed
capacitance and conductance of Fig. 6.5(a) by lumped capacitances and
a pure conductance in Fig. 6.5(d). (This language becomes even more
appropriate if we translate the diffusion problem into the heat conduction
problem, when heat capacities and thermal conductances are used.)
To make the setups comparable, let us put the same total number of
particles in both halves, namely n Al/2. Further, since the average distance
that a molecule has to diffuse in Fig. 6.5(a) is something like 1/2, and it has
to travel through an area of cross section A, let the narrow tube have the
same ratio (area)/(l#ngth), namely 2A/1 ; the rate of diffusion will then be
the same. We have :
number of molecules leaving P per second
= number entering Q per second
= (flux J) x (area of narrow tube).
To the approximation that we are dealing with "lumped" components we
can use Eq. (6.1) so that
J = D(conc'n in P — conc'n in Q)/(length of narrow tube).
Therefore
Al drip Al dn Q AD(n P — n Q )
~Y~di = ~2~~dT = 1/2
where n P + n Q = n . The solution is
»P = ^(l + en « Q = ^(le</<)
2
where
T
Thus the concentrations in P and Q approach their final values exponen
tially, Fig. 6.5(e), with time constant l 2 /SD which is not very different from
the l 2 /n 2 D quoted before.
This square law can have quite startling effects. Diffusion coefficients
of small molecules in liquids like water at ordinary temperatures are of
the order of 10 ~ 3 cm 2 s _1 . Given a tube 1 cm long, concentrations will tend
to equality in times of the order of 20 minutes. But the time for a 1 m tube
would be reckoned in months and for a 10 m tube it is decades. A famous
example of this is a very tall vertical tube, fixed to the wall of a lecture
6.2 Solutions of the diffusion equation : the Jt law 149
theatre in Glasgow University. Eighty years ago it was filled by Lord
Kelvin, the lower half with bluegreen copper sulphate solution and the
upper half with water. It is still very far from uniform in concentration.
6.2.1 Measurement of the diffusion coefficients of gases
Diffusion coefficients of one gas through another (or with certain cor
rections, of a gas through itself), can be measured using similar arrange
ments. In the experiments of Mifflin and Bennet to measure the D of
argon through argon at room temperature at very high pressures for
example, two volumes V, each 36 cm 3 , were connected by a bar of length
/ = 3.8 cm made of porous bronze (Fig. 6.6(a)). The pores were of average
diameter 2x lCT 4 cm.* The total crosssectional area of the pores was
A = 0.36 cm 2 , about ^ of the area of the bar. One volume, which we will
call Q, was filled with ordinary argon, the other, P, with argon at the same
pressure containing a small concentration of 37 A. This is a radioactive
isotope whose presence could be detected by the ionization it produced —
each volume V was in the form of an ionization chamber in which the
current was porportional to the concentration of 37 A. The halflife of the
37 A was large compared with the time taken by the experiment.
Using the same notation as in section 6.2, the equations are
V
.drip
dF
V
dn Q _ AD(n P — n Q )
~d7~ /
Steel casing Porous bronze
Ionization
chamber P
(o)
Ionization
chamber Q
30 40
Time in hours
(6)
Fig. 6.6. (a) Apparatus to measure diffusion of gases at high pressures, (b)
Difference of currents in P and Q in units of 10 ~ 14 amp, plotted on log scale, as
function of time on a linear scale.
* It is shown later in this chapter that it is important that the diameter of the pores should
be greater than the mean free path between collisions in order to measure D correctly. This
condition was in fact satisfied here.
150 Transport properties of gases Chap. 6
so that (« P — n Q ) varies as exp( — tjx), where x = V 1/2 AD. A plot of log(i P — i Q )
against time was, therefore, a straight line of negative slope 2AD/VI, the
i's being the currents. Figure 6.6(b) shows the results of one run. The
currents are in units of 10~ 14 amp. The slope of this graph is
(In 49  lnl7)/40 hours" l = (3.89  2.77)/(40 x 3,600)
= 0.78xl0 _5 s 1 .
Hence D = 1.48 x 10" 3 cm 2 s" *. This is the diffusion constant of 37 A in
40 A. A small correction must be applied in order to calculate the self
diffusion constant of 40 A through 40 A; it will be mentioned here although
it can only be understood in the light of the discussion of section 6.5.
We assume that the interatomic potential energies of the two kinds of
atom are the same but that their mean speeds c are different, because by
the equipartition law the mean kinetic energy is the same for both. This
gives a 2 % correction to c and hence to D. The results of these experiments
are quoted in section 6.5.4.
6.3 DIFFUSION AND THE RANDOM WALK PROBLEM
We have so far dealt with diffusion in macroscopic terms with little
reference to the paths followed by individual molecules. In fact, each mole
cule follows a random path, moving in more or less straight lines between
collisions with other molecules, but travelling backwards almost as often
as forwards (Fig. 6.7). It is the purpose of this section to show that the yjt
law and other characteristics of diffusion are merely consequences of the
'random walk' of each molecule.
Fig. 6.7. A random path followed by a molecule, after Perrin.
The distance that a molecule travels between collisions is called the/ree
path. It may be of any length, in any direction.
Let us, however, make a crude model of the random path by saying
(a) that each free path is of the same length, and (b) that the molecules can
6.3 Diffusion and the random walk problem 151
only move parallel to the +xor — x direction — that we are dealing with
a sort of 'onedimensional gas'. It may be guessed that these simplifications
allow us a considerable insight into the diffusion process although the
value of the diffusivity so calculated is likely to be wrong ; even then, it is
not likely to be wrong by a large factor such as 10 but rather by a factor
like 2 or 3.
The simplified problem is this. A molecule starts from the origin and
moves a distance ± / along the xaxis ; having done so, it can then move a
further distance ± /. Thus at the end of two such moves, it may have followed
one of four possible sequences of +/ or — / moves: namely, ( + /+/) or
( I / — /) or ( — /+/) or ( — /—/). All of these are equally likely to occur. The
nett distance travelled may, therefore, be + 2/ (which may be achieved in
only one way and, therefore, has probability 1/4) or (reached in two ways ;
probability 1/2) or — 2/ (probability 1/4). At the end of three moves, there
are eight possible sequences which may have been followed, all equally
likely. The end point may be +3/(1 sequence only), or +/ (there are 3
sequences with 2 positivegoing moves and 1 negativegoing), or — / (again
three ways of achieving this) and — 3/ (one way only). Thus the probabilities
are 1/8, 3/8, 3/8 and 1/8 respectively. Notice that the numbers 1, 2, 1 and
1, 3, 3, 1 occur as coefficients in the expansions of (x + y) 2 and (x + y) 3 . Let
us now generalize our results to sequences of N moves (Figure 6.8 shows
two typical sequences of 10 moves, for illustration.) We can say that the
total number of possible sequences is 2 N , and the number of ways of
achieving 2 positivegoing steps and {N — Z) negativegoing ones is the
+
Fig. 6.8. There are 2 10 different sequences of 10 moves,
all equally probable. 10 !/7 !3 ! = 120 of these consist of
7 positivegoing and 3 negativegoing moves; two of
these are illustrated (slightly displaced in a vertical
direction for clarity).
152 Transport properties of gases Chap. 6
coefficient of x z y N ~ L in the expansion of (x + y) N , namely
N\
Z!(NZ)T
These coefficients are called the binomial coefficients.
In slightly different terms, the probability that at the end of N moves
the molecule will have travelled a nett distance x — SI (that is, S steps in
the +x direction), by having made j(Af + S) positivegoing steps and
%{N — S) negativegoing steps is
We may assume that the molecule moves with speed c during its free path
so that the time required to make the N steps is t = Nl/c. Thus Eq. (6.11)
gives the probability that a molecule will have travelled a nett distance x
in time t.
We can now refer back to the problem of which Eq. (6.8) is the solution
and alter it slightly to make it correspond exactly to the present one —
namely by making it refer to 1 molecule instead of N and to diffusion in
the ±x direction instead of +x only. We can interpret the solution to
mean that the probablity that a molecule diffuses a distance between x
and (x + dx) in time t is
1 e* 2/4Dt dx. (6.12)
2(nDt)
1/2'
We can now show, by purely mathematical manipulations, that — unlikely
though this might seem — this is identical with Eq. (6.11), in the limit when
N tends to infinity, when t = Nl/c and x = SI. We need to make one
additional intuitive statement (which can, however, be properly proved),
that if N is very large, there is a great probability that S is small compared
with N; this corresponds to the fact that the binomial coefficients are
small at the beginning and end of the expansion and largest in the middle.
The key is to use Stirling's formula for N ! when N is large :*
log AT! = (N + ±)\ogNN + log{27z) 112 . (6.13)
* Natural logarithms to base e are meant, of course. For many purposes
logiV! = JV log JVJV
is a good enough approximation, for large JV. But with all the terms present, it is remarkably
accurate even for small JV. It gives 10! as 3.60 x 10 6 instead of 3.63 x 10 6 , for example. For
JV = 10 10 it is very accurate indeed.
6.4 Distribution of free paths 153
Eq. (6.11) can be written
log P = log NllogP^jllogP^I 1108 2"
and after some tedious algebra this gives
, D , /2\ 1/2 /N + S+ll / S\ lNS+l\. I. S
On the assumption that S/N is small we can write
, m s \ s s2
togll+ riv^ + '
Then
or
2 \ 1/2 S 2
l0gF = l0g U 2»
which in terms of distance x and times t may be written
1/ 2 \ 1/2
P =  —  e* 2/2cl, 2l. (6.14)
2\7icltJ
Comparing with Eq. (6.12), 2/ takes the place of dx (since in this simple
model there can be particles only at points separated by 21). The two
equations are identical in form and
D = \c\.
Thus the essential features of the diffusion process are reproduced by
this simplified 'random walk' problem. A more accurate analysis
(Appendix B) gives a coefficient in the expression for D of ^ instead of \\ as
mentioned above, this kind of discrepancy is to be expected of the over
simplified onedimensional model.
6.4 DISTRIBUTION OF FREE PATHS
The free paths of the molecules in a gas are not, of course, all equal in
length. In fact it is most probable that the free path is short and quite
154 Transport properties of gases Chap. 6
improbable that it should be long. The average or mean free path will be
denoted by X. The distribution of free paths follows the law :
(probability of free path between x, x + dx) = e
_ ~.x/X
dx
T*
(6.15)
That the exponential form of this law is correct, is shown by the following
argument.
(Probability of collision between x, x + dx)
= (probability of no collision in x) (probability of collision in dx)
and identifying the factors in this and the identical Eq. (6.15),
probability of no collision in x = e~ x/x (a),
probability of collision in dx = dx/X (b).
1 2 3 x/X
Fig. 6.9. The function exp( — x/X) as a function of (x/X).
But if (a) is correct, we would expect
probability of no collision in dx = e~ dx ^
or
probability of collision in dx = l—e~ dx/x
= dx/X
6.4 Distribution of free paths 155
for small dx. But this is exactly what we found in (b) ; therefore, the law is
selfconsistent, and satisfies the conditions imposed by the random prob
ability of collisions. Further, it can be checked by integrating Eq. (6.15)
that the probability that the free path lies between and oo is unity, as
expected ; and the mean free path, given by
f
00 dx
xe
X
does work out at X as required. Eq. (6.15), therefore, has all the desired
properties. Figure 6.9 is a graph of exp( — x/X) as a function of x/X. It is
identical in form to Fig. 3.2(b) and Fig. 4.5.
6.4.1 Mean free path and collision crosssection
Imagine all the molecules in a gas to be at rest, except one, which is
moving with velocity v in a certain direction.
This molecule will collide with another if the two centres get within
a distance a of one another (where a is the diameter of one molecule,
although we will see in section 6.4.3 that this quantity needs careful
definition). In time t, the moving molecule travels a distance vt. Any other
molecules that happen to be inside a cylinder of length vt and area na 2
will collide with it.
If there are n molecules/cm 3 , the moving molecule will, therefore, make
nna 2 vt collisions. (Of course, the molecule would be deflected but we
imagine the path to be straightened out.)
Therefore, the mean distance between collisions is
X = ix. (6.16)
nna
The quantity na 2 is called the collision crosssection of a molecule, denoted
by a. It is equal to 4 times the geometrical crosssection of one molecule,
Fig. 6.10. We can write
X = —. (6.17a)
na
In a real gas, the molecules are not all stationary in this way, but are
coming from all directions. This introduces a numerical factor : it is found
that
X = 4— (6.17b)
y/2 na
156 Transport properties of gases Chap. 6
Collision crosssection
Molecule ,
Second molecule
Fig. 6.10. The collision crosssection of a molecule of diameter a is na 2 . If the centre
of a second molecule lies within a distance a of the centre of the first, a collision occurs.
6.4.2 Estimates of mean free paths
At 1 atmosphere pressure (that is, 760 mm of mercury), at 0°C, 1 mole
of any ordinary gas occupies about 20 litres so that n = 6 x 10 23 /2 x 10 4 =
3 x 10 19 per cm 3 . If the diameter of a molecule is 4 x 10" 8 cm, its area of
crosssection is about 1.2 x 10" 15 cm 2 so the collision crosssection
' = 5 x 10" 15 cm 2 . Thus X is of the order of 7 x 10" 6 cm, which is about
200 diameters.
The pressure of a gas is proportional to n ; hence, reducing the pressure
by a certain factor increases X by the same factor. At 1/15 mm pressure, the
mean free path in air is 1/15 mm— this is a useful datum for remembering
the order of magnitude. At 10" 6 mrn pressure, the mean free path would
be 5 m, which is bigger than most ordinary containers.
6.4.3 The dynamics of collisions
The exact calculation of the collision crosssection of the molecules of a
gas is a major problem (even when purely classical laws are assumed as
they will be here). If molecules were like billiard balls with a definite
diameter, as we have tacitly assumed in the previous section, there would be
no problem. There would be no doubt when two molecules collided and
their trajectories were deflected. But with real interactomic potentials,
two effects arise which have no counterparts in the billiardball model.
Firstly, there is an effect due to the attractive forces ; this is important
at low temperatures when the mean thermal energy kT is comparable
with the depth e of the potential well. The forces cause the trajectories of
two molecules to bend towards one another— the paths are correlated in
the sense that the presence of one molecule influences the trajectory of the
other. Thus, two molecules which in the absence of attractions might just
6.4 Distribution of free paths 157
miss one another, may just hit. Collisions are, therefore, more frequent ;
the collision crosssection is increased and the mean free path decreased
compared with a gas with no attractions. We will describe later, while
dealing with van der Waals' equation, a crude method of allowing for this
effect.
Secondly, the repulsive forces (the r~ 12 term of the LennardJones
potential) can be interpreted to mean that molecules act rather like elastic
spheres which can be compressed together. Two colliding headon at high
speed are momentarily pushed together so that the distance apart of their
centres is smaller than when the two collide at low speed. We can, there
fore, not talk in a precise way about the diameter of a molecule which
behaves like this. We can, however, take the distance of closest approach
as a measure of the effective diameter. To see how this varies with the
kinetic energy of either molecule, we will state some of the more important
results of Newton's laws of motion applied to collisions and then see how
these apply to molecular collisions.
When two bodies collide, the centre of mass of the system is particularly
important. It moves with unchanged velocity at all times during the
collision. Thus, if the total mass M of the system is imagined to be concen
trated at the centre of mass, and this moves with velocity U G , then the
quantity ^MU G is conserved — in other words, U G and jMU G are un
changed, before, during and after the collision (and this holds whether the
collision is elastic or inelastic). From the point of view of an observer
situated at the centre of mass and moving with it (i.e. in a frame of reference
fixed with respect to the centre of mass), an elastic collision between equal
spheres looks symmetrical. The two approach one another with equal and
opposite velocities along parallel paths ; they are deflected, each with its
speed unchanged and they go off again on parallel paths, Fig. 6.11(a).
From the point of view of any other observer moving with constant
velocity, but not situated at the centre of mass (for example, an observer
who is stationary in the laboratory, Fig. 6.11(b)), the total kinetic energy of
the system is equal to the kinetic energies of the two particles with respect
to the centre of mass plus the term \MU G *
During an elastic collision, some of the kinetic energy with respect to the
centre of mass is momentarily transformed into potential energy of elastic
* This can be checked by measurement of the vectors in Fig. 6.11, which is drawn to scale.
There is a wellknown paradox of two trains each of mass m approaching one another, each
with speed v. A stationary observer sees their kinetic energy as 2(\mv 2 ). An observer on one
train sees the speed of the other as (2v) and its energy as jm{2v) 2 , which is different. The
question is, how much energy is dissipated during an inelastic headon collision. There is
no paradox if it is remembered that the centre of mass of the system (stationary with respect
to the earth but moving with speed v with respect to either train) conserves its speed after the
collision. With respect to the observer in the train, energy jMUq = j(2m)v 2 is conserved and
not dissipated.
158 Transport properties of gases Chap. 6
deformation. For a headon collision, all of this kinetic energy is stored
as potential energy at the instant where the spheres are reversing their
velocities ; but for a glancing collision the tangential velocity of one sphere
past the other is never zero and the spheres are less deformed.
U G
(b)
Fig. 6.11. Velocity vectors for an elastic collision of two spheres of equal mass.
(a) In the centreofmass frame of reference, (b) in the laboratory frame of reference,
where one sphere is 'chasing' the other. The velocity of the centre of mass is
shown separately.
To calculate the distance of closest approach between the centres of two
molecules, we therefore have to know the speeds of the two, which may
have any value from zero upwards (as given by the Maxwell distribution)
and their relative directions and whether the collision is headon or glanc
ing, Fig. 6.12(a). The mean distance of closest approach is governed by a
complicated averaging process, and it appears that only a fraction (about
5 or ^) of the mean kinetic energy goes into squeezing the atoms together.
It is certain, however, that the collision crosssection gets smaller with
increasing temperature because the colliding atoms on average 'climb'
farther up the potential energy curve. The overall result of these two
effects of the interatomic potentials is that the effective collision cross
section varies with temperature as shown in Fig. 6.12(b) — which is
schematic and not to be taken too literally.
It is convenient here to mention another result which is applicable to the
6.5 Calculation of transport coefficients
159
detailed discussion of the transport coefficients. Imagine one sphere aimed
at another with a given velocity, but hitting it somewhere at random so
that in a large number of throws it bounces off in all possible directions.
Then in the centreofmass frame of reference, all directions are equally
probable. But in the laboratory frame, after the velocity of the centre of
mass has been added to all the velocity vectors, there is a greater probability
that the direction of the initial velocity will be favoured. Referring back
to Fig. 4.1, for example, there is a greater probability that the molecule
will get through the layer than that it will be reflected back out. This holds
even if both molecules in a collision are moving and the effect is referred
to as the persistence of velocities.
I Headon
\collision
b A
AT~<
Temperature
id)
Fig. 6.12. (a) Distance of closest approach, (b) Variation of collision
crosssection with temperature.
6.5 CALCULATION OF TRANSPORT COEFFICIENTS
We will now calculate the steadystate transport coefficients — diffusion
coefficient, coefficient of viscosity and thermal conductivity — for gases,
using the concept of the mean free path which we have developed.
It is important to notice that whenever there exists a concentration
gradient or a velocity or temperature gradient in a gas, whenever any
transport process is taking place, then the system is, strictly speaking, not
in equilibrium. Conditions may be steady, but the speed distribution will
not follow Maxwell's law. The exact solution of all problems of this sort
then becomes difficult. Here we will assume that any departures from
equilibrium and from the Maxwell speed distribution are small, that drift
velocities are small compared with the velocity of sound, for example, and
that any gradients are small.
160 Transport properties of gases Chap. 6
6.5.1 Diffusion coefficients
We have already shown in section 6.3 that a solution of the diffusion
equation
dn ^d 2 n
dt dx<
= D ^2 (63)
can be reproduced on the simple assumption that the molecules follow a
random walk.
The object of this section is to deduce the other equation
/ = D ,6.1)
relating the flux J (the number of particles crossing unit area per second)
to the concentration gradient. We will assume steadystate conditions, so
that J does not depend on time and can be written J(z) only.
We will begin by calculating an important quantity for a gas in equili
brium — the number of molecular impacts which occur per second on a
wall (or on an area inside the gas, the molecules coming from one side
only). Let the area of the wall be A.
Let the wall be in the xy plane. Then any molecule which has a com
ponent of velocity v z normal to it and is contained in a volume of cross
sectional area A and length v z , hits the wall within one second. Therefore
the number of impacts per second from molecules whose zcomponent of
velocity lies between v z and (v z + dv z ) is
nAv z P[v z ] dv z
where n is the total number of molecules per unit volume and P[v z ] is the
probability function of v z . (This result was written down in section 5.1.2.)
Therefore the total number of impacts from all molecules moving
towards the wall (that is, v z going from to oo) is
nA
/•co I m \ 1/2 f °° I kT \ 1/2
using one of the integrals quoted on page 72 and the expression for
the probability function from Eq. (5.3). We can rewrite this in more
compact form by quoting another result, the expression for the mean speed
c, namely
2 l2kT\ l/2
V 71 ! m /
6.5 Calculation of transport coefficients 161
Hence it follows that the number of molecular impacts per second on an
area A of a wall exposed to a gas is
impacts/s = \nAc. (6.18)
This result only assumes that there is equilibrium so that the Boltzmann
distribution is followed.
Let us now select a small area dS normal to z in the gas — it can be
imagined as a kind of little pictureframe suspended in the gas— and let
us calculate the numbers of particles going through it per second, arriving
from the +z and z directions, Fig. 6.13. In equilibrium they would be
equal, but in the presence of a small concentration gradient there is a nett
flux. We will assume that nearequilibrium conditions hold.
concentration n + Xgj
oreads
concentration n
concentration n  X^
Fig. 6.13. A small area dS in a gas with molecules
arriving from the +z and — z directions.
We will now make an approximation, which leads ultimately only to an
error by a numerical factor of order unity. It is, that every molecule which
passes through dS made its last collision with another molecule in the
plane parallel to dS at a distance of one mean free path X from it. This is
not correct : on the average, each molecule makes its last collision a radial
distance X from dS, but not a distance X in the z direction because many
molecules travel obliquely to dS ; the average distance in the z direction
is something smaller than X. Nevertheless we will make this assumption.
Let the concentration in the plane of dS be n molecules per cm 3 .
The molecules coming from the + z direction come from a region where
the concentration is (n + Xdn/dz) molecules per cm 3 ; those arriving from
below come from a region with (nXdn/dz) molecules per cm 3 . Hence
more molecules come from above than below, and the nett flux of mole
cules per unit area is, in the direction of increasing z,
1 / ,dn\ 1 / ,dn\ l..dn
162 Transport properties of gases Chap. 6
which is of the same form as Eq. (6.1) expressing Fick's law. Comparing
this with Eq. (6.1),
D = \ck. (6.19a)
This is the same expression as we arrived at by considering the linear
random walk* — and the error is the same. When the distribution of free
paths is properly taken into account, together with the fact that the
molecules can pass through d5 from all angles, the result is
D = %cL (6.19b)
This exact expression is deduced in Appendix B.
Before comparing this result with experimental data, we will calculate
the coefficients of viscosity and thermal conductivity for gases.
6.5.2 Viscosity coefficient
In the arrangement of Fig. 6.2, when the top plate is moving at constant
velocity parallel to x and the gas is in a steady state, molecules continually
diffuse across the gap and carry their xdrift momentum with them. Each
molecule has drift momentum mU x , where U x is the velocity in the x
direction and is a function of z, the distance from the lower plate.
Again we image a small area dS inside the gas, and calculate the nett
flux of drift momentum through it. Since a force is a rate of change of
momentum, the nett flux is
Force/area =\n~c
mU + A d{mU >
dz
1
nc
4
mU A x
dz
where we have written m for the mass of one molecule, U for the drift
velocity in the plane of dS and we are assuming that the concentration n
is constant. Hence
Force/area  il^"
dz
This is identical in form with Eq. (6.7). We may justifiably alter the \ to
j, and write the coefficient of viscosity
rj = \nmlc. (6.20)
6.5.3 Thermal conductivity coefficient
Exactly the same methods can be used to calculate the conduction of
thermal energy across an area normal to a temperature gradient. The mean
* Presumably we identify the steplength I with the mean free path X.
6.5 Calculation of transport coefficients 163
translational kinetic energy of molecules which collide in a region where
the temperature T is given by
m? = \kT (5.12)
and the nett rate of transport of energy across unit area in the direction of
z increasing is
Q = nc\ [ kT + X^ r c^kT X\J } j
1 ,3.dr
2 2 dz
This is identical in form with Eq. (6.4). If we again alter the i to  to com
pensate for our crude averaging,
k = \nkkc.
This is correct for a monatomic gas for which the specific heat per molecule
is f/c. Remembering that n is the number of molecules/cm 3 we can write
f k as nCJN, where C v is the specific heat per mole, (f R J mol~ i deg~ x for a
monatomic gas) or alternatively as C' v which means the specific heat per
unit volume (in J cm 3 deg 1 ). For polyatomic gases, rotational energy
is transported together with translational energy, so that in general
k = ^C v Xc=^C v Xc. (6.21)
In words, the thermal conductivity of a gas is the specific heat per unit
volume, times the mean free path, times the mean speed (which is nearly
equal to the speed of sound), times a numerical factor of order unity.
6.5.4 Comparison with experiment
We have deduced expressions for the three transport coefficients for
gases :
D = ^c, (6.19b)
y\ = ^nmXc, (6.20)
where
k = i~C v Ac, (6.21)
1 1
A = ^— (6.17b)
y/2 no
164 Transport properties of gases Chap. 6
We can compare these with the results of experiments. As may be expected,
the qualitative agreement is good but the actual numbers are only correct
within a factor 2 or 3.
First, a remarkable fact can be deduced if we write the viscosity as
1
m
rj =
3V2 a
(6.22)
Here, m and a are both constants characteristic of a given gas, and c
depends only on the temperature. Thus the viscosity of a gas should be
independent of pressure. If the number of molecules/cm 3 is reduced, the
mean free path increases and their product remains constant. This predic
tion was made by Maxwell and he undertook the experiments described
in 6.1.6 to prove it. Modern measurements for argon show (Fig. 6.14(a))
that at 40°C it holds between 0.01 atmosphere and about 50 atmospheres
pressure. The constancy is astonishing when one realizes that the lefthand
side of the graph is a rough vacuum and the righthand represent a gas at
high pressure that needs a steel vessel to contain it.* The simple "billiard
ball" model seems to be adequate over this range. At extremely high
pressures it breaks down, of course, Fig. 6.14(6). This is not unexpected — at
IS)
I 3
9 2
0.001 0.01 0.1 1 10 100
pressure (atm)
(a)
F I
i
w 10
E
^ a
o> 8
.
O
^ 6

4

2
_i 1 1 — »~
500
1000 1500
pressure (atm)
ib)
Fig. 6.14. The viscosity of argon as function of pressure (a) at low and
moderate pressure (T = 313°K) and (b) up to high pressures (T = 298°K).
Dotted line calculated for a = 22 x 10" 16 cm 2 . Data from Michels, Botzen
and Schuurman, Physica 20, 1141 (1954).
* In a wellknown demonstration experiment (the 'guinea and feather' experiment invented
by Boyle) a light object, which descends only very gently when falling in air, is seen to drop
much more rapidly through a rough vacuum. Yet the viscosity of the air is the same in the
two cases. The paradox can be solved after reading section 9.7.1.
6.5 Calculation of transport coefficients
165
1,500 atmospheres and room temperature, gaseous argon is denser than
water and the mean free path is less than an atomic diameter.
Similarly we can write the thermal conductivity
1 C
K —
3^2 No
(6.23)
where the quantities N, a, C v are all constants for a given gas and c depends
on temperature ; hence k should be independent of the pressure. Figure
6.15(a) shows that this is so over the same wide range of pressures when rj
is constant — although at extremely high pressures it breaks down again,
Fig. 6.15(6). This constancy at ordinary pressures is one of the most
unexpected predictions of kinetic theory.
a> z
o •>
E
1
b 1
0.0010.01 0.1
1 10 100
pressure (atm)
500
\a)
1000 1500
pressure (atm)
(£)
Fig. 6.15. The thermal conductivity of argon as function of pressure (a) at
low and moderate pressures (T = 313°K) and (b) up to high pressures
(T = 398°K). Dotted line calculated for a = 8.8 x l(T 16 cm 2 . Data from
Waelbroeck and Zuckerbrodt, J. Chem. Phys. 28, 523 (1958), and Michels,
Botzen, Friedman and Sengers, Physica 22, 212 (1956).
The fact that both r\ and k are proportional to c means that they should
vary as y/T, the square root of the absolute temperature. Figure 6.16
shows that this is roughly true. The decrease of collision crosssection at
high temperature, described in section 6.4.3, is evident. It is perhaps un
expected that a gas should become more viscous when it is hotter ; one's
ordinary experience is limited to liquids, which show the opposite be
haviour.
By contrast to rj and k, the selfdiffusion coefficient D of a gas is directly
proportional to X by itself, and hence should be inversely proportional
to the density p: the product Dp should be constant at all pressures.
Figure 6.17 shows that this is roughly so in argon up to moderate pressures
— at 300 atmospheres the density is 0.44 g/cm 3 .
166 Transport properties of gases Chap. 6
500 1000 1500 2000 K
temperature
(a)
500
1000 1500"K
temperature
(b)
Fig. 6.16. (a) The viscosity and (b) the thermal conductivity of argon at 1
atmosphere pressure as functions of temperature. Dotted curves calculated
for a = 22 and 8.8 x 10" 16 cm 2 respectively. Data from Vasilesco, Ann Phys.
(Paris) 20, 292 (1945), (Fig. 10); Kannuluik and Carman, Proc. Phys. Soc.
65B, 701 (1952) and Schafer and Reiter, Naturwissenschaften 43, 296 (1956).
Qualitatively, then, the theory holds very well over a wide range of
temperatures and pressures. In terms of absolute values, the fit is less good.
To get the curve of Fig. 6.16(a) for the viscosity, the crosssection a must
be chosen as 22 x 10" 16 cm 2 . For the thermal conductivity, Fig. 6.16(b),
the effective a = 8.8 x 1(T 16 cm 2 . For selfdiffusion, Fig. 6.17, a =
16 x 10~ 16 cm 2 . These give the diameter of a single molecule as 2.6 A,
1.7 A and 2.25 A respectively. These are certainly of the same order of
magnitude as the diameter 3.35 A, deduced from the density of the solid
at absolute zero, Fig. 3.13(a); the discrepancy is in the expected direction.
$
0.010
0.008
0.006
0004
0.002
D in cm 2 /s
p in mol/l
100
200 300
pressure (atm)
Fig. 6.17. Selfdiffusion coefficient multiplied by density
for argon as function of pressure (T = 323°K). Dotted
line calculated for a = 16 x 10 16 cm 2 . Data from Mifflin
and Bennett, J. Chem. Phys. 29, 975 (1959).
6.5 Calculation of transport coefficients 167
It is perhaps not surprising that the transport processes in the gas and
the density of the solid do not give exactly the same values for the molecular
diameter. But the lack of selfconsistence between the collision cross
sections themselves is worth trying to explain. One approach is as follows.
In calculating the number of molecular impacts on an area of wall, we
assumed implicitly that all directions of motion were equally likely. It is
this step which is not strictly correct It is true for a gas in equilibrium,
but during transport there is some sort of drift in a special direction. In
diffusion, for example, there is a nett drift of molecules in the z direction.
Because of the persistence of velocity of the centre of mass of a colliding
pair (see the footnote to section 6.4.3), more molecules tend to travel
parallel to  z after collision than in other directions ; all directions are
not equally likely before collision and so they are not equally likely after
collision. Of course, this nonuniformity is small, but it is the transfer of
molecules in just this z direction that we are calculating. Similar con
siderations apply to heat conduction. But when we deal with viscous flow,
we are concerned with xrriomentum being transported in the  z direction
(Fig. 6.2). It is the xdirection which is favoured before and after collisions
inside dV, and the number passing through dS in Fig. 6.13 is altered. When
the very difficult averaging processes are carried through, much of the
discrepancy disappears.
6.5.5 Effusion
Suppose we have a tiny area dS which is part of a wall of a vessel con
taining gas. In Fig. 6.13, the xy plane can now be thought of as a wall, with
the gas above it. Using Eq. (6.18), which states that when there are n
molecules per cm 3 the number of impacts on unit area is \ric per second,
No. molecules hitting area dS per second = \ric dS
Pc
4 kT
, P 2 (2RT\ 112
= i *r>M dS(6  24)
where n is the number per cm 3 and c the mean speed, and we have used
the relation P = nkT and Eq. (5.10) for c.
Consider now a thin membrane with a hole cut in it, whose diameter
is comparable with the thickness of the membrane. Let there be gas at a
certain pressure on one side of this partition, and a lower pressure on the
other. Gas must diffuse through the hole but we must distinguish two sets of
conditions. If the diameter is large (more precisely, if it is large compared
168 Transport properties of gases Chap. 6
with the mean free path between collisions), any molecule suffers many
collisions while going through the hole, and the description of the diffusion
process as a random walk (superimposed on the bulk flow through the
hole) is applicable. But if instead the dimensions are small compared to
the mean free path— conditions which can be achieved using a tiny hole
with gas at low pressure— then a typical molecule suffers its last collision
some distance in front of the membrane and then goes straight through
without further collision. In fact the number going through per second
from one side to the other is equal to the number of impacts per second
on an area equal to that of the hole. If n represents the difference of
numberdensities between the two sides, and P the difference of pressure,
then the expressions given just about give the nett rate of transport of
molecules from one side to the other.
The process of diffusion through a small hole is called 'effusion'. It is
important to notice that the rate of effusion at a given temperature is
proportional to (1/M) 1/2 . A light gas, therefore, effuses more rapidly
through a tiny hole than a heavy one.
This fact has been used as the basis of a process of great technological
importance to separate gases of different molecular weights. In particular,
it can be used for enriching rare isotopes found in gases consisting of
mixtures of isotopes.
Imagine two compartments separated by a membrane with many small
holes of suitable dimensions (Fig. 6.18). One compartment contains gas
at a relatively high pressure— though it must be low enough to produce
a sufficiently long mean free path— and the other side is continuously
pumped to maintain a very much lower steady pressure. The 'partial
pressures' exerted by the gases individually will be denoted by P lh , P 2h
on the high pressure side and P u , P 2l on the low pressure side.
Now the pressure of gas 1 on the low pressure side is proportional to
the number of molecules effusing into it per second ; similarly for gas 2.
Hence
1 P lh /2RT\ 1/2
P u _ 2jnkf\M, )
P21 1 P U [2RTV 12
Ijn kT\ M 2 J
(where we have written P lh in place of (P lh P u ) for the pressure causing
gas 1 to effuse through, and similarly for gas 2). Hence
Pu (M 2 \ ll2 P lh
(6.25)
Pu Wi Pih
6.6 Knudsen gases
169
This equation says that the fractional concentration of gas 1 on the low
pressure side is a factor (M 2 /M l ) 112 times as great as on the high pressure
side. There is an enrichment of the lighter gas on the low pressure side.
high pressure
low pressure
Fig. 6.18. Effusion of gas mixture through a
membrane.
This is the basis of one method for enriching the rare isotope 235 U.
Natural uranium is mostly 238 U with 0.7 % of the lighter isotope. From the
metal, the gas uranium hexafluoride UF 6 can be produced. Its molecular
weight is about 350, so the two isotopes give molecules differing by about
1 % in mass. It follows that the concentration of the lighter isotope is
increased by a factor 1.005— that is, from 0.7% to 0.7035 %— by a single
passage through a membrane. The process can be made regenerative or
many membranes can be used in cascade^ so that useful concentrations of
the rare isotope can be obtained.
6.6 KNUDSEN GASES
It is not difficult to see why the viscosity and thermal conductivity of
gases cease to be independent of pressure when the pressure is low (Figs.
6.14(a) and 6.15(a)). It has already been pointed out (in section 6.4.2) that
at extremely low pressures the mean free path X between collisions becomes
very long. Already at 10~ 2 mm (easily attainable with a rotary pump) X
is about 1 mm for a typical gas; at 10" 4 mm it is 10 cm. Ordinary pieces
of apparatus commonly have dimensions of the order of millimetres or
centimetres so that at low pressures the calculated mean free path may be
larger than the apparatus. What this means in practice is that a molecule
can go from one side of the apparatus to the other without making any
collisions at all ; the mean distance it travels is dictated by the size of the
apparatus and not by the properties of the gas.
Gases at such low pressures are called 'Knudsen gases' after the scientist
who first investigated them systematically. They are said to exhibit
'molecular flow' instead of viscous flow.
170 Transport properties of gases Chap. 6
6.6.1 Viscous forces in a Knudsen gas
As an example of Knudsentype behaviour, consider the force between
a moving and a stationary plate immersed in a lowpressure gas (Fig. 6.19 ;
in contrast with Fig. 6.2, there is no velocity gradient in the medium
between the plates). Consider a molecule which has struck the stationary
surface and remained long enough to come to rest. It jumps off — and for
simplicity we will imagine it to be emitted normally, like molecule P in
the diagram. It then travels with its thermal speed all the way across the
gap without colliding, until it hits the other surface. After this, one of two
things might happen. It may stick for a long time and eventually be emitted
in some direction at random, or it might act as if it were specularly reflected.
moving plate
/
Q/ \R
stationary plate
Fig. 6.19. Viscous drag with molecular flow.
The experimental evidence suggests that the molecules mostly stick and
for simplicity we will assume that they all do so. Each such molecule is
travelling with xcomponent of relative velocity — U with respect to the
moving plate and therefore transfers momentum — mil directly to it. The
same overall result holds for molecules which travel obliquely across the
gap. Molecules like Q in the diagram bring some thermal momentum to
the right but this is cancelled out by molecules like R, of which there are
an equal number. The net result is that momentum — mil is transferred
from every molecule.
The total rate of transfer of momentum — the force exerted by one plate
on the other — is therefore equal to the number of molecules striking the
moving plate per second, multiplied by mil.
The number striking unit area per second is \nc. The derivation in
section 6.5.1 is independent of considerations of mean free path; the result
is true as long as there is equilibrium, or nearequilibrium. Therefore the
rate of transfer of momentum (the force on area A) is ^AnmcU*
* We could arrive at practically the same result if we started from the equations
AU
F = r] t] = jnmcl
a
and put the mean free path X equal to the separation d.
Appendix B 171
The drag is therefore proportional to the speed of the moving plate but
does not depend on the separation (provided of course that this is small).
Two oscillating disc arrangements, with different spacings between the
discs (Fig. 6.3) would have just the same damping; a coefficient of viscosity
based on Eq. (6.7) cannot be defined. At the same time, the dependence
of the viscous drag on n alone, and not on the product (nk), means that
the force is proportional to the pressure instead of being independent of it.
This behaviour is beginning to be shown at the left of Fig. 6.14(a).
For the flow of gas down a long circular tube, the analysis for viscous
flow shows that the mass of gas transported per second is proportional
to (j> A /r] where r\ is the viscosity coefficient and (f) the diameter of the tube.
In the molecular flow region, where the effective mean free path is dictated
by the diameter, the mass per second is proportional to 3 .
Similar results hold for the other transport properties. Gases become
thermal insulators, for example, at very low densities. An ordinary
vacuum flask (dewar vessel) has double walls enclosing space evacuated to
a sufficiently high vacuum for the mean free path to be limited by the
spacing. Another interesting way of providing thermal insulation is to
fill the interspace with a fine powder (derived from silica, and cheap to
produce) which is in the form of tiny thinwalled hollow spheres, loosely
packed. If the space is only roughly evacuated, the mean free path can be
limited by the diameter of the spheres or the spaces between them,
dimensions much smaller than the spacing between the walls of the vessel ;
the thermal conduction falls below the value expected if there were no
powder. Powderpacked vessels are much stronger mechanically than
dewar vessels ; tank wagons can be insulated in this way.
APPENDIX B
B.l. Diffusion coefficient in gases
In this Appendix we calculate the diffusion coefficient in a gas taking
into account the distribution of free paths — that is, not making the approxi
mation which was made in section 6.5.1.
Imagine a small area dS, normal to the zaxis (Fig. B.l), located some
where inside a large volume of gas. Consider a small volume dV located
at distance r at an angle 9 to z. We will first calculate the number of
molecules which pass through dS per second, having made their last
collision inside dV.
If there are n molecules/cm 3 in the neighbourhood of dV, there are ndV
molecules inside this little volume.
Each one, on average, undergoes one collision every time it travels a
distance k ; that is, once in every (k/c) seconds.
172 Transport properties of gases Chap. 6
Fig. B.l. Coordinate system for calculating transport coefficients.
Therefore, the number of molecules which suffer collisions inside dV in
one second is (nc/X) dV.
Each one of these molecules then goes off in some direction, and we
assume that all directions are equally likely. The fraction which start off
in the direction of dS is
solid angle subtended by dS _ dS cos
4tt 4?tr 2
Notice that by writing the solid angle in this way, we are implicitly counting
molecules that come from below as negative in number. This is because
when 9 is greater than 90°, cos 6 is negative. We are eventually, therefore,
going to calculate the nett flux of molecules going through dS from above
to below— that is, the number going downwards minus the number
going upwards, the nett rate of diffusion downwards. (If we wanted to
calculate the total number of molecules going through dS irrespective of
direction we would have to take cos 6\ instead of cos in the expression
for the solid angle.)
Of this fraction that start off in the direction of dS, some will suffer a
collision on the way. The fraction passing straight through dS will be the
fraction whose mean free path is equal to or exceeds r. Now the probability
of a free path between r and (r + dr) is
exp
r\dr
IT
Appendix B 173
from Eq. (6.15), so the probability that it exceeds r, i.e., that it lies between
r and oo, is
1 f 00
, >r ,/A dr = e" r/A .
A
Collecting these results, and using spherical polar coordinates as in Fig.
5.6(6) so that
dV= r 2 sin dr dd d(f),
we calculate that the number of molecules which collide inside dV and
then travel all the way to dS and go through it in one second is
COS C YiC
dS—  T e~ r/x dV = dS— d</3 cos 6 sin 6 dd e" r/A dr
4nr 2 X 4nX
where we are counting molecules coming from below as negative. It is
convenient now to write cos 6 = \x, so that dfi = sin 6 d6. The number is
nc
dS— d</3^d/ie~ r/A dr. (B.l)
471/1
If we integrate over all space — all values of r, 6 and <f> — we take account of
all molecules passing through dS, wherever their last collision takes place.
It is obvious, however, that only values of r comparable with the mean free
path A are important, because the factor exp( — r/X) means that there are a
negligible number of molecules whose last collision took place far from dS.
The n which appears in this expression is the number of molecules per
cm 3 inside dV. Let us now assume that there is a concentration gradient
in the direction of z — that is, that n depends on z.
If n(0) is the value of n at z = 0, the value of n at z is
™ l dn \ ! il 82n \
n(z)^ ( ) + z() o+ z^^) o + ...
and higher terms are negligible because only small values of r are important.
Writing z = r cos 6 = rfi,
dn\ 1  ^ld 2 n
„(z) = „(o, +r/i ^j o+  r V(^) o +
We can substitute this in (B.l) and integrate to get the nett number of
molecules travelling through dS from above to below per second. To cover
all possible locations of dV, r goes from to oo, </3 from to 2n and from
to n (fi from 1 to  1). If we divide through by dS, we get the nett flux
174 Transport properties of gases Chap. 6
J of molecules per unit area per second from above to below. Arranging
the terms :
Ank
/»2ti pec /»1
« d(j) e~ r/x dr /xd/i
Jo Jo J  1
d</> re" rM dr ^ 2 dji
0^0 ♦'O J I
l/^n
Let us first concentrate on the integrals with respect to p. The first and third
give even powers of pL and when limits are put in, they give zero. Only the
middle term survives :
J.
1 t i 2 d f i = ^fi 3 ]l l =i
 i
In this surviving term, the r integral
re^dr = X 2
f
and the 4> integral gives In.
Hence the nett flux from above to below is
4nX\dzJ \3/ 3 \dzj
This is a flux of molecules downwards, in the direction of z decreasing. To
put this in exactly the same form as Eq. (6.1), we have to calculate the flux
in the direction of z increasing. We must, therefore, change the sign of J.
The equation is then identical with (6.1) and
D = Uc (6 19b)
We have performed the averaging over all directions and all lengths of
free path much more precisely than in section 6.5.1, and the overall result
is to justify the factor j rather than \.
PROBLEMS
6.1. A cylindrical dewar vessel, containing water at 0°C, stands in a room where the
temperature is 17°C. The glass walls are silvered to reduce heat input; the outer
diameter of the inner wall is 10 cm and the inner diameter of the outer wall is
10.6 cm, the space between being filled with nitrogen gas at 1 cm pressure,
(a) Estimate the thermal conductivity of nitrogen gas.
Problems 175
(b) Calculate approximately the heat influx per cm height of the flask due to
heat conduction.
(c) Estimate the value (in mm Hg) to which the pressure must be reduced before
the heat influx begins to fall off.
(d) Deduce an approximate expression for the thermal conductivity of a
Knudsen gas (compare footnote, section 6.6.1) and estimate the pressure at
which the heat influx falls to ^th of its original value.
6.2. A testtube contains a liquid whose level falls slowly by evaporation. If the
liquid is very volatile, the rate of evaporation is limited by the rate at which
vapour molecules can diffuse through the air molecules. Assume that (i) the rate
of fall of level is so slow that conditions are practically 'steady state' conditions
(see section 6.1.1); (n) the number of vapour molecules/cm 3 is n v at the liquid
surface and zero at the top end of the tube which is open to an infinite atmosphere.
(a) Write down the steady state equation for the number of molecules diffusing
across any plane per second. Hence, the mass per second crossing any plane.
(b) Write down a simple equation giving the rate of loss of mass of the liquid
in terms of the rate of fall of the liquid level, dh/dt .
(c) Hence show that the distance h of the level below the open end is propor
tional tO y/t.
(d) In a narrow tube (1 mm diameter), initially full of ether, the level falls by
about 1 cm in 30 minutes, at room temperature. Deduce D for ether through
air. Make a crude estimate of the diameter of an ether molecule. (Saturated
vapour pressure of ether at room temperature = 40 cm ; it obeys PV = RT
roughly. Molecular weight = 74. Density of liquid = 0.7g/cm 3 .)
6.3. The pressure of a gas in a thermionic vacuum tube must be such that the electron
mean free path is substantially larger than the linear dimensions. If the collision
crosssection of an electron with a molecule is of the order of the geometrical
crosssection of the molecule, estimate the pressure required in mm of mercury.
Estimate also the maximum size of pinhole that may be tolerated in the envelope
if the tube is to last for at least one year. Rate of flow (g/s) through tube of
diameter D, length /, between pressure P and a much smaller pressure is
(nD*/256rjl)(M/RT)P 2 for viscous flow and ( > /27t/6)(Af//?7 , ) 1/2 (D 3 /0^ for
Knudsen flow.
6.4. A gas at a low pressure and at a temperature T is contained in a vessel from
which it effuses through a small hole whose dimensions are small compared
with the mean free path. Show that the number of molecules with speeds between
c and (c + dc) leaving the vessel per second is GcP[c] dc (where G is a geometrical
factor which need not be evaluated. See section 5.2.4.) Hence show that the mean
kinetic energy of the molecules leaving the vessel is 2kT. This is greater than the
energy, 3kT/2, of the molecules inside the vessel. Is the equipartition law
violated? Explain this result qualitatively. (It is useful to ask whether the beam
is in thermal equilibrium.)
Note
f
Jo
dx = —5.
a 5
CHAPTER
Liquids and imperfect gases
7.1 RELATIONS BETWEEN SOLID, LIQUID AND GAS
Transitions between the solid, liquid and gaseous states of any one
substance — solidification, melting, evaporation, sublimation and so on —
can be brought about by varying the temperature T, pressure P and
volume V. In any experiment to study these changes, the substance must
(in principle) be placed inside a cylinder as in Fig. 4.2, so that the pressure
acting and the volume occupied can be altered and measured. There must
also be a thermostat for controlling the temperature.
The results can be displayed in several ways. In Fig. 7.1, the axes are
pressure and temperature, in Fig. 7.2 pressure and volume. The lines are
called phase boundary lines and represent the conditions when transitions
take place. The ranges of the variables where the solid, liquid and gas
phases can exist are shown as areas, on both diagrams. At the pressure
and temperature represented by TP, called the triple point (a point on
Fig. 7.1, a line on Fig. 7.2), all three phases can exist together.
In Fig. 7.1, a vertical line represents the course of an experiment at
constant temperature. Three such lines are shown at low, medium and high
temperature, labelled a, p and y respectively. The same lines are shown on
Fig. 7.2 ; they are called isotherms.
Let us follow the isotherm a, from low to high pressure — that is, moving
upwards in Fig. 7.1 or across Fig. 7.2 starting from the low pressure, large
volume region at the bottom right. At first, the substance behaves more or
7.1 Relations between solid, liquid and gas
177
less like a perfect gas, obeying PV = constant, so that the isotherm in
Fig. 7.2 is part of a rectangular hyperbola. However, when the curve
reaches the phaseboundary line, the gas suddenly begins to condense to a
solid, which is of course much denser. If the piston which applies the pres
sure is moved so as to decrease the volume, there is no rise of pressure but
more and more gas solidifies so as to keep the pressure constant. The
isotherm in Fig. 7.2 therefore turns horizontal. When the substance inside
the cylinder has all solidified, it becomes relatively hard to compress and
the isotherm turns almost vertical.
Temperature
Fig. 7.1. Pressuretemperature diagram, sometimes
called the phase diagram showing the relations between
the solid, liquid and gas phases of the same substance.
Full lines are phaseboundary lines, broken lines are
isotherms. C is the critical point, TP the triple point.
The boundary between solid and gas is called the
sublimation curve, between liquid and gas the vapour
pressure curve, and between solid and liquid the
melting curve.
178
Liquids and imperfect gases Chap. 7
Volume
Fig. 7.2. Pressurevolume diagram for the same sub
stance as in Fig. 7.1. Full lines are phase boundary
lines, broken lines are isotherms. C is the critical point,
the triple point is the horizontal full line.
The isotherm p is of a different kind. As the pressure is raised, the gas
condenses to a liquid. In Fig. 7.1, the lower branch of the curve is crossed.
In Fig. 7.2 the isotherm turns horizontal. When the gas has all been
liquefied and becomes relatively incompressible, the isotherm in Fig. 7.2
turns steeply upwards. At some high pressure, the upper branch of the
curve in Fig. 7.1 is reached, and the liquid begins to solidify, that is, to
freeze. There is a second horizontal part in the isotherm of Fig. 7.2,
corresponding to the contraction in volume from liquid to solid. When
solidification is complete, the isotherm rises steeply again. Of course this
or any other isotherm can be followed in the reverse direction ; then the
solid would melt to a liquid, the liquid would boil to form the gas.
The high temperature isotherm y is different again. The gas can be
compressed to very high density (comparable with that of the solid) before
it condenses and when it does so, it goes straight to the solid.
7.1 Relations between solid, liquid and gas 179
One isotherm, between (3 and y, has a special importance. It is the one
going through the point C, called the critical point. C is at the abrupt end
of the lower branch in Fig. 7.1, and the top of the phase boundary curve in
Fig. 7.2. This critical temperature is the highest temperature at which the
liquid can exist.
7.1.1 Data for argon
The isotherms for argon are plotted in Fig. 7.3. Both coordinates are
logarithmic, since this allows great changes of conditions to be represented
Molar volume ■
Fig. 7.3. P, Fdiagram for argon. Pressure and volume axes are on logarithmic
scales. In this diagram, full lines are isotherms, broken lines are phase boundary
lines. Sources of data : 600° isotherm : Lecocq, J. Rech. Centre Nat. Rech. Sci.
p. 55 (1960) ; 400° isotherm : Michels, Wijker and Wijker, Physica 15, 627 (1949)
200° and 1 50° isotherms : Michels, Levelt and de Graaff, Physica 24, 659 (1958) ,
100° isotherm : Holborn and Otto, Z. Physik 33, 1 (1925). Liquid and vapour
densities : Mathias et al., Leiden Comm. 131a (1912), and Michels, Wijker and
Wijker, as above. Vapour pressures : Clark, Din et al., Physica 17, 876 (1951).
Solid density : Dobbs and Jones, Rept. Prog. Phys. 20, 516 (1957).
180 Liquids and imperfect gases Chap. 7
in one diagram. A pressure at the bottom of the graph (0.1 atmospheres or
7.6 cm mercury) is a partial vacuum; at the top, 1,000 atmospheres is an
extremely high pressure. On this loglog scale, a perfect gas isotherm
becomes a straight line at 45°.
The triple point is at 83.3°K, the critical temperature is 150.9°K. The
80° isotherm is of the type a, the 100° isotherm resembles p. The isotherm
labelled 150° is actually just 0.2° below the critical temperature, which
is why it just misses the point C. The other isotherms, at 200° and above,
are of the type y. At 200° K and 1,000 atmospheres, the molar volume
occupied by the gas is about 34 cm 3 and its density 1.2 gm/cm 3 , which is
comparable with the solid density of 1.6 gm/cm 3 . This isotherm does not
meet the solidphase boundary line till a pressure of 6,000 atmospheres,
above the top of the diagram, is reached.
7.1.2 Metastable states
In this section we will consider the processes of boiling, condensation
and freezing. The P, V diagram refers strictly to equilibrium conditions,
where in the horizontal sections of the isotherms the coexisting phases
must be at exactly equal temperatures. In a liquidvapour transition for
example, this implies that all the evaporation must take place slowly at the
surface. But in practice the heat input into a liquid may greatly exceed the
energy that can be carried away per second in this manner and when a
liquid boils at a finite rate, bubbles arise in it. In fact, their appearance is
popularly supposed to be essential to the process of boiling. We will see
however that this means that the liquid must be hotter than the vapour
and — more important — it implies that the liquid can exist under conditions
not shown in Fig. 7.2.
It is not easy to produce a bubble. Let us begin by considering one which
arises in the middle of the liquid, beginning as a small hole of atomic size
and then growing bigger. We can show that if such a hole or incipient
bubble is too small it will collapse again ; only if its radius exceeds a certain
critical value will it grow. The argument is as follows. A spherical hole of
radius r has surface area Anr 2 and energy 4nr 2 y, where y is the surface
energy in erg/cm 2 . Therefore, the energy per unit volume is 4wr 1 yj%wr' =
3y/r erg/cm 3 . By arguments which have been sketched in section 3.5.1
and will be reiterated in section 7.3, we can identify this with the pressure
inside the bubble— actually a more precise analysis gives 2y/r where y
means the surface tension, in dyn/cm. This estimate is not accurate for a
hole of a few atomic diameters in size where only a few bonds have to
be broken but it is correct in order of magnitude even then — and the
dependence on 1/r shows that enormous pressures must exist in order to
create the hole initially. These can only be produced if the liquid is super
7.1 Relations between solid, liquid and gas 181
heated above its expected boiling temperature. Let us assume that local
superheating raises the vapour pressure by AP. Then bubbles of radius
less than 2y/AP cannot be sustained and will collapse, larger bubbles can
grow. Let us assume that we have argon at around 100°K, but that the
conditions of heating are such that locally the temperature might reach
103°K — a moderate but not untypical degree of superheating under
ordinary conditions. The vapour pressure curve gives AP equal to 1 atmo
sphere, 10 6 dyn/cm 2 . The critical radius is then 2,000 A, corresponding to a
relatively large bubble occupying the volume of a million atoms of the
liquid. Big bubbles of this kind can only be produced from smaller ones
requiring greater superheating, so that it is obvious that in ordinary
boiling liquids, the bubbles must be started in some other way.
In fact most bubbles form or nucleate at solid surfaces in contact with the
liquid. The vessel which holds the liquid is likely to be hotter than else
where, so that bubbles would in any case be expected to form on its walls.
Thin flat layers of vapour are produced and these can collect together and
balloon upwards as bubbles. If the surface is rough, less superheating is
needed before the bubbles can swell up in this way, and then detach them
selves, though it is not at all obvious why this is so. Pieces of solid inside
the liquid can also act as nucleation centres — glass seems to discharge a
continuous stream of tiny particles into water so that there is never any
shortage of them. Once a bubble does form, the liquid cools itself by
evaporating into it and the temperature falls locally towards the equilibrium
value.
Under exceedingly clean conditions however, the only way the bubbling
can start is in the body of the liquid so that great superheating can occur.
One can achieve this by using chemically polished tubes of newly drawn
glass. More simply, it is possible to suspend liquids in oil of the same
density but having a higher boiling point — water, for example, can con
veniently be suspended in oil of cloves. Each liquid has a fairly well
defined temperature 7^ of maximum superheating. 10° below 7^,, it will
last for hours ; 5° below T m it can be held for a few seconds ; at T m it explodes
immediately. It will be seen later (section 7.8.1) that T m is quite close to the
critical temperature T c .
Thus superheated liquids can be kept for long periods and studied at
unexpectedly high temperatures. These are called metastable conditions.
Instead of following the expected isotherm GADF in Fig. 7.4, comparable
with the isotherm /? in Fig. 7.2, isotherms like GAB can be produced.
(AB does not represent the course of a superheating experiment but is a
plot of the properties of the metastable liquid at constant temperature.)
Similarly, gases may be made to exist at unexpectedly low temperatures
or high pressures where the P, V diagram would indicate that they should
182 Liquids and imperfect gases Chap. 7
liquefy. They are called supersaturated vapours, and they exist because
droplets have to exceed a certain critical size before they will grow;
smaller droplets evaporate again. Condensation takes place most easily
on rough solid surfaces or around small solid particles. In the absence of
these, supercooled vapours can be kept for long periods and isotherms like
DE of Fig. 7.4 can be plotted.
Fig. 7.4. Part of a P, Fisotherm for liquidgas
transition, showing metastable states. FDAG
is similar to part of the isotherm /? of Fig. 7.2.
AB: superheated liquid. X: liquid under
tension. DE : supersaturated vapour.
Finally, liquids can be supercooled below their expected freezing points
before they begin to solidify. Here the role of nuclei is to provide surfaces
with the same crystal spacing as the solid so that this can grow by the
addition of atoms to it one at a time instead of many atoms having to
arrange themselves into a regular array all at once.
7.1 Relations between solid, liquid and gas 183
7.1.3 The tensile strength of liquids
The tensile strength of a substance is the tension (measured in units like
dyn/cm 2 ) which must be applied in order to break it. For a liquid, the
stretching and breaking are represented by ABX of Fig. 7.4 : a liquid under
tension is, of course, in a metastable state.
The tensile strength of a metal or other solid specimen can be measured
by gripping its ends in some way and then pulling them apart, but this
method obviously cannot be used with a liquid. Instead, the liquid must
be put in some sort of tube so that when tension is applied in some way
to the liquid and a bubble appears in it and it breaks, one can usually not
be sure whether the break occurred inside the liquid or between the
liquid and the walls. In other words, either the tensile strength of the
liquid itself or the adhesion of liquid to wall may have been measured —
presumably the smaller of the two.
It is observed that the tensile strength of liquids is much less than that
predicted by theory. Part of the discrepancy may be due to the presence
of tiny cracks in the walls of the tube which harbour minute bubbles of gas
and prevent the liquid from entering — the very kind of nuclei which are
postulated to account for ordinary boiling— and these constitute points of
weak adhesion to the walls. It is indeed observed that if high pressure is
applied for a short time to a liquid just before the tensile test begins, the
observed strength is usually enhanced. This is presumably because the
high pressure makes the bubbles dissolve in the liquid so that the molecules
diffuse away, and do not come out again when the pressure is released.
One simple method of demonstrating the breaking of liquids is to use a
hypodermic syringe to suck up, say, water through the needle. If the
plunger is pulled rapidly so that the liquid rushes in, then where it leaves the
narrow needle and enters the wide barrel the liquid must be under tension :
bubbles can be seen to be created there. The liquid is said to cavitate.
Cavitation is responsible for a good deal of noise in plumbing systems, and
for great losses of energy from ships' propellors. Under controlled con
ditions, cavitation can be produced inside the body of a liquid, far from
any solid surfaces, by passing sound waves of high amplitude through it.
The breaks take place during the rarefactions. Unfortunately, this is not a
good way of measuring the tensile strength of a liquid because of the large
changes of temperature which also occur.
The best and most direct measurements have been taken with apparatus
of the kind shown in Fig. 7.5(a), though without elaborate techniques for
cleaning and degassing the inside of the tube the results are often a factor
10 lower than accepted values. The Zshaped tube is made of glass and is
open at both ends. It is kept in a horizontal plane by fixing it to a horizontal
184 Liquids and imperfect gases Chap. 7
disc which is itself mounted at the end of a vertical shaft of a variablespeed
motor. The tube is filled with liquid using a syringe, making sure that there
is enough to extend round the bends into the arms. Then the tube is
rotated. The liquid (surprisingly) is not thrown out but remains stably
inside the tube. (This is because the centrifugal force on the liquid at A of
Fig. 1.5{b) is less than at B because the distance OA is less than OB.)
__ B
(a)
Fig. 7.5. (a) Tube for measuring the tensile strength of a
liquid by centrifugal breaking, (b) Pressures at A, A', B, B'
are all equal to atmospheric ; hence if the liquid is originally
unsymmetrical, as shown, it must move back to a sym
metrical position.
Solid impurities should of course be centrifuged outwards, and bubbles,
similarly, collect at the centre. At low speeds, while the nett pressure is still
positive, small air bubbles sometimes collect and do not grow. It is best to
stop the rotation and coax them out of the tube and to hope that some of
the bigger nuclei have been got rid of.
At a distance r from the centre, a slice of thickness dr has mass (pot dr)
where p is the density, a the area of cross section ; the centrifugal force on it
is co 2 par dr, where co is the angular velocity in rad/s. Each slice of the
liquid exerts an outward force on the slice next to it and the total force is
greatest at the centre. Integrating between the two open ends, the tension
(per unit area) is co 2 prl, where r is the radial distance to the free meniscus.
The (positive) pressure of the atmosphere is added to this. The speed of
rotation must be increased slowly till the liquid breaks, an event which can
be easily seen because the liquid at the centre is always visible even when the
tube is rotating fast. It is convenient to use a flashing stroboscope for
7.2 The approach to the liquid state 185
illumination, however, because this allows both r and the frequency of
rotation to be measured easily.
With ordinary tap water and no special cleaning of the tube, the liquid
breaks when the frequency is about 5,00010,000 rev/min, with r about
23 cm. This corresponds to a tensile strength of 310 atmospheres. With
elaborate cleaning, values of the order of 10 times higher are found. The
tensile strength decreases with temperature.
At room temperature, carbon tetrachloride has a tensile strength of
276 atmospheres, mercury 425 atmospheres; for liquid argon at 85°K the
figure is 12 atmospheres.
7.2 THE APPROACH TO THE LIQUID STATE
It can be seen that the range of temperature over which the liquid can
exist at all is a very narrow one. It is bounded at the lower end by the
temperature of the triple point and at the upper end by the temperature
of the point C, the critical temperature. Typically, this is only a 2 : 1 range
in absolute temperature. By contrast, the gas can certainly exist at all
temperatures, and there is no evidence to suggest that the solid cannot
exist at any arbitrarily high temperature provided the pressure is high
enough. This distinguishes the liquid state sharply from the solid and gas —
it requires explanation.
An assembly of atoms in a rarefied gas is simple to treat mathematically,
because each atom is effectively isolated and moves independently of all
the others. At the other end of the scale, a perfectly regular solid can also be
treated by comparatively simple mathematical methods — at any rate, if the
amplitudes of vibration are not large — because all atoms are in identical,
highly symmetrical environments. By contrast, the disordered array of
atoms in a liquid is difficult to describe mathematically. Each atom has a
large potential energy, comparable with that in a solid, due to its inter
action with a number of neighbours. But its environment is continually
changing with time, and an atom is neither completely caged in by its
neighbours as in a solid, nor perfectly free to move as in a gas.
Now liquids can be derived from solids by melting, and this suggests
the following approach. We can consider a single atom inside an otherwise
regular solid to be displaced from its lattice position, Fig. 7.6(a). This
produces a small region where there is disorder, a region of high density
near a hole in the lattice. When we write down the energy of such a con
figuration, the kinetic term is of course unchanged, and the potential term
— the difficult one to calculate because it depends on the distances between
pairs of atoms — is also not too complicated because only a few pairs of
atoms are concerned. Thus we can deal with a single displaced atom. Then
186
Liquids and imperfect gases Chap. 7
we can imagine many such disordered regions to be produced. Fig. 1.6(b).
As long as they are far enough away from one another, each region will be
independent of all the others and the energy required to produce n of them
will be just n times the energy to produce a single one so that it can still be
calculated. When the degree of disorder becomes too great, however, the
problem becomes intractable. But we can in this way produce a first
approximation to a liquid — or at any rate, to a solid which is showing
signs of melting. Certainly, though such an assembly with a small number
of displaced atoms is really quite far from being a liquid, Fig. 7.6(c), it can
be expected to indicate relations to look for in a real liquid.
oooooo
oooooo
oooooo
OCKpOO
oooooo
(a)
icf)
Fig. 7.6. (a) One atom out of place in an otherwise regular solid. (Compared
with Fig. 2.5 the coordination number is low.) (b) Several atoms are displaced
but the disordered regions are still more or less independent, (c) Atoms in a
liquid, (d) A compressed gas at high density with several small clusters of
atoms. Both (£>) and {d) begin to resemble (c).
Alternatively, we can concentrate on the fact that liquids can be derived
from gases by condensation. We can, therefore, begin with a gas at very
low pressure and imagine the atoms brought closer and closer together so
that they spend progressively more of their time near to one or more
7.2 The approach to the liquid state 187
neighbours and the potential energy of the clusters of atoms cannot be
neglected. The gas is then said to be imperfect. Again, only an assembly
with a small number of small clusters of atoms can be dealt with simply,
but even this is a rough approximation to the onset of condensation and
the formation of the liquid, Fig. 1.6(d).
In Chapter 9 we will be concerned with imperfect solids, the onset of
melting, and liquids in so far as they resemble solids. In this chapter, we
will deal with imperfect gases, the beginning of condensation, and liquids
as derived from gases.
7.2.1 Laws of corresponding states
We have emphasized the difficulties of writing down the potential
energy term in liquids and have stressed that the procedure of regarding
them as derived from imperfect gases cannot be pushed too far. We then
meet a paradox. The simplest considerations allow us to derive an equation
— van der Waals' equation of state — whose basis is particularly crude, but
which is found to predict many properties of liquids themselves with
surprising accuracy. By rights, van der Waals' equation should only
succeed in describing the behaviour of gases which are still far from
condensing ; yet it is capable of predicting, in order of magnitude, such
properties as the latent heat of the liquid.
There is no doubt that this surprising power is in part due to the
felicitous analytical form of the equation, which is simple but adaptable.
But the important fact is that once the interaction energy e between a
pair of atoms is known, many of the properties of all the phases can be
estimated, as we have already seen. In van der Waals' equation, we
express energies not in terms of e itself but in terms of the critical tempera
ture T c (or rather kT c ), which is directly related to ^. The pressure and
volume at the critical point C are also taken as standard parameters.
Using the simple approach of Chapter 3, we were able to estimate the
equilibrium properties of substances only at very low temperatures. Van
der Waals' equation, however, suggests that we can legitimately compare
the properties of two substances at their critical points — or at some given
fraction of their critical temperatures, or under corresponding pressures or
volumes. Conditions of this sort are called corresponding states.
The real significance of van der Waals' equation is therefore that it
predicts certain laws of corresponding states. As a real description of phase
changes, it fails badly — it does not predict the existence of the solid, for
example — and even to make it refer to liquids is to use a wild and un
justifiable extrapolation. Nevertheless, more accurate theories of con
densation are extremely difficult to construct, and do not lead much
further. With these caveats, van der Waals' equation will now be derived.
188 Liquids and imperfect gases Chap. 7
7.3 VAN DER WAALS' EQUATION
The object of the section is to develop a simple theory of imperfect gases
and their relation to liquids. This means that we will try to explain as much
as possible of Fig. 7.7, which has been dissected out of Fig. 7.2.
Fig. 7.7. P, V isotherms for gas and liquid, extracted
from Fig. 7.2.
For an imperfect gas, we have to take some account of the potential
energy of the atoms due to their interactions. These are of the form given
by Fig. 7.8(a) which is identical with Fig. 3.4. r is the distance between the
centres of two atoms.
Since our theory will perforce only be crude, we will simplify the
algebra by substituting a rough approximation for this curve — the square
well potential of Fig. 7.8(b). This has the form
Potential energy i^(r) = oo
r < a
= — s a < r < oca
= aa < r.
(7.1)
a is a number greater than 1 which presumably ought to be chosen so that
the volume integral of the potential energy,
f
v a
V{r)Anr 2 dr
7.3 Van der Waals 1 equation 189
n
A
B C
r
7
aa
Fig. 7.8. (a) Interatomic potential energy as function of distance between
the centres of two atoms ; identical with Fig. 3.4. {b) Squarewell potential
energy, an approximation to (a), adequate for many purposes. A, B and C
correspond to Fig. 7.9.
£
'c
(c)
/A B C
(of)
Fig. 7.9. Interactions of molecules (each of diameter a) having squarewell poten
tials, (a) r A is just greater than a and the system has energy £. (b) r B is just less than
aa and the energy is again e. (c) r c is just greater than aa and the energy is zero.
(d) Shows the centre of the second atom in relation to the first; the interaction volume
v, lies between the two dashed outlines.
190 Liquids and imperfect gases Chap. 7
has the same value for the 612 and the square well potentials.* In this way
one obtains
a 3 l=f i.e. a = 1.54.
The potential implies that the atoms are incompressible spheres of
diameter a ; the centre of two spheres cannot therefore approach more
closely than a. At distances between a and aa, the potential energy is — e,
as shown in Fig. 7.9(a) and (b). At distances greater than oca, there is no
interaction, Fig. 7.9(c). Another way of describing this is to imagine each
atom surrounded by an 'interaction volume', bounded by spheres of
radii a and aa, as in Fig. 1.9(d), so that if the centre of another atom lies
inside this volume the energy is — e. The size of this interaction volume is
v . = f7ta 3 (a 3 l) (7.2)
If we write v for the volume of a single molecule, then we have
and
v { = 8(a 3 l)i; . (7.3)
Now the problem is to calculate the mean energy of attraction of the
assembly of molecules. To do this exactly, we should write down Boltz
mann factors representing the probability of different values of the
energy. But this becomes impossibly complicated, so we will adopt a
simple approximate procedure. Consider two atoms located somewhere
inside a volume V. Assuming that all positions are equally probable
(which is not strictly true at such short distances), the probability that
one lies within the interactionvolume of the other is (vJV), and the po
tential energy of the pair is, on average, —{v i /V)s. If there are N atoms
randomly distributed in this volume, we can select jN(N—1)~jN 2
pairs of atoms.
Therefore the potential energy of the assembly is, on average :
ir= ~\ N2 [v) e  {1A)
So the total energy, kinetic plus potential energy, for N molecules of the
imperfect gas is
jf + r = l N kT l N 2 \^\z. (7.5)
In a perfect gas, the second term is zero.
* We identify a with the a of Eq. (3.5).
7.3 Van der Waals' equation 191
Now we have already mentioned, in section 3.5.1, that an energy
density or an energy per unit volume is equivalent to a pressure, although
if we use the kinetic plus potential energy this relation is only true for
adiabatic changes when no heat flows into the system. But for rough
calculations, the error introduced is not too serious. For example, for a
perfect gas the energy is all kinetic and has the value \NkT so that the
rough rule would give P = \NkT/V, or in other words PV = \RT,
instead of RT. Here, for the imperfect gas, we are calculating the effect of a
small additional potential energy term, so it will serve our purpose to get
the major term in the pressure correct (by omitting the factor f ) and use the
rough rule for the extra term ; this is likely to introduce a numerical error
but to leave the form of the expression correct. Then, deliberately leaving
the denominator a little vague,
p = RTmVj/V)* (7 . 6)
volume
In this way we have accounted roughly for the attractive forces between the
atoms. If we seek a dynamical interpretation of the reduction in pressure, it
is that the atoms spend more time near one another than they would if
there were no attractive forces, and this reduces the number of impacts on
the walls.
There is a second effect due to the interatomic potential energy, this
time the repulsions which give the atoms their finite size. The volume
available to the atoms to move freely about in is not V but something
smaller. We should therefore, write the denominator on the right hand
side as (Vb), not V, where b is a volume presumably of the same order
of magnitude as the volume of all the molecules.
It is not easy to decide the value of b exactly. One extreme argument is to
imagine all the molecules gathered together in one lump— of solid or
liquid— except for one single one, which would then obviously find a
volume Nv not available for moving about in, where v is the volume of a
single atom, f?r(a/2) 3 . Another extreme argument is to say that the centres
of two gas molecules cannot approach more closely than a, so that a
volume Sv is excluded around each molecule, making b eight times bigger
than before. But this can be rejected as an overestimate because for a
grazing collision between two molecules the trajectories are hardly
deviated so that the centres come within just the same distance of one
another as they would have done if the atoms were points ; there are many
more glancing than headon collisions. Presumably, then, b lies some
where between Nv and SNv . The most detailed analyses, taking
proper account of collision dynamics show that 4Nv is the best estimate
ofb.
192 Liquids and imperfect gases Ch. 7
It is convenient to write a in place of^Nhfi. It follows immediately from
the connection between v it the interaction volume, and v , the volume of a
molecule (Eq. 7.3) that
a = N(ix 3  l)be (7.7)
where a is the measure of the range of the interatomic forces as shown in
Fig. 7.8(b), and s is the depth of the potential well. We have also seen that
(a 3  1) is about 8/3 so that a/b « 2.7 Ne. We have also seen, in section
3.3.1, that the molar binding energy of the condensed phase at low
temperatures is L = ^Nne (where n is the coordination number, about
8 or 10). Hence roughly,
a/b a £L . (7.8)
Eq. (7.6) now reads
„ RTa/V
p = r=r (7  9)
RT o_
since the second term is already a small correction;
so that l P +JLy V b) = RT. (7.10)
This is van der Waals' equation of state. Eq. (7.9) differs from (7.10) in
one important respect: (7.9) is quadratic in V, (7.10) is cubic, and this
latter form is essential to the ability of van der Waals' equation to interpret
many phenomena.
We can plot P, Fisotherms by assigning a value to Tand calculating P as
a function of V as in Fig. 7.10(a). These isotherms are of two kinds. At
high temperatures, P decreases monotonically with V while at low
temperatures the curves have maxima and minima, as expected of a cubic
equation. We identify these two regions as above and below the critical
temperature. It is obvious that the Sshaped curves are unphysical, but it
is tempting to identify parts of them as representing the supersaturated
vapour and the superheated liquid, and to draw horizontal lines across to
represent the equilibrium mixtures, Fig. 7.10(b). No good reason for doing
this can be adduced from the mode of derivation — the Sshapes are merely
a result of imagining only pairs of atoms being near to one another — but if
this procedure is accepted, then plausible (though not rigorous) arguments
suggest that the line should be drawn so as to make the area under it equal
to that under the curve. In other words, the areas of the two loops should
be equal.
7.4 Application to gases 193
Fig. 7.10. (a) van der Waals' isotherms, (b) Horizontal line drawn so as to
equalize the areas of the loops.
7.4 APPLICATION TO GASES
We have stressed that van der Waals' equation should be valid only when
the density is not too great. We will therefore discuss its application to
gases at low and moderate densities where the equation is applicable and
at high densities where, as expected, it breaks down. We can regard a and b
in the equation as constants whose values can be chosen to fit experimental
data It will emerge that values of a and b can be chosen which allow
several gas phenomena to be correlated. But when we compare these
values of a and b with those expected from independent estimates of the
sizes of molecules and the depth of the potential well, the agreement is in
order of magnitude only— in particular, the value of b differs by about a
factor 3 from that expected from the solid density.
Finally we will consider briefly how the theory could be improved.
7.4.1 The second virial coefficient B{T)
One of the most powerful methods of displaying the way a real gas
deviates from a perfect gas in its behaviour is to plot the ratio PV/RT as a
function of increasing pressure or decreasing volume. There are theoretical
reasons for preferring 1/Fas the variable, and curves for argon are shown in
Fig. 7.11. Each curve refers to a fixed temperature. They are called virial
plots.
194 Liquids and imperfect gases Chap. 7
Most of this graph refers to small volumes— this is the effect of using
1/Kas the variable. When V is infinite, 1/Fis zero. When V is 100 cm 3 ,
which is quite small compared with normal conditions, 1/Fis only one
third the way along the axis in Fig. 7.1 1. The molar volume of liquid argon
is about 30 cm 3 so that all the curves must be asymptotic to l/V = 0.033.
At large volumes, that is when l/V tends to zero, PV/RT = 1 always, so that
all the curves go through one point on the vertical axis.
VV cm 1
Fig. 7.11. Virial plots for argon. Sources of data as for Fig. 7.3.
Many reasonable curves y = f{x) can be represented by a polynomial,
y = a + /foe + yx 2 \ . Here, each virial curve can similarly be represented
by
PV , B C
RT V V 2 *
The coefficients B,C ■•• are called the second, third and higher virial
coefficients. They depend only on the temperature so that they should be
7.4 Application to gases 195
written B{T), C(T) and so on — in general,
PV B(T) C(T)
For the polynomial v = a + fix + yx 2 • • • , the gradient of the curve when
x is small is equal to /? (since higher terms in the expression for the gradient
are negligible). Thus the gradients of the virial plots when 1/7 is small, the
initial gradients of the curves, are equal to B(T). A graph of B as a function
of T for argon is given in Fig. 7.12. At low temperatures, when the virial
curves start downward, B is negative. At temperatures near 410°K for
argon, the virial curve starts horizontally, so that B = 0. Around this
temperature, over a considerable range of pressures, the gas obeys the
perfect gas law PV = constant (Boyle's law) with accuracy, whereas at
other temperatures it deviates significantly at much smaller pressures. This
is therefore called the Boyle temperature, denoted by T B . At high tempera
tures where the curves start upwards, B is positive and tends to a constant
value.
B(T) can be determined from experiments while PV/RT does not deviate
too much from unity — under conditions, in fact, when van der Waals'
equation should be valid. We will, therefore, rearrange (7.10) as a poly
nomial in the form of Eq. (7.11):
P =
RT a
Vb V 2
so
PV = (l_ h \~ 1 a
RT \ Vj RTV
.2
1 .t. « \1 &
= 1 + \ h Rf)v + V 2+ ' (7  12 >
and comparing with Eq. (7.11)
a
B(T) = h ~Rf (713)
Van der Waals' equation therefore predicts that at very high tem
peratures B(T) tends asymptotically to b, while at low temperatures
B(T) becomes large and negative, following a rectangular hyperbola.
The Boyle temperature T B is evidently a/Rb.
The detailed course of B(T) as calculated agrees quite well with experi
ment, as shown by the dashed curve of Fig. 7.12. This has been drawn with
196 Liquids and imperfect gases Chap. 7
b = 42cm 3 and a = 1.42 x 10 12 erg cm 3 /mol— values which give an
adequate fit over the whole range and also give a/Rb close to the observed
value, 410°K, of the Boyle temperature.^
Fig. 7.12. Second virial coefficient B{T) for argon. Dashed curve : van der
Waals' curve calculated with a = 1.42 x 10 12 erg cm 3 /mol, b = 42 cm :
B = [42 (1 71 x 10 4 )/T] cm 3 . Data from Lecocq, J. Rech. Centre Nat.
Reck Sci., p. 55 (1960).
We can now apply a stringent test of the whole theory as constructed so
far, by testing the prediction that
o/b « \L ( 7  8 )
where L is the binding energy at low temperatures. Here a/b = 3.4 x
10 10 erg/mol, compared with L = 7 x 10 10 erg/mol given in Fig. 3.13(b).
The agreement is remarkable.
The absolute value of b is not very good, however. Since we have seen
that b should be about 4 times the volume of the molecules it would imply
a molar volume of about 10 cm 3 for the solid instead of 26 cm 3 . A plausible
7.4 Application to gases 197
reason for believing that the value of B at high temperatures does never
theless depend on the volume of the molecules is provided by substances
such as helium where the a term is small. Following the arguments of
section 6.4.3, the value of b for a real gas would be expected to decrease at
high temperatures because the molecules are not really hard spheres. In
argon however, any small decrease of the b term in the expression
(ba/RT) is swamped by the change in the a/RT term. But in helium
where a is small, a small decrease in B(T) at high temperatures is evident
in the measurements — although the absolute value of b remains too
small, even then.
7.4.2 Specific heats of imperfect gases
The specific heat C v of a gas obeying van der Waals' equation is the same
as if the interaction terms a and b were removed and the gas became perfect.
This is because during a heating at constant volume, the mean distance
between molecules (and hence the potential energy) remains unchanged.
Alternativejy we can argue that since
C.[§) f (5,5)
(where we have written E for the mean value of the total kinetic plus
potential energy) and
 3 a
E = 2 RT ~V (7  5)
for one mole of an imperfect monatomic gas, C v = f R as for a perfect
monatomic gas.
But if a gas is heated and expands to keep the pressure constant, a
quantity of heat C p dTmust be supplied to raise the temperature by dT—
not only to increase the kinetic energy of the molecules and to supply the
work done, but also to increase their interatomic potential energy. For a
monatomic gas :
= ^RdT+^dV+PdV. (7.14a)
In general, for a gas whose specific heat at constant volume is C v ,
C p dT= C v dT+ (P + ^2) dV. (7.14b)
C p dT=d\RT)+PdV
198 Liquids and imperfect gases Chap. 7
Here, dTand dFmust be related because the pressure must be constant.
We can find this relation as follows. Since
small variations of P, V and T must obey
k + 172 d(Vb) + (Vb)d\P + 1 ^\ = RdT
a \ ... / „ . _a
V
P^ 2  7 0jdV+(Vb)dP = RdT.
We can neglect the term in ab/V 3 . In the present case, P is held constant,
dP = 0, so that
P — % )dV= RdT.
V )
This is the special relation required to substitute in (7.14b) above. After
simplifying, we get
P + a/V 2
C p C » R p_ a/V 2
= R 1 +
RVTj
to sufficient accuracy. Thus for monatomic gas obeying van der Waals'
equation
5 2a C_ 5 4 a
C„ = R; C p = R +Vf ; 7 = ^ = 5 + 5 ^^ (715)
This means that the ratio y of specific heats is no longer a constant even
if the classical equipartition of energy holds, y should increase as the
density and temperature are decreased, and this variation allows us to
measure a again.
The ratio y was introduced into the discussion of perfect gases (section
5.4.2) because their adiabatic elasticity is equal to yP, and the speed of
sound is equal to ^{yRT/M); this allows y to be measured easily, which in
turn allows both C p and C v to be calculated. For imperfect gases, however,
* The same result can be deduced with more sophistication using the relation stated in
section 5.4.1 :
C„C v = [P + (dE/dV) T ](dV/dT) p .
7.4 Application to gases
199
all these results require modification. For gases obeying van der Waals'
equation, it may be shown that the adiabatic elasticity is equal to yP 2 V/RT
(which reduces to yP if PV = RT, as expected) and the speed of sound is
c. =
PV yRT
RT\ M
(7.16)
Thus y can still be calculated from the measured speed of sound but with
an additional factor which can be read off Fig. 7.11. For example, for
argon at 200°K, at a pressure such that V = 200 cm 3 and l/V =
0.005 cm 3 , Fig. 7.11 gives PV/RT equal to 0.79. The speed of sound is
equal to 256 cm/s. From these data, y = 2.52 — quite a different value
from the 5/3 = 1.667 found at low densities.
200
.200° K
H*
100
0O1
002 1/1/
1000 °K
(o)
(d)
Fig. 7.13. (a) Ratio of specific heats y for argon, as a function of 1/Fat 200°K.
When V = 200 cm 3 , l/V = 0.005 cm" 3 , y = 2.52. Hence the initial gradient
of this graph is (2.521.67)/0.005 = 170 cm 3 . Data from Michels, Levelt
and Wolkers, Physica 24, 769 (1958). (b) Initial gradient of this type of
graph, as a function of temperature. Dashed curve : %a/RT; a = 1.42 x 10 12
ergcm 3 /mol. Data for gradients taken from Nat. Bur. Std. (U.S.) Circ,
564 (1955).
A graph of y as a function of l/V at 200° K is given for argon in Fig.
7.13(a). y begins by increasing linearly, as van der Waals' equation predicts,
but then the curve turns downwards again when the density becomes large.
The situation is analogous to that in the virial plots — van der Waals'
equation predicts only the initial gradient. Proceeding as we did before,
we can plot this quantity over a wide temperature range, and compare it
with the predicted 4a/3RT. The two curves are shown in Fig. 7.13(b), using
the same value of a equal to 1.42 x 10 12 erg cm 3 per mole as in Fig. 7.12
for the van der Waals' curve. The agreement is good. Thus even the crude
200
Liquids and imperfect gases Chap. 7
representation of interatomic forces provided by van der Waals' equation
is enough to account qualitatively for the specific heat variations.
7.4.3 Free expansion of gases
In section 5.4.2 we discussed the temperature change when a perfect
gas (with a = and b = 0) performs work by pushing a piston back — an
adiabatic expansion with the performance of external work. If however a
perfect gas undergoes an expansion without performing any external
work, no energy is expended and there is no temperature change. This is
called a free expansion and we will describe how it can be performed in
principle although measurements of this kind are rarely performed in
practice.
Imagine a vessel constructed of rigid material, thermally insulating and
of negligible heat capacity (Fig. 7.14). Inside, it is divided into two com
partments one of which contains a gas under pressure, while the other is
empty. The wall dividing the compartments is then broken and gas flows
so as to equalize the pressure throughout. Then, while the gas is flowing,
one compartment gets hot because the gas there is being compressed more
or less adiabatically, while the other side gets cold because it is doing the
compressing. Imagine then that the two halves later reach equilibrium by
exchanging heat with one another, and come to equal temperatures. If the
gas is perfect, this final temperature will be exactly the same as the initial
temperature.
Fig. 7.14. Free expansion of a gas inside a rigid, insulated vessel.
During the whole process, the gas is isolated from its surroundings by
the rigid vessel so that energy neither enters nor leaves. The total internal
energy, kinetic plus potential, of the gas must be constant. Let us therefore
consider one mole of an imperfect monatomic gas, whose energy is given
by the van der Waals' expression
*!"•£
(7.5)
7.4 Application to gases 201
where V is the volume it occupies. This is conserved. If the subscript i
denotes the initial and f the final conditions, then
\ RT 'vrl RT 'y' ■ ,7  17)
For 1 mole of argon expanding from 1 litre to 2 litres, that is from about
20 to 10 atmospheres, the change of temperature would be 4° (taking a
to be 10 12 ergcm 3 /mol). This is a large change but in practice the heat
capacity of the vessel — necessarily thick walled — would decrease its
magnitude and we will attempt no comparisons with the meagre experi
mental data.
7.4.4 JouleThomson coefficient
The JouleThomson porous plug experiment is a much more sensitive
method of measuring the change of internal energy of a gas with pressure.
It is a continuous process (as opposed to a "oneshot" process like the
one just described) so that the temperatures ultimately reached by different
parts of the apparatus do not depend on their heat capacities.
In principle, Fig. 7.15, a gas is maintained at pressure P t (by an external
compressor) and is brought to a known temperature T x . It is forced
through a device which can maintain a pressure difference, and does not
allow heat to be conducted across it. In the original experiments, carried
out in 1852, this was a silk handkerchief. Nowadays, a cottonwool plug
or a porous ceramic plug is used ; often, just a long length of narrowbore
tubing with a small hole in the end. The gas then emerges into a space
m
mm
pressure /° pressure P z
temperature T A temperature T z
Fig. 7.15. The flow of a gas from high to low pressure
through a porous plug.
* The general expression for the temperature change in a free expansion is
C v (dT/dV) E = P T(dP/dT) v
which for a gas obeying van der Waals' equation is a/V 2 . This gives
C„dr= (a/V 2 )dV
in agreement with the above expression.
202 Liquids and imperfect gases Chap. 7
which is maintained at another known pressure P 2 . The temperature T 2
at the exit side is measured. The gas is continuously forced through the
plug, and measurements are made only when the parameters are steady.
In such a process, the gas is certainly not isolated from its surroundings,
so that the internal energies of 1 mole of the gas on one side and the other
are not equal. However, we can still find a quantity which is conserved.
First, we note that in the steady state there is no nett interchange of heat
with the walls. Secondly, we can assume that the velocity of bulk move
ment of the gas is so small that its bulk kinetic energy can be neglected.
Finally, we concentrate on the balance between the internal energy of the
gas and the work performed on it or by it.
Let 1 mole of the gas occupy volume V r on the entrance side. The work
done on it to force it through the plug is jP . dV where P is constant at
P t and V is changed from V^ to zero ; that is, (P 1 V t ). On the other side, the
same gas performs work P 2 V 2 on the pump. The nett amount of work must
come from the internal energy, so that
E, +P.V, = E 2 + P 2 V 2 .
In thermodynamics, E + PV is called the enthalpy, and it is this quantity
which is conserved here.
Expanding van der Waals' equation and neglecting the very small
term in ab/V 2 :
PV= RT + bP.
We also have, for a monatomic gas,
E = \ RT ~V (75)
So
5 2a
E + PV = RT + bP.
We can now calculate the temperature change dT accompanying a small
change of pressure dP. First we can, with little loss of accuracy, substitute
PV = RT in the small a/ V term:
E + PV = RT — — + bP.
2 RT
Since this is conserved,
d(E + PV) = 0.
7.5 Refinements to van der Waals' equation 203
This gives
l RdT+ **dT^dP+bdP = Q
2 RT 2 RT
dT 2a/RTb
 for a JT process = & + 2«I7K7* (7.19a)
Writing 2a/VTin place of 2aP/RT 2 , the denominator is seen to be C p for
the monatomic gas, Eq. (7.15). Generalizing, one obtains for the Joule
Thomson effect in any gas obeying van der Waals' equation :
dT 2a/RTb ^^
— for a JT process = — . (7.19b)
dP F C p
This is called the differential JouleThomson coefficient. Both dTand dP
represent increases. In a real experiment, dP is always negative. Therefore,
if (2a/ RT— b) is positive, the drop of pressure will cool the gas. Evidently
this should occur if the temperature is low, because then the first term is
large. Conversely, a heating should occur at high temperatures. The
changeover from heating to cooling, at a temperature called the inversion
temperature T h occurs when T = 2a/ Rb. Van der Waals' equation predicts
that the inversion temperature should be twice the Boyle temperature T B .
Agreement with these predictions is surprisingly good. Figure 7.16
shows the observed differential JouleThomson coefficients for argon
(extrapolated to zero pressures where C p is accurately 5R/2). The observed
inversion temperature T t = 785°K; thus the ratio TJT B = 785/410 = 1.92,
close to the predicted value 2. The dashed curve is that predicted from van
der Waals' equation with a = 1.42 x 10 12 ergcm 3 /mol, b = 42 cm 3 , the
same values used for the virial coefficients and y curves, Figs. 7.12 and
7.13.
* 7.5 REFINEMENTS TO VAN DER WAALS' EQUATION
The problem of deducing an accurate equation of state, valid at all
densities, is basically one of writing down the potential energy of an
enormous number of interacting molecules. Van der Waals' equation
takes only the first step in this direction. In the following sections we will
consider some important effects which have not been taken into account,
and some ways in which van der Waals' equation might be refined.
* The general expression is (dT/dP) H = {T(dV/dT) p  V}/C p where H stands for enthalpy.
For a gas obeying a virial equation, this reduces to { TB'(T) — B(T)}/C P to a first approxima
tion.
204 Liquids and imperfect gases Chap. 7
Fig. 7.16. The differential JouleThomson coefficient for argon as a
function of temperature. Since 1 atmosphere = 1.01 x 10 6 dyn/cm 2 , 1°K/
atm ~ 10~ 6 °K/dyn.cm" 2 . Dashed curve : van der Waals' curve calculated
with a = 1.42 x 10 12 erg cm 3 /mol, b = 42 cm 3 : JouleThomson coefficient
= ( .203)°K/atm. Continuous curve: experimental results. Sources
of data : as for Fig. 7.13(a), {b).
7.5.1 Dimers
When two atoms in a gas are a small distance apart comparable with
their diameters, they have an appreciable (negative) potential energy.
In an ordinary encounter between two atoms, they approach one another
and then fly past ; it is only while they are close together that they have
potential energy comparable with  &. The pairs of atoms which we have
considered so far are not bound permanently together in any way ; they
are merely pairs of atoms which happen for a short time (a fraction vJV
of the total) to be in one anothers' vicinity.
But the existence of the minimum in the T^"(r) curve means that it is
possible to form pairs of atoms which are loosely bound together. These
are called dimers. It can be shown theoretically that the commonest form
of dimer is not two atoms statically stuck to one another, or oscillating
as if joined by a spring; instead, they rotate round one another — in orbit
round one another, like a double star. The two molecules 'touch' one
7.5 Refinements to van der Waals' equation 205
another and the mean frequency of rotation is about 10 11 rev/s.
To form such a pair out of two isolated atoms requires rather special
circumstances. As has been mentioned, two atoms which come together
will usually fly apart again ; but if by chance there is a third atom in the
vicinity at the right moment which can take away enough kinetic energy,
then the pair can be left in a bound orbit. This is called a threebody
encounter. Equally well, another collision with a sufficiently energetic
atom can knock them apart again. The energy required to disrupt a dimer
is practically equal to e if they are orbiting only very slowly, but it is
reduced if they are orbiting fast. The 'centrifugal potential' L 2 /2I has to
be added to the interatomic potential energy, where L is the angular
momentum and / the moment of inertia of the system. There is therefore a
limiting angular momentum above which the potential well is filled up and
the dimer cannot exist at all.
During the whole of its existence, a dimer makes an appreciable con
tribution to the potential energy of the gas. At high temperatures, dimers
are likely to be knocked apart again after a short time, but at low tempera
tures when they are longer lived, their energy can dominate the second
virial coefficient B(T).
Dimers have been detected and their masses measured experimentally.
This was first done by Leckenby and Robbins in 1965. The basic idea was
to produce a narrow beam of gas atoms, bombard it with electrons so as
to produce ions and then to analyse the masses present in the beam by
passing it through a mass spectrometer. The biggest technical difficulty
was to produce the narrow beam inside the high vacuum required for
the mass spectrometer to function. This was done by allowing the gas to
escape through a tiny hole from a reservoir where the pressure was of the
order of 1 mm of mercury ; the thickness of the diaphragm and the dia
meter of the hole in it were both of the order of 10" 4 cm, comparable with
the mean free path of the molecules in the gas. Under these conditions, the
beam effused through without change of temperature or mean energy,
a true sample of the molecules inside the reservoir.
With argon (atomic weight 40) molecules of mass 80 were detected.
They were shown unambiguously to have originated inside the reservoir
and not spuriously as the result of any process inside the mass spectro
meter. In the gas at room temperature at 10 cm pressure the dimer
concentration was found to be 1 in 10 4 , in agreement with theoretical
estimates. Of course these experiments can only be conducted with low
pressures in the reservoir ; dimer concentrations are greatly increased at
high pressures.
One unsuspected fact was revealed in these experiments. It has been
mentioned in section 5.4.4 that specific heat measurements indicate that
206 Liquids and imperfect gases Chap. 7
monatomic molecules in gases at ordinary temperatures do not rotate
about their centres, whereas polyatomic molecules (such as N 2 or C0 2 )
do rotate. Now a dimer of N 2 or C0 2 has three sorts of rotation going
on inside it — the spinning of each molecule about its centre and the
orbiting of the molecules round one another. Within this system, angular
momentum must be conserved. If therefore one of the molecules stops
spinning for any reason, the orbiting must speed up — and this will pro
bably cause the pair to fly apart. In monatomic gases however, this effect
cannot occur. The result is that polyatomic gases contain fewer dimers
than monatomic gases.
7.5.2 Higher clusters
The pairs of molecules which are not bound together but nevertheless
possess some potential energy because they happen to be in one anothers'
vicinity for a short time are called clusters of two. Bigger clusters are of
course possible, clusters of three or more. It was thought for a long time
that three molecules meeting at a point would be so rare an event that its
probability could be neglected: but this is not true at high densities.
Taking into account the potential energy of the higher clusters will
obviously alter the equation of state. How it will do so can be guessed from
the following line of argument. Let us first consider a gas of hardsphere
molecules, but with no attractive forces : that is, e is zero but a is not zero,
or in other words a is zero but b is not zero. Then the equation of state
becomes
P(Vb) = RT.
Expanding this as a virial equation
PV _ V b b 2 b 3
RT~ Vb ~ 1 + v + v* + V* + """
In other words, for the hardsphere model, the second virial coefficient is
b, the third b 2 and so on. However, we have seen that when we put in the
attractive forces and consider clusters of two, the second virial coefficient
becomes (ba/RT) instead of b, while the others are left unchanged. We
may therefore guess that if we consider clusters of 3 the third virial co
efficient will be modified and so on. The higher clusters are therefore
important because they allow the higher virial coefficients to be calculated
properly.
We will now outline how the clusters of 3 could be dealt with, using the
same rough methods as we did for the clusters of 2. The argument will
not be followed through to the end because the approximations are too
crude; the purpose of this calculation is to indicate some interesting
7.5 Refinements to van der Waals' equation 207
features of the argument which have their analogies in more sophisticated
treatments.
Consider a volume V containing a single molecule and a cluster of 2
molecules. The interaction volume of the cluster is shown in Fig. 7.17.
If the third molecule enters either of the volumes v A , it is near only one
neighbour and loses energy e only. But if it comes within the volume u B ,
it interacts with both of them and loses energy 2s. Thus there are two
distinct types of clusters of 3.
Fig. 7.17. Interaction volume of a cluster of 2.
Assuming that the distribution is still random, the probability that the
third molecule finds itself within one or other of the volumes v A is 2vJV;
within v B , the probability is v B /V. Thus the average extra energy lost by
the cluster of 2 because of the presence of the third molecule is
2v A v B
E f 2s —
V V
which can be written 2vJV because
2v A + v B = 2v t .
We must now put in the probability that the clusters of 2 was formed
originally, and the fact that in an assembly of N molecules there are
N(JV— 1)(N — 2)/3! « N 3 /3\ ways of selecting three of them. The average
potential energy of the assembly is thus
2! \Vj 3! \VJ
We can then write down the energy density and get a small pressure term
in V~ 3 as well as the V~ 2 term of van der Waals' equation. Expanded as a
virial equation, the third virial coefficient can be picked out. It depends
on temperature but is still of order b 2 .
208 Liquids and imperfect gases Chap. 7
The problem of computing the energy of all the possible types of
cluster becomes rapidly more complex with the size of cluster, if more
realistic potentials are used in place of the square well. Calculations of
this type have nevertheless been pursued because it was hoped that the
Sshaped curves of Fig. 7.10(a) would be eliminated if all clusters could be
included. In fact, dimers and higher bound groups of molecules probably
play a dominant role in the process of condensation. They have been
detected by the same kind of experiment as that described in section 7.5.1,
but allowing the gas to enter the vacuum through a comparatively wide
nozzle so that it cooled to a low temperature by expansion. Dimers,
trimers and all degrees of association were found up to 40molecule
aggregates, the upper limit of the instrument. Some of the biggest of these
might almost be thought of as small droplets.
7.6 CRITICAL CONSTANTS
Having emphasized the inadequacies of van der Waals' equation and
the fact that it cannot be expected to be valid beyond moderate densities,
we will nevertheless apply it to high densities. Whereas practically all the
former results could be deduced equally well from Eq. (7.9) instead of
Eq. (7.10), the cubic form of van der Waals' equation is now essential.
We have already noted that the isotherms of a van der Waals gas have
an Sshaped form below a certain temperature, which we identify with
the critical temperature.
We can select the critical isotherm by first finding the locus of maxima
and minima, and then finding the maximum of this curve.
Starting from van der Waals' equation
RT ji_
V^b~V :
P = 7TT772> ( 7 10)
we find the equation of the curve on which all turning points lie by differ
entiating with respect to volume, keeping the temperature constant,
dP\ RT 2a
+ TTT
dVj T (Vb) 2 V 3
= at the turning points.
We can get a more useful expression by using van der Waals' equation to
eliminate RT. We get
2a _ RT _ P + a/V 2
V*~ (Vb) 2 ~ (Vb)
7.6 Critical constants
209
or
P = a
(V2b)
K 3
(7.20)
as the equation of the locus of maxima and minima, Fig. 7.18. The maxi
mum of this curve is given by equating the gradient to zero :
— = ~(3bV) = 0.
ii i
ii i
ii i
n i
11 i
ii
ii
ii
H
ii
ii
ii
u
n
i!
i
Fig. 7.18. Full curve: locus of maxima and minima
of van der Waals' isotherms, Eq. (7.20). Isotherms
also appear in Fig. 7.10(a). The isotherm labelled T m
is referred to in section 7.8.1.
Thus at the critical point :
V c = 3b. (7.21a)
Substituting this value back in the equation for the locus of maxima and
minima,
a
Pr.=
21b 1
(7.21b)
210 Liquids and imperfect gases Chap. 7
and putting both these in van der Waals' equation,
It has already been shown, when a and b were originally defined, that
we expect (from Eq. (7.8)) :
alb = Ne.
1 3
This gives
c 81*
= 0.79 e/k.
Roughly speaking therefore, the critical temperature T c occurs when
kT c ~ e (7.22)
that is, when kT becomes comparable to the energy of interaction of two
molecules. When the thermal energy exceeds this, the gas cannot liquefy.
We could proceed to compare the critical constants of a number of gases
with the values of a and b which we derive from the JouleThomson
coefficient or from knowledge of the radii of the molecules and the form of
the squarewell potential. But it is more realistic to use van der Waals'
equation to provide laws of corresponding states — for example, to
compare liquids at their critical points. We can write for the van der
Waals gas
RT C 8a 27b 2 1 _ 8
YJ C ~ 21Rb ~a~ 3b ~ 3' (? " 23)
This relation is obeyed remarkably well. For argon, T c = 150°K,P C = 48.3
atmospheres = 49 x 10 7 dyn/cm 2 , V c = 74.6 cm 3 , so that this ratio is
3.41. This is within 30% of the predicted value, 2.67. Data for other gases
are:
T
•*c
°K
atm
cm 3
RT C
Nitrogen
126
33.5
90
3.43
Carbon dioxide
304
73
94
3.63
Water
647
218
56
4.35
7.7 Fluctuation phenomena 211
7.7 FLUCTUATION PHENOMENA
It is convenient at this stage to introduce an important aspect of
statistical theory, namely the fluctuations that occur in any statistical
quantity. It is a concept which is applicable to any branch of statistics
and it could have been introduced in section 4.2 where statistical ideas
were discussed. However, the optical phenomena which occur at the
critical points of liquids are among the most striking manifestations of
density fluctuations in fluids. We will therefore concentrate on phenomena
near the critical point and then extend the discussion to fluctuations in
general.
7.7.1 Critical point phenomena : critical opalescence
The region of the critical point has been much studied and some peculiar
effects seem to occur there. It seems that the temperatures at which the
isotherm becomes horizontal, at which the meniscus disappears and at
which the properties of the two phases become identical may all be
different. Thus, over a narrow interval of temperature the fluid can exist
in a tube as two layers of different density even though there is no sharply
defined meniscus separating them. At the same time, it is known that the
properties of the liquid and vapour in the region of the critical point are
strongly influenced by minute traces of impurity and it is difficult to
decide whether the observations are highly significant for a full under
standing of the process of condensation or whether they are in some sense
spurious.
One phenomenon however certainly uncovers some interesting
physical ideas. The appearance of a normally colourless fluid at its critical
point is remarkable. Illuminated by a beam of light, it looks diffuse and
shimmering and intensely white, so that one instinctively thinks that a
cloud fills the whole space. If the temperature is raised or lowered by as
little as a fraction of a degree away from the critical point, the whiteness
disappears and the gas or liquid appears colourless again as one thinks
it should. This phenomenon is called 'critical opalescence'.
The process undergone by the light in its passage through the fluid,
going in as a beam but coming out diffusely in all directions, is called
scattering. Now it can be shown that a large block of a perfectly regular
crystalline solid at absolute zero scatters no light at all — it would be
invisible except for reflections from the surface — provided that the wave
length of the light is much greater than the interatomic spacing, a condition
which is always satisfied. The fact that the fluid scatters so strongly near
the critical point indicates that it is in some way far from homogeneous.
We will be able to show (in section 7.7.3) that this is indeed so, and that its
212 Liquids and imperfect gases Chap. 7
density varies appreciably from point to point at the critical point. We
cannot however, give a satisfactory account of the optical effects, for the
following reasons. It can be shown that when the density of a medium
varies from point to point in an entirely random way, the variation being
appreciable over distances small compared with the wavelength, then
short wavelengths are scattered more than long ones and as a result the
medium looks blue. The blue of the sky, for example, originates in the
light scattered by the great thickness of air above one's head ; this in itself
is sufficient proof that the air is composed of a completely random
arrangement of small molecules. Evidently, a medium where the variations
of density are random but particularly large would scatter blue light
with great intensity. But in a highly compressed gas or a liquid the mole
cules are not arranged completely at random. There are regions where
several molecules are nearly closepacked, whose arrangement over a
short distance is fairly regular. It is this fact that causes the scattered light
near the critical point to be white rather than blue. Measurements of the
intensity at any angle can indeed give information about the scale of
distance over which the molecules are ordered — which bears a distant
relation to the average size of the clusters. Experiments show that at the
critical pressure but 10 ~ 3 deg above the critical temperature, this 'correla
tion length' is about 1000 atomic diameters; to deg away it is about
10 diameters. We cannot follow these arguments, however, but will
content ourselves by showing that the variations in density which occur
near the critical point are particularly large, without attempting to
estimate the intensity of the scattered light or to predict its colour.
7.7.2 Concepts of probability theory — II . fluctuations
The variations of density from point to point which occur in every
system but which are very great in a liquid at its critical point are a
fluctuation phenomenon. We will now consider fluctuations in general.
It must be emphasized that they occur in all systems and that they are not
due to gross effects like unequal heating or nonuniform external pressures :
they occur in systems in thermal equilibrium. Their origin lies in certain
aspects of statistical theory which we have so far ignored.
In section 4.2 we called attention to the fact that when we deal statistic
ally with small numbers of people, the characteristics of a single individual
can quite upset the shape of a histogram. Thus, Fig. 4.3(a) which refers to
a small sample of 100 people with at most 15 in any range, is not a regularly
stepped histogram ; Fig. 4.3(b) which differs from it merely in the much
larger numbers in any range, is regular. Such deviations from the most
probable value occur in all types of statistical phenomena. We will,
however, concentrate on physical examples.
7.7 Fluctuation phenomena 213
The most probable situation inside a gas or liquid in thermal equilib
rium is that (in the absence of external fields) its density is uniform at all
points, on the average. But instantaneously this is not so, as can be seen
by referring to the pictures of the molecular arrangements in gases,
liquids and solids, Figs. 2.3, 2.4, 2.5. Liquids under ordinary conditions
contain 'holes' whose size is comparable with 2 A cubed, though there are
many regions where the packing is close ; and the pattern changes all the
time. Obviously, then, the density can fluctuate over a wide range if we
concentrate on small volumes. This can be seen by making a mask of
size corresponding to 3 A square, and laying it down over Fig. 2.4 : the
number of molecules encompassed varies from about 1 (near the hole in the
lower righthand corner) to about 2 wherever the packing is close. This is
a 100% variation. If on the other hand we deal with large volumes, the
fluctuations get relatively smaller; a mask corresponding to 10 A square
encloses between about 20 and 24 molecules in Fig. 2.4, only a 20%
variation. It is indeed a general principle that the smaller the average
number of molecules enclosed contained in any arbitrarily selected
volume, the proportionally larger are the fluctuations of density.
Usually, in ordinary laboratory experiments, we cannot detect the effect
of density fluctuations in gases or liquids, because we deal with large
numbers of molecules, and most instruments cannot respond to the
fluctuations. However, the blueness of the sky and the critical opalescence
of fluids do detect them. In addition, other quantities can also fluctuate.
For example, the movement of a small particle undergoing Brownian
motion in a liquid (section 4.4.2) is determined by the mean momenta of the
molecules within a small volume of liquid, comparable with the volume of
the particle — and its jerky movement shows that this quantity fluctuates.
In other words, the Brownian motion detects the fact that though the mean
momentum crossing any plane in the interior of a stationary fluid averages
out to zero over a long time, it departs from zero at any instant ; it fluctuates
about the value of zero. In an analogous sort of way, the pressure exerted
by a gas on a very small area of wall fluctuates about its mean value and
this can also be detected — not using ordinary sluggish pressure gauges
but rapidlyresponding ones.
7.7.3 Fluctuations of volume in an elastic system
We can most simply calculate the probability of finding a certain density
near a given point in any fluid by fixing attention on a given number of
molecules and finding what volume they occupy. Consider therefore a
fixed point X and select the n molecules which at any instant are to be
found nearest to it. On the average they must occupy a volume v such
that n/v is equal to the mean numberdensity averaged over the whole
214 Liquids and imperfect gases Chap. 7
volume. But at any instant they may occupy a volume v, greater or less
than v ; of course, the numberdensity n/v is less probable than n/v .*
We may calculate the difference in probability of these two states by
calculating the difference of energy between them and then using the
Boltzmann factor.
Consider therefore a volume v of fluid in equilibrium at a certain
pressure and alter its volume to v by expanding or contracting it, keeping
the temperature constant. Following an argument similar to that in 3.5.1,
we define
(isothermal) bulk modulus K =  W^ (3.10)
dP\
=  Vo{ Tv
T
nearly, if the change of volume is not too great. Therefore when the
volume has increased from v to v the extra pressure acting is
The energy required to increase the volume further by dv is
so that the total energy required is
AE = 5 f V {v  Vo )dv = U^^ = l Kv s 2 (7.24)
vo
where s is the fractional change of volume, (v — v )/v , Eq. (3.24). This
means that the energy is increased whenever it deviates from its equilibrium
value, because the squared term must be positive. This is reasonable : the
fluid pressure resists any change from the equilibrium value.
The ratio
probability of volume v = e _ Af/kT = e  (K *o/2iW (7.25)
probability of volume v
and this gives the probability of a volume fluctuation of magnitude s.
* It does not matter if any one of the original n molecules diffuses away from the vicinity
of X and its place is taken by another because (according to both classical and quantum
mechanics but contrary to intuition) molecules are indistinguishable from one another.
We are justified in thinking that j; or v is always occupied by the same molecules.
7.7 Fluctuation phenomena 215
We could proceed to work out the meansquare volume fluctuation
? from the correctly normalized probability by averaging s 2 . But we can
avoid all this by noting that the energy A£ is a squared term of exactly
the same form as the \I(o 2 or \mv\ terms in the kinetic energy of a single
molecule. Using the same terminology as before (section 5.3), the com
pressibility of a volume v of any substance confers on it one degree of
freedom. However large or small v is, however many molecules it contains,
its fluctuation of volume confers one single degree of freedom.
Therefore the meansquare volume fluctuation is given by
^Kv 7 = ^kT
that is
s 2 =
v — v\ kT
v / Kv
(7.26)
It is obvious that if v is small, the r.m.s. value of s is large and this agrees
with the statement that fluctuations are largest in small volumes.
Eq. (7.26) says that the smaller the bulk modulus K, the greater
the fluctuations. The reason is simply that little energy is then required
to cause a change of volume so that large changes of volume become
probable. Now, the isothermal bulk modulus K is proportional to the
slope of the PV isotherm and at the critical point this is zero. Eq. (7.26)
could then indicate infinite fluctuations. Actually these cannot occur
because higher derivatives of the slope have to be taken into account — but
the fact remains that fluctuations of volume (or density) are large and
light is strongly scattered. As explained above, the analysis of the optical
effects will not be taken further.
It is worth noting that results analogous to Eq. (7.26) for other elastic
systems can be written down at once. For example, the length of a rod
fluctuates because the atoms in it are in motion. If Y is Young's modulus,
its increase of potential energy
A£ = ^Yl s 2
where / is its mean length, s the fractional change of length A/// . Hence
? = fi <7  27 >
For a rod 1 m long with Y = 10 11 dyn/cm 2 , the r.m.s. fluctuation of
length is of order 10 11 cm at room temperature, which conforms with
ordinary experience by being negligible.
216 Liquids and imperfect gases Chap. 7
7.7.4 Fluctuation in a perfect gas
An important result will be derived as a particular case of Eq. (7.26).
Consider the fluctuations of a volume containing n molecules of a perfect
gas. Since it obeys PV = RT,
K =  V (%) T = P (Z28)
— the isothermal bulk modulus is equal to the pressure. Thus
y kT
s = ^r
Pv
where v is the mean value of the volume. Now n/v — N/V , where N is
Avogadro's number and V is the molar volume at pressure P and tempera
ture T. This gives
s = L (7.29)
(This value of s is the r.m.s. fluctuation.) This means that n molecules are
contained in a volume which fluctuates between the values
1+H and v \l — r
in the sense that these limits define the r.m.s. fluctuation. We can express
this slightly differently by calculating the number of molecules contained
in a. fixed volume v . If that number on average is n, it fluctuates between
probable limits
nlH — t\ and "I 1 — r '
that is, between n±y/n. For example, a gas under standard conditions
contains 3 x 10 19 molecules in 1 cm 3 on average ; in fact the number has a
r.m.s. fluctuation of roughly 5 x 10 9 . Expressed differently, this is a fluctua
tion of order 1 in 10 10 which is negligible. This is because the number of
molecules is so large. But in a volume of 100 A cubed, there are only 30
molecules on average, and the number therefore fluctuates between about
25 and 35, which is roughly a 15% variation. Once again, the smaller
the number, the relatively larger the fluctuations.
This yjn law for a gas of independent particles has been derived in a
rather roundabout way. It could have have been deduced more directly
from first principles by expressing the fact that the probability of finding
a molecule inside v is independent of where all the other molecules are.
7.8 Properties of liquids estimated on van der Waals' equation 217
Many statistical systems obey this kind of law. The problem of counting
the heights of members of a population, referred to at the beginning of
section 7.7.2, is typical ; numerical examples of the ^Jn rule have been
given in section 4.2.
7.8 PROPERTIES OF LIQUIDS ESTIMATED ON VAN DER
WAALS' EQUATION
Extending the application of van der Waals' equation even below the
critical point into the liquid region is of course quite unwarranted. But
it must be remembered that, following the discussion of section 7.2.1,
useful laws of corresponding states can be written down. With this in
mind we can estimate the tensile strength of a liquid using van der Waals'
equation. We will also discuss certain aspects of the boiling of liquids.
7.8.1 Tensile strength and superheating of liquids
We have seen in Fig. 7.10(a) that van der Waals' equation gives isotherms
at low temperatures which go below the P = axis, corresponding to
states of tension. The minimum value of P on any isotherm corresponds
to the tensile strength at that temperature.
We have already calculated this value as
a(V2b) ,„„ rtX
P = ^yT 2 ( 7  20 )
(with V < 3b to make sure we are at a minimum). The volume of the
condensed phase is roughly equal to b, so the tensile strength at low
temperatures should be roughly
P = £ (730)
which is equal to — 27P C , where P c is the critical pressure. For liquid argon
this is badly wrong. We have already quoted (section 7.1.3) that liquid
argon has a tensile strength of 12 atmospheres ; van der Waals' equation
predicts 1,300 atmospheres.
We can interpret this equation by referring back to the relation a/b « ?L
[Eq. (7.8)], where L is the molar binding energy at low temperatures.
Our expression for the tensile strength is therefore equal to the energy
required to vaporize a mole of liquid divided by the molar volume. Since
pressure or tension is an energy density, we must be implying that when
the liquid breaks, it half vaporizes. This is not the mechanism at all,
and the van der Waals' approach is unrealistic.
218 Liquids and imperfect gases Chap. 7
By contrast, van der Waals' equation predicts with surprising accuracy
the maximum temperature of superheating, possibly because now the
liquid does all vaporize. This temperature was called T m and was defined
in section 7.1.2. Referring to the complete family of isotherms, Fig. 7.10(a),
we can say that if the branch corresponding to superheated liquid goes
below the P = axis, then in any experiment at this temperature con
ducted at low (approximately zero) external pressure, the liquid can be
contained. But if the minimum of the isotherm lies above the Faxis, then
the liquid cannot exist under low external pressure. T m is therefore the
temperature of that isotherm whose minimum first touches the P =
axis. This isotherm is labelled T m in Fig. 7.18. Putting P = in Eq. (7.20),
the condition is that V — 2b, and this in turn gives
T m = ^T c . (7.31)
In other words, van der Waals' equation predicts that liquids can be
superheated, under small external pressures, to 0.85 of their critical
temperatures.
In Fig. 7.19, T m is plotted against T Q for a number of liquids. Data for
some of the points are given here :
Sulphur dioxide 323°K 430°K
Ether 416°K 466°K
Alcohol 477°K 516°K
Water 543°K 647°K
The line of gradient 0.85 passes quite well through the points and those for
a number of other liquids.
7.8.2 Vapour pressure of liquids
In principle we can calculate the vapour pressure of a liquid once we
know where the horizontal part of an isotherm is to be drawn so as to
equalize the areas of the loops, as shown in Fig. 7.10(b). Unfortunately
it is not possible to do this analytically because the cubic form of the
equation makes it impossible to express the integrals in closed form.
However, we may guess that the probability that a molecule can
escape into the gas phase will contain the factor exp(L /#T), and that
therefore this factor will appear in the equation for the vapour pressure.
7.8 Properties of liquids estimated on van der Waals' equation
219
o
600
500
400
300
200
100
oj
100
200 300 400 500 600
700
T
Fig. 7.19. Plot of maximum temperature of superheating T m against
critical temperature T c for several liquids. Data from Kennick, Gilbert
and Wisner, J. Phys. Chem. 28, 1297 (1924).
Thermodynamic and statistical arguments show that a better equation
is of the form
P = (const.) x T n e' LolRT (7.32)
where n is equal to the difference of specific heats of vapour and liquid (in
units of R), and is therefore a small number. It is quite reasonable that this
factor T" should enter, because the latent heat varies with temperature.
Over small temperature ranges, the exponential factor varies much
more rapidly than the T" term and this allows a quick estimate of L
to be made if vapour pressures are roughly known. Comparing vapour
pressures at T x and T 2 , and taking only the exponential term into account,
~ R\T 2 TA r\ Tl
In
(7.33)
where T av is a mean between 7\ and T 2 . Water, for example, has a vapour
pressure of about 2 cm at room temperature, that is about 300° K, while
at its boiling point 373° K its vapour pressure is atmospheric, 76 cm.
Simple arithmetic gives L « 5,700 R, close to the correct value of
220 Liquids and imperfect gases Chap. 7
4 x 10 4 J/mol, since R = 8.3 J/mol deg. Alternatively, since L for most
common liquids which boil at ordinary temperatures is always of the same
order of magnitude, the same data show that the vapour pressure of
common liquids roughly doubles itself for a rise of temperature around
10°K at ordinary temperatures.
7.8.3 Supersaturated vapours
We can now return to the nucleation of liquid droplets in vapours, and
discuss how they grow, and why vapours can be supercooled.
Consider a droplet of radius r. Compared with the same number of
molecules in the interior of a large mass of liquid, the molecules of the
droplet have greater energy. This is because some of them are at the surface
and do not have their full coordination number or, in macroscopic terms,
the surface tension gives the droplet a surface energy. Or, to state just the
same result differently again, a pressure 2y/r is exerted on the molecules
in the droplet (where r is its radius, y the surface tension). The distance
between molecules in the droplet is slightly decreased because of this.
Since pressure is an energy density, the extra energy is 2yV/r where V
is the volume of the droplet (where, as usual, this is only rough but is
good enough for orderofmagnitude estimates). Thus the activation
energy for vaporization, the latent heat (which is proportional to the
energy required to separate a pair of atoms) is reduced. Consequently
the vapour pressure of the droplet, compared with that of a large mass of
the liquid with a plane surface, is increased. Instead of a term exp( — L /R T)
in the vapour pressure, we have exp( — L + 2yV/r)/R T where L , R and V
must all refer to N molecules, that is V must be the molar volume. Thus
compared with a mass of liquid with a plane surface (r = oo), the vapour
pressure is increased by the factor exp(2y V/rRT) = exp(2yv/rkT), where
v is the volume of a single molecule, k is Boltzmann's constant, r is the
radius of the droplet. It is easy to see why the vapour pressure increases
compared with a plane surface. A molecule just inside a curved surface
finds it easier to escape because it does not have to break n/2 bonds (see
section 3.4) but some smaller number, which is reduced all the more as the
surface becomes more strongly curved.
Consider now a vapour at pressure P in equilibrium with a liquid with
a flat surface. This means, of course, that the liquid is just boiling or the
vapour just condensing. The rate of arrival of molecules, ^nc/s.cm 2 ,
must be equal to the rate of escape ; therefore since n is proportional to
the vapour pressure, both rate of arrival and rate of escape must be
proportional to the vapour pressure.
Next consider a droplet inside a vapour maintained at a certain pressure
P, greater than P so that the vapour would be called supersaturated. Let
7.8 Properties of liquids estimated on van der Waals' equation 221
us calculate the radius of droplet which is in equilibrium. Compared with
a plane surface, the rate of evaporation is increased by the factor
exp(2y v/rkT), while the increased vapour density means that the rate of
arrival of molecules is increased by the factor P/P . Thus for equilibrium
— = G 2yvlrkT ( 7>34 )
Thus inside a supersaturated vapour at pressure P, a droplet of radius r c
given by
2yv (7.35)
is in equilibrium. If (P/P ) is less than unity, r c would be negative which
means that no droplet is in equilibrium. If P = P the surface must be
plane. If (P/P ) = 1.1, ln(P/P ) « 0.1 and r c is of order 10" 7 cm so that the
equilibrium droplet contains the order of 100 atoms.
But we can show that this equilibrium is an unstable one. Imagine a
vapour held at a pressure P but the radius of the droplet to be slightly
decreased from its equilibrium value. Then the number of molecules
condensing per second remains the same but the number evaporating
increases. This tends to make the droplet shrink even further. Conversely,
a droplet whose radius slightly exceeds r c must grow even bigger.
P/R >1
radius r
Fig. 7.20. Energy of a droplet of radius r as a function of r. The energy
of the same molecules inside a large mass of liquid, with no surface, is
taken as zero.
222 Liquids and imperfect gases Chap. 7
In all the equilibria we have discussed so far, the condition that the
forces are zero has been interpreted to mean that the potential energy is a
minimum (section 3.2.1). Such a system, displaced, will return to equilib
rium. Here however the potential energy of the droplet expressed as a
function of its radius must go through a maximum. There is certainly
equilibrium when the radius has exactly the required value corresponding
to the given pressure P, but displaced from this condition (i.e. slightly
increased or decreased in radius) the droplet will evaporate or expand
further and not return to equilibrium.
Figure 7.20 expresses this. Each curve refers to a given degree of super
saturation, a given {P/P ). It expresses the surprising fact that, at a given
temperature with a given pressure of vapour, the molecules condensed as a
small droplet can actually have a higher energy than when they are in the
vapour. The molecules inside the droplet have indeed each lost their
energy of condensation (not the full e per pair but something less because
of the finite temperature) ; but when the droplet is small there are, pro
portionally, so many molecules at the surface that the droplet as a whole
actually has more energy than in the corresponding vapour. Only when
the droplet is quite large does the process of condensation reduce the
energy of the assembly.
Thus there is an activation energy for droplets to form, and this at once
explains why vapours must be supersaturated before they can condense.
The surface energy of the critical droplet is Anrly but the activation energy
(by thermodynamic arguments which we cannot reproduce here) is equal
to onethird of this,
A = Wh (7.36)
Thus the probability of forming a droplet contains the factor
exp(— jnrly/kT). Expressed explicitly in terms of (P/P ) this is a com
plicated function, but it increases dramatically with (P/P ). For (P/P ) = 1.1,
the activation energy is of order 10  12 ergs or 1 eV, and the Boltzmann
factor is exp( — 40) at room temperature. If (P/P ) is 1.2, r c is reduced
by a factor 2, the activation energy by a factor 4 and the Boltzmann factor
becomes exp(— 10) which is a factor 10 10 times larger than before. This
example, though incomplete, demonstrates the interplay of the different
factors.
PROBLEMS
7.1. You are given that the critical temperature of hydrogen is 33°K.
(a) Write down a relation between the inversion temperature and the critical
temperature for a gas obeying van der Waals' equation. What is the inversion
temperature of hydrogen?
Problems 223
(b) Show that at room temperature hydrogen gets hotter when it undergoes a
JouleThomson expansion. j<,,„+„
(c) In the 1840's, Regnault found that at room temperature and moderate
pressures, PV for hydrogen was greater than RT He called ^t a more than
perfect gas' because all other gases known at the time had PV less than RT
Explain this observation.
7.2. Find the critical constants for Dieterici's equation
P{yb) = RTexpi^a/RTV)
where a and b are constants and all other symbols have their usual meaning.
What is the second virial coefficient and the Boyle temperature for Dietenci s
equation?
7.3. Methyl chloride CH 3 C1 has a critical temperature of 416°K. Tte hqiudhas
density = 1 g/cm 3 at room temperature. The molecular weight is 50.5. The second
virial coefficient is
239 255 311 366 422 450 °K
764 637 401 265 184 155 cm 3 /mol
Data from J. S. Rowlinson, Trans. Faraday Soc. 45, 974 (1 949).
(a) Estimate b and the diameter of a molecule from the molar volume of the
(b) IfTan der Waals' equation held, what function of Tplotted against B would
give a straight line graph? Plot such a graph and though it gives a pronounced
curve, strike a reasonable straight line and estimate a. Hence estimate e.
(c) Estimate £ from the critical data.
Id) The molecule has a CI" ion at one end and a concentration of positive charge
at the other. It therefore acts like an electric dipole ; the dipole moment n o\
charges +e and e separated by distance / is el. In addition to the 612
potential energies, two molecules therefore have an additional dipole
dipole interaction. This may be attractive or repulsive depending on relative
orientation but on the whole it is attractive of order of magnitude n /Am ? r
(J if a is in Coulomb metres and r is in metres). This swamps the r attraction
so we are left with a 312 potential energy. Sketch the shape of the Y (r)
curve. Estimate fi. .
le) Compare it with the moment of charges ± e (where e is the electronic charge
section 2.1.2) separated by 1 A. Sketch the charge distribution in the methyl
chloride molecule.
7 4 One mole of a van der Waals' gas is kept in a vessel at its critical volume but its
temperature T is greater than T c . Show that its isothermal bulk modulus is
3R(T T)/4b Hence show that the mean square volume fluctuation is given by
S" =
9N TT
1 5 A random walker takes N steps in the +x or x direction (see section 6.3). He
takes one step every x seconds and each is of length /. What is (a) the most
224 Liquids and imperfect gases Chap. 7
probable nett distance travelled? (b) the likely fluctuation from this? (c) Verify
the yjt law for his diffusion and write down an approximate expression for his
diffusion coefficient.
7.6. A very sensitive spring balance consists of a quartz spring suspended from a
fixed support. The spring constant is a, i.e. the restoring force of the spring is
 ax if the spring is stretched by an amount x. The balance is at a temperature T
in a location where the acceleration due to gravity is g.
(a) If a very small object of mass M is suspended from the spring, what is the
mean resultant elongati on x of t he spring?
(b) What is the magnitude ((xx) 2 ) 1/2 of the thermal fluctuations of the object
about its equilibrium position?
(c) It becomes impractica ble to m easure the mass of an object when the fluctua
tions are so large that ((x  x) 2 ) 1/2 = x. What is the minimum mass M which
can be measured with this balance?
CHAPTER
Thermal properties of solids
8.1 THE EXTERNAL FORMS OF CRYSTALS
To the physicist, intent on understanding the forces which bind matter
together, the ideal form of solid is the crystal. There, the molecules are
regularly arranged, so that the environment of any one of them is well
denned and the problem of computing the energy of the assembly is
comparatively simple. (This enthusiasm for crystalline solids is not shared
by engineers, who ask for materials which are mechanically strong.
Typically, they use materials which are deliberately made impure and
contain more than one phase, and if they are crystalline at all, the crystals
are very small.)
The inner regularity of the molecular arrangement in crystals is made
manifest by the regularity of their external forms. There is an obvious
tendency for crystals to be bounded by plane faces. Any one substance
usually forms crystals of a particular habit (that is, general shape),
whether cubes or needles or hexagonal prisms or flat plates. No two
crystals of the same substance are identical in shape ; certain facets in
one specimen, compared with another, may be enlarged at the expense
of others. But the angles between corresponding faces are remarkably
reproducible.
In Fig. 8. 1 for example, one cannot help feeling that the two crystals
(a) and (b) have an underlying identity and that the different relative sizes
of the faces is somehow accidental and unimportant. We can express this
226
Thermal properties of solids Chap. 8
as follows. We construct normals to each face and then let each normal be
moved parallel to itself so that it passes through the centre of a sphere.
Each normal intersects the sphere in a point and the arrangement of these
points on the sphere expresses the angular relationship of the faces.*
The two crystals (a) and (b) give identical patterns, independent of the
relative sizes of the faces, as shown in Fig. 8.1(c), except of course that by
chance certain facets might be absent from one.
N.
X,>!
(a)
{b
c)
id)
Fig. 8. 1 . (a) and (b) : Crystals with faces of different sizes but the same angular relation
ships. Normals to each face are shown. Three cube faces and four octahedral faces
are visible, (c) : When the normals are translated to pass through the centre of a
sphere, (a) and (b) give identical patterns, (d) : Faces near an octahedral face.
In the very simple crystal forms shown, there are only two different
kinds of face — those which are all at 90° to one another and would form a
cube if the oblique faces were absent, and the oblique faces themselves
which if the cube faces were absent would form an octahedron. We shall
refer to these as 'cube faces' and 'octahedral faces'. Faces at other angles
* It is possible to project these points on to a plane surface. Such projections are easier to
handle than the 3dimensional spherical patterns.
8.1 The external forms of crystals 227
are possible and indeed frequent, and some which might occur near an
octahedral corner are illustrated in Fig. 8.1(d). But we shall not pursue
this topic because most of the facts about the growth of crystals can be
typified by referring only to the simplest types of face.
It is observed that many edges of a crystal may be parallel to one
another. Rotation of the crystal about an axis parallel to these edges
therefore brings successive faces into parallelism. Such faces constitute
a zone and the axis is called a zone axis.
8.1.1 Optical goniometry
The angles between the faces of any crystal can be measured using a
reflecting goniometer. This can be constructed from a spectrometer or
any other optical instrument which provides a parallel beam of white
light from a collimator, and a telescope focused for parallel light, with a
crosswire. Telescope and collimator are placed roughly at rightangles to
one another and then clamped and left fixed. At the centre of the system
the crystal is mounted, carefully aligned with a zone axis vertical. Each
face can act like a tiny mirror, and when a face is set so that it reflects
light into the telescope, an image of the slit of the collimator can be
seen. If the crystal is turned about a vertical axis so that another face
throws up an image in the same way, then the angle turned through by the
crystal is equal to the angle between the faces.
The construction of a typical goniometer head, with a crystal mounted
on it, is shown in Fig. 8.2. The crystal is stuck on to a metal point so as to
be near the optical centre of the system. The head must have sufficient
adjustments* to allow the crystal to be accurately aligned. Two horizontal
screws at rightangles are necessary to bring it to the centre of the instru
ment ; two concentric circular scales at rightangles are needed to align
Telescope
Fig. 8.2. A crystal holder for an optical goniometer.
* Instrument makers call them 'degrees of freedom'.
228 Thermal properties of solids Chap. 8
certain edges (and a selected zone axis) of the crystal vertically. The precise
adjustments require considerable patience but eventually it is possible
to take measurements of the interfacial angles of a whole zone of perhaps
a dozen faces. Then the crystal has to be remounted to take measurements
on another suitably chosen zone, so that eventually all the angles can be
measured and related.
It is advisable to use a white light source, because internal refractions
and reflections can give spurious images. With white light these are
coloured because of dispersion, and are easily distinguishable from direct
reflections from the faces. With a monochromatic source, this would not
be so.
With care, measurements accurate to a few minutes of arc can be taken,
and the extraordinary symmetry of crystalline form can be revealed,
using crystals little bigger than a pinhead. Optical goniometry is one of the
most rewarding of practical exercises.
8.1.2 Molecular arrangements; unit cells
We will now show how the faces of a crystal can be related to the arrange
ment of the molecules inside it. We will in fact study solid argon : other
molecular crystals and several metals have similar structures. It will
emerge that the crystals are highly symmetrical ; cubes and other similar
shapes are common. It must be understood that in nature such very high
symmetry is not at all frequent. Of all the tens of thousands of crystals
which have been catalogued, only about 5 % have this high symmetry.
Nevertheless, many principles are illustrated by considering these sub
stances, and low symmetries will only be briefly mentioned.
Solid argon crystallizes into a dense lattice (called 'cubic close packed'
or 'face centred cubic') where each atom in the interior touches 12 nearest
neighbours. Figure 8.3 shows the arrangement; this picture shows a
brickshaped external form with one corner removed.
The most obvious feature of this or any other lattice is its repetitiveness
or periodicity. To be precise, there is a unit of repetition of the three
dimensional pattern called the unit cell. For ease of illustration however,
we will for the moment concentrate on twodimensional patterns and
their unit cells.
Figure 8.4(a) shows a twodimensional lattice of atoms. This pattern
is not related to Fig. 8.3; it has been chosen only as an example of a two
dimensional lattice. A regular network, all of whose cells are identical,
has been drawn on the lattice. These are unit cells for the lattice. Each
contains just one atom. The shape of the cells is such that they can be
packed together to cover the area entirely. Further, one unit cell can be
translated parallel to one of its edges through a distance equal to the
8. 1 The external forms of crystals
229
length of that edge and the new position of the atom noted. If the process
is repeated parallel to both edge directions, the complete lattice can be
built up. (Note that multiples of this unit cell could be used in the same way
but one usually chooses the simplest unit.)
Fig. 8.3. Atoms in solid argon (cubic closepacked structure). Some have been
removed from a corner so as to form an octahedral face.
There is in fact no unique way of choosing the unit cell for any lattice.
Two other possible ones are shown in Figs. 8.4(6) and (c). They each still
contain only one atom and indeed each has the same area as that in Fig.
8.4(a). All o.ther possible simple unit cells for this lattice have the same
characteristics.
These ideas can readily be extended to threedimensional lattices. The
unit cell may contain several atoms or molecules : for example, the unit
oooo
oooo
oooo
oooo
oeeQ
oeee
(a)
(6)
(c)
Fig. 8.4 (a), (b) and (c). A twodimensional lattice, not related to the argon
lattice, showing possible twodimensional unit cells.
230
Thermal properties of solids Chap. 8
cell of the argon structure shown in Fig. 8.3 contains 4 atoms. Of the
infinity of possible unit cells, one is usually preferred because its symmetry
is the same as that of the crystal itself. For the argon crystal, this is a
cubic cell with atoms at each vertex and in the centre of each face.
Another obvious feature of the lattice shown in Fig. 8.3, as for any
3dimensional lattice, is that the atoms are ordered in planes. Further,
in this cubic lattice each horizontal plane (for example, the topmost
one) is exactly equivalent to the vertical planes which outline the shape
and these in fact are parallel to the cube faces of Fig. 8.1(a). Within these
planes, the arrangement of atoms is comparatively open, as seen on the
lefthand face of Fig. 8.3. One of these cube faces has been dissected out in
Fig. 8.5(a). Each atom has 4 neighbours within the plane and it touches
another 4 in each of the adjacent parallel planes. If we call the diameter
of one atom a , then there is one atom inside a square of area a% in these
planes.
In Fig. 8.3, some corner atoms have been removed to leave an oblique
octahedral face, exactly corresponding to the octahedral faces of Fig. 8.1.
Such a surface would appear flat on the macroscopic scale and this
demonstrates how plane faces can develop on crystals and why the angular
relations are maintained.
(a) (d)
Fig. 8.5 (a) A cube plane and (b) an octahedral plane of atoms.
Within this octahedral plane (dissected out in Fig. 8.5(b)), the atoms are
close packed. Each atom has 6 neighbours within the plane and touches
another 3 in each of the adjacent planes. There is one atom in a rhombus
whose angles are 60° and 120° and whose side is a — whose area, that
is, is {J?>l2)al = 0.866ag.
8.1 The external forms of crystals 231
8.1.3 Surface energies of crystal faces
Let us now calculate the surface energy of a crystal face, in much the
same way as we calculated the surface tension of a liquid in section 3.4.
Consider first a cube face. Each atom has only 8 neighbours (4 in its
plane and 4 in the next plane) instead of 12. To make a new surface
we have to cut 4 bonds per atom, each demanding energy e/2 (energy e
is needed to cut one bond but two surfaces are formed). There are \ja\
atoms in 1 cm 2 so that the surface energy of this face is le/al erg/cm 2 .
Similarly, the surface energy of an octahedral face is y/3s/al = \.11&ja\
erg/cm 2 because each atom has to have three bonds cut and there are
2/y/3a.o atoms/cm 2 . The surface energy of the cube faces is therefore
higher than that of the octahedral faces.
Imagine now a crystal growing with plane faces. Ideally, it ought to
grow so as to minimize its surface energy. This means that the octahedral
faces should be prominent, the cube faces small. In different words, the
cube faces will tend to grow fastest outwards so that they eliminate
themselves. This paradoxical statement is illustrated rather sketchily in
Fig. 8.6. The shapes are meant to indicate the form of the crystal at suc
cessive times. The cube faces grow fastest outwards but get progressively
smaller in area, leaving the octahedral faces predominant, as required.
Fig. 8.6. Stages in the growth of a
crystal.
In the growth of large crystals, surface energies seem in practice to
play little part, though they are important for small ones. In any case,
small temperature inhomogeneities which may be caused by the release
of energy during the act of crystallization itself, have a profound effect
232 Thermal properties of solids Chap. 8
on growth. We will consider some details of the mechanism of growth
in section 9.4.1.
Another consequence follows from the different closeness of packing
within different planes. Densely packed planes must be further apart
(because the total number of atoms in a crystal must be the same, which
ever way we count them). Now many crystals can be cleaved by being
struck sharply in certain directions using a heavy blade. If this is well
done, the faces of the two halves are bright and plane. Planes of easy
cleavage should — according to these simplified considerations — be those
of large spacing, which automatically have small surface energies as we
have shown, and are the ones which occur in natural crystals. Frequently
this is so.
8.1.4 Singlecrystal specimens
Unless they are specially prepared, solid specimens do not usually
consist of single crystals, with their atomic planes parallel to one another
in all directions. Rather they are polycrystalline with different orientations
in different regions so that they can be considered to consist of numbers
of small crystals of arbitrary shape all packed together, separated by grain
boundaries. Certain properties of solids, notably thermal conductivities
at low temperature, vary markedly if grain boundaries are present, and
to investigate them it is necessary to convert polycrystalline specimens into
single crystals.
A single crystal may not have visible plane faces. It might be, say,
in the form of a rod of circular crosssection. However, very often the sur
face has a shimmering appearance because it consists of little steps parallel
to certain crystallographic directions which catch the light, and is not a
curved surface at all. But whatever the external shape, the planes of atoms
must all be parallel to one another throughout the specimen.
In order to grow single crystals, one always begins with a 'seed' crystal,
as perfect as possible. Using this as nucleus, single crystals can be grown
provided the crystallization proceeds very slowly. In extreme cases,
crystals a few millimetres in dimension may take months to form.
Temperatures must be controlled within fine limits over these times and
all surfaces must be clean so that no other centres of nucleation are present.
If the solid is soluble in water, the seed can be put into a saturated
solution from which the water is allowed to evaporate slowly. Molecules
crystallize most easily on seeds and the procedure is mainly a matter of
common sense— removing other nuclei, keeping the temperature constant
and so on.
Growing crystals by cooling the molten liquid is also possible, and is
nowadays much more generally used. One method, capable of great
8.2 Xray structure analysis 233
variation of detail, is to fill a tube with liquid at a temperature just above
the melting point. The tube has a pointed end, and cooling the point
causes the liquid to solidify there. In the restricted space, there is probably
only room for one nucleus to grow (or else the seed can be put there),
so that there is a tendency for a single crystal to form. Subsequently, a
carefully controlled temperature gradient is maintained so that the crystal
grows along progressively. The alkali halides, like NaCl and LiF, can
be grown in this way. They melt at temperatures of the order of 1,000°K.
Their vapour pressures are not negligible and the biggest technical
problem is to stop them distilling away and depositing on cooler parts
of the apparatus. In another temperature range, solid argon rods have
been grown by just the same method ; this is described in some detail
in section 8.5.1.
Another method is 'crystal pulling', where the seed is dipped into the
molten liquid and is gradually raised into a cooler part of the apparatus,
solidifying and pulling a single crystal rod after it. Finally, in zone melting,
the starting point is a polycrystalline rod. By some method of localized
heating (such as a narrow beam of radiation, or induction heating), a
narrow transverse section is melted. If conditions are right, the surface
tension holds the liquid in place so that it does not run away. It is arranged
that this melted zone can pass along the crystal, from one end to the other.
If the seed is at the starting end, the whole rod eventually becomes
crystalline.
8.2 XRAY STRUCTURE ANALYSIS
The atomic arrangements which have been described have all been
elucidated by Xray diffraction experiments. When a beam of radiation
falls on a periodic structure, a grating or a crystal, it is diffracted through
large angles if the wavelength A is comparable with the repetition length d
of the structure. Thus with crystals we are limited to waves whose A is
a few Angstrom units. Any type of waves can be used for diffraction
experiments ; electrons or other particles of the same wavelength would
act in the same way. But there are experimental complications when
charged particles are used, and also with neutrons which have a magnetic
moment. In any case, the principles governing the diffraction are exactly
the same for all types of waves so we will concentrate on Xrays.
When a parallel beam of Xrays of given wavelength falls on a small
crystal, most of it travels straight through, but some of the energy is
diffracted into a number of beams (Fig. 8.7(a)). These can be recorded
on film, or detected by counters and their intensities and angular distribu
tion measured. The scattering is done by the electrons in the atoms of the
234
Thermal properties of solids Chap. 8
lattice (the nuclei are so heavy that they are not affected by the Xrays and
do not scatter them). The Xrays therefore 'see' spheres whose size is
comparable with their own wavelength, and the scattering power of an
atom therefore depends on its size as well as the number of electrons in it.
All the diffracted beams from a given crystal are part of the one family.
The angles at which the beams occur depend on the periodicity of the
lattice, that is, on the size of the unit cell, while the intensities of the beams
depend on the arrangement within the unit cell. The full family of diffracted
beams, expressed as an appropriate function of angles, is called the Fourier
transform of the electron density in the crystal.
(b)
Fig. 8.7. (a) A beam of Xrays (covering a wide range of wavelengths) incident
on a crystal emerges as a family of diffracted beams, (b) Bragg reflection from
a plane.
Although it is wrong in principle to take any one diffracted ray and to
treat it in isolation from the rest of the family, nevertheless this was done
in the early days of Xray crystallography and has persisted because it is
useful in interpreting diffraction patterns. As the method gives quantitative
results about angles of diffraction using only simple mathematics, we will
give the main result.
8.2 Xray structure analysis 235
We isolate a set of parallel planes of atoms. In general, these are inclined
at some angle to the cube faces but, within the planes, the centres of the
atoms lie on some regular pattern, like those of Fig. 8.5. There are many
directions in a plane which pass through the centres of atoms, and they
do so at regular intervals. Two such lines of atoms (drawn for simplicity
as dots) are shown in Fig. 8.7(b), an upper and a lower plane, separated by
a distance d. The spacing along the lines is not equal to the distance
between planes, and the atoms in the lower plane are not necessarily
exactly below the upper ones but are displaced to one side. The third row
is similarly disposed with respect to the second, and so on.
Imagine now a beam of Xrays of wavelength A incident on an infinite
plane of atoms. It can be shown that it diffracts radiation with a strong
maximum of intensity in the direction of specular reflection, so that the
glancing angle of reflection is equal to the glancing angle of incidence,
just as if it were a continuous reflecting plane. We will only consider the
intensity in this direction, and we can speak of the Xrays being reflected
from the planes of atoms.
Let us consider interference between radiation scattered by the top
plane of atoms and the next plane. The atoms P and Q of Fig. 8.7(b) are
typical and we know that if Xrays scattered from P reinforce those
scattered from Q (because the phase relation between them is correct)
then all other similar pairs of atoms throughout all the planes will also
reinforce and the reflected wave will be of finite intensity.
The condition for constructive interference is
path difference XQ + Q Y = n A
where n is an integer, 1, 2, etc. We must therefore calculate the distances
XQ and QY. Call the angle PQX equal to a. Then angle PQY is equal to
(18O°a20). We have (from APXQ)
XQ = PQ cos a
and (from APYQ)
YQ = PQcos(18O°a20)
= PQcos(20 + a).
Therefore
XQ + YQ = PQ{cosacos(20+a)}
which by a trigonometrical identity is equal to 2PQ sin sin(0 + a). If we
drop a perpendicular from P to the lower plane, we also have
d = PQsin(0 + oc).
236 Thermal properties of solids Chap. 8
Hence the condition for the presence of a diffracted beam is
2dsind = nA. (8.1)
Notice that d is the glancing angle with respect to the plane, but neither the
spacing within the plane nor the lateral displacement of one set of atoms
with respect to the other enters the expression ; only the distance d between
parallel planes enters.
Whereas a single plane of atoms reflects Xrays of any wavelength
specularly, a stack of atomic planes only reflects a finite intensity at this
angle if, in addition, the condition (8.1) is satisfied. Other wavelengths
are cancelled out by interference. Eq. (8.1) is called the Bragg law.
We can now take a single spot of the Xray diffraction pattern and regard
it as having been produced by a Bragg reflection from a stack of parallel
planes of atoms, and this enables us to find the spacing d in terms of the
wavelength A *. For example, with Xrays of wavelength 1.541 A (pro
duced from a copper target) one particular reflection from an argon crystal
was observed to be produced at a glancing angle of 16.52°. Thus
2xdx sin(16.52°) = n x 1.541 A,
whence d = 2.71 A if n = 1, 5.42 A if n = 2, and so on. By itself, a single
reflection does not allow n to be decided, nor is it possible to say which
plane did the reflecting. These can only be decided by correlating all the
reflections from the whole pattern — intensities as well as angles — and
working out just what the unit cell is. (With substances which produce
wellshaped crystals, the process is simplified if the reflections can be
correlated with the external form.) A check on the value of n comes from
comparing the atomic diameter deduced from other methods with that
deduced from the unit cell ; an error of a factor of 2 is easily detected.
In this way it can be decided that n = 1 for this reflection. Thus the side
of the unit cell, from Fig. 8.5(a) twice the distance between neighbouring
planes, is 5.42 A, and the diameter of an argon atom is 5.42/^2 = 3.83 A.
Having used other estimates of the diameter to decide which multiple of
this figure to use, the Xray measurement is by far the most accurate.
8.3. AMPLITUDE OF ATOMIC VIBRATIONS IN SOLIDS
Consider a system in a parabolic potential well, whose equation (referred
to axes through an origin at the bottom of the well) is 'V = jax 2 . If the
system is given energy E it oscillates with an amplitude x given by
* Xray wavelengths were originally measured in absolute units using ruled diffraction
gratings. Though comparatively coarse in spacing, they were held at very small glancing
angles of incidence, so that the diffraction occurred.
8.3 Amplitudes of atomic vibrations in solids
237
E — j<xxq. A geometrical construction for finding x is shown in Fig. 8.8.
A line drawn at height E above the minimum intercepts the parabola at
± x , because at x = x all the energy is potential energy.
Now consider an atom inside a solid. We will assume that it has n
nearest neighbours, bound to it by a potential of the 612 type. We have
already seen (when we estimated the Einstein frequency of vibration
in section 3.6.1, Eq. (3.20)) that the atom 'sees' a potential well, due to all
its neighbours, which is nearly parabolic and has a curvature near the
minimum given approximately by 24ne/al, where a is the separation
between atoms.
Fig. 8.8. Amplitude of vibration with
energy £ in a parabolic potential well
If we now extrapolate this assuming that the well is accurately parabolic,
its equation referred to axes through the minimum is
TT(x) =
12ne
2
at
(8.2)
where x is the displacement from the minimum. (This result follows from
the fact that the parabola jax 2 has curvature a at the minimum.)
With these preliminaries, we can now estimate the greatest amplitude
of vibration which the atom in a solid can have under ordinary conditions
— namely, at the melting point. When the solid is at absolute zero, the
atom is (according to the ideas of classical physics) at the bottom of the
well. When it is at temperature T and the atom has a mean energy kT,
it oscillates within the well — though not, of course, with a strict periodicity
because it interchanges energy all the time with its surroundings. Now
we know that the melting temperature (at ordinary pressures) is not very
238 Thermal properties of solids Chap. 8
different from the temperature of the triple point — and this in turn is
always roughly half the critical temperature (80° K and 150°K for argon).
We already know that the mean thermal energy of an atom at its critical
temperature is about e, so that at the melting point it is about ^s. Thus
the amplitude is given by
12„e^j *. ,8.3)
that is, (x /a ) « 0.07, if n = 10. Thus we can say that in any substance
with a 612 potential, the amplitude at the melting point is less than 1/10
of the interatomic separation, which is quite small.
For metals, the ratio of triple point temperature to e/k is nearer 1 or 3
than 2 For potassium, for example, the melting point is 335°K, and from
the data of section 3.7, e/k is about 0.16 eV so that the ratio is about 0.2.
For mercury using data from the same table and a melting point of 234° K,
the ratio is about 0.15. But the amplitude at melting is not greatly different
from before.
For ionic solids, the calculation can be followed through by noting that
an ion 'sees' a potential whose curvature is about 4ne/al if the neighbours
are spherically disposed about it, as we assumed in the discussion of the
Einstein frequency, sections 3.6. 1 and 3.8.2. For sodium chloride, e ~ 9 e V
from the data of section 3.8.1, the coordination number is 6, the melting
point is 1,073°K. Hence the amplitude at melting is onethirtieth of the
interatomic spacing. Looked at from this point of view, NaCl melts more
easily than molecular solids or metals. This is caused by the presence of
two sizes of ion in the lattice — small positive ions — and negative ions
of about twice their radius. The small ions can slip easily between the
large ones, more easily than in lattices where all the members are the
same size. Thus though ionic solids like sodium chloride melt at high
absolute temperatures, these are really quite low on this proportional
scale.
We return to the onset of melting and to some premelting phenomena,
considered from a rather different point of view, in section 9.5.2.
8.4 THERMAL EXPANSION AND ANHARMONICITY
It is well known that solids expand when heated. The coefficient of linear
thermal expansion is defined by
1 / dl \
8.4 Thermal expansion and anharmonicity
239
where the subscript P denotes an expansion at constant pressure, usually
atmospheric pressure. The coefficient of volume expansion /? is defined
by a similar equation with volume V in place of linear dimension /
and since V = Z 3 , it follows that /? = 3a. Typically a is of order 10 ~ 5 per
degree for hard solids, 10 3 per degree for soft ones, at ordinary tem
peratures. We will now relate these quantities to the V{r) curve.
It must be realised at once that if the forces binding one atom to another
were purely harmonic — if the potential well between a pair of atoms were
exactly parabolic, even for large amplitudes of vibration — then the mean
separation of two atoms would always be the same, whatever the ampli
tude, Fig. 8.9(a). (In the same way, the mean position of a simple pendulum
remains fixed, whatever its amplitude of swing.) Thus, a solid bound by
purely harmonic interatomic forces would not expand with temperature.
The origin of the finite expansion coefficient must lie in the asymmetry
of the i^(r) curve, which expresses the fact that two atoms can more easily
be pulled apart from one another than pushed together. We can see this
graphically, Fig. 8.9(b), by drawing horizontal lines across the well,
representing different mean energies and hence different temperatures.
With increasing energy, the mean separation tends towards greater
separation, and the solid must expand.
Systems in nonparabolic potential wells are said to execute anharmonic
motion.
(a)
ib)
Fig. 8.9. (a) The mean displacement of a simple harmonic motion is always
zero, whatever the amplitude, (b) With an unsymmetrical potential energy
curve the mean displacement increases with amplitude.
8.4.1 Thermal expansion coefficient
To calculate the magnitude of a, the thermal expansion coefficient, we
will first find a convenient approximate equation for the interatomic
240 Thermal properties of solids Chap. 8
potential energy curve near the minimum, referred to axes through the
minimum. If a is the value of r at the minimum and therefore the separa
tion of a pair of atoms at T = 0, we will denote {ra ) by x, the displace
ment from the minimum. Then we will find the limits of x between which
a pair of atoms vibrate when their energy is given, and since the positive
swing is greater than the negative one, the mean value x is greater than
zero. x/a is the total linear expansion between any pair of atoms at temper
ature T, and differentiating with respect to T gives the linear expansion
coefficient. This procedure is not exact; it does not properly take into
account the effect of the atoms at the side of any pair, nor is the midpoint
of the swing an exact measure of the mean position ; but it is not far wrong.
First we will seek an equation for the interatomic potential energy near
the minimum — something better than the parabolic approximation which
we used previously (page 45). The anharmonic terms, the extra ones not
proportional to x 2 , will be the ones responsible for the thermal expansion.
We start by quoting Taylor's theorem. If we have a function V whose
value at r = a is i^(a ), then the value at some other point is
ldV\ 1 ld 2 i^\
i . .Jd 3 r
+ ^<">> 3 M +  (8  5)
where we use the subscript to denote that the term must be evaluated at
r = a . This expansion is true for any reasonable curve. Now let the point
a be a minimum so that (d^/dr) = 0. Let us write ArV for the increase
of potential energy compared with the minimum value, and x for (r — a ).
Then
.. x 2 /d 2 ^\ x 3 /d 3 ^\
W = ArA +77ho +■■■ (86)
2!\ dr 2 / 3!\ dr /0
The relation between the V, r coordinates and the i r , x coordinates is
made clear in Fig. 8.10.
This holds near the minimum of any curve. Let us concentrate on the
612 potential which is appropriate for molecular solids like argon and
with less accuracy (see section 3.7) for metals. Starting with Eq. (3.4) :
V = lie
dT _ _12e
dr a
a \ 13 \Oq
r
(3.4)
= when r = a
8.4 Thermal expansion and anharmonicity
241
d 2 r
dr 2
12e
[»(?)"'(?)']
72e v.
= — r when r = a
dr 3
L
a
2e
T
"l3xl4^)' 5 7
x8 —
\ H _
1,512c
al
(3.15)
when r = a .
i\W
iUV
Fig. 8.10. Change of coordinates to an origin through the minimum.
Thus, using the minimum as origin, as in Fig. 8.10, the interatomic potential
energy is given by
2
M^ = 36e
a
252e
W '
(8.7)
Since we have neglected higher terms and the coefficients are increasing,
this approximation is only good up to about (x/a ) = 0.1. But, as we have
seen, this is adequate even up to the melting point. The effect of the
anharmonic x 3 term is clearly seen in Fig. 8.11.
A{x/aJ
,/ ^A(x/o Y  B{x/af
(x/a)
Fig. 8.1 1. A curve of the type A(x/a ) 2  B(x/a ) 3 ,
for small (x/a Q ).
242 Thermal properties of solids Chap. 8
For simplicity, we will write this equation
Go \«0
Let us now choose a value of the total (kinetic plus potential) energy E t
We find the limits of swing in the usual way, solving
x' 2
and we can do this by successive approximation. First, we neglect the B
term entirely. This gives
Next, we can use these rough values to calculate the small extra term and
hence solve the equation more exactly. We write
x \ 2 I x \ 2 l x \ I x \ 2
E x =A\—\ B\— I , , ,
\a J \ a ol \«o/ \a
so that approximately
£1
^ x
AB
A + BiEJA) 1 ' 2
where the minus sign is associated with the positive square root and vice
versa. Taking the square root :
These are the second approximations to the solutions, good enough for
present purposes. The mean of the two values is found by adding and
dividing by 2 :
* i/£i\ 1/2 /B£ 1/2 \ = be 1
a 2\AJ U 3/2 / 2A 2 '
Putting in the values of A and B and simplifying :
 = ^£i. (88)
a 72e
8.4 Thermal expansion and anharmonicity 243
This gives the total mean expansion, when the energy of the oscillating
atom along the xaxis is E t . The coefficient of linear expansion is the
differential coefficient of this with respect to temperature
Now we have seen that when the total energy of N atoms (oscillating in 3
dimensions) is E, the molar specific heat C p is given by
C = ^
p dT
for any experiment which takes place at constant pressure (including
the expansion of a solid). Here, E x is the energy of a single atom oscillating
in one dimension only, so E Y = E/3N. Therefore
"^ (8 ' 10)
At high temperatures, in the equipartition region, C p is not very different
from 3Nk energy units/deg so a ~ ^ • k/e. For argon, e/k ~ 120 degrees,
whence a ~ 1CT 3 per degree which is the right order for a solid. For
metals, for which as we have seen the 612 potential can be used for rough
estimates, e/k is of order 3,000 degrees, and a ~ 10" 5 per degree, again of
the right order. Thus the identification of the mechanism causing thermal
expansion is correct.
*k 8.4.2 The Griineisen relation
There is no need to restrict the discussion to high temperatures. The
relation (8.10) shows that the expansion coefficient is proportional to the
specific heat — which implies that a falls off at low temperatures and this
is in fact observed. The expression indeed predicts that a/C p is a constant
for any substance at all temperatures. As it stands, however, the ratio
incorporates e, the interaction energy of a pair of atoms, whereas a and
C p are macroscopic properties. It is more convenient to use one of the
relations deduced in Chapter 3 to eliminate e in favour of a macroscopic
quantity, and the one usually chosen is the bulk modulus (because it
leads eventually to a dimensionless ratio, a pure number) ;
4JVne
K = — (3.16)
y o
where n is the coordination number and V the molar volume. It must be
noted that, strictly speaking, this K refers to adiabatic conditions; we
244 Thermal properties of solids Chap. 8
will write it K ad , although previously we did not emphasize the difference
between it and the isothermal bulk modulus K T because we limited the
discussion to low temperatures where differences vanish.
Finally, having eliminated e and having incorporated both a molar
volume and an elastic modulus which refers to volume changes, it is
reasonable to refer to the volume coefficient of thermal expansion /?
which is equal to 3a. In these terms our expression is
This ratio should be a constant, independent of temperature, called
Griineisen's constant y G for the solid.
We may now invoke a thermodynamic result, namely that the ratio
Kad/Cp, that is the adiabatic bulk modulus over the specific heat at
constant pressure, is identical with K T /C V , the isothermal bulk modulus
over the specific heat at constant volume. Thus we can write
/ G s~i fi \ ■ )
This is a surprising relation; one would hardly expect the thermal
expansion coefficient to be related to the specific heat. In 1908, when
Griineisen first announced his empirical law (in the form <x/C„ = constant
for any metal) he was unable to account for it. In its complete form,
Griineisen's equation relates changes of pressure, volume and temperature
and is often referred to as an equation of state for solids.
The absolute value of y G which we have calculated is roughly correct.
We have already given sufficient data for computing it for solid argon —
K T as its reciprocal the isothermal compressibility in Fig. 3.13(c), C v and
the expansion coefficient in Fig. 5.10(b). Experimentally, therefore, y G
has the value 2.8 all the way from 20° to about 60° K, falling to about 2.4
at the melting point, 80° K. The simple theory presented here predicts
about 4.5 for a close packed crystal structure with n = 12.
For metals, the Griineisen y G is usually about 1.4. Ionic crystals usually
have y G between about 1.5 and 2; the method of calculation for simple
ionic crystals is outlined in a problem at the end of the chapter.
8.5 THERMAL CONDUCTION IN SOLIDS
The problem of describing the mechanism of the conduction of heat in
solids is one of extreme difficulty. Here we can only try to give a brief
sketch of some of the physical phenomena.
= D— 2 (6.3)
(6.6)
8.5 Thermal expansion in solids 245
We have seen, in sections 6.1.2, 6.1.3 of the chapter on transport pro
cesses, that the equations describing the diffusion of molecules and those
describing the conduction of heat are formally very similar :
dn d 2 n
ct ox
and
dT _ I k \d 2 T
dt \C v pJ dx
Here, n is the concentration of a substance, measured in mol/cm 3 , diffusing
with time t in a direction x, D is the diffusion coefficient ; T is the tempera
ture, (k/C v p) is called the thermal diffusivity. These equations were derived
by general arguments and are valid for all states of matter. The symbols
represent macroscopic properties of the substance and not the properties
of the individual molecules.
The formal analogy between the two equations justifies our using the
expression that heat diffuses into a body. But we are justified in going
further than that. We can look at the process on the atomic scale and
bear in mind our success in describing diffusion in gases in terms of the
random walk executed by each molecule, in terms of the mean free path.
We can therefore try to bring together the concepts of the thermal motion
of the molecules of a solid and the ideas of the random walk and the mean
free path.
Imagine one small region of a solid to be heated. The additional thermal
energy must diffuse to distant regions. Molecules have large amplitudes
of vibration there and oscillate violently about their mean positions. Now
because of the interactions between molecules in the lattice, this must
set their neighbours into oscillation and the result is that a wave dis
turbance is propagated outwards. It travels with the speed of sound.
(The only difference between this kind of disturbance and an audible
sound wave travelling through the solid is that the typical frequency, the
Einstein frequency, is 10 10 times greater for the thermal vibrations.)
Now a wave disturbance of this kind can have its direction of energy
flow altered — that is, it can be scattered, just as a light beam can be
scattered — by a number of processes. For example, it can meet the bound
ary of the solid when it will be internally reflected or it can meet an im
perfection of some kind such as a region of high or low density or an
impurity atom of different mass from the bulk. In any case, the energy in
the disturbance cannot travel unimpeded through the lattice. Every so
often it will have its direction of propagation altered. The average distance
that a disturbance travels in a solid before it is scattered is analogous to
246 Thermal properties of solids Chap. 8
the average distance travelled by a molecule in a gas, namely the mean
free path. We shall denote this average distance in a solid by /. It is a
property of the wave motion and has nothing to do with the distance
moved by an individual molecule.
Let us take our analogy between diffusion and thermal conduction
one stage further. Let us say that the diffusion of heat is caused by the
scattering in random directions, the random walk, of the disturbance
carrying the additional thermal energy. Then quoting a result deduced
in section 6.5.1, for the diffusion coefficient of a gas of molecules:
D = \cX (6.19b)
where c is the mean speed of the molecules (practically the speed of sound)
and X is the mean free path. Therefore we expect that for the thermal
diffusivity of a solid
~^ = jc s l (8.13)
CvP
where c s is the speed of sound, X the mean distance travelled by a wave
disturbance before being deviated, and the factor \ is perhaps a little
arbitrary.
As we have presented it here, we have perforce left the description of
the flow of energy rather vague. When it is described precisely, in quantum
terms so that the energy can be considered to have particlelike aspects,
the analogy becomes exact and the equation for thermal diffusivity can be
rigorously justified.
This relation predicts that the thermal conductivity of a solid should
be given by
K = iC v pc s l. (8.14)
In this expression, all the quantities are known — C v the molar specific
heat, p the density in mol/cm 3 , c s the speed of sound — except a the 'mean
free path'.
We can therefore compare this equation with experimental measure
ments and use it to deduce the 'mean free path' ; then we can see what the
results mean in atomic terms. It will emerge that in a given specimen of
material at any given temperature there are several mechanisms for
scattering the flow of energy and all of these operate at once although with
varying effectiveness at different temperatures. Under any given set of
conditions, the mechanism which causes the strongest scattering is the one
which limits the mean free path and determines the thermal conductivity.
8.5 Thermal conduction in solids 247
8.5.1 Measurements on solid argon
The techniques of making specimens of solid argon and of measuring
their thermal conductivity are sufficiently unusual to be worth describing.
The difficulty is that solid argon exists (under ordinary pressures) only
below 83°K, the triple point, which is just above the normal boiling point
of liquid nitrogen, 77° K. A rod of solid argon has to be grown, consisting
of large crystals if possible (because these turn out to be the most interesting
specimens to study) ; then heat has to be put into one end of the rod and the
temperature gradient measured. All these manipulations have to be done
inside Dewar vessels at liquid nitrogen temperatures and below, in an
argon atmosphere out of contact with air. Then conductivity measure
ments must be made down to a few degrees absolute, using liquid helium
as the refrigerant. The techniques used by Berne, Boato and de Paz will
be described.
The rod was grown inside a pointed glass tube G (Fig. 8.12) as described
in section 8.1.4. This tube was surrounded by a copper sheath S, so
arranged that its absolute temperature could be accurately controlled
while in addition a small temperature gradient could be superimposed.
Small electrical heating elements Hj and H 2 were wound for this purpose
on the copper sheath, one at either end, while its lower end dipped into a
bath of liquid nitrogen. The level of this bath was kept constant within ± 1
mm, by an automatic toppingup arrangement. The power to H t was
adjusted to keep the temperature of the lower end of the sheath just above
the triple point, while H 2 caused the upper end of the sheath to be a few
degrees hotter. Thus when pure argon gas was let into the apparatus, it
liquefied at the bottom of the glass tube. Next, the power in H t — elec
tronically controlled — was gradually reduced so that the temperature fell
steadily, at about 0.1° per hour. As a result, the temperature at a certain
level in the glass tube fell to the triple point and this level travelled gradually
upwards — inside the tube the argon solidified up to this level, covered with
a thin layer of liquid. The rods, examined after the experiment, were found
to consist of comparatively large crystals between 1 mm and 4 mm in size.
Into the top of the tube there hung a long thin plastic rod R with a bead
on the end, and the argon crystals grew round this and enclosed it. When
the argon rod was long enough, 6 cm long, no more gas was introduced ;
instead, the surrounding argon atmosphere was slowly pumped away.
The argon rod began to evaporate a little from its surface and by gently
pulling on the plastic rod (from outside the apparatus) the argon rod
could therefore be detached from its glass mould and pulled into the
upper part of the apparatus. Next, the liquid nitrogen was pumped
away and liquid helium (4°K) syphoned in. At this very low temperature
the rod became considerably harder, capable of withstanding the hazards
of the next operation.
248
Thermal properties of solids Chap. 8
Fig. 8.12. Apparatus for measuring the
thermal conductivity of a crystalline solid
argon rod A. The rod is shown raised into the
upper part of the apparatus, clamped to the
copper block B and with its heater H 3 and gas
thermometers T t and T 2 all attached. G is the
pointed glass tube in which the rod was
grown, S the copper sheath, extended at the
bottom and dipping into liquid refrigerant.
H x and H 2 are the heaters for controlling
the temperature while growing the rod. R is
the plastic rod for raising the argon rod out
of its mould. The whole apparatus is enclosed
in a Dewar vessel. Details have been con
siderably simplified and no connecting tubes
or electrical leads are shown.
8.5 Thermal conduction in solids 249
Under vacuum the top end of the rod was clamped to a copper block B
and a heater H 3 attached to the other end, with two thermometers T* and
T 2 at points in between. All these attachments were ready in position,
with springloaded copper clamps held open by nylon strings. The argon
rod was slid into position and the strings cut with a blade controlled from
outside, so that the clamps closed. If the springs were too strong, the argon
rod broke ; if they were too loose the thermal contact was poor and the
temperatures were measured incorrectly. Of 50 experiments which were
started, 12 were carried out to completion, and of these 4 gave results
which were selfconsistent and judged to be significant.
T x and T 2 were heliumfilled gas thermometers connected to a sensitive
differential pressure gauge, so that the temperature difference (a few
tenths of a degree) could be measured directly. The temperature of the
copper block at the cold end of the specimen could be varied between
3°K and 15°K, being surrounded either by liquid helium or 'warm'
helium gas. The power fed into the specimen was of the order of milli
watts or tens of milliwatts, and the thermometers took several minutes to
come to equilibrium.
8.5.2 Impurity scattering of energy flow
The results of measurements of the thermal conductivity of argon on
four specimens are given in Fig. 8.13(a). The conductivity is plotted
vertically and the temperature horizontally, both on logarithmic scales.
At the left of the curves, the lowest measurements were taken near 3°K;
the triplepoint, 83°K, is towards the right. At the bottom of the curves,
the lowest conductivity measured, 0.004 watt/cm.deg., is comparable with
that of stone or glass at room temperature which are normally regarded
as heat insulators. At the top, the best conductivity, 0.6 watt/cm.deg., is
better than that of aluminium at room temperature, which is certainly
regarded as a good conductor of heat. The 30:1 range of temperature
pictured in this graph encompasses a 150 : 1 range of thermal conductivity
and only logarithmic plots can display these wide variations.
Above about 10°K, the curves are pretty well coincident for all speci
mens. Therefore, in this 'hightemperature' region, the dominant mechan
ism for scattering the energy flow must be one which does not depend
on some variable quality of the specimens (such as their shape or size
or purity) but must depend on the bulk properties of solid argon itself.
We will return to this region later.
Below 10° K however, different specimens have different conductivities,
as evidenced by the displacement of the curves parallel to one another.
The experimenters in fact noted that the specimens looked different from
one another. For example, the bestconducting specimen was made from
250
Thermal properties of solids Chap. 8
spectroscopically pure argon and was fairly transparent. The next one
was grown from less pure gas, although it still looked clear; the worst
specimens (actually prepared by a slightly different method from that
described) were 'quite cloudy and opaque'. Since the transparency of a
crystal is an indication of its perfection (see the discussion on the scattering
of light, section 7.7.1), we deduce that the mean free path for diffusion
of the thermal energy is limited, below 10°K, by imperfections in the
crystals. Indeed, we expect wave motions going through regions of
imperfection to be deviated. The imperfections may be grain boundaries,
or small regions of different density caused by strains, or of different
composition caused by impurities.
E
o
>10
<J* 1
10'
10'
1CT
1d 2
10
100 °K 1
Temperature
10
100 °K
Temperature
(a)
(£>)
Fig. 8.13. (a) The thermal conductivity of four specimens of solid argon
as a function of temperature, on logarithmic scales. Data from Berne,
Boato and de Paz, Nuovo Cimento 46B, 182 (1966). (b) Specific heat C v
plotted on similar logarithmic scales and expressed as specific heat per
cm 3 . Data from Fig. 5.10(b).
An analogous situation has already been encountered, in section 6.6.1
when discussing thermal insulation with silica powder. In a real gas at
low pressures, the mean free path of the molecules would normally be
large and the thermal conductivity would have a certain value. But if
we fill the space with a fine powder, the mean free path is limited by the
distance between the grains ; the conductivity is reduced — and so, presum
ably, is the diffusion coefficient. In imperfect crystals, the imperfections
8.5 Thermal conduction in solids 251
(some of which also scatter light) similarly limit the mean free path of the
energy flow.
In any given specimen, these imperfections should not change in any
way with temperature so that X should be constant. The density p and
the speed of sound are also constant ; hence the thermal conductivity
should be proportional to the specific heat C v . In Fig. 8.13(6), C v has
been plotted on logarithmic scales— the data of Fig. 5.10 replotted and
converted into specific heat per cm 3 using a molar volume of 22.6 cm 3 .
Now if the conductivity k is proportional to C v and bpth are functions
of T, then log k and log C v plotted against T (or log T) should be parallel
curves. It can be seen that below 10° K this is indeed so. This then allows
us to measure X the mean free path. The speed of sound c s is about 10 5
cm/s. At 5°K, C v is equal to 0.0164 J/cm 3 .deg. For the best specimen, the
thermal conductivity is equal to 0.55 watt/cm.deg, and for worst 0.015
watt/cm.deg. From these data, X is equal to 10~ 3 cm and 3 x 10~ 5 cm
respectively. These are reasonable figures ; for the transparent specimens,
X is many times the wavelength of light, for the cloudy ones it is comparable
with the wavelength, as we would expect.
8.5.3 Long mean free paths : the Knudsen region
One's belief in the validity of the energyflow model is reinforced by the
fact that it is possible to prepare crystals (not of argon but of other sub
stances) which are practically free from impurities, so that at low tempera
tures where the energy density is small, the mean free paths become
extremely long, comparable with the dimensions of the crystal itself.
The energy wavetrains then behave Hke a Knudsen gas. This means that
when they diffuse down a rod they collide only with the surface of the
specimen ; there is nothing else to collide with.
Two stringent conditions must be fulfilled before this behaviour can be
observed — one concerning the surface of the specimens and the other
concerning their physical and chemical purity.
In discussing the molecular flow of gases down tubes, it was not possible
to predict whether the molecules stick to the surface and are reemitted at
a random angle ('diffuse reflection') or whether they are reflected specularly.
Here, we meet an analogous problem. If the surface of the crystal is smooth
on the atomic scale, the energy flow will be reflected specularly and the
flux of energy down the rod will be unaltered by the collision ; on the other
hand, if it is desired to make the energy flow undergo a random walk,
it must be reflected diffusely and the surface must be rough on the atomic
scale.
When a crystal is to be prepared having a long enough mean free
path to observe Knudsentype behaviour, an unusual degree of purity
252 Thermal properties of solids Chap. 8
is demanded. This is because the energy flow can be scattered by any
departure from perfect regularity in the lattice. Now a foreign atom, of
different mass from the rest, can act as a scattering centre. (In the same
way, a wave passing down a stretched string is partly reflected at a knot
or any small section of different density from the rest.) But any chemical
element as found naturally consists of a mixture of isotopes, atoms of
different masses. To prepare crystals which are pure enough for the
present purpose, the elements therefore have to be pure isotopes.
The most convincing measurements have been made on lithium
fluoride, LiF, of which large singlecrystal ingots can be grown, using
the pointedtube technique (section 8.1.4). Lithium as it occurs naturally
is predominantly 7 Li with about 7.5% of 6 Li; fluorine is pure 19 F. The
lighter 6 Li was therefore extracted before the LiF was prepared (and a
later set of experiments showed that the thermal conductivity was
thereby increased). Several specimens were cut from the same big crystal,
in the form of squaresection rods of different sizes. Finally, the surfaces
were sandblasted in an attempt to make them rough on the atomic
scale. This treatment did not so much roughen the surface as produce
a thin layer of damaged crystal structure just below the surface and a
detailed analysis of the measurements showed that the most of the
reflection was diffuse.
The results are shown in Fig. 8. 14(a), together with specific heat measure
ments in (b), both on loglog scales. Below 20°K, the curves are parallel
showing that k is proportional to C v , but the conductivity is larger the
bigger the crosssection of the specimens. Though the curves look as if
they are close together, they in fact show that k for the 7 mm specimen is
between 5 and 10 times that for the 1 mm specimen. Note the enormous
value of k at 20° K — about 200 times better than that of copper at room
temperature, an astonishing fact when one considers that LiF is an
insulator, since one is used to the fact that they are worse conductors of
heat than metals.
The speed of sound is 5 x 10 5 cm/s. Using all the data, it can be cal
culated that the mean free path in the 7 mm x 7 mm specimen is about
3 mm ; in the 1 mm x 1 mm rod it is about 0.6 mm. Thus the mean free path
is certainly of the same order as the dimensions, as predicted by the
Knudsentype theory.
8.5.4 High temperature behaviour
Above about 10°K for solid argon or 20° K for lithium fluoride, the
thermal conductivity decreases with rising temperature. At high tempera
tures in the equipartition region, the graphs on log/log scales tend to
become straight lines at 45° showing that kccI/T. Since C v is constant, this
8.5 Thermal conduction in solids 253
implies that the mean free path Xccl/T. As remarked previously, the fact
that impurities or the mode of preparation of the specimens have no
effect on this part of the curve— all graphs for different specimens of any
substance are coincident— shows that the mean free path is dictated by
some property of the lattice itself.
10
10
10
10
10
!io 3
10
10 100 °K
Temperature
(a)
^6 5
10 100
Temperature
Fig. 8.14. (a) The thermal conductivity of four extremely pure LiF rods
as a function of temperature on logarithmic scale. The rods were nearly
square in cross section of side approximately 7 mm (specimen a), 4 mm
(specimen b), 2 mm (specimen c) and 1mm (specimen d). Data from
Thacher, Phys. Rev. 156, 975 (1967). (b) Specific heat per unit volume
below 20° K, the low temperature tail of the curve, far below the equi
partition region. Data from Scales, Phys. Rev. 112, 49 (1958).
It is now accepted that this is the anharmonicity of the lattice vibrations.
When two wave disturbances pass through one another, they create new
frequencies which were not originally present in either. This process
has no analogy under ordinary conditions for light waves, for example ;
if it did, then a red beam passing through a green beam might generate
ultraviolet and this does not occur. The generation of new frequencies
from two wave motions is called 'frequency mixing' and electronic
engineers are familiar with related effects using nonlinear circuit elements ;
because of anharmonicity, a solid is similarly said to be a nonlinear
254 Thermal properties of solids Chap. 8
medium to the passage of sound waves. When a wave passes over another
and generates a third frequency, this can undergo a sort of Bragg reflection
with the nett effect that the direction of energy flow is altered. It is plausible
that the effect should become more important as the amplitude of the
atomic motions is increased — that is, as the temperature is raised. This is
why the thermal conductivity decreases as the temperature increases.
8.6 ELECTRONS IN METALS
Metals can be described as lattices of ions, permeated by a kind of
gas of electrons. We have already described (sections 2.1.2, 2.1.4) how
some electrons inside atoms are tightly bound, while others are only
loosely bound. It is the loosely bound ones which can wander through the
lattice. They can be accelerated by electric fields and their energy can be
increased by raising the temperature.
In many metals such as copper, silver or sodium, the number of free
electrons is about one per atom. In a molar volume of metal, therefore,
there are about 6 x 10 23 free electrons. We shall call this number '1 mole
of electrons'.
The most obvious property of metals is that they are good electrical
conductors. But not only do metals conduct electricity, they are also much
better conductors of heat than insulators. For example, at room tempera
ture, sodium chloride which is an electrical insulator has a thermal
conductivity of 0.06 watt/cm.deg, while metallic copper has k = 0.92
watt/cm.deg. Among liquids, alcohol at room temperature and sodium
at 200° C have thermal conductivities of 0.0017 and 0.82 watt/cm.deg
respectively ; again the metal has the much higher thermal conductivity.
Further, the thermal conductivity among metals increases roughly in
proportion to the electrical conductivity a. For example, copper and zinc
have electrical conductivities of 0.63 x 10~ 6 and 0.16 x 10~ 6 (ohm cm)' l
respectively, that is in the ratio 3.8:1. Their thermal conductivities are
respectively 0.92 and 0.265 watt/cm.deg, which are in the ratio 3.5:1.
This rough proportionality holds for all metals. This fact was discovered
experimentally by Wiedemann and Franz in 1853, who stated that
k/o = constant for many metals at room temperature. In 1881, the
Danish physicist Lorentz made measurements between 0°C and 100°C
and was able to restate the law with another factor present — in the form
\=^> (814)
G 1
a constant for all metals for all temperatures. It is called the Lorentz
ratio or the Lorentz constant.
8.6 Electrons in metals 255
Typical values of k and a for a number of metals at 20° C (293° K) are
given in the table. The conductivities cover a sevenfold range but the
Lorentz ratio agree within 10% of one another— although some metals
which are poor conductors do not agree so well.
a k k/oT,
Metal (ohm cm) 1 watt/cm.deg T=293°K
Tin
0.087 x 10 6
0.64
2.51 xlO 8
Zinc
0.174
1.18
2.32
Aluminium
0.354
2.36
2.28
Copper
0.591
3.96
2.28
If we study one metal, copper, over a wide range of temperatures, we
find that the Lorentz ratio remains fairly constant within 20% or so
from 1,000°K down to 100°K. Below that temperature it changes quite
rapidly, however (Fig. 8.15).
From the proportionality of electrical to thermal conductivity over a
wide range of temperature, we can surmise that the electrons are respon
sible not only for the transport of electricity, but also for heat conduction—
and that this mode of heat transport is far more effective than other
modes of heat transport through the lattice of a metal.
There is one further experimental fact which intimately concerns these
electrons in metals. Surprisingly, their specific heat is extremely small.
If they behaved like a classical gas, and if 1 mole of metal contained about
1 mole of mobile or conduction electrons, then they would be expected to
contribute about f R to the measured molar specific heat of the metal.
Copper for example would be expected to have a specific heat of fK = 25
J/moldeg from the lattice and fR = 12.5J/moldeg from the electrons,
a total of 37.5 J/moldeg. But at ordinary temperatures the observed
value is not very different from that expected for the lattice alone so that
all that can be said is that the contribution from the electrons is small.
At extremely low temperatures however, below about 2°K, the lattice
specific heat is expected to be vanishingly small yet the measured specific
heat is not small and varies proportionally to T, Fig. 8.16(a). It can be
represented by
C v = 7.5 x 10 ~ 4 T J/moldeg
for copper. This is interpreted to be the contribution due to the electrons.
If we extrapolate it still using the linear law up to room temperature, it
gives only a small contribution of 1 % to the total instead of the expected
33% and this at least is selfconsistent, (Fig. 8.16(b)). It follows inevitably
that electrons in a metal do not behave like a classical gas.
256
Thermal properties of solids Chap. 8
i
i
X.
b
10
\ o
en
T3
\ x
F ,
\
^b
■ V
o
F
3
o
\
o
C
n
E
i 1 ^~
500 1000
Temperature
(a)
'O
500 1000
Temperature
500 1000 °K
Temperature
(c)
Fig. 8.15. Thermal conductivity, electrical conductivity and the Lorentz
ratio of copper as functions of temperature.
So far, we have assembled some experimental facts about metals. The
plan for the following section will be to show how some of the properties
of the electron gas can be deduced by harmonizing the Wiedemann
Franz law with the unexpected observation about the specific heat. The
result will be in the form of an equation for the energy of the electron gas
as a function of temperature — which is certainly nonclassical in form
and can only be understood in terms of quantum mechanics. Historically,
the problem was not approached in this way, but the electrons were as
sumed to behave like a classical gas ; it was found possible to explain the
electrical and thermal conductivities of metals but their specific heats
remained mysterious. Because of its historical interest, we will reproduce
this calculation also.
8.6 Electrons in metals
257
0.0003
: ^0.0002
<o"
1 2 3 °K 100 200 300 °K
Temperature Temperature
(a) (b)
Fig 8.16. (a) Low temperature measurements, below 2°K, of the specific heat of
copper. The predicted contribution from the lattice is shown as a shaded area,
the linear part is interpreted to be the contribution from the electrons, (b) Measure
ments up to room temperature. The lower line is the extrapolated contribution
from the electrons.
* 8.6.1 Thermal and electrical conductivities of metals
We calculate the thermal conductivity of a metal by imagining the
lattice not to exist, but the space occupied by the metal to be filled with
the electron 'gas'. This transports heat like an ordinary gas, although with
the appropriate values of mean speed and specific heat. The electrons
must be presumed to have a finite mean free path. Just what mechanism
is responsible for altering the trajectories of the electrons need not be
specified ; it might be collisions between electrons (a plausible suggestion
but not a correct one) or collisions with imperfections or impurities or
other departures from perfect regularity in the lattice such as thermal
vibrations. Then
k = %C' v cA
(8.15)
where c is the mean speed of the electrons, X the mean free path and C' v is
the specific heat per cm 3 of electron 'gas'. If there are n electrons/cm 3
inside the metal, then
1 n o~ ,
k=C v cX
where C v is the specific heat of 1 mole of electrons.
Next we will calculate the electrical conductivity of the same 'gas'.
Imagine the metal in the form of a rod or wire of crosssectional area
258 Thermal properties of solids Chap. 8
A and length /. Let a voltage V act between the ends. Then the electric
field is V//, and if the charge on the electron is e, the force on it is e\/l.
This causes the electron to accelerate, with an acceleration eV/lm, where
m is the mass of the electron.
If the mean free path is A and the mean speed is c, then the mean time
between collisions is A/c. We will assume that after a collision, the electron
is brought to rest. During its mean free time it accelerates and reaches
a final velocity equal to the acceleration multiplied by the time. It is
brought to rest by the next collision and then the process repeats itself.
Thus the mean velocity with which the electron drifts along the wire is
_ _ 1 VeA
2 mcl '
If there are n electrons per cm 3 , the number drifting across any plane in
the wire is vn A, so the charge transported per second is
_ A 1 \e 2 Xn A
Current = vn Ae = — .
2 mcl
Ohm's law states
A\
current = (conductivity a) — —
so that we have derived Ohm's law and
1 n e 2 X
o =  ^r. (8.16)
2 mc
Let us compare this expression with that for the thermal conductivity.
Both contain the mean free path A, which is not surprising for transport
processes with the same carriers. If we form the ratio, A cancels out :
k ^ Un /N)C v cX = 2 C v mc 2
o ^{n e 2 /mc)X 3 Ne 2 ' { }
Now let us rewrite this expression in slightly more general terms, \rn~c 1
is the mean kinetic energy of an electron — and we can assume that it is
not very different from \mc 2 (notice the different averaging). Let us
denote the mean energy of N electrons by E :
E = \Nmc^
The specific heat C v is equal to dE/dT. Therefore
k _ 4 E dE/dT
a ~ 3 (Ne) 2 '
8.6 Electrons in metals 259
Experimentally, k/<jT = Z£ the Lorentz constant. Hence,
EdE/dT= l(Ne) 2 ^T
so that
E 2 = E 2 +l(Ne) 2 £eT 2
where the term Eq is an arbitrary additive constant. We can take the
square root of both sides, making the assumption (which we will justify
shortly) that Eq is much larger than the T 2 term under our conditions.
Then
E = E
\ 3 (Ne) 2 J?
S E
1/2
= £o 3^ r2+ (g]8)
to h
This is the variation of energy of the electron gas with temperature which
we deduce from the observation that the Lorentz ratio is a constant.
Let us see what specific heat it predicts. We write
_dE _3(Ne) 2 J?
v dT 8 E
It can be seen at once that at any rate this is of the correct form, in the sense
that it agrees with observations at very low temperatures. If we put in
numbers, we can find the one unknown E .
Putting C v = 0.002 T J/moldeg deduced from measurements at low
temperatures, Ne = !F the faraday equal to 10 5 coulombs, the Lorentz
constant equal to 2.5 x 10 ~ 8 watt ohm/deg 2 , then E = 50,000 J/mol — an
enormous value equivalent to 1 eV for each electron. (At least our assump
tion that El > the term in T 2 is justified and our method of taking the
square root is selfconsistent.)
Thus the discussion of the WiedemannFranz law and the small
specific heat of the electron gas leads us to conclude that even at absolute
zero the electrons have enormous energies, the £ term. Being a constant,
E does not show up in the specific heat measurements which only
measure dE/dT: all these detect is the coefficient of the small T 2 term in
the energy, and at ordinary temperatures the energy of the electrons
changes by very little.
These facts can only be explained in terms of quantum mechanics —
in particular the uncertainty relation and the exclusion principle — which
260 Thermal properties of solids Chap. 8
we will not attempt to do. All we have achieved, while suggesting a mechan
ism for the conduction of electricity and heat in metals, is to point out yet
another system to which the Maxwell distribution and the law of equi
partition of energy are not applicable. For completeness, we will mention
that the specific heat of the electron gas as a function of temperature
has been calculated quantummechanically and is shown in Fig. 8.17.
It does eventually reach the equipartition value of R expected of a
monatomic gas, but only at the impossibly high temperature of 10 4 °K
— when the metal would have ceased to exist.
10
Fig. 8.17. Predicted specific heat of 1 mole of electron
gas.
8.6.2 The classical calculation
We have already explained that the approach which we have just
adopted was not taken by the pioneers in the subject. It was at the very
end of the nineteenth century that electrons ('atoms of electricity') were
discovered in ionized gases and their elementary charge roughly measured
by J. J. Thomson. Within a few years, Lorentz and Drude were proposing
that electrons formed a gas inside metals, in much the same way as we have
done. But it was natural in those days, before the advent of even the most
rudimentary form of quantum theory, to assume that the electrons really
did behave like an ordinary gas obeying the laws of classical physics.
The ratio of conductivities can be written
k 4 C V E
a
3 {Nef
Problems 261
on either theory. If now we assume the classical equipartition laws :
C v = $Nk, E = ^NkT, (5.12)
then we find
This is a very precise prediction — and it almost agrees with experiment.
Putting R the gas constant equal to 8.31 J/mol deg and J* the faraday to
10 5 coulomb (or k, Boltzmann's constant, equal to 1.38 x 1CT 23 J/deg, e,
the electron charge, to 1.6 x 10~ 19 C) the Lorentz number is predicted
to be about 2.1 x 10 ~ 8 watt ohm/deg 2 which is remarkably good. But of
course the specific heat is completely incorrect and it was because of
this that the whole theory had eventually to be demolished and rebuilt in
quantum terms — a process which took more than 20 years.
PROBLEMS
8.1. An imaginary element of atomic weight 80 has density 1.2 g/cm 3 . Its crystal
structure is known to be simple cubic and an experiment is carried out to
determine its lattice spacing accurately. Using Cu Ka radiation (A = 1.54 A),
a reflection from a cube plane is observed at 6 = 73° 45'. What value of n
must be used in Bragg's equation for this reflection and what is the lattice
spacing? At what angles 6 would reflections from octahedral planes be expected?
8.2. An Xray reflection from a crystal occurs at 6 = 45° when the crystal is main
tained at 0°C. When it is heated to 100°C, 9 decreases by 3.42 minutes of arc.
What is the expansion coefficient of the substance? How is the argument affected
if the crystal is not cubic in symmetry?
8.3. Calculate the expansion coefficient and Griineisen constant for an ionic crystal.
Start with the pair potential of Eq. (3.27).
(a) Using Taylor's theorem, show that the equation near the minimum is
A f = JPA ag2 l x \ 2 (P1)(P + 4 ) «e 2 (*\ 3
\ 2! /47te ao\«o/ 3! 47r£ ao\ao/
(Here a is the Madelung constant, not the expansion coefficient.)
(b) Show that in the equipartition region, the linear expansion coefficient is
equal to
(p + 4\/4m: a \
p— lj\ ae 2
where k is Boltzmann's constant. Estimate it for KC1 for which a = 3.1 A
and Madelung constant = 1.75.
(c) Use expressions for the compressibility and molar volume taken from
section 3.8.2. Hence show that the Griineisen constant is (p + 4)/9.
262 Thermal properties of solids Chap. 8
(d) Compare this with the experimental measurements on KCL (Data from
G. K. White, Phil. Mag. 6, 1425 (1961)).
T
°K
deg '
J/mol. deg.
V
cm 3
dyn/cm 2
30
65
283
9.90 x 10" 6
51.6
111.3
8.31
28.6
48.7
36.7
36.75
37.4
1.95 xlO 11
1.93
1.69
CHAPTER
Defects in solids: Liquids as
disordered solids
9.1 DEFORMATION OF SOLIDS
In section 3.7.1 we calculated the deformation produced in a solid by
the application of pressures or tensions to it. We will now extend the
discussion to deformations which are so great that the solid breaks.
The conventional nomenclature is to call the fractional deformation
the strain (as already mentioned) and the force per unit area the stress.
Previously, we considered only hydrostatic stresses, that is pressures
or tensions acting uniformly over the whole surface of the body (like those
which exist inside liquids, which cannot support shear strains). In these
the atoms in the body become uniformly squeezed or separated from
their neighbours in all directions equally. Now in practice, one is very
often concerned not with these hydrostatic forces but with tensions or
pressures acting along a line, or else with twisting forces. However,
we have noted (in section 3.5) that the order of magnitude of the bulk
modulus — the initial slope of the stress/strain curve — is of the same
order of magnitude as the other elastic moduli so we will continue to deal
only with uniform hydrostatic stresses.
We showed in section 3.7.1 that the strain s induced by a stress P is
given approximately by
P= K(sh 2 + ) (325)
264 Defects in solids : Liquids as disordered solids Chap. 9
assuming that higher terms can be neglected for the small values of s
we deal with in practice. (A positive stress is a pressure which produces
compression, a positive strain is an increase of volume ; whence the minus
sign in front of the leading term.) If we plot this function up to large values
of s, beyond the range shown in Fig. 3.15, P should go through a maximum.
In practice, this result can only mean that there is a maximum tension
which the solid can withstand, and when this tension is exceeded the solid
breaks.
Therefore the tensile strength of a material should be given by the con
dition
£*
that is,
K(l9s) =
s m * 10%
where s m is the tensile strength.
Our simple theory predicts that a body should be able to withstand a
10% strain. Though we have derived this result only for hydrostatic ten
sions, the same result should hold for stretching or twisting strengths :
a wire ought ideally to be able to be stretched by 10% in length before
breaking, a rod should be capable of being sheared through ^ radian
before shearing off.
9.1.1 Ductile, brittle and plastic materials
The behaviour of real metals is in strong contrast to this result. Some
typical stress/strain graphs are shown in Fig. 9.1. In (a), the body is
strained by only a small amount— say 0.001 % to 0.01 %. Then, when the
stresses are removed, the body returns more or less exactly to its original
shape and size. However, there exists a strain (typically of order 0.01 % to
0.1%) called the elastic limit, marked E on Fig. 9.1(b); if the body is
strained beyond E, and the forces then removed, it does not return to its
original shape and size but remains permanently distorted. Subsequent
stressings follow a complicated pattern, but when the strain is of the
order 0.1 % to 1 %, the metal breaks. This means that metals are a factor
of 10 or 100 times weaker than our simple theory predicts.
Materials which behave in this way are called ductile. This refers to
their property of distorting permanently before breaking.
Metals can be prepared as single crystals and these are often very soft
and ductile. They can be deformed by squeezing in one's fingers and
in extreme cases will even flow under their own weight. If pulled, they can
be extended to many times their own length— the strain can reach 10 or
265
10 to 10 Strain
(c)
Fig. 9.1. Stress (tension) strain curves for (a) ductile material in the
elastic range, (b) ductile material strained beyond the elastic limit E,
(c) brittle material. The curves are not to scale.
266 Defects in solids : Liquids as disordered solids Chap. 7
20 — though all these deformations are inelastic in the sense that they are
permanent and the metal does not regain its original shape when the
stresses are removed. These deformations are again quite different from
those predicted in section 9.1.
Some metals, however, like cast iron and many nonmetals like glass
or stone are brittle. Under small tensions they distort but then quite
suddenly they break. There is no previous warning in the form of perma
nent stretching as with ductile metals and the halves or fragments can be
fitted together afterwards to reconstruct the original shape. The stress/
strain graph is shown in Fig. 9.1(c), on greatly enlarged scales. This refers
to tension: many brittle materials can, however, withstand quite large
compressions. Houses and bridges can be built of brick and stone provided
those materials are not called upon to withstand tension.
Plastic materials (polythene, nylon, etc.) are different again in their
behaviour. The molecules of many of them are long chains which may be
crumpled or coiled and when the substance is stretched the chains may
straighten ; later they may slip slowly over one another under the action
of a constant force, as the weak bonds between the molecules break and
form again. As a result, many plastics can suffer elongations of several
times their original length without breaking. We will not however study
these materials but will concentrate on substances built of simple mole
cules.
9.1.2 Friction of metals
The empirical laws of friction are well known. When a body rests on a
horizontal plane, the horizontal force required to move it at constant low
speed is proportional to the mass of the body but independent of the area
of contact, Fig. 9.2(a) and (b). A brickshaped object requires the same
force to move it whether it is resting on a face of large area or one of small
area. Over a wide range of conditions, the frictional force is almost
independent of the speed of relative movement.
We will now show that though friction is a force which acts at the surface
of a body, it can be related to the bulk properties of the substance.
It is known that when one body rests on another, the area of real contact
between the surfaces is extremely small. Though each body has a macro
scopic area of several square centimetres, say, the area where atoms of
one metal actually touch atoms of the other is only a tiny fraction of this.
(The experimental proof of this statement came originally from measure
ments of the electrical resistance between two pieces of metal placed in
contact under only small forces.) The reason is that no metal surface is
plane on the atomic scale. There are always large bumps or asperities
even on surfaces which appear to be smooth— at least, if one uses the
9.1 Deformation of solids 267
F
(o)
(b)
(c)
Weight Mg
(d)
Force F ■
Fig. 9.2. (a), (b) The force required to move a given object at low speeds against
friction is independent of the area of contact with the plane, (c) Points of real
contact between two bodies, (d) Simplified model of contacts. Metal flows
until the pressure due to the weight is equal to the yield strength of the metal.
(e) Shearing of contacts by horizontal force.
268 Defects in solids : Liquids as disordered solids Chap. 9
word large' to mean large on the atomic scale. When two such surfaces
are brought together, it is probable that only at points like P and Q (Fig.
9.2(c)) will the two really touch.
If this does occur, then the weight of the upper body is borne on a
very small area of contact so that the pressure, the weight per unit area,
is very large. As a result, the metal in these regions is crushed beyond its
yield point and it flows. The two bodies may even weld together there.
Let us call the maximum stress that the metal can bear in compression
without flowing the yield strength, and let us denote it by S Y .
Then the contacts will go on flowing and increasing their crosssectional
areas till the weight Mg of the body can be supported ; then movement
will stop. Let the total area of real contact be A when this happens (Fig.
9.2(d)). Then A is determined by
^=S Y , (9.1)
A
Typically, S Y is of the order of 10 10 dyn/cm 2 for a metal. If the mass of
the body is 100 g, Mg is 10 5 dyn and A is of the order of 10" 5 cm 2 —
though a body weighing 100 g is likely to have dimensions of a few centi
metres.
Now imagine a horizontal force F to act on the upper body. The
contacts experience a shearing stress, Fig. 9.2(e), equal to this force
divided by the area tangential to it, that is F/A. They can withstand this
deformation until the stress reaches a limiting value called the shear
strength, denoted by S s , when the contact snaps. In a real situation there
are of course many contacts which break at different times and reform
elsewhere, but the force required to move the body is given by
 = S s . (9.2)
A
Eliminating A from these two equations,
— = ^ = fi (9.3)
Mg Sy
where \i is called the coefficient of static friction, the ratio of the horizontal
force acting on the body to its weight. Thus we have shown that \i is
independent of the size or shape of the bodies and is equal to the ratio
of the shear strength to the yield strength. Roughly, this is equal to the
ratio of the rigidity modulus to Young's modulus, and this from the table
in section 3.5 is roughly \. Indeed it is observed that the coefficient of
friction between metals is of this order.
9.2 Brittle materials 269
In addition to the welding and snapping of the regions of contact,
the coefficient of friction is affected in practice by a number of other effects
which are of technical importance. For example, the ploughing of one
asperity through another as the body moves sideways can be important.
Lubricants and films of oxide on the surface can have a profound effect
on friction. Further, when two different metals are in contact, the fact that
one metal melts more easily than the other can alter the details of the
mechanism. We will content ourselves however with having related the
phenomena at the surface with the properties of the bulk material.
9.2 BRITTLE MATERIALS
Glasses and ceramics (like pottery and bricks, which are mostly metallic
oxides) are brittle materials. They have disordered, noncrystalline
structures: Fig. 9.3 is an attempt to picture some neighbouring atoms
in such a solid. The interatomic bonds are highly directional in nature,
being of the electronsharing type (section 2.1.4). The coordination
number of each atom is small and the structure more or less rigid. As
mentioned previously (section 2.2.4), amorphous solids can be pictured
as liquids 'frozen' into a particular disordered configuration which does
not change with time. For a molecular solid, like solid argon in a non
crystalline state, the molecules are arranged as in Fig. 2.4 ; the coordination
number is high, the bonds directed at all angles. Amorphous regions in
metals have similar structures. Fig. 9.3 and Fig. 2.4 resemble one another
Fig. 9.3. Atomic structure of an oxide glass. The atoms or ions have been drawn rather
small for clarity. The lines joining them represent directional covalent bonds.
270 Defects in solids : Liquids as disordered solids Chap. 9
in that both are disordered, but they differ in that one lattice is much more
rigid than the other.
To explain their stress/strain curves, it has been proposed that brittle
materials contain cracks, which may be of microscopic or even atomic
dimensions, and which may either be wholly inside the material or else
originate on the surface. Such imperfections weaken the material and it
breaks when one crack suddenly spreads. A familiar example of such
behaviour is the way one can break glass in a controlled way by first
scoring the surface with a diamond and then bending the glass so as to
open the crack.
We can simulate the stresses round a crack of known shape but of
large size, and study them using an optical method. Materials like glass
or plastics when deformed exhibit double refraction. This means that the
refractive index for visible light is different for light polarized parallel to
the direction of the stretching, from that for light polarized at rightangles
to it. Using polarizing plates and a quarterwave plate in a way which is
described in standard texts on optics, the strain in a photoelastic material
can actually be seen and measured through the intensity of light it trans
mits. In particular, lines of constant strain can be made visible as dark
lines, interference fringes, on an illuminated background.
Cut
Tension
Fig. 9.4. (a) Stress pattern round the end of a cut in a slab of photoelastic material.
(b) End of the cut showing semicircular end of radius p.
Figure 9.4(a) shows a photograph of part of a slab of plastic (Columbia
resin, CR 39) 6 mm thick, 19 mm wide and 7 cm long being stressed
horizontally with a tension of several hundreds of kilograms. A parallel
sided cut, 0.4 mm wide and 1 1 mm long was made in it, transversely, with
the ends carefully machined to be semicircular Fig. 9.4(b). The photograph
concentrates on the end of the cut and it can be seen that the dark lines all
crowd in there.
9.2 Brittle materials 271
In some preliminary experiments with a specimen of plastic of the same
thickness but with no cut in it, the optical components were set so that
the specimen was uniformly dark when there was no tension on it. When
the tension was increased, the field lightened gradually and then went
dark again; this occurred with a tension of 1,200 kg/cm 2 . Subsequently,
the field went dark again every time the tension was increased by about
this amount. Now each fringe on the pattern represents a line of constant
strain : we can therefore say that if Hooke's law holds the tension at all
points along one fringe differs from that at points in the next fringe by
1,200 kg/cm 2 . The fact that all the dark lines crowd together therefore
means that the tension varies very rapidly from point to point, and that it
is a maximum at the end of the cut.
From the intensity at the top and bottom of the specimen, the applied
tension was estimated to be about 600 kg/cm 2 . There are 7 fringes between
these regions and the end of the cut, so that at that point it is 8,400 kg/cm 2
higher. Hence the tension at the end of the cut is about 15 times as great
as the tension far from it.
The theory of these stress/strain patterns in irregularly shaped bodies
is complex : we will however quote the result that if a transverse cut is of
total length / and the radius of the end is p, the stress in the neighbourhood
of the end of the cut is enhanced by a factor
Here, this is 2 x (11/0.2) 1/2 = 15, in good agreement with the measure
ments on the photoelastic specimen.
It is obvious, then, that a crack inside a solid with a small radius p at
the end — that is, a sharp pointed crack — concentrates the stress at the
end. Whereas the tension far from the crack may be quite small, well
within the tensile strength of the material, the stress at the point may
become too great and the material will fail there. This occurs when the
stress s is given by
s m = s.2l l  (9.4)
where s m is the expected breaking stress, in the absence of cracks. As the
crack extends, / becomes larger and p remains the same or gets smaller,
so that it continues to grow.
There are two points to note. The first is that cuts or cracks which are
parallel to the line of the tension have no effect as stress concentrators.
This is the reason for the great strength of freshly drawn fibres of quartz
or glass — they can withstand elongations of several percent and their
272 Defects in solids : Liquids as disordered solids Chap. 9
Young's moduli are higher than that of steel so that their breaking stresses
are higher than that of steel. They owe this property to the fact that when
they are prepared by pulling a filament of melted but highly viscous liquid
just above the melting point, all cracks become pulled out also, but parallel
to the length of the fibre. Until the surface becomes pitted by chemical
action through exposure to the atmosphere, they retain their strength.
The second point is that only the ratio of the length of the crack to its
width enters into the stress ratio. A crack 4 A wide and 110 A long would
be just as effective as the one in the specimen of Fig. 9.4. Now most
ceramics are found to be inhomogeneous in their crystal structure when
examined by Xray diffraction. It is quite possible therefore that in a
structure like Fig. 9.3 with rigid, directed bonds there may be misfitting
atomic planes with few bonds crossing them. These behave like long
narrow cracks, and can therefore act as stress concentrators. Glass on the
other hand usually seems to break because of defects on the surface —
scratches or particles of dust picked up during manufacture. The evidence
for this comes from the study of the fractured surfaces. There is always
a small area with a very smooth surface — so smooth that no roughness
can be seen even under a microscope. But surrounding this small area,
the surface is covered with small ridges. These different areas are thought
to be caused firstly by the rapid tearing of the glass and secondly a forking
of the crack as it accelerates. When a rod is deliberately notched and then
bent and broken, so that there is no doubt where the break originated, the
mirrorlike surface is always found near the origin and the ridges radiate
roughly from the same point. Thus we have a good method for locating
the origin of any crack. All cracks seem to start from the surface, not from
inside.
The great strength of brittle materials when they are compressed is
explained by the closing of the cracks and notches. Glass can be toughened
using this effect, by arranging that the surface layers of a sheet of molten
glass cool more quickly than the inside ; the inside of the sheet is the last
to solidify. This has the effect of compressing the outer layers, that is, the
atoms in the outer layers are closer together than those in the inner layers.
Therefore when the surfaces are stressed, these tensions must first overcome
the lockedin compressive stresses before any cracks can begin to open
so the glass is stronger than when prepared normally. Concrete can
similarly be prestressed by allowing it to harden under great pressure.
9.2. 1 Dynamic behaviour of brittle materials
It is good common sense to argue that if a number of cracks begin to
spread through a solid, only one of them can 'win', and that once the
solid has broken the stress is immediately relieved and the other cracks
9.2 Brittle materials
273
must stop growing. In other words, a solid ought to break only at one
place at once. It would be a gross and improbable coincidence if there were
two cracks of exactly the same rate of growth so that a solid broke at
two places at once.
Yet if one bends a glass rod (say 25 cm long and 3 mm diameter) rapidly
but firmly so that it breaks, it may fly into several pieces, three or four or
more (Fig. 9.5.). This is seemingly impossible behaviour. But it is not
difficult to demonstrate that the cracks occur one after the other and not
simultaneously, so that the commonsense argument is not incorrect.
Similar things happen when a rod is broken by straight tension.
' : %TiMmif i fflMiiiiimiiiiil^F "*"*
Photos by Colin King
Fig. 9.5. Flash photographs of two different glass rods breaking under the action of
shear — the rods were bowed upwards.
The key to these phenomena is the presence of stress waves which are
propagated along the rod. In a rod broken by bending, they start as a
flexural pulse which whips back and forth along the rod. This is a compli
cated motion in which short wavelengths travel fastest and an initially
compact pulse straggles ; later, the shortest wavelengths can start bouncing
around from side to side inside the rod. Or if a rod is broken by pulling,
then after the break the surfaces spring back and a compression pulse is
propagated along each half. When it meets the far end it is reflected.
If that end is free to move, being undamped, then the pulse is reflected
as a tension pulse. For a short time, while the incident compression pulse
is passing over the reflected tension pulse, the stresses cancel, but later
274 Defects in solids : Liquids as disordered solids Chap. 9
the tension appears — and the rod may break again, a little away from
the end.
9.2.2 Twocomponent materials
Fibreglass is a typical twocomponent material. It consists of glass
fibres (which in the absence of surface flaws have great tensile strength)
embedded in a matrix of soft, nonbrittle plastic. Consider the fibres all to
be parallel and the tension to be applied in their direction. Then practically
all the stress is borne by the fibres. Some may crack across, but because
they are separated from each other a crack cannot spread from one to
another. If the matrix material adheres strongly to the fibres, then one
broken fibre will be held in place through the strength of its neighbours.
The tensile strength of the material is therefore extremely high.
Its shear strength is small, however, because the fibres can be easily
bent, so that aligned fibre glass is not a good structural material. However,
if the fibres are tangled up and are aligned in all directions, the strength
for all kinds of stress can be quite high.
Wood is another common twocomponent material — cellulose fibres
in a lignin matrix. Again, it is extremely strong in tension but its shear
strength is low. This can be improved by laminating it to form ply
wood, in which successive layers have their fibres oriented in different
directions.
9.3 DEFORMATION OF DUCTILE METALS
In section 9.1.1 we defined the elastic limit of a substance, and stated
that soft metals prepared as single crystals can be strained beyond their
elastic limits by the action of quite small forces. Cadmium notably can
be pulled to 10 or 20 times its original length.
When the surface of a singlecrystal rod is examined after it has been
pulled beyond its elastic limit, it is seen that its surface is covered with
fine lines. These show that the material has divided itself into bands which
have slipped or glided with respect to one another, Fig. 9.6. The direction
of easy glide are closepacked crystallographic planes. Each section has
slithered sideways, pulled over by the component of the tension parallel
to the glide direction, but as a result the specimen as a whole has become
longer. The elongation has taken place because each section undergoes
shear with respect to the neighbouring ones.
Each unslipped section has an almost perfect crystal structure: the
deformation is located inside regions which are only a few atomic planes
thick. The width of the unslipped regions may be anything upwards from
a few thousand atomic spacings and the steps on the surface may be a few
9.3 Deformation of ductile metals
275
hundred or thousand atomic spacings high, so that the structure is usually
visible under a microscope.
Thus the process of deformation in a ductile metal is quite different
from that which we considered in section 9.1 where we worked out the
tensile strength of a material — on the assumption that the deformation
was elastic and would return to zero when the stresses were removed.
There, we assumed that the strain was homogeneous, that the distance
between every atom and its nearest neighbours increased by the same
amount. But it appears that, beyond the elastic limit, the strain is far
from homogeneous : large movements take place in relatively few planes.
Fig. 9.6. Stretching of a singlecrystal rod under tension
by slipping in sections along direction of easy glide. The
sheared rod is slightly longer than it was originally.
9.3.1 Dislocation lines
It is useful to study the strains in a metal where the slip has travelled
only part way across the thickness. Fig. 9.7(a) shows this diagrammatically.
The whole block, originally rectangular, is now distorted and out of true.
Part of the lefthand front face has been pushed in a short distance, and
an incipient step is visible on this face. With this deformation, part of the
whole block has also slipped and the distortion of the atomic planes is
visible on the righthand face. The boundary between the slipped and the
unslipped region is marked by a curve inside the block, shown by the
shading on the slip plane. This boundary is called a dislocation line.
The strain, the displacement of an atom relative to its neighbours, is
greater near the dislocation line. But elsewhere, the crystal is almost
perfect.
It is these relative displacements in the vicinity of the dislocation line
which we will have to study in some detail. Now when the dislocation line
is curved, as in the diagram, the strains are difficult to describe. But two
basic types of straight dislocation lines can be distinguished, and any
curved line can be regarded as a combination of these.
276
Defects in solids : Liquids as disordered solids Chap. 9
°o o o o
; 1 1 \ I \ i \
aOOCxWOO
OtyOO
p6 6 6 6 6
R
id)
OQ9Q0
0<>c6
i ! ! ■ ■'
a666
c)
Fig. 9.7. (a) A strained cube with a slipped region bounded by the shaded
arc. At A a screw dislocation emerges, at B an edge dislocation. The deforma
tion is in the direction RP. (6) Looking down on a screw dislocation ; A, P,
Q, R correspond to A, P, Q, R in the cube above. In a perfect lattice the
atoms would lie on separate lines, but here they lie on a helical surface, (c)
Looking along an edge dislocation, (d) Smallangle grain boundary ; D and D'
are edge dislocations.
9.3 Deformation of ductile metals 277
In the diagram, the slip is in the direction of RP. When the dis
location is a straight line parallel to the slip, as it is near A, it is called a
screw dislocation. When it is perpendicular, as it is near B, it is called an
edge dislocation.
The arrangement of atoms near to a screw dislocation line is shown in
Fig. 9.7(b). What were originally parallel planes of atoms have now be
come distorted into a helical surface, like that generated by a propellor
rotating slowly as it moves forward. Starting on one plane at the top of
the diagram and moving from atom to atom to the bottom and then up
again, one arrives without discontinuity at the next plane ; in fact, there is
strictly speaking only one single sheet of atoms, instead of parallel planes.
The lefthand face, with the step, in Fig. 9.7(a) is the outermost layer of
this sheet.
The atoms near an edge dislocation are shown in Fig. 9.7(c). In the upper
half, there are five columns of atoms, in the lower half only four. Similar
patterns are visible on the righthand face of Fig. 9.7(a), and also in Fig. 2.5.
Dislocations may be produced by any irregularity during the growth
of a crystal. They occur at grain boundaries, the junctions between
crystals which do not fit perfectly because there is an angle between the
sets of crystallographic planes. The resulting misfit can be described as
a row of edge dislocations, regularly spaced, as in Fig. 9.1(d). Dislocations
may also be clustered round any misfitting inclusions or foreign atoms
inside a lattice.
Dislocations can be seen most easily in specimens made in the form of
very thin films, say 1,000 A thick, examined in an electron microscope
capable of magnifying by a factor of the order of 10 5 . Dislocation lines
which run more or less vertically through the thickness are visible. Points
like A and B in Fig. 9.7(a) where dislocations emerge at surfaces of crystals
are chemically active, because of the high energy of atoms with the wrong
coordination. Thus they are vulnerable to chemical attack by appropriate
reagents. Little holes called etch pits appear at odd places on the surface.
Another chemical method for seeing dislocations is the 'decoration'
of the lines inside transparent crystals like silver bromide and silver
chloride. These substances are used in photographic plates because, under
the action of light, free silver is formed which is opaque. Not surprisingly,
it is formed preferentially on the dislocation lines so that these can be
seen under a microscope.
In a crystal of metal prepared under ordinary conditions, there are
typically about 10 7 dislocations in any square centimetre of cross section.
This sounds an enormous number, but expressed rather differently the
same datum shows that such a crystal is highly regular. For if the inter
atomic spacing is 2 A, there are 5 x 10 7 atoms in 1 cm of length; 10 7
278 Defects in solids : Liquids as disordered solids Chap. 9
dislocations per cm 2 means one dislocation every 3 x 10 4 cm, that is one
every 15,000 atoms. For comparison, it may be noted that in the most
perfect of single crystals which can be prepared the density is around 10 3
dislocations/cm 2 . At the other end of the scale in heavily damaged metals
where the crystal structure has been broken up there might be 10 12
dislocations/cm 2 , one every 50 atoms. Even in the most disordered
crystals, the assumption that most of the atoms are in their regular lattice
sites is a valid one for many purposes.
9.3.2 Movement of dislocations
Having described what dislocations are, we can now show how they
can produce slip in a crystal under the influence of small forces — tensions
which are small compared with the elastic moduli.
The essential process is the movement of a dislocation through the
lattice. We will concentrate on edge dislocations. Similar arguments
apply to screw dislocations but it will be left as an exercise for the reader
to construct these.
Imagine a block sheared as in Fig. 9.8. A dislocation is created on the
left, runs from left to right and emerges from the block. The net result
is that the top half of the crystal has sheared over. By itself, this process
causes only a very small slip, only one atomic spacing high. But it is the
basic process which is responsible for the inelastic extension of metals
under small forces. Given 10 7 dislocations per cm 2 , a movement of each
one of them would add a 2 A step to one side, and if all of them moved
a 1 cm cube would shear over a distance of 2 mm, a 20 % strain.
We can now look at Fig. 9.6 with much more understanding than before.
One would guess at first sight that each slip band has slipped bodily
across, like a penny in a pile which has been pushed askew. But now we
can see that each section has not been moved like a rigid body — dis
locations have moved across instead — and when we say that a dislocation
has moved, Fig. 9.9 shows that all we mean is that all the atoms in the
slip plane have moved a small fraction of an atomic spacing, one after the
other.
It is not difficult to see qualitatively that dislocations, once created,
can move through a lattice very easily. Not only are the individual
displacements of the atoms very small, which implies small activation
energies, but when one atom gains potential energy (by moving a little
further from its neighbours) another loses an almost equal amount (by
moving closer). Therefore during the movement the potential energy of
the system hardly changes.
Quantitatively, however, the mobility depends critically on the geometry
of the lattice. The dislocations which we have pictured have all been narrow
9.3 Deformation of ductile metals
279
ones, in the sense that there are only two or three atoms in a line which are
badly, displaced from their regular positions. The number w of such
atoms in a line is a useful measure of the width. But it is possible to produce
wide dislocations, as in Fig. 9.9(b). Here there are many more displaced
atoms in a line but the change of position of each one when the dislocation
moves is much smaller than before so that wide dislocations are more
mobile than narrow ones. A detailed analysis shows that if s d is the stress
required to move a dislocation and K is one of the elastic moduli, then in
order of magnitude
K 6 *
(9.5)
e 2 * is equal to 500 so that the width co of the dislocation should have a
dominant effect on the yield strength of a material.
Fig. 9.8. Shearing of a block by the movement of a dislocation across it.
Wide dislocations are favoured by large spacings between atomic
planes ; such planes themselves are closepacked crystallographic planes.
Hence one would expect that dislocations can move most easily in close
280 Defects in solids : Liquids as disordered solids Chap. 9
packed crystallographic planes. This result was observed experimentally
in metals and has already been mentioned in section 9.3.
Wide dislocations are also most easily formed in substances like metals
where the bonds are nondirectional. Indeed, very pure metals are ex
tremely soft. However, another effect enters as soon as a metal has been
repeatedly deformed: it is observed that its hardness increases. This is
called work hardening. It is interpreted to mean that the metal has become
full of dislocation lines — and it is difficult for one line to pass through
another. The lines repel one another because of the stresses round them ;
thus the movement of dislocations is impeded, and the metal hardens.
Ordinary metals also owe some of their hardness to another effect, the
presence of impurities which can also impede the motion of dislocations.
(a)
OCCCOOCOD
GCCODOCCCO
U>)
(c)
id)
Fig. 9.9. (a) and {b) : Dislocation moves one atomic space, (c) Relative positions
of atoms in top line in (a) and (b), showing that each atom moves only a small
distance, (d) Same diagram for a wider dislocation : co is about 8 whereas co
is about 5 or 6 in (c).
Hard metals like steel owe their combination of ductility and hardness to
the judicious introduction of other elements such as carbon, and of
several types of crystal modifications dispersed throughout the material.
Narrow dislocations are characteristic of crystalline substances with
covalent bonds, which are rigid and difficult to distort. Covalent materials
9.4 Growth of crystals 281
in any case have high elastic moduli, but in addition we can now see why
they are not ductile. Diamond, for example, is the hardest substance
known.
* 9.4 GROWTH OF CRYSTALS
We will now discuss the rate of growth of crystals and to simplify matters
we will consider growth from the vapour phase. The classic studies were
done on iodine crystals; iodine has a vapour pressure of 0.1 mm at room
temperature. The rates of growth were typically about 0.1 mm per hour
when the vapour pressure was 1.05 times the saturated vapour pressure —
that is, when the vapour was 5 % supersaturated. In this section we will
outline a simple theory and demonstrate that it is wrong by many orders
of magnitude. We will assume that the initiation of a new layer of molecules
on a plane crystal face is much like the nucleation of a droplet of liquid in
a supersaturated vapour (see section 7.8.3). We will also assume that the
crystal face is perfectly plane to begin with. If a single molecule arrives
from the vapour and sticks to the plane, the number of bonds which it
forms is only a small fraction of its full coordination number, about \.
So this lone molecule probably evaporates again after a short but finite
time. If, however, there is an 'island' already formed on the face (a two
dimensional droplet, as it were), the newly arrived molecule might lodge
at the edge of it and then its coordination number approaches n/2 and
there is a much greater chance of it sticking permanently ; the island acts
as a nucleus, Fig. 9.10. Let us for simplicity assume that the island is
circular. Then we can treat it like a cylindrical droplet of finite radius and
Fig. 9.10. The formation of an 'island' from the aggregation
of newlyarrived molecules landing anywhere on the
crystal face.
282 Defects in solids : Liquids as disordered solids Chap. 9
only one molecule high. Using exactly the same methods as in section
7.8.3 for spherical droplets we can calculate the critical radius for a
given degree of supersaturation of the vapour, the radius of island which
can grow instead of evaporating. We do this by balancing the rate of
evaporation from the curved surface against the rate of arrival of new
molecules which might have landed anywhere on the crystal face and
diffused to the island. (This last is different from the corresponding
conditions during the formation of a liquid droplet). Finally we can
calculate the activation energy A for forming such an island ; the prob
ability of a critical island forming contains the Boltzmann factor
exp( — A /kT). The steps are as follows. Inside a cylindrical droplet of
any length but of radius r, the excess pressure is y/r. Its vapour pressure
is therefore increased over that of a plane surface by the factor
P
_ p yvlrkT
Po~
where v is the volume of one molecule, y the surface tension, k Boltzmann's
constant ; compare Eq. (7.34). The critical radius is
yv
c kT\n(P/P Y
compare Eq. (7.35). The activation energy A for forming such an island
is (following precise thermodynamic arguments) equal to onehalf of the
surface energy of the edge :
A = nr c dy
(compare and contrast Eq. (7.36)) where d is the diameter of a molecule
(the height of the cylinder).
Let us simplify these expressions by writing y = j J/ "m (see section 3.4)
where A" = \ja% . The volume v of one molecule is al • Then :
A  n ( U£ ) 2 / QA v
Ao ~ l6kTln(P/P ) ( * 6)
for an island of critical radius to be nucleated in a supersaturated vapour
of pressure P.
Let us now put in numbers. For a molecular crystal, let us take n = 10,
£ = 0.1 eV and for room temperature kT = ^ eV. ; whence
9.4 Growth of crystals 283
Suppose the area of face is 1 mm square, 1CT 2 cm 2 . At a vapour pressure
of 0.1 mm, the number of molecules arriving at unit area, \nc, is 10 18 per
second. Further, the critical radius is always small and a new nucleus
might be formed anywhere on the crystal face. Thus the number of
critical nuclei formed per second is equal to the number of molecules
arriving per second on the whole face multiplied by the probability
that one of them has enough energy to form the nucleus :
10 16 exp(/4 //cT) = 10 16 exp[400/ln(P/P )].
Consider a crystal growing at 0.1 mm per hour, which (for a = 3 A) is
3 x 10 6 layers per hour or 1,000 layers per second. For this
10 3 = 10 16 exp[400/ln(P/P )].
This tells us the supersaturation required; it gives ln(P/P ) = 13, which
means that the vapour pressure should be e 13 or 5x 10 5 times super
saturated. Experiments gave 5 % supersaturation for this rate of growth.
Thus the theory is badly wrong.
The weak point of this theory is that after an island has grown and
covered the whole crystal face, it produces a perfect plane ; a finite time,
very long on the molecular timescale, has to elapse before another
fluctuation occurs which is big enough to form another island. In order
to explain the observed rate of growth, we need to postulate some sort of
step against which newly arrived molecules can stick themselves, but a
step which remains in existence even after the face has been covered
with a new layer.
9.4.1 Growth spirals
The point of emergence of a screw dislocation at a crystal face provides
just such a step — Fig. 9.11. We can imagine atoms arriving more or less
uniformly all along the line and the crystal can grow. But the step does not
advance as a straight line. Near the centre of the screw there are some sites
where the step is not of the full depth, and the atoms lodging there soon
convert the end of the straight ledge into a tight spiral. This spiral then
grows upwards like a ziggurat, and the crystal grows as the turns of the
spiral sweep round over the crystal surface — and they continue to advance
however much the crystal grows. This is ultimately a consequence of the
fact that there is only one sheet of atoms in a crystal containing a single
screw dislocation.
Spiral markings can be found on many crystals under a microscope,
Fig. 9.11(e). The faces of an average crystal have many screw dislocations
emerging on them, and no point of a face is more than a few hundred
atomic distances from a spiral step. This fact obviously increases the
284 Defects in solids : Liquids as disordered solids Chap. 9
chance of permanently capturing a molecule from the vapour. But at
first sight, it might appear that such a molecule would have to land right
on the step, or within one atomic distance or thereabouts from the step,
in order to be captured. But this is not so : we can show that a lone molecule,
landing far from a step, sticks for a comparatively long time and diffuses
an appreciable distance over the surface before being thrown off again.
In its wandering it can meet a step and stick there, so that the probability
of capture is greatly enhanced.
We show this as follows. For an isolated molecule on a flat plane, the
number of bonds it forms is about rt where n is its full coordination
number. The probability of its evaporating therefore contains the factor
exp(^ne//cT).
Fig. 9.11. First four pictures show the early stages in the growth of a spiral
step on a crystal face round the point of emergence of a screw dislocation.
(e) shows the appearance of a growth spiral in the electron microscope : the
steps are typically 100 A to 1,000 A across.
9.4 Growth of crystals 285
Now the molecules on which it is sitting are vibrating at the Einstein
frequency (see section 3.6.1), which we call v E . One would guess that there
is a maximum probability of the lone molecule being thrown off once
per vibration. Therefore the chance per second that the molecule evapo
rates is v E exp( — Ine/kT). With reasonable numbers for a molecular solid,
the 'sticking time' is about 10~ 8 s. The molecule can jump to a neigh
bouring site without evaporating, and the energy needed to do this is
very small indeed compared with e. One can imagine the molecule sitting
in a dimple on the surface and the activation energy required to make it
jump to a neighbouring dimple is obviously very small. Since the cor
responding Boltzmann factor is almost equal to unity, it probably jumps
once every vibration, so that its jump frequency is v E . Therefore it performs
exp(ine/kT) jumps before evaporating. Each jump moves a distance a ,
but it performs a random walk. The net distance it goes in s steps is there
fore only a Js (see section 6.3). Hence it diffuses an average distance
a exp(fne//cT)
(where the effect of taking the square root of the exponential is to alter the
I to ). For ns = 1 eV, kT = ^ eV, this is about 150 atomic diameters.
This is comparable with the distance between the spiral steps which are
to be found on the average crystal face. Therefore we can say that practi
cally every molecule that arrives from the vapour is eventually captured.
Of course the crystal also loses molecules by evaporation. The number
lost per cm 2 in one second is (\nc) evaluated at P , the equilibrium vapour
pressure (because this must be equal to the rate of arrival of molecules,
in equilibrium). If, however, the crystal is exposed to vapour at super
saturated pressure P, the rate of arrival of molecules is (i«c) evaluated
at P. Therefore the net rate of gain of molecules is
1
~P ~
nc
1
4
Po
per cm 2 in one second, where the first factor is about 10 1 8 for iodine vapour
at room temperature. Putting in numbers, the supersaturation required
for a 1 mm square crystal face to advance at a rate of 0.1 mm per hour is
1 % — in good agreement with the 5 % observed.
This detailed study of the mechanism of crystal growth is not incom
patible with the account given in section 8.1.3. If screw dislocations inside
a growing crystal are oriented in all directions at random, then the growth
will proceed as in Fig. 8.6.
286 Defects in solids : Liquids as disordered solids Chap. 9
* 9.4.2 Singlecrystal whiskers
Under the right conditions, crystals can grow in the form of tiny
whiskers, typically a few thousand atoms in diameter and anything up
to millimetres in length. NaCl and KC1 can, for example, be grown out
of the sides of big crystals by exposing them to the vapour at tempera
tures of the order of 1,000° K. Xray examination shows that these
filaments are single crystals. It is thought that they are produced when
the outer turns of the growth spirals on the surface of the big crystal
are stopped from advancing because foreign impurity atoms have lodged
there. This allows only the tight central turns of the spiral to grow up
wards and the result is a filament. Singlecrystal whiskers are extra
ordinarily strong. For example, the specimen of iron capable of with
standing a 4% elongation which was elastic and reversible (see section
3.7.1) was a singlecrystal whisker. This strength is in marked contrast
to the ductility of large single crystals of metals. But we expect a
whisker to have only one single screw dislocation running down the
middle of it. Such a line defect does not weaken a lattice when tension is
applied parallel to it — in much the same way as a crack does not weaken
a specimen of brittle material if it is parallel to the stress (section 9.2).
The observed strength of singlecrystal whiskers therefore confirms the
proposed explanation for their growth. It should be mentioned for
completeness that other mechanisms of growth are possible — some
whiskers actually seem to grow from the base, pushing their way upwards.
* 9.5 POINT DEFECTS
We have described dislocations in solids, but there exists a much
simpler kind of imperfection, the point defect. This is the single atom (or
molecule or ion) which has been displaced from its lattice position. If it
has lodged in a nearby site, wedged in between a near set of neighbours
which have to move a little out of the way in order to accommodate it,
it becomes an interstitial atom; it leaves a vacancy behind, Fig. 9.12.
Vacancies can exist by themselves without any corresponding interstitial
atom : in that case we can regard the displaced atom as having been
removed to the surface of the crystal.
Vacancies are important because their presence greatly enhances the
speed of diffusion of atoms in solids, and this is because the activation
energy for one to jump one atomic space is not large. We can follow a line
of argument suggested in section 7.2 and regard liquids as resembling
highly disordered solids containing many vacancies and interstitial atoms ;
we can then gain new insights into diffusion mechanisms in liquids and
thence into other transport processes.
9.5 Point defects 287
oooooooo
O xQ o o o o 0,0
o o y o o ""o 6
oooooooo
Fig. 9.12. Atoms removed to an interstitial position and
to the surface, leaving vacancies behind.
9.5.1 Concentration of vacancies
First we will estimate the number of vacancies in a solid as a function
of temperature. Atoms can jump out of place as a result of thermal agita
tion so that the displacing of an atom is thermally activated. Let us
compare the energy of a perfect lattice with that of a lattice containing
one single vacancy, with the displaced atom removed to the outer surface.
We can imagine the atom to be cut out of the lattice by breaking n bonds
(n the coordination number). This requires energy ne. The atoms around
the vacancy then relax a little towards one another and this reduces their
energy. Finally, putting the one atom back on the surface reduces the
energy a little further still. The net result is that the energy of the whole
crystal is increased by something like ins. For solid argon this is about
0.02 eV, for sodium chloride it is about 3 eV, a much larger value because
of the long range forces.
Let the increase of energy of the crystal due to the presence of one
vacancy be denoted by AE. The Boltzmann factors for the crystal with
one vacancy and for the perfect crystal can be written down ; the ratio of
probabilities of the two states is exp( — AE/kT).
Thus for a solid in equilibrium the fraction of vacancies is
exp( AE/kT). This function increases with temperature and for
a solid reaches its greatest value when the solid melts. We there
fore expect some warning of the onset of melting to be given in
the last 10° or 100° before the solid melts, in the form of large numbers of
vacancies. In section 8.3 we noted that the amplitude of atomic vibrations
in a solid at the melting point was surprisingly small — onetenth of the
interatomic spacing in molecular solids, onethirtieth in ionic solids. We
will now be able to show that under these same conditions, vacancies are
produced in large numbers. The smallness of the amplitude at melting
therefore appears a little less mysterious, because we can accept that a
lattice which is full of holes can collapse more easily than a perfect one.
288 Defects in solids : Liquids as disordered solids Chap. 9
There are indeed a number of 'premelting' phenomena which can be
observed in many substances. But the fact that we can explain these
does not imply that we can explain the phenomenon of melting itself.
The sharpness of the melting point, the suddenness of the onset of the
transition, cannot be explained in simple terms. All that we can show is
that because of thermal equilibrium a substance can be quite inhomoge
neous on the microscopic scale.
9.5.2 Premelting phenomena
For rough calculations on molecular substances and metals near their
melting point let us take the melting temperature 7} to be half the critical
temperature T c :
K If = 2^ l c = 2 £
(see section 8.3). Then AE/kT f must be about 4 for a close packed lattice
with n = 12. Now exp( — 4) = j$ ; thus at the melting point we would
expect about 2 % of the atoms to be displaced. This means that about
1 atom in 4 linearly is out of place and really it represents a highly dis
ordered lattice. Vacancies are therefore very close to one another and the
energy required to produce a new one is thereby reduced. So our estimate
is, if anything, likely to be too low. Further, we have not taken the expan
sion of the lattice into account ; this would make it easier for atoms to
slip through their neighbours. In other words, the activation energy
should decrease with temperature. Again, this means that we are under
estimating the number of vacancies. Since we are dealing with exponentials,
we may be wrong by orders of magnitude. In fact it will emerge that,
numerically, any simple theory of this sort is not very useful.
In thermal equilibrium there must be vacancies present. Perhaps
unexpectedly, a perfect crystal cannot exist at a finite tempera
ture. Vacancies in turn affect the specific heat. In other words, if heat
energy is put into a body, not all of it goes into raising the temperature
but some goes into creating defects. The specific heat is thereby increased.
We will calculate this with our simple theory.
If there are N atoms, N exp( — AE/kT) are displaced and each one has
an extra energy A£ in addition to the 3/cT it has (if it is a monatomic
molecule) in the perfect lattice. So the mean energy is
E = 3NkT+AE . N Q ^ kT . (9.7)
The specific heat is given by
Cv = of = lNk + Nk(^\\^ kT . (9.8)
9.5 Point defects
289
At the melting temperature, (AE/kT) is about 4, so that the extra term is
about j R which is appreciable. We have just seen that this is likely to be
an underestimate.*
Some such rise, of about \ R, is present in the curve for argon, Fig. 5.10(6),
but it is difficult to disentangle it from the variation at low temperatures
due to quantization of the lattice energy. A comparable rise, of about $R,
is present in the specific heat of krypton (melting point 115°K) which is
clearly separated from the fall at low temperatures; but the difficulty
with this substance is that the expansion coefficient is not known with
sufficient accuracy to allow C v to be calculated. Exact comparisons are
therefore not possible for either of these two simple substances. Potassium
shows a similar rise before it melts at 335° K, Fig. 9.13.
AR
~0 100 200 300 °K
Fig. 9.1 3. Specific heat C v of potassium as function of temperature.
For ionic solids, we have already mentioned (section 8.3) when
discussing NaCl that small positive ions can probably slip more
easily through a vibrating lattice than larger negative ions; in other
words, their activation energy should be small. The concentration of
vacancies is of course very sensitive to the exact value of the activation
energy and as a result, extreme variations in concentration occur from
substance to substance. In NaCl, it is very small. In AgBr, however, it
* The term is of the form x 2 . e x and this goes through a maximum when x
decrease of A£ with temperature alters this to an increasing function.
The
290 Defects in solids : Liquids as disordered solids Chap. 9
is about 2% at the melting point, although the crystal structure is the
same as for NaCl and the ratio of radii is similar. However in AgBr the
bonds are not wholly ionic in character but partly covalent; perhaps
this decreases the activation energy significantly. The effect of the presence
of 2% of vacancies is very marked. The rise in specific heat is large,
the value just below the melting point being almost three times the
expected equipartition value for the perfect lattice.
Another premelting phenomenon will also be mentioned. The
volume becomes greater when vacancies are formed. If there is no relaxa
tion of the atoms round a vacancy, each one adds a volume v to the volume
of the crystal. The total increase of volume due to this effect is therefore
AV= Nve AE/kT .
Nv is equal to the molar volume of the substance. This expansion must
be added to that due to the anharmonicity. The extra term in the expansion
coefficient is
dT K kT 2
Let us put AE/kT equal to about 4 at the melting point. For solid argon,
7} is about 80° K and since exp( —4) is roughly 0.02, the extra term in the
coefficient is about 1 x 10 ~ 3 . Now it is observed experimentally that the
expansion coefficient at the melting point is 1.8 x 10~ 3 so that our simple
theory implies that over 50 % of the measured effect is due to the vacancy
contribution. But we have already seen (in section 8.4.2) that the Griineisen
ratio (whose constancy implies that only the anharmonicity of lattice
vibrations contributes to the expansion) decreases only by about 15 % as
the melting point is approached in solid argon. Thus our value for the
vacancy contribution must be an overestimate. In turn, this can only
mean that the extra volume created by a vacancy is much less than the
volume of one atom. It appears in fact that the neighbours relax a good
deal and only about \v is added per vacancy as the melting point is
approached.
* 9.6 DIFFUSION IN SOLIDS
We usually consider every atom in a solid to be bound to its own lattice
site and never to move from it. But this is not so ; diffusion can take place
in solids, albeit very slowly. An atom in a perfect lattice can change its
position by jumping to an interstitial position and thence to a neighbour
ing interstitial position and so on. Each jump requires a high activation
9.6 Diffusion in solids 291
energy and so this process is comparatively rare. However, in an imperfect
lattice where there are defects, diffusion is much easier. An atom can jump
into a vacancy much more easily than into an interstitial position : the
process of squeezing past its neighbours into an already existing hole
requires rather less energy than that needed to create the vacancy initially
— not much less, because of the relaxation of the atoms round the vacancy.
After the vacancy has been occupied, another is left behind, Fig. 9.14,
and the process can happen again. This mechanism must dominate the
diffusion process and is the only one we will consider.
ooooo ooooo
ooo o oo oo
ooooo ooooo
Fig. 9.14. The movement of a vacancy caused by an atom jumping into it.
The probability of an atom at A being able to jump into a neighbouring
site B is equal to
(probability of a vacancy existing at B)
x (probability of jump into vacancy).
The first of these two factors we have already calculated as exp( — A£//cT)
where A£ is the energy required to create the vacancy.
The second factor we can calculate by noting the resemblance to the
situation shown in Fig. 5.12(a) and (b), namely that in order to jump into
a hole, an atom has to squeeze past an intervening position of higher
potential energy. We need to know this activation energy. Let us call it
A£j, which it would be reasonable to expect to be about half of A£. The
Boltzmann factor is exp( — AE } /kT).
The lattice is vibrating at about the Einstein frequency v £ , and one
would expect an atom to have a maximum chance of jumping once per
cycle. So the number of jumps per second is
v e A£/fcT Q AEj/kT
In a time t it makes
y e (A£ + A£j)/kT t
jumps. Each one moves the atom a distance a . But the path is a random
292 Defects in solids : Liquids as disordered solids Chap. 9
walk and the net distance moved in s jumps is a ^/s. Hence
distance moved = a (v E e (A£+A£j)/fcT . t) 1 ' 2 . (9.9)
Now when we discussed the connection between the random walk problem
and diffusion, in sections 6.2 and 6.3, we showed that the mean distance
travelled in a medium of diffusion coefficient D in time t is, neglecting
numerical constants, (Dt) 1 ' 2 . We can therefore say that the coefficient
of selfdiffusion in a solid is given by
Dxa 2 v E e {AE+AE ' ),kT . (9.10)
Our theory therefore predicts that diffusion should become more rapid
as the temperature is raised. This is in complete contrast to diffusion in
gases, which becomes more rapid when the temperature is lowered. We
will see that this prediction agrees with observation and that the form of
the variation with temperature is correct — but quantitatively, agreement
is not good.
9.6.1 Comparison with experiment
In section 6.2 we described two solutions of the diffusion equation which
are the bases of all measurements of coefficients of selfdiffusion. One was
used in a set of experiments to measure D for copper. The idea was to
follow the diffusion of the radioactive isotope 64 Cu through a single crystal
of 63 Cu. The isotope was converted into copper nitrate and then electro
deposited on to a flat surface of the single crystal as a layer about 50 A
thick. 64 Cu, the only available radioactive isotope, has a halflife of only
13 hours. This meant that the temperature had to be raised so that it
diffused a measurable distance in a time not too long compared with 13
hours. The specimen was therefore kept at as high a temperature as
possible, in the region of 1,000°K, not very far below the melting point,
where D is of the order of 10" J1 cm 2 sec" l so that in 1 day (10 5 s), the
mean depth of diffusion y/(Dt) was about 10" 3 cm. To find out how far
the isotope had actually travelled, the specimen was put in a precision
grinding machine and extremely thin layers ground off the surface. The
accuracy of this operation was about 10 ~ 4 cm. The amount of radioactive
tracer in the metal taken from each layer was determined by counting the
betaray activity. Knowing the depth of the layer below the original
surface and allowing for the decay of radioactivity with time, D could be
calculated. A plot of log D against 1/T is a straight line, as predicted,
Fig. 9.15.
Measurements were also made on solid argon by following the progress
of the stable isotope 36 A through large crystals of 40 A. The isotope was
introduced into the vapour — at the temperature chosen, the vapour
9.6 Diffusion in solids
293
pressure is large— and some atoms entered the solid and diffused inwards.
Thus the concentration of 36 A in the vapour decreased as time went on,
and this was followed by continuously analyzing the vapour with a mass
spectrometer. In a typical run, the vapour pressure was several centimetres,
the initial concentration of 36 A was 12% and it decreased to about half
in an hour.
10 8
10"
10'
10
10"
1300 1200
1100
1000
900 °K
0.0008
0.0009
0001
00011
VT
Fig. 9.15. Plot of log D against 1/Tfor copper. The temperature is marked
along the top. Data from Kuper and others, Phys. Rev. 96, 1224 (1954).
Diffusion can take place rapidly along grain boundaries, since these are
liable to be regions with large numbers of defects. Diffusion through
polycrystalline specimens is therefore likely to be spuriously large, and
in all these experiments the specimens were examined to make sure that
they were single crystals.
For metals, the experimental result is indeed of the form
D = D Q AolkT (9.11)
294 Defects in solids : Liquids as disordered solids Chap. 9
where the constant D and the activation energy A are typically of the
orders of magnitude l10cm' 2 /sec and 1 eV respectively. (Copper, for
example, from Fig. 9.15 has D = 0.5 cm 2 /s, A — 2.0 eV). These magni
tudes give D ~ e" 40 = 10 17 cm 2 /s at room temperature, which means
diffusion through a mean distance of 1 mm in 10 15 seconds, 30 million
years, at room temperature.
For comparison, our expression gives a^v^ for the constant, and we
can take a , the lattice spacing, to be 2.5 A and v E , the Einstein frequency,
to be of the order of 10 13 cycles per second. Hence we would predict
D ~ 6 x 10 ~ 3 cm 2 /sec, which is 100 times too small. For the activation
energy, we would expect about 3e or 0.4 eV, which is a factor 2 or 3 too
small. (This has a catastrophic effect on the value of the exponential term.)
The experiments on argon reveal even worse discrepancies. They give
D ~ 4 cm 2 sec" l , A ~ 0.11 eV. For an atomic spacing of 2 A and an
Einstein frequency of 10 12 cycles per second we would expect D to be a
factor 10 4 smaller and an activation energy 3 times smaller than that
observed. It is however possible to advance plausible reasons for these
discrepancies. The activation energy can be expected to decrease if the
mean distance between atoms is increased ; this happens as the temperature
is raised. We ought therefore to get somewhat better agreement with
observation if we compared our predictions with measurements at
constant density — measurements, that is, at high pressures which counter
act the thermal expansion. We return to this point in section 9.7.3. For
solids, this would require the application of extremely high pressures
and no measurements have been made. But in the absence of suitable
measurements we can postulate some sort of decrease of the activation
energy with temperature. A decrease by a factor 2 by the time the melting
point is reached would indeed greatly reduce the discrepancies by orders
of magnitude. However, we will not pursue this but will be content that
the form of the expression for the diffusion coefficient is correct.
9.6.2 Diffusion and electrical conduction in ionic crystals
A most interesting related phenomenon is that ionic crystals, usually
considered to be insulators, can conduct electricity because of the diffusion
of ions under the influence of an electric field. (In metals the same phe
nomenon occurs but it contributes a negligible current compared with
that carried by the electrons : in ionic substances the movement of the ions
is the only possible process.)
The potential energy of a vacancy or an interstitial ion as it moves
through the lattice is shown in Fig. 9. 16(a). The situation is analogous to that
in Fig. 9.14 except that we expect the changes of energy to be higher than
those in a lattice of neutral atoms because of the electrostatic attractions
9.6 Diffusion in solids 295
and repulsions. When an electric field acts across the crystal, the energy
of the ion is changed. If a voltage V acts between the faces of a slab of
thickness /, the electric field is V// and the potential energy of a charge e
coulombs changes by V ea /l when it moves a distance a . If V// is expressed
in volts/metre and a is expressed in metres then the energy is measured
in joules. The potential energy of an ion or a vacancy as a function of
position is then shown in Fig. 9.16(6).
i i AVeaJ ,.. „
Fig. 9.16. (a) Potential energy of a vacancy or interstitial ion at different
positions in a lattice, (b) The same in the presence of a potential gradient.
If the vacancy is in a potential well, the height of the barrier it must
overcome in order to jump 'downhill' in the direction of the field is
(AE j — ^Vea /l) where a is the interatomic spacing. To j ump in the opposite
direction against the field, the activation energy is (AEj+^Veao/l).
Therefore the probabilities of jumping in the two directions are respectively
proportional to
exp (AE } ^Vea /l)kT and exp (AE } +^\ea /l)/kT
which makes it more probable that the ion will move in the direction
of the field than against it. In other words, an electric current i flows, pro
portional to the difference between the two probabilities :
i oc exp(AE } /kT)[exp$Vea /lkT)exp(±Yea /lkT)]. (9.12)
Now \ea /l is very small compared with kT under normal conditions.
For example, if \/l is large, a kilovolt per centimetre, then putting
e~l(T 19 C, a ~2A, Yea /l~ 10" 24 J, whereas /cT~lCr 21 J at
room temperature. Since e* is equal to (1 +x) when x is small, the current
ix j^f e ~ AEjlkT  (913)
To find a complete expression for the current, we should find the total
number of ions (or vacancies) jumping in the direction of the field per
second, and then multiply by the charge on the ion. The important results
can however be seen at once : that the current is proportional to the voltage
296
Defects in solids : Liquids as disordered solids Chap. 9
gradient (Ohm's law holds) and that the conductivity (which is the ratio
of the current to the voltage) contains the same Boltzmann factor as the
diffusion coefficient.
It should be remarked that both positive and negative interstitial ions
and also positive and negative vacancies can contribute to the conductivity,
but it is to be expected that the mechanism with the smallest activation
energy is likely to dominate the conductivity.
We can therefore find the activation energy for diffusion of ions by
two quite independent methods. We can use tracer methods to determine
the diffusion coefficient, or we can find the electrical conductivity as a
function of temperature. A plot of logD against 1/T and a plot of
(log cr + log T) against 1/Tshould be straight lines with the same gradient.
In practice, the conductivity of many ionic solids changes by a factor 10 6
when the temperature changes by a factor 2 ; in other words, if we use logs
to the base 10, log a changes by 6 while log Tonly changes by 0.3. Thus it is
accurate enough to plot log a (by itself) against 1/T over a restricted tem
perature range and to take the mean gradient of that graph as the activa
tion energy for electrical conduction.
Curves for NaCl are shown in Fig. 9.17. The resistivity is of
the order of 10 3 ohm cm near the melting point, 10 6 ohm cm at 800° K.
To measure high resistivities of this kind accurately, a high voltage is
applied as a pulse and the current is measured as quickly as possible.
In the diagram, only high temperature observations have been plotted,
above about 800° K (1/Tless than about 0.0013). Below that temperature
the behaviour changes and both the conductivity and the diffusion
coefficient become dependent on the grain size and the presence of
~ E 10°
o
**■
o v
MO
M.P
^10 3
itf
10"
'10°
0001 0.0011 0.0012 0.0013 1/7"
MP
0.001 ooon 0.0012 0.0013 vr
1000 900 800 °K
1000 900 800 °K
Fig. 9.17. Diffusion coefficient and electrical conductivity of NaCl as functions
of temperature, (log 10 l/T plots). Data from Mapother, Crooks and Maurer,
J. Chem. Phys. 18, 1231 (1950).
9.7 Diffusion in liquids 297
impurities. Further, it is thought that positive and negative vacancies
can migrate together, acting as an electrically neutral pair whose move
ment contributes to the diffusion process but not to the conductivity.
Concentrating, then, on the high temperature region, we see that the lines
are almost parallel. The activation energy is nearly 2 eV per ion, and we
get a consistent picture of both the conduction and the diffusion if we
assume that the small Na + ions are the most mobile and dominate both
processes.
* 9.7 DIFFUSION IN LIQUIDS
We will now apply this theory, developed to account for transport
through disordered solids, to liquids. This follows the plan outlined in
section 7.2, Fig. 7.6.
If this approach is correct, we would expect the coefficient of self
diffusion through liquids to be of the same form as that for solids,
D = D Qxp(A /kT). Of course, whenever we find it convenient we can
regard liquids as derived from dense gases. This approach would lead us
to expect D to be proportional to T, which is quite different behaviour.
It is in fact reasonable to expect that when the density of the liquid is
low, some features of its behaviour might be 'gaslike', but at high densities
some features might be 'solidlike'. In the meantime, let us see whether
the solidlike model of a liquid ever applies at all.
We would expect the activation energy AE } for an atom to jump into a
vacancy in a liquid to be much smaller than in the corresponding solid,
because there is a 10 % expansion in volume between liquid and solid ;
small changes of interatomic distance have a large effect on the activation
energy. We would also expect AE, the energy of creation of a vacancy,
to be smaller than in the solid.
The diffusion coefficient of liquid argon (radioactive 37 A through 40 A)
has been measured using yet another procedure, appropriate to liquids,
based on section 6.2. A narrow capillary was made, closed at both ends
by needle valves. The top one was opened and pure liquid 40 A was con
densed in; the valve was closed again. Then liquid argon containing a
small percentage of 37 A was condensed outside. The bottom valve was
opened for a known time, during which some 37 A diffused into the capil
lary. Convection, which might have disturbed the results grossly, was
discouraged because the capillary was very narrow. Afterwards, the
liquid in it was pumped away and the total amount of 37 A determined
from its radioactivity. Knowing the time and the concentration in the
surrounding liquid, D could be calculated. The measurements give
D = 6.1 x 1(T 4 exp(0.027//cT)
298
Defects in solids : Liquids as disordered solids Chap. 9
where (kT) is to be measured in electron volts. The activation energy is
indeed about 5 times smaller than in the solid and the form of the
expression justifies our regarding a liquid as resembling a highly dis
ordered solid in some of its properties. We will now construct a theory of
the viscosity of liquids, and we will do this by showing that there is a
connection between the viscosity n and the diffusion coefficient D. Since
we already know D, this relation allows us to estimate n.
* 9.7.1 Stokes' law
As a preliminary study, interesting for its own sake, we will estimate the
force on a sphere of radius r which moves at constant velocity through a
liquid of viscosity n. We can concentrate on a few little specks of liquid and
follow their displacements as they go past the sphere. These define stream
lines and they represent the way the liquid flows past the sphere, or the
way the sphere pushes the liquid apart in order to pass through it. Fig. 9.18
shows some of these. It implies that the flow pattern is symmetrical and
that at some distance away from the sphere the liquid is hardly disturbed.
This distance is called the penetration depth. We will make the assumption
that the penetration depth is the same order as the radius of the sphere,
which is reasonable and is what the diagram implies. We saw in section
6.1.5 that the pressure on a moving object is given by (Eq. (6.7))
pressure = rj x velocity gradient.
Here, the velocity gradient is of the order of magnitude v/r.
Fig. 9.18. Stream lines round a sphere moving
through a liquid.
9.7 Diffusion in liquids 299
Also implied in Fig. 9.18 is that the flow is nonturbulent. In the language
of section 6.1.5, the kinetic energy of the mass in motion— the inertial
energy— must be small compared with the energy dissipated against
viscosity. This means we have to compare terms of the type \pv 2 (kinetic
energy per unit volume, p being the density and v the velocity of the fluid
in motion) with the viscous pressure nv/r (which is also an energy per unit
volume). Thus pv 2 must be less than nv/r to make sure that the flow is not
turbulent ; in other words, v must be less than rjr/p. For a sphere of radius
10 cm moving through water for which n = 10~ 2 gm/cms, v should not
exceed 0.1 cm/s; a spherical particle of radius 10" 3 cm can travel at 10
cm/s and, extrapolating, a sphere of atomic dimensions can travel at
10 6 cm/s and the flow will still be nonturbulent.* With these assumptions,
the pressure on the sphere is nv/r. The surface area of the sphere is Anr 2 .
Therefore the force on the sphere should be roughly Annrv.
An exact analysis with proper regard to the flow pattern shows that
force = 6nnrv (9.14)
This is called Stokes' law.
9.7.2 Einstein's relation between viscosity and diffusion
We establish the relation between viscosity and diffusion by studying
the Brownian motion of an assembly of particles. For clarity, we will
consider the special situation of the fluid in the field of the earth's gravity,
although it is not in fact necessary to restrict the calculation in this way.
Further, for the sake of illustration we will first consider a gas and having
shown that the results agree with previous calculations we will extend the
method to liquids. We begin, then, by considering a gas in a gravitational
field, as we did in section 4.4. There, we established the fact that the
concentration n at a height z is given by
n = noQ m^T ( 413)
where n is the concentration at zero height and m is the mass of one
molecule. The new idea is that we will regard the equilibrium as being
produced by two opposing motions. First, we have the tendency of each
atom to fall downwards. Secondly, opposing this, we have the tendency
of each atom to diffuse along the concentration gradient (which means a
tendency to move upwards, because n decreases with height). We express
this balance as follows. Let us take z to be positive upwards, and con
sistent with this let us take the drift velocity v z of the atom to be positive
* The ratio vrp/rj where v is the limiting velocity for the onset of turbulence is called the
Reynolds number. It is about 1 for a sphere. It is much smaller for a cylinder because the
penetration depth is much greater.
300 Defects in solids : Liquids as disordered solids Chap. 9
upwards ; we will also take all fluxes of atoms (numbers crossing 1 cm 2
in 1 second) as positive upwards. Then the net flux is given by
J = nv z D—. (9.15)
The first term on the right hand side is the number of atoms per unit
area falling downwards under gravity — they are accelerating of course,
but are brought to rest at intervals, and v z is their mean velocity. The
second term is the rate of diffusion down the concentration gradient.
For equilibrium, J = 0.
Let us next evaluate v z , the mean drift velocity. If X is the mean free path
between collisions, the mean free time between collisions is (X/c) av which
is nearly equal to X/c, where c is the mean speed. In this time, every atom
acquires an extra downward velocity equal to (acceleration) x (time), that
is gX/c. Let us assume that after every collision, each atom is stationary.
This is rarely likely to be true, because of the persistence of velocities (sec
tion 6.4.3) but it only introduces an error by a factor of order unity. With
this assumption, the mean velocity downwards is half the terminal velocity,
that is gX/2c. Further, let us put in the expression for n. This allows us
to evaluate the nv z term completely. Finally, by differentiating we evaluate
the dn/dz term. We find
0= ~n e mgz/kT + D^n e mgzlkT
2c kT
which gives
D = 2mc < 9 " 16 >
If we put kT = \mc 2 (not paying too much attention to the distinction
between mean and r.m.s. speeds), we get
D = ±Xc.
The more exact expression arrived at in Appendix B is ^Xc. This is practi
cally our result and the discrepancy is due to our crude averaging. This
digression has therefore justified the idea of regarding the equilibrium of
the gas as a balance between a drift velocity in one direction and a diffusion
in the opposite direction down a concentration gradient.
Let us now return to the real subject of this section and use the same
method for particles suspended in a liquid, undergoing Brownian motion —
resin particles in water for example. We have already studied this system in
9.7 Diffusion in liquids 301
section 4.4.2, when we concluded that the concentration is given by
exactly the same law :
n = n exp(m*gz/kT) (4.17)
with m* the effective mass, corrected for the buoyancy of the surrounding
liquid.
Again in equilibrium we have a balance between drift downwards and
diffusion upwards :
= _„„ X D^. (9.17)
dz
Let us assume here that the force on the particle is given by Stokes' law.
Then the particle (of radius r) reaches a terminal velocity when the viscous
force (6nnrv z ) is equal to the weight (m*g) : that is,
m*g
6nnr
Then our equation reads
= ^n e^ T + D^n e'"^
onrjr k 1
which gives
kT
D = — . (918)
dnnr
This is the diffusion coefficient of the particles of radius r through a
medium of viscosity n at temperature T. Finally, let us assume that the same
approach is valid when the particle is an atom of the liquid itself. As we
have presented it, this is a rather controversial step to take. There are two
points which we have to consider. First, are we justified in applying Stokes'
law for the force on the moving atom? In deriving this law we clearly
regarded the sphere as immersed in a continuum, a medium without any
atomic structure, in which it generated stream lines. This hardly seems
justified if the sphere itself is of atomic size. But the mean free time between
collisions is certainly very short— in contrast to the situation in a gas— so
that a continuum is not too bad an approximation. Further, we have seen
that Stokes' law implies that the penetration depth of the disturbance
caused by the passage of the sphere is between 1 and 2 atomic radii : this
is likely to be roughly correct, so that perhaps Stokes' law can be applied.
Secondly, are we justified in talking of the effective mass m* of an atom
inside a liquid of identical atoms? An incompressible liquid in a gravita
tional field has the same density at all heights (that is, the scale height is
302 Defects in solids : Liquids as disordered solids Chap. 9
infinite) and this is equivalent to saying that m* is zero. Admittedly, m*
does not appear in the final relation, but it is always dangerous to cancel
both sides of an equation by a quantity equal to zero. Perhaps the best
justification of this step is to say that real liquids are compressible so that
there is a small change of density with height and m*, though very small,
is not zero.
We say then that the relation
kT
6nrjr ^ ' '
where r is the radius of an atom, relates the diffusion coefficient and the
viscosity of a liquid. This is called Einstein's relation between the two
quantities.
^k 9.7.3 Viscosity of liquids
Using the expression for the variation of diffusion coefficient with
temperature
D = D e~ Ao/kT (9.11)
where A is the activation energy per atom, Einstein's relation gives
kT
We can test this relation by comparing with experiment. It predicts
that the viscosity decreases as the temperature is raised, in contrast
to the behaviour of gases.
Among the most interesting measurements of the viscosity of liquid
argon are those shown in Fig. 9.19. They were performed in a very simple
way. The liquid was contained in a metal tube, through which a cylindrical
weight could fall very slowly, the clearance between tube and cylinder
being only small. The cylinder had a magnetic core inside it and its
position could be detected magnetically. The time required to fall a known
distance was determined. This was calibrated by timing the fall of the
cylinder using liquids of known viscosity. The apparatus was robust and,
as we shall see, was used up to very high pressures.
On the diagram, the two vertical dashed lines represent the triple point
(83° K) and the critical temperature (150°K). The region between them
represents the range of temperature in which the liquid can exist.
The full curve AB represents the viscosity of argon in equilibrium with
its vapour pressure. At A, the liquid is at a low pressure, has a density
of 1.37 g/cm 3 , at B it is under 48 atmospheres and its density is 0.70 g/cm 3 .
9.7 Diffusion in liquids
303
100
200 300 °K
Temperature
Fig. 9.19. Viscosity of liquid argon as a function of temperature. Line AB:
measurements on the liquid under its saturated vapour pressure, from the
triple point to the critical point. Dashed line: calculated from D using
Einstein's relation, Eq. (9.18). Line AC: measurements at constant density,
1.37 g/cm 3 — the part to the right of the critical point refers to the gas. The
descending curve shows liquidlike behaviour. Line BD; measurements on
gas at constant density, p = 0.70 g/cm 3 . Data from Zhdanova, Soviet Phys.
J.E.T.P. 4, 749(1957).
The dashed curve close to it is calculated using the experimental values
of D (section 9.7) and Einstein's relation, assuming an atomic radius of
2 A (or, what comes to the same thing, a rather smaller radius together
with a larger number than 671 in the expression for the force on a moving
atom). The agreement is good, and it shows that Einstein's relation is
valid within a factor close to unity. The upper curve AC represents
measurements at constant density equal to 1.37 g/cm 3 (the density at the
triple point temperature). For each measurement, the temperature was
first raised to a chosen value and then the pressure was increased until
the density reached the value 1.37 g/cm 3 ; the viscosity was then deter
mined. Thus for all points along this curve, the mean distance between
304 Defects in solids : Liquids as disordered solids Chap. 9
atoms was the same. This is the kind of experiment, mentioned in section
9.6.1, which might be expected to give results more consistent with
theoretical estimates. The activation energy calculated from this curve*
is a factor 2 or smaller than that from the curve AB where the density
varies — this is certainly a change in the right direction. The measure
ments at this density, 1.37 g/cm 3 were continued above the critical
temperature ; thus the part of the curve AC to the right of the dashed line
at 150°K represents measurements on the gas. The pressure needed at
250°K to keep the density at 1.37 g/cm 3 is about 2,000 atmospheres.
There is no discontinuity in viscosity when the liquid changes to the very
dense gas — which means that we have a gas in which the transport of
momentum proceeds by the same mechanism as in a liquid. This must
mean that in these circumstances the atoms progress by a series of short
jumps into holes in the assembly, and not by a series of long free flights
terminated by collisions.
By contrast, the line BD is the viscosity of the gas held at the lower
constant density of 0.70 g/cm 3 , the density at the critical point. The
viscosity now increases with temperature, which we might call gaslike
behaviour although y\ is not proportional to ^/T as in a wellbehaved gas.
This must mean that the atoms now spend a considerable fraction of their
time in free flight between encounters.
Measurements were also taken of the viscosity of the liquid at small
densities, just above 0.70 g/cm 3 . The curves, above BD and almost parallel
to it, have not been shown. They also tend upwards which means that in
the liquid near the critical point there are sufficient large holes in the
structure for the motion of the molecules to be gaslike.
PROBLEMS
9.1. (a) Estimate the extra energy per unit length (in erg/cm) of an edge dislocation
in a metal due to the low coordination and increased interionic spacing
of some of the ions.
(6) A specimen of metal has 10 9 dislocations per cm 2 of cross section. How
much energy is stored in the dislocations per cm 3 ? Will this extra energy
appear as a reproducible contribution to the specific heat, like that due to
vacancies — in other words, is the number of dislocations a unique, function
of the temperature? If not, why not?
(c) Work hardening is due to the presence of large numbers of dislocations
tangled together and repelling one another, so that large stresses must be
applied before they can move. The extra stress (dyn/cm 2 ) must be of the same
order as the energy density (erg/cm 3 ) due to the dislocations. Estimate the
concentration of dislocations at which significant work hardening takes
place in a typical metal at a strain of 1 %.
* Determined by plotting log(>j/r) against l/T.
Problems 305
9.2. Xray examination of a small region of a crystal of cubic symmetry and lattice
parameter 4 A shows that it consists of two slightly misaligned crystals. The
angle between the planes is 4o° What pattern of etch pits would be expected
along the grain boundary? See Fig. 9.7(d).
9.3. The mobility q of a charged particle (such as a vacancy or an interstitial ion in
an ionic solid) is defined by the equation
mean drift velocity
electric force Ee acting on charge
where E is the electric field and e is the charge on the particle. The problem is to
show that there is a relation between q and the diffusion coefficient D, using a
method analogous to that in section 9.7.2.
(a) Consider a steady electric field E acting across a slab of material, considered
to be a continuum without atomic structure. A positive charged particle
inside the slab is attracted to the negative face and vice versa. What is its
potential energy at a distance h from the face compared with the value at the
face? In equilibrium, what is the probability of finding the particle between
h and (h + dh)l Show that the charge is distributed exponentially with h,
with a scale length equal to kT/Ee. Estimate this length at room temperature
in a field of 1 volt/cm for e equal in magnitude to the electronic charge,
remembering that kT at room temperature is 1/40 eV.
(b) Does this result conflict with the predictions of elementary electrostatic
theory about charge distributions?
(c) Set up a differential equation expressing the equilibrium as a balance
between diffusion down the concentration gradient and a steady drift down
the electric field. Apply this equation to the system discussed in (a). Deduce
Einstein's relation : q = D/kT.
306 Solutions to problems
SOLUTIONS TO PROBLEMS
Chapter 3
3.1 (a)
3.2 (a)
GMm
2 HqM 2
3 h*
(b)
GMm
(d) See Fig. P.2 (e)
(c) See Fig. P. 1 (d)
2fi M 2 \ 1/4
GM
mg
CO 2tt
,5/4
3.3 (d) See Fig. P.3.
2 5/4 /^ M 2 \ 1/4 _ / 2 /^
fc
a{a+h)
Fig. P.l
Fig. P.2
Fig. P.3
(*)
surface tension energy 47t 2 y
, where y ~ 10 dyn/cm, p ~ 1.4 g/cm 3 ,
gravitational energy A z pg
g ~ 10 3 cm/s 2 ; 210 c/s ; 21 cm/s.
3.4 28 ; 0.81 g/cm 3 ; 34.5 cm 3 ; ~ 3.9 A ; 8.8 dyn/cm ; 0.0034 eV
3.5 (a) 28.5 cm 3 , 3.55 A (b) 3.95 A (c) 0.0134 eV (note : n = 12)
{d) 0.01 14 eV (e) 0.0124 eV
(/) 2.5 x 10 10 dyn/cm 2 , compared with ~2.2x 10 10 erg/cm 3 or 1.7xl0 10 erg/
cm 3 according to value of L .
(g) 83 or 66 (h) ~l.lxl0 12 c/s
3.6 Assume closepacked discs in planes, n = 6, e = £/30 within planes ; n = 2 ,
e — between planes.
(a) L = l.lNEpermol
{b) If discs parallel to surface, JT = {l/ljlr 2 ) (see Fig. P.4; each triangle con
tains \ disc and has area V3r 2 ) and y ~ 0.1£/r 2 . If perpendicular to surface,
jV = l/(2r x r/10), and y = 0.25 E/r z . Surface tension somewhere between
these two.
Fig. P.4
(c) Tendency for parallel orientation because lower surface energy.
3.7 a = 2.32 A N = 5.8 x 10 23 assuming r" 10 repulsions.
Solutions to problems
307
3.8 Binding energy =
Nae 2
0.37 A
3.9 (a) 4.55 eV
(b) If solvents were structureless media, binding energy would be reduced by
factor 80, 25, 18 respectively. But to screen one ion from another, solvent
molecules must crowd round ions and size is important.
3.10 (a) 2.404 (b) 1.803
3.11 12A;9A;3.3A;7A
3.12 K = 22.1 xlO 4 kg/cm 2 .
Comparison with
1 (M 14/3
10 I V\
>V 4/3
v
shown in Fig. P. 5. Theory should be quite accurate.
PfK
Fig. P. 5
3.13 _L[6a y/m] 1/2 gives 3.4 x 10 13 , 0.86 x 10 13 and 1.56 x 10 12 c/s.
2n
Chapter 4
4.1 Effective mass 3.8 x 10" 14 g, scale height 10.0 fi. N = 6.47 x 10 23 .
Inverted concentration gradient with same scale height.
Chapter 5
5.1 (a) 0.0022 A
(b) 0.0104 A if other causes of broadening are independent of temperature.
, x GMm GMm „, / GMm \
5.2 (a) ,; — (6)exp +w
308 Solutions to problems
(c) A4nr 2 exp(GMm/rkT) dr, where A is a normalizing factor.
(d) exp factor *■ 1, so that whole expression » oo when r *■ oo.
(e) (ii) ; (c) cannot be normalized.
5.3 Centre of gravity rises by R/Mg for 1° increase of temperature so that gravi
tational potential energy increases by R.
5.4 31 % assuming that at 1 atm, both N 2 4 and N0 2 act like perfect gases.
5.5 11.2 km/sec; 1.37 km/sec.
Highspeed 'tail' of Maxwell distribution allows more He atoms than 2 or N 2
atoms to escape ; see problem 2 above, part (e).
5.6 (a) (u x — 2£) ; \mu 2 x  ImuJ,
(b) nAu x di • P[u x ] du x ; nAu x dx • P[u x ] du x (  ImuJ,) ;
2nmdV J£ u 2 x P[u x ] du x = nkTdV
(c) iknV;dT= dVkTftknV
(d) TV 213 = const., becomes PV 513 = const, since PV = RT
(e) dT= dVkT/C v V where C v is specific heat per molecule.
(/) See discussion in section 5.1.2.
5.7 (0 A sin 6 dd d</> exp(((iH/kT) cos 0), A a normalizing factor.
(ii) Integral over gives lit. Integral over 6 gives
2sinhx , fiH x I
where x — r=, whence A =
x
kT' 4n sinhx
(Hi) — r: — exp(  x cos 6) ■ sin 9 dd
2 sinhx
x
(iv) Nu cos 6 ^— — — exp(  x cos 6) d(  cos 6)
2 sinhx
(v) Integral over 6 gives iv>[coth x 1/x]
5.8 (a) 5 x 10 12 c/s. (b) 1.89 x 10~ 3 <5 2 J, where 8 in A
(c) p[S] dd = 0.304 exp( 34.2<5 2 ) d^ where 8 in A
(d) 0.12 A
5.9 (a) 2 (b) kT (c) 2.9 x 10" 4 rad (d) (v) (e) (Hi)
(f) Bombardment by photons of thermal radiation.
5.10(a) 6 x KT 39 gem 2 (b) \kT (c) 3 x 10 12 c/s
(d) 6 vibrational + 3 rotational degrees of freedom ; f R
(e) 3 translational + 3 rotational degrees of freedom ; f R
(/)f
(g) Rotations (c) take place, rotation about dumbell axis does not ; 5 degrees of
freedom ; y near ^.
5.11 1.4
5.12 0.3 eV (b)
Chapter 6
6.1 (a) C v  20.8 J/mol deg, a ~ 35 x 10" 16 cm 2 , c ~ 4.5 x 10 4 cm/sec ;
k; ~ 1 x 10 ~ 4 watt/cm deg
(b) 0.175 watt (c) 2xKT 3 m
(d) k = %n(CJN)cd where d is spacing ; 2 x 10 ~ 4 m
6.2 (a) (DA/h)mn v
Solutions to problems
309
(/,) _ Pl a dh/dt g/s where A is area of tube and p L is density of liquid
(c) h 2 =
2Dn„m
Pl
t
(d) 0.11 cm 2 /s. Assume collisions with air molecules are much more frequent
than with other ether molecules; then X = l/n A a E , where subscript A means
air, E ether; D EA = %Xc E ; whence a E = 10" 15 cm 2 .
6.3 Assume only collisions with air molecules are important ; X = l/n A <r where n A
= number of air molecules/cm 3 . Take a = 10 x 10" 16 cm 2 . For small thermionic
tube, dimension ~ 1 cm, volume
10 3 cm 3
1 cm 3 ; for large transmitting tube, dimen
Then P ~ 0.02 mm, 0.002 mm and area of
sion ~ 10 cm. volume
hole ~ 10" n cm 2 , 10" 12 cm 2 respectively (Knudsen flow).
6.4 Maxwell distribution inside vessel. All molecules within cone of height c and
solid angle determined by shape and size of hole will leave vessel in 1 sec;
whence factors c and G respectively.
G[*\mc 2 cP[c]dc
Mean kinetic energy = ° G ^ cP[c]dc •
Denominator is equivalent to inc.
Expression GcP[c] dc for number escaping per sec shows that fast molecules
predominate and distribution is nonMaxwellian.
Chapter 7
7.1 (a) T i = 2 iT c ;220°K
(c) T B = $Ti = 110°K, so that at room temperature PV > RT.
7.2 V. = 2b ■ T. = ^; P. = ^ ~ 3^; BiT) = (fc^LJ ; % = ^
7.3 (a) 200 cm 3 ; 4.4 A
(b) B plotted against 1/T; B = 280 (2.18 x 10 5 )/Tis a rough fit— see Fig. P.6. ;
a ~ 1.83 x 10 6 J cm 3 /mol ; using a/b ~ 2.7Ne, e ~ 0.025 eV.
Fig. P.6
310 Solutions of problems
(c) 0.036 eV
W " = (^4^]^^= — l^MxlOCo,
(f) 1.6xlO" 20 Cm; +5 electron charge 2 A apart, or larger charges nearer
together.
7.5 (a) zero (b) r.m.s. distance = ^/N/ (c) D = ?(/ 2 /t)
7.6 (a) x = (b) — (c) *
a V a z
Chapter 8
8.1 6; 4.82 A; 16.1°, 33.6°, 56.1°
8.2 Bragg's law gives dd/d — — d0/tan0 for small changes of d and 0; a = 10~ 5 .
Expansion coefficient varies with direction.
8.3 (b) 16xl0 _6 ,forp= 11
(d) y = 0.85 at 30° K, 1.28 at 65°K, 1.45 at 283°K compared with 1.67 for p = 1 1.
Chapter 9
9.1 (a) ~ 10" 4 erg/cm
(6) 10 5 erg/cm 3 . No, number depends on mechanical and thermal treatment of
specimen,
(c) Elastic energy = (^/2)s 2 ~ 10 8 erg/cm 3 ; 10 12 dislocations/cm 2
9.2 DD 1 (Fig. 9.7(d)) is given by a/DD 1 = sin where 6 is angle between grains,
a is lattice parameter. Line of etch pits 4460 A apart.
9.3 (a) Veh/l ; Fe//feTexp(  Veh/lkT) dh, if / > h ; ^ cm.
(6) Yes. Elementary electrostatics neglects the diffusion of electrons, because
electric charge is treated as a continuous fluid.
(c) Ddn/dh + qn(Ve/t) — 0, where n = no. of charged particles/cm 3 ; this gives
n oc exp[ — q(Veh/lD)].
Reading List
The following list contains a few suggestions for complementary and further
reading.
GENERAL
R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics,
Vols. 1 and 2, AddisonWesley, Reading, Mass., 1963.
T. L. Hill, Lectures on Matter and Equilibrium, Benjamin, New York, 1966.
F. Mandl, Statistical Physics (Manchester Physics Series), to be published by John
Wiley, London.
F. Reif, Statistical Physics (Berkeley Physics Course, Vol. 5), McGrawHill, New
York, 1967. (Mainly relevant to chapters 49 of this book.)
F. O. Rice and E. Teller, The Structure of Matter, Science Editions, New York, 1966.
D. Tabor, Gases, Liquids and Solids, Penguin Books, Harmondsworth, Middlesex,
1969.
M. W. Zemansky, Heat and Thermodynamics, 5th ed., McGrawHill, 1968.
CHAPTER 3
A. H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964.
C. Kittel, Introduction to Solid State Physics, 3rd ed., Wiley, New York, 1966.
CHAPTER 4
R. D. Present, Kinetic Theory of Gases, McGrawHill, New York, 1958.
E. Schrodinger, Statistical Thermodynamics, Cambridge University Press, Cam
bridge, 1964.
H. D. Young, Statistical Treatment of Experimental Data, McGrawHill, New York,
1962.
312 Reading list
CHAPTERS 5, 6 and 7
R. D. Present, Kinetic Theory of Gases, McGrawHill, New York, 1958.
CHAPTER 8
B. E. Chalmers, Principles of Solidification, Wiley, New York, 1964.
C. Kittel, Introduction to Solid State Physics, 3rd ed., Wiley, New York, 1966.
F. C. Phillips, An Introduction to Crystallography, 3rd ed., Longmans, London, 1966.
CHAPTER 9
F. P. Bowden and D. Tabor, Friction and Lubrication, Methuen, London, 1967.
B. E. Chalmers, Principles of Solidification, Wiley, New York, 1964.
A. H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964.
J. Frenkel, Kinetic Theory of Liquids, Dover, New York, 1955.
C. Kittel, Introduction to Solid State Physics, 3rd ed., Wiley, New York, 1966.
The following films deal with experiments which are described in this book.
During the course of each film, measurements can be taken from the screen so that
students can write down their own readings and later work out their own results.
A booklet is available giving essential numerical and other data about each piece of
apparatus.
The series is called 'experiment'.
No. 5 The Determination of Boltzmann's Constant (Chapter 4)
No. 3 C p /C v for Helium, Nitrogen and Carbon Dioxide (Chapter 5)
No. 4 The Effect of Pressure on the Thermal Conductivity
of a Gas (Chapter 6)
Nos. 1 and 2 pV Isotherms of Carbon Dioxide (Chapter 7)
Films (16 mm sound colour, 15 minutes each) can be purchased from Granada
International Productions Ltd., 36 Golden Square, London Wl, or hired from The
British Film Institute Distribution Library, 42/43 Lower Marsh, London SE1.
ndex
References such as 304(9.1) are to Problems, where information is given which is not in
the text.
Activation energy 127, 222, 296
Adiabatic changes 120
Adsorption 35
Amorphous solids 16, 20, 269
Anharmonicity 239, 253
Argon (Ar)
critical constants 180, 210, 238
density 48
latent heats 48
pVT diagram 179
triple point 180, 238
Argon gas
Boyle temperature 195
C p /C v 123, 199
diffusion 166
dimers 205
inversion temperature 203
JouleThomson coefficient 204
2nd virial coefficient 196
thermal conductivity 165
virial plots 194
viscosity 164, 166, 304
Argon liquid
diffusion 297
Argon liquid — continued
surface tension 48
tensile strength 185
viscosity 302
Argon solid
crystal structure 228, 236
diffusion 292, 294
elastic moduli 41, 48
Griineisen constant 244
melting 289
single crystals 233, 247
specific heats 124, 289
thermal conductivity 247
thermal expansion 124, 243
Atomic mass 13
units 14
Atomic number 9
Avogadro's number 14
early measurements 15, 35
Binding energy 6, 31
ionic crystals 55
metals 50
molecular crystals 32
314
Index
Boiling 180
Boltzmann factor 77, 80, 90, 94
separability 78, 81,97
Boltzmann 's constant 87
Boyle temperature 195, 203
Bragg reflections 234, 252
Brittle materials 266, 272
Brownian movement 87, 213, 299
Bubbles 180
Bulk modulus 40, 41, 50, 57, 215, 244, 263
and latent heat 43
and speed of sound 41, 46, 120
gases 16
ionic solids 57
metals 50
molecular solids 43, 48
Capillary rise 1, 35
Carbon compounds 13
Carbon dioxide (C0 2 )
critical constants 210
dimers 206
Carbon tetrachloride (CC1 4 )
bulk modulus 44
Einstein frequency 46
latent heat 33
surface tension 35
tensile strength 185
Ceramics 269
Characteristic speeds 109
Characteristic temperatures i
Characteristic times 21, 147
Chemical reactions 13, 127
Cleavage, crystal 232
Clusters 187, 207
Collision
cross section 155
dynamics 156
three body 205
Compressibility 16, 41
see bulk modulus
Coordination number 18
Copper (Cu)
diffusion 292, 294
electrons in 254
Lorentz ratio 255
specific heat 123, 255, 257
Corresponding states 187
Coulomb forces 10, 53
Covalent bonds, see electron sharing ;
see diamond, glass, silver bromide
Cracks 270
Critical constants 179, 208
Critical opalescence 210
Critical temperature 89, 179, 210, 238
Crystal
cleavage 232
faces 225
growth 231, 281
habit 225
• structure 20, 228
unit cell 228
surface energy 231
zone 227
Crystals
single 232, 247, 264, 281
whiskers 286
Degree of freedom 116
Density
gases 16
liquids 17
solids 19
Diameter of molecules 22, 29, 155, 157,
190
Diamond (C) specific heat 123
Diffusion 136, 138, 145
and random walk 150
gases 149, 161, 166, 171, 175 (6.2), 299
liquids 297, 299
of heat 245
solids 290
Dimers 204
Dislocation
edge 277
lines 275
screw 277
width 279
Doppler
broadening 101
shift 102
Droplets, condensation on 220
Ductile materials 264, 274
Effusion 167
Einstein frequency 46, 47, 50, 57, 245
and specific heat 123
Einstein relation 299
Elastic limit 264
Index
315
Elastic moduli 40
Electrical conductivity
ionic crystals 294
metals 257
Electron
charge 9
cloud 9
diffraction 233
'gas' in metals 10, 12, 254, 257, 260
sharing 10, 11
volt 7
Element 9
Energy
density 42, 191,220,299
equipartition 114, 116, 127
units 6
internal (£) 118, 197, 243, 258, 288
Enthalpy 202
Equilibrium, thermal 74, 159
Equipartition of energy 114, 116, 127
Expansion coefficient see thermal ex
pansion
Expansion of gases
adiabatic 121, 132 (5.6)
free 200
JouleThomson 201
Faraday (unit) 15
Fibre glass 274
Fick's law 138, 162
Fluctuations 63, 68, 75, 211
elastic solid 213
perfect gas 216
Force
Coulomb 10
gravitational 25
interatomic 23
short and long range 24
and potential energy 26
Fourier transform 234
Free expansion of gases 200
Free path 153
mean 155, 156, 169,246
Frequency
Einstein 46
harmonic motion 46
mixing 253
Friction 266
internal 142
Gas constant (R) 87
diffusion 149, 161, 166, 171, 299
free expansion 200
imperfect 187, 188
JouleThomson process 201
Knudsen 169
Maxwell distribution 112
specific heats 118, 123, 197
speed of sound 112, 120, 199
structure 16, 17, 23, 185, 212
thermal conductivity 140, 162, 165,
171
viscosity 16, 143, 162, 164, 170, 304
Glass
brittleness 269, 273
fibres 271
melting 16
structure 269
Goniometer 227
Grain boundaries 277, 293
Gram molecule 14
Growth spirals 283
Griineisen constant 244, 261 (8.3)
Habit, crystal 225
Hamiltonian equations 76
Harmonic
motion 44
oscillator 115, 123
Histogram 68
Hooke's law 50
departures from 50
Ice 20, 21, 135 (5.12)
Impacts, molecular on wall 105, 160, 285
Internal friction 142
Interstitial atom 286
Inversion temperature 203
Ionic substance 12, 13, 53
binding energy 55
diffusion 294
Einstein frequency 57
elasticity 57
electrical conductivity 294
Griineisen constant 261 (8.3)
melting 238
see lithium chloride, lithium fluoride,
silver bromide, sodium chloride
316
Index
Ionization
by collision 90
energy 33, 54
Ions 10,11
Iron (Fe) stress/strain curve 52
Isotherms 176
Isotope 14
enrichment 169
Joule 4, 62
(unit) 6
JouleThomson process 201
Kelvin 5, 35, 149
Knudsen gas 169, 251
Latent heat 6
evaporation 31, 192, 196, 219
melting, solidification 31
sublimation 31
Lattice, crystal 20, 228
Lead (Pb)
flow of metal 21
specific heat 123
LennardJones potential 29
used for metals 49
Liquid
adhesion to solids 35
boiling 178, 180
capillary rise 35
compressibility 17
condensation 178, 182, 220
diffusion 297
droplets 208, 220
freezing 178
latent heats 31, 192, 196
range of existence 185
rigidity 17
structure 18, 185
surface tension 34
tensile strength 183, 217
vapour pressure 177, 218
viscosity 16, 19, 142, 298, 302
see argon, carbon tetrachloride, nitro
gen
Lithium chloride (LiCl) structure 12
Lithium fluoride (LiF)
single crystals 233
thermal conductivity 253
Lorentz ratio 254, 260
Macroscopic variables 66
Madelung constant 56
Mass spectrometer 13, 205
Maxwell 143, 164
Maxwell distribution 107, 112, 260
Mean free path 155, 156, 169, 246
Mean molecular speeds 109
and speed of sound 111
Mean values 71
Melting 16, 19, 177
amplitude of vibration 238, 287
Mercury (Hg) 11, 50
adhesion to glass 37
capillary fall 38
latent heat 50
Maxwell distribution 113
melting point 238
surface tension 50
tensile strength 185
Metals 10, 12, 48, 254, 260
binding energy 50
bulk modulus 41, 50
ductility 264, 274
Einstein frequency 50
electrical conductivity 257
electrons in 10, 12, 48, 254, 260
friction 266
internal 142
Griineisen constant 244
LennardJones potential 49
Lorentz ratio 254
single crystals 264
specific heats 123, 289
surface tension 50
thermal conductivity 257
thermal expansion 243
whiskers 286
work hardening 280
see copper, iron, lead, mercury, potas
sium, steel
Metastable states 180
Mobility 305 (9.3)
Molar volume 22
Mole see gram molecule
Molecular flow 169
Molecular weight 14
Molecules 11
Mossbauer effect 102
Nearest neighbour interactions 30
Index
317
Nitrogen (N) liquid 59 (3.4)
binding energy 33
bulk modulus 44
dimers 206
Einstein frequency 46
surface tension 35
Nucleus 9
Nuclear reactions 13, 90
Ordered molecular motion 63
degradation 64, 66
Orders of magnitude 5
Penetration depth 298
Perfect gas 86, 103
pressure fluctuations 216
temperature scale 77, 86
Perrin 88, 150
Persistence of velocities 159, 167
Phase boundary lines 176
Phase diagram 177
Planck's constant 123
Plasticity 266
Point defects 286
Porous plus experiment 201
Potassium (K)
bulk modulus 50
latent heat 50
melting 238, 289
specific heat 289
surface tension 50
Potential
centrifugal 205
energy 25, 31
intermolecular 27
interionic 56
LennardJones 29, 49
square well 188
well 27
anharmonic 239
parabolic 46, 237, 239
square 188
Premelting phenomena 288, 290
Pressure
as energy density 42, 191, 220, 299
perfect gas 86, 103, 191
imperfect gas 188
Probability 69, 70
function 70
independent 73
Quantum theory 9, 123, 259
Random
molecular motion 63
walk 150
Reynolds number 299
Rigidity 16, 40
gases 16, 21
liquids 17, 21
solids 19, 21, 41
Ripples 38
Rotator
energy 115, 125
Riichhardt's method 121
Sample average 71
Scale height
atmosphere 84, 87
suspended particles 88
Scattering
energy flow in solids 249, 251
light 211
Xrays 233
Shock waves 21, 53
Silver bromide (AgBr)
specific heat 290
Simple harmonic motion
see harmonic motion
Size effects in thermal conduction 253
Sky, blue of 212
Sodium chloride (NaCl)
binding energy 55
bulk modulus 41, 57, 61 (3.12)
diffusion 296
Einstein frequency 57
elastic moduli 41
electrical conductivity 296
melting 238, 289
single crystals 233
solubility 60 (3.9)
Solids
brittle 266, 272
compressibility 19, 20, 40
density 19, 20
ductility 264, 274
flow 21
plastic 266
rigidity 19, 20, 40
structure 20, 185
Young's modulus 40
318
Index
Sound, speed of 16, 41, 46, 120
imperfect gases 199
and molecular speeds 112
Specific heat 117, 197
(CpCJ 118, 119, 198
C p /C„ 119, 121, 198
solid near melting point 288
Spectral lines, broadening 101
Stars 13, 90
Statistical specifications 67
Steady state 138, 159
Steel 41, 280
Stokes' law 298
Strain 51, 263
Streamline flow 142, 299
Stress 263
concentration 270
Sun, surface temperature 102
Superheating 181, 217
Supersaturation 182, 220, 281
Surface
of crystal 231
energy 33
ripples 38
tension 34, 35
Temperature
absolute 77
Boyle 195, 203
characteristic 88
critical 179, 210
inversion 203
maximum superheating 218
melting 177, 238
perfect gas scale 77
and random motion 64
triple point 176
Tensile strength
liquids 183, 217
solids 264
Thermal conductivity 140
effect of impurities 249
gases 140, 162, 165, 171
metals 257
size effects 251
solids 244
Thermal diffusivity 140
Thermal equilibrium
see equilibrium, thermal
Thermal expansion 1 19, 238
Thixotropic liquids 142
Time average 71
Transport coefficients 136
Triple point 176, 238
Turbulent flow 142, 299
Two component materials 274
Unit cell 228
Uranium hexafluoride (UF 6 ) 169
Vacancy 286
and diffusion 291
Van der Waals
equation 188
refinements to 203
forces 25
Vapour
pressure 218
supersaturation 182, 220
Velocity coefficient of chemical reaction
130
Velocitycomponent distribution 98, 103
Virial coefficients 193
2nd coefficient 195, 205
higher coefficients 206
Viscosity 16, 141
gases 16, 143, 162, 164, 170, 304
liquids 16, 19, 142, 298, 302
solids — see internal friction
Vorticity 143
Water 12, 22
critical constants 210
maximum superheating 218
tensile strength 185
Whiskers, single crystal 286
WiedemannFranz law 254
Wood 16, 274
Work hardening 280, 304 (9.1)
Xrays
characteristic 9
diffraction 233
structure analysis 233
Yield strength 268, 279
Young's modulus 40
and fluctuations 215
Zone
crystal 225
melting 233
Physical constants and conversion factors
Physical constants
Avogadro's number
Boltzmann's constant
Gas constant
Molar volume of perfect
gas at STP
Planck's constant
Elementary charge
Electron rest mass
Proton rest mass
Speed of light
Conversion factors
1 angstrom (A)
1 electron volt (eV)
kT = 1 eV
1 atmosphere (atm)
Value
6.02 x 10 23 mol" 1
1.38 x 10" 23 J deg" 1
8.62 x 10" 5 eVdeg" 1
8.31 J deg" 1 mor 1
22.4 x 10 3 cm 3 mol" 1
6.626 x 10" 34 J sec
1.60 x 10 ~ 19 coulomb
9.11 x 10" 28 gm
1.67 x 10" 24 gm
2.998 x 10 8 msec" 1
10
cm
1.60 x 10" 19 J
96.5 kJ mol" 1
when 7" = 1.16 x 10*deg K
1.01 x 10 6 dyn cm" 2
1.01 x 10 5 Nm" 2
Symbol
N
k
R
h
e
The Manchester Physics Series will be a collection of textbooks suitable for
an undergraduate degree course in Physics,
Each book will have been Individually developed, through a preliminary
edition, to provide a reliable, selfcontained text for an uptodate course.
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The Manchester Physics Series
General Editors
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