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ARTISANS AND STUDENTS IN PUBLIC AND SCIENCE SCHOOLS
THEORY OF HEAT
THEORY OF HEAT
BY
J. CLERK MAXWELL, M.A.
LL.D. EIMN., F.R.SS. L. & E.
Honorary Fellow of Trinity College
Professor of Experimental Physics in the University of Cambridge
WITH CORRECTIONS AND ADDITIONS (1891)
BY
LORD RAYLEIGH, M.A., D.C.L., LL.D.
Secretary of the Royal Society, Professor of Natural Philosophy in the
Royal Institution, and late Professor of Experimental Physics
in the University of Cambridge
NEW "IMPRESSION
LONGMANS, GREEN, AND CO.
39 PATERNOSTER ROW, LONDON
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1902
All rights reserved
PREFACE.
THE AIM of this book is to exhibit the scientific
connexion of the various steps by which our know-
ledge of the phenomena of heat has been extended.
The first of these steps is the invention of the thermo-
meter, by which the registration and comparison
of temperatures is rendered possible. The second
step is the measurement of quantities of heat, or
Calorimetry. The whole science of heat is founded
on Thermometry and Calorimetry, and when these
operations are understood we may proceed to the
third step, which is the investigation of those relations
between the thermal and the mechanical properties of
substances which form the subject of Thermodynamics.
The whole of this part of the subject depends on the
consideration of the Intrinsic Energy of a system of
bodies, as depending on the temperature and physical
state, as well as the form, motion, and relative position
of these bodies. Of this energy, however, only a
part is available for the purpose of producing me-
chanical work, and though the energy itself is inde-
structible, the available part is liable to diminution by
the action of certain natural processes, such as con-
399323
vi Preface.
duction and radiation of heat, friction, and viscosity.
These processes, by which energy is rendered unavail-
able as a source of work, are classed together under
the name of the Dissipation of Energy, and form the
subjects of the next division of the book. The last
chapter is devoted to the explanation of various
phenomena by means of the hypothesis that bodies
consist of molecules, the motion of which constitutes
the heat of those bodies.
In order to bring the treatment of these subjects
within the limits of this text-book, it has been found
necessary to omit everything which is not an essential
part of the intellectual process by which the doctrines
of heat have been developed, or which does not
materially assist the student in forming his own judg-
ment on these doctrines.
For this reason, no account is given of several very
important experiments, and many illustrations of the
theory of heat by means of natural phenomena are
omitted. The student, however, will find this part of
the subject treated at greater length in several excel-
lent works on the same subject which have lately
appeared.
A full account of the most important experiments
on the effects of heat will be found in Dixon's
'Treatise on Heat' (Hodges & Smith, 1849).
Professor Balfour Stewart's treatise contains all that
is necessary to be known in order to make experi-
ments on heat. The student may be also referred to
Deschanel's 'Natural Philosophy/ Part Untranslated
by Professor Everett, who has added a chapter on
Thermodynamics ; to Professor Rankine's work on the
Steam Engine, in which he will find the first systematic
Preface. vii
treatise on thermodynamics to Professor Tait's * Ther-
modynamics,' which contains an historical sketch of
the subject, as well as the mathematical investigations ;
and to Professor Tyndall's work on ' Heat as a Mode
of Motion,' in which the doctrines of the science are
forcibly impressed on the mind by well-chosen illus-
trative experiments. The original memoirs of Pro-
fessor Clausius, one of the founders of the modern
science of Thermodynamics, have been edited in
English by Professor Hirst
NOTE BY LORD RAYLEIGH.
In the tenth edition, printed in 1891, only such
corrections and additions were introduced as seemed
really called for. It is believed that they would have
commended themselves to the Author, and, indeed,
they are in great measure derived from his later
writings. In all cases the authorship of an addition
is indicated by the signature ' R.,' and by enclosure
within square brackets.
CONTENTS.
CHAPTER I
INTRODUCTION.
PACK
Meaning of the word Temperature .1
The Mercurial Thermometer 5
Heat as a Quantity 6
Diffusion of Heat by Conduction and Radiation ... .10
The three Physical States of Bodies i
CHAPTER II.
THERMOMETRY, OR THE REGISTRATION OF TEMPERATURE.
Definition of Higher and Lower Temperature . - . . 32
Temperatures ol Reference 34
Different Thermometric Scales ....... 37
Construction of a Thermometer 40
The Air Thermometer . . . .... .46
Other Methods of Ascertaining Temperatures . . . .51
CHAPTER IIL
CALORIMETRY, OR THE MEASUREMENT OF HEAl.
Selection of a Unit of Heat 54
All Heat is of the same Kind 56
Ice Calorimeters 58
Bunsen's Calorimeter . 6l
Method of Mixture 63
Definitions of Thermal Capacity and Specific Heat ... 65
Latent Heat of Steam 69
x Contents.
CHAPTER IV.
ELEMENTARY DYNAMICAL PRINCIPLES.
PAGE
Measurement of Quantities ....... 74
The Units of Length, Mass, and Time, and their Derived Units . 76
Measurement of Force . 83
Work and Energy . . 87
Principle of the Conservation of Energy 92
CHAPTER V.
MEASUREMENT OF INTERNAL FORCES AND THEIR EFFECTS.
Longitudinal Pressure and Tension . . . . . . 94
Definition of a Fluid. Hydrostatic Pressure .... 95
Work done by a Piston on a Fluid 101
Watt's Indicator and the Indicator Diagram .... 102
Elasticity of a Fluid .107
CHAPTER VL
LINES OF EQUAL TEMPERATURE ON THE INDICATOR
DIAGRAM.
Relation between Volume, Pressure, and Temperature . .108
Isothermal Lines of a Gas . . . . . . .no
Isothermal Lines of a Vapour in Contact with its Liquid . . 113
Steam Line and Water Line 117
Continuity of the Liquid and Gaseous States. Experiments of
Cagniard de la Tour and Andrews . . . . 118
CHAPTER VII.
ADIABATIC LINES.
Properties of a Substance when heat is prevented from entering or
leaving it . .127
The Adiabatic Lines are Steeper than the Isothermals . .130
Diagram showing the Effects of Heat on Water. . . .134
Contents. x i
CHAPTER VIII.
HEAT ENGINES.
PAG
Carnot's Engine ... 138
Second Law of Thermodynamics 153
Carnot's Function and Thomson's Absolute Scale of Temperature 155
Maximum Efficiency of a Heat Engine 158
Thermodynamic Scale of Temperature 160
Entropy 162
Fictitious Thermal Lines 164
CHAPTER IX.
RELATIONS BETWEEN THE PHYSICAL PROPERTIES
OF A SUBSTANCE.
Four Thermodynamic Relations 165
The two Modes of Defining Specific Heat . . . .169
The two Modes of Defining Elasticity 171
CHAPTER X.
LATENT HEAT.
Relation between the Latent Heat and the Alteration of the Volume
of the Substance during a Change of State . . * . 173
Lowering of the Freezing Point by Pressure . . . 1 76
CHAPTER XI.
THERMODYNAMICS OF GASES.
Cooling by Expansion 180
Calculation of the Specific Heat of Air 183
CHAPTER XII.
ON THE INTRINSIC ENERGY OF A SYSTEM OF BODIES.
Intrinsic Energy defined 185
Available Energy 187
Dissipation of Energy 192
Mechanical and Thermal Analogies 193
Prof. Gibbs' Thermodynamic Model 195
xii Contents.
CHAPTER XIIL
ON FREE EXPANSION.
PAGE
Theory of a Fluid rushing through a Porous Plug . . . 209
Determination of the Dynamical Equivalent of Heat . . .211
Determination of the Absolute Scale of Temperature . . . 213
CHAPTER XIV.
DETERMINATION OF HEIGHTS BY THE BAROMETER.
Principle of the Barometer . . . .... .217
The Barometer in a Diving Bell .218
Height of the ' Homogeneous Atmosphere ' . . . .220
Height of a Mountain found by the Barometer . . . . 221
CHAPTER XV.
ON THE PROPAGATION OF WAVES OF LONGITUDINAL
DISTURBANCE.
Waves of Permanent Type 223
Velocity of Sound . 228
CHAPTER XVI.
ON RADIATION.
Definition of Radiation 230
Interference 234
Different Kinds of Radiation ....... 237
Prevost's Theory of Exchanges c . 240
Rate of Cooling 246
Effects of Radiation on Thermometers . , 248
Contents xiii
CHAPTER XVII.
ON CONVECTION CURRENTS.
PAGE
How they are Produced 250
\ ouie's Determination of the Point of Maximum Density of Water 252
CHAPTER XVIII.
ON THE DIFFUSION OF HEAT BY CONDUCTION.
Conduction through a Plate 253
Different Measures of Conductivity 255
Conduction in a Solid 255
Sketch of Fourier's Theory 259
Harmonic Distributions of Temperature 263
Steady and Periodic Flow of Heat 265
Determination of the Thermal Conductivity of Bodies . 268
Applications of the Theory 2J2
CHAPTER XIX.
ON THE DIFFUSION OF FLUIDS.
Coefficient of Diffusion 277
Researches of Graham and Loschmidt 278
i
CHAPTER XX.
ON CAPILLARITY.
Superficial Energy and Superficial Tension . . . .281
Rise of a Liquid in a Tube .*..... 288
Evaporation and Condensation as Affected by Capillarity . . 289
Table of Superficial Tension 295
[Superficial Viscosity] 298
xiv Contents.
CHAPTER XXI.
ON ELASTICITY AND VISCOSITY.
PAGE
Biffeient Kinds of Stress and Strain. . . . . ,301
Coefficient of Viscosity . . . . . . . . 34
CHAPTER XXII.
MOLECULAR THEORY OF THE CONSTITUTION OF BODIES.
Kinetic and Potential Energy 38
Evidence that Heat is the Kinetic Energy of the Molecules of a
Body 3ic
Kinetic Theory of Gases 3 12
Deduction of the Laws of Gases . . . . . . 3 21
Equilibrium of a Vertical Column 3 2 9
Diffusion, Viscosity, and Conduction 33 l
Evaporation and Condensation ....... 333
Electrolysis . . . . . . . . . 335
Radiation 33 6
Limitation of the Second Law of Thermodynamics . . . 33 8
The Properties of Molecules ... .... 340
INDEX ...*..' 345
A TREATISE
ON
HEAT.
CHAPTER I.
INTRODUCTION.
THE DISTINCTION between hot bodies and cold ones is
familiar to all, and is associated in our minds with the
difference of the sensations which we experience in touching
various substances, according as they are hot or cold. The
intensity of these sensations is susceptible of degrees, so that
we may estimate one body to be hotter or colder than
another by the touch. The words hot, warm, cool, cold,
are associated in our minds with a series of sensations which
we suppose to indicate a corresponding series of states of
an object with respect to heat.
We use these words, therefore, as the names of these
states of the object, or, in scientific language, they are the
names of Temperatures, the word hot indicating a high
temperature, cold a low temperature, and the intermediate
terms intermediate temperatures, while the word temperature
itself is a general term intended to apply to any one of these
states of the object.
Since the state of a body may vary continuously from
cold to hot, we must admit the existence of an indefinite
number of intermediate states, which we call intermediate
2 ^ ; 3* y Introduction.
temperatures. We may give names to any number of
particular degrees of temperature, and express any other
temperature by its relative place among these degrees.
The temperature of a body, therefore, is a quantity which
indicates how hot or how cold the body is.
When we say that the temperature of one body is higher
or lower than that of another, we mean that the first body is
hotter or colder than the second, but we also imply that we
refer the state of both bodies to a certain scale of tempe-
ratures. By the use, therefore, of the word temperature,
we fix in our minds the conviction that it is possible, not
only to feel, but to measure, how hot a body is.
Words of this kind, which express the same things as
the words of primitive language, but express them in a way
susceptible of accurate numerical statement, are called
scientific l terms, because they contribute to the growth of
science.
We might suppose that a person who has carefully cul-
tivated his senses would be able by simply touching an
object to assign its place in a scale of temperatures, but it is
found by experiment that the estimate formed of temperature
by the touch depends upon a great variety of circumstances,
some of these relating to the texture or consistency of the
object, and some to the temperature of the hand or the
state of health of the person who makes the estimate.
For instance, if the temperature of a piece of wood were
the same as that of a piece of iron, and much higher than
that of the hand, we should estimate the iron to be hotter
than the wood, because it parts with its heat more readily to
the hand, whereas if their temperatures were equal, and
much lower than that of the hand, we should estimate the
iron to be colder than the wood.
There is another common experiment, in which we place
one hand in hot water and the other in cold for a sufficient
' ' Scientifick, adj. Producing demonstrative knowledge. ' Johnsorii
Vift.
Temperature. 3
time. If we then dip both hands in the same basin of
lukewarm water alternately, or even at once, it will appear
cold to the warmed hand and hot to the cooled hand.
In fact, our sensations of every kind depend upon so
many variable conditions, that for all scientific purposes we
prefer to form our estimate of the state of bodies from their
observed action on some apparatus whose conditions are
more simple and less variable than those of our own senses.
The properties of most substances vary when their tem-
perature varies. Some of these variations are abrupt, and
serve to indicate particular temperatures as points of re-
ference; others are continuous, and serve to measure other
temperatures by comparison with the temperatures of refer-
ence.
For instance, the temperature at which ice melts is found
to be always the same under ordinary circumstances, though,
as we shall see, it is slightly altered by change of pressure.
The temperature of steam which issues from boiling water
is also constant when the pressure is constant.
These two phenomena therefore the melting of ice and
the boiling of water indicate in a visible manner two tempe-
ratures which we may use as points of reference, the position
of which depends on the properties of water and not on the
conditions of our senses.
Other changes of state which take place at temperatures
more or less definite, such as the melting of wax or of
lead, and the boiling of liquids of definite composition, are
occasionally used to indicate when these temperatures are
attained, but the melting of ice and the boiling of pure
water at a standard pressure remain the most important
temperatures of reference in modern science.
These phenomena of change of state serve to indicate
only a certain number of particular temperatures. In
order to measure temperatures in general, we must avail
ourselves of some property of a substance which alters
continuously with the temperature.
v 2
4 Introduction,
The volume of most substances increases continuously
as the temperature rises, the pressure remaining constant.
There are exceptions to this rule, and the dilatations of
different substances are not in general in the same propor-
tion ; but ahy substance in which an increase of temperature,
however small, produces an increase of volume may be used
to indicate changes of temperature.
For instance, mercury and glass both expand when heated,
but the dilatation of mercury is greater than that of glass.
Hence if a cold glass vessel be filled with cold mercury, and
if the vessel and the mercury in it are then equally heated,
the glass vessel will expand, but the mercury will expand
more, so that the vessel will no longer contain the mercury.
If the vessel be provided with a long neck, the mercury
forced out of the vessel will rise in the neck, and if the neck
is a narrow tube finely graduated, the amount of mercury
forced out of the vessel may be accurately measured.
This is the principle of the common mercurial thermo-
meter, the construction of which will be afterwards more
minutely described. At present we consider it simply as an
instrument the indications of which vary when the tempe-
rature varies, but are always the same when the temperature
of the instrument is the same.
The dilatation of other liquids, as well as that of solids and
of gases, may be used for thermometric purposes, and the
thermo-electric properties of metals, and the variation of their
electric resistance with temperature, are also employed in
researches on heat. We must first, however, study the theory
of temperature in itself before we examine the properties of
different substances as related to temperature, and for this
purpose we shall use the particular mercurial thermometer
just described.
The Thermometer.
THE MERCURIAL THERMOMETER.
This thermometer consists of a glass tube terminating in
a bulb, the bulb and part of the tube bein^ filled with
mercury, and the rest of the tube being empty. We shall
suppose the tube to be graduated in any manner so that the
height of the mercury in the tube may be observed and
recorded. We shall not, however, assume either that the
tube is of uniform section or that the degrees are of equal
size, so that the scale of this primitive thermometer must be
regarded as completely arbitrary. By means of our thermo-
meter we can ascertain whether one temperature is higher or
lower than another, or equal to it, but we cannot assert that
the difference between two temperatures, A and B, is greater
or less than the difference between two other temperatures,
c and D.
We shall suppose that in every observation the temperature
of the mercury and the glass is equal and uniform over the
whole thermometer. The reading of the scale will then
depend on the temperature of the thermometer, and, since
we have not yet established any more perfect thermometric
scale, we shall call this reading provisionally * the temperature
by the arbitrary scale of the thermometer.'
The reading of a thermometer indicates primarily its own
temperature, but if we bring the thermometer into intimate
contact with another substance, as for instance if we plunge
it into a liquid for a sufficient time, we find that the reading
of the thermometer becomes higher or lower according as
the liquid is hotter or colder than the thermometer, and that
if we leave the thermometer in contact with the substance for
a sufficient time the reading becomes stationary. Let us
call this ultimate reading ' the temperature of the substance.'
We shall find as we go on that we have a right to do so.
Let us now take a vessel of water which we shall suppose
fro be at the temperature of the air, so that if left to itself it
6 Introduction.
would remain at the same temperature. Take anothei
smaller vessel of thin sheet copper or tin plate, and fill it
with water, oil, or any other liquid, and immerse it in the
larger vessel of water for a certain time. Then, if by means
of our thermometer we register the temperatures of the
liquids in the two vessels before and after the immersion of
the copper vessel, we find that if they are originally at the
same temperature the temperature of both remains the same,
but that if one is at a higher temperature than the other, that
which has the higher temperature becomes colder and that
which has the lower temperature becomes hotter, so that if
they continue in contact for a sufficient time they arrive at
last at the same temperature, after which no change of tem-
perature takes place.
The loss of temperature by the hot body is not in general
equal to the gain of temperature by the cold body, but it is
manifest that the two simultaneous phenomena are due to
one cause, and this cause may be described as the passage
of Heat from the hot body to the cold one.
As this is the first time we have used the word Heat, let us
examine what we mean by it.
We find the cooling of a hot body and the heating of
a cold body happening simultaneously as parts of the same
phenomenon, and we describe this phenomenon as the pas-
sage of heat from the hot body to the cold one. Heat, then,
is something which may be transferred from one body to
another, so as to diminish the quantity of heat in the first
and increase that in the second by the same amount.
When heat is communicated to a body, the temperature
of the body is generally increased, but sometimes other
effects are produced, such as change of state. When heat
leaves a body, there is either a fall of temperature or a
change of state. If no heat enters or leaves a body, and
if no changes of state or mechanical actions take place
in the body, the temperature of the body will remain
constant.
Heat as a Quantity. 7
Heat, therefore, may pass out of one body into another
just as water may be poured from one vessel into another,
and it may be retained in a body for any time, just as water
may be kept in a vessel. We have therefore a right to speak
of heat as of a measurable quantity, and to treat it mathema-
tically like other measurable quantities so long as it continues
to exist as heat. We shall find, however, that we have no
right to treat heat as a substance, for it may be transformed
into something which is not heat, and is certainly not a
substance at all, namely, mechanical work.
We must remember, therefore, that though we admit heat
to the title of a measurable quantity, we must not give it
rank as a substance, but must hold our minds in suspense
till we have further evidence as to the nature of heat.
Such evidence is furnished by experiments on friction, in
which mechanical work, instead of being transmitted from
one part of a machine to another, is apparently lost, while
at the same time, and in the same place, heat is generated,
the amount of heat being in an exact proportion to the
amount of work lost. We have, therefore, reason to believe
that heat is of the same nature as mechanical work, that is,
it is one of the forms of Energy.
In the eighteenth century, when many new facts were
being discovered relating to the action of heat on bodies,
and when at the same time great progress was being made
in the knowledge of the chemical action of substances, the
word Caloric was introduced to signify heat as a measurable
quantity. So long as the word denoted nothing more than
this, it might be usefully employed, but the form of the word
accommodated itself to the tendency of the chemists of that
time to seek for new 'imponderable substances,' so that
the word caloric came to connote * not merely heat, but heat
as an indestructible imponderable fluid, insinuating itself
into the pores of bodies, dilating and dissolving them, and
1 ' A connotative term is one which denotes a subject and implies an
attribute.' MilFs Logic , book i. chap. ii. 5.
8 Introduction.
ultimately vaporising them, combining with bodies in definite
quantities, and so becoming latent, and reappearing when
these bodies alter their condition. In fact, the word caloric,
when once introduced, soon came to imply the recognised
existence of something material, though probably of a more
subtle nature than the then newly discovered gases. Caloric
resembled these gases in being invisible and in its property
of becoming fixed in solid bodies. It differed from them
because its weight could not be detected by the finest
balances, but there was no doubt in the minds of many
eminent men that caloric was a fluid pervading all bodies,
probably the cause of all repulsion, and possibly even of the
extension of bodies in space.
Since ideas of this kind have always been connected
with the word caloric, and the word itself has been in no
slight degree the means of embodying and propagating
these ideas, and since all these ideas are now known to be
false, we shall avoid as much as possible the use of the
word caloric in treating of heat. We shall find it useful,
however, when we wish to refer to the erroneous theory
which supposes heat to be a substance, to call it the
' Caloric Theory of Heat.'
The word heat, though a primitive word and not a
scientific term, will be found sufficiently free from ambiguity
when we use it to express a measurable quantity, because it
will be associated with words expressive of quantity, indi-
cating how much heat we are speaking of
We have nothing to do with the word heat as an abstract
term signifying the property of hot things, and when we
might say a certain heat, as the heat of new milk, we shall
always use the more scientific word temperature, and speak
of the temperature of new milk.
We shall never use the word heat to denote the sensation
of heat In fact, it is never so used in ordinary language,
which has no names for sensations, unless when the sensation
itself is of more importance to us than its physical cause, as
Measurement of Heat. 9
in the case of pain, &c. The only name we have for this
sensation is ' the sensation of heat.'
When we require an adjective to denote that a phe-
nomenon is related to heat we shall call it a thermal
phenomenon, as, for instance, we shall speak of the thermal
conductivity of a substance or of thermal radiation to dis-
tinguish the conduction and radiation of heat from the
conduction of electricity or the radiation of light. The
science of heat has been called (by Dr. Whewell and others)
Thermotics, and the theory of heat as a form of energy is
called Thermodynamics. In the same way the theory of the
equilibrium of heat might be called Thermostatics, and that
of the motion of heat Thermokinematics.
The instrument by which the temperature of bodies is
registered is called a Thermometer or measurer of warmth,
and the method of constructing and using thermometers may
be called Thermometry.
The instrument by which quantities of heat are measured
is called a Calorimeter, probably because it was invented at
a time when heat was called Caloric. The name, however,
is now well established, and is a convenient one, as its form
is sufficiently distinct from that of the word Thermometer.
The method of measuring heat may be called Calorimetry.
A certain quantity of heat, with which all other quantities
are compared, is called a Thermal Unit. This is the quantity
of heat required to produce a particular effect, such as to
melt a pound of ice, or to raise a pound of water from one
defined temperature to another defined temperature. A par-
ticular thermal unit has been called by some authors a Calorie.
We have now obtained two of the fundamental ideas
of the science of heat the idea of temperature, or the
property of a body considered with reference to its power of
heating other bodies ; and the idea of heat as a measurable
quantity, which may be transferred from hotter bodies to
colder ones. We shall consider the further development of
these ideas in the chapters on Thermometry and Calorimetry,
1C Introduction.
but we must first direct our attention to the process by which
heat is transferred from one body to another.
This process is called the Diffusion of Heat. The diffusion
of heat invariably transfers heat from a hotter body to a colder
one, so as to cool the hotter body and warm the colder body.
This process would go on till all bodies were brought to the
same temperature if it were not for certain other processes
by which the temperatures of bodies are changed inde-
pendently of any exchange of heat with other bodies, as, for
instance, when combustion or any other chemical process
takes place, or when any change occurs in the form, structure,
or physical state of the body.
The changes of temperature of a body arising from other
causes than the transfer of heat from other bodies will be
considered when we come to describe the different physical
states of bodies. We are at present concerned only with
the passage of heat into the body or out of it, and this
always takes place by diffusion, and is always from a hotter
to a colder body.
Three processes of diffusion of heat are commonly recog-
nised Conduction, Convection, and Radiation.
Conduction is the flow of heat through an unequally heated
body from places of higher to places of lower temperature.
Convection is the motion of the hot body itself carrying its
heat with it. If by this motion it is brought near bodies colder
than itself it will warm them faster than if it had not been
moved nearer to them. The term convection is applied to
those processes by which the diffusion of heat is rendered
more rapid by the motion of the hot substance from one
place to another, though the ultimate transfer of heat may
still take place by conduction.
In Radiation, the hotter bod> loses heat, and the colder
body receives heat by means of a process occurring in some
intervening medium which does not itself become thereby hot.
In each of these three processes of diffusion of heat the
temDeratures of the bodies between which the process takes
Diffusion of Heat. 1 1
place tend to become equal. We shall not at present discuss
the convection of heat, because it is not a purely thermal
phenomenon, since it depends on a hot substance being
carried from one place to another, either by human effort,
as when a hot iron is taken out of the fire and put into the
tea-urn, or by some natural property of the heated substance,
as when water, heated by contact with the bottom of a
kettle placed on the fire, expands as it becomes warmed,
and forms an ascending current, making way for colder and
therefore denser water to descend and take its place. In
every such case of convection the ultimate and only direct
transfer of heat is due to conduction, and the only effect of
the motion of the hot substance is to bring the unequally
heated portions nearer to each other, so as to facilitate the
exchange of heat. We shall accept the conduction of heat
as a fact, without at present attempting to form any theory
of the details of the process by which it takes place. We
do not even assert that in the diffusion of heat by conduc-
tion the transfer of heat is entirely from the hotter to the
colder body. All that we assert is, that the amount of heat
transferred from the hotter to the colder body is invariably
greater than the amount, if any, transferred from the colder
to the hotter.
ON CONDUCTION.
In the experiments which we have described, heat passes
from one body into another through an intervening sub-
stance, as from a vessel of water through the glass bulb of a
thermometer into the mercury inside the bulb.
This process, by which heat passes from hotter to colder
parts of a body, is called the conduction of heat. When
heat is passing through a body by conduction, the tem-
perature of the body must be greater in the parts from
which the heat comes than in those to which it tends,
and the quantity of heat which passes through any thin
layer of the substance depends on the difference of the
12 Introduction.
temperatures of the opposite sides of the layer. For instance,
if we put a silver spoon into a cup of hot tea, the part
of the spoon in the tea soon becomes heated, while the
part just out of the tea is comparatively cool. On ac-
count of this inequality of temperature, heat immediately
FlG> x - begins to flow along the metal from
A to B. The heat first warms B a
little, and so makes B warmer than
c, and then the heat flows on from
B to c, and in this way the very
end of the spoon will in course of
time become warm to the touch.
The essential requisite to the con-
duction of heat is, that in every part of its course the heat
must pass froir hotter to colder parts of the body. No
heat can be conducted as far as E till A has been made
hotter than B, B than c, c than D, and D than E. To do
this requires a certain amount of heat to be expended in
warming in succession all these intermediate parts of the
spoon, so that for some time after the spoon is placed in
the cup no alteration of temperature can be perceived at
the end of the spoon.
Hence we may define conduction as the passage of heat
through a body depending on inequality of temperature in
adjacent parts of the body.
When any part of a body is heated by conduction, the
parts of the body through which the heat comes to it must
be hotter than itself, and the parts higher up the stream of
heat still hotter.
If we now try the experiment of the spoon in the teacup
with a German silver spoon along with the silver one, we
shall find that the end of the silver spoon becomes hot long
before that of the German silver one ; and if we also put in a
bone or horn spoon, we shall not be able to perceive any
varmth at the end of it, however long we wait.
This shows that silver conducts heat quicker than German
Radiation, 1 3
silver, and German silver quicker than bone or horn. The
reason why the end of the spoon never gels as hot as the
tea is, that the intermediate parts of the spoon are cooling,
partly by giving their heat to the air in contact with them,
and partly by radiation out into space.
To show that the first effect of heat on the thermometer
is to warm the material of which the bulb is composed, and
that the heat cannot reach the fluid inside till the bulb has
been warmed, take a thermometer with a large bulb, watch
the fluid in the tube, and dash a little hot water over the
bulb. The fluid will fall in the tube before it begins to
rise, showing that the bulb began to expand before the fluid
expanded.
ON RADIATION.
On a calm day in winter we feel the sun's rays warm even
when water is freezing and ice is hard and dry.
If we make use of a thermometer, we find that if the
sun's rays fall on it, it indicates a temperature far above
freezing, while the air immediately surrounding the bulb is
at a temperature below freezing. The heat, therefore, which
we feel, and to which the thermometer also responds, is not
conveyed to it by conduction through the air, for the air
is cold, and a cold body cannot make a body warmer than
itself by conduction. The mode in which the heat reaches
the body which it warms, without warming the air through
which it passes, is called radiation. Substances which
admit of radiation taking place through them are called
Diathermanous. Those which do not allow heat to pass
through them without becoming themselves hot are called
Athermanous. That which passes through the medium
during this process is generally called Radiant Heat,
though as long as it is radiant it possesses none of the
properties which distinguish heat from other forms of energy,
since the temperature of the body through which it passes,
14 Introduction.
and the other physical properties of the body, are in no way
affected by the passage of the radiation, provided the body
is perfectly diathermanous. If the body is not perfectly
diathermanous it stops more or less of the radiation, and
becomes heated itself, instead of transmitting the whole
radiation to bodies beyond it.
The distinguishing characteristic of radiant heat is, that
it travels in rays like light, whence the name radiant. These
rays have all the physical properties of rays of light, and are
capable of reflexion, refraction, interference, and polarisation.
They may be divided into different kinds by the prism, as
light is divided into its component colours, and some of the
heat-rays are identical with the rays of light, while other
kinds of heat-rays make no impression on our eyes. For in-
stance, if we take a glass convex lens, and place it in the sun's
rays, a body placed at the focus where a small image of the
sun is formed will be intensely heated, while the lens itself
and the air through which the rays pass remain quite cold.
If we allow the rays before they reach the focus to fall on the
surface of water, so that the rays meet in a focus in the inte-
rior of the water, then if the water is quite clear at the focus
it will remain tranquil, but if we make the focus fall upon a
mote in the water, the rays will be stopped, the mote will be
heated and will cause the water next it to expand, and so an
upward current will be produced, and the mote will begin to
rise in the water. This shows that it is only when the radia-
tion is stopped that it has any effect in heating what it falls on.
By means of any regular concave piece of metal, such as
the scale of a balance, pressed when hot against a clear
sheet of ice, first on one side and then on the other, it is easy
to make a lens of ice which may be used on a sunny day as
a burning glass ; but this experiment, which was formerly
in great repute, is far inferior in interest to one invented by
Professor Tyndall, in which the heat, instead of being con-
centrated by ice, is concentrated in ice. Take a clear block
of ice and make a flat surface on it,- parallel to the original
Radiation. 1 5
surface of the lake, or to the layers of bubbles generally
found in large blocks ; then let the converging rays of the
sun from an ordinary burning glass fall on this surface, and
come to a focus within the ice. The ice, not being per-
fectly diathermanous, will be warmed by the rays, but much
more at the focus than anywhere else. Thus the ice will
begin to melt at the focus in the interior of its substance,
and, as it does so, the portions which melt first are regu-
larly formed crystals, and so we see in the path of the beam
a number of six-rayed stars, which are hollows cut out of
the ice and containing water. This water, however, does not
quite fill them, because the water is of less bulk than the ice
of which it was made, so that parts of the stars are empty.
Experiments on the heating effects of radiation show
that not only the sun but all hot bodies emit radiation. When
the body is hot enough, its radiations become visible, and
the body is said to be red hot. When it is still hotter it
sends forth not only red rays, but rays of every colour, and
it is then said to be white hot. When a body is too cold to
shine visibly, it still shines with invisible heating rays, which
can be perceived by a sufficiently delicate thermometer, and
it does not appear that any body can be so cold as not to
send forth radiations. The reason why all bodies do not
appear to shine is, that our eyes are sensitive only to parti-
cular kinds of rays, and we only see by means of rays of
these kinds, coming from some very hot body, either directly
or after reflexion or scattering at the surface of other bodies.
We shall see that the phrases radiation of heat and radiant
heat are not quite scientifically correct, and must be used
with caution. l Heat is certainly communicated from one body
to another by a process which we call radiation, which takes
place in the region between the two bodies. We have no
I 1 It is interesting to note Newton's language in the sixth query appended
to his ' Opticks.' ' Do not black bodies conceive heat more easily from
light than those of other colours do, by reason that the light falling on them
is not reflected outwards, but enters the bodies, and is often reflected and
icfracied within them, until it be stifled and lost ? ' R.]
\ 6 Introduction.
right, however, to speak of this process of radiation as heat
We have defined heat as it exists in hot bodies, and we have
seen that all heat is of the same kind. But the radiation
between bodies differs from heat as we have defined it ist,
in not making the body hot through which it passes ; 2nd
in being of many different kinds, Hence we shall generally
speak of radiation, and when we speak of radiant heat we
do not mean to imply the existence of a new kind of heat
but to consider radiation in its thermal aspect.
ON THE DIFFERENT PHYSICAL STATES OF BODIES.
Bodies are found to behave in different ways under the
action of forces. If we cause a longitudinal pressure to act
on a body in one direction by means of a pair of pincers or
a vice, the body being free to move in all other directions,
we find that if the body is a piece of cold iron there is very
little effect produced, unless the pressure be very great ; if
the body is a piece of india-rubber, it is compressed in the
direction of its length and bulges out at the sides, but it
soon comes into a state of equilibrium, in which it continues
to support the pressure ; but if we substitute water for the
india-rubber we cannot perform the experiment, for the
water flows away laterally, and the jaws of the pincers come
together without having exerted any appreciable pressure.
Bodies which can sustain a longitudinal pressure, however
small that pressure may be, without being supported by a
lateral pressure, are called solid bodies. Those which
cannot do so are called fluids. We shall see that in a fluid
at rest the pressure at any point must be equal in all direc-
tions, and this pressure is called the pressure of the fluid.
There are two great classes of fluids. If we put into a
closed vessel a small quantity of a fluid of the first class, such
as water, it will partly fill the vessel, and the rest of the vessel
may either be empty or may contain a different fluid. Fluids
having this property are called liquids. Water is a liquid,
and if we put a little water into a bottle the water will lie at
Solids, Liquids, and Gases. 17
the bottom of the bottle, and will be separated by a distinct
surface from the air or the gaseous water- substance above it.
If, on the contrary, the fluid which we put into the closed
vessel be one of the second class, then, however small a
portion we introduce, it will expand and fill the vessel, or at
least as much of it as is not occupied by a liquid.
Fluids having this property are called gases. Air is a
gas, and if we first exhaust the air from a vessel and then
introduce the smallest quantity of air, the air will immediately
expand till it fills the whole vessel so that there is as much
air in a cubic inch in one part of the vessel as in another.
Hence a gas cannot, like a liquid, be kept in an open-
mouthed vessel.
The distinction, therefore, between a gas and a liquid is
that, however large the space may be into which a portion of
gas is introduced, the gas will expand and exert pressure on
every part of its boundary, whereas a liquid will not expand
more than a very small fraction of its bulk, even when the
pressure is reduced to zero ; and some liquids can even
sustain a hydrostatic tension, or negative pressure, without
their parts being separated.
The three principal states in which bodies are found are,
therefore, the solid, the liquid, and the gaseous states.
Most substances are capable of existing in all these states,
as, for instance, water exists in the forms of ice, water, and
steam. A few solids, such as carbon, have not yet been
melted ; and a few gases, such as oxygen, hydrogen, and
nitrogen, have not yet been liquefied or solidified, but these
may be considered as exceptional cases, arising from the
limited range of temperature and pressure which we can
command in our experiments. 1
The ordinary effects of heat in modifying the physical
state of bodies may be thus described. We may take water
[* In consequence of the experiments of Pictet and Cailletet, the excep-
tions referred to in the text must now be removed. Considerable quantities
of oxygen and nitrogen have been prepared in the liquid state. R.]
C
1 8 Introduction.
as a familiar example, and explain, when it is necessary, the
different phenomena of other bodies.
At the lowest temperatures at which it has been observed
water exists in the solid form as ice. When heat is com-
municated to very cold ice, or to any other solid body not
at its melting temperature
1. The temperature rises.
2. The body generally expands (the only exception among
solid bodies, as far as I am aware, is the iodide of silver,
which has been found by M. Fizeau to contract as the
temperature rises).
3. The rigidity of the body, or its resistance to change of
form, generally diminishes. This phenomenon is more
apparent in some bodies than in others. It is very con-
spicuous in iron, which when heated but not melted becomes
soft and easily forged. The consistency of glass, resins, fats,
and frozen oils alters very much with change of temperature.
On the other hand, it is believed that steel wire is stiffer at
100 C. than at o C., and it has been shown by Joule and
Thomson that the longitudinal elasticity of caoutchouc
increases with the temperature between certain limits of
temperature. When ice is very near its melting point it
becomes very soft.
4. A great many solid bodies are constantly in a state of
evaporation or transformation into the gaseous state at their
free surface. Camphor, iodine, and carbonate of ammonia
are well-known examples of this. These solid bodies, if not
kept in stoppered bottles, gradually disappear by evapora-
tion, and the vapour which escapes from them may be
recognised by its smell and by its chemical action. Ice,
too, is continually passing into a state of vapour at its
surface, and in a dry climate during a long frost large
pieces of ice become smaller and at last disappear.
There are other solid bodies which do not seem to lose
any of their substance in this way; at least, we cannot
detect any loss. It is probable, however, that those solid
Fusion. 19
bodies which can be detected by their smell are evaporating
with extreme slowness. Thus iron and copper have each a
well-known smell. This, however, may arise from chemical
action at the surface, which sets free hydrogen or some
other gas combined with a very small quantity of the
metal.
FUSION.
When the temperature of a solid body is raised to a
sufficient height it begins to melt into a liquid. Suppose a
small portion of the solid to be melted, and that no more heat
is applied till the temperature of the remaining solid and of
the liquid has become equalised ; if a little more heat is then
applied and the temperature again equalised there will be
more liquid matter and less solid matter, but since the liquid
and the solid are at the same temperature, that temperature
must still be the melting temperature.
Hence, if the partly melted mass be kept well mixed
together, so that the solid and fluid parts are at the same
temperature, that temperature must be the melting tempera-
ture of the solid, and no rise of temperature will follow from
the addition of heat till the whole of the solid has been con-
verted into liquid.
The heat which is required to melt a certain quantity of
a solid at the melting point into a liquid at the same
temperature is called the latent heat of fusion.
It is called latent heat, because the application of this
heat to the body does not raise its temperature or warm the
body.
Those, therefore, who maintained heat to be a substance
supposed that it existed in the fluid in a concealed or latent
state, and in this way they distinguished it from the heat
which, when applied to a body, makes it hotter, or raises the
temperature. This they called sensible heat A body, there-
fore, was said to possess so much heat. Part of this heat was
called sensible heat, and to it was ascribed the temperature
oi
2O Introduction.
of the body. The other part was called latent heat, and
to it was ascribed the liquid or gaseous form of the body.
The fact that a certain quantity of heat must be applied
to a pound of melting ice to convert it into water is all that
we mean in this treatise when we speak of this quantity
of heat as the latent heat of fusion of a pound of water.
We make no assertion as to the state in which the heat
exists in the water. We do not even assert that the heat
communicated to the ice is still in existence as heat.
Besides the change from solid to liquid, there is generally
a change of volume in the act of fusion. The water formed
from the ice is of smaller bulk than the ice, as is shown by
ice floating in water, so that the total volume of the ice and
water diminishes as the melting goes on.
On the other hand, many substances expand in the act of
fusion, so that the solid parts sink in the fluid. During the
fusion of the mass the volume in these cases increases.
It has been shown by Prof. J. Thomson, 1 from the
principles of the dynamical theory of heat, that if pressure is
applied to a mixture of ice and water, it will not only compress
both the ice and the water, but some of the ice will be
melted at the same time, so that the total compression will
be increased by the contraction of bulk due to this melting
The heat required to melt this ice being taken from the rest
of the mass, the temperature of the whole will diminish.
Hence the melting point is lowered by pressure in the
case of ice. This deduction from theory was experimentally
verified by Sir W. Thomson.
If the substance had been one of those which expand in
melting, the effect of pressure would be to solidify some of
the mixture, and to raise the temperature of fusion. Most of
the substances of which the crust of the earth is composed
expand in the act of melting. Hence their melting points
will rise under great pressure. If the earth were throughout
1 Transactions of the Jtoyal Society of Edinburgh^ 1849.
Fusion. 2 1
in a state of fusion, when the external parts began to solidify
they would sink in the molten mass, and when they had
sunk to a great depth they would remain solid under the
enormous pressure even at a temperature greatly above the
point of fusion of the same rock at the surface. It does not
follow, therefore, that in the interior of the earth the matter
is in a liquid state, even if the temperature is far above that
of the fusion of rocks in our furnaces.
It has been shown by Sir W. Thomson that if the earth, as
a whole, were not more rigid than a ball of glass of equal size,
the attraction of the moon and sun would pull it out of shape,
and raise tides on the surface, so that the solid earth would
rise and fall as the sea does, only not quite so much. It is
true that this motion would be so smooth and regular that
we should not be able to perceive it in a direct way, but its
effect would be to diminish the apparent rise of the tides of
the ocean, so as to make them much smaller than they
actually are.
It appears, therefore, from what we know of the tides of
the ocean, that the earth as a whole is more rigid than glass,
and therefore that no very large portion of its interior can
be liquid. The effect of pressure on the melting point of
bodies enables us to reconcile this conclusion with the
observed increase of temperature as we descend in the
earth's crust, and the deductions as to the interior tempera-
ture founded on this fact by the aid of the theory of the
conduction of heat.
EFFECT OF HEAT ON LIQUIDS.
When heat is applied to a liquid its effects are
i. To warm the liquid. The quantity of heat required to
raise the liquid one degree is generally greater than that
required to raise the substance in the solid form one degree,
and in general it requires more heat at high than at low
temperatures to warm the liquid one degree.
a. To alter its volume. Most liquids expand as their
2.2 introduction.
temperature rises, but water contracts from o C. to 4 C C.
and then expands, slowly at first, but afterwards more
rapidly.
3. To alter its physical state. Liquids, such as oil, tar,
&c, which are sluggish in their motion, are said to be
viscous. When they are heated their viscosity generally
diminishes and they become more mobile. This is the case
even with water, as appears by the experiments of M. O. E
Meyer.
When sulphur is heated, the melted sulphur undergoes
several remarkable changes as its temperature rises, being
mobile when first melted, then becoming remarkably viscous
at a higher temperature, and again becoming mobile when
still more heated.
4. To convert the liquid or solid into gas When a liquid
or a solid body is placed in a vessel the rest of which is
empty, it gives off part of its own substance in the form of
gas. This process is called evaporation, and the gas given
off is commonly called the vapour of the solid or liquid sub-
stance. The process of evaporation goes on till the density
of the vapour in the vessel has reached a value which de-
pends only on the temperature.
If in any way, as by the motion of a piston, the vessel De
made larger, then more vapour will be formed till the density
is the same as before. If the piston be pushed in, and the
vessel made smaller, some of the vapour is condensed into
the liquid state, but the density of the remainder of the
vapour still remains the same.
If the remainder of the vessel, instead of containing
nothing but the vapour of the liquid, contains any quantity
of air or some other gas not capable of chemical action on
the liquid, then exactly the same quantity of vapour will be
formed, but the time required for the vapour to reach the
further parts of the vessel will be greater, as it has to
diffuse itself through the air in the vessel by a kind of
percolation.
These laws of evaporation were discovered by Dalton.
Evaporation. 23
The conversion of the* liquid into vapour requires an
amount of ' latent heat ' which is generally much greater
than the latent heat of fusion of the same substance.
In all substances, the density, pressure, and temperature
are so connected that if we know any two of them the value
of the third is determinate. Now in the case of vapours in
contact with their own liquids or solids, there is for each
temperature a corresponding density, which is the greatest
density which the vapour can have at that temperature,
without being condensed into the liquid or solid form.
Hence for each temperature there is also a maximum
pressure which the vapour can exert.
A vapour which is at the greatest density and pressuie
corresponding to its temperature is called a saturated vapour.
It is then just at the point of condensation, and the slightest
increase of pressure or decrease of temperature will cause
some of the vapour to be condensed. Professor Rankine
restricts the use of the word vapour by itself to the case of a
saturated vapour, and when the vapour is not at the point of
condensation he calls it superheated vapour, or simply gas.
BOILING.
When a liquid in an open vessel is heated to a tempera-
ture such that the pressure of its vapour at that tempera-
ture is greater than the pressure at a point in the interior
of the liquid, the liquid will begin to evaporate at that
point, so that a bubble of vapour will be formed there.
This process, in which bubbles of vapour are formed in
the interior of the liquid, is called boiling or ebullition.
When water is heated in the ordinary way by applying
heat to the bottom of a vessel, the lowest layer of the water
becomes hot first, and by its expansion it becomes lighter
than the colder water above, and gradually rises, so that a
gentle circulation of water is kept up, and the whole water
is gradually warmed, though the lowest layer is always the
hottest. As the temperature increases, the absorbed air,
24 Introduction.
which is generally found in ordinary water, is expelled, and
rises in small bubbles without noise. At last the water in
contact with the heated metal becomes so hot that, in spite
of the pressure of the atmosphere on the surface of the
water, the additional pressure due to the water in the
vessel, and the cohesion of the water itself, some of the
water at the bottom is transformed into steam, forming a
bubble adhering to the bottom of the vessel. As soon as a
bubble is formed, evaporation goes on rapidly from the water
all round it, so that it soon grows large, and rises from the
bottom. If the upper part of the water into which the
bubble rises is still below the boiling temperature, the
bubble is condensed, and its sides come together with a
sharp rattling noise, called simmering. But the rise of the
bubbles stirs the water about much more vigorously than
the mere expansion of the water, so that the water is soon
heated throughout, and brought to the boil, and then the
bubbles enlarge rapidly during their whole ascent, and
burst into the air, throwing the water about, and making
the well-known softer and more rolling noise of boiling.
The steam, as it bursts out of the bubbles, is an invisible
gas, but when it comes into the colder air it is cooled below
its condensing point, and part of it is formed into a cloud
consisting of small drops of water which float in the air.
As the cloud of drops disperses itself and mixes with dry
air the quantity of water in each cubic foot diminishes as
the volume of any part of the cloud increases. The little
drops of water begin to evaporate as soon as there is suffi-
cient room for the vapour to be formed at the temperature
of the atmosphere, and so the cloud vanishes again into
thin air.
The temperature to which water must be heated before it
boils depends, in the first place, on the pressure of the
atmosphere, so that the greater the pressure, the higher the
boiling temperature. But the temperature requires to be
raised above that at which the pressure of steam is equal to
Boiling. 2 5
that of the atmosphere, for in ordei to form bubbles the
pressuie of the steam has to overcome not only the pressure
due to the atmosphere and a certain depth of water, but that
cohesion between the parts of the water of which the effects
are visible in the tenacity of bubbles and drops. Hence it
is possible to heat water 20 F. above its boiling point with-
out ebullition. If a small quantity of metal-filings are now
thrown into the water, a little air will be carried down on
the surface of the filings, and the process of evaporation will
take place at the interface between this air and the hot water
with such rapidity as to produce a violent boiling, almost
amounting to an explosion.
If a current of steam from a boiler is passed into a vessel
of cold water, we have first the condensation of steam,
accompanied with a very loud simmering or rattling noise, and
a rapid heating of the water. When the water is sufficiently
heated, the steam is not condensed, but escapes in bubbles,
and the water is now boiling.
As an instance of a different kind, let us suppose that
the water is not pure, but contains some salt, such as
common salt, or sulphate of soda, or any other substance
which tends to combine with water, and from which the
water must separate before it can evaporate. Water con-
taining such substances in solution requires to be brought
to a temperature higher than the boiling point of pure
water before it will boil. Water, on the other hand, con-
taining air or carbonic acid, will boil at a lower temperature
than pure water till the gas is expelled.
If steam at 100 C. is passed into a vessel containing a
strong solution of one of the salts we have mentioned,
which has a tendency to combine with water, the conden-
sation of the steam will be promoted by this tendency,
and will go on even after the solution has been heated far
above the ordinary boiling point, so that by passing steam
at 100 C. into a strong solution of nitrate of soda, Mr. Peter
Spence ' has heated it up to i2i'i C.
1 Transactions of the British Association, 1869, p. 75.
26 Introduction.
If water at a temperature below 100 C. be placed in a
vessel, and if by means of an air-pump we reduce the pres-
sure of the air on the surface of the water, evaporation goes
on and the surface of the water becomes colder than the
interior parts. If we go on working the air-pump, the
pressure is reduced to that of vapour of the temperature of the
interior of the fluid. The water then begins to boil, exactly
as in the ordinary way, and as it boils the temperature
rapidly falls, the heat being expended in evaporating the
water.
This experiment may be performed without an air-pump
in the following way : Boil water in a flask over a gas-
flame or spirit-lamp, and while it is boiling briskly cork the
flask, and remove it from the flame. The boiling will soon
cease, but if we now dash a little cold water over the flask,
some of the steam in the upper part will be condensed, the
pressure of the remainder will be diminished, and the water
will begin to boil again. The experiment may be made
more striking by plunging the flask entirely under cold
water. The steam will be condensed as before, but the
water, though it is cooled more rapidly than when the cold
water was merely poured on the flask, retains its heat longer
than the steam, and continues to boil for some time.
Laws of Gases. 27
ON THE GASEOUS STATE.
The distinguishing property of gases is their power of
indefinite expansion. As the pressure is diminished the
volume of the gas not only increases, but before the pressure
has been reduced to zero the volume of the gas has become
greater than that of any vessel we can put it in.
This is the property without which a substance cannot
be called a gas, but it is found that actual gases fulfil with
greater or less degrees of accuracy certain numerical laws,
which are comirionly referred to as the ' Gaseous Laws.'
LAW OF BOYLE.
The first of these laws expresses the relation between the
pressure and the density of a gas, the temperature being
constant, and is usually stated thus : ' The volume of a
portion of gas varies inversely as the pressure.'
This law was discovered by Robert Boyle, and published
by him in 1662, in an appendix to his ' New Experiments,
Physico-mechanical, &c., touching the Spring of the Air.'
Mariotte, about 1676, in his treatise ' De la Nature de
1'Air,' enunciated the same law, and carefully verified it, and it
is generally referred to by Continental writers as Mariotte's
law.
This law may also be stated thus :
The pressure of a gas is proportional to its density.
Another statement of the same law has been proposed by
Professor Rankine, which I think places the law in a very
clear light.
If we take a closed and exhausted vessel, and introduce
into it one grain of air, this air will, as we know, exert a
certain pressure on every square inch of the surface of the
vessel. If we now introduce a second grain of air, then this
second grain will exert exactly the same pressure on the
sides of the vessel that it would have exerted if the first grain
28 Introduction.
had not been there before it, so that the pressure will now
be doubled. Hence we may state, as the property of a
perfect gas, that any portion of it exerts the same pressure
against the sides of a vessel as if the other portions had not
been there.
Dalton extended this law to mixtures of gases of different
kinds.
We have already seen that if several different portions of
the same gas are placed together in a vessel, the pressure on
any part of the sides of the vessel is the sum of the pres-
sures which each portion would exert if placed by itself in
the vessel.
Dalton's law asserts that the same is true for portions of
different gases placed in the same vessel, and that the
pressure of the mixture is the sum of the pressures due to the
several portions of gas, if introduced separately into the
vessel and brought to the same temperature.
This law of Dalton is sometimes stated as if portions of
gas of different kinds behave to each other in a different
manner from portions of gas of the same kind, and we are
told that when gases of different kinds are placed in the
same vessel, each acts as if the other were a vacuum.
This statement, properly understood, is correct, but it
seems to convey the impression that if the gases had been
of the same kind some other result would have happened,
whereas there is no difference between the two cases.
Another law established by Dalton is that the maximum
density of a vapour in contact with its liquid is not affected
by the presence of other gases. It has been shown by
M. Regnault that when the vapour of the substance has a
tendency to combine with the gas, the maximum density
attainable by the vapour is somewhat increased.
Before the time of Dalton it was supposed that the cause
of evaporation was the tendency of water to combine with
air, and that the water was dissolved in the air just as salt is
dissolved in water.
Gases and Vapours. 29
Dalton showed that the vapour of water is a gas, which
just at the surface of the water has a certain maximum
density, and which will gradually diffuse itself through the
space above, whether filled with air or not, till, if the space is
limited, the density of the vapour is a maximum throughout,
or, if the space is large enough, till the water is all dried up.
The presence of air is so far from being essential to this
process that the more air there is, the slower it goes on,
because the vapour has to penetrate through the air by the
slow process of diffusion.
The phenomenon discovered by Regnault that the density
of vapour is slightly increased by the presence of a gas
which has a tendency to combine with it, is the only instance
in which there is any truth in the doctrine of a liquid being
held in solution by a gas.
The law of Boyle is not perfectly fulfilled by any actual
gas. It is very nearly fulfilled by those gases which we are
not able to condense into liquids, and among other gases it
is most nearly fulfilled when their temperature is much above
their point of condensation.
When a gas is near its point of condensation its density
increases more rapidly than the pressure. When it is
actually at the point of condensation the slightest increase of
pressure condenses the whole of it into a liquid, and in the
liquid form the density increases very slowly with the
pressure.
LAW OF CHARLES.
The second law of gases was discovered by Charles, 1 but
is commonly referred to as that of Gay-Lussac or of Dalton. 2
It may be stated thus :
1 Professor of Physics at the Conservatoire des Arts et Metiers, Pans
Born 1746. Died 1823. Celebrated as having first employed hydrogen
in balloons.
2 Dalton, in 1801, first published this law. Gay-Lussac published
it, in 1802, independently of Dalton. In his memoir, however (Ann.
3O Introduction.
The volume of a gas under constant pressure expands
when raised from the freezing to the boiling temperature by
the same fraction of itself, whatever be the nature of the gas.
It has been found by the careful experiments of M.
Regnault, M. Rudberg, Prof. B. Stewart, and others that the
volume of air at constant pressure expands from i to i '3665
between o C. and 100 C. Hence 30 cubic inches of
air at o C. would expand to about 41 cubic inches at
100 C.
If we admit the truth of Boyle's law at all temperatures,
and if the law of Charles is found to be true for a particular
pressure, say that of the atmosphere, then it is easy to show
that the law of Charles must be true for every other pressure.
For if we call the volume v and the pressure P, then we
may call the product of the numerical value of the volume
and pressure v P, and Boyle's law asserts that this pro-
duct is constant, provided the temperature is constant. If
then we are further informed that when p has a given
value v is increased from i to 1*3665 when the temperature
rises from the freezing point to the boiling point, the product
v p will be increased in the same proportion at that particular
pressure. But v p we know by Boyle's law does not depend on
the particular pressure, but remains the same for all pressures
when the temperature remains the same. Hence, whatever
be the pressure, the product v p will be increased in the
proportion of i to 1*3665 when the temperature rises from
o C. to 100 C.
The law of the equality of the dilatation of gases, which, as
originally stated, applied only to the dilatation from o C.
to 100 C., has been found to be true for all other tempera-
tures for which it has hitherto been tested.
de Chimie, xliii. p. 157 [1802]), he states that Citizen Charles had
remarked, fifteen years before the date of his memoir, the equality of
the dilatation of the principal gases ; but, as Charles never published
these results, he had become acquainted with them by mere chance.
The Gaseous State. 31
It appears, therefore, that gases are distinguished from
other forms of matter, not only by their power of indefinite
expansion so as to fill any vessel, however large, and by the
great effect which heat has in dilating them, but by the
uniformity and simplicity of the laws which regulate these
changes. In the solid and liquid states the effect of a
given change of pressure or of temperature in changing the
volume of the body is different for every different substance.
On the other hand, if we take equal volumes of any two
gases, measured at the same temperature and pressure,
their volumes will remain equal if we afterwards bring them
both to any other temperature and pressure, and this
although the two gases differ altogether in chemical nature
and in density, provided they are both in the perfectly gaseous
condition.
This is only one of many remarkable properties which
point out the gaseous state of matter as that in which its
physical properties are least complicated.
In our description of the physical properties of bodies as
related to heat we have begun with solid bodies, as those
which we can most easily handle, and have gone on to
liquids, which we can keep in open vessels, and have now
come to gases, which will escape from open vessels, and
which are generally invisible. This is the order which is
most natural in our first study of these different states. But
as soon as we have been made familiar with the most prominent
features of these different conditions of matter, the most
scientific course of study is in the reverse order, beginning
with gases, on account of the greater simplicity of their laws,
then advancing to liquids, the more complex laws of which
are much more imperfectly known, and concluding with
the little that has been hitherto discovered about the con-
stitution of solid bodies.
32 Tkermometry.
CHAPTER II.
ON THERMOMETRY-, OR THE THEORY OF TEMPERATURE.
Definition of Temperature. The temperature of a body
is its thermal state considered with reference to its power of
communicating heat to other bodies.
Definition of Higher and Lower Temperature. If when
two bodies are placed in thermal communication, one of the
bodies loses heat, and the other gains heat, that body which gives
out heat is said to have a higher temperature than that which
receives heat from it.
Cor. If when two bodies are placed in thermal communica
tion neither of them loses or gains heat, the two bodies are
said to have equal temperatures or the same temperature. The
two bodies are then said to be in thermal equilibrium. We
have here a means of comparing the temperature of any
two bodies, so as to determine which has the higher
temperature, and a test of the equality of temperature
which is independent of the nature of the bodies tested.
But we have no means of estimating numerically the differ-
ence between two temperatures, so as to be able to assert
that a certain temperature is exactly halfway between two
other temperatures.
Law of Equal Temperatures. Bodies whose temperatures
are equal to that of the same body have themselves equal tem-
peratures. This law is not a truism, but expresses the fact
that if a piece of iron when plunged into a vessel of water
is in thermal equilibrium with the water, and if the same
piece of iron, without altering its temperature, is transferred
to a vessel of oil, and is found to be also in thermal equi-
librium with the oil, then if the oil and water were put
into the same vessel they would themselves be in thermaJ
Comparison of Temperatures. 33
equilibrium, and the same would be true of any other three
substances.
This law, therefore, expresses much more than Euclid's
axiom that ' Things which are equal to the same thing are
equal to one another/ and is the foundation of the whole
science of thermometry. For if we take a. thermometer,
such as we have already described, and bring it into in-
timate contact with different bodies, by plunging it into
liquids, or inserting it into holes made in solid bodies, we
find that the mercury in the tube rises or falls till it has
reached a certain point at which it remains stationary. We
then know that the thermometer is neither becoming hotter
nor colder, but is in thermal equilibrium with the surround-
ing body. It follows from this, by the law of equal tem-
peratures, that the temperature of the body is the same as
that of the thermometer, and the temperature of the thermo-
meter itself is known from the height at which the mer-
cury stands in the tube.
Hence' the reading, as it is called, of the thermometer
that is, the number of degrees indicated on the scale by the
top of the mercury in the tube informs us of the tem-
perature of the surrounding substance, as well as of that of
the mercury in the thermometer. In this way the thermo-
meter may be used to compare the temperature of any
two bodies at the same time or at different times, so as
to ascertain whether the temperature of one of them is
higher or lower than that of the other. We may compare
in this way the temperatures of the air on different days ;
we may ascertain that water boils at a lower temperature at
the top of a mountain than it does at the sea-shore, and that
ice melts at the same temperature in all parts of the world.
For this purpose it would be necessary to carry the same
thermometer to different places, and to preserve it with
great care, for if it were destroyed and a new one made,
we should have no certainty that the same temperature is
indicated by the same reading in the two thermometers,
D
34 Thermometry.
Thus the observations of temperature recorded during
sixteen years by Rinieri l at Florence lost their scientific
value after the suppression of the Accademia del Cimento,
and the supposed destruction of the thermometers with
which the observations were made. But when Antinori IP
1829 discovered a number of the very thermometers usec
in the ancient observations, Libri 2 was able to compare them
with Reaumur's scale, and thus to show that the climate of
Florence has not been rendered sensibly colder in winter
by the clearing of the woods of the Apennines.
In the construction of artificial standards for the measure-
ment of quantities of any kind it is desirable to have the
means of comparing the standards together, either directly,
or by means of some natural object or phenomenon which
is easily accessible and not liable to change. Both methods
are used in the preparation of thermometers.
We have already noticed two natural phenomena which
take place at definite temperatures the melting of ice and
the boiling of water. The advantage of employing these
temperatures to determine two points on the scale of the
thermometer was pointed out by Sir Isaac Newton (' Scala
Graduum Caloris,' Phil. Trans. 1701).
The first of these points of reference is commonly called
the Freezing Point. To determine it, the thermometer i?
placed in a vessel filled with pounded ice or snow thorough!
moistened with water. If the atmospheric temperature be
above the freezing point, the melting of the ice will ensure,
the presence of water in the vessel. As long as every part
of the vessel contains a mixture of water and ice its tem-
perature remains uniform, for if heat enters the vessel it
can only melt some of the ice, and if heat escapes from
the vessel some of the water will freeze, but the mixture can
be made neither hotter nor colder till all the ice is melted
or all the water frozen.
1 Pupil of Galileo ; died 1647.
* dnnqles de Chimie ef dc Physique^ xly.
Temperatures of Reference.
35
FIG.
The thermometer is completely immersed in the mixture
of ice and water for a sufficient time, so that the mercury
has time to reach its stationary point. The position of the
top of the mercury in the tube is
then recorded by making a scratch
on the glass tube. We shall call
this mark the Freezing Point. It
may be determined in this way with
extreme accuracy, for, as we shall
see afterwards, the temperature of
melting ice is very nearly the same
under very different pressures.
The other point of reference is
called the Boiling Point. The tem-
perature at which water boils de-
pends on the pressure of the atmo-
sphere. The greater the pressure of
the air on the surface of the water,
the higher is the temperature to
which the water must be raised
before it begins to boil.
To determine the Boiling Point, the stem of the thermo-
meter is passed through a hole in the lid of a tall vessel,
in the lower part of which water is made to boil briskly, so
that the whole of the upper part, where the thermometer is
placed, is filled with steam. When the thermometer has
acquired the temperature of the current of steam the stem
is drawn up through the hole in the lid of the vessel till the
top of the column of mercury becomes visible. A scratch
is then made on the tube to indicate the boiling point
In careful determinations of the boiling point no part oi
the thermometer is allowed to dip into the boiling water,
because it has been found by Gay-Lussac that the temperature
of the water is not always the same, but that it boils at
different temperatures in different kinds of vessels. It has
been shown, however, by Rudberg that the temperature of
Thermometry
FIG. 3.
the steam which escapes from boiling water is the same in
every kind of vessel, and depends only on the pressure at
the .surface of the water. Hence the thermometer is not
dipped in the water, but suspended in the issuing steam. To
ensure that the temperature of the steam shall be the same
when it reaches the thermometer as when it issues from the
boiling water, the sides of the vessel are sometimes protected
by what is called a steam-jacket. A current of steam is
made to play over the out-
side of the sides of the
vessel. The vessel is thus
raised to the same tempe-
rature as the steam itself, so
that the steam cannot be
cooled during its passage
from the boiling water to
the thermometer.
For instance, if we take
any tall narrow vessel, as
a coffee-pot, and cover its mouth and part of its sides
with a wider vessel turned upside down, taking care that
there shall be plenty of room for the steam to escape, then
if we boil- a small quantity of water in the coffee-pot, a thermo-
meter placed in the steam above will be raised to the
exact temperature of the boiling point of water corresponding
to the state of the barometer at the time.
To mark the level of the mercury on the tube of the
thermometer without cooling it, we must draw it up through
a cork or a plug of india-rubber in the steam-jacket through
which the steam passes till we can just see the top of the
column of mercury. A mark must then be scratched on the
glass to register the boiling point. This experiment of
exposing a thermometer to the steam of boiling water is an
important one, for it not only supplies a means of gradu-
ating thermometers, and testing them when they have been
graduated, but, since the temperature at which water boils
Scale of the Thermometer. 37
depends on the pressure of the air, we may determine the
pressure of the air by boiling water when we are not able to
measure it by means of the appropriate instrument, the
barometer.
We have now obtained two points of reference marked by
scratches on the tube of the thermometer the freezing point
and the boiling point. We shall suppose for the present
that when the boiling point was marked the barometer
happened to indicate the standard pressure of 29*905
inches of mercury at o c C. at the level of the sea in the
latitude of London. In this case the boiling point is
the standard boiling point. In any other case it must be
corrected.
Our thermometer will now agree with any other properly
constructed thermometer at these two temperatures.
In order to indicate other temperatures, we must construct
a scale that is, a series of marks either on the tube itself or
on a convenient part of the apparatus close to the tube and
well fastened to it.
For this purpose, having settled what values we are to give
to the freezing and the boiling points, we divide the space
between those points into as many equal parts as there are
degrees between them, and continue the series of equal divi-
sions up and down the scale as far as the tube of the thermo-
meter extends.
Three different ways of doing this are still in use, and,
as we often find temperatures stated according to a
different scale from that which we adopt ourselves, it is
necessary to know the principles on which these scales are
formed.
The Centigrade scale was introduced by Celsius. 1 In it
the freezing point is marked o and called zero, and the
boiling point is marked 100.
The obvious simplicity of this mode of dividing the space
between the points of reference into 100 equal parts and
1 Professor of Astronomy in the University of Upsala.
38 Thermometry.
calling each of these a degree, and reckoning all temperatures
in degrees from the freezing point, caused it to be very
generally adopted, along with the French decimal system of
measurement, by scientific men, especially on the Continent
of Europe. It is true that the advantage of the decimal
system is not so great in the measurement of temperatures as
in other cases, as it merely makes it easier to remember the
freezing and boiling temperatures, but the graduation is not
too fine for the roughest purposes, while for accurate
measurements the degrees may be subdivided into tenths and
hundredths.
The other two scales are called by the names of those who
introduced them.
Fahrenheit, of Dantzig, about 1714, first constructed
thermometers comparable with each other. In Fahrenheit's
scale the freezing point is marked 32, and the boiling point
212, the space between being divided into 180 equal parts,
and the graduation extended above and below the points of
reference. A point 32 degrees below the freezing point is
called zero, or o, and temperatures below this are indicated
by the number of degrees below zero.
This scale is very generally used in English-speaking
countries for purposes of ordinary life, and also for those of
science, though the Centigrade scale is coming into use
among those who wish their results to be readily followed by
foreigners.
The only advantages which can be ascribed to Fahrenheit's
scale, besides its early introduction, its general diffusion, and
its actual employment by so many of our countrymen, are
that mercury expands almost exactly one ten-thousandth of
its volume at 142 F. for every degree of Fahrenheit's scale,
and that the coldest temperature which we can get by
mixing snow and salt is near the zero of Fahrenheit's
scale.
To compare temperatures given in Fahrenheit's scale with
temperatures given in the Centigrade scale we have only to
Thermometric Scales. 39
remember that o Centigrade is 32 Fahrenheit, and that five
degrees Centigrade are equal to nine of Fahrenheit.
The third thermometric scale is that of Rdaumur. In this
scale the freezing point is marked o and the boiling point
80. I am not aware of any advantage of this scale. It is
used to some extent on the Continent of Europe for medical
and domestic purposes. Four degrees of Reaumur corre-
spond to five Centigrade and to nine of Fahrenheit.
The existence of these three thermometric scales furnishes
an example of the inconvenience of the want of uniformity in
systems of measurement. The whole of what we have said
about the comparison of the different scales might have
been omitted if any one of these scales had been adopted by
all who use thermometers. Instead of spending our time in
describing the arbitrary proposals of different men, we should
have gone on to investigate the laws of heat and the pro-
perties of bodies.
We shall afterwards have occasion to use a scale differing
in its zero-point from any of those we have considered, but
when we do so we shall bring forward reasons for its adoption
depending on the nature of things and not on the predilec-
tions of men.
If two thermometers are constructed of the same kind of
glass, with tubes of uniform bore, and are filled with the same
liquid and then graduated in the same way, they maybe con-
sidered for ordinary purposes as comparable instruments;
so that though they may never have been actually com-
pared together, yet in ascertaining the temperature of any-
thing there will be very little difference whether we use the
one thermometer or the other.
But if we desire great accuracy in the measurement of
temperature, so that the observations made by different
observers with different instruments may be strictly com-
parable, the only satisfactory method is by agreeing to
choose one thermometer as a standard and comparing all the
others with it.
4-O Thermometry.
All thermometers ought to be made with tubes of as
uniform bore as can be found ; but for a standard thermometer
the bore should be calibrated that is to say, its size should be
measured at short intervals all along its length.
For this purpose, before the bulb is blown, a small quantity
of mercury is introduced into the tube and moved along the
tube by forcing air into the tube behind it. This is done by
squeezing the air out of a small india-rubber ball which is
fastened to the end of the tube.
If the length of the column of mercury remains exactly
the same as it passes along the tube, the bore of the tube
must be uniform ; but even in the best tubes there is always
some want of uniformity.
But if we introduce a short column of mercury into the
tube, then mark both ends of the column, and move it on its
own length, till one end comes exactly to the mark where
the other end was originally, then mark the other end, and
move it on again, we shall have a series of marks on the tube
such that the capacity of the tube between any two consecu-
tive marks will be the same, being equal to that of the
column of mercury.
By this method, which was invented by Gay-Lussac, a
number of divisions may be marked on the tube, each of
which contains the same volume, and though they will pro-
bably not correspond to degrees when the tube is made up
into a thermometer, it will be easy to convert the reading of
this instrument into degrees by multiplying it by a proper
factor, and in the use of a standard instrument this trouble is
readily undertaken for the sake of accuracy.
The tube having been prepared in this way, one end is
heated till it is melted, and it is blown into a bulb by forcing
air in at the other end of the tube. In order to avoid
introducing moisture into the tube, this is done, not by the
mouth, but by means of a hollow india-rubber ball, which is
fastened to the end of the tube.
FIG. 4-
Constriction of a Thermometer. 41
The tube of a thermometer is generally so narrow thai
mercury will not enter it, for a reason which we shall explain
when we come to the properties of liquids. Hence the
following method is adopted to fill the thermometer. By
rolling paper round the open end of
the tube, and making the tube thus
formed project a little beyond the
glass tube, a cavity is formed, into
which a little mercury is poured.
The mercury, however, will not run
down the tube of the thermometer,
partly because the bulb and tube are
already full of air, and partly because
the mercury requires a certain pres-
sure from without to enter so narrow
a tube. The bulb is therefore gently
heated so as to cause the air to ex-
pand, and some of the air escapes
through the mercury. When the bulb
cools, the pressure of the air in the
bulb becomes less than the pressure
of the air outside, and the difference
of these pressures is sufficient to
make the mercury enter the tube,
when it runs down and partially fills
the bulb.
In order to get rid of the remainder of the air, and of any
moisture in the thermometer, the bulb is gradually heated
till the mercury boils. The air and steam escape along
with the vapour of mercury, and as the boiling continues the
last remains of air are expelled through the mercury at the
top of the tube. When the boiling ceases, the mercury runs
back into the tube, which is thus perfectly filled with mercury.
While the thermometer is still hotter than any temperature
at which it will afterwards be used, and while the mercury or
42 Thermometry.
its vapour completely fills it, a blowpipe flame is made to
play on the top of the tube, so as to melt it and close the end
of the tube. The tube, thus closed with its own substance,
is said to be ' hermetically sealed/ l
There is now nothing in the tube but mercury, and when
the mercury contracts so as to leave a space above it, this
space is either empty of all gross matter, or contains only
the vapour of mercury. If, in spite of all our precautions,
there is still some air in the tube, this can easily be ascertained
by inverting the thermometer and letting some of the mer-
cury glide towards the end of the tube. If the instrument
is perfect, it will reach the end of the tube and completely
fill it. If there is air in the tube the air will form an elastic
cushion, which will prevent the mercury from reaching the
end of the tube, and will be seen in the form of a small
bubble.
We have next to determine the freezing and boiling points,
as has been already described, but certain precautions have
still to be observed. In the first place, glass is a substance
in which internal changes go on for some time after it
has been strongly heated, or exposed to intense forces.
In fact, glass is in some degree a plastic body. It is
found that after a thermometer has been filled and sealed
the capacky of the bulb diminishes slightly, and that this
change is comparatively rapid at first, and only gradually
becomes insensible as the bulb approaches its ultimate con-
dition. It causes the freezing point to rise in the tube to
o'3 or o'5, and if, after the displacement of the zero, the
mercury be again boiled, the zero returns to its old place
and gradually rises again.
This change of the zero-point was discovered by M.
Flaugergues. 2 It may be considered complete in from four to
1 * From Hermes or Mercury, the imagined inventor of chemistry,'
Johnson's Diet.
' Ann. de Chimie et de Physique, xxi. p. 333 (1822).
Comparison of Thermometers. 43
six months. 1 In order to avoid the error which it would
introduce into the scale, the instrument should, if possible,
have its zero determined some months after it has been
filled, and since even the determination of the boiling point
of water produces a slight depression of the freezing point
(that is, an expansion of the bulb), the freezing point should
not be determined after the boiling point, but rather
before it.
When the boiling point is determined, the barometer is
probably not at the standard height. The mark made on
the thermometer must, in graduating it, be considered to
represent, not the standard boiling point, but the boiling
point corresponding to the observed height of the baro-
meter, which may be found from the tables.
To construct a thermometer in this elaborate way is by
no means an easy task, and even when two thermometers have
been constructed with the utmost care, their readings at
points distant from the freezing and boiling points may not
agree, on account of differences in the law of expansion of
the glass of the two thermometers. These differences, how-
ever, are small, for all thermometers are made of the same
description of glass.
But since the main object of thermometry is that all
thermometers shall be strictly comparable, and since thermo-
meters are easily carried from one place to another, the
best method of obtaining this object is by comparing all
thermometers either directly or indirectly with a single
standard thermometer. For this purpose, the thermometers,
after being properly graduated, are all placed along with the
standard thermometer in a vessel, the temperature of which
can be maintained uniform for a considerable time. Each
thermometer is then compared with the standard thermometer.
1 Dr. Joule, however, finds that the rise of the freezing point of a
delicate thermometer has been going on for twenty-six years, though the
changes are now exceedingly minute. Phil. Soc. Manchester, Feb. 23,
1870.
44 Thermometry.
A table of corrections is made for each thermometer
by entering the reading of that thermometer, along with
the correction which must be applied to that reading to
reduce it to the reading of the standard thermometer.
This is called the proper correction for that reading. If
it is positive it must be added to the reading, and if negative
it must be subtracted frorn it.
By bringing the vessel to various temperatures, the cor-
rections at these temperatures for each thermometer are
ascertained, and the series of corrections belonging to each
thermometer is made out and preserved along with that
thermometer.
Any thermometer may be sent to the Observatory at
Kew, and will be returned with a list of corrections, by the
application of which, observations made with that thermo-
meter become strictly comparable with those made by the
standard thermometer at Kew, or with any other thermometer
similarly corrected. The charge for making the comparison
is very small compared with the expense of making an
original standard thermometer, and the scientific value of
observations made with a thermometer thus compared is
greater than that of observations made with the most elabo-
rately prepared thermometer which has not been compared
with some existing and known standard instrument.
I have described at considerable length the processes by
which the thermometric scale is constructed, and those by
which copies of it are multiplied, because the practical
establishment of such a scale is an admirable instance of
the method by which we must proceed in the scientific
observation of a phenomenon such as temperature, which, for
the present, we regard rather as a quality, capable of greater
or less intensity, than as a quantity which may be added to
or subtracted from other quantities of the same kind.
A temperature, so far as we have yet gone in the science
of heat, is not considered as capable of being added to
another temperature so as to form a temperature which is
Temperature considered as a Quality. 45
the sum of its components. When we are able to attach a
distinct meaning to such an operation, and determine its
result, our conception of temperature will be raised to the
rank of a quantity. For the present, however, we must be
content to regard temperature as a quality of bodies, and be
satisfied to know that the temperatures of all bodies can be
referred to their proper places in the same scale.
For instance, we have a right to say that the temperatures
of freezing and boiling differ by 180 Fahrenheit ; but we
have as yet no right to say that this difference is the same
as that between the temperatures 300 and 480 on the
same scale. Still less can we assert that a temperature of
244 F. = 32 + 2 12
is equal to the sum of the temperatures of freezing and
boiling. In the same way, if we had nothing by which to
measure time except the succession of our own thoughts,
we might be able to refer each event within our own ex-
perience to its proper chronological place in a series, but
we should have no means of comparing the interval of time
between one pair of events with that between another pair,
unless it happened that one of these pairs was included
within the other pair, in which case the interval between the
first pair must be the smallest It is only by observation of
the uniform or periodic motions of bodies, and by ascertain-
ing the conditions under which certain motions are always
accomplished in the same time, that we have been enabled
to measure time, first by days and years, as indicated by
the heavenly motions, and then by hours, minutes, and
seconds, as indicated by the pendulums of our clocks, till
we are now able, not only to calculate the time of vibration
of different kinds of light, but to compare the time of vibra-
tion of a molecule of hydrogen set in motion by an electric
discharge through a glass tube, with the time of vibration
of another molecule of hydrogen in the sun, forming part of
some great eruption of rosy clouds, and with the time of
vibration of another molecule in Sirius which has not
46 Thermometry.
transmitted its vibrations to our earth, but has simply
prevented vibrations arising in the body of that star from
reaching us.
In a subsequent chapter we shall consider the further
progress of our knowledge of Temperature as a Quantity.
ON THE AIR THERMOMETER.
The original thermometer invented by Galileo was an
air thermometer. It consisted of a glass bulb with a long
neck. The air in the bulb was heated, and then the neck
was plunged into a coloured liquid. As the air in the bulb
cooled, the liquid rose in the neck, and the higher the
liquid the lower the temperature of the air in the bulb.
By putting the bulb into the mouth of a patient, and noting
the point to which the liquid was driven down in the tube, a
physician might estimate whether the ailment was of the
nature of a fever or not. Such a thermometer has several
obvious merits. It is easily constructed, and gives larger
indications for the same change of temperature than a thermo-
meter containing any liquid as the thermometric sub-
stance. Besides this, the air requires less heat to warm it
than an equal bulk of any liquid, so that the air thermo-
meter is very rapid in its indications. The great incon-
venience of the instrument as a means of measuring tem-
perature is, that the height of the liquid in the tube depends
on the pressure of the atmosphere as well as on the tem-
perature of the air in the bulb. The air thermometer cannot
therefore of itself tell us anything about temperature. We
must consult the barometer at the same time, in order to
correct the reading of the air thermometer. Hence the air
thermometer, to be of any scientific value, must be used
along with the barometer, and its readings are of no use
till after a process of calculation has been gone through.
This puts it at a great disadvantage compared with the
mercurial thermometer as a means of ascertaining tempera-
T tie Air Thermometer. 47
tures. But if the researches on which we are engaged are
of so important a nature that we are willing to undergo the
labour of double observations and numerous calculations,
then the advantages of the air thermometer may again pre-
ponderate.
We have seen that in fixing a scale of temperature after
marking on our thermometer two temperatures of reference
and filling up the interval with equal divisions, two thermo-
meters containing different liquids will not in general agree
except at the temperatures of reference.
If, on the other hand, we could secure a constant pressure
in the air thermometer, then if we exchange the air for any
other gas, all the readings will be exactly the same provided
the reading at one of the temperatures of reference is the
same. It appears, therefore, that the scale of temperatures
as indicated by an air thermometer has this advantage over
the scale indicated by mercury or any other liquid or solid,
that whereas no two liquid or solid substances can be made to
agree in their expansion throughout the scale, all the gases
agree with one another. In the absence of any better
reasons for choosing a scale, the agreement of so many
substances is a reason why the scale of temperatures fur-
nished by the expansion of gases should be considered as of
great scientific value. In the course of our study we
shall find that there are scientific reasons of a much higher
order which enable us to fix on a scale of temperature,
based not on a probability of this kind, but on a more inti-
mate knowledge of the properties of heat. This scale, so
far as it has been investigated, is found to agree very closely
with that of the air thermometer.
There is another reason, of a practical kind, in favour of
the use of air as a thermometric substance, namely, that air
remains in the gaseous state at the lowest as well as the
highest temperatures which we can produce, 1 and there are
no indications in either case of its approaching to a change
of state. Hence air, or one of the permanent gases, is of
[' See note to p. 17. R.]
4 8
Thermometry.
FIG. 5.
AIR THERMOMETER.
Tin
45i
100
the greatest use in estimating temperatures lying far outside
of the temperatures of reference, such, for instance, as the
freezing point of mercury or the melting point of silver.
' We shall consider the practical method of using air as a
thermomecric substance when we come to Gasometry. In
the meantime let us consider the air thermometer in its
simplest form, that of a long tube of uniform bore closed at
one end, and containing air or some other gas which is
separated from the outer air by a short column of mercury,
oil, or some other liquid which is
capable of moving freely along the
tube, while at the same time it pre-
, vents all communication between the
'Jjj confined air and the atmosphere.
J I We shall also suppose that the pres-
sure acting on the confined air is in
212 Boiling some wa y maintained constant dur-
ing the course of the experiments
we are going to describe.
The air thermometer is first sur-
rounded with ice and ice-cold water.
Let us suppose that the upper surface
-38-8 -| -3?'9 Mer- of the air now stands at the point
marked ' Freezing. 5 The thermometer
is then surrounded with the steam
rising from water boiling under an
atmospheric pressure of 29*905 inches
of mercury. Let the surface of the
enclosed air now stand at the point
marked ' Boiling.' In this way, the
two temperatures of reference are to
-^ 6o be marked on the tube.
To complete the scale of the
thermometer we must divide the distance between boiling and
freezing into a selected number of equal parts, and carry
this graduation up and down the tube beyond the freezing
and boiling points with degrees of the same length.
Natterer's
observed
-140
-273
32Freezing.
~ 220
A bsolute Zero. 49
Of course, if we carry the graduation far enough down
the tube, we shall at last come to the bottom of the tube.
What will be the reading at that point 1 and what is meant
by it?
To determine the reading at the bottom of the tube is a
very simple matter. We know that the distance of the
freezing point from the bottom of the tube is to the distance
of the boiling point from the bottom in the proportion of
i to 1*3665, since this is the dilatation of air between the
freezing and the boiling temperatures. Hence it follows, by
an easy arithmetical calculation, that if, as in Fahrenheit's
scale, the freezing point is marked 32, and the boiling
point 212, the bottom of the tube must be marked
459 0tI 3- If> as m tne Centigrade scale, the freezing point
is marked o, and the boiling point 100, the bottom of the
tube will be marked 272 0> 85. This, then, is the reading at
the bottom of the scale.
The other question, What is meant by this reading?
requires a more careful consideration. We have begun by
denning the measure of the temperature as the reading
of the scale of our thermometer when it is exposed to that
temperature. Now if the reading could be observed at the
bottom of the tube, it would imply that the volume of the
air had been reduced to nothing. It is hardly necessary to
say that we have no expectation of ever observing such a
reading. If it were possible to abstract from a substance all
the heat it contains, it would probably still remain an
extended substance, and would occupy a certain volume.
Such an abstraction of all its heat from a body has never
been effected, so that we know nothing about the tem-
perature which would be indicated by an air thermometer
placed in contact with a body absolutely devoid of heat.
This much we are sure of, however, that the reading would
be above 459'i3 F.
It is exceedingly convenient, especially in dealing with
questions relating to gases, to reckon temperatures, not from
5O Thermometry.
the freezing point, or from Fahrenheit's zero, but rrorn the
bottom of the tube of the air thermometer.
This point is then called the absolute zero of the air
thermometer, and temperatures reckoned from it are called
absolute temperatures. It is probable that the dilatation of
a perfect gas is a little less than 1-3 665. If we suppose it
1*366, then absolute zero will be 460 on Fahrenheit's
scale, or 273^ Centigrade.
If we add 460 to the ordinary reading on Fahrenheit's
scale, we shall obtain the absolute temperature in Fahren-
heit's degrees.
If we add 273^ to the Centigrade reading, we shall obtain
the absolute temperature in Centigrade degrees.
We shall often have occasion to speak of absolute
temperature by the air thermometer. When we do so we
mean nothing more than what we have just said namely,
temperature reckoned from the bottom of the tube of the air
thermometer. We assert nothing as to the state of a body
deprived of all its heat, about which we have no experimental
knowledge.
One of the most important applications of the conception
of absolute temperature is to simplify the expression of the
two laws discovered respectively by Boyle and by Charles.
The laws may be combined into the statement that the
product of the volume and pressure of any gas is proportional
to the absolute temperature.
For instance, if we have to measure quantities of a gas by
their volumes under various conditions as to temperature
and pressure, we can reduce these volumes to what they
would be at some standard temperature and pressure.
Thus if v, P, T be the actual volume, pressure, and absolute
temperature, and V the volume at the standard pressure P O ,
and the standard temperature T O , then
JVP__ VQ PQ
T " T
V O = V!- T O
P T
Absolute Temperatures. 51
If we have only to compare the relative quantities of the
gas in different measurements in the same series of experi-
ments, we may suppose P O and T O both unity, and use the
quantity V without always multiplying it by , which is
T P
a constant quantity. 1
The great scientific importance of the scale of temperature
as determined by means of the air or gas thermometer arises
from the fact, established by the experiments of Joule and
Thomson, that the scale of temperature derived from the
expansion of the more permanent gases is almost exactly the
same as that founded upon purely thermodynamic considera-
tions, which are independent of the peculiar properties of the
thermometric body. This agreement has been experimentally
verified only within a range of temperature between o C.
and 100 C. If, however, we accept the molecular theory of
gases, the volume of a perfect gas ought to be exactly pro-
portional to the absolute temperature on the thermodynamic
scale, and it is probable that as the temperature rises the
properties of real gases approximate to those of the theo-
retically perfect gas.
All the thermometers which we have considered have
been constructed on the principle of measuring the expansion
of a substance as the temperature rises. In certain cases it is
convenient to estimate the temperature of a substance by the
heat which it gives out as it cools to a standard temperature.
Thus if a piece of platinum heated in a furnace is dropped
into water, we may form an estimate of the temperature of
the furnace by the amount of heat communicated to the
water. Some have supposed that this method of estimating
temperatures is more scientific than that founded on expan-
sion. It would be so if the same quantity of heat always
caused the same rise of temperature, whatever the original
1 For a full account of the methods of measuring gases the student ia
referred to Bunsen's Gasometty > translated by Roscoe.
E 2
52 Thermometry.
temperature of the body. But the specific heat of most
substances increases as the temperature rises, and it in-
creases in different degrees for different substances, so that
this method cannot furnish an absolute scale of temperature.
It is only in the case of gases that the specific heat of a given
mass of the substance remains the same at all temperatures.
There are two methods of estimating temperature which
are founded on the electrical properties of bodies. We
cannot, within the limits of this treatise, enter into the
theory of these methods, but must refer the student to works
on electricity. One of these methods depends on the fact
that in a conducting circuit formed of two different metals,
if one of the junctions be warmer than the other, there will
be an electromotive force which will produce a current of
electricity in the circuit, and this may be measured by
means of a galvanometer. In this way very minute differences
of temperature between the ends of a piece of metal may be
detected. Thus if a piece of iron wire is soldered at both
ends to a copper wire, and if one of the junctions is at a place
where we cannot introduce an ordinary thermometer, we may
ascertain its temperature by placing the other junction in a
vessel of water and adjusting the temperature of the water
till no current passes. The temperature of the water will
then be equal to that of the inaccessible junction.
Electric currents excited by differences of temperature in
different parts of a metallic circuit are called thermo-electric
currents. An arrangement by which the electromotive forces
arising from a number of junctions may be added together
is called a thermopile, and is used in experiments on the
heating effect of radiation, because it is more sensitive to
changes of temperature caused by small quantities of heat
than any other instrument.
Professor Tait 1 has found that if ^ and / 2 denote the
temperatures of the hot and cold junction of two metals,
1 Proceedings of the Royal Society of Edinburgh, 1870-71.
Electrical Thermometric Methods. 53
the electromotive force of the circuit formed by these two
metals is A (/j _ /) [ T - i (t, + /,)],
where A is a constant depending on the nature of* the metals,
and T is a temperature also depending on the metals,
such that when one junction is as much hotter than T as the
other is colder, no current is produced. T may be called the
neutral temperature for the two metals. For copper and
iron it is about 284 C.
The other method of estimating the temperature of a place
at which we cannot set a thermometer is founded on the in-
crease of the electric resistance of metals as the temperature
rises. This method has been successfully employed by Mr.
Siemens. 1 Two coils of the same kind of fine platinum wire
are prepared so as to have equal resistance. Their ends are
connected with long thick copper wires, so that the coils may
be placed if necessary a long way from the galvanometer.
These copper terminals are also adjusted so as to be of the
same resistance for both coils. The resistance of the termi-
nals should be small as compared with that of the coils. One
of the coils is then sunk, say to the bottom of the sea, and
the other is placed in a vessel of water, the temperature of
which is adjusted till the resistance of both coils is the same.
By ascertaining with a thermometer the temperature of the
vessel of water, that of the bottom of the sea may be deduced.
Mr. Siemens has found that the resistance of the metals
may be expressed by a formula of the form 2
R = V~r + /3 T + 7,
where R is the resistance, T the absolute temperature, and
a /3 y coefficients. Of these a is the largest, and the re-
sistance depending on it increases as the square root of the
absolute temperature, so that the resistance increases more
1 Proceedings of the Royal Society, April 27, 1871.
[* Calendar's experiments (Phil. Mag., July 1891) lead him to prefer the
simple parabolic formula, R/R = i + *t + 0t 2 . R.1
54 Calorimetry.
slowly as the temperature rises. The second term, /3 T, is
proportional to the temperature, and may be attributed to
the expansion of the substance. The third term is con-
stant.
CHAPTER III.
CALORIMETRY.
HAVING explained the principles of Thermometry, or the
method of ascertaining temperatures, we are able to under-
stand what we may call Calorimetry, or the method of
measuring quantities of heat.
When heat is applied to a body it produces effects of
various kinds. In most cases it raises the temperature of
the body ; it generally alters its volume or its pressure, and in
certain cases it changes the state of the body from solid to
liquid or from liquid to gaseous.
Any effect of heat may be used as a means of measuring
quantities of heat by applying the principle that when two
equal portions of the same substance in the same state are
acted on by heat in the same way so as to produce the
same effect, then the quantities of heat are equal.
We begin by choosing a standard body, and defining the
standard effect of heat upon it. Thus we may choose a
pound of ice at the freezing point as the standard body, and
we may define as the unit of heat that quantity of heat which
must be applied to this pound of ice to convert it into a
pound of water still at the freezing point. This is an
example of a certain change of state being used to define
what is meant by a quantity of heat. This unit of heat was
brought into actual use in the experiments of Lavoisier and
Laplace.
In this system a quantity of heat is measured by the
number of pounds (or of grammes) of ice at the freezing
The Unit of Heat. 55
point which that quantity of heat would convert into water
at the freezing point.
We might also employ a different system of measurement
by denning a quantity of heat as measured by the number of
pounds of water at the boiling point which it would convert
into steam at the same temperature.
This method is frequently used in determining the amount
of heat generated by the combustion of fuel.
Neither of these methods requires the use of the thermo-
meter.
Another method, depending on the use of the thermo-
meter, is to define as the unit of heat that quantity of heat
which if applied to unit of mass (one pound or one gramme)
of water at some standard temperature (that of greatest
density, 39 F. or 4 C., or occasionally some temperature
more convenient for laboratory work, such as 62 F. or 15 C.),
will raise that water one degree (Fahrenheit or Centigrade)
in temperature.
According to this method a quantity of heat is measured
by the quantity of water at a standard temperature which that
quantity of heat would raise one degree.
All that is assumed in these methods of measuring heat is
that if it takes a certain quantity of heat to produce a certain
effect on one pound of water in a certain state, then to produce
the same effect on another similar pound of water will
require as much heat, so that twice th,e quantity of heat
is required for two pounds, three times for three pounds,
and so on.
We have no right to assume that because a unit of heat
raises a pound of water at 39 F. one degree, therefore two
units of heat will raise the same pound two degrees ; for the
quantity of heat required to raise the water from 40 to 41
may be different from that which raised it from 39 to 40.
Indeed, it has been found by experiment that more heat
is required to raise a pound of water one degree at high
temperatures than at low ones.
56 Calorimetry.
But if we measure heat according to either of the methods
already described, either by the quantity of a particular kind
of matter which it can change from one easily observed state
to another without altering its temperature, or by the
quantity of a particular kind of matter which it can raise
from one given temperature to another given temperature
we may treat - quantities of heat as mathematical quantities,
and add or subtract them as we please.
We have, however, in the first place to establish that the
heat which by entering or leaving a body in any manner
produces a given change in it is a quantity strictly com-
parable with that which melts a pound of ice, and differs
from it only by being so many times greater or less.
In other words, we have to show that heat of all kinds,
whether coming from the hand, or hot water, or steam, or red-
hot iron, or a flame, or the sun, or from any other source, can
be measured in the same way, and that the quantity of each
required to effect any given change, to melt a pound of ice,
to boil away a pound of water, or to warm the water from one
temperature to another, is the same from whatever source the
heat comes.
To find whether these effects depend on anything except
the quantity of heat received for instance, if they depend in
any way on the temperature of the source of heat suppose
two experiments tried. In the first a certain quantity of heat
(say the heat emitted by a candle while an inch of candle is
consumed) is applied directly to melt ice. In the second the
same quantity of heat is applied to a piece of iron at the
freezing point so as to warm it, and then the heated iron is
placed in ice so as to melt a certain quantity of ice, while the
iron itself is cooled to its original temperature.
If the quantity of ice melted depends on the temperature
of the source from whence the heat proceeds, or on any
other circumstance than the quantity of the heat, the quan-
tity melted will differ in these two cases ; for in the first the
heat comes directly .from an exceedingly hot flame, and in
AH Heat is' of the same kind. 57
the second the same quantity of heat comes from compara-
tively cool iron.
It is found by experiment that no such difference exists,
and therefore heat, considered with respect to its power of
warming things and changing their state, is a quantity strictly
capable of measurement, and not subject to any variations
in quality or in kind.
Another principle, the truth of which is established by
calorimetrical experiments, is, that if a body in a given state
is first heated so as to make it pass through a series of states
denned- by the temperature and the volume of the body
in each state, and if it is then allowed to cool so as to
pass in reverse order through exactly the same series of
states, then the quantity of heat which entered it during the
heating process is equal to that which left it during the
cooling process. By those who regarded heat as a sub-
stance, and called it Caloric, this principle was regarded
as self-evident, and was generally tacitly assumed. We shall
show, however, that though it is true as we have stated it,
yet, if the series of states during the process of heating is
different from that during the process of cooling, the quan-
tities of heat absorbed and emitted may be different. In
fact heat may be generated or destroyed by certain pro-
cesses, and this shows that heat is not a substance. By
finding what it is produced from, and what it is reduced to,
we may hope to determine the nature of heat.
In most of the cases in which we measure quantities of
heat, the heat which we measure is passing out of one body
into another, one of these bodies being the calorimeter
itself. We assume that the quantity of heat which leaves
the one body is equal to that which the other receives,
provided, ist, that neither body receives or parts with heat
to any third body ; and, 2ndly, that no action takes place
among the bodies except the giving and receiving of heat.
The truth of this assumption may be established ex-
perimentally by taking a number of bodies at different
58 Calorim'etry.
temperatures, and determining first the quantity of heat re-
quired to be given to or taken from each separately to bring
it to a certain standard temperature. If the bodies are
now brought to their original temperatures, and allowed to
exchange heat among themselves in any way, then the total
quantity of heat required to be given to the system to bring
it to the standard temperature will be found to be the same
as that which would be deduced from the results in the first
case.
We now proceed to describe the experimental methods
by which these results may be verified, and by which quanti-
ties of heat in general may be measured.
In some of the earlier experiments of Black on the heat
required to melt ice and to boil water, the heat was applied
by means of a flame, and as the supply of heat was assumed
to be uniform, the quantities of heat supplied were inferred
to be proportional to the time during which the supply
continued. A method of this kind is obviously very im-
perfect, and in order to make it at all accurate would need
numerous precautions and auxiliary investigations with
respect to the laws of the production of heat by 'jie flame
and its application to the body which is heated. Another
method, also depending on the observation of time, is more
worthy of -confidence. We shall describe it under the name
of the Method of Cooling.
ICE CALORIMETERS.
Wilcke, a Swede, was the first who employed the melting
of snow to measure the heat given off by bodies in cooling.
The principal difficulty in this method is to ensure that all
the heat given off by the body is employed in melting the
ice, and that no other heat reaches the ice so as to melt it,
or escapes from the water so as to freeze it. This condition
was first fulfilled by the calorimeter of Laplace and La-
voisier, of which the description is given in the Memoirs of
The Ice Calorimeter.
59
FIG. 6.
the French Academy of Sciences for 1780. The instrument
itself is preserved in the Conservatoire des Arts et Metiers
at Paris.
This apparatus, which
afterwards received the
name of Calorimeter, con-
sists of three vessels, one
within another.
The first or innermost
vessel, which we may call
the receiver, is intended to
hold the body from which
the heat to be measured
escapes. It is made of
thin sheet copper, so that
the heat may readily pass
into the second vessel. The
second vessel, or calorimeter proper, entirely surrounds the
first. The lower part of the space between the two vessels is
filled with broken ice at the freezing (or melting) point, and
the first vessel is then covered by means of a lid, which is
itself a vessel full of broken ice. When the ice melts in this
vessel, whether in the lower part or in the cover of the first
vessel, the water trickles down and passes through a drainer,
which prevents any ice from escaping, and so runs out into a
bottle set to catch it. The third vessel, which we may call
the ice jacket, entirely surrounds the second, and is furnished,
like the second, with an upper lid to cover the second. Both
the vessel and the lid are full of broken ice at the freezing
point, but the water formed by the melting of this ice is
carried off to a vessel distinct from that which contains the
water from the calorimeter proper.
Now, suppose that there is nothing in the receiver, and
that the temperature of the surrounding air is above the
freezing point. Any heat which enters the outer vessel
will melt some of the ice in the jacket, and will not pass on,
60 Calorimetry.
and no ice will be melted in the calorimeter. As long as
there is ice in the jacket and in the calorimeter the tem-
perature of both will be the same, that is, the freezing point,
and therefore, by the law of equilibrium of heat, no heat
will pass through the second vessel either outwards o*
inwards. Hence, if any ice is melted in the calorimeter,
the heat which melts it must come from the receiver.
Let us next suppose the receiver at the freezing tempera-
ture ; let the two lids be carefully lifted off for an instant, and
a body at some higher temperature introduced into the re-
ceiver ; then let the lids be quickly replaced. Heat will pass
from the body through the sides of the receiver into the
calorimeter, ice will be melted, and the body will be cooled,
and this process will go on till the body is cooled to the
freezing point, after which there will be no more ice
melted.
If we measure the water produced by the melting of the
ice, we may estimate the quantity of heat which escapes
from the body while it cools from its original temperature to
the freezing point. The receiver is at the freezing point at
the beginning and at the end of the operation, so that the
heating and subsequent cooling of the receiver does not
influence the result
Nothing can be more perfect than the theory and design
of this apparatus. It is worthy of Laplace and of Ijavoisier,
and in their hands it furnished good results.
The chief inconvenience in using it arises from the fact
that the water adheres to the broken ice instead of draining
away from it completely, so that it is impossible to estimate
accurately how much ice has really been melted.
To avoid this source of uncertainty, Sir John Herschel
proposed to fill the interstices of the ice with water at the
freezing point, and to estimate the quantity of ice melted by
the contraction which the volume of the whole undergoes,
since, as we shall afterwards see, the volume of the water is
less than that of the ice from which it was formed. I am
Bunsen's Calorimeter. 6l
not aware that this suggestion was ever developed into an
experimental method.
Bunsen, 1 independently, devised a calorimeter founded on
the same principle, but in the use of which the sources
of error are eliminated, and the physical constants deter-
mined with a degree of precision seldom before attained
in researches of this kind.
Bunsen's calorimeter, as devised by its author, is a small
instrument. The body which is to Fia 7
give off the heat which is to be
measured is heated in a test-tube
placed in a current of steam of
known temperature. It is then
dropped, as quickly as may be, into
the test-tube T of the calorimeter,
which contains water at o C. The
body sinks to the bottom and gives
off heat to the water. The heated water does not rise in the
tube, for the effect of heat on water between o C. and 4 C.
is to increase its density. It therefore remains surrounding
the body at the bottom of the tube, and its heat can escape
only by conduction either upwards through the water, or
through the sides of the tube, which, being thin, afford a
better channel. The tube is surrounded by ice at o C. in
the calorimeter, c, so that as soon as any part of the water
in the tube is raised to a higher temperature, conduction
takes place through the sides, and part of the ice is melted.
This will go on till everything within the tube is again
reduced to o C., and the whole quantity of ice melted by
heat from within is an accurate measure of the heat which
the heated body gives out as it cools to o C.
To prevent any exchange of heat between the calorimeter
c and surrounding bodies, it is placed in a vessel s filled with
snow gathered when new fallen and free from smoke. This
Ann. Sept. 1870, and Phil. Mag. 1871.
62 Calorimetry.
substance, unless the temperature of the room is below o C,
soon acquires and long maintains the temperature of o C.
In preparing the calorimeter, it is filled with distilled water,
from which every trace of air must be expelled by a careful
process of boiling. If there is air in the water, tie process
of freezing expels it and produces bubbles of air, the volume
of which introduces an error of measurement. The lower
part of the calorimeter contains mercury, and communicates
with a bent tube also containing mercury. The upper part
of this tube is bent horizontally, and is carefully calibrated
and graduated. As the mercury and the vessel are always
at the temperature o C., they are of constant volume, and
any changes in the position of the mercury in the graduated
tube are due to the melting of ice in the calorimeter, and
the consequent diminution of volume of the mass of ice and
water in it.
The motions of the extremity of the column of mercury
being proportional to the quantities of heat emitted from
the test-tube into the calorimeter, it is easy to see how
quantities of heat may be compared. In fact, Bunsen has
made satisfactory determinations of the specific heat of those
rare metals, such as indium, of which only a few grammes
have been obtained.
To prepare the calorimeter for use, ice must be formed
in the calorimeter round the test-tube. For this purpose,
Bunsen causes a current of alcohol, cooled below o C. by a
freezing mixture, to flow to the bottom of the test-tube and
up along its sides. In this way the greater part of the water
in the calorimeter is soon frozen. When the apparatus has
been left for a sufficient time in the vessel containing snow,
the temperature of this ice rises to o C., and the apparatus
is ready for use. A great many experiments may be made
after one freezing of the water. 1
' See Pogg. Ann. Sept. 1870, or Phil. Mag. 1871.
Experiments for the Student. 63
METHOD OF MIXTURE.
The second calorimetric method is usually called the
Method of Mixture. This name is given to all the processes
in which the quantity of heat which escapes from one body
is measured by the increase of temperature it produces in
another body into which it escapes. The most perfect
method of ensuring that all the heat which escapes from the
one body passes into the other is to mix them, but in many
cases to which the method is now applied this cannot be
done.
We shall illustrate this method by a few experiments,
which can be performed by the student without any special
apparatus. A few experiments of this kind actually per-
formed by himself will give the student a more intelligent
interest in the subject, and will give him a more lively faith
in the exactness and uniformity of nature, and in the inac-
curacy and uncertainty of our observations, than any reading
of books, or even witnessing elaborate experiments performed
by professed men of science.
1 shall suppose the student to have a thermometer, the
bulb of which he can immerse in the liquids of which the
temperature is to be measured, and I shall suppose the
graduation of the thermometer to be that of Fahrenheit, as
it is the most common in this country.
To compare the effects of heat on water and on lead, take
a strip of sheet lead, weighing, say, one pound, and roll it
into the form of a loose spiral, so that when it is dropped
into water the water may play round every part of it freely.
Take a vessel of a convenient shape, such that the roll of
lead when placed in the vessel will be well covered with a
pound of water.
Hang up the lead by a fine string and dip it in a saucepan
of boiling water, and continue to boil it till it is thoroughly
heated. While this is going on weigh out a pound of cold
64 Calorimetry.
water in your vessel, and ascertain its temperature with
the thermometer. Then lift the roll of lead out of the
boiling water, hold it in the steam till the water is drained
off, and immerse it as quickly as possible in the cold water
in the vessel. By means of the string you may stir it about in
the water so as to bring it in contact with new portions of the
water, and to prevent it from giving its heat directly to the
sides of the vessel.
From time to time observe the temperature of the water
as indicated by the thermometer. In a few minutes the
temperature of the water will cease to rise, and the experi-
ment may then be stopped and the calculation begun.
I shall suppose (for the sake of fixing our ideas) that the
temperature of the water before the hot lead was put in was
57 R, and that the final temperature, when the lead ceased
to impart heat to the water, was 62 F. If we take as our
unit of heat that quantity of heat which would raise a pound
of water at 60 F. one degree, we have here five units of heat
imparted to the water by the lead.
Since the lead was for some time in boiling water, and
was afterwards held in the steam, we may assume its original
temperature to be 212 (this, however, should be tested by the
thermometer). During the experiment the lead cooled 150
from 212 to 62 and gave out, as we have seen, five units
of heat to the water. Hence the difference of the heat of a
pound of lead at 212 and at 62 is five units ; or the same
quantity of heat which will heat a pound of water five degrees
from 57 to 62 will heat a pound of lead 150 degrees from
62 to 212. If we assume, what is nearly though not
exactly true, that the quantity of heat required to heat the
lead is the same for each degree of rise of temperature, then
we might say that to raise a pound of lead five degrees
requires only one thirtieth part of the heat required to raise
a pound of water five degrees.
We have thus made a comparison of the effects of heat on
lead and on water. We have found that the same quantity
Thermal Capacity of a Body. 65
of heat would raise a pound of lead through thirty times as
many degrees as it would raise a pound of water, and we
have inferred that to produce any moderate change of
temperature on a pound of lead requires one-thirtieth of the
heat required to produce the same change on an equal weight
of water.
This comparison is expressed in scientific language by
saying that the capacity cf lead for heat is one-thirtieth of
that of an equal weight of water.
Water is generally taken as a standard substance with
which other substances are compared, and the fact which we
have stated above is expressed in a still more concise mannei
by saying that the specific heat of lead is -g^.
The fact that when equal weights of quicksilver and water
are mixed together the resulting temperature is not the mean of
the temperatures of the ingredients was known to Boerhaave
and Fahrenheit. Dr. Black, however, was the first to explain
this phenomenon and many others by the doctrine which he
established, that the effect of the same quantity of heat in
raising the temperature of the body depends not only on the
amount of matter in the body, but on the kind of matter of
which it is formed. Dr. Irvine, Black's pupil and assistant,
gave to this property of bodies the name of Capacity for
Heat. The expression Specific Heat was afterwards intro-
duced by Gadolin, of Abo, in 1784.
I think we shall secure accuracy, along with the greatest
conformity to established custom, by defining these terms
thus:
DEFINITION OF THE CAPACITY OF A BODY.
The capacity of a body for heat is the number of units of
heat required to raise that body one degree of temperature.
We may speak of the capacity for heat of a particular
thing, such as a copper vessel, in which case the capacity
depends on the weight as well as on the kind of matter*
F
66 Calorimetry.
The capacity of a particular thing is often expressed by
stating the quantity of water which has the same capacity.
We may also speak of the capacity for heat of a substance,
such as copper, in which case we refer to unit of mass of the
substance.
DEFINITION OF SPECIFIC HEAT.
The Specific Heat of a body is the ratio of the quantity of
heat required to raise that body one degree to the quantity
required to raise an equal weight of water one degree.
The specific heat therefore is a ratio of two quantities of
the same kind, and is expressed by the same number, what-
ever be the units employed by the observer, and whatever
therrnometric scale he adopts.
It is very important to bear in mind that these phrases
mean neither more nor less than what is stated in these defi-
nitions.
Irvine, who contributed greatly to establish the fact that
the quantity of heat which enters or leaves a body depends
on its capacity for heat multiplied by the number of degrees
through which its temperature rises or falls, went on to
assume that the whole quantity of heat in a body is equal to
its capacity multiplied by the total temperature of the body,
reckoned from a point which he called the absolute zero.
This is equivalent to the assumption that the capacity of the
body remains the same from the given temperature down-
wards to this absolute zero. The truth of such an assump-
tion could never be proved by experiment, and its falsehood
is easily established by showing that the specific heat of
most liquid and solid substances is different at different
temperatures.
The results which Irvine, and others long after him,
deduced by calculations founded on this assumption are not
only of no value, but are shown to be so by their incon-
sistency with each other.
We shall now return to the consideration of the experiment
Specific Heat of a Substance. 67
witft the lead and water, in order to show how it can be
made more accurate by attending to all the circumstances of
the case. I have purposely avoided doing so at first, as my
object was to illustrate the meaning of ' Specific Heat'
In the former description of the experiment it was
assumed, not only that all the heat which escapes from the
lead enters the water in the vessel, but that it remains in
the water till the conclusion of the experiment, when the
temperatures of the lead and water have become equalised.
The latter part of this assumption cannot be quite true,
for the water must be contained in a vessel of some kind,
and must communicate some of its heat to this vessel, and
also must lose heat at its upper surface by evaporation, &c.
If we could form the vessel of a perfect non-conductor of
heat, this loss of heat from the water would not occur j but
no substance of which a vessel can be formed can be con-
sidered even approximately a non-conductor of heat ; and if
we use a vessel which is merely a slow conductor of heat, it is
very difficult, even by the most elaborate calculations, to
determine how much heat is taken up by the vessel itself
during the experiment.
A better plan is to use a vessel which is a very good
conductor of heat, but of which the capacity for heat is
small, such as a thin copper or silver vessel, and to prevent
this vessel from parting rapidly with its heat by polishing
its outer surface, and not allowing it to touch any large
mass of metal, but rather giving it slender supports and
placing it within a metal vessel having its inner surface
polished.
In this way we shall ensure that the heat shall be quickly
distributed between the water and the vessel, and may con-
sider their temperatures at all times nearly equal, while the
loss of heat from the vessel will take place slowly and at a
rate which may be calculated when we know the temperature
of the vessel and of the air outside.
For this purpose, if we intended to make a very elaborate
w3
68 Calorimetry.
experiment, we should in the first place determine the
capacity for heat of the vessel by a separate experiment, and
then we should put into the vessel about a pound of warm
water and determine its temperature from minute to minute,
while at the same time we observe with another thermometer
the temperature of the air in the room. In this way we should
obtain a set of observations from which we might deduce the
rate of cooling for different temperatures, and compute the
rate of cooling when the vessel is one, two, three, &c. 5
degrees hotter than the air ; and then, knowing the tempe-
rature of the vessel at various stages of the experiment for
finding the specific heat of lead, we should be able to calcu-
late the loss of heat from the vessel due to the cooling during
the continuance of the experiment.
But a much simpler method of getting rid of these diffi-
culties is by the method of making two experiments the first
with the lead which we have described, and the second with
hot water, in which we endeavour to make the circumstances
which cause the loss of heat as similar as we can to those in
the case of the lead.
For instance, if we suppose that the specific gravity of lead
is about eleven times that of water, if instead of a pound of
lead we use one-eleventh of a pound of water, the bulk of the
water will be the same as that of the lead, and the depth of
the water in the vessel will be equally increased by the lead
and the water.
If we also suppose that the specific heat of lead is one-
thirtieth of that of water, then the heat given out by a pound
of lead in cooling 150 will be equal to the heat given out
by one-eleventh of a pound of water in cooling 55.
Hence, if we take one-eleventh of a pound of water at 55
above 62, that is at 117, and pour it into the vessel with
the water as before at 57, we may expect that the level of
the water will rise as much as when the hot lead was put in,
and that the temperature will also rise to about the same
degree. The only difference between the experiments, as
Method of Double Experiments. 69
far as the loss of heat is concerned, is, that the warm water
will raise the temperature of the cold water in a much
shorter time than the hot lead did, so that if we observe the
temperature at the same time after the mixture in both
cases, the loss by cooling will be greater with the warm water
than with the hot lead.
In this way we may get rid of the chief part of the diffi-
culty of many experiments of comparison. Instead of
making one experiment, in which the cooling of the lead is
compared with the heating of the water and the vessel,
including an unknown loss of heat from the outside of the
vessel, we make two experiments, in which the heating
of the vessel and the total loss of heat shall be as nearly as
possible the same, but in which the heat is furnished in the
one case by hot lead, and in the other by warm water.
The student may compare this method with the method of
double weighing invented by Pere Amiot, but commonly
known as Borda's method, in which first the body to be
weighed, and then the weights, are placed in the same scale,
and weighed against the same counterpoise.
We shall illustrate this method by finding the effect of steam
in heating water, and comparing it with that of hot water.
Take a kettle, and make the lid tight with a little flour and
water, and adapt a short india-rubber tube to the spout, and
a tin or glass nozzle to the tube. Make the water in the kettle
boil, and when the steam comes freely through the nozzle
dip it in cold water, and you will satisfy yourself that the
steam is rapidly condensed, every bubble of steam as it
issues collaps.ing with a sharp rattling noise.
Having made yourself familiar with the general nature of
the experiment of the condensation of steam, you may
proceed to measure the heat given out to the water. For
this purpose, put some cold water in your vessel, say about
three-quarters of a pound. Weigh the vessel and water
carefully, and observe the temperature of the water ; then,
while the steam flows freely from the nozzle, condense steam
70 Calorimetry.
in the water for a short time, and remove the nozzle ; observe
the temperature and weigh the water in its vessel again,
taking note of the time of the experiment.
Let us suppose the original weight . . 5,ooo grains
Weight after the condensation of steam . 5, 100 grains
Hence the weight of steam condensed is . 100 grains
Temperature of water at first . . . 55 F.
Temperature at the end of experiment . 77 F.
Rise of temperature 22
Let us now make a second experiment, as like the first
as we can, only differing from it by the use of hot watei
instead of steam to produce the rise of temperature.
It is impossible in practice to ensure that everything shall
be exactly the same, but after a few trials we may select a
method which will nearly, if not quite, fulfil the conditions.
Thus it is easy to bring the vessel and cold water to the
same weight as before, namely, 5,000 grains ; but we shall
suppose the temperature now to be 56 F. instead of 55.
We now pour in water at 176 F. gradually, so as to make this
experiment last about as long as the first, and we find that
the temperature is now 76, and the weight 6,000 grains.
Hence 1,000 grains of water cooling 100 raise the vessel
and its contents 22.
Assuming that the specific heat of water is the same at
all temperatures, which is nearly, though by no means
exactly, true, the quantity of' heat given out by the water
in the second experiment is equal to what would raise
100,000 grains of water one degree.
In the experiment with the steam the temperatures were
nearly though not exactly equal, but the rise o'f temperature
was greater in the proportion of 22 to 20. Hence we may
conclude that the quantity of heat which produced this
heating effect in the experiment with steam was greater than
in the experiment with water in the same proportion. This
makes the heat given out by the steam equal to that which
would raise 110,000 grains of water one degree.
Latent Heat of Steam. 71
This was done by the condensation and subsequent
cooling of 100 grains of steam. Let us begin with the heat
given out by the 100 grains of water at 212 F., into which
the steam is condensed. It is cooled from 212 to 77 or
135, and gives out therefore an amount of heat which
would raise 13,500 grains of water one degree. But the
whole effect was 110,000, so that there is an amount of
heat which would raise 96,500 grains of water one degree,
which must be given out during the condensation of the
steam, and before the cooling begins. Hence each grain
of steam in condensing gives out as much heat as would
raise 965 grains of water i F. or 536 grains i Centi-
grade.
The fact that steam at the boiling point gives out a large
quantity of heat when it is condensed into water which is
still at the same temperature, and the converse fact that in
order to convert water at the boiling temperature into steam
of th same temperature a large quantity of heat must
be communicated to it, was first clearly established by
Black in 1757.
He expressed it by saying that the latent heat of steam
is 965 F., and this form of expression is still in use, and
we should take it to mean neither more nor less than what
we have just stated.
Black, however, and many of his followers, supposed heat
to be a substance which when it makes a thing hot is
sensible, but which when it is not perceived by the hand
or the thermometer still exists in the body in a latent or
concealed state. Black supposed that the difference between
boiling water and steam is, that steam contains a great deal
more caloric than the hot water, so that it may be con-
sidered a compound of water and caloric ; but, since this
additional caloric produces no effect on the temperature,
but lurks concealed in the steam ready to appear when it is
condensed, he called this part of the heat latent heat.
In considering the scientific value of Black's discovery of
JT2 Calorimetry.
latent heat, and of his mode of expressing it, we should
recollect that Black himself in 1754 was the discoverer of the
fact that the bubbles formed when marble is put into an acid
consist of a real substance different from air, which, when free,
is similar to air in appearance, but when fixed may exist in
liquids and in solids. This substance, which we now call
carbonic acid, Black called fixed air, and this was the first
gaseous body distinctly recognised as such. Other airs or
gases were afterwards discovered, and the impulse given to
chemistry was so great, on account of the extension of the
science to these attenuated bodies, that most philosophers
of the time were of opinion that heat, light, electricity, and
magnetism, if not the vital force itself, would sooner or later
be added to the list. Observing, however, that the gases
could be weighed, while the presence of these other agents
could not be detected by the balance, those who admitted
them to the rank of substances called them imponderable
substances, and sometimes, on account of their mobility,
imponderable fluids.
The analogy between the free and fixed states of carbonic
acid and the sensible and latent states of heat encouraged
the growth of materialistic phrases as applied to heat ; and
it is evident that the same way of thinking led electricians to
the notion of disguised or dissimulated electricity, a notion
which survives even yet, and which is not so easily stripped
of its erroneous connotation as the phrase ' latent heat.'
It is worthy of remark that Cavendish, though one of the
greatest chemical discoverers of his time, would not accept
the phrase 'latent heat' He prefers to speak of the
generation of heat when steam is condensed, a phrase
inconsistent with the notion that heat is matter, and
objects to Black's term as relating 'to an hypothesis
depending on the supposition that the heat of bodies is
owing to their containing more or less of a substance
called the matter of heat ; and, as I think Sir Isaac Newton's
opinion that heat consists in the internal motion of the
Latent Heat. 7 3
particles of bodies much the most probable, I chose to use
tne expression, " heat is generated." ' l
We shall not now be in danger of any error if we use
latent heat as an expression meaning neither more nor less
than this :
DEFINITION. Latent heat is the quantity of heat which
must be communicated to a body in a given state in order
to convert it into another state without changing its tempera-
ture.
We here recognise the fact that heat when applied to a
body may act in two ways by changing its state, or by
raising its temperature and that in certain cases it may act
by changing the state without increasing the temperature.
The most important cases in which heat is thus employed
are
1. The conversion of solids into liquids. This is called
melting or fusion. In the reverse process of freezing or
solidification heat must be allowed to escape from the body
to an equal amount
2. The conversion of liquids (or solids) into the gaseous
state. This is called evaporation, and its reverse condensa-
tion.
3. When a gas expands, in order to maintain the tem-
perature constant, heat must be communicated to it, and
this, when properly defined, may be called the latent heat of
expansion.
4. There are many chemical changes during which heat is
generated or disappears.
In all these cases the quantity of heat which enters or
leaves the body may be measured, and in order to express
the result of this measurement in a convenient form, we
may call it the latent heat required for a given change in the
substance.
We must carefully remember that all that we know about
heat is what occurs when it passes from one body to another,
1 Phil. Trans. 1783, quoted by Forbes. Dissertation VI. Encyc. Brit,
74 Elementary Dynamical Principles.
and that we must not assume that after heat has entered
a substance it exists in the form of heat within that
substance. That we have no right to make such an
assumption will be abundantly shown by the demonstration
that heat may be transformed into and may be produced
from something which is not heat.
Regnault's method of passing large quantities of the
substance through the calorimeter will be described in
treating of the properties of gases, and the Method oi
Cooling will be considered in the chapter on Radiation.
CHAPTER IV.
ELEMENTARY DYNAMICAL PRINCIPLES.
IN the first part of this treatise we have confined ourselves
to the explanation of the method of ascertaining the tem-
perature of bodies, which we call thermornetry, and the
method of measuring the quantity of heat which enters or
leaves a body, and this we call calorimetry. Both of these
are required in order to study the effects of heat upon bodies;
but we cannot complete this study without making measure-
ments of a mechanical kind, because heat and mechanical
force may act on the same body, and the actual result
depends on both actions. I propose, therefore, to recall to
the student's memory some of those dynamical principles
which he ought to bring with him to the study of heat, and
which are necessary when he passes from purely thermal
phenomena, such as we have considered, to phenomena in-
volving pressure, expansion, &c., and which will enable him
afterwards to proceed to the study of thermodynamics
proper, in which the relations of thermal phenomena among
themselves are deduced from purely dynamical principles.
The most important step in the progress of every
Measurement of Quantities. 75
science is the measurement of quantities. Those whose
curiosity is satisfied with observing what happens have
occasionally done service by directing the attention of others
to the phenomena they have seen ; but it is to those who
endeavour to find out how much there is of anything that
we owe all the great advances in our knowledge.
Thus every science has some instrument of precision,
which may be taken as a material type of that science which
it has advanced, by enabling observers to express their
results as measured quantities. In astronomy we have
the divided circle, in chemistry the balance, in heat the
thermometer, while the whole system of civilised life may
be fitly symbolised by a foot rule, a set of weights, and a
clock. I shall, therefoie, make a few remarks on the
measurement of quantities.
Every quantity is expressed by a phrase consisting of two
components, one of these being the name of a number, and
the other the name of a thing of the same kind as the
quantity to be expressed, but of a certain magnitude agreed
on among men as a standard or unit.
Thus we speak of two days, of forty-eight hours.
Each of these expressions has a numerical part and a
denominational part, the numerical part being a number,
whole or fractional, and the denominational part being the
name of the thing, which is to be taken as many times as is
indicated by the number.
If the numerical part is the number one, then the quantity
is the standard quantity itself, as when we say one pound,
or one inch, or one day. A quantity of which the numerical
part is unity is called a unit. When the numerical part is
some other number, the quantity is still said to be referred to,
or to be expressed in terms of that quantity which would be
denoted if the number were one, and which is 'called the unit.
In all cases the unit is a quantity of the same kind as the
quantity which is expressed by means of it.
In many cases several units of the same kind are in use,
76 Elementary Dynamical Principles.
as miles, yards, feet, and inches, as measures of length ; cubic
yards, gallons, and fluid ounces, as measures of capacity ;
besides the endless variety of units which have been adopted
by different nations, and by different districts and different
trades in the same nation.
When a quantity given in terms of one unit has to be ex-
pressed in terms of another, we find the number of times
the second unit is contained in the first, and multiply this
by the given number.
Hence the numerical part of the expression of the same
quantity varies inversely as the unit in which it is to be ex-
pressed, as in the example, two days and forty-eight hours,
which mean the same thing.
There are many quantities which can be defined in terms
of standard quantities of a different kind. In this case we
make use of derived units. For instance, as soon as we
have fixed on a measure of length, we may define by means
of it not only all lengths, but also the area of any surface,
and the content of any space. For this purpose, if the foot
is the unit of length, we construct, by Euclid I. 46, a square
whose side is a foot, and express all areas in terms of this
square foot, and by constructing a cube whose edge is
a foot we have defined a cubic foot as a unit of capacity.
We also express velocities in miles an hour, or feet in a
second, &c.
In fact, all quantities with which we have to do in dynamics
may be expressed in terms of units derived by definition from
the three fundamental units of Length, Mass, and Time.
STANDARD OF LENGTH.
It is so important to mankind that these units should be
well defined that in all civilised nations they are defined by
the State with' reference to material standards, which are pre-
served with the utmost care. For instance, in this country
it was enacted by Parliament } ' that the straight line or
1 18 & 19 Viet. c. 72, July 30, 1855.
Units of Length. 77
distance between the centres of the transverse lines in the
two gold plugs in the bronze bar deposited in the office
of the Exchequer shall be the genuine standard yard
at 62 F., and if lost it shall be replaced by means of its
copies.'
The authorised copies here referred to are those which are
preserved at the Royal Mint, the Royal Society of London,
the Royal Observatory at Greenwich, and the New Palace
at Westminster. Other copies have been made with great
care, and with these all measures of length must be com-
pared.
The length of the Parliamentary standard was chosen so
as to be as nearly as possible equal to that of the best
standard yards formerly used in England. The State, there-
fore, endeavoured to maintain the standard of its ancient
magnitude, and by its authority it has defined the actual
magnitude of this standard with all the precision of which
modern science is capable.
The metre derives its authority as a standard from a law
of the French Republic in 1795. '
It is defined to be the distance between the ends of a rod
of platinum made by Borda, 1 the rod being at the tempera-
ture of melting ice. This distance was chosen without
reference to any former measures used in France. It was
intended to be a universal and not a national measure, and
was derived from Delambre and Mechain's measurement of
the size of the earth. The distance measured along the
earth's surface from the pole to the equator is nearly ten
million of metres. If, however, in the progress of geodesy, a
different result should be obtained from that of Delambre,
the metre will not be altered, but the new result will be
expressed in the old metres. The authorised standard of
length is therefore not the terrestrial globe, but Borda's
1 Mtre conforme a la loi du 18 Germinal, an III. Pr^sent^ le
4 Messidor, an VII.
78 Elementary Dynamical Principles.
platinum rod, which is much more likely to be accurately
measured.
The value of the French system of measures does not
depend so much on the absolute values of the units adopted
as on the fact that all the units of the same kind are
connected together by a decimal system of multiplication
and division, so that the whole system, under the name of
the metrical system, is rapidly gaining ground even in
countries where the old national system has been carefully
defined.
The metre is 39*37043 British inches.
STANDARD OF MASS.
By the Act above cited a weight of platinum marked
' P. S, 1844, i lb./ deposited in the office of the Exchequer,
' shall be the legal and genuine standard measure of weight,
and shall be and be denominated the Imperial Standard
Pound Avoirdupois, and shall be deemed to be the only
standard measure of weight from which all other weights and
other measures having reference to weight shall be derived,
computed, and ascertained, and one equal seven-thousandth
part of such pound avoirdupois shall be a grain, and five
thousand seven hundred and sixty such grains shall be and
be deemed to be a pound troy. If at any time hereafter the
said Imperial Standard Pound Avoirdupois be lost or in any
manner destroyed, defaced, or otherwise injured, the Com-
missioners of Her Majesty's Treasury may cause the same to
be restored by reference to or adoption of any of the copies
aforesaid, 1 or such of them as may remain available for that
purpose/
The construction of this standard was entrusted to Pro-
fessor W. H. Miller, who has given an account of the
methods employed in a paper, 2 which may be here referred
to as a model of scientific accuracy.
1 In the same places as the Standards of Length.
2 Phil Trans. 1856, p. 753.
Units of Mass. 79
The French standard of mass is the Kilogramme des
Archives, made of platinum by Borda, and is intended to
represent the mass of a cubic decimetre of distilled water
at the temperature 4 C.
The actual determination of the density of water is an
operation which requires great care, and the differences
between the results obtained by the most skilful observers,
though small, are a thousand times greater than the differ-
ences of the results of a comparison of standards by weighing
them. The differences of the values of the density of water
as found by careful observers are as much as a thousandth
part of the whole, whereas the method of weighing admits
of an accuracy of within one part in five millions.
Hence the French standards, though originally formed
to represent certain natural quantities, must be now con-
sidered as arbitrary standards, of which copies are to be
taken by direct comparison. The French or metric system
has the advantage of a uniform application of the decimal
method, and it is also in many cases convenient to remember
that a cubic metre of water is a tonne, a cubic decimetre a
kilogramme, a cubic centimetre a gramme, and a cubic
millimetre a milligramme, the water being at its maximum
density or at about 4 C.
In 1826 the British standard of mass was defined by
saying that a cubic inch of water at 62 F. contains 252-458
grains, and though this is no longer a legal definition, we
may take it as a rough statement of a fact, that a cubic inch
of water weighs about 252-5 grains, a cubic foot about 1,000
ounces avoirdupois, and a cubic yard about three-quarters of
a ton. Of these estimates the second is the furthest from
the truth.
Professor Miller has compared the British and French
standards, and finds the Kilogramme des Archives equal to
i543 2 '34874 grains.
From these legal definitions it will be seen that what is
generally called a standard of weight is a certain piece of
8o Elementary Dynamical Principles.
platinum that is, a particular body the quantity of matter in
which is taken and denned by the State to be a pound or a
kilogramme.
The weight strictly so called that is, the tendency ofthi?
body to move downwards is not invariable, for it depends
on the part of the world where it is placed, its weight being
greater at the poles than at the equator, and greater at the
level of the sea than at the top of a mountain.
What is really invariable is the quantity of matter in the
body, or what is called in scientific language the mass of the
body, and even in commercial transactions what is generally
aimed at in weighing goods is to estimate the quantity of
matter, and not to determine the force with which they tend
downwards.
In fact, the only occasions in common life in which it is
required to estimate weight considered as a force is when we
have to determine the strength required to lift or carry
things, or when we have to make a structure strong enough
to support their weight. In all other cases the word weight
must be understood to mean the quantity of the thing as
determined by the process of weighing against ' standard
weights!
As a great deal of confusion prevails on this subject in
ordinary language, and still greater confusion has been
introduced into books on mechanics by the notion that a
pound is a certain force, instead of being, as we have seen, a
certain piece of platinum, or a piece of any other kind of
matter equal in mass to the piece of platinum, I have
thought it worth while to spend some time in defining
accurately what is meant by a pound and a kilogramme.
ON THE UNIT OF TIME.
All nations derive their measures of time from the
apparent motions of the heavenly bodies. The motion of
rotation of the earth about its axis is very nearly indeed
uniform, and the measure of time in which one day is equal
Unit of Time. 81
to the time of revolution of the earth about its axis, or more
exactly to the interval between successive transits of the first
point of Aries, is used by astronomers under the name of
sidereal time.
Solar time is that which is given by a sun-dial, and is
not uniform. A uniform measure of time, agreeing with
solar time in the long run, is called mean solar time, and is
that which is given by a correct clock. A solar day is longer
than a sidereal day. In all physical researches mean solar
time is employed, and one second is generally taken as the
unit of time.
The evidence upon which we form the conclusion that
two different portions of time are or are not equal can only
be appreciated by those who have mastered the principles
of dynamical reasoning. I can only here assert that the
comparison, for example, of the length of a day at present
with the length of a day 3,000 years ago is by no means
an unfruitful enquiry, and that the relative length of these
days may be determined to within a small fraction of a
second. This shows that time, though we conceive it
merely as the succession of our states of consciousness, is
capable of measurement, independently, not only of our
mental states, but of any particular phenomenon whatever.
ON MEASUREMENTS FOUNDED ON THE THREE
FUNDAMENTAL UNITS.
In the measurement of quantities differing in kind from
the three units, we may either adopt a new unit independently
for each new quantity, or we may endeavour to define a unit
of the proper kind from the fundamental units. In the latter
case we are said to use a system of units. For instance, if
we have adopted the foot as a unit of length, the systematic
unit of capacity is the cubic foot.
The gallon, which is a legal measure in this country, is
unsystematic considered as a measure of capacity, as it
G
82 Elementary Dynamical Principles.
contains the awkward number of 277*274 cubic inches. The
gallon, however, is never tested by a direct measurement of
its cubic contents, but by the condition that it must contain
ten pounds of water at 62 F.
DEFINITION OF DENSITY. The density of a body is
measured by the number of units of mass in unit of volume,
of the substance.
For instance, if the foot and the pound be taken as
fundamental units, then the density of anything is the
number of pounds in a cubic foot. The density of water
is about 62*5 pounds to the cubic foot. In the metric
system, the density of water is one tonne to the stere, one
kilogramme to the litre, one gramme to the cubic centi-
metre, and one milligramme to the cubic millimetre.
We shall sometimes have to use the word rarity, to
signify the inverse of density, that is, the volume of unit of
mass of a substance.
DEFINITION OF SPECIFIC GRAVITY. The specific gravity
of a body is the ratio of its density to that of some standard
substance, generally water.
Since the specific gravity of a body is the ratio of two
things of the same kind, it is a numerical quantity, and has
the same value, whatever national units are employed by
those who determine it. Thus, if we say that the specific
gravity of mercury is about 13*5, we state that mercury
is about thirteen and a half times heavier than an equal bulk
of water, and this fact is independent of the way in which
we measure either the mass or the volume of the liquids.
DEFINITION OF UNIFORM VELOCITY. The velocity of a
body moving uniformly is measured by the number of units of
Imgth travelled over in unit of time.
Thus we speak of a velocity of so many feet or metres
per second.
DEFINITION OF MOMENTUM. The momentum of a body is
measured by the product of the velocity of the body into the
number of units of mass in the body
Measurement of Force. 83
DEFINITION OF FORCE. Force is whatever changes or
tends to change the motion of a body by altering either its direc-
tion or its magnitude; and a force acting on a body is measured
by the momentum it produces in its own direction in unit
of time.
The unit of force is that force which if it acted on unit of
mass for unit of time would produce in it unit of velocity.
For the British unit of force the name of Poundal has been
proposed by Prof. James Thomson. It is that force which,
if it acted for a second on a pound, would produce in it a
velocity of one foot per second.
In the centimetre-gramme-second system, adopted by the
Committee on Units of the British Association, the unit of
force is the Dyne. A dyne acting for one second on a
gramme would give it a velocity of one centimetre per
second.
The weight of any body at London, acting on that body
for a second, would produce in it a velocity of 32-1889 feet
per second. Hence the weight of a pound at London is
32-1889 poundals.
At Paris the velocity of a body after falling freely for one
second is 980*868 centimes per second. Hence the weight
of a gramme at Paris is 980*868 dynes.
It is so convenient, especially when all our experiments
are conducted in the same place, to express forces in terms
of the weight of a pound or a gramme, that in all countries
the first measurements of forces were made in this way, and
a force was described as a force of so many pounds weight
or grammes weight. It was only after the measurements of
forces made by persons in different parts of the world had
to be compared that it was found that the weight of a
pound or a gramme is different in different places, and
depends on the intensity of gravitation, or the attraction of
the earth ; so that for purposes of accurate comparison all
forces must be reduced to absolute or dynamical measure
as explained above. We shall distinguish the measure by
01
84 Elementary Dynamical Principles.
comparison with weight as the gravitation measure of force.
To reduce forces expressed in gravitation measure to abso-
lute measure, we must multiply the number denoting the
force in gravitation measure by the value of the intensity of
gravity expressed in the same metrical system. The value
of the intensity of gravity is a very important number in all
scientific calculations, and it is generally denoted by the
letter g. The number g may be defined in any of the
following ways, which are all equivalent :
g is a number expressing the velocity produced in a falling
body in unit of time.
g is a number expressing twice the distance through which a
body falls in unit of time.
g is a number expressing the weight of unit of mass in
absolute measure.
The value of g is generally determined at any place by
experiments with the pendulum. These experiments re-
quire great care, and the description of them does not
belong to our present subject. The value of g may be
found with sufficient accuracy for the present state of science
by means of the formula,
g= G (1 0-0025659 cos 2 X) Ji /2 $- CjJ
In this formula, G is the intensity of gravity a the mean
level of the sea in latitude 45 :
0=32-1703 poundals to the pound, or 9*80533 dynes to the
gramme.
\ is the latitude of the place. The formula shows that the
force of gravity at the level of the sea increases from the
equator to the poles. The last factor of the formula ex-
presses, according to the calculations of Poisson, 1 the
effect of the height of the place of observation above
the level of the sea in diminishing the force of gravity.
The symbol p represents the mean density of the whole
earth, which is probably about 5 J times that of water, p'
1 Traitt de Mttcaniqut t t. ii. p. 629.
Weight. 85
represents the mean density of the ground just below the
place of observation, which may be taken at about 2^
times the density of water, so that we may write
2 =1*32 nearly.
2 p
z Is the height of the place above the level of the sea, in
feet or metres, and r is the radius of the earth :
r 20,886,852 feet, or 6,366,198 metres.
For rough purposes it is sufficient to remember that in
Britain the intensity of gravity is about 32*2 poundals to the
pound, and in France about 980 dynes to the gramme.
The reason why, in all accurate measurements, we have
to take account of the variation of the intensity of gravity in
different places is, that the absolute value of any force, such
as the pressure of air of a given density and temperature,
depends entirely on the properties of air, and not on
the force of gravity at the place of observation. If,
therefore, this pressure has been observed in gravitation
measure, that is, in pounds on the square inch, or in inches
of mercury, or in any way in which the weight of some
substance is made to furnish the measure of the pressure, then
the results so obtained will be true only as long as the
intensity of gravity is the same, and will not be true without
correction at a place in a different latitude from the place of
observation. Hence the use of reducing all measures of
force to absolute measure.
In a rude age, before the invention of means for
overcoming friction, the weight of bodies formed the chief
obstacle to setting them in motion. It was only after
some progress had been made in the art of throwing
missiles, and in the use of wheel- carriages and floating
vessels, that men's minds became practically impressed
with the idea of mass as distinguished from weight. Ac-
cordingly, while almost all the metaphysicians who dis-
cussed the qualities of matter assigned a prominent place to
86 Elementary Dynamical Principles.
weight among the primary qualities, few or none of them
perceived that the sole unalterable property of matter is its
mass. At the revival of science this property was expressed
by the phrase ' the inertia of matter ; ' but while the men of
science understood by this term the tendency of the body
to persevere in its state of motion (or rest), and considered
it a measurable quantity, those philosophers who were un-
acquainted with science understood inertia in its literal
sense as a quality mere want of activity or laziness.
Even to this day those who are not practically familiar
with the free motion of large masses, though they all admit
the truth of dynamical principles, yet feel little repugnance
in accepting the theory known as Boscovich's that sub-
stances are composed of a system of points, which are
mere centres of force, attracting or repelling each other. It
is probable that many qualities of bodies might be explained
on this supposition, but no arrangement of centres of force,
however complicated, could account for the fact that a body
requires a certain force to produce in it a certain change
of motion, which fact we express by saying that the body
has a certain measurable mass. No part of this mass can
be due to the existence of the supposed centres of force.
I therefore recommend to the student that he should
impress his mind with the idea of mass by a few experiments,
such as setting in motion a grindstone or a well-balanced
wheel, and then endeavouring to stop it, twirling a long
pole, &c., till he comes to associate a set of acts and sensa-
tions with the scientific doctrines of dynamics, and he will
never afterwards be in any danger of loose ideas on these
subjects. He should also read Faraday's essay on Mental
Inertia, 1 which will impress him with the proper meta-
phorical use of the phrase to express, not laziness, but
habitude.
1 Life, by Dr. Bence Jones, vol. i. p. 268.
Work. 87
ON WORK AND ENERGY.
Work is done when resistance is overcome, and the quantity
of work done is measured by the product of the resisting
force and the distance through which that force is over-
come.
Thus, if one pound is lifted one foot high in opposition to
the force of gravity, a certain amount of work is done, and
this quantity is known among engineers as a foot-pound.
If a body whose mass is twenty pounds is lifted ten feer,
this might be done by taking one of the pounds and raising it
first one foot and then another till it had risen ten feet, and
then doing the same with each of the remaining pounds, so
that the quantity of work called a foot-pound is performed
200 times in raising twenty pounds ten feet. Hence the
work done in lifting a body is found by multiplying the weight
of the body in pounds by the height in feet. The result
is the work in foot-pounds.
The foot-pound is a gravitation measure, depending on
the intensity of gravity at the place. To reduce it to absolute
measure we must multiply the number of foot-pounds by the
intensity of gravity at the place to get the number of foot-
poundals.
The work done when we raise a heavy body is done in
overcoming the attraction of the earth. Work is also done
when we draw asunder two magnets which attract each
other, when we draw out an elastic cord, when we compress
air, and, in general,* when we apply force to anything which
moves in the direction of the force.
There is one case of the application of force to a moving
body which is of great importance, namely, when the force
is employed in changing the velocity of the body.
Suppose a body whose mass is M (M pounds or M grammes)
to be moving in a certain direction with a velocity which
we shall call v, and let a force, which we shall call F, be
88 Elementary Dynamical Principles.
applied to the body in the direction of its motion. Let us
consider the effect of this force acting on the body for a
very small time T, during which the body moves through
the space s, and at the end of which its velocity is v'.
To ascertain the magnitude of the force F, let us consider
the momentum which it produces in the body, and the time
during which the momentum is produced.
The momentum of the beginning of the time T was MZ/,
and at the end of the time T it was MZ/, so that the momentum
produced by the force F acting for the time T is uv' uv.
But since forces are measured by the momentum produced
in unit of time, the momentum produced by F in one unit
of time is F, and the momentum produced by F in T units of
time is FT. Since the two values are equal,
FT = M(Z>' v).
This is .one form of the fundamental equation of dynamics.
If we define the impulse of a force as the average value of
the force multiplied by the time during which it acts, then
this equation may be expressed in words by saying that
the impulse of a force is equal to the momentum produced
by it.
We have next to find s, the space described by the body
during the -time T. If the velocity had been uniform, the
space described would have been the product of the time
by the velocity. When the velocity is not uniform the time
must be multiplied by the mean or average velocity to get
the space described. In both these cases in which average
force or average velocity is mentioned, the time is supposed
to be subdivided into a number of equal parts, and the
average is taken of the force or of the velocity for all these
divisions of the time. In the present case, in which the
time considered is so small that the change of velocity is also
small, the average velocity during the time T may be taken
as the arithmetical mean of the velocities at he beginning
and at the end of the time, or J (v + v').
Kinetic Energy. 89
Hence the space described is
s = \(v + 2/)T.
This may be considered as a kinematical equation, since
it depends on the nature of motion only, and not on that
of the moving body.
If we multiply together these two equations we get
and if we divide by T we find
FS
Now FS is the work done by the force F acting on the
body while it moves in the direction of F through a space s.
If we also denote -^Mz/ 2 , the mass of the body multiplied by
half the square of its velocity, by the expression the kinetic
energy of the body, then ^Mz/ 2 will be the kinetic energy
after the action of the force F through a space s.
We may now express the equation in words by saying
that the work done by the force F in setting the body in
motion is measured by the increase of kinetic energy during
the time that the force acts.
We have proved that this is true when the interval of time
during which the force acts is so small that we may consider
the mean velocity during that time as equal to the arithme-
tical mean of the velocities at the beginning and end of the
time. This assumption, which is exactly true when the
force is uniform, is approximately true in every case when
the time considered is small enough.
By dividing the whole time of action of the force into
small parts, and proving that in each of these the work done
by the force is equal to the increase of kinetic energy of the
body, we may, by adding the different portions of the work
and the different increments of energy, arrive at the result
that the total work done by the force is equal to the total
increase of kinetic energy.
If the force acts on the body in the direction opposite to
the motion, the kinetic energy of the body will be diminished
yO Elementary Dynamical Principles.
instead of increased, and the force, instead of doing work on
the body, will be a resistance which the body in its motion
overcomes. Hence a moving body can do work in over-
coming resistance as long as it is in motion, and the work
done by the moving body is equal to the diminution of its
kinetic energy, till, when the body is brought to rest, the
whole work it has done is equal to the whole kinetic energy
which it had at first.
We now see the appropriateness of the name kinetic
energy, which we have hitherto used merely as a name for
the product ^Mz/ 2 . For the energy of a body may be
defined as the capacity which it has of doing work, and is
measured by the quantity of work which it can do. The
kinetic energy of a body is the energy which it has in
virtue of being in motion, and we have just shown that its
value may be found by multiplying the mass of the body by
half the square of the velocity.
In our investigation we have, for the sake of simplicity,
supposed the force to act in the same direction as the
motion. To make the proof perfectly general, as it is given
in treatises on dynamics, we have only to resolve the actual
force into two parts, one in the direction of the motion and
the other at right angles to it, and to observe that the part
at right angles to the motion can neither do any work on the
body nor change the velocity or the kinetic energy, so that
the whole effect, whether of work or of alteration of kinetic
energy, depends on the part of the force which is in the
direction of the motion.
The student, if not familiar with this subject, should refer
to some treatise on dynamics, and compare the investigation
there given with the outline of the reasoning given above.
Our object at present is to fix in our minds what is meant
by Work and Energy.
The great importance of giving a name to the quantity
which we call Kinetic Energy seems to have been first recog-
nised by Leibnitz, who gave to the product of the mass b^
Kinetic and Potential Energy. 91
the square of the velocity the name of Vis Viva. This is
twice the kinetic energy.
Newton, in a scholium to his Third Law of Motion, has
stated the relation between work and kinetic energy in a
manner so perfect that it cannot be improved, but at the
same time with so little apparent effort or desire to attract
attention that no one seems to have been struck with the
great importance of the passage till it was pointed out
recently by Thomson and Tait.
The use of the term Energy, in a scientific sense, to express
the quantity of work a body can do, was introduced by Dr.
Young (' Lectures on Natural Philosophy,' Lecture VIII.).
The energy of a system of bodies acting on one another
with forces depending on their relative positions is due partly
to their motion, and partly to their relative position.
That part which is due to their motion was called Actual
Energy by Rankine, and Kinetic Energy by Thomson and
Tait.
That part which is due to their relative position depends
upon the work which the various forces would do if the
bodies were to yield to the action of these forces. This is
called the Sum of the Tensions by Helmholtz, in his cele-
brated memoir on the ' Conservation of Force.' * Thomson
called it Statical Energy, and Rankine introduced the term
Potential Energy, a very felicitous name, since it not only
signifies the energy which the system has not in possession,
but only has the power to acquire, but it also indicates that
it is to be found from what is called (on other grounds) the
Potential Function.
Thus when a heavy body has been lifted to a certain
height above the earth's surface, the system of two bodies, it
and the earth, have potential energy equal to the work
which would be done if the heavy body were allowed to
descend till it is stopped by the surface of the earth.
If the body were allowed to fall freely, it would acquire
1 Berlin, 1847. Translated in Taylor's Scientific Memoirs, Feb. 1853.
92 Elementary Dynamical Principles,
velocity, and the kinetic energy acquired would be exactly
equal to the potential energy lost in the same time.
It is proved in treatises on dynamics that if, in any system
of bodies, the force which acts between any two bodies is in
the line joining them, and depends only on their distance,
and not on the way in which they are moving at the time,
then if no other forces act on the system, the sum of the
potential and kinetic energy of all the bodies of the system
will always remain the same.
This principle is called the Principle of the Conservation
of Energy ; it is of great importance in all branches of science,
and the recent advances in the science of heat have been
chiefly due to the application of this principle.
We cannot indeed assume, without evidence of a satis-
factory nature, that the mutual action between any two parts
of a real body must always be in the line joining them, and
must depend only on their distance. We know that this is
the case with respect to the attraction of bodies at a distance,
but we cannot make any such assumption concerning the
internal forces of bodies of whose internal constitution we
know next to nothing.
We cannot even assert that all energy must be either
potential or kinetic, though we may not be able to conceive
any other form. Nevertheless, the principle has been de-
monstrated by dynamical reasoning to be absolutely true for
systems fulfilling certain conditions, and it has been proved
by experiment to be true within the limits of error of obser-
vation, in cases where the energy takes the forms of heat,
magnetisation, electrification, &c., so that the following state-
ment is one which, if we cannot absolutely affirm its neces-
sary truth, is worthy of being carefully tested, and traced
into all the conclusions which are implied in it.
GENERAL STATEMENT OF THE CONSERVATION OF ENERGY.
* 77ie total energy of any body or system of bodies is a
quantity which can neither be increased nor diminished by any
Conservation of Energy. 93
mutual action of these bodies, though it may be transformed
into any of the forms of which energy is susceptible!
If by the application of mechanical force, heat, or any
other kind of action to a body, or system of bodies, it is
made to pass through any series of changes, and at last to
return in all respects to its original state, then the energy
communicated to the system during this cycle of operations
must be equal to the energy which the system communicates
to other bodies during the cycle.
For the system is in all respects the same at the beginning
and at the end of the cycle, and in particular it has the same
amount of energy in it ; and therefore, since no internal
action of the system can either produce or destroy energy,
the quantity of energy which enters the system must be
equal to that which leaves it during the cycle.
The reason for believing heat not to be a substance
is that it can be generated, so that the quantity of it may
be increased to any extent, and it can also be destroyed,
though this operation requires certain conditions to be
fulfilled.
The reason for believing heat to be a form of energy is
that heat may be generated by the application of work, and
that for every unit of heat which is generated a certain
quantity of mechanical energy disappears. Besides, work
may be done by the action of heat, and for every foot-
pound of work so done a certain quantity of heat is put out
of existence.
Now when the appearance of one thing is strictly con-
nected with the disappearance of another, so that the
amount which exists of the one thing depends on and can
be calculated from the amount of the other which has dis-
appeared, we conclude that the one has been formed at the
expense of the other, and that they are both forms of the
same thing.
Hence we conclude that heat is energy in a peculiar
form. The reasons for believing heat as it exists in a hot
94 Stresses and Strains.
body to be in the form of kinetic energy that is, that the
particles of the hot body are in actual though invisible
motion will be discussed afterwards.
CHAPTER V.
ON THE MEASUREMENT OF PRESSURE AND OTHER INTERNAL
FORCES, AND OF THE EFFECTS WHICH THEY PRODUCE.
EVERY force acts between two bodies or parts of bodies.
If we are considering a particular body or system of bodies,
then those forces which act between bodies belonging to this
system and bodies not belonging to the system are called
External Forces, and those which act between the different
parts of the system itself are called Internal Forces.
If we now suppose the system to be divided in imagina-
tion into two parts, we may consider the forces external to
one of the parts to be, first, those which act between that
part and bodies external to the system, and, second, those
which act between the two parts of the system. The com-
bined effect of these forces is known by the actual motion
or rest of the part to which they are applied, so that, if we
know the resultant of the external forces on each part, we
can find that of the internal forces acting between the two
parts.
Thus, if we consider a pillar supporting a statue, and
imagine the pillar divided into two parts by a horizontal
plane at any distance from the ground, the internal force
between the two parts of the pillar may be found by con-
sidering the weight of the statue and that part of the pillar
which is above the plane. The lower part of the pillar
presses on the upper part with a force which exactly counter-
balances this weight. This force is called a Pressure.
In the same way we may find the internal force acting
through any horizontal section of a rope which supports a
Pressures and Tensions. 95
heavy body to be a Tension equal to the weight of the
heavy body and of the part of the rope below the imaginary
section.
The internal force in the pillar is called Longitudinal
Pressure, and that in the rope is called Longitudinal Tension.
If this pressure or tension is uniform over the whole hori-
zontal section, the amount of it per square inch can be
found by dividing the whole force by the number of square
inches in the section.
The internal forces in a body are called Stresses, and
longitudinal pressure and tension are examples of particular
kinds of stress. It is shown in treatises on Elasticity that
the most general kind of stress at any point of a body may
be represented by three longitudinal pressures or tensions in
directions at right angles to each other.
For instance, a brick in a wall may support a vertical
pressure depending on the height of the wall above it, and
also a horizontal pressure in the direction of the length of
the wall, depending on the thrust of an arch abutting against
the wall, while in the direction perpendicular to the face of
the wall the pressure is that of the atmosphere.
In solid bodies, such as a brick, these three pressures may
be all independent, their magnitude being limited only by
the strength of the solid, which will break down if the force
applied to it exceeds a certain amount.
In fluids, the pressures in all directions must be equal,
because the very slightest difference between the pressures
in the three directions is sufficient to set the fluid in motion.
The subject of fluid pressure is so important to what
follows that I think it worth while, at the risk of repeating
what the student ought to know, to state what we mean by
a fluid, and to show from the definition that the pressures in
all directions are equal.
DEFINITION OF A FLUID. A fluid is a body the contiguous
parts of which act on one a?iother with a pressure which is
perpendicular to the interface which separates those parts.
06 Stresses and Strains.
Since the pressure is entirely perpendicular to the sur-
face, there can be no friction between the parts of a fluid
in contact.
Theorem. The pressures in any two directions at a point
of a fluid are equal. For, let the plane
FIG. 8. rt f <-T-o T-voi-k/aT- \\a 4-vn+- Q the two
directions, and draw an isosceles triangle
whose sides are perpendicular to the two
directions respectively, and consider the
equilibrium of a small triangular prism
R of which this triangle is the base. Let
p Q be the pressures perpendicular to the sides, and R
that perpendicular to the base. Then, since these three
forces are in equilibrium, and since R makes equal angles
with p and Q, p and Q must be equal. But the faces on
which p and Q act are also equal ; therefore the pressures
referred to unit of area on these faces are equal, which was
to be proved.
A great many substances may be found which perfectly
fulfil this definition of a fluid when they are at rest, and they
are therefore called fluids. But no existing fluid fulfils the
definition when it is in motion. In a fluid in motion the
pressures at a point may be greater in one direction than
in another, or, what is the same thing, the force between
two parts may not be perpendicular to the interface which
separates those parts.
If a fluid could be found which fulfilled the definition
when in motion as well as when at rest, it would be called a
Perfect Fluid. All actual fluids are imperfect, and exhibit
the phenomenon of internal friction or viscosity, by which
their motion after being stirred about in a vessel is gradually
stopped, and the energy of the motion is converted into
heat.
The degree of viscosity varies from that of tar to that of
water, or ether, or hydrogen gas, but no actual fluid is perfect
in the sense of the definition when in motion.
Pressure in a Fluid. 97
The pressure at any point of a fluid is the ratio of the
whole pressure on a small surface to the area of that surface
when the area of the surface is made to dimmish indefinitely,
but so that the centre of gravity of the surface always coincides
with the given point.
This pressure is sometimes called hydrostatic pressure, to
distinguish it from longitudinal pressure. Both kinds of
pressure are measured by the number of units of force in the
pressure on unit of area ; for instance, in pounds' weight on
the square inch or square foot, and in kilogrammes' weight
on the square metre. Both these measures are gravitation
measures, and must be multiplied by the value of the inten-
sity of gravity to reduce them to absolute measures.
Pressures are also measured in terms of the height of a
column of water or of mercury, which would produce by its
weight an equal pressure. Thus a pressure of 16 feet of
water is nearly equal to 1,000 pounds' weight on the square
foot, and a pressure of 4 inches of water is more nearly equal
to 101 grains' weight on the square inch,
In the metrical system the pressure of water on a surface
at any depth is expressed by the product of the depth into
the area of the surface. If we employ the metre as the
measure of length, the pressure will be expressed in tonnes'
weight, but if we use the decimetre, centimetre, or millimetre,
the pressure will be expressed in kilogrammes, grammes,
or milligrammes respectively, in gravitation measure.
The density of mercury at o C. is 13*596 times that of
water at 4 C. Hence the pressure due to a given depth of
mercury is about 13-6 times that of an equal depth of water.
The Barometer. The pressure of the air is generally
measured by means of the mercurial barometer. This baro-
meter consists of a glass tube closed at one end and filled
with mercury, from which all air and moisture are expelled
by boiling it in the tube. The tube is then placed with its
open end in a vessel of mercury, and its closed end raised
till the tube is vertical The mercury is found to stand at
H
98 Stresses and Strains.
a certain level in the tube, the height of which above the
level of the mercury in the vessel or cistern is called the
height of the barometer.
The surface of the mercury in the cistern is exposed to
the pressure of the air, while the surface of the mercury in
the tube is exposed only to the pressure of whatever is in
the tube above it. The only known substance which can
be there is the vapour of mercury, the pressure of which at
ordinary temperatures is so small that it may be neglected,
so that the pressure of the air may be measured by that
due to the difference of level of the mercury in the tube
and in the cistern.
The pressure of the atmosphere is, as we know, very
variable, and is different in different places ; but for various
purposes it is convenient to use, as a large unit of pressure,
a pressure not very different from the average atmospheric
pressure at the mean level of the sea. This unit of pressure
is called an atmosphere, and is used in measuring pressures
in steam-engines and boilers. Its exact value in the metrical
system is the pressure due to a depth of 760 millimetres of
mercury at o C. at Paris, where the force of gravity is
9*80868 metres. This is equal to 1*033 kilogrammes' weight
on the square centimetre. In absolute measure it is equal
to 1,013,237, the gramme, the centimetre, and the second
being the fundamental units.
In the British system an atmosphere is denned as the
pressure due to a depth of 29*905 inches of mercury at
32 F. at London, where the force of gravity is 32*1889 feet,
and is, roughly, 14! pounds' weight on the square inch. It is
therefore 0*99968 of the atmosphere of the metrical system.
ON THE ALTERATION OF THE DIMENSIONS AND VOLUME
OF BODIES BY MECHANICAL FORCES AND BY HEAT.
We have seen that effects of the same kind in changing
the form or volume of bodies are produced by. mechanical
force and by heat. We cannot therefore fully understand
Strains. 99
the effects of heat alone on these bodies without at the same
time considering those of mechanical force.
We have first to explain, from a purely geometrical point
of view, the various kinds of change of form of which a body
is capable, considering only those cases in which every part
of the body undergoes a similar change of form. We shall
use the word strain to express generally any alteration of
form of a body.
Longitudinal Strain. Suppose the body to be elongated
or compressed in one direction only, so that if two points
in the body lie in a line parallel to this direction, their
distance will be increased or diminished in a certain ratio,
but if the line joining the points be perpendicular to this
direction the length of the line will not be altered.
This is called longitudinal extension or compression, or
more generally longitudinal strain, and is measured by the
fraction of its original length by which any longitudinal line
in the body is elongated or contracted.
General Strain. Such an alteration of the form of the
body may take place simultaneously or successively in
three directions at right angles to each other. This system
of three longitudinal strains is shown in treatises on the
motion of continuous bodies to be the most general kind of
strain of which a body is capable.
We shall, however, only consider two cases in particular.
i st. Isotropic Strain. When the strains in the three
directions at right angles to each other are all equal, the
form of the body remains similar to itself, and it expands
or contracts equally in all directions, as most solid bodies do
when heated.
Since each of the three longitudinal strains of which this
strain is compounded increases the volume by a fraction
of itself equal to the value of the longitudinal strain, it
follows that when each of the strains is a very small frac-
tion, the total increment of volume is equal to the original
volume multiplied by the algebraical sum of the three strains.
IOO Stresses and Strains.
The ratio of the increment of volume to the original volume is
called the voluminal expansion when positive, or the voluminal
contraction when negative, and it appears, from what we have
said, that when the strains are small the voluminal expansion
is equal to the sum of the longitudinal extensions, or, when
these are equal, to three times the longitudinal extension.
2nd. Shearing Strain. The other particular case is when
the dimensions of the body are extended in one direction in
the ratio of a to i, and contracted in a perpendicular direc-
tion in the ratio of i to a. In this case there is no altera-
tion of volume, but the body is distorted.
WORK DONE BY A STRESS ON A BODY WHOSE FORM IS
CHANGING OR IS UNDERGOING A STRAIN.
We shall in the first place suppose that the stress con-
tinues constant during the change of form which we consider.
If during a considerable change of form the stress undergoes
considerable change, we may divide the whole operation into
parts, during each of which we may regard the stress as
constant, and find the total work by summation.
The general rule is that, if the stress and the strain are of
the same type, the work done on unit of volume during any
strain is the product of the strain into the average value of
the stress.
If, however, the stress be of a type conjugate to the strain,
no work is done.
Thus, if the stress be a longitudinal one, we must multiply
the average value of the stress by the longitudinal strain in
the same direction, and the result is not affected by the
magnitude of the longitudinal strains in directions at right
angles to the stress.
If the stress be a hydrostatic pressure, we must multiply
the average value of this pressure by the voluminal com-
pression to find the work done on the body per unit of
volume, and the result is not affected by any strain of dis-
tortion which does not change the volume of the body.
Work done on; 0, Fluids. ;' \ "j 1; \ \ jjoi
Hence the work done by external forces on a fluid when
its volume is diminished is equal to the product of the
average pressure into the diminution of volume, and if
the fluid expands and overcomes the resistance of external
forces, the work done by the fluid is measured by the pro-
duct of the increase of volume, into the average pressure
during that increase.
The consideration of the work gained or lost during the
change of volume of a fluid is so important that we shall
calculate it from the beginning.
WORK DONE BY A PISTON ON A FLUID.
Let us suppose that the fluid is in communication with a
cylinder in which a piston is free FIG. 9.
to slide.
Let the area of the face of the
piston be denoted by A.
Let the pressure of the fluid
be denoted by p on unit of area.
Then the whole pressure of the fluid on the face of the
piston will be A/, and if P is the external force which keeps
the piston in equilibrium, p = A/. Now let the piston be
pressed inwards against the fluid through a distance cc.
The volume of the cylinder occupied by the fluid will be
diminished by a volume v = AX, because the volume of a
cylinder is equal to the area of its base multiplied by its
height
If the force P continues uniform, or if p is the average
value of the external force during this motion, the work
done by the external force will be w = PX.
If we put for P its value in terms of/, the pressure of the
fluid per unit of area, this becomes
w = Apx ;
and if we remember that AX is equal to v, this becomes
w = v
'Stresses arid Strains.
or the work done by the piston against the fluid is equal to
the diminution of the volume of the fluid multiplied by the
average value of the hydrostatic pressure.
It will be observed that this result is independent of the
area of the piston, and of the form and capacity of the
vessel with which the cylinder communicates.
If, for convenience, we suppose that the area of the piston
is unity, then putting A = i we shall have P =p and v = x,
so that the linear distance travelled by the piston is nu-
merically equal to the volume displaced.
ON INDICATOR DIAGRAMS.
I shall now describe a
method of studying the action
B of a fluid of variable volume,
which was invented by James
Watt, as a practical method of
determining the work done by
the steam-engine, and of which
the construction has been
gradually perfected, till it is
v now capable of tracing every
part of the action of the steam
in the most rapidly working engines.
At present, however, I shall use this method as a means
of explaining and representing to the eye the working of a
fluid. This use of the indicator diagram, which was intro-
duced by Clapeyron, has been greatly developed by Rankine
in his work on the steam-engine.
Let o v be a horizontal straight line, and op a vertical
line. On o v (which we shall call the line of volumes) take
distances o #, o b, o c to represent the volume occupied by
the fluid at different times, and at a b c erect perpendiculars
a A, b B, c c, representing, on a convenient scale, the pressure
of the fluid at these different times.
Indicator Diagram. 103
(For instance, we may suppose that, in the scale of volumes,
one inch, measured horizontally, represents a volume equal
to a cubic foot ; and that in the scale of pressures, one inch,
measured vertically, represents a pressure of 1,000 pounds'
weight on the square foot.)
Let us now suppose that the volume increases from o a
to o , while the pressure remains constant, so that a A = b B.
Then the increase of volume is measured by a b, and the
pressure which is overcome by the expansion of the fluid by
a A or b B, so that the work done by the fluid is represented
by the product of these quantities, or a b . a A, that is, the
area of the rectangle A a b B.
On the scale which we have assumed, every square inch
of the area of the figure A D b a represents 1,000 foot-pounds
of work.
We have supposed the pressure to remain constant during
the change of volume. If this is not the case, but if the
pressure changes from b B to c c, while the volume changes
from o b to o c, then if we take b c small enough, we may
suppose the pressure to change uniformly from the one
value to the other, so that we may take the mean value of
the pressure to be -|(B b + c c}. Multiplying this by b c>
we get ^(B b + c c} b c, which is the well-known expression
for the area of the strip B c c b, supposing B c a straight
line.
The work done by the fluid is therefore still equal to the
area enclosed by B c, the two vertical lines from its extre-
mities, and the horizontal line o v.
In general, if the volume and pressure of the fluid are made
to vary in any manner whatever, and if a point P be made at
the same time to move so that its horizontal distance from the
line o p represents the volume which the fluid occupies at
that instant, while its vertical distance from o v represents
the hydrostatic pressure of the fluid at the same instant, and
if, at the beginning and end of the path traced by P, vertical
lines be drawn to meet o z>, then, if the path of P does not
IO4
Stresses and Strains.
intersect itself, the aioa between these boundaries represents
the work done by the fluid against external forces, if it
lies on the right-hand side of the path of the tracing
point. If the area lies on the left-hand side of the path, it
represents the work done by the external forces on the
fluid.
If the path of p returns into itself so as to form a loop or
Fie. w.
Richards's Indicator.
closed figure, then the vertical lines at the beginning and end
of the path will coincide, so that it is unnecessary to draw
them, and the work will be represented by the area of the
loop itself. If P in its circuit goes round the loop in the
direction of the hands of a watch, then the area represents
the work done by the fluid against external forces ; but if p
goes round the loop in the opposite direction, the area of
Action of the Indicator. 105
the loop represents the work done by the external forces on
the fluid.
In the indicator as constructed by Watt and improved by
McNaught and Richards, the steam or other fluid is put in
connection with a small cylinder containing a piston. When
the fluid presses this piston and raises it, the piston presses
against a spiral spring, so constructed that the distance
through which the spring is compressed is proportional
to the pressure on the piston. In this way the height of the
piston of the indicator is at all times a measure of the pressure
of the fluid.
The piston also carries a pencil, the point of which presses
lightly against a sheet of paper which is wrapped round a
vertical cylinder capable of turning round its axis.
This cylinder is connected with the working piston of the
engine, or with some part of the engine which moves along
with the piston, in such a way that the angle through which
the cylinder turns is always proportional to the distance
through which the working piston has moved.
If the indicator is not connected with the steam pipe,
the cylinder will turn beneath the point of the pencil, and
a horizontal line will be drawn on the paper. This line
corresponds to o v, and is called the line of no pressure.
But if the steam be admitted below the indicator piston,
the pencil will move up and down, while the paper moves
horizontally beneath it, and the combined motion will trace
out a line on the paper, which is called an indicator diagram.
When the engine works regularly, so that each stroke is
similar to the last, the pencil will trace out the same curve
at every stroke, and by examining this curve we may learn
much about the action of the engine. In particular, the area
of the curve represents the amount of work done by the
steam at each stroke of the engine.
If the indicator had been connected with a pump, in
which the external forces do work on the fluid, the tracing
point would move in the opposite direction round the
io6 Stresses and Strains.
diagram, and its area would indicate the amount of work
done on the fluid during the stroke.
Hitherto we have confined our attention to the work done
by the pressure on the piston, and have not been concerned
with the cause of the alteration of volume of the fluid. The
increase of volume may, for anything we know, arise from
an additional supply being introduced into the cylinder, as
when steam is introduced from the boiler, and the dimi-
nution of volume may arise from the escape of the fluid
from the cylinder.
As we are now going to use the diagram for the purpose
of explaining the properties of bodies when acted on by heat
and by mechanical force, we shall suppose that the body,
whether fluid or partly solid, is placed in a cylinder with
one end closed, and that its volume is measured by the
distance of the piston from the closed end of the cylinder.
If at any instant the volume
FlG * I2t of the body is v and its pres-
sure/, we represent this fact
by means of the point P in the
\ diagram, drawing o L along
\ the line of volumes to reprc-
\ sent v, and L P vertical to re-
present /.
In this way the position of
a point in the diagram may be
made to indicate the volume
and the pressure of a body at
any instant.
Now let the pressure be increased, the temperature re-
maining the same, then the volume of the fluid will be
diminished. (It is manifest that an increase of pressuie can
never produce an increase of volume, for in that case the
force would produce a motion in the contrary direction to
that in which it acts, and we should have a source oi inex-
haustible energy.)
Elasticity. 107
Let the pressure, therefore, increase from o F to o G, and
let the consequent diminution of volume be from o L to
o M, and complete the rectangle o G Q M.
Then the point p indicates the original and Q the final
condition of the fluid with respect to pressure and volume,
and all the intermediate states of the fluid will be repre-
sented by points in a line, straight or curved, which joins p
and Q.
The work done by the pressure on the fluid is represented
by the area of the figure P Q M L, which is on the left hand
of the tracing point as it moves along p Q.
If p r and Q M intersect in R, then p R represents the
actual diminution of volume, and R Q the actual increase of
pressure. The actual volume is represented by F p, so that
the voluminal compression is represented by the ratio of p R
to FP.
DEFINITION OF THE ELASTICITY OF A FLUID. The
elasticity of a fluid under any given conditions is the ratio
of any small increase of pressure to the voluminal compression
hereby produced.
Since the voluminal compression is a numerical quantity,
the elasticity is a quantity of the same kind as a pressure.
To express the elasticity of the fluid by means of the
diagram, join p Q by a straight line, and produce it till it
meets the vertical line o p in E ; then F E is a pressure equal
to the elasticity of the fluid in the state represented by p f
and under conditions which cause its state to vary in 9
manner represented by the line P Q.
For it is plain that F E is to R Q in the ratio of p F to p R,
or F E = L2 = _mcrement of pressure =
PJR voluminal compression
p F
Hence if the relation between the volume and the pies-
sure of a fluid under certain conditions, as for instance at a
given temperature, is represented by a curve traced out by p,
the elasticity of the fluid when in the state represented by P
io8 Isothermal Curves.
may be found by drawing p E a tangent to the curve at p,
and P F a horizontal line. The portion F E of the vertical
line o / cut off between these lines represents, on the scale
of pressures, the elasticity of the fluid.
We have hitherto supposed that the temperature of the
body remains the same during its compression from the
volume P F to the volume Q G. This is the most common
supposition when the elasticity of a fluid is to be measured.
But in most bodies a compression produces a rise of tempe-
rature, and if the heat is not allowed to escape, the effect of
this will be to make the increment of pressure greater than
in the case of constant temperature. Hence every substance
has two elasticities, one corresponding to constant tempera-
ture, and the other corresponding to the case where no heat
is allowed to escape. The first value is applicable to stresses
and strains which are long continued, so that the substance
acquires the temperature of surrounding bodies. The
second value is applicable to the case of rapidly changing
forces, as in the case of the vibrations of bodies which
produce sounds, in which there is not time for the tempe-
rature to be equalised by conduction. The elasticity in
these cases is always greater than in the case of uniform
temperature.
CHAPTER VI.
ON LINES OF EQUAL TEMPERATURE, OR ISOTHERMAL LINES
ON THE INDICATOR DIAGRAM.
IF the pressure is made to vary while the temperature re-
mains constant, the volume will diminish as the pressure
increases, and the point p will trace out a line in the diagram
which is called a line of equal temperature, or an isothermal
line. By means of this line we can show the whole behaviour
Their Construction. 109
of the substance under various pressures at that particular
temperature.
By making experiments on the substance at other tem-
peratures, and drawing the isothermal lines belonging to
these temperatures, we can express all the relations between
the pressure, volume, and temperature of the substance.
In the diagram, each isothermal line should be marked
with the temperature to which it corresponds in degrees,
and the lines should be drawn for every degree, or for every
ten or every hundred degrees, according to the purpose for
which the diagram is intended.
When the volume and the pressure are known, the
temperature is a determinate quantity, and it is easy to see
how from any two of these three quantities we can deter-
mine the third. Thus if the curved lines in the diagram
are the lines of equal temperature, the temperature cor-
responding to each being indicated by the numeral at the
end of the line, we can solve three problems by means of
this diagram.
1. Given the pressure and the volume, to find the tempe-
rature.
Lay off o L on the line of volumes to represent the given
volume, and o F on the line of pressures to represent the
given pressure, then draw F p horizontal and L p vertical, to
determine the point P. If the point p falls on one of the
lines of equal temperature, the numeral attached to that line
indicates the temperature. If the point p falls between two
of the lines, we must estimate its distance from the two
nearest lines, and then as the sum of these distances is to the
distance from the lower line of temperature, so is the dif-
ference of temperature of the two lines to the excess of the
true temperature above that of the lower line.
2. Given the volume and temperature to find the pres-
sure.
Lay off o L to represent the volume and draw L p vertical,
and let p be the point where this line cuts the line of the
no
Isothermal Curves.
given temperature. Then L P represents the required
pressure.
3. Given the pressure and temperature, to find the
volume.
FIG.
Lay off o F to represent the pressure and draw F p hori-
zontal till it meets the line of the given temperature in p,
then F p represents the required volume.
ON THE FORM OF THE ISOTHERMAL CURVES IN DIFFERENT
CASES.
The Gaseous State.
If the substance is in the gaseous state, then it is easy to
draw the isothermal curves by taking account of the laws of
Boyle and Charles.
By Boyle's law the product of the volume and the pres-
Their Characteristics. Ill
sure is always the same for the same temperature. Hence,
in the curve, the area of the rectangle o L p F will be the
same provided p be a point in the same isothermal curve.
The curve which has this property is known in geometry
by the name of the rectangular hyperbola, the lines o v and
o/ being the asymptotes of the hyperbolas in fig. 13. The
asymptotes are lines such that a point travelling along the
curve in either direction continually approaches one or
other of the asymptotes, but never reaches it. The physical
interpretation of this is that if a gas fulfils Boyle's law, and
if the temperature remain the same
1. Suppose we travel along the curve in the direction
leading toward o /, that is to say, suppose the pressure
is gradually increased, then the volume will continually
diminish, but always slower and slower; for, however much
we increase the pressure, we can never reduce the volume to
nothing, so that the isothermal line will never reach the line
o /, though it continually approaches it. At the same time,
if Boyle's law is fulfilled we can always, by doubling the
pressure, reduce the volume to one half, so that by a suffi-
cient increase of pressure the volume may be reduced till it
is smaller than any prescribed quantity.
2. Suppose we travel in the other direction along the
curve, that is to say, suppose we increase the volume of the
vessel which contains the gas, then the point / approaches
nearer and nearer to the line o v, but never actually reaches
it. This shows that the gas will always expand so as to fill
the vessel, and press upon it with a force represented by the
distance from o v, and this pressure, though it diminishes as
the vessel is enlarged, will never be reduced to nothing,
however large the vessel may become.
Elasticity of a Perfect Gas. Another property of the
hyperbola is that if p E be drawn a tangent to the curve
at P till it meets the asymptote, F E = o F. Now F E
represents the elasticity of the substance, and o F the pres-
sure. Hence the elasticity of a perfect gas is numerically
112 Isothermal Curves.
equal to the pressure, when the temperature is supposed to
remain constant during the compression.
The Liquid State.
In most liquids, the compression produced by the pres-
sures which we are able to apply is exceedingly small. In
the case of water, for example, under ordinary circumstances
as to temperature, the application of a pressure equal to one
atmosphere produces a compression of about 46 millionth
parts of the volume, or 0*000046. Hence in drawing an
indicator diagram for a liquid we must represent changes of
volume on a much larger scale than in the case of gases, if
the diagram is to have any visible features at all. The
most convenient way is to suppose the line o L to represent,
not the whole volume, but the excess of the volume above a
thousand or a million of the units we employ.
It is manifest that the relation between the pressure and
the volume of any substance must be such that no pressure,
however great, can reduce the volume to nothing. Hence
the isothermal lines cannot be straight lines, for a straight
line, however slightly inclined to the line of no volumes o F,
and however distant from it, must cut that line somewhere.
The limited range of pressures which we are able to produce
does not in some cases cause sufficient change of volume to
indicate the- curvature of the isothermal lines. We may
suppose that for the small portion we are able to observe
they are nearly straight lines.
The expansion due to an increase of temperature is also
much smaller in the case of liquids than in the case of
gases.
If, therefore, we were to draw the indicator diagram of a
liquid on the same scale as that of a gas, the isothermal
lines would consist of a number of lines very close together,
nearly vertical, but very slightly inclined towards the line o F.
If, however, we retain the scale of pressures and greatly
magnify the scale of volumes, the isothermal lines will be
Saturated Vapour 113
more inclined to the vertical and wider apart, but still very
nearly straight lines. Liquids, however, which are near the
critical point described at the end of this chapter are more
compressible than even a gas.
The Solid State.
In solid bodies the compressibility and the expansion by-
heat are in general smaller than in liquids. Their indicator
diagrams will therefore have the same general characteristics
is those of liquids.
INDICATOR DIAGRAM OF A SUBSTANCE PART OF WHICH
IS LIQUID AND PART VAPOUR.
Let us suppose that a pound of water is placed in a vessel
and brought to a given temperature, say 212 R, and that
by means of a piston the capacity of the vessel is made
larger or smaller, the temperature remaining the same. If
we suppose the vessel to be originally very large, say 100 cubic
feet, and to be maintained at 212 F., then the whole of the
water will be converted into steam, which will fill the vessel
and will exert on it a pressure of about 575 pounds' weight
on the square foot. If we now press down the piston, and
so cause the capacity of the vessel to diminish, the pressure
will increase nearly in the same proportion as the volume
diminishes, so that the product of the numbers representing
the pressure and volume will be nearly constant. When,
however, the volume is considerably diminished, this product
begins to diminish, that is to say, the pressure does not in-
crease so fast as it ought to do by Boyle's law if the steam
were a perfect gas. In the diagram, fig. 14, p. 114, the
relations between the pressure and volume of steam at 212
are indicated by the curve a b. The pressure in atmo-
spheres is marked on the right hand of the diagram, and the
volume of one pound, in cubic feet, at the bottom.
When the volume is diminished to 26-36 cubic feet the
i
Isothermal Curves.
FIG. 14.
302 c
Isothennals for Steam and Water.
Water and Steam. 115
pressure is 2,116 lb., so that the product of the volume
and pressure, instead of 57,500, is now reduced to 55,770.
This departure from the law of Boyle, though not very large,
is quite decided. The pressure and volume of the steam in
this state are indicated by the point b in the diagram.
If we now diminish the volume and still maintain the
same temperature, the pressure will no longer increase, but
part of the steam will be converted into water ; and as the
volume continues to diminish, more and more of the steam
will be condensed into the liquid form, while the pressure
remains exactly the same, namely, 2,116 pounds' weight on
the square foot, or one atmosphere. This is indicated by
the horizontal line b c in the diagram.
This pressure will continue the same till all the steam is
condensed into water at 212, the volume of which will be
o-o 1 6 of a cubic foot, a quantity too small to be represented
clearly in the diagram.
As soon as the volume, therefore, is reduced to this value
there will be no more steam to condense, and any further
reduction of volume is resisted by the elasticity of water,
which, as we have seen, is very large compared with that of
a gas.
We are now able to trace the isothermal line for water
corresponding to the temperature 2 1 2. When v is very
great the curve is nearly of the form of an hyperbola for
which v P = 5 7,500. As v diminishes, the curve falls slightly
below the hyperbola, so that when v = 26-36, v P = 55,770.
Here, however, the line suddenly and completely alters its
character, and becomes the horizontal straight line b <r, for
which p = 2,116, and this straight line extends from
v = 26^36 to v = o'oi6, when another equally sudden
change takes place, and the line, from being exactly horizon-
tal, becomes nearly but not quite vertical, nearly in the
direction c p, for the pressure must be increased beyond
the limits of our experimental methods long before any
very considerable change is made in the volume of the water.
I Q
n6 Isothermal Curves.
The isothermal line in a case of this kind consists of three
parts. , In the first part, ab, it resembles the isothermal lines of a
perfect gas, but as the volume diminishes the pressure begins
to be somewhat less than it should be by Boyle's law. This
however, is only when the line approaches the second part
of its course, be, in which it is accurately horizontal. This part
corresponds to a state in which the substance exists partly
in the liquid and partly in the gaseous state, and it extends
from the volume of the gas to the volume of the liquid at
the same temperature and pressure. The third part of the
isothermal line is that corresponding to the liquid state of
the substance, and it may be considered as a line which on
the scale of our diagrams would be very nearly vertical,
and so near to the line c p that it cannot be distinguished
from it.
In the diagram, fig. 14, the isothermal line of water for
the temperature 212 R, the ordinary boiling point, is re-
presented by a b cp, and that for 302 F. by d ef p.
At the temperature of 302 F. the pressure at which con-
densation takes place is much greater, being 9,966 pounds'
weight on the square foot; and the volume to which the
steam is reduced before condensation begins is much
smaller, being 6-153 cubic feet. This is indicated by the
point e. At this point the product v p is 61,321, which is
considerably less than 65,209, its value when the volume is
very great.
At this point condensation begins and goes on till the
whole steam is condensed into water at 302 F., the volume
of which is 0-0166 cubic feet. This volume is somewhat
greater than the volume of the same water at 212 F.
It appears, therefore, that as the temperature rises the
pressure at which condensation occurs is greater. It also
appears that the diminution of volume when condensation
takes place is less than at low temperatures, and this for
two reasons. The first is, that the steam must be reduced
to a smaller volume before condensation begins ; and the
Steam Line and Water Line. 117
second is, that the volume of the liquid when condensed is
greater.
The dotted line in the diagram indicates the pressures
and the volumes at which condensation begins at the
various temperatures marked on the horizontal parts of the
isothermal lines.
When the pressure and volume are those indicated by
points above or on the right hand of this curve the whole
substance is in the gaseous state. We may call this line the
steam line. It is not an isothermal line.
If the scale of the diagram had been large enough to have
represented the volume of the condensed water, we should
have had another dotted line near the line o/, such that for
points on the left hand of this line the whole substance is in
the liquid state. We may call this the water line. For
conditions of pressure and volume indicated by points
between the two dotted lines, the substance is partly in the
liquid and partly in the gaseous state. If we draw a hori-
zontal line through the given point till it meets the two
dotted lines, then the weight of steam is to the weight of
water as the segment between the point and the water line
is to the segment between the point and the steam line. In
the lower part of the diagram for carbonic acid, fig. 15,
p. 1 20, the isothermal lines are seen to consist of a curved
portion on the right hand representing the gaseous
state, a horizontal portion representing the process of con-
densation, and a nearly vertical portion representing
the liquid state. The right-hand branch of the dotted
line, which we must here call the gas line, corresponds
to the steam line ; and the left-hand branch, or liquid line,
corresponds to the water line, which was not distinguish-
able in fig. 14.
Since these two lines, which we have called the steam line
and the water line, continually approach each other as the
temperature is raised, the question naturally arises, Do they
ever meet 1 The peculiarity of the conditions indicated by
1 1 8 Isothermal Curves.
points between these lines is that the liquid and its vapour
can exist together under the same conditions as to tempera-
ture and pressure without the vapour being liquefied or the
liquid evaporated. Outside of this region the substance
must be either all vapour or all liquid.
If the two lines meet, then at the pressure indicated by
the point of meeting there is no temperature at which the
substance can exist partly as a liquid and partly as a vapour,
but the substance must either be entirely converted from
the state of vapour into the state of liquid at once and with-
out condensation, or, since in this case the liquid and the
vapour have the same density, it may be suspected that the
distinctions we have been accustomed to draw between
liquids and vapours have lost their meaning.
The answer to this question has been to a great extent
supplied by a series of very interesting researches.
In 1822 M. Cagniard de la Tour l observed the effect of
a high temperature upon liquids enclosed in glass tubes of a
capacity not much greater than that of the liquid itself. He
found that when the temperature was raised to a certain
point, the substance, which till then was partly liquid and
partly gaseous, suddenly became uniform in appearance
throughout, without any visible surface of separation, or any
evidence that the substance in the tube was partly in one
state and partly in another.
He concluded that at this temperature the whole became
gaseous. The true conclusion, as Dr. Andrews has shown, is
that the properties of the liquid and those of the vapour
continually approach to similarity, and that, above a certain
temperature, the properties of the liquid are not separated
from those of the vapour by any apparent distinction be.
tween them.
In 1823, the year following the researches of Cagniard
de la Tour, Faraday succeeded in liquefying several bodies
hitherto known only in the gaseous form, by pressure alone,
1 Annales de Chimie, 2 m serie, xxi. et xxii.
Carbonic Acid. 119
and in 1826 he greatly extended our knowledge of the
effects of temperature and pressure on gases. He considers
that above a certain temperature, which, in the language of
Dr. Andrews, we may call the critical temperature for the
substance, no amount of pressure will produce the pheno-
menon which we call condensation, and he supposes that the
temperature of 166 F. below zero is probably above the
critical temperature for oxygen, hydrogen, and nitrogen.
Dr. Andrews has examined carbonic acid under varied
conditions of temperature and pressure, in order to ascertain
the relations of the liquid and gaseous states, and has
arrived at the conclusion that the gaseous and liquid states
are only widely separated forms of the same condition of
matter, and may be made to pass one into the other with-
out any interruption or breach of continuity. 1
Carbonic acid is a substance which at ordinary tempera-
tures and pressures is known as a gas. The measurements
of Regnault and others show that as the pressure increases
the volume diminishes faster than that of a gas which obeys
the law of Boyle, and that as the temperature rises the ex-
pansion is greater than that assigned by the law of Charles.
The isothermal lines of the diagram of carbonic acid at
ordinary temperatures and pressures are therefore somewhat
flatter and also somewhat wider apart than those of the
more perfect gases.
The diagram (p. 120) for carbonic acid is taken from Dr.
Andrews's paper, with the exception of the dotted line
showing the region within which the substance can exist
as a liquid in presence of its vapour. The base line of the
diagram corresponds, not to zero pressure, but to a pressure
of 47 atmospheres.
The lowest of the isothermal lines is that of 13*! C. or
<5-6 F.
This line shows that at a pressure of about 47 atmospheres
condensation occurs. The substance is seen to become
1 Phil. Trans. 1869, p. 575.
I2O
Isothermal Curves.
FIG. 15.
Isothermal s of Carbonic Acid.
Experiments of A ndrews. 1 2 1
separated into two distinct portions, the upper portion being
in the state of vapour or gas, and the lower in the state of
liquid. The upper surface of the liquid can be distinctly
seen, and where this surface is close to the sides of the glass
containing the substance it is seen to be curved, as the
surface of water is in small tubes.
As the volume is diminished, more of the substance is
liquefied, till at last the whole is compressed into the liquid
form.
I have described this isothermal line at greater length,
that the student may compare the properties of carbonic acid
at 55'6 F. with those of water at 212 F.
1. The steam before condensation begins has properties
agreeing nearly, though not quite, with those of a perfect gas.
In carbonic acid the volume just before liquefaction com-
mences is little more than three-fifths of that of a perfect
gas at the same temperature and pressure. The corresponding
isothermal lines for air are given in the diagram, and it
will be seen how much the carbonic acid isothermal has
fallen below that of air before liquefaction begins.
2. The steam when condensed into water occupies less
than the sixteen-hundredth part of the volume of the steam.
The liquid carbonic acid, on the other hand, occupies nearly
a fifth part of its volume just before condensation. We are
therefore able to draw the dotted line of complete conden-
sation in this diagram, though in the case of water it would
have required a microscope to distinguish it from the line of
no volume.
3. The steam when condensed into water at 212 has
properties not differing greatly from those of cold water.
Its dilatability by heat and its compressibility by pressure
are probably somewhat greater than when cold, but not
enough to be noticed when the measurements are not very
precise.
Liquid carbonic acid, as was first observed by Thilorier,
dilates as the temperature rises to a greater degree than even
122 Isothermal Curves.
a gas, and, as Dr. Andrews has shown, it yields to pressure
much more than any ordinary liquid. From Dr. Andrews's
experiments it also appears that its compressibility dimi-
nishes as the pressure increases. These results are apparent
even in the diagram. It is, therefore, far more compressible
than any ordinary liquid, and it appears from the experi-
ments of Andrews that its compressibility diminishes as the
volume is reduced.
It appears, therefore, that the behaviour of liquid carbonic
acid under the action of heat and pressure is very different
from that of ordinary liquids, and in some respects approaches
to that of a gas.
If we examine the next of the isothermals of the diagram,
that for 2i'5 C. or 7o7 F., the approximation between the
liquid and the gaseous states is still more apparent. Here
condensation takes place at about 60 atmospheres of pres-
sure, and the liquid occupies nearly a third of the volume of
the gas. The exceedingly dense gas is approaching in its
properties to the exceedingly light liquid. Still there is a
distinct separation between the gaseous and liquid states,
though we are approaching the critical temperature. This
critical temperature has been determined by Dr. Andrews to
be 3o'92 C. or Sy -; F. At this temperature, and at a
pressure of from 73 to 75 atmospheres, carbonic acid appears
to be in the critical condition. No separation into liquid and
vapour can be detected, but at the same time very small
variations of pressure or of temperature produce such great
variations of density that flickering movements are observed
in the tube c resembling in an exaggerated form the appear-
ances exhibited during the mixture of liquids of different
densities, or when columns of heated air ascend through
colder strata.' v
The isothermal line for 31-! C. or 88 F. passes above
this critical point. During the whole compression the sub-
stance is never in two distinct conditions in different parts of
the tube. When the pressure is less than 73 atmospheres
the isothermal line, though greatly flatter than that of a perfect
Continuity of the Liquid and Gaseous States. 123
gas, resembles it in general features. From 73 to 75 atmo-
spheres the volume diminishes very rapidly, but by no means
suddenly, and above this pressure the volume diminishes
more gradually than in the case of a perfect gas, but still
more rapidly than in most liquids.
In the isothermals for 32 0< 5 C. or 90^5 F. and for 35'5 C.
or 95'9 F. we can still observe a slight increase of compres-
sibility near the same part of the diagram, but in the
isothermal line for 48 i C. or n8'6 F. the curve is con-
cave upwards throughout its whole course, and differs from
the corresponding isothermal line for a perfect gas only by
being somewhat flatter, showing that for all ordinary pres-
sures the volume is somewhat less than that assigned by
Boyle's law.
Still at the temperature of n8'6 F. carbonic acid has all
the properties of a gas, and the effects of heat and pressure on
it differ from their effects on a perfect gas only by quantities
requiring careful experiments to detect them.
We have no reason to believe that any phenomenon
similar to condensation would occur, however great a
pressure were applied to carbonic acid at this temperature.
In fact, by a proper management we can convert car-
bonic acid gas into a liquid without any sudden change
of state.
If we begin with carbonic acid gas at 50 F. we may first
heat it till its temperature is above 88 F., the critical point
We then gradually increase the pressure to, say, 100 atmo-
spheres. During this process no sign of liquefaction occurs.
Finally we cool the substance, still under the pressure of
100 atmospheres, to 50 F. During this process no sudden
change of state can be observed, but carbonic acid at 50 F.
and under a pressure of 100 atmospheres has all the pro-
perties of a liquid. At the temperature of 50 F. we cannot
convert carbonic acid gas into a liquid without a sudden
condensation, but by this process, in which the pressure is
applied at a high temperature, we have caused the substance
to pass from an undoubtedly gaseous to an undoubtedly
t24 Isothermal Curves.
liquid state without at any time undergoing an abrupt change
similar to ordinary liquefaction.
I have described the experiments of Dr. Andrews on car-
bonic acid at greater length because they furnish the most
complete view hitherto given of the relation between the
liquid and the gaseous state, and of the mode in which the
properties of a gas may be continuously and imperceptibly
changed into those of a liquid.
The critical temperatures of most ordinary liquids are
much higher than that of carbonic acid, and their pressure
in the critical state is very great, so that experiments on the
critical state of ordinary liquids are difficult and dangerous.
M. Cagniard de la Tour estimated the temperature and pres-
sure of the critical state to be :
Temperature Pressure
Fahr. (Atmospheres)
Ether . , . . . . . 3690-5 37-5
Alcohol 497 '5 119-0
Bisulphide of Carbon . . . 504 -5 66-5
Water ... . . . 773-o
In the case of water the critical temperature was so
high that the water began to dissolve the glass tube which
contained it
The critical temperature of what are called the permanent
gases is probably exceedingly low, so that we cannot by any
known method produce a degree of cold sufficient, even
when applied along with enormous pressure, to condense
them into the liquid state.
It has been suggested by Professor James Thomson * that
the isothermal curves for temperatures below the critical
temperature are only apparently, and not really, discon-
tinuous, and that their true form is somewhat similar in its
general features to the curve ABCDEFGHK.
The peculiarity of this curve is, that between the pressures
indicated by the horizontal lines B F and D H, any horizon-
tal line such as c E G cuts the curve in three different
points. The literal interpretation of this geometrical cir-
1 Proceedings of the Royal Society, 1871, No. 130.
Retardation of Boiling and of Condensation. 125
cumstance would be that the fluid at this pressure, and at
the temperature of the isothermal line, is capable of existing
in three different states. One of these, indicated by c,
evidently corresponds to the liquid state. Another, indi-
cated by G, corresponds to the gaseous state. At the inter-
mediate point E the slope of the curve indicates that the
volume and the pressure increase and diminish together.
FIG. x&
No substance having this property can exist in stable equili-
brium, for the very slightest disturbance would make it rush
into the liquid or the gaseous state. We may therefore
confine our attention to the points c and G.
According to the theory of exchanges, as explained at p. 303,
when the liquid is in contact with its vapour the rate of evapo-
ration depends on the temperature of the liquid, and the rate
of condensation on the density of the vapour. Hence for
every temperature there is a determinate vapour-density, and
therefore a determinate pressure, represented by the horizon-
tal line CG, 1 at which the evaporation exactly balances the con-
f 1 The precise position of the horizontal line C G is determined by the
condition that it cuts off equal areas from the curve above and below.
Maxwell, Nature, xi. p. 357, 1875. R.]
126 Isotliermal Curves.
densation. At the pressure indicated by this horizontal line
the liquid will be in equilibrium with its vapour. At all greater
pressures the vapour, if in contact with the liquid, will be con-
densed ; and at all smaller pressures the liquid, if in contact
with its vapour, will evaporate. Hence the isothermal line, as
deduced from experiments of the ordinary kind, will consist of
the curve ABC, the straight line c G, and the curve G K.
But it has been pointed out by Prof. J. Thomson that
by suitable contrivances we may detect the existence of
other parts of the isothermal curve. We know that the
portion of the curve corresponding to the liquid state ex-
tends beyond the point c; for if the liquid is carefully freed
from air and other impurities, and is not in contact with
anything but the sides of a vessel to which it closely ad-
heres, the pressure may be reduced considerably below that
indicated by the point c, till at last, at some point between
C and D, the phenomenon of boiling with bumping com-
mences, as described at p. 25.
Let us next consider the substance wholly in the state of
vapour, as indicated by the point K, and let it be kept at the
same temperature and gradually compressed till it is in the
state indicated by the point G. If there are any drops of
the liquid in the vessel, or if the vessel is capable of being
wetted by the liquid, condensation will now begin. But if
there are no facilities for condensation, the pressure may be
increased and the volume diminished till the state of the
vapour is that which is represented by the point F. At this
point condensation must take place if it has not begun
before. 1
The existence of this variability in the circumstances of
condensation, though seemingly probable, is not as yet
established by experiment, like that of the variability in the
circumstances of evaporation. Prof. J. Thomson suggests
that by investigating the condensation produced by the
rapid expansion of vapour in a vessel provided with a
1 See the chapter on Capillarity.
Adiabatic Curves. 127
steam-jacket, the existence of this part of the isothermal
curve might be established.
The state of things, however, represented by the portion
of the isothermal curve D E F, can never be realised in a
homogeneous mass, for the substance is then in an essentially
unstable condition, since the pressure increases with the
volume. We cannot, therefore, expect any experimental
evidence of the existence of this part of the curve, unless, as
Prof. J. Thomson suggests, this state of things may exist
in some part of the thin superficial stratum of transition
from a liquid to its own gas, in which the phenomena of
capillarity take place.
CHAPTER VII.
ON THE PROPERTIES OF A SUBSTANCE WHEN HEAT
IS PREVENTED FROM ENTERING OR LEAVING IT.
HITHERTO we have considered the properties of substance
only with respect to the volume occupied by a pound of the
substance, the pressure acting on every square foot or inch,
and the temperature of the substance, which we have assumed
to be uniform. We suppose the temperature measured by a
thermometer, and when, in order to change the state of the
body, heat must be supplied to it or taken from it, we have
supposed this to be done without paying any attention to
the quantity of heat required in each case. For the actual
measurements of such quantities of heat we must employ the
processes described in our chapter on Calorimetry, or others
equivalent to them. Before entering on these considerations,
however, we shall examine the very important case in which
the changes which take place are effected without any
passage of heat either into the substance from without or out
of the substance into other bodies.
For the sake of associating the statement of scientific facts
128 Adiabatic Curves.
with mental images which are easily formed, and which pre-
serve the statements in a form always ready for use, we shall
suppose that the substance is contained in a cylinder fitted
with a piston, and that both the cylinder and the piston are
absolutely impermeable to heat, so that not only is heat
prevented from getting out or in by passing completely
through the cylinder or piston, but no heat can pass between
the enclosed substance and the matter of the cylinder or
piston itself.
No substance in nature is absolutely impermeable to heat,
so that the image we have formed can never be fully realised ;
but it is always possible to ascertain, in each particular case,
that heat has not entered or left the substance, though the
methods by which this is done and the arrangements by
which the condition is fulfilled are complicated. In the
present discussion it would only distract our attention from
the most important facts to describe the details of physical
experiments. We therefore reserve any description of actual
experimental methods till we can explain them in connexion
with the principles on which they are founded. In explain-
ing these principles we make use of the most suitable illus-
trations, without assuming that they are physically possible.
We therefore suppose the substance placed in a cylinder,
and its volume and pressure regulated and measured by a
piston, and we suppose that during the changes of volume
and pressure of the substance no heat either enters it or
leaves it.
In order to represent the relation between the volume and
the pressure, we suppose a curve traced on the indicator
diagram during the motion of the piston, exactly as in the
case of the isothermal lines formerly described. The only
difference is that whereas in the case of the isothermal
lines the substance was maintained always at one and the
same temperature, in the present case no heat is allowed
to enter or leave the substance, which, as we shall see, is
a condition of quite a different kind.
Their Definition. 129
The line drawn on the indicator diagram in the latter case
has been named by Professor Rankine an Adiabatic line,
because it is defined by the condition that heat is not allowed
to pass through (m/3a/vcu') the vessel which confines the
substance.
Since the properties of the substance under this condition
are completely defined by its adiabatic lines, it will assist us
in understanding these properties if we associate them with
the corresponding features of the adiabatic lines.
The first thing to be observed is that as the volume dimi-
nishes the pressure invariably increases. In fact, if under any
circumstances the pressure were to diminish as the volume
diminishes, the substance would be In an unstable state, and
would either collapse or explode till it attained a condition
in which the pressure increased as the volume diminished.
Hence the adiabatic lines slope downwards from left to
right in the indicator diagram as we have drawn it.
If the pressure be continually increased, up to the greatest
pressure which we can produce, the volume continually
diminishes, but always slower and slower, so that we cannot
tell whether there is or is not a limiting volume such that no
pressure, however great, can compress the substance to a
smaller volume.
We cannot, in fact, trace the lines upward beyond a
certain distance, and therefore we cannot assert anything of
the upper part of their course, except that they cannot recede
from the line of pressures, because in that case the volume
would increase on account of an increase of pressure.
If, on the other hand, we suppose the piston to be drawn
out so as to allow the volume to increase, the pressure will
diminish.
If the substance is in the gaseous form, or assumes that
form during the process, the substance will continue to exert
pressure on the piston even though the volume is enormously
increased, and we have no experimental reason to believe
that the pressure would be reduced to nothing, however much
130 Adiabatic Curves.
the volume were increased. For gaseous bodies, therefoie.
the lines extend indefinitely in the direction of the line oi
volumes, continually approaching but never reaching it.
With respect to substances which are not originally in the
gaseous form, some of them, when the pressure is sufficiently
diminished, are known to assume that form, and it is plausibly
argued that we have no evidence that any substance, however
solid and however cold, if entirely free from external pres-
sure, would not sooner or later become dissipated through
space by a kind of evaporation.
The smell by which such metals as iron and copper may
be recognised is adduced as an indication that bodies,
apparently veiy fixed, are continually throwing off portions
of themselves in some very attenuated form, and if in these
cases we have no means of detecting the effluvium except by
the smell, in other cases we may be deprived of this evidence
by the circumstance that the effluvium does not affect our
sense of smell at all.
Be this as it may, there are many substances the pressure
of which seems to cease entirely when the volume has
FIO. 17. reached a certain value. Be-
yond this the pressure, if it
exists, is far too small to be
\ measured. The lines of such
\- x substances may without sen-
V*^, isothermal s ible error be considered as
X.^ Adiabatic meeting the line of volumes
^ v within the limits of the diagram.
The next thing to be observed about the adiabatic lines is
that where they cross the isothermal lines they are always
inclined at a greater angle to the horizontf] line than the
isothermal lines.
In other words, to diminish the volume of a substance by
a given amount requires a greater increase of pressure when
the substance is prevented from gaining or losing heat than
when it is kept at a constant temperature.
Their Relation to the Isothermals. 131
This is an illustration of the general principle that when
ihe state of a body is changed in any way by the application
of force in any form, and if in one case the body is subjected
to some constraint, while in another case it is free from this
constraint but similarly circumstanced in all other respects,
then if during the change the body takes advantage of this
freedom, less force will be required to produce the change
than when the body is subjected to constraint.
In the case before us we may suppose the condition of
constant temperature to be obtained by making the cylinder
of a substance which is a perfect conductor of heat, and
surrounding it with a very large bath of a fluid which is also a
perfect conductor of heat, and which has so great a capacity
for heat that all the heat it receives from or gives off to the sub-
stance in the cylinder does not sensibly alter its temperature.
The cylinder in this case is capable of constraining the
substance itself, because it cannot get through the sides of
the cylinder; but it is not capable of constraining the heat of
the substance, which can pass freely out or in through the
walls of the cylinder.
If we now suppose the walls of the cylinder to become
perfect non-conductors of heat, everything remains the same,
except that the heat is no longer free to pass into or out of
the cylinder.
If in the first case the motion of the piston gives rise to
any motion of the heat through the walls, then in the
second case, when this motion is prevented, more force will
be required to produce a given motion of the cylinder on
account of the greater constraint of the system on which the
force acts.
From this we may deduce the effect which the compression
of a substance has on its temperature when heat is prevented
from entering or leaving the substance.
We have seen that in every case the pressure increases
more than it does when the temperature remains constant, or
if the increase of pressure be supposed given, the diminution
1*2
Adiabatic Curves.
FIG 18
\ / \\\ \ \ \ \ \ \ ^
\ M \ V \ X Vx \ \ \
\ I . \ \ \ *,\ \ ^ \ * \ \ %
\ \ \\\.\\\ Vv \ ^
\ 4 \ \ \ \\ \ \ V \ \ X
\ I \\ \ \\ \ V\\ \
?. \\ \\ \ \\ \ \\ \
\ \ \ \ ^ "^
*>, V ^-^^
Thermal Lines for Air
Isothermals
Adiabatics
Effect of Pressure on Temperature. 133
of volume is less when the heat is confined. Hence the
volume after the pressure is applied is greater when the heat
is confined than when the temperature is constant.
Far the greater number of substances expand when their
temperature is raised, so that for the same pressure a greater
volume corresponds to a higher temperature. In these sub-
stances, therefore, compression produces a rise of temperature
if heat is not allowed to escape ; but if the walls of the
cylinder permit the passage of heat, as soon as the tempe-
rature has begun to rise heat will begin to flow out, so that
if the compression is effected slowly the principal thermal
effect of the compression will be to make the substance part
with some of its heat. The isothermal and adiabatic lines
of air are given in fig. 1 8, p. 132. The adiabatic lines are
more inclined to the horizontal than the isothermal lines.
There are, however, certain substances which contract
instead of expanding when their temperature is raised.
When pressure is applied to these substances the compression
produced is, as in the former case, less when heat is pre-
vented from passing than when the temperature is maintained
constant. The volume after the application of pressure is
therefore, as before, greater than when the temperature is con-
stant ; but since in these substances an increase of volume
indicates a fall of temperature, it follows that, instead of being
heated, they are cooled by compression, and that, if the walls
of the cylinder permit the passage of heat, heat will flow in
from without to restore the equilibrium of temperature.
During a change of state, when, at a given pressure, the
volume alters considerably without change of temperature, as
successive portions of the substance pass from the one state
to the other, the isothermal lines are, as we have already
remarked, horizontal. The adiabatic lines, however, are
inclined downwards from left to right. Any increase of
pressure will cause a portion of the substance to pass into
that one of the two states in which its volume is least. In
BO doing it will give out heat if, as in the case of a liquid and
its vapour, the substance gives out heat in passing into the
134 Adiabatic Curves.
denser state ; but if, as in the case of ice and water, the ice
requires heat to melt it into the denser form of water, then
an increase of pressure will cause some of the ice to melt,
and the mixture will become colder.
The isothermal and adiabatic lines for steam in presence
of water are given in fig. 19, p. 135. The isothermal lines
are here horizontal.
The steam line v v, which indicates the volume of one
pound of saturated steam, is also drawn on the diagram. Its
inclination to the horizontal line is less than that of the
adiabatic lines. Hence when no heat is allowed to escape,
an increase of pressure causes some of the water to become
steam, and a diminution of pressure causes some of the
steam to be condensed into water. This was first shown by
Clausius and Rankine.
By means of diagrams of the isothermal and adiabatic
lines the thermal properties of a substance can be com
pletely defined, as we shall show in the subsequent chapters
As a scientific method, this mode of representing the pro-
perties of the substance is by far the best, but in order to
interpret the diagrams, some knowledge of thermodynamics
is required. As a mere aid to the student in remembering
the properties of a substance, the following mode of tracing
the changes of volume and temperature at a constant pres-
sure may be found useful, though it is quite destitute of
those scientific merits which render the indicator diagrams
so valuable in the investigation of physical phenomena.
The diagram on p. 137 represents the effect of the appli-
cation of heat to a pound of ice at o F. The quantity of
heat applied to the ice is indicated by the distance measured
along the base line marked 'units of heat.' The volume
of the substance is indicated by the length ot the per-
pendicular from the base line cut off by the 'line of
volume,' and the temperature is indicated by the length
cut off by the dotted ' line of temperature.'
The specific heat of ice is about 0*5, so that it requires
1 6 units of heat to raise its temperature from o F. to 32 F.
The specific gravity of ice at 32 F. is, according to Bunsen,
Adiabatic Curves. 135
FIG. 19.
\ V \
1 \ \ \ \
\ \
\ \ \ \ \
_ _ -
Thermal Lines of Steam and SYater.
Tsothermals
Adiabatics - - - - - -
Steam Line v v
136 Diagram of Effects of Heat.
0-91674, so that its volume, as compared with water at 39* i>
is i '0908.
The ice now begins to melt, the temperature remains
constant at 32 F., but the volume of ice diminishes and
the volume of water increases, as is represented by the
line marked * volume of ice.' The latent heat of ice is
144 F., so that the process of melting goes on till 144 units
of heat have been applied to the substance, and the whole
is converted into water at 32 F.
The volume of the water at 32 F. is, according to
M. Despretz, i '000127. Its specific heat is at this tem-
perature a very little greater than unity ; it is exactly unity
at 3 9 'i F., and as the temperature rises the specific heat
increases, so that to heat the water from 32 F. to 212 F.
requires 182 units instead of 180. The volume of the
water diminishes as the temperature rises from 32 F. to
3 9 *i F., where it is exactly i. It then expands, slowly at
first, but more rapidly as the temperature rises, till at 212 F
the volume of the water is 1-04315.
If we continue to apply heat to the water, the pressure
being still that of the atmosphere, the water begins to boil.
For every unit of heat, one nine hundred and sixty-fifth
part of the pound of water is boiled away and is converted
into steam, the volume of which is about 1,700 times that of
the water from which it was formed. The diagram might be
extended on a larger sheet of paper to represent the whole
process of boiling the water away. This process would re-
quire 965 units of heat, so that the whole length of the base
line from o would be 1 1 '07 inches. At this point the water
would be all boiled away, and the steam would occupy a
volume of 1,700 times that of the water. The vertical line
on the diagram which would represent the volume of the
steam would be 3,400 inches, or more than 286 feet long.
The temperature would be still 212 F. If we continue to
apply heat to the steam, still at the atmospheric pressure,
its temperature will rise in a perfectly uniform manner at
Diagram of Effects of Heat.
FIG. 20.
CE BEGINS TO MELT
138 Heat Engines.
the rate of 2*o8 degrees for every unit of heat, the specific
heat of steam being 0*4805.
The volume of the superheated steam also increases in a
regular manner, being proportional to its absolute tempe-
rature reckoned from 460 F.
CHAPTER VIII.
ON HEAT ENGINES.
HITHERTO the only use we have made of the indicatoi
diagram is to. explain the relation between the volume and the
pressure of a substance placed in certain thermal conditions.
The condition that the temperature is constant gave us the
isothermal lines, and the condition that no communication
of heat takes place gave us the adiabatic lines. We have
now to consider the application of the same method to the
measurements of quantities of heat and quantities of me-
chanical work.
At p. 102 it was shown that if the pencil of the indicator
moves from B to c, this shows that the volume of the sub-
stance has increased from o b to o c, under a pressure which
was originally B b and finally c c.
The work done by the pressure of the substance against
the piston during this motion is represented by the area
B c c , and since the volume increases during the process,
it is the substance which does the work on the piston,
and not the piston which does the work on the substance.
In heat engines of ordinary construction, such as steam
engines and air engines, the form of the path described by
the pencil depends on the mechanical arrangements of the
engine, such as the opening and shutting of the valves which
admit or carry off the steam.
For the purposes of scientific illustration, and for obtaining
clear views of the dynamical theory of heat, we shall describe
Carnot 1 s Engine.
139
the working of an engine of a species entirely imaginary
one which it is impossible to construct, but very easy to
understand.
This engine was invented and described by Sadi Carnot,
in his ' Reflexions sur la Puissance motrice du Feu.' pub-
lished in 1824. It is called Carnot's Reversible Engine for
reasons which we shall explain.
All the arrangements connected with this engine are con-
trived for the sake of being explained, and are not intended
to represent anything in the working of real engines.
Carnot himself was a believer in the material nature of
heat, and was in consequence led to an erroneous statement
of the quantities of heat which must enter and leave the
engine. As our object is to understand the theory of heat,
and not to give an historical account of the theory, we shall
avail ourselves of the important step which Carnot made,
while we avoid the error into which he fell.
FIG. 21.
T
B
COLD
Let D be the working substance, which may be any
stance whatever which is in any way affected by heat, but,
for the sake of precision, we shall suppose it to be either air
or steam, or partly steam and partly condensed water at the
same temperature.
The working substance is contained in a cylinder fitted
with a piston. The walls of the cylinder and the piston are
140
Heat Engines.
FIG.
supposed to be perfect non-conductors of heat, but the
bottom of the cylinder is a perfect conductor of heat, and has
so small a capacity for heat that the amount of heat required
to raise its temperature may be left out of account. All the
communication of heat between the working substance and
things outside the cylinder is supposed to take place
through this conducting bottom, and the quantities of heat
are supposed to be measured as they pass through.
A and B are two bodies the temperatures of which are
maintained uniform. A is kept always hot, at a temperature
s, and B is kept always cold, at a temperature T. c is a
stand to set the cylinder on, the upper surface of which is u
perfect non-conductor of heat.
Let us suppose that the working substance is at the tem-
perature T of the cold body B, and that its volume and
pressure are represented in the in-
dicator diagram by o a and a A, the
point A being on the isothermal line
A D corresponding to the lower tem-
perature T.
First Operation. We now place
the cylinder on the non-conducting
stand c, so that no heat can escape,
and we then force the piston down,
so as to diminish the volume of the
substance. As no heat can escape,
the temperature rises, and the rela-
tion between volume and pressure
at any instant will be expressed by
the pencil tracing the adiabatic line A B.
We continue this process till the temperature has risen to
s, that of the hot body A. During this process we have ex-
pended an amount of work on the substance which is re-
presented by the area A B b a. If work is reckoned negative
when it is spent on the substance, we must regard that
employed in this first operation as negative.
Car no? s Four Operations. 141
Second Operation. We now transfer the cylinder to the
hot body A, and allow the piston gradually to rise. The
immediate effect of the expansion of the substance is to
make its temperature fall, but as soon as the temperature
begins to fall, heat flows in from the hot body A through the
perfectly conducting bottom, and keeps the temperature from
falling below the temperature s.
The substance will therefore expand at the temperature s,
and the pencil will trace out the line B c, which is part of the
isothermal line corresponding to the upper temperature s.
During this process the substance is doing work by its
pressure on the piston. The amount of this work is re-
presented by the area B c c b, and it is to be reckoned
positive.
At the same time a certain amount of heat, which we shall
call H, has passed from the hot body A into the working
substance.
TJiird Operation. The cylinder is now transferred from the
hot body A to the non-conducting body c, and the piston is
allowed to rise. The indicating pencil will trace out the
adiabatic line c D, since there is no communication of heat,
and the temperature will fall during the process. When
the temperature has fallen to T, that of the cold body,
let the operation be stopped. The pencil will then have
arrived at D, a point on the isothermal line for the lower
temperature T.
The work done by the substance during this process is
represented by the area c D d c, and is positive.
Fourth Operation. The cylinder is placed on the cold
body B. It has the same temperature as B, so that there is no
transfer of heat. But as soon as we begin to press down the
piston heat flows from the working substance into B, so that
the temperature remains sensibly equal to T during the
operation. The piston must be forced down till it has
reached the point at which it was at the beginning of the
first operation, and, since the temperature is also the same,
142 Heat Engines.
the pressure will be the same as at first. The working
substance, therefore, after these four operations, has returned
exactly to its original state as regards volume, pressure, and
temperature.
During the fourth operation, in which the pencil traces the
portion D A of the isothermal line for the lower temperature,
the piston does work on the substance, the amount of which
is to be reckoned negative, and which is represented by the
area D A a d.
At the same time a certain amount of heat, which we shall
denote by -#, has flowed from the working substance into the
cold body B.
DEFINITION OF A CYCLE. A series of operations by which
the substance is finally brought to the same state in all respects
as at first is called a Cycle of operations.
Total Work done during the Cycle. When the piston is
rising the substance is giving out work ; this is the case in
the second and third operations. When the piston is sinking it
is performing work on the substance which is to be reckoned
negative. Hence, to find the work performed by the substance
we must subtract the area D A B b d, representing the negative
work, from the positive work, B c D d b. The remainder,
A B c D, represents the useful work performed by the sub-
stance during the cycle of operations. If we have any diffi-
culty in understanding how this amount of work can be
obtained in a useful form during the working of the engine,
we have only to suppose that the piston when it rises is
employed in lifting weights, and that a portion of the weight
lifted is employed to force the piston down again. As the
pressure of the substance is less when the piston is sinking
than when it is rising, it is plain that the engine can raise a
greater weight than that which is required to complete the
cycle of operations, so that on the whole there is a balance
of useful work.
Transference of Heat during the Cycle. It is only in the
second and fourth operations that there is any transfer of
Comparison of Thermal and Mechanical Effects. 1 4 3
heat, for in the first and third the heat is confined by the
non-conducting stand.
In the second operation a quantity of heat represented by
H passes from the hot body A into the working substance at
the upper temperature s, and in the fourth operation a
quantity of heat represented by h passes from the working
substance into the cold body B at the lower temperature T.
The working substance is left after the cycle of operations
in precisely the same state as it was at first, so that the whole
physical result of the cycle is
1. A quantity, H, of heat taken from A at the temperature s.
2. The performance by the substance of a quantity of
work represented by A B c D.
3. A quantity, //, of heat communicated to B at the tem-
perature T.
APPLICATION OF THE PRINCIPLE OF THE CONSERVATION
OF ENERGY.
It has long been thought by those who study natural
forces that in all observed actions among bodies the work
which is done is merely transferred from one body in which
there is a store of energy into another, so as to increase the
store of energy in the latter body.
The word energy is employed to denote the capacity
which a body has of performing work, whether this capacity
arises from the motion of the body, as in the case of a cannon-
ball, which is able to batter down a wall before it can be
stopped ; or from its position, as in the case of the weight of a
clock when wound up, which is able to keep the clock going
for a week ; or from any other cause, such as the elasticity of
a watch-spring, the magnetisation of a compass needle, the
chemical properties of an acid, or the heat of a hot body.
The doctrine of the conservation of energy asserts that all
these different forms of energy can be measured in the same
way that mechanical work is measured, and that if the whole
energy of any system were measured in this way the mutual
1 44 Heat Engines.
action of the parts of the system can neither increase not
diminish its total stock of energy.
Hence any increase or diminution of energy in a system
must be traced to the action of bodies external to the
system.
The belief in the doctrine of the conservation of energy
has greatly assisted the progress of physical science, especially
since 1840. The numerous investigations which have been
made into the mechanical value of various forms of energy
were all undertaken by men who believed that in so doing
they were laying a foundation fora more accurate knowledge
of physical actions considered as forms of energy. The fact
that so many forms of energy can be measured on the
hypothesis that they are all equivalent to mechanical energy,
and that measurements conducted by different methods are
consistent with each other, shows that the doctrine con-
tains scientific truth.
To estimate its truth from a demonstrative point of view
we must consider, as we have always to do in making such
estimates, what is involved in a direct contradiction of the
doctrine. If the doctrine is not true, then it is possible for
the parts of a material system, by their mutual action alone,
and without being themselves altered in any permanent way,
either to do work on external bodies or to have work done
on them by external bodies. Since we have supposed the
system after a cycle of operations to be in exactly the same
state as at first, we may suppose the cycle of operations to
be repeated an indefinite number of times, and therefore the
system is capable in the first case of doing an indefinite
quantity of work without anything being supplied to it, and
in the second of absorbing an indefinite quantity of work
without showing any result.
That the doctrine of the conservation of energy is not
self-evident is shown by the repeated attempts to discover
a perpetual motive power, and though such attempts have
been long considered hopeless by scientific men, these men
Conservation of Energy. 145
themselves had repeatedly observed the apparent loss of
energy in friction and other natural actions, without making
any attempt or even showing any desire to ascertain whal
becomes of this energy.
The evidence, however, which we have of the doctrine is
nearly if not quite as complete as that of the conservation ol
matter the doctrine that in natural operations the quantity
of matter in a system always remains the same though it may
change its form.
No good evidence has been brought against either of these
doctrines, and they are as certain as any other part of our
knowledge of natural things.
The great merit of Carnofs method is that he arranges his
operations in a cycle, so as to leave the working substance
in precisely the same condition as he found it. We are
therefore sure that the energy remaining in the working
substance is the same in amount as at the beginning of the
cycle. If this condition is not fulfilled, we should have to
discover the energy required to change the substance from
its original to its final state before we could make any
assertion based upon the conservation of energy.
We have therefore got rid of the consideration of the
energy residing in the working substance, which is called its
intrinsic energy, and we have only to compare
1. The original energy, which is a quantity H of heat at the
temperature s of the hot body. This being communicated to
the working substance, we get for the resulting energy
2. A quantity of work done, represented by A B c D ; and
3. A quantity h of heat at the temperature T of the cold
body.
The principle of the conservation of energy tells us that
the energy of the heat H at the temperature s exceeds that
of the heat h at the temperature T by a quantity of n?e-
chanical energy represented by A B c D, which can be easily
expressed in foot-pounds. This is admitted by all.
Now Carnot believed heat to be a material substance,
L
146 Heat Engines.
called caloric, which of course cannot be created or destroyed
He therefore concluded that, since the quantity of heat re-
maining in the substance is the same as at first, H, the quantity
of heat communicated to it, and h, the quantity of heat
abstracted from it, must be the same.
These two portions of heat, however, are, as Carnot
observed, in different conditions, for H is at the temperature
of the hot body, and h at that of the cold body, and Carnot
concluded that the work of the engine was done at the
expense of the fall of temperature, the energy of any
distribution of heat being greater the hotter the body which
contains it.
He illustrated this theory very clearly by the analogy of a
water-mill. When water drives a mill the water which enters
the mill leaves it again unchanged in quantity, but at a lower
level. Comparing heat with water, we must compare heat
at high temperature with water at a high level. Water tends
to flow from high ground to low ground, just as heat tends to
flow from hot bodies to cold ones. A water-mill makes use
of this tendency of water, and a heat engine makes use of the
corresponding property of heat.
The measurement of quantities of heat, especially when it
has to be done in an engine at work, is an operation of great
difficulty, and it was not till 1862 that it was shown experi-
mentally by Hirn that h, the heat emitted, is really less than
H, the heat received by the engine. But it is easy to see
that the assumption that H is equal to h must be wrong.
For if we were to employ the engine in stirring a liquid,
then the work A B c D spent in this way would generate an
amount of heat which we may denote by in the liquid.
The heat H at the high temperature has therefore been
used, and we find instead of it a quantity h at the low
temperature, and also at the temperature of the liquid,
whatever it is.
But if heat is material, and therefore H = h, then h + $
u gi eater than the original quantity H, and heat has been
Heat ts not a Substance. 147
created, which is contrary to the hypothesis that it is
material.
Besides this, we might have allowed the heat H to pass
from the hot body to the cold body by conduction, either
directly or through one or more conducting bodies, and in
this case we know that the heat received by the cold body
would be equal to the heat taken from the hot body, since
conduction does not alter the quantity of heat. Hence in
this case H = ^, but no work is done during the transfer of
heat. When, in addition to the transfer of heat, work is done
by the engine, there ought to be some difference in the final
result, but there will be no difference if h is still equal to H.
The hypothesis of caloric, or the theory that heat is a kind
of matter, is rendered untenable, first by the proof given by
Rumford, and more completely by Davy, that heat can be
generated at the expense of mechanical work ; and, second,
by the measurements of Hirn, which show that when heat
does work in an engine, a portion of the heat disappears.
The determination of the mechanical equivalent of heat by
Joule enables us to assert that the heat which is required to
raise a pound of water from 39 F. to 40 F. is mechanically
equivalent to 772 foot-pounds of work.
It is to be observed that in this statement nothing is said
about the temperature of the body in which the heat exists.
The heat which raises the pound of water from 39 F. to
40 F. may be taken from a vessel of cold water at 50 F.,
from a red-hot iron heater at 700 F., or from the sun at a
temperature far above any experimental determination, and
yet the heating effect of the heat is the same whatever be the
source from which it flows. When heat is measured as a
quantity, no regard whatever is paid to the temperature of
the body in which the heat exists, any more than to the size,
weight, or pressure of that body, just as when we deter-
mine the weight of a body we pay no attention to its other
properties.
Hence if a body in a certain state, as to temperature, &c.j
148 Heat Engines.
is capable of heating so many pounds of water from 39 F. to
40 F. before it is itself cooled to a given temperature, say
40 F., then if that body, in its original state, is stirred about
and its parts rubbed together so as to expend 772 foot-pounds
of work in the process, it will be able to heat one pound
more of water from 39 F. to 40 F. before it is cooled to the
given temperature.
Carnot, therefore, was wrong in supposing that the
mechanical energy of a given quantity of heat is greater
when it exists in a hot body than when it exists in a cold
body. We now know that its mechanical energy is exactly
the same in both cases, although when in the hot body it is
more available for the purpose of driving an engine.
In our statement of the four operations of Carnot's engine
we arranged them so as to leave the result in a state in
which we can interpret it either as Carnot did, or according
to the dynamical theory of heat. Carnot himself began with
the operation which we have placed second, the expansion
at the upper temperature, and he directs us to continue the
fourth operation, compression at the lower temperature, till
exactly as much heat has left the substance as entered during
the expansion at the upper temperature. The result of this
operation would be, as we now know, to expel too much
heat, so that after the substance had been compressed on
the non-conducting stand to its original volume, its tempera-
ture and pressure would be too low. It is easy to amend the
directions for the extent to which the outflow of heat is to be
permitted, but it is still easier to avoid the difficulty by
placing this operation last, as we have done.
We are now able to state precisely the relation between ^,
the quantity of heat which leaves the engine, and H, the
quantity received by it. H is exactly equal to the sum of //,
and the heat to which the mechanical work represented by
A B c D is equivalent.
In all statements connected with the dynamical theory of
heat it is exceedingly convenient to state quantities of heat
Heat expressed in Foot-pounds. 149
in foot-pounds at once, instead of first expressing them in
thermal units and then reducing the result to foot-pounds by
means of Joule's equivalent of heat In fact, the thermal
unit depends for its definition on the choice of a standard
substance to which heat is to be applied, on the
choice of a standard quantity of that substance, and
on the choice of the effect to be produced by the heat
According as we choose water or ice, the grain or the
gramme, the Fahrenheit or the Centigrade scale of tempera-
tures, we obtain different thermal units, all of which have
been used in different important researches. By expressing
quantities of heat in foot-pounds we avoid ambiguity, and,
especially in theoretical reasonings about the working of
engines, we save a great deal of useless phraseology.
As we have already shown how an area on the indicator
diagram represents a quantity of work, we shall have no
difficulty in understanding that it may also be taken to re-
present a quantity of heat equivalent to the same quantity of
work, that is the same number of foot-pounds of heat
We may therefore express the relation between H and h
still more concisely thus :
The quantity, H, of heat taken into the engine at the
upper temperature s exceeds the quantity, ^, of heat given
out by the engine at the lower temperature T by a quantity
of heat represented by the area A B c D on the indicator
diagram.
This quantity of heat is, as we have already shown, con-
verted into mechanical work by the engine.
ON THE REVERSED ACTION OF CARNOT'S ENGINE.
The peculiarity of Carnot's engine is, that whether it is
receiving heat from the hot body, or giving it out to the
cold body, the temperature of the substance in the engine
differs extremely little from that of the body in thermal
communication with it. By supposing the conductivity of
1 50 Heat Engines.
the bottom of the cylinder to be sufficiently great, or by
supposing the motions of the piston to be sufficiently
slow, we may make the actual difference of temperature
which causes the flow of heat to take place as small as we
please.
If we reverse the motion of the piston when the substance
is in thermal communication with A or B, the first effect will
be to alter the temperature of the working substance, but
an exceedingly small alteration of temperature will be suf-
ficient to reverse the flow of heat, if the motion is slow
enough.
Now let us suppose the engine to be worked backwards
by exactly reversing all the operations already described.
Beginning at the lower temperature and volume o 0, let it
be placed on the cold body and expand from volume o a to
o d. It will receive from the cold body a quantity of heat
h. Then let it be compressed without losing heat to o c.
It will then have the upper temperature s. Let it then be
placed on the hot body and compressed to volume o b. It
will give out a quantity of heat H to the hot body. Finally,
let it be allowed to expand without receiving heat to volume
o #, and it will return to its original state. The only difference
between the direct and the reverse action of the engine is,
that in the direct action the working substance must be a
little cooler than A when it receives its heat, and a little
warmer than B when it gives it out ; whereas in the reverse
action it must be warmer than A when it gives out heat, and
cooler than B when it takes heat in. But by working the
engine sufficiently slowly these differences may be reduced
within any limits we please to assign, so that for theo-
retical purposes we may regard Carnot's engine as strictly
reversible.
In the reverse action a quantity h of heat is taken from
the cold body B, and a greater quantity H is given to the
hot body A, this being done at the expense of a quantity of
work measured by the area A D c B, which also measures
Carnofs Engine Reversed 151
the quantity of heat into which this work is transformed
during the process.
The reverse action of Carnot's engine shows us that it is
possible to transfer heat from a cold body to a hot one,
but that this operation can only be done at the expense of
a certain quantity of mechanical work.
The transference of heat from a hot body to a cold one
may be effected by means of a heat engine, in which case
part of it is converted into mechanical work, or it may
take place by conduction, which goes on of itself, but
without any conversion of heat into work. It appears,
therefore, that heat may pass from hot bodies to cold ones
in two different ways. One of these, in which a highly
artificial engine is made use of, is nearly, but not quite
completely, reversible, so that by spending the work we
have gained, we can restore almost the whole of the heat
from the cold body to the hot. The other mode of trans-
fer, which takes place of itself whenever a hot and a cold
body are brought near each other, appears to be irreversible,
for heat never passes from a cold body to a hot one of
itself, but only when the operation is effected by the artificial
engine at the expense of mechanical work.
We now come to an important principle, which is en-
tirely due to Carnot. If a given reversible engine, working
between the upper temperature s and the lower tempera
ture T, and receiving a quantity H of heat at the upper
temperature, produces a quantity w of mechanical work,
then no other engine, whatever be its construction, can
produce a greater quantity of work, when supplied with
the same amount of heat, and working between the same
temperatures.
DEFINITION OF EFFICIENCY. If H is the supply of heat,
and w the work done by an engine, both measured in foot-
pounds, then the fraction - is called the Efficiency of the
TT
engine.
r 5 2 Heat Engines.
Garnet's principle, then, is that the efficiency of a rever
sible engine is the greatest that can be obtained with a given
range of temperature.
For suppose a certain engine, M, has a greater efficiency
between the temperatures s and T than a reversible engine
N, then if we connect the two engines, so that M by its
direct action drives N in the reverse direction, at each stroke
of the compound engine N will take from the cold body
B the heat //, and by the expenditure of work w give to the
hot body A the heat H. The engine M will receive this
heat H, and by hypothesis will do more work while trans-
ferring it to B than is required to drive the engine N.
Hence at every stroke there will be an excess of useful
work done by the combined engine.
We must not suppose, however, that this is a violation of
the principle of conservation of energy, for if M does more
work than N would do, it converts more heat into work in
every stroke, and therefore M restores to the cold body a
smaller quantity of heat than N takes from it. Hence, the
legitimate conclusion from the hypothesis is, that the com-
bined engine will, by its unaided action, covert the heat
of the cold body B into mechanical work, and that this
process may go on till all the heat in the system is converted
into work.
This is manifestly contrary to experience, and therefore
we must admit that no engine can have an efficiency greater
than that of a reversible engine working between the same
temperatures. But before we consider the results of Car-
not's principle we must endeavour to express clearly the
law which lies at the bottom of the reasoning.
The principle of the conservation of energy, when applied
to heat, is commonly called the First Law of Thermo-
dynamics. It maybe stated thus : When work is transformed
into heat, or heat into work, the quantity of work is
mechanically equivalent to the quantity of heat.
The application of the law involves the existence of the
mechanical equivalent of heat-
First and Second Laws of Thermodynamics. 153
Carnot's principle is not deduced from this law, and
indeed Carnot's own statement involved a violation of it.
The law from which Carnot's principle is deduced has been
called the Second Law of Thermodynamics.
Admitting heat to be a form of energy, the second law
asserts that it is impossible, by the unaided action of natural
processes, to transform any part of the heat of a body into
mechanical work, except by allowing heat to pass from that
body into another at a lower temperature. Clausius, who
first stated the principle of Carnot in a manner consistent
with the true theory of heat, expresses this law as follows :
It is impossible for a self-acting machine, unaided by any
external agency, to convey heat from one body to another
at a higher temperature.
Thomson gives it a slightly different form :
It is impossible, by means of inanimate material agency,
to derive mechanical effect from any portion of matter by
cooling it below the temperature of the coldest of the sur-
rounding objects.
By comparing together these statements, the student will
be able to make himself master of the fact which they em-
body, an acquisition which will be of much greater import-
ance to him than any form of words on which a demon-
stration may be more or less compactly constructed.
Suppose that a body contains energy in the form of heat,
what are the conditions under which this energy or any
part of it may be removed from the body ? If heat in a
body consists in a motion of its parts, and if we were able
to distinguish these parts, and to guide and control their
motions by any kind of mechanism, then by arranging our
apparatus so as to lay hold of every moving part of the
body, we could, by a suitable train of mechanism, transfer
the energy of the moving parts of the heated body to any
other body in the form of ordinary motion. The heated
body would thus be rendered perfectly cold, and all its
thermal energy would be converted into the visible motion
ot some other body.
1 54 Heal Engines.
Now this suppositic n involves a direct contradiction to
the second law of thermodynamics, but is consistent with
the first law. The second law is therefore equivalent to a
denial of our power to perform the operation just described,
either by a train of mechanism, or by any other method yet
discovered. Hence, if the heat of a body consists in the
motion of its parts, the separate parts which move must
be so small or so impalpable that we cannot in any way lay
hold of them to stop them.
In fact, heat, in the form of heat, never passes out of a
body except when it flows by conduction or radiation into a
colder body.
There are several processes by which the temperature of
a body may be lowered without removing heat from it, such
as expansion, evaporation, and liquefaction, and certain
chemical and electrical processes. Every one of these,
however, is a reversible process, so that when the body is
brought back by any series of operations to its original state,
without any heat being allowed to enter or escape during
the process, the temperature will be the same as before, in
virtue of the reversal of the processes by which the tempera-
ture was lowered. But if, during the operations, heat
has passed from hot parts of the system to cold by con-
duction, or if anything of the nature of friction has taken
place, then to bring the system to its original state will
require the expenditure of work, and the removal of heat.
We must now return to the important result demonstrated
by Carnot, that a reversible engine, working between two
given temperatures, and receiving at the higher temperature
a given quantity of heat, performs at least as much work
as any other engine whatever working under the same
conditions. It follows from this that all reversible engines,
whatever be the working substance employed, have the
same efficiency, provided they work between the same
temperature of the source of heat A and the same tempera-
ture of the refrigerator B.
Hence Carnot showed that if we choose two tempera-
Carnofs Function. 155
cures differing very slightly, say by y^Vs of a degree, the
efficiency of an engine working between these temperatures
will depend on the temperature only, and not on the sub-
stance employed, and this efficiency divided by the differ-
ence of temperatures is the quantity called Carnof s function,
a quantity depending on the temperature only.
Carnot, of course, understood the temperature to be
estimated in the ordinary way by m'eans of a thermometer
of a selected substance graduated according to one of the
established scales, and his function is expressed in terms of
the temperature so determined. But W. Thomson, in 1848,
was the first to point out that Carnot's result leads to a
method of denning temperature which is much more
scientific than any of those derived from the behaviour of
one selected substance or class of substances, and which
is perfectly independent of the nature of the substance
employed in defining it.
THOMSON'S ABSOLUTE SCALE OF TEMPERATURE.
Let T A B c represent the isothermal line corresponding
to temperature T for a certain substance. For the sake of
distinctness in the figure, I have supposed the substance to
be partly in the liquid and partly in the gaseous state, so
that the isothermal lines are horizontal, and easily dis-
tinguished from the adiabatic lines, which slope downwards
to the right. The investigation, however, is quite indepen-
dent of any such restriction as to the nature of the working
substance. When the volume and pressure of the substance
are those indicated by the point A, let heat be applied
and let the substance expand, always at the temperature T,
till a quantity of heat H has entered, and let the state of
the substance be then indicated by the point B. Let
the process go on till another equal quantity, H, of heat has
entered, and let c indicate the resulting state. The process
may be carried on so as to find any number of points on
156
Heat Engines,
Now let A A' A", B B' B'
FIG. 23.
tne isothermal line, such that for each point passed during
the expansion of the substance a quantity H of heat has been
communicated to it.
c c' c" be adiabatic lines drawn
through ABC, that is, lines
representing the relation be-
tween volume and pressure
when the substance is allowed
to expand without receiving
heat from without.
LetT / A'B'c'andT"A"B"c"
be isothermal lines corre-
sponding to the temperatures
T' and T".
We have already followed
Carnot's proof that in a re-
versible engine, working from
the temperature T of the source of heat to the temperature T'
of the refrigerator, the work w produced by the quantity of
heat H drawn from the source depends only on T and T'.
Hence, since A B and B c correspond to equal quantities
of heat H received from the source, the areas A B B' A' and
B c c' B', which represent the corresponding work performed,
must be equal.
The same is true of the areas cut off by the adiabatic lines
from the space between any other pair of isothermal lines.
Hence if a series of adiabatic lines be drawn so that the
points at which they cut one of the isothermal lines corre-
spond to successive equal additions of heat to the substance
at that temperature, then this series of adiabatic lines will cut
off a series of equal areas from the strip bounded by any two
isothermal lines.
Now Thomson's method of graduating a scale of tempera-
ture is equivalent to choosing the points A A' A", from which
to draw a series of isothermal lines, so that the area A B B' A'
contained between two consecutive isothermals T and T f shall
A bsolute Scale of Temperature. 157
be equal to the area A' B' B'' A 1 ' contained between any other
pair of consecutive isothermals T' T".
It is the same as saying that the number of degrees between
the temperature T and the temperature T" is to be reckoned
proportional to the area A B B'' A".
Of course two things remain arbitrary, the standard tem-
perature which is to be reckoned zero, and the size of the
degrees, and these may be chosen so that the absolute scale
corresponds with one of the ordinary scales at the two
standard temperatures, but as soon as these are determined
the numerical measure of every other temperature is settled,
in a manner independent of the laws of expansion of any
one substance by a method, in fact, which leads to the same
result whatever be the substance employed.
It is true that the experiments and measurements required
to graduate a thermometer on the principle here pointed out
would be far more difficult than those required by the
ordinary method described in the chapter on Thermometry.
But we are not, in this chapter, describing convenient methods
or good working engines. Our objects are intellectual,
not practical, and when we have established theoretically
the scientific advantages of this method of graduation, we
shall be better able to understand the practical methods by
which it can be realised.
We now draw the series of isothermal and adiabatic lines
in the following way :
A particular isothermal line, that of temperature T, is cut
by the adiabatic lines, so that the expansion of the substance
between consecutive adiabatic lines corresponds to successive
quantities of heat, each equal to H, applied to the substance.
This determines the series of adiabatic lines.
The isothermal lines are drawn so that the successive
isothermals cut off from the space between the pair of
adiabatic lines A A' A" and B B' B" equal areas A B B 7 A',
i' B' B" A", &c.
The isothermal lines so determined cut off equal area?
158 Heat Engines.
from every other pair of adiabatic lines, so that the two
systems of lines are such that all the quadrilaterals formed
by two pairs of consecutive lines are equal in area.
We have now graduated the isothermals on the diagram
by a method founded on Carnot's principle alone, and in-
dependent of the nature of the working substance, and it is
easy to see how by altering, if necessary, the interval between
the lines and the line chosen for zero, we can make the
graduation agree, at the two standard temperatures, with
the ordinary scale.
EFFICIENCY OF A HEAT ENGINE.
Let us now consider the relation between the heat supplied
to an engine and the work done by it as expressed in terms
of the new scale of temperature.
If the temperature of the source of heat is T, and if H is
the quantity of heat supplied to the engine at that tempera-
ture, then the work done by this heat depends entirely on
the temperature of the refrigerator. Let i" be the tempera-
ture of the refrigerator, then the work done by H is represented
by the area A B B" A", or, since all the areas between the
isothermals and the adiabatics are equal, let H c be the area
of one of the quadrilaterals, then the work done by H will be
H c (T x"). The quantity c depends only on the tem-
perature T. It is called Carnot's Function of the tempera-
ture. We shall find a simple expression for it at page 160.
This, therefore, is a complete determination of the work
done when the temperature of the source of heat is T. It
depends only on Carnot's principle, and is true whether we
admit the first law of thermodynamics or not.
If the temperature of the source is not T, but T 7 , we must
consider what quantity of heat is represented by the expan-
sion A' B' along the isothermal T'. Calling this quantity of
heat H', the work done by an engine working between the
temperatures T 7 and T" is
w = H c (T' T").
Their Efficiency. 1 59
Now Carnot supposed that H' = H, which would make
the efficiency of the engine simply - = c (T' - T"), where C
H
is Carnot's function, a constant quantity on this supposition.
But according to the dynamical theory of heat, we get by the
first law of thermodynamics
H' = H A B B' A',
the heat being measured as mechanical work, or
H' = H H c (T T').
On this theory, therefore, the efficiency of the engine
working between T' and T" is
w__ H c (T 7 T")
H'~ H H c (T T 7 )
T' - T"
ON ABSOLUTE TEMPERATURE.
We have now obtained a method of expressing differences
of temperature in such a way that the difference of two
temperatures may be compared with the difference of two
other temperatures. But we are able to go a step farther
than this, and to reckon temperature from a zero point
denned on thermodynamic principles independently of the
properties of a selected substance. We must carefully
distinguish between what we are doing now on really scientific
principles from what we did for the sake of convenience in
describing the air thermometer. Absolute temperature on
the air thermometer is merely a convenient expression of the
laws of gases. The absolute temperature as now defined
is independent of the nature of the thermometric substance.
It so happens, however, that the difference between these
two scales of temrfcrature is very small. The reason of this
will be explained afterwards.
160 Heat Engines.
It is plain that the work which a given quantity of heat
H can perform in an engine can never be greater than the
mechanical equivalent of that heat, though the colder the
refrigerator the greater proportion of heat is converted into
work. It is plain, therefore, that if we determine T" the
temperature of the refrigerator, so as to make w the work
mechanically equivalent to H, the heat received by the
engine, we shall obtain an expression for a state of things in
which the engine would convert the whole heat into work,
and no body can possibly be at a lower temperature than
the value thus assigned to T".
Putting w = H', we find T" = T -.
This is the lowest temperature any body can have. Call-
ing this temperature zero, we find
or the temperature reckoned from absolute zero is the
reciprocal of Carnot's function c.
We have therefore arrived at a complete definition of the
measure of temperature, in which nothing remains to be
determined except the size of the degrees. Hitherto the
size of the degrees has been chosen so as to be equal to the
mean value of those of the ordinary scales. To convert the
ordinary expressions into absolute temperatures we must add
to the ordinary expression a constant number of degrees,
which may be called the absolute temperature of the zero of
the scale. There is also a correction varying at different
parts of the scale, which is never very great when the tem-
perature is measured by the air thermometer. We may now
express the efficiency of a reversible heat engine in terms of
the absolute temperature s of the source of heat, and the
absolute temperature T of the refrigerator. If H is the
quantity of heat supplied to the engine, and w is the quantity
of work performed, both estimated in dynamical measure,
w s T
Absolute Temperature. Ibl
The quantity of heat which is given out to the refrigerator
at temperature T is /& = H w = H J , whence
5 = *or"=I
S T k T
that is, in a reversible engine the ratio of the heat received to
the heat rejected is that of the numbers expressing on an abso-
lute scale the temperatures of the source and the refrigerator.
This relation furnishes us with a method of determining
the ratio of two temperatures on the absolute scale. It is
independent of the nature of the substance employed in the
reversible engine, and is therefore a perfect method con-
sidered from a theoretical point of view. The practical
difficulties of fulfilling the required conditions and making
the necessary measurements have not hitherto been over-
come, so that the comparison of the absolute scale of tem-
perature with the ordinary scale must be made in a different
way. (See p. 213.)
Let us now return to the diagram fig. 23 (p. 156), on which
we have traced two systems of lines, the isothermals and
the adiabatics. To draw an isothermal line through a given
point requires only a series of experiments on the substance
at a given temperature, as shown by a thermometer of any
kind. To draw a series of these lines to represent succes-
sive degrees of temperature is equivalent to fixing a scale of
temperature.
Such a scale might be defined in many different ways,
each of which depends on the properties of some selected
substance. For instance, the scale might be founded on the
expansion of a particular substance at some standard pressure.
In this case, if a horizontal line is drawn to represent the
standard pressure, then the isothermal lines of the selected
substance will cut this line at equal intervals. If, however,
the nature of the substance or the standard pressure be
different, the thermometric scale will be in general different
The scale might also be founded on the variation of pressure
M
1 62 Thermodynamics.
of a substance confined in a given space, as in the case of
certain applications of the air thermometer.
It has also been proposed to define temperature so that
equal increments of heat applied to a standard substance
will produce equal increments of temperature. This method
also fails to give results consistent for all substances, because
the specific heats of different substances are not in the same
ratio at different temperatures.
The only method which is certain to give consistent re-
sults, whatever be the substance employed, is that which is
founded on Carnot's Function, and the most convenient
form in which this method can be applied is that which de-
fines the absolute temperature as the reciprocal of Carnot's
Function. We shall see afterwards how a comparison can
be made between the absolute temperature on the thermo-
dynamic scale and the temperature as indicated by a
thermometer of a particular kind of gas. (See p. 213.)
ON ENTROPY.
We have next to consider the series of adiabatic lines as
indicating a series of degrees of another property of the
body, expressed as a measurable quantity, such that when
there is no communication of heat this quantity remains
constant, but when heat enters or leaves the body the quan-
tity increases or diminishes.
We shall adopt the name given by Clausius to this quan-
tity, and call it the entropy of the body. Rankine, in whose
investigations this quantity also plays an important part, calls
it the thermodynamic function. This term, however, is not
so appropriate, as the name might have been assigned to any
one of several important quantities in thermodynamics.
We must regard the entropy of a body, like its volume,
pressure, and temperature, as a distinct physical property of
the body depending on its actual state.
The proper zero of entropy is that of the body when entirely
deprived of heai, but as we cannot bring the body into this
condition it is more convenient to reckon entropy from a
standard state defined by a standard temperature and pressure.
Entropy. 163
The entropy of the body in any other condition is then
measured thus. Let the body expand (or contract) without
communication of heat till it reaches the standard tempera-
ture, the value of which, on the thermodynamic scale, is T.
Then let the body be kept at trie standard temperature and
brought to the standard pressure, and let H be the number
of units of heat given out during this process. Then the
TT
entropy of the body in its original state is .
We shall use the symbol to denote the entropy.
If the body, in order to arrive at the standard state,
requires to absorb heat, then its original entropy must be
reckoned negative with respect to the standard state.
When heat enters a body at the temperature and causes
the entropy to increase from fa to fa, the amount of heat
which enters the body is 0(0 2 ^i)-
The entropy of a body in a given state is proportional to
the mass of the body, so that the entropy of two pounds of
water is double that of one pound in the same state.
We often, however, speak of the entropy of a substance,
by which we mean the entropy of unit of mass of that sub-
stance in the given state.
The entropy of a system of bodies in different states is
the sum of the entropies of each of the bodies.
When a quantity, H, of heat passes from a body at tempera-
ture 0j to a body at temperature 2 , the entropy of the first body
TT
is diminished by , while that of the second is increased by
#i
TT a /
, so that the entropy of the system increases by H * 2 .
2 0! 2
Now it is the condition of the transfer of heat that it
passes from the hotter to the colder body, and therefore 0,
must be greater than 2 .
The transference of heat, therefore, from one body of the
system to another always increases the entropy of the system.
Clausius expresses this by saying that the entropy of the
system always tends towards a maximum value.
164 Thermodynamics.
The heat which enters the body during any very small
change of state is represented, as we have seen, by 0(</> 2 <M,
where 6 is the mean temperature of the body during the
process, and t and </> 2 represent the entropy at the beginning
and the end of the process.
If we suppose the two isentropic lines ^ and 2 to be
continued in the direction of decreasing temperatures down
to the temperature T, then the area included between the
two isentropic lines between the temperatures and T will
be (d-i) (^-^
If we could draw the isentropic and isothermal lines cor-
rectly for all temperatures down to the absolute zero of the
thermodynamic scale, then the whole area included between
the isentropic lines and the isothermals for 6 and zero would
be 0(02 0i)> an d this area would represent the heat which
enters the body during the process.
But though it is impossible to conjecture the properties
of a body at absolute zero or to draw on a diagram the true
forms of the thermal lines near that temperature, it is easy,
after we have constructed the thermodynamic diagram foi
that part of the field which is known by observation,
to draw lines in the unknown part of the field, by means of
which we may still represent quantities of heat by areas.
If the known part of the field is bounded by the isother-
mal T, and if we draw from the extremities of the known
parts of the isentropic lines a series of lines of any form
which do not intersect each other, and if we draw anothei
line, z z', so that the space included between this line, two
neighbouring isentropics ty^ and 2 > an d the isothermal line
T is T(0 2 0i), we may, in calculating quantities of heat, treat
the line z z' as the fictitious isothermal of absolute zero, and
the series of lines as a fictitious isentropic series.
For the area between the two isentropic lines from tem-
perature 6 to temperature T is (0 T) (0 2 tyj. This area is
within the known part of the field. The continuation of
this area in the unknown part of the field down to the ficti-
tious isothermal of absolute zero is T^ U . The whole
Fictitious Thermal Lines. 165
area therefore is Q(fa<t>\), and it therefore represents the
quantity of heat absorbed in passing at the temperature Q
from the line <f> l to the line fa.
The whole heat absorbed by a body in passing from a
state A to a state B through a definite series of intermediate
steps represented by
the path AB, may be
called the 'heat of
the path A B.' By
dividing AB into a
sufficient number of
small parts, and con-
sidering the area re-
presenting the heat
absorbed during the f i fa
passage of the body
over each of these divisions, we find that the sum of these
areas is the area included by the path AB, the isentropics
through A and B including their fictitious parts, and the ficti
tious isothermal of absolute zero.
CHAPTER IX.
ON THE RELATIONS BETWEEN THE PHYSICAL
PROPERTIES OF A SUBSTANCE.
LET T! T! and T 2 T 2 represent two isothermal lines corre-
sponding to two consecutive degrees of temperature. Let
0! (j) l and (f> 2 02 represent two consecutive adiabatic lines.
Let A BCD be the quadrilateral which lies between both
these pairs of lines. If the lines are drawn close enough to
each other we may treat this quadrilateral as a parallelogram.
The area of this parallelogram is, as we have already
shown, equal to unity.
Draw horizontal lines through A and D to meet the line
B c produced in K and Q, then, since the parallelograms
A B c D and A K Q D stand on the same base and are between
the same parallels, they are equal. Now draw the vertical
1 66
Thermodynamics.
lines A k and K P to meet Q D, produced if necessary. Then
the rectangle A K p k is equal to the parallelogram A K Q D,
because they stand on the same base A K, and are between
the same parallels A K and k Q. Hence the rectangle A K p k
FIG. 24.
\N
is also equal to the original parallelogram A B c D. If,
therefore, we draw A K from A horizontally to meet the
isothermal T 2 , and A k vertically to meet a horizontal line
through D, we shall have the following relation :
In the same way, if the horizontal line through A cuts the
adiabatic line 2 in L and the verticals through D and sjn
m and n, and if the vertical line through A cuts the isothermal
line T 2 in M, the adiabatic line <p 2 in N, and the horizontal
line through B in /, we shall get the following four values of
the area of A BCD, including that which we have already
investigated :
ABCD = AK.A^ = AL.A/=AM.A/ = AN.A= I.
We have next to interpret the physical meaning of the
four pairs of lines which enter into these products.
We must remember that the volume of the substance is
measured horizontally to the right, and its pressure vertically
Four Thermodynamical Relations, 1 67
upwards ; that the interval between the isothermal lines
represents one degree of temperature, the graduation of the
scale being as much subdivided as we please ; and that the
interval between the adiabatic lines represents the addition
of a quantity of heat whose numerical value is T, the
absolute temperature.
(1) A K represents the increase of volume for a rise of
temperature equal to one degree, the pressure being main-
tained constant. This is called the cftlatability of the
substance per unit of mass, and if we denote the dilatability
per unit of volume by a, A K will be denoted by v a.
A k represents the diminution of pressure corresponding
to the addition of a quantity of heat represented numerically
by T, the temperature being maintained constant.
If the pressure is increased by unity, the temperature
remaining constant, the quantity of heat which is emitted by
the substance is ~. Since A K . A k = i, -- = T . A K.
A k A. k
Hence the following relation between the dilatation under
constant pressure and the heat developed by pressure.
First Thermodynamic Relation. If the pressure of a sub-
stance be increased by unity while the temperature is main-
tained constant, the quantity of heat emitted by the sub-
stance is equal to the product of the absolute temperature
into the dilatation for one degree of temperature under
constant pressure.
Hence, if the temperature is maintained constant, those
substances which increase in volume as the temperature
rises give out heat when the pressure is increased, and
those which contract as the temperature rises absorb heat
when the pressure is increased.
(2) A L represents the increase of volume under constant
pressure when a quantity of heat numerically equal to T is
communicated to the substance.
A / represents the increase of pressure required to raise
1 68 Thermodynamics.
the substance one degree of temperature when no heat is
allowed to escape.
Second TJiermodynamic Relation. The quantity re-
A L
presents the heat which must be communicated to the sub-
stance in order to increase its volume by unity, the pressure
being constant. This is equal to the product of the ab-
solute temperature into the increase of pressure required
to raise the temperature one degree when no heat is allowed
to escape.
(3) A M represents the increase of pressure corresponding
to a rise of one degree of temperature, the volume being
constant. (We may suppose the substance enclosed in a
vessel the sides of which are perfectly unyielding.)
A m represents the increase of volume produced by the
communication of a quantity of heat numerically equal to
T, the temperature being maintained constant.
The heat given out by the substance when the volume is
diminished by unity, the temperature being maintained con-
stant, is therefore -^ . This quantity is called the latent
A m
heat of expansion.
Since A M . A m = i, we may express the relation between
these lines thus : ~^- = T . A M, or, in words :
A m
Third Thermo dynamic Relation. The latent heat of ex-
pansion is equal to the product of the absolute temperature
and the increment of pressure per degree of temperature at
constant volume.
(4) A N represents the increase of the pressure when a
quantity, T, of heat is communicated to the substance, the
volume being constant.
A n represents the diminution of volume when the sub-
stance, being prevented from losing heat, is compressed till
the temperature rises one degree. Hence :
Specific Heat. 169
Fourth Thermodynamic Relation. - - represents the
rise of temperature due to a diminution of the volume
by unity, no heat being allowed to escape, and this is equal
to A N, the increase of pressure at constant volume due to
a quantity of heat, numerically equal to T, communicated to
the substance.
We have thus obtained four relations among the physical
properties of the substance. These four relations are not
independent of each other, so as to rank as separate truths.
Any one might be deduced from any other. The equality
of the products A K, A /, &c., to the parallelogram A B c D
and to each other is a merely geometrical truth, and does
not depend upon thermodynamical principles. What we
learn from thermodynamics is that the parallelogram and
the four products are each equal to unity, whatever be the
nature of the substance or its condition as to pressure and
temperature. *
ON THE TWO MODES OF MEASURING SPECIFIC HEAT.
The quantity of heat required to raise unit of mass of the
substance one degree of temperature is called the specific
heat of the substance.
1 These four relations may be concisely expressed in the language of
the Differential Calculus as follows:
dv _ d _$ /,\
de(P const.) ~ dp (9 const.) '
dv dB
d $(p const.) = Tp($ const.) '
dp
dp d 6
d (v const.) ~ d v (0 const.)
(3)
d<p(v const.) " dv (9 const.) v4)
Here v denotes the volume.
p pressure.
B ,, absolute temperature.
$> ,, thermodynamic function, or entropy.
1 7 TJiermoaynam ics .
At p. 66 this quantity of heat is expressed in terms of the
thermal unit, or the heat required to raise unit of mass of
\vater one degree. To reduce this to dynamical measure we
must multiply by Joule's mechanical equivalent of the thermal
anit. The quantity thus found is no longer a mere ratio, as
at p. 66, but depends on the thermometric scale which we
select and also on the unit of work.
But the specific heat of a substance depends on the mode
in which the pressure and volume of the substance vary
during the rise of temperature.
There are, therefore, an indefinite number of modes of
defining the specific heat. Two only of these are of any
practical importance. The first method is to suppose the
volume to remain constant during the rise of temperature.
The specific heat under this condition is called the specific
heat at constant volume. We shall denote it by K V .
In the diagram the line A M N represents the different
states of the substance when the volume is constant, A M
represents the increase of pressure due to a rise of one
degree of temperature, and A N that due to the application
of a quantity of heat numerically equal toT. Hence to find
the quantity of heat, K T , which must be communicated to
the substance in order to raise its temperature one degree,
and so increase the pressure by A M , we have
A N : A M : : T : K T
K,=T.^.
A N
The second method of defining specific heat is to suppose
the pressure constant. The specific heat under constant
pressure is denoted by K P .
The line A L K in the diagram represents the different states
of the substance at constant pressure, A K represents the in-
crease of volume due to a rise of one degree of temperature,
and A L represents the increase of volume due to a quantity
of heat numerically equal to T. Now the quantity K P of
heat raises the substance one degree, and therefore increases
the volume by A K.
Relations of Specific Heat and Elasticity. 171
Hence
A L : A K : : T : K P
or
K p = T^.
A L
(A third mode of defining specific heat is sometimes
adopted in the case of saturated steam. In this case the
steam is supposed to remain at the point of saturation as
the temperature rises. It appears, from the experiments of
M. Regnault, as shown in the diagram at p. 135, that heat
leaves the saturated steam as its temperature rises, so that
its specific heat is negative, a result pointed out by Clausius
and Rankine.)
ON THE TWO MODES OF MEASURING ELASTICITY.
The elasticity of*a substance was defined at p. 107 to
be the ratio of the increment of pressure to the com-
pression produced by it, the compression being defined
to be the ratio of the diminution of volume to the original
volume.
But we require to know something about the thermal
conditions under which the substance is placed before we
can assign a definite value to the elasticity. The only two
conditions wliich are of practical importance are, first,
when the temperature remains constant, and, second, when
there is no communication of heat.
(1) The elasticity under the condition that the temperature
remains constant may be denoted by E0.
In this case the relation between volume and pressure is
defined by the isothermal line D A. The increment of
pressure is k A, and the diminution of volume is m A.
Calling the volume v, the elasticity at constant tempera-
ture is
E, = vAi = y.-i*.
Am A K
(2) The elasticity under the condition that heat neither
enters nor leaves the substance is denoted by E^.
In this case the relation between volume and pressure is
Thermodynamics.
defined by the adiabatic line A B. The increment of pressure
is A /, and the decrement of volume is A n. Hence the
elasticity when no heat escapes is
, = V. A _L.V.*f.
A n A L
There are several important relations among these
quantities. In the first place, we find for the ratio of the
specific heats,
T AK y AN
K p _ A L _ * A L __ E$
*v ~~ T . "" V. AM "" ^
'AN ' A K
or the ratio of the specific heat at constant pressure to that
at constant volume is equal to the ratio of the elasticity
when no heat escapes to the elasticity at constant tempera-
ture. This relation is quite independent of the principles of
thermodynamics, being a direct consequence of the defini-
tions.
The ratio of K P to K V , or of E^ to E^is commonly denoted
by the symbol y : thus K P = yK y , and E^ = yE0.
Let us next determine the difference between the two
elasticities
A m . A n
The numerator of the fraction is evidently, by the geo-
metry of the figure, equal to the parallelogram A B c D.
Multiplying by K T , we find
Am AN. An
since A n . A N = A B c D, as we have shown.
Since K Y E$ = K P E0, we also find
These relations are independent of the principles of
thermodynamics.
Latent Heat. 173
If we now apply the thermodynamical equation A M . A m
=. i, each of these quantities becomes equal to
T v . (A i.i) 2 .
Now A M is the increment of pressure at constant volume
per degree of temperature, a very important quantity. The
results therefore may be written
K (E^ - E0) = T V A M* = E0 (K p K T ).
CHAPTER X.
ON LATENT HEAT.
A VERY important class of cases is that in which the sub-
stance is in two different states at the same temperature and
pressure, as when part of it is solid and part liquid, or part
solid or liquid and part gaseous.
In such cases the volume occupied by the substance must
be considered as consisting of two parts, v l being that of the
substance in the first state, and # 2 that of the substance in
the second state. The quantity of heat necessary to convert
unit of mass of the substance from the first state to the
second without altering its temperature is called the Latent
Heat of the substance, and is denoted by L.
During this process the volume changes from v l to z> 2 at
the constant pressure/.
Let P s be an isothermal Fias.
line, which in this case is hori-
zontal, and let it correspond to
the pressure P and the tempe-
rature s.
Let Q T be another iso-
thermal line corresponding to
the pressure Q and the tempe-
rature T.
\ A
\
1/4 Latent Heat.
Let B A and c D be adiabatic lines cutting the isothermals
in A B c D.
Then the substance, in expanding at the temperature s
from the volume P B to the volume P c, will absorb a
quantity of heat equal to L B c , where L is the latent
t - V
heat at temperature s.
When the substance is compressed from Q D to Q A at
temperature T it will give out a quantity of heat equal to
AD
where the accented quantities refer to the temperature T.
The quantity of work done by an engine when the indi-
cating point describes the figure A B c D on the diagram is
represented by the area of this figure, and if the temperatures
s and T are so near each other that we may neglect the
curvature of the lines A B and c D, this area is
\ (B c + A D) P Q.
If the difference of pressures P Q is very small, B c = A D
Dearly, so that we may write the area thus :
B c (P - Q).
But we may calculate the work in another way. It is
equal to the heat absorbed at the higher temperature,
multiplied by the ratio of the difference of the temperatures
to the higher temperature. This is
B c s T
z/ 2 - v l s
Equating the two values of the work, we find the latent
heat
where it is to be remembered that in calculating the frac-
tion P "" Q the difference of the pressures P and Q and the
s T
difference of the temperatures s and T are to be supposed
Latent Heat. 175
very small. In fact, this, fraction is that which in the lan-
guage of the differential calculus would be denoted by .
The student may deduce the equation at once from the
third thermodynamic relation at p. 168.
The most important case of a substance in two different
states is that in which the substance is partly water and
partly steam at the same temperature.
The pressure of steam in a vessel containing water at the
same temperature is called the pressure of saturated steam
or aqueous vapour at that temperature.
The value of this pressure has been determined for a great
number of temperatures as measured on the ordinary scales.
The most complete determinations of this kind are those of
Regnault Regnault has also determined L, the latent heat
of unit of mass of steam, for many different temperatures.
Hence, if we also knew the value of z> 2 v l} or the
difference of volume between unit of mass of water and the
same when converted into steam, we should have all the
data for determining s, the absolute temperature on the
thermodynamic scale.
Unfortunately there is considerable difficulty in ascer-
taining the volume of steam at the point of saturation. If
we place a known weight of water in a vessel, the capacity
of which we can adjust, and determine either the capacity
corresponding to a given temperature at which the whole is
just converted into steam, or the temperature corresponding
to a given capacity, we may obtain data for determining
the density of saturated steam, but it is exceedingly difficult
to observe either the completion of the evaporation or the
beginning of the condensation, and at the same time to
avoid other causes of error. It is to be hoped that these
difficulties will be overcome, and then our knowledge of the
other properties of saturated steam will enable us to compare
the ordinary scales of temperature with the thermodynamic
scale through a range extending from 30 F. to 432 F.
In the meantime Clausius and Rankine have made use of
176 Latent Heat.
the formula in order to calculate the density of saturated
steam, assuming that the absolute temperature is equal to the
temperature reckoned from 460 of Fahrenheit's scale.
The same principle enables us to establish relations
between the physical properties of a substance at the point
at which it changes from the solid to the liquid state.
The temperature of melting ice was always supposed to be
absolutely constant till it was pointed out by Professor James
Thomson ! that it follows from Carnot's principle that the
melting point must be lowered when the pressure increases ;
for if v l is the volume of a pound of ice, and z> 2 that of a
pound of water, both being at 32 F., we know that the
volume of the ice is greater than that of the water. Hence
if s be the melting point at pressure P, and T the melting
point at pressure Q, we have, as at p. 1 74,
S T / .8
F^Q =<".-".) i-
If we make P = h, the pressure of one atmosphere, and
s = 32 F., then the melting temperature at pressure Q is
Now the volume of a pound of ice at 32 F. is 0-0174
cubic feet = v lt and that of a pound of water at the same
temperature is 0*016 cubic feet = v 2 . s, the absolute tempe-
rature, corresponding to 32 F., is 492. L, the latent heat
required to convert a pound of ice into a pound of water,
= 142 thermal units =142 x 772 foot-pounds. Hence T,
the temperature of melting, corresponding to a pressure of
Q pounds weight per square foot, is
T = 32 o-ooooo63 x (Q h).
If the pressure be that of n atmospheres, each atmosphere
being 2,116 pounds weight per square foot,
T = 32 o-oi33 (n i).
1 Transactions of the Royal Society of Edinburgh % vol. xvi. p. 575,
January 2, 1849.
Freezing Point altered by Pressure. 177
Hence the melting point of ice is lowered by about the
seventy-fifth part of a degree of Fahrenheit for every
additional atmosphere of pressure. This result of theory
was verified by the direct experiments of Professor W.
Thomson. 1
Professor J. Thomson has also pointed out the importance
of the unique condition as to temperature and pressure under
which water or any other substance can permanently exist
in the solid, liquid, and gaseous forms in the same vessel
This can only be at the freezing temperature corresponding
to the pressure of vapour at this freezing point. He calls
this the triple point, because three thermal lines meet in it
(i) the steam line, which divides the liquid from the gaseous
state ; (2) the ice line, which divides the liquid from the solid
state ; (3) the hoar-frost line, which divides the solid from the
gaseous state.
Whenever the volume of the substance is, like that of
water, less in the liquid than in the solid state, the effect of
pressure on a vessel containing the substance partly in a
liquid and partly in a solid state is to cause some of the
solid portion to melt, and to lower the temperature of the
whole to the melting point corresponding to the pressure.
If, on the contrary, the volume of the substance is greater in
the liquid than in the solid state, the effect of pressure is to
solidify some of the liquid part, and to raise the temperature
to the melting point corresponding to the pressure. To
determine at once whether the volume of the substance is
greater in the liquid or the solid state, we have only to
observe whether solid portions of the substance sink or swim
in the melted substance. If, like ice in water, they swim,
the volume is greater in the solid state, and pressure causes
melting and lowers the melting point. If, like sulphur, wax,
and most kinds of stone, the solid substance sinks in the
liquid, then pressure causes solidification and raises the
melting point.
1 Proceedings of the Royal Society of Edinburgh, 1850.
1 78 Application of Thermodynamics to Gases.
When two pieces of ice at the melting point are pressed
together, the pressure causes melting to take place at die
portions of the surface in contact. The water so formed
escapes out of the way and the temperature is lowered.
Hence as soon as the pressure diminishes the two parts are
frozen together with ice at a temperature below 32. This
phenomenon is called Regelation.
It is well known that the temperature of the earth increases
as we descend, so that at the bottom of a deep boring it is
considerably hotter than at the surface. We shall see that,
unless we suppose the present state of things to be of no
great antiquity, this increase of temperature must go on to
much greater depths than any of our borings. It is easy on
this supposition to calculate at what depth the temperature
would be equal to that at which most kinds of stone melt in
our furnaces, and it has been sometimes asserted that at this
depth we should find everything in a state of fusion. But
we must recollect that at such depths there is an enormous
pressure, and therefore rocks which in our furnaces would
be melted at a certain temperature may remain solid even at
much greater temperatures in the heart of the earth.
CHAPTER XI.
ON THE APPLICATION OF THE PRINCIPLES OF
THERMODYNAMICS TO GASES,
THE physical properties of bodies in the gaseous state are
more simple than when they are in any other state. The
relations of the volume, pressure, and temperature are
then more or less accurately represented by the laws of
Boyle and Charles, which we shall speak of, for brevity, as
Thermodynamics of Gases. 179
the 'gaseous laws.' We may express them in the following
form:
Let v denote the volume of unit of mass, / the pressure,
/ the temperature measured by an air thermometer and
reckoned from the absolute zero of that instrument, then
the quantity ?-- remains constant for the same gas.
We here use the symbol / to denote the absolute tempera-
ture as measured by the air thermometer, reserving the
symbol to denote the temperature according to the
absolute thermodynamic scale.
We have no right to assume without proof that these two
quantities are the same, although we shall be able to show
by experiment that the one is nearly equal to the other.
It is probable that when the volume and the temperature
are sufficiently great all gases fulfil with great accuracy the
gaseous laws ; but when, by compression and cooling, the
gas is brought near to its point of condensation into the
liquid form, the quantity ^- becomes less than it is for
the perfectly gaseous state, and the substance, though still
appapently gaseous, no longer fulfils with accuracy the
gaseous laws. (See pp. 116, 119.)
The specific heat of a gas can be determined only by a
course of experiments involving considerable difficulty and
requiring great delicacy in the measurements. The gas
must be enclosed in a vessel, and the density of the
gas itself is so small that its capacity for heat forms but
a small part of the total capacity of the apparatus. Any
error, therefore, in the determination of the capacity either
of the vessel itself or of the vessel with the gas in it will
produce a much larger error in the calculated specific heat of
the gas.
Hence tne determinations of the specific heat of gases
were generally very inaccurate, till M. Regnault brought
all the resources of his experimental skill to bear on the
N 2
I So Application of Thermodynamics to Gases.
investigation, and, by making the gas pass in a continuous
current and in large quantities through the tube of his calori-
meter, deduced results which cannot be far from the truth.
These results, however, were not published till 1853, but in
the meantime Rankine, by the application of the principles
of thermodynamics to facts already known, determined
theoretically a value of the specific heat of air, which he
published in 1850. The value which he obtained differed
from that which was then received as the best result of direct
experiment, but when Rcgnault's result was published it
agreed exactly with Rankine's calculation.
We must now explain the principle which Rankine
applied. When a gas is compressed while the temperature
remains constant, the product of the volume and pressure
remains constant. Hence, as we have shown, the elasticity
of the gas at constant temperature is numerically equal to its
pressure.
But if the vessel in which the gas is contained is incapable
of receiving heat from the gas, or of communicating heat to
it, then when compression takes place the temperature will
rise, and the pressure will be greater than it was in the
former case. The elasticity, therefore, will be greater in the
case of no thermal communication than in the case of
constant temperature.
To determine the elasticity under these circumstances in
this way would be impossible, because we cannot obtain a
vessel which will not allow heat to escape from the gas
within it. If, however, the compression is effected rapidly,
ihere will be very little time for the heat to escape, but
then there will be very little time to measure the pressure
in the ordinary way. It is possible, however, after com-
pressing air into a large vessel at a known temperature, to
open an aperture of considerable size for a time which is
sufficient to allow the air to rush out till the pressure is the
same within and without the vessel, but not sufficient to
allow much heat to be absorbed by the air from the sides of
Cooling of Air by Expansion. 1 8 j
the vessel. When the aperture is closed the air is somewhat
cooler than before, and though it receives heat from the
sides of the vessel so fast that its temperature in the cooled
state cannot be accurately observed with a thermometer, the
amount of cooling may be calculated by observing the
pressure of the air within the vessel after its temperature has
become equal to that of the atmosphere. Since at the
moment of closing the aperture the air within was cooler than
the air without, while its pressure was the same, it follows
that when the temperature within has risen so as to be
equal to that of the atmosphere its pressure will be greater.
Let/! be the original pressure of the air compressed in a
vessel whose volume is v ; let its temperature be T, equal to
that of the atmosphere.
Part of the air is then allowed to escape, till the pressure
within the vessel is P, equal to that of the atmosphere ; let
the temperature of the air remaining within the vessel be /.
Now let the aperture be closed, and let the temperature of
the air within become again T, equal to that of the atmosphere,
and let its pressure be then / 2 -
To determine /, the absolute temperature of the air when
cooled, we have, since the volume of the enclosed air
is constant, the proportion
or
This gives the cooling effect of expansion from the
pressure p\ to the pressure P. To determine the corre-
sponding change of volume we must calculate the volume
originally occupied by the air which remains in the vessel.
At the end of the experiment it occupies a volume v, at a
pressure / 2 an d a temperature T. At the beginning of the
experiment its pressure was p l and its temperature T :
hence the volume which it then occupied was v O v, and
1 82 Application of TJiermodynamics to Gases.
a sudden increase of volume in the ratio of p^ to/ t corre-
sponds to a diminution of pressure from / 1 to p. Since / a
is greater thanyp, the ratio of the pressures is greater than
the ratio of the volumes.
The elasticity of the air under the condition of no thermal
communication is the value of the quantity
when the expansion is very small, or when/, is very little
greater than p.
But we know that the elasticity at constant temperature
is numerically equal to the pressure (see p. in). Hence we
find for the value of y, the ratio of the two elasticities,
or, more exactly,
= log /i - log P
log/, - log / 2 *
Although this method of determining the elasticity in the
case of no thermal communication is a practicable one, it is
by no means the most perfect method. It is difficult, for
instance, to arrange the experiment so that the pressure
may be completely equalised at the time the aperture is
closed, while at the same time no sensible portion of heat
has been communicated to the air from the sides of the
vessel. It is also necessary to ensure that no air has en-
tered from without, and that the motion within the vessel has
subsided before the aperture is closed.
But the velocity of sound in air depends, as we shall after-
wards show, on the relation between the variations of its
density and its pressure during the rapid condensations and
rarefactions which occur during the propagation of sound. As
these changes of pressure and density succeed one another
several hundred, or even several thousand, times in a second,
the heat developed by compression in one part of the air has no
Ratio of Elasticities. 183
time to travel by conduction to parts cooled by expansion,
even if air were as good a conductor of heat as copper is.
But we know that air is really a very bad conductor of heat,
so that in the propagation of sound we may be quite certain
that the changes of volume take place without any appreci-
able communication of heat, and therefore the elasticity, as
deduced from measurements of the velocity of sound, is
that corresponding to the condition of no thermal communi-
cation.
The ratio of the elasticities of air, as deduced from experi-
ments on the velocity of sound, is
y = 1-408.
This is also, as we have shown, the ratio of the specific
heat at constant pressure to the specific heat at constant
volume.
These relations were pointed out by Laplace, long before
the recent development of thermodynamics.
We now proceed, following Rankine, to apply the thermo-
dynamical equation of p. 173 :
E0 (K P - K V ) = T v (A M).
In the case of a fluid fulfilling the gaseous laws, and
also such that the absolute zero of its thermometric scale
coincides with the absolute zero of the thermodynamic scale,
we have
i
and
E* = A
Hence
flv
v K Jr v
IVp IVy - - .,
O
a constant quantity.
Now at the freezing temperature, which is 492-6
on Fahrenheit's scale from absolute zero, / v = 26,214
1 84 Application of Thermodynamics to Gases.
foot-pounds by Regnaulfs experiments on air, so that R
is 53*21 foot-pounds per degree of Fahrenheit.
This is the work done by one pound of air in expanding
under constant pressure while the temperature is raised one
degree Fahrenheit.
Now K T is the mechanical equivalent of the heat required
to raise one pound of air one degree Fahrenheit without
any change of volume, and K P is the mechanical equivalent
of the heat required to produce the same change of tempera-
ture when the gas expands under constant pressure, so that
Kp K V represents the additional heat required for the ex-
pansion. The equation, therefore, shows that this additional
heat is mechanically equivalent to the work done by the
air during its expansion. This, it must be remembered,
is not a self-evident truth, because the air is in a different
condition at the end of the operation from that in which
it was at the beginning. It is a consequence of the fact,
discovered experimentally by Joule (p. 216), that no change
of temperature occurs when air expands without doing
external work.
We have now obtained, in dynamical measure, the differ-
ence between the two specific heats of air.
We also know the ratio of K P to K T to be 1-408. Hence
K T = 53 2 . = 130-4 foot-pounds per degree Fahrenheit,
408
and
Kp K T + 53*21 = 183-6 foot-pounds per degree Fah.
Now the specific heat of water at its maximum density is
Joule's equivalent of heat : for one pound it is 772 foot-
pounds per degree Fahrenheit.
Hence if Cp is the specific heat of air at constant pressure
referred to that of water as unity,
Cp = ^ = 0-2378.
This calculation was published by Rankine in 1850.
Energy. 185
The value of the specific heat of air, determined directly
from experiment by M. Regnault and published in 1853, is
Cp = 0-2379.
CHAPTER XII.
ON THE INTRINSIC ENERGY OF A SYSTEM OF BODIES.
THE energy of a body is its capacity for doing work, and
is measured by the amount of work which it can be made
to do. The Intrinsic energy of a body is the work which it
can do in virtue of its actual condition, without any supply
of energy from without.
Thus a body may do work by expanding and overcoming
pressure, or it may give out heat, and this heat may be
converted into work in whole or in part. If we possessed a
perfect reversible engine, and a refrigerator at the absolute
zero of temperature, we might convert the whole of the heat
which escapes from the body into mechanical work. As we
cannot obtain a refrigerator absolutely cold, it is impossible,
even by means of perfect engines, to convert all the heat
into mechanical work. We know, however, from Joule's
experiments, the mechanical value of any quantity of heat,
so that if we know the work done by expansion, and the
quantity of heat given out by the body during any alteration
of its condition, we can calculate the energy which has been
expended by the body during the alteration.
As we cannot in any case deprive a body of all its heat,
and as we cannot, in the case of bodies which assume the
gaseous form, increase the volume of the containing vessel
sufficiently to obtain all the mechanical energy of the ex-
pansive force, we cannot determine experimentally the whole
energy of the body. It is sufficient, however, for all
practical purposes to know how much the energy exceeds
or falls short of the energy of the body in a certain definite
1 86
Energy y Entropy, and Dissipation.
condition- -for instance, at a standard temperature and a
standard pressure.
In all questions about the mutual action of bodies we are
concerned with the difference between the energy of each
body in different states, and not with its absolute value, so
that the method of comparing the energy of the body at
any time with its energy at the standard temperature and
pressure is sufficient for our purpose. If the body in its
actual state has less energy than when it is in the standard
state, the expression for the relative energy will be nega-
tive. This, however, does not imply that the energy of
a body can ever be really negative, for this is impossible.
It only shows that in the standard state it has more energy
than in the actual state.
Let us compare the energy of a substance in two different
states. Let the two states be indicated in the diagram by
the points A and B, and let the intermediate states through
which it passes be indicated by the line, straight or curved,
which is drawn from A to B.
The work of the path, or the work which the body does
while passing from the state A to the state B along the path
A B, is represented, as we
have shown at p. 103, by
the area included between
the path A B, the line of
equal volume, B#, the line
of zero pressure, ba, and
the line of equal volume,
a A, and it is to be reckoned
positive when this area is
described in the direction
of the hands of a watch.
The heat of the path, or
the heat absorbed by the
body during its passage
along A B, is represented by the area included between the
FIG. 26.
T
Available Energy. 187
path A B, the isentropic B /3, the fictitious zero isothermal /3 a,
and the isentropic a A. (See page 164.)
This area is to be reckoned positive when it lies on the
right hand of A B. In the figure, in which it lies on the left
hand of AB, it must be reckoned negative, or, in other words,
it represents heat given out by the body.
The sum of the work done and of heat given out by the
body, both in dynamical measure, is the whole energy given
out by the body during its passage from the state A to the
state B. It is represented by the whole area 0Aa/3B0, and
this area, therefore, represents the diminution of the energy
of the body, which is evidently independent of the form of
the path between A and B. Now this area is the difference
between the areas AaZtzA and B/3z^B, which are bounded
by the line of zero pressure, the fictitious line of zero tempe-
rature, and the lines of equal volume and of equal entropy.
If we suppose the fictitious line of zero temperature joined
to the line of zero pressure by a line of any form, /3z, we
may consider the area bounded by these lines and by the
lines of equal volume and of equal entropy through A as
representing that part of the energy of the body in the
state A the variations of which we are dealing with, for if
the body passes into the state B, by doing work and giving
out heat, the energy given out is represented by the excess
of the area Aaz#A above B/}ZB, and this, therefore, re-
presents the excess of the energy in the state A above that
in the state B.
Hence, in discussing the variations of the energy, we may
consider them represented by the variations of the area
Aaz0A, or, what is the same thing, we may suppose the
energy to be represented by this area together with an
unknown constant.
AVAILABLE ENERGY.
The sum of the work done by the body and the dynamical
equivalent of the heat which it gives out during its passage
FIG. 26*.
188 Energy, Entropy, and Dissipation.
from the state A to the state B is, as we have seen, the
same whatever be the path by which the body passes from
the state A to the state B. If, however, we suppose that
the body is surrounded by a medium, the temperature of
which is maintained con-
stant, so that the body can
give out heat only when its
temperature is higher than
that of the medium, and
can take in heat only when
its temperature is lower
than that of the medium,
then these conditions will
confine the path within
certain limits.
Draw the isothermal TT ,
representing the constant temperature of the surrounding
medium. Then since the temperature of the body at A and
at all points above the line T T 7 is higher than that of the
medium, the body cannot receive heat from the medium.
Hence its entropy cannot increase, and the path cannot rise
above the adiabatic or isentropic A a, drawn through A.
Again, when the body gives out heat to the medium, its
temperature must be higher than that of the medium.
Hence the -path must be above the isothermal T T'.
The path formed by the isentropic A T and the isothermal
T B is therefore the limiting form of the path, and is that
wherein tfie work done by the body is a maximum, and the
heat given ont by it a minimum.
If we denote; the energy of the body in the state A by e,
and its entropy by 0, and the energy and entropy of the
body at the temperature and pressure of the surrounding
medium (represented by B) by <? and , then the total
energy given out as work and heat during the passage from
the state A to the state B is e e .
Available Energy, 189
The amount of heat which the body gives out during the
process cannot be less than that corresponding to the path
A T B, which is
where T is the absolute temperature of the surrounding
medium.
The amount of work done by the body during the process
cannot, therefore, be greater than
This, therefore, is the part of the energy which is available
for mechanical purposes under the circumstances in which
the body is placed, namely, when surrounded by a medium
at temperature T and pressure P.
It appears, therefore, that the greater the original entropy,
the smaller is the available energy of the body. 1
If the system under consideration consists of a number of
bodies at different pressures and temperatures contained
within a vessel from which neither matter nor heat can
escape, then the amount of energy converted into work will
be greatest when the system is reduced to thermal and
mechanical equilibrium by the following process.
i st. Let each of the bodies be brought to the same tem-
perature by expansion or compression without communica-
tion of heat.
2nd. The bodies being now at the same temperature, let
those which exert the greatest pressure be allowed to expand
1 In former editions of this book the meaning of the term Entropy,
as introduced by Clausius, was erroneously stated to be that part of the
energy which cannot be converted into work. The book then proceeded
to use the term as equivalent to the available energy ; thus introducing
great confusion into the language of thermodynamics. In this edition
1 have endeavoured to use the word Entropy according to its original
definition by Clausius.
190 Energy, Entropy, and Dissipation.
and to compress those which exert less pressure, till the
pressures of all the bodies in the vessel are equal, the process
being conducted so slowly that the temperatures of all the
bodies remain sensibly equal to each other throughout the
process.
During the first part of this process, in which there is no
communication of heat between the bodies, the entropy of
each body remains constant. During the second part, the
bodies are all at the same temperature, and therefore the com-
munication of heat from one body to another diminishes
the entropy of the one body as much as it increases that of
the other, so that the sum of the entropy remains constant.
Hence the total entropy of the system remains the same
from the beginning to the end of the process. The work
done against mechanical resistances during the establishment
of thermal and mechanical equilibrium is greater when the
process is conducted in this way than when conduction of
heat is allowed to take place between bodies at sensibly
different temperatures.
Hence the final state of the system is determined by the
following conditions :
Let n be the number of bodies forming the system.
Let m { . . . m n be the masses of these bodies,
v x . . . z/ n the volume of unit of mass of each,
<pj - . . . ^> n the entropy of unit of mass of each,
e l . . . e n the energy of unit of mass of each,
/i ... / n the pressure of each,
0, ... n the temperature of each.
The volume of the whole is
m l v l + . . . + w n e' n
and since the system is contained in a vessel of volume v,
V(mv) = v
during the whole process
Available Energy. 191
The entropy of the whoie is
m \ 0i + 4- w n ^> n = 2(m(j>) =. <&.
When there is no communication of heat except between
bodies of equal temperature, 4> remains constant. When
there is communication of heat between bodies of different
temperature, < increases.
In the final state of the system
There are therefore n i conditions with respect to
pressure, and n i conditions with respect to temperature,
together with one condition with respect to volume and one
with respect to entropy, or, in all, 2 n conditions to be satis-
fied by the n bodies ; and since the state of each body is a
function of two variables, the conditions are necessary and
sufficient to determine the final state of each of the n bodies.
The work done against resistances external to the system
may be determined by comparing the total energy at the
beginning of the process with the final energy ; for, since no
heat is allowed to escape, any diminution of energy must
arise from work being done.
The total energy is
E.
If E be the original and E' the final value of this quantity,
the energy available to produce mechanical work is
E - E'.
If during any part of the process by which the system
reaches its final state of thermal and mechanical equilibrium
there takes place a communication of a quantity H of heat
from a body at temperature 0j to a body at temperature 2 >
the increase of the total entropy of the system arising from
the communication is, as we have shown (at p. 163),
192 Energy, Entropy, and Dissipation.
and the final entropy, instead of being equal to the original
entropy $, becomes
This increase of the final entropy involves a corresponding
increase in the final temperature and the final energy.
If the rise of the final temperature is small, then, since the
volume is constant, the increase of the final energy is
and the available energy is therefore diminished by this
quantity on account of the passage of the quantity H of
heat from a body at temperature 6 l to a body at tem-
perature 6 2 .
Processes of this kind, by which, while the total energy
remains the same, the available energy is diminished, are
instances of what Sir W. Thomson has called the Dissipa-
tion of Energy. The doctrine of the dissipation of energy
is closely connected with that of the growth of entropy, but
is by no means identical with it.
The increment of the total entropy of a system arising
from the communication of a given amount of heat, H, from
a body at one given temperature, 1? to another given tem-
perature, 2 , is, as we have seen,
a -
a quantity completely determined by the state of the system
when this communication takes place.
The energy dissipated or rendered unavailable as a source
of mechanical work is
into which a new factor, 9, enters, and this fector denotes
Dissipation of Energy. 193
the final temperature of the system when it has reached the
state of thermal and mechanical equilibrium. 0, therefore,
since it depends on the final state of the system, can
only be calculated when we know not only the relations
between the thermodynamic variables for all the bodies, but
the volume which they occupy in their final state.
The calculation of the amount of energy dissipated during
any process is therefore much more difficult than that of the
increase of the total entropy.
If the system is allowed to reach its final state of thermal
and mechanical equilibrium, in such a manner that no ex-
ternal work is done, and no heat is allowed to leave or enter
the system, the condition is that the final energy is equal to
the original energy.
Combining this with the other conditions, that the volume is
unchanged, and that the final state with respect to pressure
and temperature is common to all the bodies, we may deter-
mine the final value of the temperature, pressure, and total
entropy.
The total entropy will now have the maximum value con-
sistent with the original state of the system. The dissipation
of the available energy will be complete.
MECHANICAL AND THERMAL ANALOGIES.
In studying thermodynamics we may find considerable
assistance from a comparison between the thermal and the
mechanical phenomena.
We have to do with energy in two forms, work and heat.
When energy is being transferred from one body to another
we can always tell whether the first body is doing mechanical
work on the second or communicating heat to it. Work is
done by motion against resistance. Heat is communicated
from a hotter to a colder body.
But as soon as the energy has entered the second body,
o
194 Mechanical and Thermal Analogies.
we can no longer distinguish by any legitimate process
whether it is in the form of work or of heat. In fact we may
remove it from the body under either of these forms.
If a fluid at a pressure / increases in volume from v to z/,
it performs work against external resistance, the amount of
which work is
v) = w.
If a body at temperature increases in entropy from to
', an amount of heat must have entered it represented by
- = H.
If both these processes take place, and if the energy of
the body is thereby changed from E to E', then
E' E = H w = 6 (0' 0) / (v' - v}.
Here then we have two sets of quantities, one relating to
work, the other to heat.
w v p
H
Of these quantities Work and Heat are simply two forms
of Energy.
The volume is a quantity such that without a change of
its value no work can be done. The amount of work done.
however, is measured, not by the change of volume alone,
but by that change multiplied by another quantity the
pressure.
In the same way the entropy is a quantity such that
without a change in its value no heat can enter or leave the
body. The amount of this heat, however, is not measured
by the change of entropy, but by that change multiplied by
another quantity the absolute temperature.
Again, the pressure is a quantity such that its equality in
two communicating vessels determines their mechanical
Mechanical and Thermal Analogies. 195
equilibrium, while its excess in either determines a flow of
fluid from that vessel to the other.
In like manner the temperature is a quantity such that its
equality in two bodies in contact determines their thermal
equilibrium, while its excess in either determines a flow of
heat from that body to the other.
If we regard the energy of a body as determined by its
volume and its entropy, then the pressure may be defined as
the rate at which the energy diminishes with increase of
volume, while the entropy remains constant.
The temperature may in like manner be defined as the
rate at which the energy increases with increase of entropy,
the volume remaining constant.
REPRESENTATION OF THE PROPERTIES OF A SUBSTANCE BY
MEANS OF A SURFACE.
Professor J. Willard Gibbs, of Yale College, U.S., to whom
we are indebted for a careful examination of the different
methods of representing thermodynamic relations by plane
diagrams, has introduced an exceedingly valuable method of
studying the properties of a substance by means of a surface. 1
According to this method, the volume, entropy, and
energy of the body in a given state are represented by the
three rectangular coordinates of a point in the surface, and
this point on the surface is said to correspond to the given
state of the body. We shall suppose the volume measured
towards the east from the meridian plane corresponding to
no volume, the entropy measured towards the north from a
vertical plane perpendicular to the meridian, whose position
is entirely arbitrary, and the energy measured downwards
from the horizontal plane of no energy, the position of which
may be considered as arbitrary, because we cannot measure
the whole energy existing in a body.
1 Transactions of the Academy of Sciences of Connecticut, vol. ii.
O 2
196 Thermodynamic Stir/ace.
The section of this surface by a vertical plane perpen-
dicular to the meridian represents the relation between
volume and energy when the entropy is constant, that is,
when no heat enters or leaves the body.
If the pressure is positive, then the body, by expanding,
would do work against external resistance, and its intrinsic
energy would diminish. The rate at which the energy
diminishes as the volume increases is represented by the
tangent of the angle which the curve of section makes with
the horizon.
The pressure is therefore represented by the tangent of
the angle of slope of the curve of section. The pressure is
positive when the curve slopes downwards towards the west.
When the slope of the curve is towards the east the corre-
sponding pressure is negative.
A tension or negative pressure cannot exist in a gas. It
may, however, exist in a liquid, such as mercury. Thus, if
a barometer tube is well filled with clean mercury, and
then placed in a vertical position, with its closed end
uppermost, the mercury sometimes does not fall in the
tube to the point corresponding to the atmospheric pres-
sure, but remains suspended in the tube, so as to fill it
completely.
The pressure in this case is negative in that part of the
mercury which is above the level of the ordinary barometric
column.
In solid bodies, as we know, tensions of considerable
magnitude may exist.
Hence in our thermodynamic model the pressure of the
substance is indicated by the tangent of the slope of the
curve of constant entropy, and is reckoned positive when
the energy diminishes as the volume increases.
The section of the surface by a vertical plane parallel to
the meridian is a curve of constant volume. In this curve
the temperature is represented by the rate at which the
Representation of Pressure and Temperature. 197
energy increases as the entropy increases, that is to say, by
the tangent of the slope of the curve.
Since the temperature, reckoned from absolute zero, is an
essentially positive quantity, the curve of constant volume
must be such that the entropy and energy always increase
together.
To ascertain the pressure and temperature of the substance
in a given state, we may draw a tangent plane to the cor-
responding point of the surface. The normal to this plane
through the origin will cut a horizontal plane at unit of dis-
tance above the origin at a point whose coordinates represent
the pressure and temperature, the pressure being represented
by the coordinate drawn towards the west, and the tempera-
ture by the coordinate drawn towards the north.
The pressure and temperature are thus represented by
the direction of this normal, and if, at any two points
of the surface, the directions of the normals are parallel,
then in the two states of the substance corresponding tc
these two points the pressure and temperature must be the
same.
If we wish to trace out on a model of the surface a series
of lines of equal pressure, we have only to place it in the
sunshine and to turn it so that the sun's rays are parallel to
the plane of volume and energy, and make an angle with the
line of volume whose tangent is proportional to the pressure
Then, if we trace on the surface the boundary of light and
shadow, the pressure at all points of this line will be the
same.
In like manner, if we place the model so that the sun's
rays are parallel to the plane of entropy and energy, the
boundary of light and shadow will be a line such that the
temperature is the same at every point, and proportional to
the tangent of the angle which the sun's rays make with the
line of entropy.
In this way we may trace out on the model two series of
198 Thermodynamtc Model.
lines : lines of equal pressure, which Professor Gibbs calls
Isopiestics ; and lines of equal temperature, or Isothermals.
Besides these, we may trace the three systems of plane sec-
tions parallel to the coordinate planes, the isometrics or lines
of equal volume, the isentropics or lines of equal entropy,
which we formerly called, after Rankine, adiabatics, and
the isenergics or lines of equal energy.
The network formed by these five systems of lines will
form a complete representation of the relations between the
five quantities, volume, entropy, energy, pressure, and tem-
perature, for all states of the body.
The body itself need not be homogeneous either in
chemical nature or in physical state. All that is necessary
is that the whole should be at the same pressure and the
same temperature.
By means of this model Professor Gibbs has solved several
important problems relating to the thermodynamic relations
between two portions of a substance, in different physical
states, but at the same pressure and temperature.
Let a substance be capable of existing in two different
states, say liquid and gaseous, at the same temperature and
pressure. We wish to determine whether the substance will
tend of itself to pass from one of these states to the other.
Let the substance be placed in a cylinder, under a piston,
and surrounded by a medium at the given temperature and
pressure, the extent of this medium being so great that its
pressure and temperature are not sensibly altered by the
changes of volume of the working substance, or by the
heat which that body gives out or takes in.
The two physical states which are to be compared are re-
presented by two points on the surface of the model ; and
since the pressure and temperature are the same, the tangent
planes at these points are either coincident or parallel.
The surface representing the thermodynamic properties of
the surrounding medium must be supposed to be constructed
FIG.
Equilibrium between Two Physical States. 199
on a scale proportional to the amount of this medium ; and
as we assume that there is a very great mass of this medium,
the scale of the surface will be so great that we may regard
the portion of the surface with which we have to do as
sensibly plane ; arid since its pressure and temperature are
those of the working substance in the given state, this plane
surface is parallel to the
tangent plane at the
given point of the sur-
face of the model.
Let A B c be three
points of the model at
which the tangent planes
are parallel, the energy
being reckoned down-
wards.
Let A a a be the tangent plane at A, and let us consider it
as part of the model representing the external medium, this
model being so placed that volume, entropy, and energy
are reckoned in the opposite directions from those in the
model of the working substance.
Now let us suppose the substance to pass from the state A
to the state B, passing through the series of states repre-
sented by the points on the isothermal line joining the points
of equal temperature A and B.
Then since the working substance and the external medium
are always at the same temperature, the entropy lost by the
one is equal to that gained by the other.
Also the one gains in volume what is lost by the other.
Hence, during the passage of the working substance from
the state A to the state B, the state of the external medium
is always represented by a point in the tangent plane in the
same vertical line as the point representing the state of the
working substance.
For the same horizontal motion which represents a gain of
2OO Tkermc dynamic Model.
volume or entropy of the one substance represents an equal
loss of volume or entropy in the other.
Hence, when the state of the working substance is repre-
sented by the point B, that of the external medium will be
represented by the point a, where the vertical line through
B meets the tangent plane through A.
Now the energy is reckoned downwards for the working
substance and upwards for the external medium. Hence,
drawing A K horizontal, K B represents the gain in energy of
the working substance, and K a the loss of energy of the
external medium.
The line B a, or the vertical height of the tangent plane
above the point B, represents the gain of energy in the whole
system, consisting of the working substance and the external
medium, during the passage from the state A to the state B.
But the energy of the system can be increased only by doing
work on it.
But if the system can of itself pass from one state to
another, the work required to produce the corresponding
changes of configuration must be drawn from the energy of
the system, and the energy must therefore diminish.
The fact, therefore, that in the case before us the energy
increases, shows that the passage from the state A to the
state B in presence of a medium of constant temperature
and pressure, cannot be effected without the expenditure of
work by some external agent.
The working substance, therefore, cannot of itself pass
from the state A to the state B, if B lies below the plane
which touches the surface at A.
We have supposed the substance to pass from A to B by a
process during which it is always at the same temperature
as the external medium. In this case the entropy of the
system remains constant.
If, however, the communication of heat between the sub-
stances occurs when they are not at the same temperature.
Condition of Stability. 20 1
the entropy of the system will increase; and if in the figure
the gain of entropy of the working substance is represented
by the horizontal component of A B, the loss of entropy of
the external medium will be represented by a smaller
quantity, such as the horizontal component of A a. Hence
a' will be to the left of a, and therefore higher. The gain
of entropy of the system will therefore be represented by the
horizontal part of a a'.
Now since temperature is essentially positive, a gain of
entropy at a given volume always implies a gain of energy.
Hence the gain of energy is greater when there is a gain of
entropy than when the entropy remains constant.
There is, therefore, no method by which the change from
A to B can be effected without a gain of energy, and this
implies the expenditure of work by an external agent.
If, therefore, the tangent plane at A is everywhere above
the thermodynamic surface, the condition of the working
substance represented by the point A is essentially stable,
and the substance cannot of itself pass into any other state
while exposed to the same external influences of pressure
and temperature.
This will be the case if the surface is convexo-convex
upwards.
If, on the other hand, the surface, as at the point B, is
either concave upwards in all directions, or concave in
one direction and convex in another, it will be possible to
draw on the surface a line from the point of contact lying
entirely above the tangent plane, and therefore representing
a series of states through which the substance can pass of
itself.
In this case the point of contact represents a state of the
substance which, if physically possible for an instant, is
essentially unstable, and cannot be permanent.
There is a third case, however, in which the surface, as
at the Doint c, is convexo-convex, so that a line drawn OD
2O2 Thermo dynamic Model.
the surface from the point of contact must lie below the
tangent plane ; but the tangent plane, if produced far enough,
cuts the surface at c, so that the point A lies above the
tangent plane. In this case the substance cannot pass
through any continuous series of states from c to A, because
any line drawn on the surface from c to A begins by dipping
below the tangent plane. But if a quantity, however small,
of the substance in the state A is in physical contact with
the rest of the substance in the state c, minute portions will
pass at once from the state c to the state A without passing
through the intermediate states.
The energy set at liberty by this transformation will
accelerate the subsequent rate of transformation, so that the
process will be of the nature of an explosion.
Instances of such a process occur when a liquid not in
presence of its vapour is heated above its boiling point, and
also when a liquid is cooled below its freezing point, or when
a solution of a salt, or of a gas, becomes supersaturated.
In the first of these cases the contact of the smallest
quantity of vapour will produce an explosive evaporation t
in the second, the contact of ice will produce explosive
freezing ; in the third, a crystal of the salt will produce ex-
plosive crystallization ; and in the fourth, a bubble of any
gas will produce explosive effervescence.
Finally, when the tangent plane touches the surface at
two or more points, and is above the surface everywhere
else, portions of the substance in states corresponding to the
points of contact can exist in presence of each other, and
the substance can pass freely from one state to another in
either direction.
The state of the whole body when part is in one physical
state and part in another is represented by a point in the
straight line joining the centre of gravity of two masses equal
respectively to the masses of the substance in the two states,
and placed at the points of the model corresponding to these
states.
Primitive and Secondary Surfaces. 203
Hence, in addition to the surface already considered, which
we may call the primitive surface, and which represents the
properties of the substance when homogeneous, all the points
of the line joining the two points of contact of the same
tangent plane belong to a secondary surface, which repre-
sents the properties of the substance when part is in one
state and part in another.
To trace out this secondary surface we may suppose the
doubly tangent plane to be made to roll upon the surface,
always touching it at two points called the node-couple.
The two points of contact will thus trace out two curves
such that a point in the one corresponds to a point in the
other. These two curves are called in geometry the node-
couple curves.
The secondary surface is generated by a line which moves
so as always to join corresponding points of contact. It is
a developable surface, being the envelope of the rolling
tangent plane.
To construct it, spread a film of grease on a sheet of glass
and cause the sheet of glass to roll without slipping on the
model, always touching it in two points at least.
The grease will be partly transferred from the glass to the
model at the points of contact, and there will be traces on
the model of the node-couple curves, and on the glass of
corresponding plane curves.
If we now copy on paper the curve traced out on the
glass and cut it out, we may bend the paper so that the cut
edges shall coincide with the two node- couple curves, and
the paper between these curves will form the derived sur
face representing the state of the body when part is in one
physical state and part in another.
There is one position of the tangent plane in which it
touches the primitive surface in three points. These points
represent the solid, liquid, and gaseous states of the sub-
stance when the temperature and the pressure are such that
the three states can exist together in equilibrium.
204 Thermodynamic Model.
The plane triangle, of which these points are the angles,
represents all possible mixtures of these three states. For
instance, if there are s grammes in the solid state, L grammes
in the liquid state, and v grammes in the state of vapour,
this condition of the substance will be represented by a
point in the triangle which is the centre of gravity of masses
s, L, and v placed at the corresponding angular points.
From this position of the tangent plane it may roll on the
primitive surface in three directions so as in each case to touch
it at two points. We thus obtain three sheets of the derived
surface, the first connecting the solid and liquid states, the
second the liquid and gaseous states, and the third the gas-
eous and solid states. These three developable surfaces,
together with the plane triangle s L v, constitute what Pro-
fessor Gibbs calls the Surface of Dissipated Energy.
Of the three developable surfaces the first and third, those
which connect the solid state with the liquid and gaseous,
have been experimentally investigated only to a short dis-
tance from the triangle s L v ; but the sheet which connects
the liquid and gaseous states has been thoroughly explored.
The experiments of Cagniard de la Tour and the numeri-
cal determinations of Andrews show that the curves traced
out by the two points of contact of the doubly tangent plane
unite in a point which represents what Andrews calls the
critical state. At this point the two points of contact of the
rolling tangent plane coalesce, and if the plane continues to
roll on the surface it will touch it at one point only.
If the primitive surface forms a continuous sheet beneath
the surface of dissipated energy, it cannot be at all points
Fic y6c convexo-convex upwards. For
let AD be the line joining two
corresponding points of contact
of the doubly tangent plane, and
let A B c D be the section of the
primitive surface by a vertical plane through A D, then it is
Condition of Instability. 205
manifest that the curve A B c D must in some part of its
course be concave upwards.
Now a point on the primitive surface at which either of its
principal curvatures is concave upwards, represents a state
of the body which is essentially unstable. Part of the
primitive surface, therefore, if it is continuous, must repre-
sent states of the body essentially unstable. If, therefore,
the primitive surface is continuous, there must be a region
representing states essentially unstable, because one or both
of the principal curvatures is concave upwards. This region
is bounded by what is called in geometry the spinode curve.
Beyond this curve the surface is convexo-convex, but the
tangent plane still cuts the surface at some more or less
distant point till we come to the curve of the node-couple,
at which the tangent plane touches the surface at two points.
Beyond this the tangent plane lies entirely above the surface,
and the corresponding state of the body is essentially stable.
The region between the spinode curve and the node-
couple curve represents states of the body which, though
stable when the whole substance is homogeneous, are liable
to sudden change if a portion of the same substance in
another state is present.
Since every vertical section through two corresponding
points of contact must cut the spinode curve at the points
of inflexion B and c, the chord A D of the node-couple curve
and the chord B c of the spinode curve must coincide at the
critical point, so that at this point the spinode curve and the
two branches of the node-couple curve coalesce and have a
common tangent. This point is called in geometry the
tacnodal point.
Note. For these geometrical names I am indebted to Professor
Cayley.
206 Thermo dynamic Model.
THERMAL LINES ON THE THERMODYNAMIC SURFACE.
(F/G. 2&)
o Origin:
o v Axis of volume,
o Axis of entropy,
o e Axis of energy.
P! . . . p fi Isopiestics or lines of equal pressure.
Of these P, represents a negative pressure, or, in other
words, a tension, such as may exist in solids and in some
liquids.
T! . . . T 6 Isothermals, or lines of equal temperature.
The curves T 3 and T 4 have branches in the form of closed
loops.
F G H c. To the right of this line the substance is gaseous
and absolutely stable. To the left of F G it may condense
into the solid state, and to the left of G H c it may condense
into the liquid state.
c K L M N. Below this line the substance is liquid and
absolutely stable. To the right of L K c it may evaporate, to
the left of L M N it may solidify.
Q R s E. To the left of this line the substance is solid and
absolutely stable. To the right of s R Q it may melt, and
above s E it may evaporate.
c is the critical point of the liquid and gaseous states.
Below this point there is no discontinuity of states.
c is called in geometry the tacnodal point.
The curves F G, G H c K L, L M N, Q R s, and s E are
branches of what is called in geometry the node-couple
curve.
The curves xcx and YY are branches of the spinode
curve.
Above this curve the substance is absolutely unstable.
Between it and the node-couple curve the substance is stable,
but only if homogeneous.
Thermal Lines on the Model 2O7
FIG. 26^.
Thermodynamic Surface.
2O8 Thermodynamic Model.
The plane triangle SLG represents that state of uniform
pressure and temperature at which the substance can be
partly solid, partly liquid, and partly gaseous.
The straight lines represent states of uniform pressure and
temperature in which two different states are in equilibrium
s G and E F between solid and gaseous.
GL and KH between liquid and gaseous,
s L, R M, and Q N between solid and liquid.
The surface of dissipated energy consists of the plane
triangle SLG and the three developable surfaces of which
the generating lines are those above mentioned. This sur-
face lies above the primitive thermodynamic surface and
touches it along the node-couple curve.
Free
209
CHAPTER XIII.
ON FREE EXPANSION.
Theory of a Fluid in which no External Work is
during a Change of Pressure.
LET a fluid be forced through a small hole, or one or more
narrow tubes, or a porous plug, and let the work done by
the pressure from behind be entirely employed in over-
coming the resistance of the fluid, so that when the fluid,
after passing through the plug, has arrived at a certain point
its velocity is very small. Let us also suppose that no heat
enters or leaves the fluid, and that no sound or other
vibration, the energy of which is comparable with that
which would sensibly alter the temperature of the fluid,
escapes from the apparatus.
We also suppose that the motion is steady that is, that
the same quantity of the fluid enters and issues from the
apparatus in every second.
During the passage of unit of mass through the apparatus,
if P and v are its pressure and volume at the
section A before reaching the plug, and/, v
the same at the section B after passing through
it, the work done in forcing the fluid through
the section A is P v, and the work done by the
fluid in issuing through the section B is p v, so
that the amount of work communicated to the
fluid in passing through the plug is p v p v .
Hence, if E is the energy of unit of mass of
the fluid while entering at the section A, and e the energy of
unit of mass issuing at the section B,
e E = PV pv,
or
i + pv = <t/0 . . . (i)
FIG. 27.
210
Free Expansion.
FIG. 28.
That is to say, the sum of the intrinsic energy and the
product of the volume and the pressure remains* the same
after passing through the plug, provided no heat is lost or
gained from external sources.
Now the intrinsic energy E is indicated on the diagram
by the area between A a an
adiabatic line, A a a vertical
line, and a b v the line of no
pressure, and p v is represent-
ed by the rectangle A/ o a.
Hence the area included by
a A/ o v y the lines A a and o v
being produced till they meet,
represents the quantity which
remains the same after passing
through the plug. Hence in
the figure the area \pq *R. is
equal to the area contained
between B R and the two adiabatic lines R a and B ft.
We shall next examine the relations between the different
properties of the substance, in order to determine the rise of
temperature corresponding to a passage through the plug
from a pressure P to a pressure /, and we shall first suppose
that P is not much greater than p.
Let A c be an isothermal line through A, cutting q B in c,
and let us suppose that the passage ot the substance from
the state represented by A to the state represented by B is
effected by a passage along the isothermal line A c, followed
by an increase of volume from c to B. The smaller the
distance A B, the less will the results of this process differ
from those of the actual passage from A to B, in whatever
manner this is really effected.
In passing from A to c, at the constant temperature d, the
pressure diminishes from p to/. The heat absorbed during
this process is, by the first thermodynamic relation (p. 167),
O - P', v a,
Free Expansion. 21 1
where a is the dilatation of unit of volume at constant pres-
sure per degree of temperature.
In passing from c to B the substance expands at constant
pressure, and its temperature rises from to 6 + r.
The heat required to produce this rise of temperature is
where K P denotes the specific heat of the substance at con-
stant pressure.
The whole heat absorbed by the substance during the
passage from A to B is therefore
(P -/) V0a + K p r,
and this is the value of the area between A B and the two
adiabatic lines A a, B (3.
Now this is equal to the area A p q B or (P /) v.
Hence we have the equation
K p r = (p-/)v(i - 0) . . . (2)
where K P denotes the specific heat of unit of mass at con-
stant pressure, expressed in dynamical measure ;
r, the rise of temperature after passing through the plug ;
p p y the small difference of pressure on the two sides of
the plug ;
v, the volume of unit of mass (when p p is so great as
to cause considerable alteration of volume, this quantity
must be treated differently) ;
0, the temperature on the absolute dynamical scale ;
a, the dilatation of unit of volume at constant pressure
per degree of temperature.
There are two cases in which observations of the rise (or
fall) of temperature may be applied to determine quantities
of great importance in the science of heat.
1. To Determine the Dynamical Equivalent of Heat. The
first case is that in which the substance is a liquid such as
water or mercury, the volume of which is but slightly affected
either by pressure or by temperature. In this case v will
p 2
212 Fr& Expansion.
vary so little that the effect of its variation may be taken
into account as a correction required only in calculations of
great accuracy. The dilatation a is also very small, so much
so that the product 6 a, though not to be absolutely neglected,
may be found with sufficient accuracy without a very accurate
knowledge f the absolute value of 0.
If we suppose the pressure to be due to a depth of fluid
equal to H on one side of the plug and h on the other, then
where p is the density, and g is the numerical measure of the
force of gravity. Now
vp = i,
so that equation (2) becomes
K p r=-(H -/&)(! - 0),
an equation from which we can determine K P when we know
r the rise of temperature, and H h the difference of level
of the liquid, a its coefficient of dilatation by heat, and
(within a moderate degree of exactness) 6 the absolute tem-
perature in terms of the degrees of the same thermometer
which is used to determine r.
The quantity K P is the specific heat at constant pressure,
that is the quantity of heat which will raise unit of mass of
the substance one degree of the thermometer. It is ex-
pressed here in dynamical measure or foot-poundals.
If the specific heat is to be expressed in gravitation
measure, as in foot-pounds, we must divide by g, the intensity
of gravity. If the specific heat is to be expressed in terms
of the specific heat of a standaid substance, as, for instance,
water at its maximum density, we must divide by j, the
specific heat of this substance.
We have already shown how by a direct experiment to
compare the specific heat of any substance with that of
water. If the specific heat expressed in this way is denoted
by c p , while K P is the same quantity expressed in dynamical
Dynamical Equivalent of Heat. 213
measure, then the dynamical equivalent of the thermal
unit is
The quantity j is called Joule's Mechanical Equivalent
of Heat, because Joule was the first to determine its value
by an accurate method. It may be defined as the specific
heat, in dynamical measure, of water at its maximum
density.
It is equal to 772 foot-pounds at Manchester per pound
of water. If we alter the standard of mass, we at the same
time alter the unit of work in the same proportion, so that
we must still express j by the same number. Hence we
may express Joule's result by saying that the work done by
any quantity of water in falling 772 feet at Manchester is
capable of raising that water one degree Fahrenheit. If we
wish to render the definition independent of the value of
gravity at a particular place, we have only to calculate the
velocity of a body after falling 772 feet at Manchester. The
energy corresponding to this velocity in any mass of water
is capable when converted into heat of raising the water one
degree Fahrenheit.
There are considerable difficulties in obtaining the value of
j by this method, even with mercury, for which a pressure
of 25 feet gives a rise of one degree Fahrenheit.
2. To reduce Temperatures to the Thermodynamic Scale.
The most important application of the method is to
ascertain the temperature, 0, on the thermodynamic scale,
which corresponds to the reading, /, registered by any ordi-
nary thermometer, e.g. a centigrade thermometer.
The substance employed is air, or any other gas which
satisfies approximately the gaseous laws expressed in the
equation
vp = z'o/o (i + </)
where z/ , / ft , are the volume and pressure at the zero of the
214 free Expansion.
thermometer, and o is the voluminal dilatation per degree
at that temperature.
The voluminal dilatation, a, at the temperature / is therefore
so that the expression for K p r becomes
K P T = z/o/o - P - $ (i + a / - u 0).
This expression is strictly true only for a very small
variation of the pressure. When, as in the experiments of
Joule and Thomson, p is several times /, we must ascertain
the effect of the gradual diminution of pressure by the process
described at p. 221, which is applicable in this case, because
the variation of temperature is found to be small. The
T> _ Jy T>
result is that instead of ^- we must write log e -, where
P P
the logarithm is Napierian, or 2-3026 log - , where the log-
arithm is taken from the common tables. Hence we find
'4343
log? - log/
an expression which gives the temperature, 0, on the thermo-
dynamic scale corresponding to the reading, /, of an ordinary
thermometer, the degrees of the thermodynamic scale being
equal to those of the thermometer near the temperature of
the experiment.
In the case of most of the gases examined by Joule
and Thomson there was a slight cooling effect on the gas
passing through the plug. In other words, T was negative,
and the absolute temperature was therefore higher than
that indicated by the gaseous thermometer. The ratio,
therefore, in which the gas expanded between two standard
Determination of Absolute Temperature. 215
temperatures was greater than the true ratio of these tem-
peratures on the thermodynamic scale. The cooling effect
was much greater with carbonic acid than with oxygen,
nitrogen, or air, as was to be expected, because we know
from the experiments of Regnault that the dilatation of
carbonic acid is greater than that of air or its constituents.
It was also found, for all these gases, that the cooling effect
was less at high temperatures, which shows that as the
temperature rises the dilatation of the gas is more and
more accurately proportional to the absolute temperature
of the thermodynamic scale.
The only gas which exhibited a contrary effect was
hydrogen, in which there was a slight heating effect after
passing the plug.
The result of the experiments of Joule and Thomson
was to show that the temperature of melting ice is
2 73 7 on the thermodynamic scale, the degrees being
such that there are 100 of them between this temperature
and that of the vapour of boiling water at the standard
pressure.
The absolute zero of the thermodynamic scale is there-
fore 273*7 Centigrade, or 46o 0> 66 Fahrenheit.
It appears, therefore, that, in the more perfect gases, the
cooling effect due to expansion is almost exactly balanced
by the heating effect due to the work done by the expansion
when this work is wholly spent in generating heat in the
gas. This result had been already obtained, although by a
method not admitting of such great accuracy, by Joule, 1 who
showed that the intrinsic energy of a gas is the same at
the same temperature, whatever be the volume which it
occupies.
To test this, he compressed air into a vessel till it con-
tained about 22 atmospheres, and exhausted the air from
another vessel. These vessels were then connected by
1 Phil. Mag. May 1845.
216 Free Expansion.
means of a pipe closed by a stopcock-, and the whole placed
in a vessel of water.
After a sufficient time the water was thoroughly stirred,
and its temperature taken by means of a delicate thermo-
meter. The stopcock was then opened by means of a proper
key, and the air allowed to pass from the full into the empty
vessel till equilibrium was established between the two.
Lastly the water was again stirred and its temperature
carefully noted.
From a number of experiments of this kind, carefully
corrected for all sources of error, Joule was led to the
conclusion that no change of temperature occurs when air
is allowed to expand in such a manner as not to develop
mechanical power.
This result, as has been shown by the more accurate
experiments afterwards made by Joule and W. Thomson, is
not quite correct, for there is a slight cooling effect. This
effect, however, is very small in the case of permanent gases,
and diminishes when the gas, by rise of temperature or
diminution of pressure, approaches nearer to the condition
of a perfect gas.
We may however assert, as the result of these experiments,
that the amount of heat absorbed by a gas expanding at
uniform temperature is nearly, though not exactly, the thermal
equivalent of the mechanical work done by the gas during
the expansion. In fact, we know that in the case of air the
heat absorbed is a little greater and in hydrogen a very little
less than this quantity.
This is a very important property of gases. If we reverse
the process, we find that the heat developed by compressing
air at constant temperature is the thermal equivalent of the
work done in compressing it.
This is by no means a self-evident proposition. In fact,
it is not true in the case of substances which are not in the
gaseous state, and even in the case of the more imperfect
gases it deviates from the truth. Hence the calculation of
Measurement of Heights by the Barometer. 217
the dynamical equivalent of heat, which Mayer founded on
this proposition, at a time when its truth had not been
experimentally proved, cannot be regarded as legitimate.
CHAPTER XIV.
ON THE DETERMINATION OF HEIGHTS BY THE BAROMETER.
THE barometer is an instrument by means of which the
pressure of the air at a particular place may be measured.
In the mercurial barometer, which is the most perfect form of
the instrument, the pressure of the air on the free surface of
the mercury in the cistern is equal to that of a column of
mercury whose height is the difference between the level of
the mercury in the cistern, which sustains the pressure of the
air, and that of the mercury in the tube, which has no air
above it The pressure of the air is often expressed in terms
of the height of this column. Thus we speak of a pressure
of 30 inches of mercury, or of a pressure of 760 millimetres of
mercury.
To express a pressure in absolute measure we must
consider the force exerted against unit of area. For this
purpose we must find the weight of a column of mercury of
the given height standing on unit of area as base.
If h is the height of the column, then, since its section is
unity, its volume is expressed by h.
To find the mass of mercury contained in this volume we
must multiply the volume by the density of mercury. If this
density is denoted by /o, the mass of the column is p h. The
pressure, which we have to find, is the force with which this
mass is drawn downwards by the earth's attraction. If g
denotes the force of the earth's attraction on unit of mass,
then the force on the column will be gp h. The pressure
218 Measurement of Heights by the Barometer.
therefore of a column of mercury of height h is expressed
by
gph,
where h is the height of the column, p the density of mercury,
and g the intensity of gravity at the place. The density of
mercury diminishes as the temperature increases. It is usual
to reduce all pressures measured in this way to the height of
a column of mercury at the freezing temperature of water.
If two barometers at the same place are kept at different
temperatures, the heights of the barometers are in the pro-
portion of the volumes of mercury at the two temperatures.
The intensity of gravitation varies at different places, being
less at the equator than at the poles, and less at the top of a
mountain than at the level of the sea.
It is usual to reduce observed barometric heights to the
height of a column of mercury at the freezing point and at
the level of the sea in latitude 45, which would produce the
same pressure.
If there were no tides or winds, and if the sea and the air
were perfectly calm in the whole region between two places,
then the actual pressure of the air at the level of the sea
must be the same in these two places ; for the surface of
the sea is everywhere perpendicular to the force of gravity.
If, therefore, the pressure on its surface were different in
two places, water would flow from the place of greater pres-
sure to the place of less pressure till equilibrium ensued.
Hence, if in calm weather the barometer is found to stand
at a different height in two different places at the level of
the sea, the reason must be that gravity is more intense at
the place where the barometer is low.
Let us next consider the method of finding the depth
below the level of the sea by means of a barometer carried
down in a diving bell.
If D is the depth of the surface of the water in the diving
bell below the surface of the sea, and if/ is the pressure of
the atmosphere on the surface of the sea, then the pressure
Barometer in a Diving Bell. 2 19
of the air in the diving bell must exceed that on the surface
of the sea by the pressure due to a column of water of depth
D. If <r is the density of sea- water, the pressure due to a
column of depth D is g a D.
Let the height of the barometer at the surface of the sea
be observed, and let us suppose that in the diving bell it is
found to be higher by a height h, then the additional pres-
sure indicated by this rise is g p h, where p is the density of
mercury. Hence
or
where s = t = density of mercury = ifi<; ^ Qf
o density of water
mercury.
The depth below the surface of the sea is therefore equal
to the product of the rise of the barometer multiplied by the
specific gravity of mercury. If the water is salt we must
divide this result by the specific gravity of the salt water at
the place of observation.
The calculation of depths under water by this method is
comparatively easy, because the density of the water is not
very different at different depths. It is only at great depths
that the compression of the water would sensibly affect the
result.
If the density of air had been as uniform as that of water,
the measurement of heights in the atmosphere would have
been as easy. For instance, if the density of air had been
equal to a at all pressures, then, neglecting the variation of
gravity with height above the earth, we should find the
height <$ of the atmosphere thus : Let h be the height of
the barometer, and p the density of mercury, then the pressure
indicated by the barometer is
P = g p ^
220 Measurement of Heights by the Barometer.
If is the height of an atmosphere of density <r, it
produces a pressure
/ = g &
Hence
This is the height of the atmosphere above the place on
the false supposition that its density is the same at all heights
as it is at that place. This height is generally referred to as
the height of the atmosphere supposed of uniform density, or
more briefly and technically as the height of the homogeneous
atmosphere.
Let us for a moment consider what this height (which
evidently has nothing to do with the real height of the
atmosphere) really represents. From the equation
P = g * *,
remembering that a the density of air is the same thing as
the reciprocal of v the volume of unit of mass, we get
.
or % is simply the product / v expressed in gravitation
measure instead of absolute measure.
Now, by Boyle's law the product of the pressure and
the volume at a constant temperature is constant, and by
Charles's law this product is proportional to the absolute
temperature. For dry air at the temperature of melting ice,
and when g = 32*2,
$=*JL = 26,2 14 feet,
t
or somewhat less than five statute miles.
It is well known that Mr. Glaisher has ascended in a
balloon to the height of seven miles. This balloon was
supported by the air, and though the air at this great height
was more than three times rarer than at the earth's surface, it
was possible to breathe in it. Hence it is certain that the
Height of a Mountain. 221
atmosphere must extend above the height , which we have
deduced from our false assumption that the density is
uniform.
But though the density of the atmosphere is by no means
uniform through great ranges of height, yet if we confine
ourselves to a very small range, say the millionth part of &
that is, about 0-026 feet, or less than the third of an inch the
density will only vary one-millionth part of itself from the
top to the bottom of this range, so that we may suppose the
pressure at the bottom to exceed that at the top by exactly
one-millionth.
Let us now apply this method to determine the height of
a mountain by the following imaginary process, too laborious
to be recommended, except for the purpose of explaining
the practical method :
We shall suppose that we begin at the top of the mountain,
and that, besides our barometer, we have one thermometer
to determine the temperature of the mercury, and another to
determine the temperature of the air. We are also provided
with a hygrometer, to determine the quantity of aqueous
vapour in the air, so that by the thermometer and hygrometer
we can calculate , the height of the homogeneous atmo-
sphere, at every station of our path.
On the top of the mountain, then, we observe the height of
the barometer to be/. We now descend the mountain till
we observe the mercury in the barometer to rise by one-
millionth part of its own height The height of the baro-
meter at this first station is
p l = (roooooi)/.
The distance we have descended is one-millionth of $,
the height of the homogeneous atmosphere for the observed
temperature at the first stage of the descent. Since it is
at present impossible to measure pressures, &c, to one-
millionth of their value, it does not matter whether $ be
222 Measurement of Heights by the Barometer.
measured at the top of the mountain or one-third of an inch
lower down.
Now let us descend another stage, till the pressure again
increases one-millionth of itself, so that if / 2 is the new
pressure,
Pi = (1-000001)^1,
and the second descent is through a height equal to the
millionth of & 2 , the height of the homogeneous atmosphere
in the second stage.
If we go on in this way n times, till we at last reach the
bottom of the mountain, and if / is the pressure at the
bottom,
A = (i'ooi)A-i
= (l'OOOOOl) 9 / n _ 2
= (I'OOOOOl)"/,
and the whole vertical height will be
+ + &c. + &
1,000,000
If we assume that the temperature and humidity are the
same at all heights between the top and the bottom, then
j = $ 2 = &c. = $ n = <, and the height of the mountain
will be
1,000,000
If we Know n, the number of stages, we can determine
the height of the mountain in this way. But it is easy to
find n without going through the laborious process of
descending by distances of the third of an inch, for since
p n = P is the pressure at the bottom, and p that at the top,
we have the equation
p = (I'OOOOOl)"/.
Taking the logarithm of both sides of this equation, we
get
Waves. 223
log P aae log (I'OOOOOl) -f log /,
or
ff _ log P - log/
log ( I'OOOOOl)'
Now log i*oooooi = o'oooooo4342942648.
Substituting this value in the expression for /t, we get
k = --*_ log *
434294 /
where the logarithms are the common logarithms to base 10,
or
/&= 2 -302585 log ?.
For dry air at the temperature of melting ice = 26,214
feet : hence
h = log- x I 60360 + (0 - 32) (122-68)}
p (
gives the height in feet for a temperature 6 on Fahrenheit's
scale.
For rough purposes, the difference of the logarithms of the
heights of the barometer multiplied by 10,000 gives the
difference of the heights in fathoms of six feet.
CHAPTER XV.
ON THE PROPAGATION OF WAVES.
THE following method of investigating the conditions of the
propagation of waves is due to Prof. Rankine. 1 It involves
only elementary principles and operations, but leads to
results which have been hitherto obtained only by opera-
tions involving the higher branches of mathematics.
1 Phil. Trans. 1869: 'On the Thermodynamic Theory of Waves of
Finite Longitudinal Disturbance.'
224 Waves.
The kind of waves to which the investigation applies are
those in which the motion of the parts of the substance is
along straight lines parallel to the direction in which the
wave is propagated, and the wave is defined to be one
which is propagated with constant velocity, and the type of
which does not alter during its propagation.
In other words, if we observe what goes on in the
substance at a given place when the wave passes that place,
and if we suddenly transport ourselves a certain distance
forward in the direction of propagation of the wave, then
after a certain time we shall observe exactly the same things
occurring in the same order in the new place, when the wave
reaches it. If we travel with the velocity of the wave, we
shall therefore observe no change in the appearance pre-
sented by the wave as it travels along with us. This is the
characteristic of a wave of permanent type.
We shall first consider the quantity of the substance
which passes in unit of time through unit of area of a plane
which we shall suppose fixed, and perpendicular to the
direction of motion.
Let u be the velocity of the substance, which we shall
suppose to be uniform, then in unit of time a portion of the
substance whose length is u passes through any section
of a plane perpendicular to the direction of motion. Hence
the volume which passes through unit of area is represented
by u.
Now let Q be the quantity of the substance which passes
through, and let v be the volume of unit of mass of the
substance, then the whole volume is Q v, and this, by what
we have said, is equal to #, the velocity of the substance.
If the plane, instead of being fixed, is moving forwards with
a velocity u, the quantity which passes through it will
depend, not on the absolute velocity, u, of the substance,
but on the relative velocity, u u, and if Q is the quantity
which passes through the plane from right 'to left,
Q v = u u (i)
Waves of Longitudinal Displacement. 225
Let A be an imaginary plane moving from left to right
with velocity u, and let this be the velocity of propagation
FIG. 39.
> > *
of the wave, then, as the plane A travels along, the values of
u and all other quantities belonging to the wave at the
plane A remain the same. If u l is the absolute velocity of
the substance at A, v l the volume of unit of mass, and/! the
pressure, all these quantities will be constant, and
Qi Vi = u - ! (2)
If B be another plane, travelling with the same velo-
city u, and if Q 2 u% v 9 / 2 be the corresponding values
atu,
Q 2 z/ 2 = u - 2 (3)
The distance between the planes A and B remains in-
variable, because they travel with the same velocity. Also
the quantity of the substance intercepted between them
remains the same, because the density of the substance at
corresponding parts of the wave remains the same as the
wave travels along. Hence the quantity of matter which
enters the space between A and B at A must be equal to
that which leaves it at B, or
Qi = Q 2 = Q (say) (4)
Hence
*! = U - Q i # 2 = U - Q Z/ 2 . . (5)
so that when we know u and Q and the volume of unit of
mass, we can find u and 2 .
Let us next consider the forces acting on the matter con-
tained between A and B. If p^ is the pressure at A, and p %
Q
226 Waves.
that at B, the force arising from these pressures tending to
increase the momentum from left to right is/ 2 p v
This is the momentum generated in unit of time by the
external pressures on the portion of the substance between
A and B.
Now we must recollect that, though corresponding points
of the substance in this interval are always moving in the
same way, the matter itself between A and B is continually
changing, a quantity Q entering at A, and an equal quantity
Q leaving at B.
Now the portion Q which enters at A has a velocity u lt
and therefore a momentum Q u lt and that which issues at
B has a velocity 2 , and therefore a momentum Q 2 .
Hence the momentum of the entering fluid exceeds that
of the issuing fluid by
Q(! 2 ).
The only way in which this momentum can be produced
is by the action of the external pressures p^ and/ 2 ; for the
mutual actions of the parts of the substance cannot alter the
momentum of the whole. Hence we find
P\ -/ 2 = Q("i -a) . . . . (6)
Substituting the values of u l and u 2 from equation (5), we
find
P\ -/2 = Q 2 (^2~^l) (7)
Hence
/ 1 +Q 1 =/ i + Q 1 . ..... (8)
Now the only restriction on the position of the plane B is
that it must remain at a constant distance behind A, and
whatever be the distance between A and B, the above
equation is always true.
Hence the quantity p + Q 8 v must continue constant
during the whole process involved in the passage of the
wave. Calling this quantity p, we have
/ = P-Q (9)
Waves of Permanent Type. 227
or the pressure is equal to a constant pressure, p, diminished
by a quantity proportional to the volume v.
This relation between pressure and volume is not fulfilled
in the case of any actual substance. In all substances it is
true that as the volume diminishes the pressure increases,
but the increase of pressure is never strictly proportional to
the diminution of volume. As soon as the diminution of
volume becomes considerable, the pressure begins to in-
crease in a greater ratio than the volume diminishes.
But if we consider only small changes of volume and
pressure, we may make use of our former definition of elas-
ticity at p. 107 namely, the ratio of the number expressing
the increment of pressure to that expressing the voluminal
compression, or, calling the elasticity E,
E = v y _ ^ = v Q 2 by equation (7) (10)
where v is the volume of unit of mass, and since v l and z> 2
are very nearly equal, we may take either for the value of v.
Again, if v is the volume of unit of mass in those parts of the
substance which are not disturbed by the wave, and for
which, therefore, u 3= o,
u = Q z/ ... (n)
Hence we find
u 2 = Q* z/ a = E v (i2>
which shows that the square of the velocity of propagation
of a wave of longitudinal displacement in any substance is
equal to the product of the elasticity and the volume of unit
of mass.
In calculating the elasticity we must take into account the
conditions under which the compression of the substance
actually takes place. If, as in the case of sound-waves, it is
very sudden, so that any heat which is developed cannot be
conducted away, then we must calculate the elasticity on the
supposition that no heat is allowed to escape.
In the case of air or any other gas the elasticity at constant
Q2
228 Waves.
temperature is numerically equal to the pressure. If we
denote, as usual, the ratio of the specmc heat at constant
pressure to that at constant volume by the symbol y, the
elasticity when no heat escapes is
Hence, if u is the velocity of sound,
U 2 = y/z> .......... (14)
We know that when the temperature is the same the
product p v remains constant. Hence, the velocity of sound
is the same for the same temperature, whatever be the
pressure of the air.
If is the height of the atmosphere supposed homo-
geneous that is to say, the height of a column of the
same density as the actual density, the weight of which
would produce a pressure equal to the actual pressure then,
if the section of the column is unity, its volume is , and if
m is its mass, $ m v.
Also the weight of this column is / = m g, where g is the
force of gravity.
Hence
P v = g $
and
u 2 = g 7
The velocity of sound may be compared with that ot a
body falling a certain distance under the action of gravity.
For if v is the velocity of a body falling through a height s,
v* = 2 g s.
If we make v = u, then s = ^ y .
At the temperature of melting ice * = 26,214 feet if the
force of gravity is 32-2.
At the same temperature the velocity of sound in air is
1,090 feet per second by experiment.
The square of this is 1,188,100, whereas the square of
the velocity due to half the height of the homogeneous
Velocity of Sound. 229
atmosphere is 843,821. Hence by means of the known
velocity of sound we can determine y, the ratio of 1,188,100
to 843,821, to be 1-408.
The height of the homogeneous atmosphere is proportional
to the temperature reckoned from absolute zero. Hence the
velocity of sound is proportional to the square root of the
absolute temperature. In several of the more perfect gases
the value of y seems to be nearly the same as in air. Hence
in those gases the velocity of sound is inversely as the square
root of their specific gravity compared with air.
This investigation would be perfectly accurate, however
great the changes of pressure and density due to the passage
of the sound-wave, provided the substance is such that in the
actual changes of pressure and volume the quantity
remains constant, Q being the velocity of propagation. In
all substances, as we have seen, we may, when the values of
p and v are always very near their mean values, assume a
value of Q which shall approximately satisfy this condition ;
but in the case of very violent sounds and other disturbances
of the air the changes of p and v may be so great that this
approximation ceases to be near the truth. To understand
what takes place in these cases we must remember that the
changes of/ and v are not proportional to each other, for iit
almost all substances / increases faster for a given diminution
of v as / increases and v diminishes.
Hence Q, which represents the mass of the substance
traversed by the wave, will be greater in those parts of the
wave where the pressure is great than in those parts where
the pressure is small; that is, the condensed portions of the
wave will travel faster than the rarefied portions. The result
of this will be that if the wave originally consists of a gradual
condensation followed by a gradual rarefaction, the conden-
sation will become more sudden and the rarefaction more
gradual as the wave advances through the air, in the same
23 Radiation.
way and for nearly the same reason as the waves of the sea
on coming into shallow water become steeper in front and
more gently sloping behind, till at last they curl over on the
shore.
FIG. 30.
CHAPTER XVI.
ON RADIATION.
WE have already noticed some of the phenomena of radia-
tion, and have shown that they do not properly belong to the
science of Heat, and that they should rather be treated,
along with sound and light, as a branch of the great science
of Radiation.
The phenomenon of radiation consists in the transmis-
sion of energy from one body to another by propagation
through the intervening medium, in such a way that the
progress of .the radiation may be traced, after it has left the
first body and before it reaches the second, travelling through
the medium with a certain velocity, and leaving the medium
behind it in the condition in which it found it.
We have already considered one instance of radiation in
the case of waves of sound. In this case the energy com-
municated to the air by a vibrating body is propagated
through the air, and may finally set some other body, as the
drum of the ear, in motion. During the propagation of the
sound this energy exists in the portion of air through which
it is travelling, partly in the form of motion of the air to and
Radiation. 231
fro, and partly in the form of condensation and rarefaction.
The energy due to sound in the air is distinct from heat, be-
cause it is propagated in a definite direction, so that in a
certain time it will have entirely left the portion of air under
consideration, and will be found in another portion of air to
which it has travelled. Now heat never passes out of a hot
body except to enter a colder body, so that the energy of
sound-waves, or any other form of energy which is propa-
gated so as to pass wholly out of one portion of the medium
and into another, cannot be called heat.
There are, however, important thermal effects produced
by radiation, so that we cannot understand the science of heat
without studying some of the phenomena of radiation.
When a body is raised to a very high temperature it
becomes visible in the dark, and is said to shine, or to emit
light. The velocity of propagation of the light emitted by
the sun and by very hot bodies has been approximately mea-
sured, and is estimated to be between 180,000 and 192,000
miles per second, or about 900,000 times faster than sound
in air.
The time taken by the light in passing from one place to
another within the limited range which we have at our com-
mand in a laboratory is exceedingly short, and it is only by
means of the most refined experimental methods that it has
been measured. It is certain, however, that there is an
interval of time between the emission of light by one body
and its reception by another, and that during this time the
energy transmitted from the one body to the other has
existed in some form in the intervening medium.
The opinions with regard to the relation between light
and heat have suffered several alternations, according as
these agents were regarded as substances or as accidents.
At one time light was regarded as a substance projected
from the luminous body, which, if the luminous body
were hot, might itself become hot like any other substance.
Heat was thus regarded as an accident of the substance light.
232 Radiation.
When the progress of science had rendered the measure
ment of quantities of heat as accurate as the measurement
of quantities of gases, heat, under the name of caloric, was
placed in the list of substances. Afterwards, the independent
progress of optics led to the rejection of the corpuscular
theory of light, and the establishment of the undulatory
theory, according to which light is a wave-like motion of a
medium already existing. The caloric theory of heat, how-
ever, still prevailed even after the corpuscular theory of
light was rejected, so that heat and light seemed almost to
have exchanged places.
When the caloric theory of heat was at length demon-
strated to be false, the grounds of the argument were quite
independent of those which had been used in the case of
light.
We shall therefore consider the nature of radiation,
whether of light or heat, in an independent manner, and
show why we believe that what is called radiant heat is the
same thing as what is called light, only perceived by us
through a different channel. The same radiation which
when we become aware of it by the eye we call light, when
we detect it by a thermometer or by the sensation of heat
we call radiant heat.
In the first place, radiant heat agrees with light in always
moving in straight lines through any uniform medium. It is
not, therefore, propagated by diffusion, as in the case of the
conduction of heat, where the heat always travels from hotter
to colder parts of the medium in whatever direction this
condition may lead it.
The medium through which radiant heat passes is not
heated if perfectly diathermanous, any more than a per-
fectly transparent medium through which light passes is
rendered luminous. But if any impurity or defect of trans-
parency causes the medium to become visible when light
passes through it, it will also cause it to become hot and to
stop part of the heat when traversed by radiant heat
L igkt and Heat. 233
In the next place, radiant heat is reflected from the
polished surfaces of bodies according to the same laws as
light. A concave mirror collects the rays of the sun into a
brilliantly luminous focus. If these collected rays fall on a
piece of wood, they will set it on fire. If the luminous rays
are collected by means of a convex lens, similar heating
effects are produced, showing that radiant heat is refracted
when it passes from one transparent medium to another.
When light is refracted through a prism, so as to change
its direction through a considerable angle of deviation, it is
separated into a series of kinds of light which are easily
distinguished from each other by their various colours.
The radiant heat which is refracted through the prism is also
spread out through a considerable angular range, which shows
that it also consists of radiations of various kinds. The
luminosity of the different radiations is evidently not in the
same proportion as their heating effects. For the blue and
green rays have very little heating power compared with the
extreme red, which are much less luminous, and the heating
rays are found far beyond the end' of the red, where no light
at all is visible.
There are other methods of separating the different kinds
of light, which are sometimes more convenient than the use
of a prism. Many substances are more transparent to
one kind of light than another, and are therefore called
coloured media. Such media absorb certain rays and
transmit others. If the light transmitted by a stratum of a
coloured medium afterwards passes through another stratum
of the same medium, it will be much less diminished in
intensity than at first. For the kind of light which is most
absorbed by the medium has been already removed, and
what is transmitted by the first stratum is that which can pass
most readily through the second. Thus a very thin stratum
of a solution of bichromate of potash cuts off the whole of
the spectrum from the middle of the green to the violet, but
the remainder of the light, consisting of the red, orange,
234 Radiation.
yellow, and part of the green, is very slightly diminished in
intensity by passing through another stratum of the same
medium.
If, however, the second stratum be of a different medium,
which absorbs most of the rays which the first transmits, it
will cut off nearly the whole light, though it may be itself
very transparent for other rays absorbed by the first medium.
Thus a stratum of sulphate of copper absorbs nearly all the
rays transmitted by the bichromate of potash, except a few
of the green rays.
Melloni found that different substances absorb different
kinds of radiant heat, and that the heat sifted by a screen
of any substance will pass in greater proportion through
a screen of the same substance than unsifted heat, while it
may be stopped in greater proportion than unsifted heat by
a screen of a different substance.
These remarks may illustrate the general similarity between
light and radiant heat. We must next consider the reasons
which induce us to regard light as depending on a particular
kind of motion in the medium through which it is pro-
pagated. These reasons are principally derived from the
phenomena of the interference of light. They are explained
more at large in treatises on light, because it is much easier
to observe these phenomena by the eye than by any kind
of thermometer. We shall therefore be as brief as possible.
There are various methods by which a beam of light from
a small luminous object may be divided into two portions,
which, after travelling by slightly different paths, finally fall
on a white screen. Where the two portions of light overlap
each other on the screen, a series of long narrow stripes may
be seen, alternately lighter and darker than the average
brightness of the screen near them, and when white light is
used, these stripes are bordered with colours. By using light
of one kind only, such as that obtained from the salted wick
of a spirit-lamp, a greater number of bands or fringes may
be seen, and a greater difference of brightness between the
Interference. 235
light and the dark bands. If we stop either of the portions
of light into which the original beam was divided, the whole
system of bands disappears, showing that they are due,
not to either of the portions alone, but to both united.
If we now fix our attention on one of the dark bands, and
then cut off one of the partial beams of light, we shall
observe that instead of appearing darker it becomes actually
brighter, and if we again allow the light to fall on the screen
it becomes dark again. Hence it is possible to produce
darkness by the addition of two portions of light If light
is a substance, there cannot be another substance which
when added to it shall produce darkness. We are therefore
compelled to admit that light is not a substance.
Now is there any other instance in which the addition of
two apparently similar things diminishes the result? We
know by experiments with musical instruments that a com-
bination of two sounds may produce less audible effect than
either separately, and it can be shown that this takes place
when the one is half a wave-length in advance of the other.
Here the mutual annihilation of the sounds arises from the
fact that a motion of the air towards the ear is the exact
opposite of a motion away from the ear, and if the two in-
struments are so arranged that the motions which they tend
to produce in the air near the ear are in opposite direc-
tions and of equal magnitude, the result will be no motion
at all. Now there is nothing absurd in one motion being
the exact opposite of another, though the supposition that
one substance is the exact opposite of another substance, as
in some forms of the Two-Fluid theory of Electricity, is an
absurdity.
We may show the interference of waves in a visible
manner by dipping a two-pronged fork into water or mercury.
The waves which diverge from the two centres where the
prongs enter or leave the fluid are seen to produce a
greater disturbance when they exactly coincide than when
one gets ahead of the other.
236 Radiation,
Now it is found, by measuring the positions of the bright
and dark bands on the screen, that the difference of the
distances travelled by the two portions of light is for the
bright bands always an exact multiple of a certain very
small distance which we shall call a wave-length, whereas
for the dark bands it is intermediate between two multi-
ples of the wave-length, being \, i, <z\, &c., times that
length.
We therefore conclude that whatever exists or takes
place at a certain point in a ray of light, then, at the same
instant, at a point at \ or i^ of the wave-length in advance,
something exactly the opposite exists or takes place, so that
in going along a ray we find an alternation of conditions
which we may call positive and negative.
In the ordinary statement of the theory of undulations
these conditions are described as motion of the medium in
opposite directions. The essential character of the theory
would remain the same if we were to substitute for ordinary
motion to and fro any other succession of oppositely
directed conditions. Professor Rankine has suggested op-
posite rotations of molecules about their axes, and I have
suggested oppositely directed magnetizations and electro-
motive forces y but the adoption of either of these hypotheses
would in no way alter the essential character of the undula-
tory theory.
Now it is found that if a very narrow thermo-electric pile
be placed in the position of the screen, and moved so that
sometimes a bright band and sometimes a dark one falls on
the pile, the galvanometer indicates that the pile receives
more heat when in the bright than when in the dark band,
and that when one portion of the beam is cut off the heat in
the dark band is increased. Hence in the interference of
radiations the heating effect obeys the same laws as the
luminous effect.
Indeed, it has been found that even when the source
of radiation is a hot body which emits no luminous rays,
Polarization. 237
the phenomena of interference can be traced, showing
that two rays of dark heat can interfere no less than two
rays of light Hence all that we have said about the waves
of light is applicable to the heat-radiation, which is therefore
a series of waves.
It is also known in the case of light that after passing
through a plate cut from a crystal of tourmaline parallel to
its axis the transmitted beam cannot pass through a second
similarly cut plate of tourmaline whose axis is perpendicular
to that of the first, though it can pass through it when the axis
is in any other position. Such a beam of light, which has
different properties according as the second plate is turned
into different positions round the beam as an axis, is called
a polarized beam. There are many other ways of polarizing
a. beam of light, but the result is always of the same kind.
Now this property of polarized light shows that the motion
which constitutes light cannot be in the direction of the
ray, for then there could be no difference between different
sides of the ray. The motion must be transverse to the
direction of the ray, so that we may now describe a ray of
polarized light as a condition of disturbance in a direction
at right angles to the ray propagated through a medium, so
that the disturbance is in opposite directions at every half
wave-length measured along the ray. Since Principal J. D.
Forbes showed that a ray of dark heat can be polarized, we
can make the same assertion about the heat radiation.
Let us now consider the consequences of admitting that
what we call radiation, whether of heat, light, or invisible
rays which act on chemical preparations, is of the nature of
a transverse undulation in a medium.
A transverse undulation is completely defined when we
know
1. Its wave-length, or the distance between two places in
which the disturbance is in the same phase.
2. Its amplitude, or the greatest extent of the disturb-
ance.
238 Radiation.
3. The plane in which the direction of the disturbance
lies.
4. The phase of the wave at a particular point.
5. The velocity of propagation through the medium.
When we know these particulars about an undulation, it
is completely defined, and cannot be altered in any way
without changing some of these specifications.
Now by passing a beam consisting of any assemblage of
undulations through a prism, we can separate it into portions
according to their wave* lengths, and we can select rays of a
particular wave-length for examination. Of these we may, by
means of a plate of tourmaline, select those whose plane of
polarization is the principal plane of the tourmaline, but this
is unnecessary for oar purpose. We have now got rays of a
definite wave-length. Their velocity of propagation depends
only on the nature of the ray and of the medium, so that we
cannot alter it at pleasure, and the phase changes so rapidly
(billions of times in a second) that it cannot be directly
observed. Hence the only variable quantity remaining is
the amplitude of the disturbance, or, in other words, the
intensity of the ray.
Now the ray may be observed in various ways. We may,
if it excites the sensation of sight, receive it in to our eye. If
it affects chemical compounds, we may observe its effect on
them, or we may receive the ray on a thermo-electric pile
and determine its heating effect.
But all these effects, being effects of one and the same
thing, must rise and fall together. A ray of specified wave-
length and specified plane of polarization cannot be a
combination of several different things, such as a light-ray, a
heat-ray, and an actinic ray. It must be one and the same
thing, which has luminous, thermal, and actinic effects, and
everything which increases one of these effects must increase
the others also.
The chief reason why so much that has been written on
this subject is tainted with the notion that heat is one thing
Light and Heat. > 239
and light is another seems to be that the arrangements
for operating on radiations of a selected wave-length are
troublesome, and when mixed radiations are employed, in
which the luminous and the thermal effects are in different
proportions, anything which alters the proportion of the
different radiations in the mixture alters also the proportion
of the resulting thermal and luminous effect, as indeed it
generally alters the colour of the mixed light.
We have seen that the existence of these, radiations may
be detected in various ways by photographic preparations,
by the eye, and by the thermometer. There can be no
doubt, however, as to which of these methods gives the true
measure of the energy transmitted by the radiation. This
is exactly measured by the heating effect of the ray when
completely absorbed by any substance.
When the wave-length is greater than 812 millionths of a
millimetre no luminous effect is produced on the eye, though
the effect on the thermometer may be very great. When
the wave-length is 650 millionths of a millimetre the ray is
visible as a red light, and a considerable heating effect is
observed. But when the wave-length is 500 millionths of a
millimetre, the ray, which is seen as a brilliant green, has
much less heating effect than the dark or the red rays, and
it is difficult to obtain strong thermal effects with rays of
smaller wave-lengths, even when concentrated.
But, on the other hand, the photographic effect of the
radiation on salts of silver, which is very feeble in the red
rays, and even in the green rays, becomes more powerful
the smaller the wave-length, till for rays whose wave-length
is 400, which have a feeble violet luminosity and a still
feebler thermal effect, the photographic effect is very
powerful; and even far beyond the visible spectrum, for wave-
lengths of less than 200 millionths of a millimetre, which
are quite invisible to our eyes and quite undiscoverable by
our thermometers, the photographic effect is still observed.
This shows that neither the luminous nor the photographic
240 Radiation.
effect is in any way proportional to the energy of the radia
tion when different kinds of radiation are concerned. It
is probable that when the radiation produces the photo-
graphic effect it is not by its energy doing work on the
chemical compound, but rather by a well-timed vibration of
the molecules dislodging them from the position of almost
indifferent equilibrium into which they had been thrown by
previous chemical manipulations, and enabling them to rush
together according to their more permanent affinities, so as
to form stabler compounds. In cases of this kind the effect
is no more a dynamical measure of the cause than the effect
of the fall of a tree is a measure of the energy of the wind
which uprooted it.
It is true that in many cases the amount of the radiation
may be very accurately estimated by means of its chemical
effects, even when these chemical effects tend to diminish
the intrinsic energy of the system. But by estimating the
heating effect of a radiation which is entirely absorbed by
the heated body we obtain a true measure of the energy of
the radiation. It is found that a surface thickly coated
with lampblack absorbs nearly the whole of every kind of
radiation which falls on it. Hence surfaces of this kind are
of great value in the thermal study of radiation.
We have now to consider the conditions which determine
the amount and quality of the radiation from a heated body.
We must . bear in mind that temperature is a property of
hot bodies and not of radiations, and that qualities such as
wave-lengths, &c., belong to radiations, but not to the heat
which produces them or is produced by them.
ON PREVOST'S THEORY OF EXCHANGES.
When a system of bodies at different temperatures is left
to itself, the transfer of heat which takes place always has
the effect of rendering the temperatures of the different
bodies more nearly equal, and this character of the transfer
Theory of Exchanges. 241
of heat, that it passes from hotter to colder bodies, is the
same whether it is by radiation or by conduction that the
transfer takes place.
Let us consider a number of bodies, all at the same
temperature, placed in a chamber the walls of which are
maintained at that temperature, and through which no heat
can pass by radiation (suppose the walls of metal, for
instance). No change of temperature will occur in any of
these bodies. They will be in thermal equilibrium with
each other and with the walls of the chamber. This is a
consequence of the definition of equal temperature at p. 32.
Now if any one of these bodies had been taken out of
the chamber and placed among colder bodies there would
be a transfer of heat by radiation from the hot body to the
colder ones j or if a colder body had been introduced into
the chamber it would immediately begin to receive heat by
radiation from the hotter bodies round it. But the cold
body has no power of acting directly on the hot bodies at a
distance, so as to cause them to begin to emit radiations,
nor has the hot chamber any power to stop the radiation of
any one of the hot bodies placed within it. We therefore
conclude with Prevost that a hot body is always emitting
radiations, even when no colder body is there to receive
them, and that the reason why there is no change of tem-
perature when a body is placed in a chamber of the same
temperature is that it receives from the radiation of the walls
of the chamber exactly as much heat as it loses by radiation
towards these walls.
If this is the true explanation of the thermal equilibrium
of radiation, it follows that if two bodies have the same
temperature the radiation emitted by the first and absorbed
by the second is equal in amount to the radiation emitted
by the second and absorbed by the first during the same
time
The higher the temperature of a body, the greater its
radiation is found to be, so that when the temperatures of the
R
242 Radiation
bodies are unequal the hotter bodies will emit more radia-
tion than they receive from the colder bodies, and therefore,
on the whole, heat will be lost by the hotter and gained by
the colder bodies till thermal equilibrium is attained. We
shall return to the comparison of the radiation at different
temperatures after we have examined the relations between
the radiation of different bodies at the same temperature.
The application of the theory of exchanges has at various
times been extended to the phenomena of heat as they
were successively investigated Fourier has considered the
law of radiation as depending on the angle which the ray
makes with the surface, and Leslie has investigated its
refation to the state of polish of the surface ; but it is in
recent times, and chiefly by the researches of B. Stewart,
Kirchhoff, and De la Provostaye, that the theory of ex-
changes has been shown to be applicable, not only to the
total amount of the radiation, but to every distinction in
quality of which the radiation is capable.
For, by placing between two bodies of the same tempera-
ture a contrivance such as that already noticed at p. 238, so
that only radiations of a determinate wave-length and in a
determinate plane can pass from the one body to the other,
we reduce the general proposition about thermal equilibrium
to a proposition about this particular kind of radiation. We
may therefore transform it into the following more definite
proposition.
If two bodies are at the same temperature, the radiation
emitted by the first and absorbed by the second agrees with
the radiation emitted by the second and absorbed by the
first, not only in its total heating effect, but in the intensity,
wave-length, and plane of polarization of every component
part of either radiation. And the law that the amount of
radiation increases with the temperature must be true, not
only for the whole radiation, but for all the component parts
of it when analysed according to their wave-lengths and
planes of polarization.
and A bsorption. 243
The consequences of these two propositions, applying as
they do to every kind of radiation, whether detected by its
thermal or by its luminous effects, are so numerous and
varied that we cannot attempt any full enumeration of them
in this treatise. We must confine ourselves to a few ex-
amples.
When a radiation falls on a body, part of it is reflected,
and part enters the body. The latter part again may either
be wholly absorbed by the body or partly absorbed and
partly transmitted.
Now lampblack reflects hardly any of the radiation which
falls on it, and it transmits none. Nearly the whole is
absorbed.
Polished silver reflects nearly the whole of the radiation
which falls upon it, absorbing only about a fortieth part, and
transmitting none.
Rock salt reflects less than a twelfth part of the radiation
which falls on it; it absorbs hardly any, and transmits ninety-
two per cent.
These three substances, therefore, may be taken as types of
absorption, reflexion, and transmission respectively.
Let us suppose that these properties have been observed
in these substances at the temperature, say, of 212 F., and
let them be placed at this temperature within a chamber
whose walls are at the same temperature. Then the amount
of the radiation from the lampblack which is absorbed by
the other two substances is, as we have seen, very small.
Now the lampblack absorbs the whole of the radiation from
the silver or the salt Hence the radiation from these
substances must also be small, or, more precisely
The radiation of a substance at a given temperature is to
the radiation of lampblack at that temperature as the amount
of radiation absorbed by the substance at that temperature is to
the whole radiation which falls upon it.
Hence a body whose surface is made of polished silver
will emit a much smaller amount of radiation than one
R 2
244 Radiation
whose surface is of lampblack. The brighter the surface of
a silver teapot, the longer will it retain the heat of the tea ;
and if on the surface of a metal plate some parts are polished,
others rough, and others blackened, when the plate is made
red hot the blackened parts will appear brightest, the rough
parts not so bright, and the polished parts darkest. This is
well seen when melted lead is made red hot. When part
of the dross is removed, the polished surface of the melted
metal, though really hotter than the dross, appears of a less
brilliant red.
A piece of glass when taken red hot out of the fire appears
of a very faint red compared with a piece of iron taken from
the same part of the fire, though the glass is really hotter
than the iron, because it does not throw off its heat so fast.
Air or any other transparent gas, even when raised to a
heat at which opaque bodies appear white hot, emits so little
light that its luminosity can hardly be observed in the
dark, at least when the thickness of the heated air is not
very great.
Again, when a substance at a given temperature absorbs
certain kinds of radiation and transmits others, it emits at
that temperature only those kinds of radiation which it
absorbs. A very remarkable instance of this is observed in
the vapour of sodium. This substance when heated emits
rays of two definite kinds, whose wave-lengths are 0*00059053
and 0-00058989 millimetre respectively. These rays are
visible, and may be seen in the form of two bright lines by
directing a spectroscope upon a flame in which any com-
pound of sodium is present.
Now if the light emitted from an intensely heated solid
body, such as a piece of lime in the oxyhydrogen light, be
transmitted through sodium-vapour at a temperature lower
than that of the lime, and then analysed by the spectro-
scope, two dark lines are seen, corresponding to the two
bright ones formerly observed, showing that sodium-vapour
absorbs the same definite kinds of light which it radiates.
as depending on Temperature. 245
If the temperature of the sodium-vapour is raised, say by
using a Bunsen's burner instead of a spirit-lamp to produce
it, or if the temperature of the lime is lowered till it is
the same as that of the vapour, the dark lines disappear,
because the sodium-vapour now radiates exactly as much
light as it absorbs from the light of the lime-ball at the
same temperature. If the sodium-flame is hotter than the
lime-ball the lines appear bright.
This is an illustration of Kirchhoff's principle, that the
radiation of every kind increases as the temperature rises.
In performing this experiment we suppose the light from
the lime-ball to pass through the sodium-flame before it
reaches the slit of the spectroscope. If, however, the flame
is interposed between the slit and the eye, or the screen on
which the spectrum is projected, the dark lines may be seen
distinctly, even when the temperature of the sodium-flame is
higher than that of the lime-ball. For in the parts of the
spectrum near the lines the light is now compounded of the
analysed light of the lime-ball and the direct light of the
sodium-flame, while at the lines themselves the light of the
spectrum of the lime-ball is cut off, and only the direct light
of the sodium-flame remains, so that the lines appear darker
than the rest of the field.
It does not belong to the scope of this treatise to attempt
to go over the immense field of research which has been
opened up by the application of the spectroscope to dis-
tinguish different incandescent vapours, and which has led
to a great increase of our knowledge of the heavenly
bodies.
If the thickness of a medium, such as sodium-vapour,
which radiates and absorbs definite kinds of light, be very
great, the whole being at a high temperature, the light
emitted will be of exactly the same composition as that
emitted from lampblack at the same temperature. For,
though some kinds of radiation are much more feebly
emitted by the substance than others, these are also so
246 Radiation.
feebly absorbed that they can reach the surface from im-
mense depths, whereas the rays which are so copiously
radiated are also so rapidly absorbed that it is only from
places very near the surface that they can escape out of the
medium. Hence both the depth and the density of an
incandescent gas cause its radiation to assume more and
more of the character of a continuous spectrum.
When the temperature of a substance is gradually raised,
not only does the intensity of every particular kind of radia-
tion increase, but new kinds of radiation are produced.
Bodies of low temperature emit only rays of great wave-
length. As the temperature rises these rays are more
copiously emitted, but at the same time other rays of
smaller wave-length make their appearance. When the tem-
perature has risen to a certain point, part of the radiation is
luminous and of a red colour, the luminous rays of greatest
wave-length being red. As the temperature rises, the other
luminous rays appear in the order of the spectrum, but every
rise of temperature increases the intensity of all the rays
which have already made their appearance. A white-hot
body emits more red rays than a red-hot body, and more
non-luminous rays than any non-luminous body.
The total thermal value of the radiation. at any tempera-
ture, depending as it does upon the amount of all trie different
kinds of rays of which it is composed, is not likely to be a
simple function of the temperature. Nevertheless, Dulong
and Petit succeeded in obtaining a formula which expresses
the facts observed by them with tolerable exactness. It is
of the form
R ma e ,
where R is the total loss of heat in unit of time by radia-
tion from unit of area of the surface of the substance at the
temperature 0, m is a constant quantity depending only on
the substance and the nature of its surface, and a is a
numerical quantity which, when expresses the temperature
on the Centigrade scale, is 1-0077.
Total Quantity of Radiation. 247
If the body is placed in a chamber devoid of air, whose
walls are at the temperature /, then the heat radiated from
the walls to the body and absorbed by it will be
r = mat,
so that the actual loss of heat will be
R r = ma 9 ma 1 .
The constancy of the amount of radiation between the same
surfaces at the same temperatures affords a very convenient
method of comparing quantities of heat. This method was
referred to in our chapter on Calorimetry (p. 74), under the
name of the Method of Cooling.
The substance to be examined is heated and put into a
thin copper vessel, the outer surface of which is blackened,
or at least is preserved in the same state of roughness or of
polish throughout the experiments. This vessel is placed
in a larger copper vessel so as not to touch it, and the outer
vessel is placed in a bath of water kept at a constant tem-
perature. The temperature of the substance in the smaller
vessel is observed from time to time, or, still better, the times
are observed at which the reading of a thermometer im-
mersed in the substance is an exact number of degrees. In
this way the time of cooling, say from 100 to 90, from 90
to 80, is registered, the temperature of the outer vessel "being
kept always the same.
Suppose that this observation of the time of cooling is
made first when the vessel is filled with water, and then
when some other substance is put into it. The rate at which
heat escapes by radiation is the same for the same tempera-
ture in both experiments. The quantity of heat which
escapes during the cooling, say from 100 to 90, in the two
experiments, is proportional to the time of cooling. Hence
the capacity of the vessel and its contents in the first experi-
ment is to its capacity in the second experiment as the time
of cooling from 100 to 90 in the first experiment is to the
time of cooling from 100 to 90 in the second experiment
248 Radiation
The method of cooling is very convenient in certain cases,
but it is necessary to keep the temperature of the whole of
the substance in the inner vessel as nearly uniform as possible,
so that the method must be restricted to liquids which we
can stir, and to solids whose conductivity is great, and
which may be cut in pieces and immersed in a liquid.
The method of cooling has been found very applicable to
the measurement of the quantity of heat conducted through
a substance. (See the chapter on Conduction.)
EFFECT OF RADIATION ON THERMOMETERS.
On account of the radiation passing in all directions through
the atmosphere, it is a very difficult thing to determine the
true temperature of the air in any place out of doors by
means of a thermometer.
If the sun shines on the thermometer, the reading is of
course too high ; but if we put it in the shade, it may be too
low, because the thermometer may be emitting more radia-
tion than it receives from the clear sky. The ground, walls
of houses, clouds, and the various devices for shielding the
thermometer from radiation, may all become sources of
error, by causing an unknown amount of radiation on the
bulb. For rough purposes the effects of radiation may be
greatly removed by giving the bulb a surface of polished
silver, of which, as we have seen, the absorption is only a
fortieth of that of lampblack.
A method described by Dr. Joule in a communication to
the Philosophical Society of Manchester, November 26, 1867,
seems the only one free from all objections. The thermo-
meter is placed in a long vertical copper tube open at both
ends, but with a cap to close the lower end, which may be
removed or put on without warming it by the hand. What-
ever radiation affects the thermometer must be between it
and the inside of the tube, and if these are of the same
as affecting Thermometers. 249
temperature, the radiation will have no effect on the observed
reading of the thermometer. Hence, if we can be sure that
the copper tube and the air within it are at the temperature
of the atmosphere, and that the thermometer is in thermal
equilibrium, the thermometer reading will be the true tem-
perature.
Now, if the air within the tube is of the same temperature
as the air outside, it will be of the same density, and it will
therefore be in statical equilibrium with it. If it is warmer
it will be lighter, and an upward current will be formed in
the tube when the cap is removed. If it is colder, a down-
ward current will be formed.
To detect these currents a spiral wire is suspended in the
tube by a fine fibre, so that an upward or downward current
causes the spiral to twist the fibre, and any motion of the
spiral is made apparent by means of a small mirror attached
to it
To vary the temperature of the copper tube, it is enclosed
in a wider tube, so that vater may be placed in the space
between the tubes, and by pouring in warmer or cooler water
the temperature may be adjusted till there is no current.
We then know that the air is of the same temperature
within the tube as it is without But we know that the
tube is also of the same temperature as the air, for if it
were not it would heat or cool the air and produce a cur-
rent Finally, we know that the thermometer, if stationary,
is at the temperature of the atmosphere ; for the air in contact
with it, and the sides of the tube, which alone can exchange
radiations with it, have the same temperature as the atmo
sphere.
250 Convection,
CHAPTER XVII.
ON CONVECTION CURRENTS.
WHEN the application of heat to a fluid causes it to expand
or to contract, it is thereby rendered rarer or denser than the
neighbouring parts of the fluid ; and if the fluid is at the
same time acted on by gravity, it tends to form an upward
or downward current of the heated fluid, which is of course
accompanied with a current of the more remote parts of the
fluid in the opposite direction. The fluid is thus made to
circulate, fresh portions of fluid are brought into the neigh-
bourhood of the source of heat, and these when heated
travel, carrying their heat with them into other regions.
Such currents, caused by the application of heat, and carry-
ing this heat with them, are called convection currents.
They play a most important part in natural phenomena, by
causing a much more rapid diffusion of heat than would
take place by conduction alone in the same medium if re-
strained from moving. The actual diffusion of heat from
one part of the fluid to another takes place, of course, by
conduction,; but, on account of the motion of the fluid, the
isothermal surfaces are so extended, and in some cases con-
torted, that their areas are greatly increased while the dis-
tances between them are diminished, so that true conduction
goes on much more rapidly than if the medium were at
rest.
Convection currents depend on changes of density in a
fluid acted on by gravity. If the action of heat does not
produce a change of density, as in the case of water at a
temperature of about 39 R, no convection current will be
produced. If the fluid is not acted on by gravity, as would
Production of Currents, 251
be the case if the fluid were removed to a sufficient distance
from the earth and other great bodies, no convection cur-
rents would be formed. As this condition is not easily
realised, we may take the case of a vessel containing fluid,
and descending according to the law of motion of a body
falling freely. The pressure in this fluid will be the same
in every part, and a change of density in any part of the
fluid will not occasion convection currents.
When we wish to avoid the formation of convection
currents we must arrange matters so that during the whole
course of the experiment the density of each horizontal
stratum is the same throughout, and that the density increases
with the depth. If, for instance, we are studying the con-
duction of heat in a fluid which expands when heated, we
must make the heat flow downwards through the fluid. It
we wish to determine the law of diffusion of fluids we must
place the denser fluid underneath the rarer one.
Convection currents are produced by changes of density
arising from other causes. Thus if a crystal of a soluble
salt be suspended in a vessel of water, the water in contact
with the crystal will dissolve a portion of it, and, becoming
denser, will begin to sink, and its place will be supplied by
fresh water. Thus a convection current will be formed, a
solution of the salt will descend from the crystal, and this
will cause an upward current of purer water, and a circula-
tion will be kept up till either the crystal is entirely dissolved,
or the liquid has become saturated with the salt up to the
level of the top of the crystal. In this case it is the salt
which is carried through the liquid by convection.
A convection current may be produced in which electricity
is the thing carried. If a conductor terminating in a fine
point is strongly electrified, the particles of air near the point
will be charged with electricity, and then urged from the
point towards any surface oppositely electrified. A current
of electrified air is thus formed, which diffuses itself about
the room, and generally reaches the walls, where the electrified
252
Convection
FIG. ji.
air clings to the oppositely electrified wall, and is sometimes
not discharged for a long time.
The method of determining by convection currents the
temperature at which water has its maximum density seems
to have been first employed by Hope. He cooled the
middle part of a tall vessel of water by surrounding this part
of the vessel with a freezing mixture. As long as the tempe-
rature is above 40 F. the cooled water descends, and causes
a fall of temperature in a thermometer placed in the lower
part of the vessel. Another thermometer, placed in the
upper part of the vessel, remains stationary. But when the
temperature is below 39 F. the water cooled by the freezing
mixture becomes lighter and ascends, causing the upper
thermometer to fall, while the lower one remains sta-
tionary.
The investigation of the maximum density of water has
been greatly improved by Joule, who also
made use of convection currents. He em-
ployed a vessel consisting of two vertical
cylinders, each 4! feet high and 6 inches
diameter, connected below by a wide tube
with a cock, and above by an open trough
or channel. The whole was filled with water
up to such a level that the water could flow
freely through the channel. A glass specific
gravity bead which would just float in water
was placed in the channel, and served to
indicate any motion of the water in the
channel. The very smallest difference of
density between the portions of water in the
two columns was sufficient to produce a
current, and to move the bead in the
channel.
The cock in the connecting tube being
closed, the temperature of the water in the two tubes was
adjusted, the water well mixed in each tube by stirring,
Maximum Density of Water. 253
and when it had come to rest the temperature of each
column was observed, and the cock was opened. If a cur-
rent was then observed in the channel, it indicated that
the water in the tube towards which the current flowed was
the denser. By finding a pair of different temperatures
at which the density is exactly the same, we may be sure
that one of them is below and the other above the tempe-
rature of maximum density; and by obtaining a series of
such pairs of temperatures of which the difference is smaller
and smaller, Dr. Joule determined the temperature of maxi-
mum density to be 39'! F. within a very small fraction of a
degree.
CHAPTER XVIII.
ON THE DIFFUSION OF HEAT BY CONDUCTION.
WHENEVER different parts of a body are at different tem-
peratures, heat flows from the hotter paits to the neigh-
bouring colder parts. To obtain an FIG
exact notion of conduction, let us
consider a large boiler with a flat
bottom, whose thickness is c. The
fire maintains the lower surface
at the temperature T, and heat
flows upwards through the boiler
plate to the upper surface, which is
in contact with the water at the lower temperature, s.
Let us now restrict ourselves to the consideration of a
rectangular portion of the boiler plate, whose length is a,
its breadth , and its thickness c.
The things to be considered are the dimensions of this
portion of the body, and the nature of the material of which
it is made, the temperatures of its upper and lower surfaces,
and the flow of heat through it as determined by these
254 Diffusion of Heat by Conduction.
conditions. In the first place it is found that when the
difference of the temperatures s and T is not so great as to
make a sensible difference between the properties of the
substance at these two temperatures, the flow of heat is
exactly proportional to the difference of temperatures, other
things being the same.
Let us suppose that when a, b, and c are each equal to
the unit of length, and when T is one degree above s, the
steady flow of heat is such that the quantity which enters
the lower surface or leaves the upper surface in the unit of
time is , then k is defined as the specific thermal con-
ductivity of the substance. To find H, the quantity of heat
which flows in a time / through the portion of boiler plate
whose area is a b, and whose thickness is c, when the lower
surface is kept at a temperature T, and the upper at a
temperature s, till the flow has become steady, divide the
plate into c horizontal layers, the thickness of each layer
being unity, and divide each layer into a b cubes, the sides
of each cube being unity.
Since the flow of heat is steady, the difference of tem-
perature of the upper and lower faces of each cube will
be - - (T s). The flow of heat through each cube will
be - - (T s) in unit of time. Now, in each layer there
are a b such cubes, and the flow goes on for / units of time,
so that we obtain for the whole heat conducted in time /
where a b is the area and c the thickness of the plate, / the
time, T s the difference of temperature which causes the
flow, and k the specific thermal conductivity of the sub-
stance of the plate.
It appears, therefore, that the heat conducted is directly
proportional to the area of the plate, to the time, to the differ-
Measures of Conductivity. 255
ence of temperature, and to the conductivity, and inversely
proportional to the thickness of the plate.
ON THE DIMENSIONS OF k, THE SPECIFIC THERMAL
CONDUCTIVITY.
From the equation we find
Hence if [L] be the unit of length, [T] the unit of time,
[H] the unit of heat, and [@] the unit of temperature, the
FH!
dimensions of k will be r -L- L=p .
[LT0]
The further discussion of the dimensions of k will depend
on the mode of measuring heat and temperature.
(1) If heat is measured as energy, its dimensions are
f L 1^L 1, and those of k become [--,-5-"] This ma 7 be
called the dynamical measure of the conductivity.
(2) If heat is measured in thermal units, such that each
thermal unit is capable of raising unit of mass of a standard
substance through one degree of temperature, the dimen-
sions of H are [M ], and those of k will be [- ] This
may be called the calorimetric measure of the conductivity.
(3) If we take as the unit of heat that which will raise unit
of volume of the substance itself one degree, the dimensions
of H are [L 3 ], and those of k are _| . This may be
called the thermometric measure of the conductivity.
In order to obtain a distinct conception of the flow of
heat through a solid body, let us suppose that at a given
instant we know the temperature of every point of the body.
If we now suppose a surface or interface to be described
within the body such that at every point of this interface the
temperature has a given value T, we may call this interface
250 Diffusion of Heat by Conduction.
the isothermal interface of T. (Of course, when we suppose
this interface to exist in the body, we do not conceive the
body to be altered in any way by this supposition, as if the
body were really cut in two by it.) This isothermal interface
separates those parts of the body which are hotter than
the temperature T from those which are colder than this
temperature.
Let us now suppose the isothermal interfaces drawn for
every exact degree of temperature, from that of the hottest
part of the body to that of the coldest part. These interfaces
may be curved in any way, but no two different interfaces
can meet each other, because no part of the body can at
the same time have two different temperatures. The body
will therefore be divided into layers or shells by these inter-
faces, and the space between two isothermal surfaces differing
by one degree of temperature will be in the form of a thin
shell, whose thickness may vary from one part to another.
At every point of this shell there is a flow of heat from
the hotter surface to the colder surface through the substance
of the shell.
The direction of this flow is perpendicular to the surface
of the shell, and the rate of flow is greater the thinner the
shell is at the place, and the greater its conductivity.
If we draw a line perpendicular to the surface of the shell,
and of length unity, then if c is the thickness of the shell,
and if the neighbouring shells are of nearly the same thick-
ness, this line will cut a number of shells equal to - . This,
then, is the difference of temperature between two points in
the body at unit of distance, measured in the direction of
the flow of heat, and therefore the flow of heat along this
line is measured by , where k is the conductivity.
We can now imagine, with the help of the isothermal inter-
faces, the state of the body at a given instant. Wherever
tnere is inequality of temperature between neighbouring
Conduction in a Solid. 257
parts of the body a flow of heat is going on. This flow is
everywhere perpendicular to the isothermal interfaces, and
the flow through unit of area of one of these interfaces in unit
of time is equal to the conductivity divided by the distance
between two consecutive isothermal interfaces.
The knowledge of the actual thermal state of the body,
and of the law of conduction of heat, thus enables us to
determine the flow of heat at every part of the body. If the
flow of heat is such that the amount of heat which flows into
any portion of the body is exactly equal to that which flows
out of it, then the thermal state of this portion of the body-
will remain the same as long as the flow of heat fulfils this
condition.
If this condition is fulfilled for every part of the body, the
temperature at any point will not alter with the time, the
system of isothermal interfaces will continue the same, and
the flow of heat will go on without alteration, being always
the same at the same part of the body.
This state of things is referred to as the state of steady flow
of heat. It cannot exist unless heat is steadily supplied to
the hotter parts of the surface of the body, from some source
external to the body, and an equal quantity removed from
the colder parts of the surface by some cooling medium, or
by radiation.
The state of steady flow of heat requires the fulfilment at
every part of the body of a certain condition, similar to that
which is fulfilled in the flow of an incompressible fluid.
When this condition is not fulfilled, the quantity of heat
which enters any portion of the body may be greater or less
than that which escapes from it. In the one case heat will
accumulate, and the portion of the body will rise in tempe-
rature. In the other case the heat of the portion will
diminish, and it will fall in temperature. The amount of
this rise or fall of temperature will be measured numerically
by the gain or loss of heat, divided by the capacity for heat
of the portion considered.
s
258 Diffusion of Heat by Conduction.
If the portion considered is unit of volume, and if we
measure heat as in the third method given at p. 255 by the
quantity required to raise unit of volume of the substance,
in its actual state, one degree, then the rise of temperature
of this portion will be numerically equal to the total flow
of heat into it.
We are now able, by means of a thorough knowledge of
the thermal state of the body at a given instant, to determine
the rate at which the temperature of every part must be
changing, and therefore we are able to predict its state in
the succeeding instant. Knowing this, we can predict its
state in the next instant following, and so on.
The only parts of the body to which this method does not
apply are those parts of its surface to which heat is supplied,
or from which heat is abstracted, by agencies external to the
body. If we know either the rate at which heat is supplied
or abstracted at every part of the surface, or the actual tem-
perature of every part of the surface during the whole time,
either of these conditions, together with the original thermal
state of the body, will afford sufficient data for calculating
the temperature of every point during all time to come.
The discussion of this problem is the subject of the great
work of Joseph Fourier, Theorie de la Chaleur. It is not
possible in a treatise of the size and scope of this book to
reproduce, or even to explain, the powerful analytical methods
employed -by Fourier to express the varied conditions, as to
the form of its surface and its original thermal state, to which
the body may be subjected. These methods belong, rather,
to the general theory of the application of mathematics to
physics; for in every branch of physics, when the investiga-
tion turns upon the expression of arbitrary conditions, we
have to follow the method which Fourier first pointed out
in his 'Theory of Heat.'
I shall only mention one or two of the results given by
Fourier, in which the intricacies arising from the arbitrary
conditions of the problem are avoided.
Sketch of Fourier's Theory. 259
The first of these is the case in which the solid is supposed
of infinite extent, and of the same conductivity in every part.
The temperature of every point of this body at a given
time is supposed to be known, and it is required to deter-
mine the temperature of any given point p after a time / has
elapsed.
Fourier has given a complete solution of this problem, of
which we may obtain some idea by means of the following
considerations. Let k be the conductivity, measured by the
third method, in which the unit of heat adopted is that
which will raise unit of volume of the substance one degree ;
then if we make
k t = a ,
a will be a line the length of which will be proportional
to the square root of the time.
Let Q be any point in the body, and let its distance from
p be r. Let the original temperature of Q be 6. Now take
?*_
a quantity of matter proportional to e & and of the
temperature 0, and mix it with portions of matter taken
from every other part of the body, the temperature of each
portion being the original temperature of that point, and
_ r>
the quantity of each portion being proportional to e ***
The mean temperature of all such portions will be the
temperature of the point P after a time /.
In other words, the temperature of p after a time / may
be regarded as in some sense the mean of the original
temperatures of all parts of the body. In taking this mean,
however, different parts are allowed different weights, de-
pending on their distance from p, the parts near p having
more influence on the result than those at a greater dis-
tance.
The mathematical formula which indicates the weight to
be given to the temperature of each part in taking the
mean is a very important one. It occurs in several
s 2
260 Diffusion of Heat by Conduction
branches of physics, particularly in the theory of errors
and in that of the motions of systems of molecules.
It follows from this result that, in calculating the tem-
perature of the point P, we must take into account the
temperature of every other point Q, however distant, and
however short the time may be during which the propaga-
tion of heat has been going on. Hence, in a strict sense,
the influence of a heated part of the body extends to the
most distant parts of the body in an incalculably short time,
so that it is impossible to assign to the propagation of heat
a definite velocity. The velocity of propagation of thermal
effects depends entirely on the magnitude of the effect
which we are able to recognise ; and if there were no limit
to the sensibility of our instruments, there would be no
limit to the rapidity with which we could detect the in-
fluence of heat applied to distant parts of the body. But
while this influence on distant points can be expressed
mathematically from the first instant, its numerical value is
excessively small until, by the lapse of time, the line a has
grown so as to be comparable with r, the distance of P from
Q. If we take this into consideration, and remember that it
is only when the changes of temperature are comparable with
the original differences of temperature that we can detect
them with our instruments, we shall see that the sensible
propagation of heat, so far from being instantaneous, is an
excessively- slow process, and that the time required to
produce a similar change of temperature in two similar
systems of different dimensions is proportional to the
square of the linear dimensions For instance, if a red-hot
ball of four inches diameter firjd into a sandbank has in an
hour raised the temperature of the sand six inches from its
centre 10 R, then a red-hot ball of eight inches diameter
would take four hours to raise the temperature of the sand
twelve inches from its centre by the same number of degrees.
This result, which is very important in practical questions
about the time of cooling or heating of bodies of any form,
in an Infinite Solid. 261
may be deduced directly from the consideration of the
dimensions of the quantity k namely, the square of a length
divided by a time. It follows from this that if in two un-
equally heated systems of similar form but different dimen-
sions the conductivity and the temperature are the same at
corresponding points at first, then the process of diffusion of
heat will go on at different rates in the two systems, so that
if for each system the time be taken proportional to the
square of the linear dimensions, the temperatures of corre-
sponding points will still be the same in both systems.
The method just described affords a complete determina-
tion of the temperature of any point of a homogeneous
infinite solid at any future time, the temperature of every
point of the solid being given at the instant from which we
begin to count the time. But when we attempt to deduce
from a knowledge of the present thermal state of the body
what must have been its state at some past time, we find
that the method ceases to be applicable.
To make this attempt, we have only to make /, the
symbol of the time, a negative quantity in the expressions
given by Fourier. If we adopt the method of taking the
mean of the temperatures of all the particles of the solid, each
particle having a certain weight assigned to it in taking the
mean, we find that this weight, according to the formula, is
greater for the distant particles than for the neighbouring ones,
a result sufficiently startling in itself. But when we find
that, in order to obtain the mean, after taking the sum of
the temperatures multiplied by their proper factors, we have
to divide by a quantity involving the square root of /,
the time, we are assured that when / is taken negative the
operation is simply impossible, and devoid of any physical
meaning, for the square root of a negative quantity, though
it may be interpreted with reference to some geometrical
operations, is absolutely without meaning with reference to
time.
It appears, therefore, that Fourier's solution of this
262
Diffusion of Heat by Conduction.
problem, though complete considered with reference to future,
time, fails when we attempt to discover the state of the
body in past time.
Tn the diagram fig. 33 the curves show the distribution of
FIG. 33
SCALE OF TEMPERATURE
temperature in an infinite mass at different times, after the
sudden introduction of a hot horizontal stratum in the
midst of the infinite solid. The temperature is indicated by
the horizontal distance to the right of the vertical line, and
Harmonic Distribution of Temperature. 263
the hot stratum is supposed to have been introduced at tlie
middle of the figure.
The curves indicate the temperatures of the various strata
one hour, four hours, and sixteen hours after the intro-
duction of the hot stratum. The gradual diffusion of the
heat is evident, and also the diminishing rate of diffusion as
its extent increases.
The problem of the diffusion of heat in an infinite solid
does not present those difficulties which occur in problems
relating to a solid of definite shape. These difficulties
arise from the conditions to which the surface of the solid
may be subjected, as, for instance, the temperature may be
given over part of the surface, the quantity of heat supplied
to another part may be given, or we may only know that
the surface is exposed to air of a certain temperature.
The method by which Fourier was enabled to solve many
questions of this kind depends on the discovery of har-
monic distributions of heat.
Suppose the temperatures of the different parts of the body
to be so adjusted that when the body is left to itself under
the given conditions relating to the surface, the tempera-
tures of all the parts converge to the final temperature,
their differences from the final temperature always preserv-
ing the same proportion during the process ; then this
distribution of temperature is called an harmonic dis-
tribution. If we suppose the final temperature to be taken
as zero, then the temperatures in the harmonic distribution
diminish in a geometrical progression as the times increase
in arithmetical progression, the ratio of cooling being the
same for all parts of the body.
In each of the cases investigated by Fourier there may
be an infinite series of harmonic distributions. One of
these, which has the slowest rate of diminution, may be
called the fundamental harmonic; the rates of diminution
of the others are proportional to the squares of the natural
numbers.
264 Diffusion of Heat by Conduction.
If the body is originally heated in any arbitrary manner,
Fourier shows how to express the original temperature as the
sum of a series of harmonic distributions. When the body
is left to itself the part depending on the higher harmonics
rapidly dies away, so that after a certain time the distribu-
tion of heat continually approximates to that due to the
fundamental harmonic, which therefore represents the law
of cooling of a body after the process of diffusion of heat
has gone on for a long time.
Sir William Thomson has shown, in a paper published in
the ' Cambridge and Dublin Mathematical Journal ' in 1844,
how to deduce, in certain cases, the thermal state of a body
in past time from its observed condition at present.
For this purpose, the present distribution of temperature
must be expressed (as it always may be) as the sum of a
series of harmonic distributions. Each of these harmonic
distributions is such that the difference of the temperature of
any point from the final temperature diminishes in a geo-
metrical progression as the time increases in arithmetical
progression, the ratio of the geometrical progression being
the greater the higher the degree of the harmonic.
If we now make / negative, and trace the history of the
distribution of temperature up the stream of time, we shall
find each harmonic increasing as we go backwards, and the
higher harmonics increasing faster than the lower ones.
If the present distribution of temperature is such that it
may be expressed in a finite series of harmonics, the distri-
bution of temperature at any previous time maybe calculated;
but if '(as is generally the case) the series of harmonics is
infinite, then the temperature can be calculated only when
this series is convergent. For present and future time it is
always convergent, but for past time it becomes ultimately
divergent when the time is taken at a sufficiently remote
epoch. The negative value of /, for which the series becomes
ultimately divergent, indicates a certain date in past time
such that the present state of things cannot be deduced from
Steady and Periodic Flow of Heat. 265
any distribution of temperature occurring previously to that
date, and becoming diffused by ordinary conduction. Some
other event besides ordinary conduction must have occurred
since that date in order to produce the present state of things.
This is only one of the cases in which a consideration of
the dissipation of energy leads to the determination of a
superior limit to the antiquity of the observed order of
things.
A very important clas? of problems is that in which there
is a steady flow of heat into the body at one point of
its surface, and out of it at another part. There is a
certain distribution of temperature in all such cases, which
if once established will not afterwards change: this is
called the permanent distribution. If the original distri-
bution differs from this, the effect of the diffusion of heat will
be to cause the distribution of temperature to approximate
without limit to this permanent distribution. Questions
relating to the permanent distribution of temperature and
the steady flow of heat are in general less difficult than
those in which this state is not established.
Another important class of problems is that in which heat
is supplied to a portion of the surface in a periodic manner,
as in the case of the surface of the earth, which receives and
emits heat according to the periods of day and night, and
the longer periods of summer and winter.
The effect of such periodic changes of temperature at the
surface is to produce waves of heat, which descend into the
earth and gradually die away. The length of these waves is
proportional to the square root of the periodic time. If we
examine the wave at a depth such that the greatest heat
occurs when it is coldest at the surface, then the extent of
the variation of temperature at this depth is only -% of its
value at the surface. In the rocks of this country this depth
is about 25 feet for the annual variations.
In the diagram fig. 34 the distribution of temperature in
the different strata is represented at two different times. Il
266
Diffusion of Heat by Conduction.
we suppose the figure to represent the diurnal variation of
temperature, then the curves indicate the temperatures at
FIG. 34.
SURFACE
0.12!,
2 A.M. and 8 A.M. If we suppose it to represent the annual
variation, then the curves correspond to January and ApriL
Underground Temperature. 267
Since the depth of the wave varies as the square root of the
periodic time, the wave-length of the annual variation of
temperature will be about nineteen times the depth of those
of the diurnal variation. At a depth of about 50 feet the
variation of annual temperature is about a year in arrear.
The actual variation of temperature at the surface does
not follow the law which gives a simple harmonic wave, but,
however complicated the actual variation may be, Fourier
shows how to decompose it into a number of harmonic
waves of which it is the sum. As we descend into the earth
these waves die away, the shortest most rapidly, so that we
lose the irregularities of the diurnal variation in a few inches,
and the diurnal variation itself in a few feet. The annual
variation can be traced to a much greater depth ; but at
depths of 50 feet and upwards the temperature is sensibly
constant throughout the year, the variation being less than
the five-hundredth part of that at the surface.
But if we compare the mean temperatures at different
depths, we find that as we descend the mean temperature
rises, and that after we have passed through the upper strata,
in which the periodic variations of temperature are observed,
this increase of temperature goes on as we descend to the
greatest depths known to man. In this country the rate of
increase of temperature appears to be about i F. for 50
feet of descent.
The fact that the strata of the earth are hotter oelow than
above shows that heat must be flowing through them from
below upwards. The amount of heat which thus flows
upwards in a year through a square foot of the surface can
easily be found if we know the conductivity of the substance
through which it passes. For several kinds of rock the
conductivity has been ascertained by means of experiments
made upon detached portions of the rock in the laboratory.
But a still more satisfactory method, wh ere it can be employed,
is to make a register of the temperature at different depths
throughout the year, and from this to determine the length
268 Diffusion of Heat by Conduction.
of the annual wave of temperature, or its rate of decay.
From either of these data the conductivity of the substance
of the earth may be found without removing the rocks from
their bed.
By observations of this kind made at different points of
the earth's surface we might determine the quantity of heat
which flows out of the earth in a year. This can be done
only roughly at present, on account of the small number of
places at which such observations have been made, but we
know enough to be certain that a great quantity of heat
escapes from the earth every year. It is not probable that
any great proportion of this heat is generated by chemical
action within the earth. We must therefore conclude that
there is less heat in the earth now than in former periods of
its existence, and that its internal parts were formerly very
much hotter than they are now.
In this way Sir W. Thomson has calculated that, if no
change has occurred in the order of things, it cannot have
been more than 200,000,000 years since the earth was in
the condition of a mass of molten matter, on which a solid
crust was just beginning to form.
ON THE DETERMINATION OF THE THERMAL CONDUCTIVITY
OF BODIES.
The most obvious method of determining the conduc-
tivity of a substance is to form it into a plate of uniform
thickness, to bring one of its surfaces to a known tempera-
ture and the other to a known lower temperature, and to
determine the quantity of heat which passes through the
plate in a given time.
For instance, if we could bring one surface to the tem-
perature of boiling water by a current of steam, and keep
the other at the freezing temperature by means of ice, we
might measure the heat transmitted either by the quantity
of steam condensed, or by the quantity of ice melted.
Measurenmt of Conductivity. 269
The chief difficulty in this method is that the surface of
the plate does not acquire the temperature of the steam 01
the ice with which it is in contact, and that it is difficult to
ascertain its real temperature with the accuracy necessary
for a determination of this kind.
Most of the actual determinations of conductivity have
been made in a more indirect way by observing the per-*
manent distribution of temperature in a bar, one end of
which is maintained at a high temperature, while the rest
of its surface is exposed to the cooling effects of the atmo-
sphere.
The temperatures of a series of points in the bar are
ascertained by means of thermometers inserted into holes
drilled in it, and brought into thermal connexion with its
substance by means of fluid metal surrounding the bulbs.
In this way the rate of diminution of temperature with
the distance can be ascertained at various points on the bar.
To determine the conductivity, we must compare the
rate of variation of temperature with the flow of heat which
is due to it. It is in the determination of this flow of heat
that the indirectness of the metho \ consists. The most
trustworthy method of determining the flow of heat is that
employed by Principal Forbes in his experiments on the
conduction of heat in an iron bar. ! He took a bar of exactly
the same section and material as the experimental bar, and,
after heating it uniformly, allowed it to cool in air of the
same temperature as that surrounding the experimental bar.
By observing the temperature of the cooling bar at frequent
intervals of time, he ascertained the quantity of heat which
escaped from the sides of the bar, this heat being measured
in terms of the quantity of heat required to raise unit of
volume of the bar one degree. This loss of heat depended
of course on the temperature of the bar at the time, and a
table was formed showing the loss from a linear foot of the
bar in a minute at any temperature.
1 Trans. Roy. Sec. Edinb. 1861-2.
2/o Diffusion of Heat by Conduction.
Now, in the experimental bar the temperature of every
part was known, and therefore the loss of heat from any
given portion of the bar could be found by making use of
the table. To determine the flow of heat across any par-
ticular section, it was necessary to sum up the loss of heat
from all parts of the bar beyond this section, and when this
was done, by comparing the flow of heat across the section
with the rate of diminution of temperature per linear foot
in the curve of temperature, the conductivity of the bar
for the temperature of the section was ascertained. Prin-
cipal Forbes found that the thermal conductivity of iron
decreases as the temperature increases.
The conductivity thus determined is expressed in terms
of the quantity of heat required to raise unit of volume of
the substance one degree. If we wish to express it in the
ordinary way in terms of the thermal unit as denned with
reference to water at its maximum density, we must
multiply our result by the specific heat of the substance,
and by its density ; for the quantity of heat required to
raise unit of mass of the substance one degree is its specific
heat, and the number of units of mass in unit of volume is
the density of the substance.
As long as we are occupied with questions relating to the
diffusion of heat and the waves of temperature in a single
substance, the quantity on which the phenomena depend
is the thermometric conductivity expressed in terms of the
substance itself; but whenever we have to do with the
effects of the flow of heat upon other bodies, as in the case
of boiler plates, steam-condensers, &c., we must use a
definite thermal unit, and express the calorimetric con-
ductivity in terms of it. It has been shown by Professor
Tyndall that the wave of temperature travels faster in bis-
muth than in iron, though the conductivity of bismuth is
much less than that of iron. The reason is that the
thermal capacity of the iron is much greater than that of an
equa] volume of bismuth.
Conductivity of various Substances. 271
Forbes was the first to remark that the order in which
the metals follow one another in respect of thermal con-
ductivity is nearly the same as their order as regards electric
conductivity. This remark is an important one as regards
certain metals, but it must not be pushed too far; for
there are substances which are almost perfect insulators ot
electricity, whereas it is impossible to find a substance
which will not transmit heat.
The electric conductivity of metals diminishes as the
temperature rises. The thermal conductivity of iron also
diminishes, but in a smaller ratio, as the temperature rises.
Professor Tait has given reasons for believing that the
thermal conductivity of metals may be inversely proportional
to their absolute temperature.
The electric conductivity of most non-metallic substances,
and of all electrolytes and dielectrics, increases as the tem-
perature rises. We have not sufficient data to determine
whether this is the case as regards their thermal conduc-
tivity. According to the molecular theory of Chapter XXII.
the thermal conductivity of gases increases as the tempera-
ture rises.
ON THE CONDUCTIVITY OF FLUIDS.
It is very difficult to determine the thermal conductivity of
fluids, because the variation of temperature which is part of
the phenomenon produces a variation of density, and unless
the surfaces of equal temperature are horizontal, and the upper
strata are the warmest, currents will be produced in the fluid
which will entirely mask the phenomena of true conduction.
Another difficulty arises from the fact that most fluids
have a very small conductivity compared with solid bodies. 1
Hence the sides of the vessel containing the fluid are often
the principal channel for the conduction of heat.
In the case of gaseous fluids the difficulty is increased by
the greater mobility of their pans, and by the great variation
[ l The conductivity of water is about '0014 of that of copper. R.]
272 Diffusion of Heat by Conduction.
of density with change of temperature. Their conductivity
is extremely small, and the mass of the gas is generally small
compared with that of the vessel in which it is contained.
Besides this, the effect of direct radiation from the source
of heat through the gas on the thermometer produces a
heating effect which may, in some cases, completely mask
the effect of true conduction. For all these reasons, the
determination of the thermal conductivity of a gas is an
investigation of extreme difficulty. (See Appendix.)
APPLICATIONS OF THE THEORY.
The great thermal conductivity of the metals, especially
of copper, furnishes the means of producing many thermal
effects in a convenient manner. For instance, in order
to maintain a body at a high temperature by means of a
source of heat at some distance from it, a thick rod of copper
may be used to conduct the heat from the source to the
body we wish to heat ; and when it is desired to warm the
air of a room by means of a hot pipe of small dimensions,
the effect may be greatly increased by attaching copper
plates to the pipe, which become hot by conduction, and
expose a great heating surface to the air.
To ensure an exact equality of temperature in all the
parts of a body, it may be placed in a closed chamber formed
of thick sheet copper. If the temperature is not quite
uniform outside this chamber, any difference of temperature
between one part of the outer surface and another will
produce such a flow oi heat in the substance of the copper
that the temperature of the inner surface will be very nearly
uniform. To maintain the chamber at a uniform high tem-
perature by means of a flame, as is sometimes necessary, it
may be placed in a larger copper chamber, and so suspended
by strings or supported on legs that very little heat can
pass by direct conduction from the outer to the inner waii.
Thus we have first an outer highly conducting shell of copper;
Chamber of Uniform Temperature. 273
next a slowly conducting shell of air, which, however, tends
to equalize the temperature by convection ; then another
highly conducting shell of copper ; and lastly the inner
chamber. The whole arrangement facilitates the flow of
heat parallel to the walls of the chambers, and checks its
flow perpendicular to the walls. Now differences of tempe-
rature within the chamber must arise from the passage of
heat from without to within, or in the reverse direction, and
the flow of heat along the successive envelopes tends only
to equalize the temperature. Hence, by the arrangement of
successive shells, alternately of highly conducting and slowly
conducting matter, and still more if the slowly conducting
matter is fluid, an almost complete uniformity of temperature
may be maintained within the inner chamber, even when the
outer chamber has all the heat applied to it at one point.
This arrangement was employed by M. Fizeau in his
researches on the dilatation of bodies by heat.
CHAPTER XIX.
ON THE DIFFUSION OF FLUIDS.
THERE are many liquids which, when they are intermingled
by being stirred together, remain mixed, and, though their
densities are different, they do not separate from each other
as oil and water do. When liquids which are capable of
being permanently mixed are placed in contact with each
other, the process of mixture goes on in a slow and gradual
manner, and continues till the composition of the mixture is
the same in every part.
Thus if we put a strong solution of any salt in the lower
part of a tall glass jar, we may, by pouring water in a gentle
stream on a small wooden float, fill up the jar with water
without disturbing the solution. The process of diffusion
will then go on between the water and the solution, and will
T
274 Diffusion of Matter.
continue for weeks or months, according to the nature of
the salt and the height of the jar.
If the solution of the salt is strongly coloured, as in the
case of sulphate of copper, bichromate of potash, &c., we
may trace the process of diffusion by the gradual rise of the
colour into the upper part of the jar, and the weakening of
the colour in the lower part. A more exact method is that
employed by Sir William Thomson, of placing a number of
glass bubbles or beads in the jar, whose specific gravities
are intermediate between that of the strong solution and
that of water. At first the beads all float in the surface of
separation between the two liquids, but as diffusion goes on
they separate from each other, and indicate by their positions
the specific gravity of the mixture at various depths. It is
necessary to expel the air very thoroughly from both liquids
by boiling before commencing this experiment. If this is
not done, air separates from the liquids, and attaches itself
in the form of small bubbles to the specific gravity beads, so
that they no longer indicate the true specific gravity of the
fluid in which they float. In order to determine the strength
of the solution at any point, as indicated by one of the
beads, we have only to measure the amount of the salt
which must be added to a known quantity of pure water, in
order to make the bead swim in the mixture.
Voit has investigated the process of diffusion of a solution
of sugar by passing a ray of plane polarized light horizontally
through the liquid at various depths. The solution of sugar
causes the plane of polarization to rotate through a certain
angle, and from this angle the percentage of sugar in any
given stratum of the fluid can be determined without disturb-
ing the vessel.
There are many pairs of liquids which do not diffuse into
each other, and there are others in which the diffusion, after
going on for some time, stops as soon as a certain small
proportion of the heavier liquid has become mixed with the
lighter, and a small proportion of the lighter has become
mixed with the heavier.
Law of Diffusion. 275
In the case of gases, however, there is no such limitation.
Every gas diffuses into every other gas, so that, however
different the specific gravities of two gases may be, it is
impossible to keep them from mixing if they are placed in
the same vessel, even when the denser gas is placed below
the rarer.
[Since the distinction between gases and liquids is not
absolute, we may infer that the latter, as well as the former,
will mix in all proportions if the temperature be high enough.
Even short of the critical temperatures, heat is found to
promote solubility.
If two liquids which do not sensibly mix e.g., bisulphide
of carbon and water are in equilibrium in a closed vessel,
every cubic inch of the space not occupied by liquid con-
tains as much of the vapour of each constituent as if the
other had been absent, and the resultant pressure is the
sum of those due (at the actual temperature) to the separate
constituents. The boiling-point that is, the temperature at
which a bubble at the interface of the two liquids will
acquire the atmospheric pressure is thus lower for the
association of the two liquids than for either of them
separately.
If the liquids mix in some proportions, but not in others,
the result of shaking them together will depend upon the
proportions taken. Thus, in the case of ether and water,
if the ether be more than ^th, and the water more than
sVth, there will be separation into two layers, each of
definite composition (at a given temperature), but the
relative amounts of the two layers will depend upon the
proportion originally chosen. If, however, the original
proportion be more extreme in either direction than those
above specified, there will be no separation into two layers
that is, the composition will be uniform throughout.
The relation between the percentage composition of the
vapour and that of the liquid can only be fully determined
by special experiment, but its general character may be
T2
276 Diffusion of Matter.
sketched beforehand. Let us trace the course of things
as the proportion of ether increases. At first, when the
percentage of ether in the liquid is infinitesimal, so is
the percentage in the vapour. Both increase up to the
point at which the liquid begins to separate into two
layers. From this point onwards the composition of the
vapour remains constant, until from deficiency of water the
second point is reached where the liquid forms one mixture
only. At this stage the vapour becomes richer in ether,
until, finally, water disappears simultaneously from liquid
and vapour.
In the case of alcohol and water, which mix in all pro-
portions, the vapour and liquid become continuously richer
together.
With the aid of a third liquid e.g., alcohol two others,
ether and water, may be mixed in proportions that would
not otherwise be possible. The theory of such ternary
combinations has been given by Sir G. Stokes (' Proc. Roy.
Soc.,' vol. xlix. p. 174, 1891). R.]
The fact of the diffusion of gases was first remarked by
Priestley. The laws of the phenomena were first investigated
by Graham. The rate at which the diffusion of any substance
goes on is in every case proportional to the rate of variation
of the strength of that substance in the fluid as we pass
along the line in which the diffusion takes place. Each
substance in the mixture flows from places where it exists in
greater quantity to places where it is less abundant.
The law of diffusion of matter is therefore of exactly the
same form as that of the diffusion of heat by conduction,
and we can at once apply all that we know about the con-
duction of heat to assist us in understanding the phenomena
of the diffusion of matter.
To fix our ideas, let us suppose the fluid to be contained
in a vessel with vertical sides, and let us consider a horizontal
stratum of the fluid of thickness c. Let the composition of
the fluid at the upper surface of this stratum be denoted by
Law of Diffusion. 277
A, and that of the fluid at the lower surface of the stratum
by B.
The effect of the diffusion which goes on in the stratum
will be the same as if a certain volume of fluid of composition
A had passed downwards through the stratum while an equal
volume of fluid of composition B had passed upwards through
the stratum at the same time.
Let d be the thickness of the stratum which either of these
equal volumes of fluid would form in the vessel, then d is
evidently proportional :
i st. To the time of diffusion.
2nd. Inversely to the thickness of the stratum through
which the diffusion takes place.
3rd. To a coefficient depending on the nature of the
interdiffusing substances. Hence if t is the time of dif-
fusion and k the coefficient of diffusion,
d = k! f or k= c .
We thus find that the dimensions of k, the coefficient of
diffusion, are equal to the square of a length divided by
a time.
Hence, in the experiment with the jar, the vertical
distance between strata of corresponding densities, as indi-
cated by the beads which float in them, varies as the square
root of the time from the beginning of the diffusion.
When the mixture of two liquids or gases is effected in a
more rapid manner by agitation or stirring, the only effect
of the mechanical disturbance is to increase the area of the
surfaces through which diffusion takes place. Instead of
the surface of separation being a single horizontal plane, it
becomes a surface of many convolutions, and of great
extent, and m order to effect a complete mixture the dif-
fusion has to extend only over the distance between the
successive convolutions of this surface instead of over half
the depth of the vessel.
2? 8 Diffusion of Matter.
Since the time required for diffusion varies as the square
of trie distance through which the diffusion takes place, it
is easy to see that by stirring the solution in a jar along
with the water above it, a complete mixture may be effected
in a few seconds, which would have required months if the
jar had been left undisturbed. That the mixture effected
by stirring is not instantaneous may be easily seen by
observing that during the operation the fluid appears to
be full of streaks, which cause it to lose its transparency.
This arises from the different indices of refraction of different
portions of the mixture, which have been brought near each
other by stirring. The surfaces of separation are so drawn
out and convoluted that the whole mass has a woolly
appearance, for no ray of light can pass without being
turned many times out of its path.
The same appearance may also be observed when we
mix hot water with cold, and even when very hot air is
mixed with cold air. This shows that what is called the
equalization of temperature by convection currents really
takes place by conduction between portions of the substance
brought near each other by the currents.
If we observe the process of diffusion with our most
powerful microscopes, we cannot follow the motion of any
individual portions of the fluids. We cannot point out one
place in which the lower fluid is ascending, and another in
which the upper fluid is descending. There are no currents
visible to us, and the motion of the material substances goes
on as imperceptibly as the conduction of heat or of elec-
tricity. Hence the motion which constitutes diffusion
must be distinguished from those motions of fluids which
we can trace by means of floating motes. It may be de-
scribed as a motion of the fluids, not in mass, but by mole-
cules.
We have not hitherto taken any notice of molecular
theories, because we wish to draw a distinction between
that part of our subject which depends only on the
Molecular Motion. 279
universal axioms of dynamics, combined with observa-
tions of the properties of bodies, and the part which en-
deavours to arrive at an explanation of these properties by
attributing certain motions to minute portions of matter
which are as yet invisible to us.
The description of diffusion as a molecular motion is
one which we shall justify when we come to treat of
molecular science. At present, however, we shall use the
phrase ' molecular motion ' as a convenient mode of de-
scribing the transference of a fluid when the motion of
sensible portions of the fluid cannot be directly observed.
Graham observed that the diffusion both of liquids and
gases takes place through porous solid bodies, such as
plaster of Paris and pressed plumbago, at a rate not very much
less than when no such body is interposed, and this even
when the solid division is amply sufficient to check all
ordinary currents, and even to support considerable differ-
ences of pressure on its opposite sides.
By taking advantage of the different velocities with which
different liquids and gases pass through such substances,
he was enabled to effect many important analyses and
to arrive at new views of the constitution of various
bodies.
But there is another class of cases in which a liquid or
gas can pass through a diaphragm which is not in the
ordinary sense porous. For instance, when carbonic acid
gas is confined in a soap-bubble, it gradually escapes. The
liquid absorbs the gas at its inner surface, where it has the
greatest density ; and on the outside, where the density of
the carbonic acid is less, the gas diffuses out into the atmo-
sphere. During the passage of the gas through the film it is
in the state of solution in water. It is also found that hydrogen
and other gases can pass through a layer of caoutchouc.
The ratios in which different gases pass through this substance
are different from the ratios in which they percolate through
porous plugs. Graham shows that the chemical relations
280 Diffusion of Matter.
between the gases and the caoutchouc determine these
ratios, and that it is not through pores in the ordinary sense
that the motion takes place.
According to Graham's theory, the caoutchouc is a colloid
substance that is, one which is capable of being united, in a
temporary and very loose manner, with various proportions
of other substances, just as glue will form a jelly with
various proportions of water. Another class of substances,
which Graham calls crystalloid, are distinguished from these
by being always of definite composition, and not admitting
of these temporary associations. When a colloid substance
has in different parts of its mass different proportions of
water, alcohol, or solutions of crystalloid bodies, diffusion
takes place through the colloid substance, although no part
of it can be shown to be in a liquid state.
On the other hand, a solution of a colloid substance is
almost incapable of diffusion through a porous solid, or
through another colloid substance. Thus, if a solution of
gum in water containing salt be placed in contact with a
solid jelly of gelatine containing alcohol, salt and water
will be diffused into the gelatine, and alcohol will be diffused
into the gum, but there will be no mixture of the gum and
the gelatine.
There are certain metals whose relation to certain gases
Graham explained by this theory. For instance, hydrogen
can be made to pass through iron and palladium at a high
temperature, and carbonic oxide can be made to pass
through iron. The gases form colloidal unions with the
metals, and are diffused through them just as water is diffused
through a jelly.
Graham made many determinations of the relative diffu-
sibility of different salts. Accurate determinations of the
coefficient of diffusion of- liquids and gases are very much
wanted, as they furnish important data for the molecular
theory of these bodies. The most valuable determinations
of this kind are those of the coefficient of diffusion between
Capillarity. 281
pairs of simple gases made by Professor J. Loschmidt of
Vienna. 1
He has determined the coefficient of diffusion in square
metres per hour for ten pairs of the most important gases.
We shall consider these results when we come to the mole-
cular theory of gases.
CHAPTER XX.
CAPILLARITY.
WE have hitherto considered the energy of a body as
depending only on its temperature and its volume. The
whole of the energy of gases, and the most important part of
the energy of liquids, may be expressed in this way, but a
very important part of the energy of a solid body may
depend on the form which it is compelled to assume as
well as on its volume. We shall return to this subject
when treating of Elasticity and Viscosity, but we shall con-
sider at present that part of the energy of a liquid which
depends on the nature and extent of its surface.
Ii? many cases two substances when placed in contact do
not diffuse into each other, and when we attempt to mix
them they separate from each other when left to themselves.
Thus, if we mix water with alcohol the liquids diffuse into
each other. If we now attempt to mix oil with the alcohol
and water, the two liquids separate from each other of them-
selves, and in the act of separation sufficient force is brought
into play to set in motion considerable masses of the fluids,
especially when, as in Plateau's experiments, the mixture of
alcohol and water is of the same density as the oil.
1 Experimental-Untersuchungen tiber die Diffusion von Gasen ohne
porose Scheidewande. Sitzb. d. k. Akad. d. Wissensch. Rd. Ixi-
\ March and July 1870.) (See Appendix.)
282 Capillarity.
The work required to produce these motions must be
derived from the system itself, as no work is done on it by
external agency.
The system of two fluids must therefore have more
energy when the fluids are mixed than when they are sepa-
rated.
Now the only difference between these two states is one
of arrangement ; a greater number of particles of either fluid
being close to the surface of separation when the fluids are
mixed than when they are separate.
We therefore conclude that the energy of a particle of
either fluid is greater when it is very close to the surface
of that fluid than when it is at a greater distance from the
surface. It is probable that it is only within a distance of
a thousandth of a millimetre or less from the surface that this
increase of energy is sensible.
One effect of this will be that the particles near the sur-
face will be drawn inwards towards the mass of their own
fluid ; but as this force acts equally on all the surface par-
ticles, it will only increase the internal pressure by a constant
quantity, and no visible effect will be produced.
We may calculate the whole energy of the system of two
fluids if we know their arrangement. Each fluid occupies
the same total volume in whatever way it is arranged ; and
if the energy of every particle were the same, the total energy-
would not depend on the arrangement.
Since, however, the particles in a very thin stratum close
to the surface of separation have greater energy than those
in the interior of the fluid mass, the excess of energy due to
this cause will be proportional to the total area of the sur-
face of separation.
Hence the energy of the system consists of two parts : the
first depends on the volume, temperature, &c., of the fluids,
and is unaffected by the form of their surface. The second
is proportional to the area of the surface separating the
two fluids.
Capillarity. 283
It is on this second part of the energy that the phenomena
of what is called capillary attraction depend.
In the case of a soap-bubble the energy is greater the
greater the extent of surface exposed to air. The amount of
this energy for a soap-bubble at ordinary temperatures is,
according to Plateau, about 5*6 gramme-metres per square
metre in gravitation units. This is the amount of work
required to blow a soap-bubble whose superficial extent is
one square metre. As the soap-bubble has two surfaces
exposed to air, the energy of a single surface is only 2*8
gramme-metres per square metre.
We shall call this the superficial energy of the soap-
bubble. It is measured by the energy in unit of surface,
and its dimensions when expressed in dynamical measure
are therefore :
energy __ L 2 M i _ M
area x 2 I*~ ~" x 2 '
or it is of one dimension as regards mass, and of two dimen-
sions inversely as regards time, and it is independent of the
unit of length. Superficial energy depends on the nature
of both the media of which the surface is a boundary.
The media must be such as do not mix with each other,
otherwise diffusion occurs, and the surface of separation
becomes indefinite ; but there is a coefficient of superficial
energy^ for every surface which separates two liquids which
do not mix a liquid and a gas, or its own vapour ; and for
he surface which separates a liquid and a solid, whether it
dissolves the solid or not. There is also a coefficient of
superficial energy for the surface separating a gas and a
solid, or two solids ; but as any two gases diffuse into each
other, they can have no surface of separation.
Superficial Tension.
When the area of the surface is increased in any way, work
must be done ; and when the surface is allowed to contract,
284 Capillarity.
it does work on other bodies. Hence it acts like a stretched
sheet of india-rubber, and exerts a tension of the same kind.
The only difference is, that the tension in the sheet of
india-rubber depends on the amount of stretching, and may
be greater in one direction than in a direction at right
angles to it, whereas the tension in the soap-bubble remains
the same however much the film is extended, and the tension
at any point is the same in all directions.
If we draw a straight line, P Q, across the surface A B D c,
and if the whole tension exerted by
the surface across the line P Q is
F, then the superficial tension is
measured by the tension across unit
of length of the line P Q ; or, since *
is the tension across the whole line,
if T is the superficial tension across
unit of length,
F = T. PQ.
Now let us suppose that the lines A B and c D were
originally in contact, and that the surface A B D c was
produced by drawing c D away from A B by the action of
the force F.
If we suppose A B and B c to be rods wet with soapsuds,
placed between two parallel rods A c and B D and then
drawn asunder, the soap film A B D c will be formed. If s
is the superficial energy of the film per unit of area, then
the work done in drawing it out will be s . A B . A c. But if
F is the force required to draw A B from c D, the same work
may be written F.AC, or, putting for F its value in terms of
T, and equating the two expressions for the work,
S . AB . AC = T . PQ . AC
Or = T . AB . AC.
Hence
s = T,
or the numerical value of the superficial energy per unit
of area is equal to that of the superficial tension per unit of
Superficial Tension. 285
length. This quantity is usually called the Coefficient of
Capillarity, because it was first considered with reference to
the ascent of liquids in capillary tubes. These tubes de-
rived their name from the smallness of their bore, which
would only admit a hair (capilla). I have used the phrases
1 superficial energy ' and ' superficial tension because I think
they help us to direct our attention to the facts, and to
understand the various phenomena of liquid surfaces better
than a name which is purely technical, and which has
already done a great deal of harm when used without being
understood. If by the help of this treatise, or otherwise, any-
one has obtained a clear conception of the real phenomena
called Capillary Attraction and Capillarity, he may use
these words quite freely. The theory as we shall state it
does not differ essentially from that originally given by
Laplace, though by the free use of the idea of superficial
tension we avoid some of the mathematical operations
which are required to deduce the phenomena from the
hypothesis of molecular attractions.
We shall now suppose that the superficial tension is
known for the surfaces which bound every pair of the
media with which we have to do. For instance, we may
denote by T ofc the superficial tension of the surface which
separates the medium a from the medium b.
Let there be three fluid media, a, b, c, and let the surface of
separation between a and b meet the surface of separation
between b and c along a line of any form having continuous
curvature. Let o be a point in this line, and let the plane
of the paper represent a section perpendicular to the line.
The three tensions i ab , i bc , and i ca must be in equili-
brium along this line, and since we know these tensions,
we can easily determine the angles which they make with
each other. In fact, if we construct a triangle ABC having
lines proportional to these tensions for its sides, the exterior
angles of this triangle will be equal to the angles formed by
the three surfaces of separation which meet in a line.
t86
Capillarity.
By trigonometry, if A B c are the angles of the edges
formed by the media a b c, then
sm A
sm B
sn c
It appears from this that whenever three fluid media are
J n contact and in equilibrium, the angles between their
FIG 36.
Toft
B T,
surfaces of separation depend only on the values of the
superficial tensions of these three surfaces, and are there-
fore always the same for the same three fluids.
But it is not always possible to construct a triangle with
three given lines as its sides. If any one of the lines is
greater than the sum of the other two, the triangle cannot be
formed. ' For the same reason, if any one of the three super-
ficial tensions is greater than the sum of the other two, the
three fluids cannot be in equilibrium in contact.
For instance, if the tension of the surface separating air
and water is greater than the sum of the tensions of the
surfaces separating air and oil, and oil and water, then a
drop of oil cannot be in equilibrium on the surface of water.
The edge of the drop, where the oil meets the air and the
water, becomes thinner and thinner ; but even when the
angle is reduced to the thinnest edge, the tension of the free
Angles of Contact of Three Fluids. 287
surface of the water exceeds the tensions of the two surfaces
of the oil, so that the oil is drawn out thinner and thinner,
till it covers a vast expanse of water. In fact, the process
may go on till the oil becomes so thin, and contains so
small a number of molecules in its thickness, that it no
longer has the properties of the liquid in mass.
[There is no case known in which the triangle of tensions
is possible. The liquid of intermediate tension always
spreads upon the interface of the liquids of greatest and
least tensions. When a drop of oil stands upon water, it
is because the surface of the water is already coated with a
thin skin of oil. At one time the case of mercury, water
and air was regarded as an exception to the above rule, laid
down by Marangoni. But the surface of all ordinary
mercury is greasy, and it has been shown by Quincke that
mercury may be prepared so clean that a drop of water will
spread upon it, instead of, as usual, standing as a drop upon
the surface. R.]
When a solid body is in contact with two fluids, then if
the tension of the surface separating the solid from the first
fluid exceeds the sum of the tensions of the other two sur-
faces, the first fluid will gather itself up into a drop, and
the second will spread over the surface. If one of the
fluids is air, and the other a liquid, then the liquid, if it
corresponds to the first fluid mentioned above, will stand
in drops without wetting the surface ; but if it corresponds
to the second, it will spread itself over the whole surface,
and wet the solid.
When the tension of the surface separating the two fluids
is greater than the difference of the tensions of the surfaces
separating them from the solid, then the surface of separation
of the two fluids will be inclined at a finite angle to the
surface of the solid. Thus, if a and b are the two fluids, and c
the solid, then to find the angle of contact P o Q we must
make P o = T a6 , and o Q = T 6c T a . This angle is called
the angle of capillarity.
288
Capillarity
ON THE RISE OF A LIQUID IN A TUBE.
FIG 37.
Let a be a liquid in a tube of a substance <r, whose radius
is r. Let the fluid b be air or any
other fluid. Let a be the angle of ca-
pillarity. The circumference of the
tube is 2 TT r. All round this circum-
ference there is a tension T ab acting at
an angle inclined o to the vertical, and
therefore the whole vertical force is
2 TT r T a6 COS a.
If this force raises the liquid to a
height h, then, neglecting the weight of
the sides of the hollow portion x Y z,
the weight of fluid supported is
TT p g r 2 h,
Equating this force to the weight
which it supports, we find
Hence the height to which the fluid is drawn up is
inversely as the radius of the tube.
A liquid is drawn up in the same way in the space be-
tween two parallel plates separated by a distance d. If we
now suppose fig. 38 to represent a section of the film or liquid,
the horizontal breadth of which is /, then the surface-tension
of the liquid on the line which bounds the wet and dry
parts of each plate is T /, and this force acts at an angle
with the vertical. The whole force, therefore, arising from
the surface-tension, and tending to raise the liquid, is
2 T / cos .
The weight of the liquid raised is
p g h Id.
in Relation to Evaporation and Condensation. 289
Equating the force to the weight which it supports, we
find
FIG. 38.
This expression differs from that for the height in a
cylindrical tube only by the substitution of d, the distance
between the parallel plates, for r, the radius of the tube.
Hence the height to which a liquid will ascend between
two plates is equal to the height to which it rises in a tube
whose radius is equal to the distance between the plates,
or whose diameter is twice that distance.
A remarkable application of the principles of thermo-
dynamics to capillary phenomena has recently been made
by Sir W. Thomson. 1 Let a fine tube be
placed in a liquid, and let the whole be
placed in a vessel from which air is ex-
hausted, so that the whole space above
the liquid becomes filled with its vapour
and nothing else.
Let the permanent level of the liquid
be at A in the small tube, and at B in the
vessel, and let us suppose the tempera-
ture the same throughout the apparatus.
There is a state of equilibrium between
the liquid and its vapour, both at A and at
B ; otherwise evaporation or condensation
would occur, and the permanent state
would not exist.
Now the pressure of the vapour at B exceeds that at A by
the pressure due to a column of the vapour of the height
A B.
It follows that the vapour is in equilibrium with the
liquid at a lower pressure where the surface of the liquid is
concave, as at A, than where it is plane, as at B.
Now let the lower end of the tube be closed, and let
1 Proceedings of the Royal Society of Edinburgh, Feb. 7, 1870.
U
2go Capillarity.
some of the liquid be taken out of it, so that the liquid in
the tube does not reach up to the point A.
Then vapour will condense inside the tube, owing to the
concavity of its surface, and this will go on till it is filled
with liquid up to the level A, the same as if it had been
open at the bottom.
Hence, if at any point of a concave liquid surface r and
r 1 are the principal radii of curvature of the surface, and if
the pressure of vapour in equilibrium with a plane surface
of its liquid at the given temperature is *, and if p is the
pressure of equilibrium of the vapour in contact with the
curved surface,
p-ar
where a is the density of the vapour, and p that of the liquid.
If h is the height to which the liquid would rise in virtue
of the curvature of its surface in a capillary tube, and if <>
is the height of a homogeneous atmosphere of the vapour,
--(-I)
Sir W. Thomson has calculated that in a tube whose
radius is about a thousandth of a millimetre, and in which
water would rise about thirteen metres above the plane
level, the equilibrium pressure of aqueous vapour would be
less than that on a plane surface of water by about a thou-
sandth of its own amount.
He thinks it probable that the moisture which vegetabl(
substances, such as cotton, cloth, &c., acquire from air at
temperatures far above the dew point may be explained by
the condensation of water in the narrow tubes and cells of
the vegetable structure.
In the case of a spherical bubble of steam in water, the
increase or diminution of the diameter depends on the
temperature and pressure of the vapour within ; and the
condition that ebullition may take place is that the pres-
Conditions of Boiling. 291
sure of saturated vapour at the temperature of the liquid
must exceed the actual pressure of the liquid by a pressure
equal to that of a column of the liquid of the height to
which it would ascend in a tube whose section is equal to
that of the bubbles.
If the liquid contains any gas in solution, or any liquid
more volatile than itself, or if air or steam is made to
bubble up through the liquid, then bubbles will be formed
of a visible diameter, and the ebullition will be kept up by
evaporation at the surface of these bubbles. But if, by long
boiling or otherwise, the liquid is deprived of any substance
more volatile than itself, and if the sides of the vessel in which
it is contained are such that the liquid adheres closely to
them, so that bubbles, if formed at the surface of the vessel,
will rather collect into a spherical form that spread along
the surface, then the temperature of the liquid may be
raised far above the boiling point, and when boiling at
last occurs, it goes on in an almost explosive manner,
and the liquid ' bumps ' violently on the bottom of the
vessel.
The highest temperature to which water may be raised
under the atmospheric pressure without ebullition cannot be
said to be accurately known, for every improvement in the
arrangements for getting rid of condensed air, &c., has made
it possible to raise liquid water to a higher temperature.
In an experiment due to Dufour, the water, instead of being
allowed to touch the sides of the vessel, is dropped into a
mixture of linseed oil and oil of cloves, which has nearly
the same density as itself. By this means, drops of liquid
water may sometimes be observed swimming in the mixture
at a temperature of 356 F. The pressure of aqueous
vapour is at this temperature nearly ten atmospheres, or
about 147 pounds weight on the square inch. Hence the
cohesion of the water must be able to support at least 132
pounds weight on the square inch.
[According to Laplace's theory, the cohesion of a liquid is
o a
292 Capillarity.
measured by the internal pressure K, due to the mutual
attraction of its parts. This quantity has been estimated,
in the first instance by Young, at the enormous figure of
20,000 atmospheres. The relation between K and T, the
surface-tension, may be illustrated by considering the pres-
sure p in the interior of a small spherical cavity of radius r.
So long as r is not very small, / is given by the usual
formula,
, 2T
/=-.
and it increases as r diminishes. If the above law held
good without limit, / would become infinite. In this case
the initiation of a bubble of steam in a boiling liquid would
be opposed by infinite force. In reality the law changes as
soon as r falls below the range of the cohesive forces, and
the ultimate value of/ is not infinite, but equal to K, which
may thus be regarded as the pressure, due to the cohesive
forces, within an infinitely small cavity.
The above argument shows that the range of the forces
must be of the order of magnitude K/T, a conclusion first
drawn by Young.
In the experiments of Berthelot water was subjected to
an actual tension estimated at 50 atmospheres.
The connexion of the capillary quantities with the
latent heat of evaporation has been pointed out by Water -
ston. 1 As was first shown by Dupre, the work required to
divide a unit of volume of liquid into very small parts, and
to separate these parts to such distances that they no longer
act sensibly upon one another, is measured by K. This is
substantially what occurs during evaporation, so that K
represents the work equivalent to the latent heat of evapo-
ration of unit, volume. A calculation on this basis led
Dupre" to the conclusion that in the case of water K is
about 25,000 atmospheres. R.]
We may also apply Sir W. Thomson's principle to the
i Phil. Mag: xv. p. i, 1858.
Formation of Fog. 293
case of evaporation from a small drop. In this case the
surface of the liquid is convex, so that if r is the radius of
the drop,
Here ta is the pressure of saturated vapour corresponding
to the temperature when the surface of the liquid is plane,
and p is the pressure of vapour required to prevent the
drop from evaporating. A small drop will therefore evapo-
rate in air containing so much moisture that condensation
would take place on a flat surface.
Hence, if a vapour free from suspended particles, and
not in contact with any solid body except such as are
warmer than itself, is cooled by expansion, it is probable
that the suggestion of Prof. J. Thomson at p. 126 might
be verified, and that the vapour might be cooled below its
ordinary point of condensation without liquefaction, for the
first effect of condensation would be to produce excessively
small drops, and these, as we have seen, would not tend to
increase unless the vapour surrounding them were more
than saturated.
[By a series of beautiful experiments Aitken has shown
that when ordinary moist air is cooled so as to form fog,
each aqueous spherule founds itself upon a minute particle
of foreign matter suspended as dust. When the air is
nearly freed from dust by filtration through cotton wool, or
otherwise, expansion produces a fine rain, consisting of com-
paratively few spheres of large diameter. The passage of a
single electric spark between platinum points is sufficient
to re-charge the air with nuclei, so that on repeating the
experiment the previous rain is replaced by a dense fog
containing innumerable fine particles. R.]
The formation of cloud in vapour often appears very
sudden, as if it had been at first retarded by some cause of
this kind, so that when at last the cloud is formed conden-
sation occurs with great rapidity, reminding us of the con-
294 Capillarity.
verse phenomenon of the rapid boiling of an overheated
liquid.
The drops in a cloud, for the same reason, cannot remain
of the same size, even if they are not jostled against each
other, for the smaller drops will evaporate, while the larger
ones are increased by condensation, so that visible drops
will be formed by pure condensation without any necessity
for the coalescence of smaller drops.
Up to this point we have not considered the effect of
heat on the superficial tension of liquids. In all liquids on
which experiments have been made the superficial tension
diminishes as the temperature rises, being greatest at the
freezing point of the substance, and vanishing altogether at
the critical point where the liquid and gaseous states become
continuous.
It appears, therefore, that the phenomenon is intimately
related to the apparent discontinuity of the liquid and
gaseous states, and that it must be studied in connexion
with the conditions of evaporation and the phenomenon
called latent heat. Much light will probably be thrown on
all these subjects by investigations which as yet can hardly
be said to be begun.
Sir W. Thomson has applied the principles of thermo-
dynamics to the case of a film of water extended by a force
applied to it, and has shown that in order to maintain the
temperature of the film constant an amount of heat must
be supplied to it nearly equal in dynamical measure to half
the work done in stretching the film.
In fact, the third thermodynamical relation (p. 168) may
be applied at once to the case by making the following
substitutions: for 'pressure' put 'superficial tension,' and
for ' volume ' put ' area.'
We thus find that the latent heat of extension of unit of
area is equal to the product of the absolute temperature
and the decrement of superficial tension per degree of tem-
perature. At ordinary temperatures it appears from experi-
Table of Tensions.
295
ment that this product is about half the superficial ten-
sion. Hence the latent heat of extension in dynamical
measure is about half the work spent in producing the ex-
tension.
The student may also adapt the investigation of latent
heat as given at p. 173 to the case of the extension of a
liquid film.
The following table, taken from the memoir of M. Quincke,
gives the superficial tension of different liquids in contact
with air, water, and mercury. The tension is measured in
grammes weight per linear metre, and the temperature is
20 C.
Table of Superficial Tension at 20 C.
Liquid
Sp. gravity
Tension of surface separating
the liquid from
Air
Water | Mercury
Water
1-0
8-253
o 42-58
Mercury
I3'5432
5 5 '03
42-58 o
Bisulphide of Carbon
I-2687
3-274
4-256 37-97
Chloroform
1-4878
3-120
3'OIQ 4071
Alcohol
0-7906
2-599
40-71
Olive Oil .
0-9136
3-760
2-096 34-19
Turpentine *
0-8867
3-030
I-I77 25-54
Petroleum .
07977
3-233
2-834 28-94
Hydrochloric Acid
I-I
7-15
38-41
Solution of Hyposulphite of
Soda . . . .
I'I248
7-903
45-u
It appears from this table that water has the greatest
superficial tension of all ordinary liquids. For this reason
it is very difficult to preserve a surface of pure water, It is
sufficient to touch any part of the surface of pure water
with a greased rod to reduce its tension considerably. The
smallest quantity of any kind of oil immediately spreads
itself over the surface, and completely alters the superficial
tension. Hence the importance in all experiments on super-
ficial tension of having the vessel thoroughly clean. This
296 Capillarity.
has been well pointed out by Mr. Tomlinson in his researches
on the ' cohesion figures of liquids.'
[It seems doubtful whether the tension of water is really
so high as that recorded in the table. Observations upon
very clean surfaces, in which the tension was determined
from its effect upon the propagation of ripples, gave y^. 1
A convenient test for ascertaining whether a water sur-
face is moderately clean is afforded by camphor. If a
wineglass, after thorough rinsing under a tap, be allowed to
fill with water, the surface will probably be clean enough for
the experiment. In this case small fragments of camphor
scraped off with a penknife, and allowed to fall upon the
surface, will at once assume vigorous movements, principally
of rotation. If now the surface of the water be touched
with the finger, the motion of the camphor fragments will
probably be arrested in consequence of the grease com-
municated to the water. The movements upon a clean
surface are due to the gradual solution of the camphor, and
to the fact that the solution has a smaller tension than pure
water. In consequence, the part of the surface immediately
surrounding a fragment is constantly being drawn outwards,
while the radial outflow of the camphor tends to be com-
pensated by the entrance of fresh material into solution.
If this action took place with perfect symmetry, the frag-
ment would remain at rest ; but in consequence of irregu-
larities of- outline, the strength, and therefore the tension,
of the surface is not the same on all sides, and there remain
residual forces competent to set these small masses into
rotation.
The vigorous movement of camphor fragments does not
require an absolutely clean surface, and in fact we may
experimentally determine the amount of any kind of oil
necessary to stop them. In one trial it appeared that about
*8 milligram of olive oil was required upon a circular water
1 ' On the Tension of Water Surfaces, clean and contaminated, Phil.
Mag. Nov. 1890.
Camphor Movements. 297
surface 84 cm. in diameter. If from these data we calculate
the thickness of the oil film upon the supposition that its
density is the same ('9) as usual, we find i*6xio~ 7 cm.
Allowing a little for the imperfect purity of the surface
before the addition of the oil, we may conclude that an oily
film 2 millionths of a millimetre in thickness suffices to
arrest the camphor movements.
If the oily film be less than the above, the tension of the
contaminated surface, though reduced, is still sufficient to
overcome that of the camphor solution which may be
supposed to have developed itself round the fragment, and
thus the action continues. But if the grease be present in
such quantity that the tension of the contaminated surface
is less than that of a saturated solution of camphor, it is no
longer possible for the latter to spread along the surface,
and then the movements cease.
If we call the tension of a clean surface 100, that of a
saturated solution of camphor is 72. A surface upon which
there is an excess of olive oil has a tension of 54, while that
of a solution of soap is only about 34.
As a check upon the correctness of the explanation just
given, we may compare the behaviour of camphor fragments
upon surfaces greased with different materials, but of the
same tension ; and the easiest way to secure the desired
equality of tensions is to use different parts of the same
surface. A line of dust, such as sulphur or lycopodium, is
distributed upon the surface of water in a large flat dish,
so as to divide it into equal parts. If a small chip of wood
greased, for example, with olive oil .be allowed to touch one
part of the surface, the line of dust is repelled by the ex-
pansion of that part, but the effect may be compensated by
a slight greasing of the other side with oil of cassia. By
careful alternate additions the line of dust may be kept
central, while the two halves become increasingly greased
with the two sorts of oil. At every stage of this process,
so long as the surface is at rest, the tension of all parts is
298 Capillarity.
necessarily the same. Experiments of this kind with a
large variety of oils showed that the effect upon camphor
of the different parts of the surface was indistinguishable,
in spite of the different sorts of grease in operation.
There is an important difference in the mechanical be-
haviour of clean and contaminated surfaces. In the case
of the former no force opposes the expansion of one part
of the surface and the contraction of another. But if there
is a film of grease, the thickness of the film is increased by
any contraction and diminished by any expansion. These
differences of thickness entail corresponding differences
of surface-tension, so that if a greasy surface be moved in
such a way as to expand or contract any part, forces are
called into play tending to restore the original situation.
This is the origin of the ' superficial viscosity ' of Plateau,
which is thus a property of contaminated, and not of clean,
surfaces. By suitable methods water may be prepared
devoid of superficial viscosity. 1
A like explanation, first correctly given by O. Reynolds,
applies to the effect of oil upon waves. The ordinary
propagation of waves imposes upon the surface periodic
local expansions and contractions. To these a greasy surface
offers opposition. It is to be understood that the calming
effect of oil applies in the first instance only to small waves
and ripples, but it appears to be by means of these that the
crests of the large waves are driven forward and rendered
dangerous.
We have seen that the tension of a greasy surface in-
creases under extension. The same principle applies to a
soap film. If a film be horizontal and at rest, all parts must
exercise the same tension ; and the fact that such a film may
exhibit various colours at different parts shows that the
tension may be the same in spite of great relative alterations
of thickness. Again, in this position no force opposes the
substitution of a thin for a thick part at any place, provided
1 Proc. Roy. Soc., vol. xlviii. p. 127.
Tension of Soap Films. 299
there be no extension or contraction of either. If, however,
the film be raised from the horizontal to the vertical position,
it is observed that, as shown by the colour, the thick parts
find their way to the bottom and the thin parts to the top.
The result is attained by an actual transfer of parts, and
not by a thickening of those which may accidentally find
themselves at the bottom, and a thinning of those acciden-
tally at the top. After a short time all is sensibly at rest,
and this proves, contrary to what is often asserted, that the
tension of the film is greater above than below. Were it
not so, the intermediate parts of the film, being under the
influence of gravity alone, would fall sixteen feet in the first
second of time. The stability of the film requires that the
tension be not absolutely constant, but liable to augment
under extension. If the central parts of a vertical film were
suddenly displaced downwards, an increase of tension above,
and a decrease below, would be called into play, and the
original condition would be restored.
The greatly diminished tension of soapy water is doubt-
less due to a film upon the surface. This film is evolved
from the interior, and is probably capable of reabsorp-
tion. It has been proved by Duprelf and the present writer
that at the very first moment of their formation surfaces
of soapy water have hardly less tension than those of pure
water. R.]
When one of-the liquids is soluble in the other, the effects
of superficial tension are very remarkable. For instance, if
a drop of alcohol be placed on the surface of a thin layer of
water, the tension is immediately reduced to 2 '6, where the
alcohol is pure, and varies from this value to 8-25, where the
water is pure. The result is that the equilibrium of the sur-
face is destroyed, and the superficial film of the liquid is
set in motion from the alcohol towards the water, and if
the water is shallow this motion of the surface will drag
the whole of the water with it, so as to lay bare part of the
bottom of the vessel. A dimple may be formed on the
3OO Capillarity.
surface of water by bringing a drop of ether close to the sur
face. The vapour of the ether condensed on the surface
of the water is sufficient to cause the outward current
mentioned above.
Wine contains alcohol and water, and when it is exposed to
the air the alcohol evaporates faster than the water, so that
the superficial layer becomes weaker. When the wine is in a
deep vessel, the strength is rapidly equalized by diffusion ;
but in the case of the thin layer of wine which adheres to
the sides of a wineglass, the liquid rapidly becomes more
watery. This increases the superficial tension at the sides
of the glass, and causes the surface to be dragged from the
strong wine to the weak. The watery portion is always
uppermost, and creeps up the sides of the glass, dragging the
stronger wine after it till the quantity of the fluid becomes so
great that the different portions mix, and the drop runs down
the side.
This phenomenon, known as the tears of strong wine, was
first explained on these principles by Professor James Thom-
son. It is probable that it is referred to in Proverbs xxiii.
31, as an indication of the strength of the wine. The motion
ceases in a stoppered bottle as soon as enough of vapour of
alcohol has been formed in the bottle to be in equilibrium
with the liquid alcohol in the wine.
The fatty oils have a greater superficial tension than tur-
pentine, benzol, or ether. Hence if there is a greasy spot on
a piece of cloth, and if one side of it is wetted with one of
these substances, the tension is greatest on the side of the
grease, and the portions consisting of mixtures of benzol and
grease move from the benzol towards the grease.
If in order to cleanse the grease-spot we begin by wetting
the middle of the spot with benzol, we drive away the grease
into the clean part of the cloth. The benzol should there-
fore be applied first in a ring all round the spot, and gradu-
ally brought nearer to the centre of the spot, and a fibrous
substance, such as blotting-paper, should be placed in contact
Grease Spots. 301
with the cloth, so that when the grease is chased by the
benzol to the middle of the spot it may make its escape into
the blotting-paper, instead of remaining in globules on the
surface, ready to return into the cloth when the benzol
evaporates.
Another very effectual method of getting rid of grease-
spots is founded on the fact that the superficial tension of a
substance always diminishes as the temperature rises. If,
therefore, the temperature is different at different parts of a
greasy cloth, the grease tends to move from the hot parts to
the cold. We therefore apply a hot iron to one side of the
cloth, and blotting-paper to the other, and the grease is
driven into the blotting-paper. If there is blotting-paper on
both sides it will be found that the grease is driven mainly
into that on the opposite side from the hot iron.
CHAPTER XXI.
ON ELASTICITY AND VISCOSITY.
On Stresses and Strains.
WHEN the form of a connected system is altered in any
way, the alteration of form is called a Strain. The force
or system of forces by which this strain is produced or
maintained is called the Stress corresponding to the strain.
There are different kinds of strains, and different kinds of
stresses corresponding to them.
The only case which we have hitherto considered is that
in which the three longitudinal stresses are equal. This
kind of stress is called Hydrostatic Pressure, and is the
only kind which can exist in a fluid at rest. The pressure
is the same in whatever direction it is estimated.
3O2 Stresses and Strains.
A very important kind of stress is called Shearing Stress :
it is compounded of two equal longi- FIG. 39 .
tudinal stresses, one being a tension
and the other a pressure acting at
right angles to each other. When a
pair of scissors is employed to cut
anything, the two blades produce a
shearing stress in the material be-
I
tween them, tending to make one /
portion slide over the other.
We have now to consider the properties of bodies when
acted on by this kind of stress.
A body which when subjected to a stress experiences no
strain would, if it existed, be called a Perfectly Rigid Body.
There are no such bodies, and this definition is given only to
indicate what is meant by perfect rigidity.
A body which when subjected to a given stress at a given
temperature experiences a strain of definite amount, which
does not increase when the stress is prolonged, and which
disappears completely when the stress is removed, is called
a Perfectly Elastic Body.
Gases and liquids, and perhaps most solids, are perfectly
elastic as regards stress uniform in all directions, but no sub-
stance which has yet been tried is perfectly elastic as regards
shearing stress, except perhaps for exceedingly small values
of the stress.
Now suppose that stresses of the same kind, but of con-
tinually increasing magnitude, are applied to a body in
succession. As long as the body returns to its original
form when the stress is removed it is said to be perfectly
elastic.
If the form of the body is found to be permanently altered
when the stress exceeds a certain value, the body is said
to be soft, or plastic, and the state of the body when the
alteration is just going to take place is called the Limit of
Perfect Elasticity.
Definition of Solidity. 303
If the stress be increased till the body breaks or gives way
altogether, the value of the stress is called the Strength of
the body for that kind of stress.
If breaking takes place before there is any permanent
alteration of form, the body is said to be Brittle.
If the stress, when it is maintained constant, causes
a strain or displacement in the body which increases
continually with the time, the substance is said to be
Viscous.
When this continuous alteration of form is only produced
by stresses exceeding a certain value, the substance is called
a solid, however soft it may be. When the very smallest
stress, if continued long enough, will cause a constantly
increasing change of form, the body must be regarded as
a viscous fluid, however hard it may be.
Thus, a tallow candle is much softer than a stick of
sealing-wax ; but if the candle and the stick of sealing-wax
are laid horizontally between two supports, the sealing-wax
will in a few weeks in summer bend with its own weight,
while the candle remains straight. The candle is therefore
a soft solid, and the sealing-wax a very viscous fluid.
What is required to alter the form of a soft solid is a
sufficient force, and this, when applied, produces its effect
at once. In the case of a viscous fluid it is time which is
required, and if enough time is given, the very smallest
force will produce a sensible effect, such as would require a
very large force if suddenly applied.
Thus a block of pitch may be so hard that you cannot
make a dint in it by striking it with your knuckles ; and
yet it will, in the course of time, flatten itself out by its
own weight, and glide down hill like a stream of water.
A glass fibre was found by M. F. Kohlrausch * to be-
come more and more twisted when constantly acted on by
the small twisting force arising from the action of the earth
on a little magnet suspended by the fibre. I have found slow
i Pogg.
304 Viscosity.
changes in the torsion of a steel wire going on for many days
after it had received a slight permanent twist, and Sir W.
Thomson l has investigated the viscosity of other metals.
There are instances of viscosity among very hard bodies.
Returning to our former example, pitch : we may mix it in
various proportions with tar so as to form a continuous
series of compounds passing from the apparently solid
condition of pitch to the apparently fluid condition of tar,
which may be taken as a type of a viscous fluid. By
mixing the tar with turpentine the viscosity may be still
further reduced, and so we may form a series of fluids of
diminishing viscosity till we arrive at the most mobile fluids,
such as ether.
DEFINITION OF THE COEFFICIENT OF VISCOSITY.
Consider a stratum of the substance of thickness <r, con
tained between the horizontal fixed plane FlG 40
A B and the plane c D, which is moving c - -i>
horizontally from c towards D, with the
velocity v. Let us suppose that the substance
between the two planes is also in motion, the stratum in
contact with c D moving with velocity v, while the velocity
of any intermediate stratum is proportional to its height
above A B.
The substance between the planes is undergoing shearing
strain, and the rate at which this strain is increasing is measured
by the velocity v of the upper plane, divided by the distance
y
c between the planes, or .
The stress F is a shearing stress, and is measured by the
horizontal force exerted by the substance on unit of area
of either of the planes, and acting from A to B on the lower
plane, and from D to c on the upper.
The ratio of this force to the rate of increase of the shear-
1 Proc. Roy. Soc. May 18, 1865.
Dimensions of Viscosity. 305
ing stress is called the coefficient of viscosity, and is denoted
by the symbol /*. We may therefore write F = /x -.
If R is the amount of this force on a rectangular area of
length a and breadth b,
R = a b?
^v
c
and Rf
v ab'
If v, 0, b y and c are each unity, then p. = R.
Definition. The viscosity of a substance is measured by
the tangential force on the unit of area of either of two hori-
zontal planes at the unit of distance apart, one of which is
fixed, while the other moves with the unit of velocity, the
space between being filled with the viscous substance.
The dimensions of // may be easily determined. If R is the
moving force which would generate a certain velocity v in the
AT 7) At 7) r
mass M in the time /, then R = , and
,
Here 0, b, c are lines, and v and v are velocities, so that
the dimensions of p. are [M L" 1 T" 1 ], where M, L, and T are the
units of mass, length, and time.
When we wish to express the absolute forces called into
play by the viscosity of a substance, we must use the ordi-
nary unit of mass (a pound, a grain, or a gramme) ; but if we
wish only to investigate the motion of the viscous substance,
it is convenient to take as our unit of mass that of unit of
volume of the substance itself. If p is the density of the
substance, or the mass of unit of volume, the viscosity v
measured in this kinematic way is related to ^, its value by
the former, or dynamical method, by the equation p. = v p.
The dimensions of v, the kinematic viscosity, are [L 2 T~ ! ].
Investigations of the value of viscosity have been made,
for solids by Sir W. Thomson ; for liquids by Poiseuille,
X
306 Viscosity.
Graham, O. E. Meyer, and Helmholtz ; and for gases by
Graham, Stokes, O. E. Meyer, and myself.
I find the value of p for air at Centigrade to be
p = -0001878 (i + -003660),
the centimetre, gramme, and second being units.
[Recent observers have found lower numbers for the vis-
cosity of air. The value for o C. would seem to be about
-000168.
For water at o C, n = 'oi3i.
In the case of liquids the viscosity diminishes as the tem-
perature rises.
The kinematic measure, r, of the viscosity is less in the
case of water than in the case of air. R.]
In British measure, using the foot, the grain, and the
second, and Fahrenheit's thermometer, this becomes
fj, = '000179 (46* + 0)-
The viscosity /u is proportional to the absolute tempera-
ture, and independent of the pressure, being the same for a
pressure of half an inch as for a pressure of thirty inches of
mercury. The significance of this remarkable result will be
seen when we come to the molecular theory of gases.
The kinematic measure, r, of the viscosity is found by
dividing p, by the density. It is therefore directly propor-
tional to the square of the absolute temperature, and in-
versely proportional to the pressure.
The value of p. for hydrogen is less than half that for
air. Oxygen, on the other hand, has a viscosity greater than
that of air. That of carbonic acid is less than that of air.
It appears, from the calculations of Professor Stokes,
combined with the value of the viscosity of air given above,
that a drop of water falling through air one thousand times
rarer than itself (which we may suppose to be the case
at the ordinary height of a cloud) would fall about ^
of an inch in a second if its diameter were the thousandth
part of an inch. If the diameter of the drop were only one
Subsidence of Clouds 307
ten-thousandth of an inch the rate at which it would make
its way through the air would be a hundred times smaller,
or half an inch in a minute. If a cloud is formed of little
drops of water of this size, their motion through the air
would be so slow that it would escape observation, and the
motion of the cloud, so far as it can be observed, would be
the same as that of the air in that place. In fact, the
settling down through the air of any very small particles,
such as the fine spray of waves or waterfalls, and all kinds
of dust and smoke, is a very slow process, and the time of
settling down through a given distance varies inversely as
the square of the dimensions of the particles, their density
and figure being the same. If, however, a cloud of fine
dust contains so many particles that the mass of a cubic
foot of dusty air is sensibly greater than that of a cubic
foot of pure air, the dusty air will descend in mass below
the level of the pure air like a fluid of greater density, so
that a room may have its lower half filled with dusty air
separated by a level surface from the pure air above.
There are some kinds of fogs the mean density of which
is greater than that of the purer air in the neighbourhood,
and these lie like lakes in hollows, and pour down valleys
like streams. On the other hand, the mean density of a
cloud may be less than that of the surrounding air, and it
will then ascend.
In the case of smoke, both the air and the sooty particles
are heated by the fire before they escape into the atmo-
sphere, but, independently of this kind of heating, if the 'sun
shines on a cloud of dust or smoke, the particles absorb
heat, which they communicate to the air round them, and
thus, though the particles themselves remain much denser
than the air in the neighbourhood, they may cause the cloud
which they form to appropriate so much of the sun's heat
that it becomes lighter as a whole than the surrounding pure
air, and so rises.
In the case of a cloud of watery particles, besides this
X 2
308 Molecular Theory.
kind of action, there is another, depending on the evapora-
tion from the surface of the little drops. The vapour of
water is much rarer than air. and damp air is lighter than
dry air at the same temperature and pressure. Hence the
little drops make the air of the cloud damp, and if the
mean density of the cloud is by this means made less than
that of the surrounding air, the cloud will ascend.
CHAPTER XXII.
ON THE MOLECULAR THEORY OF THE CONSTITUTION OF
BODIES.
WE have already shown that heat is a form of energy that
when a body is hot it possesses a store of energy, part at
least of which can afterwards be exhibited in the form of
visible work.
Now energy is known to us in two forms. One of these
is Kinetic Energy, the energy of motion. A body in motion
has kinetic energy, which it must communicate to some
other body during the process of bringing it to rest. This
is the fundamental form of energy. When we have acquired
the notion of matter in motion, and know what is meant by
the energy of that motion, we are unable to conceive that
any possible addition to our knowledge could explain the
energy of motion, or give us a more perfect knowledge of it
than we have already.
There is another form of energy which a body may have,
which depends, not on its own state, but on its position
with respect to other bodies. This is called Potential
Energy. The leaden weight of a clock, when it is wound
up, has potential energy, which it loses as it descends. It
is spent in driving the clock. This energy depends, not on
the piece of lead considered in itself, but on the position of
Is Heat Motion? 309
the lead with respect to another body the earth which
attracts it.
In a watch, the mainspring, when wound up, has poten-
tial energy, which it spends in driving the wheels of the
watch. This energy arises from the coiling up of the
spring, which alters the relative position of its parts. In
both cases, until the clock or watch is set agoing, the
existence of potential energy, whether in the clock-weight
or in the watch-spring, is not accompanied with any visible
motion. We must therefore admit that potential energy can
exist in a body or system all whose parts are at rest.
It is to be observed, however, that the progress of science
is continually opening up new views of the forms and
relations of different kinds of potential energy, and that
men of science, so far from feeling that their knowledge of
potential energy is perfect in kind, and incapable of essential
change, are always endeavouring to explain the different
forms of potential energy ; and if these explanations are in
any case condemned, it is because they fail to give a suffi-
cient reason for the fact, and not because the fact requires
no explanation.
We have now to determine to which of these forms of
energy heat, as it exists in hot bodies, is to be referred. Is
a hot body, like a coiled-up watch-spring, devoid of motion
at present, but capable of exciting motion under proper
conditions ? or is it like a fly-wheel, which derives all its
tremendous power from the visible motion with which it is
animated ?
It is manifest that a body may be hot without any motion
being visible, either of the body as a whole, or of its parts
relatively to each other. If, therefore, the body is hot
in virtue of motion, the motion must be carried on by parts
of the body too minute to be seen separately, and within
limits so narrow that we cannot detect the absence of any
part from its original place.
The evidence for a state of motion, the velocity of which
3io Molecular Theory.
must far surpass that of a railway train, existing in bodies
which we can place under the strongest microscope, and in
which we can detect nothing but the most perfect repose,
must be of a very cogent nature before we can admit that
heat is essentially motion.
Let us therefore consider the alternative hypothesis that
the energy of a hot body is potential energy, or, in other
words, that the hot body is in a state of rest, but that this
state of rest depends on the antagonism of forces which
are in equilibrium as long as all surrounding bodies are
of the same temperature, but which as soon as this equi-
librium is destroyed are capable of setting bodies in
motion. With respect to a theory of this kind, it is to be
observed that potential energy depends essentially on the
relative position of the parts of the system in which it exists,
and that potential energy cannot be transformed in any
way without some change of the relative position of these
parts. In every transformation of potential energy, therefore,
motion of some kind is involved.
Now we know that whenever one body of a system is
hotter than another, heat is transferred from the hotter to
the colder body, either by conduction or by radiation. Let
us suppose that the transfer takes place by radiation.
Whatever theory we adopt about the kind of motion which
constitutes radiation, it is manifest that radiation consists of
motion of some kind, either the projection of the particles
of a substance called caloric across the intervening space, or
a wave-like motion propagated through a medium filling that
space. In either case, during the interval between the time
when the heat leaves the hot body and the time when it
reaches the cold body, its energy exists in the intervening
space in the form of the motion of matter.
Hence, whether we consider the radiation of heat as
effected by the projection of material caloric, or by the
undulations of an intervening medium, the outer surface of
a hot body must be in a state of motion, provided any cold
Molecular Motion. 311
body is in its neighbourhood to receive the radiations which
it emits. But we have no reason to believe that the pre-
sence of a cold body is essential to the radiation of heat by
a hot one. Whatever be the mode in which the hot body
shoots forth its heat, it must depend on the state of the hot
body alone, and not on the existence of a cold body at a
distance, so that even if all the bodies in a closed region
were equally hot, every one of them would be radiating
neat ; and the reason why each body remains of the same
temperature is, that it receives from the other bodies exactly
as much heat as it emits. This, in fact, is the foundation of
Prevost's Theory of Exchanges. We must therefore admit
that at every part of the surface of a hot body there is a
radiation of heat, and therefore a state of motion of the
superficial parts of the body. Now this motion is certainly
invisible to us by any direct mode of observation, and
therefore the mere fact of a body appearing to be at rest
cannot be taken as a demonstration that its parts may
not be in a state of motion.
Hence part, at least, of the energy of a hot body must be
energy arising from the motion of its parts, or kinetic energy.
The conclusion at which we shall arrive, that a very
considerable part of the energy of a hot body is in the form
of motion, will become more evident when we consider the
thermal energy of gases.
Every hot body, therefore, is in motion. We have next
to enquire into the nature of this motion. It is evidently
not a motion of the whole body in one direction, for how-
ever small we make the body by mechanical processes, each
visible particle remains apparently in the same place, how-
ever hot it is. The motion which we call heat must there-
fore be a motion of parts too small to be observed separately ;
the motions of different parts at the same instant must be
in different directions ; and the motion of any one part must,
at least in solid bodies, be such that, however fast it moves,
it never reaches a sensible distance from the point from
which it started.
312 Molecular Theory.
We have now arrived at the conception of a body as
consisting of a great many small parts, each of which is in
motion. We shall call any one of these parts a molecule of
the substance. A molecule may therefore be denned as a
small mass of matter the parts of which do not part com-
pany during the excursions which the molecule makes when
the body to which it belongs is hot.
The doctrine that visible bodies consist of a determinate
number of molecules is called the molecular theory of matter.
The opposite doctrine is that, however small the parts may
be into which we divide a body, each part retains all the
properties of the substance. This is the theory of the
infinite divisibility of bodies. We do not assert that there
is an absolute limit to the divisibility of matter : what we
assert is, that after we have divided a body into a certain
finite number of constituent parts called molecules, then
any further division of these molecules will deprive them
of the properties which give rise to the phenomena ob-
served in the substance.
The opinion that the observed properties of visible bodies
apparently at rest are due to the action of invisible mole-
cules in rapid motion is to be found in Lucretius.
Daniel Bernoulli was the first to suggest that the pressure
of air is due to the impact of its particles on the sides of
the vessel containing it ; but he made very little progress in
the theory which he suggested.
Lesage and Prevost of Geneva, and afterwards Herapath
in his ' Mathematical Physics/ made several important appli-
cations of the theory.
Dr. Joule in 1848 explained the pressure of gases by the
impact of their molecules, and calculated the velocity which
they must have to produce the observed pressure.
Kronig also directed attention to this explanation of the
phenomena of gases.
It is to Professor Clausius, however, that we owe the recent
development of the dynamical theory of gases. Since he
Solids, Fluids, and Gases. 313
took up the subject a great advance has been made by
many enquirers. I shall now endeavour to give a sketch of
the present state of the theory.
All bodies consist of a finite number of small parts called
molecules. Every molecule consists of a definite quantity
of matter, which is exactly the same for all the molecules of
the same substance. The mode in which the molecule is
bound together is also the same for all molecules of the
same substance. A molecule may consist of several distinct
portions of matter held together by chemical bonds, and
may be set in vibration, rotation, or any other kind of
relative motion, but so long as the different portions do
not part company, but travel together in the excursions
made by the molecule, our theory calls the whole connected
mass a single molecule.
The molecules of all bodies are in a state of continual
agitation. The hotter a body is, the more violently are its
molecules agitated. In solid bodies, a molecule, though in
continual motion, never gets beyond a certain very small
distance from its original position in the body. The path
which it describes is confined within a very small region
of space.
In fluids, on the other hand, there is no such restriction
to the excursions of a molecule. It is true that the mole-
cule generally can travel but a very small distance before
its path is disturbed by an encounter with some other mole-
cule ; but after this encounter there is nothing which deter-
mines the molecule rather to return towards the place from
whence it came than to push its way into new regions.
Hence in fluids the path of a molecule is not confined
within a limited region, as in the case of solids, but may
penetrate to any part of the space occupied by the fluid.
The actual phenomena of diffusion both in liquids and
in gases furnish the strongest evidence that these bodies
consist of molecules in a state of continual agitation.
But when we apply the methods of dynamics to the
314 Molecular Theory.
investigation of the properties of a system consisting of a
great number of small bodies in motion the resemblance
of such a system to a gaseous body becomes still more
apparent.
I shall endeavour to give some account of what is known
of such a system, avoiding all unnecessary mathematical
calculations.
ON THE KINETIC THEORY OF GASES.
A gaseous body is supposed to consist of a great number
of molecules moving with great velocity. During the greater
part of their course these molecules are' not acted on by any
sensible force, and therefore move in straight lines with
uniform velocity. When two molecules come within a
certain distance of each other, a mutual action takes place
between them, which may be compared to the collision of
two billiard balls. Each molecule has its course changed,
and starts on a new path. I have concluded from some
experiments of my own that the collision between two hard
spherical balls is not an accurate representation of what
takes place during the encounter of two molecules. A
better representation of such an encounter will be obtained
by supposing the molecules to act on one another in a more
gradual manner, so that the action between them goes on for
a finite time, during which the centres of the molecules first
approach each other and then separate.
We shall refer to this mutual action as an Encounter
between two molecules, and we shall call the course of a
molecule between one encounter and another the Free Path
of the molecule. In ordinary gases the free motion of a
molecule takes up much more time than that occupied by an
encounter. As the density of the gas increases, the free path
diminishes, and in liquids no part of the course of a molecule
can be spoken of as its free path.
In an encounter between two molecules we know that,
since the force of the impact acts between the two bodies,
Statistical Method. 315
the motion of the centre of gravity of the two molecules
remains the same after the encounter as it was before. We
also know by the principle of the conservation of energy that
the velocity of each molecule relatively to the centre of
gravity remains the same in magnitude, and is only changed
in direction.
Let us next suppose a number of molecules in motion
contained in a vessel whose sides are such that if any
energy is communicated to the vessel by the encounters of
molecules against its sides, the vessel communicates as
much energy to other molecules during their encounters
with it, so as to preserve the total energy of the enclosed
system. The first thing we must notice about this moving
system is that even if all the molecules have the same velo-
city originally, their encounters will produce an inequality
of velocity, and that this distribution of velocity will go on
continually. Every molecule will then change both its
direction and its velocity at every encounter; and, as we
are not supposed to keep a record of the exact particulars
of every encounter, these changes of motion must appear to
us very irregular if we follow the course of a single molecule.
If, however, we adopt a statistical view of the system, and
distribute the molecules into groups, according to the
velocity with which at a given instant they happen to be
moving, we shall observe a regularity of a new kind in the
proportions of the whole number of molecules which fall into
each of these groups.
And here I wish to point out that, in adopting this
statistical method of considering the average number of
groups of molecules selected according to their velocities, we
have abandoned the strict kinetic method of tracing the
exact circumstances of each individual molecule in all its
encounters. It is therefore possible that we may arrive at
results which, though they fairly represent the facts as long
as we are supposed to deal with a gas in mass, would cease
to be applicable if our faculties and instruments were so
316 Molecular Theory.
sharpened that we could detect and lay hold of each mole-
cule and trace it through all its course.
For the same reason, a theory of the effects of education
deduced from a study of the returns of registrars, in which no
names of individuals are given, might be found not to be
applicable to the experience of a schoolmaster who is able
to trace the progress of each individual pupil.
The distribution of the molecules according to their veloci-
ties is found to be of exactly the same mathematical form as
the distribution of observations according to the magnitude of
their errors, as described in the theory of errors of observation.
The distribution of bullet-holes in a target according to their
distances from the point aimed at is found to be of the same
form, provided a great many shots are fired by persons of
the same degree of skill.
We have already met with the same form in the case of
heat diffused from a hot stratum by conduction. Whenever
in physical phenomena some cause exists over which we
have no control, and which produces a scattering of the
particles of matter, a deviation of observations from the truth,
or a diffusion of velocity or of heat, mathematical expressions
of this exponential form are sure to make their appearance.
It appears then that of the molecules composing the
system some are moving very slowly, a very few are moving
with enormous velocities, and the greater number with inter-
mediate velocities. To compare one such system witk
another, the best method is to take the mean of the squares
of all the velocities. This quantity is called the Mean Square
of the velocity. The square root of this quantity is called
the Velocity of Mean Square.
DISTRIBUTION OF KINETIC ENERGY BETWEEN TWO
DIFFERENT SETS OF MOLECULES.
If two sets of molecules whose mass is different are in
motion in the same vessel, they will by their encounters
Internal Kinetic Energy. 317
exchange energy with each other till the average kinetic
energy of a single molecule of either set is the same. This
follows from the same investigation which determines the
law of distribution of velocities in a single set of molecules.
Hence if the mass of a molecule of one kind is M t , and
that of a molecule of the other kind is M 2 , and if their average
velocities of agitation are Vj and V 2 , then
M, V, 2 = M 2 V 2 2 (l)
The quantity \ M v 2 is called the average kinetic energy
of agitation of a single molecule. We shall return to this
result when we come to Gay-Lussac's Law of the Volumes
of Gases.
INTERNAL KINETIC ENERGY OF A MOLECULE.
If a molecule were a mathematical point endowed with
inertia and with attractive and repulsive forces, the only
kinetic energy it could possess is that of translation as a
whole. But if it be a body having parts and magnitude,
these parts may have motions of rotation or of vibration
relative to each other, independent of the motion of the
centre of gravity of the molecule. We must therefore admit
that part of the kinetic energy of a molecule may depend on
the relative motions of its parts. We call this the Internal
energy, to distinguish it from the energy due to the trans-
lation of the molecule as a whole. The ratio of the internal
energy to the energy of agitation may be different in
different gases.
DEFINITION OF THE VELOCITY OF A GAS.
It is evident that if a gas consists of a great number of
molecules moving about in all directions we cannot identify
the velocity of any one of these molecules with what we are
accustomed to consider as the velocity of the gas itselt
Let us consider the case of a gas which has remained in a
fixed vessel for a sufficient time to arrive at the normal
3i8 Molecular Theory.
distribution of velocities. This gas, according to the ordi-
nary notions, is at rest, though the molecules of which it is
composed may be flying about in all directions.
Now consider any plane area of an imaginary surface
described within the vessel. This surface does not interfere
with the motion of the molecules. Some molecules pass
through the surface in one direction, and others in the
opposite direction ; but it is evident, since the gas does not
tend to accumulate on one side rather than on the other,
that exactly the same number of molecules pass in the one
direction as in the other. If, therefore, a gas is at rest, as
many molecules pass through a fixed surface in the one
direction as in the other in the same time.
It is evident that if the vessel, instead of being at rest, had
been in a state of uniform motion, an equal number of mole-
cules would pass in both directions through any surface
fixed with respect to the vessel. Hence we find that if a
gas is in motion, and if the velocity of a surface coincides in
direction and magnitude with that of the gas, the same
number of molecules will pass through that surface in the
positive direction as in the negative.
This leads to the following definition of the velocity of a
gas:
If we determine the motion of the centre of gravity of all
the molecules within a very small region surrounding a point
in a gas, then the velocity of the gas within that region is
defined as the velocity of the .centre of gravity of all the mole-
cules within that region.
This is what is meant by the motion of a gas in common
language. Besides this motion, there are two other kinds ol
motion considered in the kinetic theory of gases. The first is
the motion of agitation of the molecules. This is the hitherto
invisible motion of the molecule considered as a whole.
Its course consists of broken portions, called free paths,
interrupted by the encounters between different molecules.
The second is the internal motion of each molecule,
Pressure of a Gas. 319
consisting partly of rotation and partly of vibrations among
the component parts of the molecule.
The velocity of the centre of gravity of a molecule is the
resultant of the velocity of the gas and the velocity of agita-
tion of the individual molecule at the given instant. The
velocity of a constituent part of a molecule is the resultant
of the velocity of its centre of gravity and the velocity of
the constituent part relatively to the centre of gravity of the
molecule.
THEORY OF THE PRESSURE OF A GAS.
Let us consider two portions of a gas separated by a plane
surface which moves with the same velocity as
the gas. We have seen that in this case the
number of molecules which pass through the
plane in opposite directions is the same.
Each molecule in crossing the plane from
the region A to the region B enters the second
region in precisely the same state as it leaves the first. It
therefore carries over into the region B, not only its mass,
but its momentum and its kinetic energy. Hence, if we
consider the quantity of momentum in a given direction
existing at any instant in the particles in the region B, this
quantity will be altered whenever a molecule crosses the
boundary, carrying its momentum along with it.
Now let us consider all the molecules whose velocity
differs by less than a certain quantity, c, from a given velocity
the components of which are u in the direction perpen-
dicular to the plane from A towards B, and v and w in two
other directions parallel to the plane. Let there be N
molecules whose velocity is within these limits in every unit
of volume, and let the mass of each of these be M.
Then the number of these molecules which will cross unit
of area of the plane from A to B in unit of time is
N U
320 Molecular Theory.
The momentum of each of these molecules resolved in
the direction A B is M u.
Hence the momentum in this direction communicated to
the region B in unit of time is
M N u 2 .
Since this bombardment of the region B does not pro-
duce motion of the gas, a pressure must be exerted on
the gas by the sides of the vessel, and the amount of this
pressure for every unit of area must be M N u 2 .
The region A loses positive momentum at the same rate,
and in order to preserve equilibrium there must be a pressure
equal to M N * on every unit of area of the surface of the
region A.
Hitherto we have considered only one group of molecules,
whose velocities lie between given limits. In every such
group that which determines the pressure in the direction A B
on the surface separating A from B is a quantity of the form
M N u 2 , where N is the number of molecules in the group, and
u is the velocity of each molecule resolved in the direction
A B. The other components of the velocity do not influence
the pressure in this direction.
To find the whole pressure, we must find the sum of all
such expressions as M N 2 for all the groups of molecules
in the system. We may write this result p = M N # 2 , where
N now signifies the total number of molecules in unit of
volume, and u 2 denotes the mean value of u* for all these
molecules. Now if v 2 is the square of the velocity without
regard to direction, v 2 u 2 -f v 2 + a/ 2 , where u v w are the
components in three directions at right angles. Hence if
u 2 , v 2 , and w 2 denote the mean square of these components,
and v 2 the mean square of the resultant, v 2 = u 2 + v 2 + w 2 .
When, as in every gas at rest, the pressure is equal in all
directions, u 2 = v 2 = w 2 , and therefore v 2 = 3 u 2 .
Hence the pressure of a gas is
/-JMNV 2 . ...... (2)
Velocity of Molecular Motion. 3 2 1
where M is the mass of each molecule, N is the number ot
molecules in unit of volume and v is the mean square of
the velocity.
In this expression there are two quantities which have
never been directly measured the mass of a single molecule,
and the number of molecules in unit of volume. But we
have here to do with the product of these quantities, which
is evidently the mass of the substance in unit of volume, or in
other words, its density. Hence we may write the expression
/ = *pv* (3)
where p is the density of the gas.
It is easy from this expression to determine, as was first
done by Joule, the mean square of the velocity of the
molecules of a gas, for
v* = 3^ (4)
where p is the pressure, and p the density, which must of
course be expressed in terms of the same fundamental units.
For instance, under the atmospheric pressure of 2116-4
pounds weight on the square foot, and at the temperature of
melting ice, the density of hydrogen is 0*005592 pounds in
a cubic foot. Hence ^-= 378470 in gravitation units, and
P
if the intensity of gravity where this relation was observed
was 32-2, we have v 2 = 36560000,0^ taking the square root
of this quantity, v = 6046 feet per second.
This is the velocity of mean square for the molecules of
hydrogen at 32 F. and at the atmospheric pressure.
LAW OF BOYLE.
Two bodies are said to be of the same temperature
when there is no more tendency for heat to pass from the
first to the second than in the reverse direction. In the
kinetic theory of heat, as we have seen, this thermal equili-
Y
322 Molecular Theory.
brium is established when there is a certain relation between
the velocities of agitation of the molecules of the two bodies.
Hence the temperature of a gas must depend on the velocity
of agitation of its molecules, and this velocity must be the
same at the same temperature, whatever be the density.
In the expression p = ^ p v 2 , the quantity v 2 depends
only on the temperature as long as the gas remains the
same. Hence when the density p varies, the pressure p
must vary in the same proportion. This is Boyle's law,
which is now raised from the rank of an experimental fact
to that of a deduction from the kinetic theory of gases.
If v denotes the volume of unit of mass, we may write this
expression
pv = V . (5)
Now/ v is proportional to the absolute temperature, as
measured by a thermometer, of the particular gas under
consideration. Hence v 2 , the mean square of the velocity of
agitation, is proportional to the absolute temperature mea-
sured in this way.
[In the preceding calculation of the pressure of a gas
it is assumed that the time during which the particles
are subject to one another's influence is negligeable in com-
parison with the time during which they are free. By means
of the equation of virial, established by Clausius, it is possible
greatly ta extend the generality of the investigation.
When an attraction or repulsion exists between two points,
half the product of the stress into the distance between the
two points is called the Virial of the stress, and is reckoned
positive when the stress is an attraction, and negative when
it is a repulsion. The virial of a system is the sum of the
virial of the stresses which exist in it.
As applied to a system of moving particles, the equation
may be written
i2/;/v 2 =:f/z; + i22(Rr).
The left-hand member denotes the kinetic energy.
Virial. 323
On the right hand, in the first term, p is the external
pressure on unit of area, and v is the volume of the vessel.
The second term represents the virial arising from the
action between every pair of particles. R is the attractior
between the particles, and r the distance between them
The double sign of summation is used because every pair o/
points must be taken into account, those between which
there is no stress contributing, of course, nothing to the
virial.
A general idea of the manner in which virial acts in oppo-
sition to kinetic energy may be obtained from the very
simple case of two equal masses m revolving in circular
paths about their centre of gravity. If p be the radius
of the circular path, r = 2 p, \ 2 2 (R r) = R p, ^ S m v 2
= m v 2 , so that the equation expresses the ordinary law of
centrifugal force,
m v*/p = R.
In gases the virial is very small compared with the kinetic
energy. Hence, if the kinetic energy is constant, the pro-
duct of the pressure and the volume remains constant. This
is the case for a gas at constant temperature.
In liquids and in highly compressed gas the virial becomes
important, and if we assume that the temperature is still
measured by the mean kinetic energy of a molecule, we
obtain the means of determining it I y observing the devia-
tion of the product of the pressure and volume from the
constant value given by Boyle's law.
It appears by Dr. Andrews' experiments that when the
volume of carbonic acid is diminished, the temperature
remaining constant, the product of the volume and pressure
at first diminishes, the rate of diminution becoming more and
more rapid as the density increases. Now, the virial depends
upon the number of pairs of molecules which are at a given
instant acting upon one another, and this number in unit
of volume is proportional to the square of the density.
Y 2
324 Molecular Theory.
Hence the part of the pressure depending on the virial
increases as the square of the density, and since in the case
of carbonic acid it diminishes the pressure, it must be of
the positive sign, that is, it must arise from attraction between
the molecules.
But if the volume is still further diminished, at a certain
point liquefaction begins, and from this point till the gas is
all liquefied no increase of pressure takes place. As soon,
however, as the whole substance is in the liquid condition,
any further diminution of volume produces a great rise of
pressure, so that the product of pressure and volume in-
creases rapidly. This indicates negative virial, and shows
that the molecules are now acting upon each other by
repulsion.
This is what takes place in carbonic acid below the tem-
perature of 30-9 C. Above that temperature there is first
a positive and then a negative virial, but no sudden lique-
faction. Similar phenomena occur in all the liquefiable
gases.
We have thus evidence that the molecules of gases attract
each other at a certain small distance, but when they are
brought still nearer they repel each other. This is quite in
accordance with Boscovitch's theory of atoms as massive
centres of force, the force being a function of the distance,
and changing from attractive to repulsive, and back again
several times, as the distance diminishes. If we suppose
that when the force begins to be repulsive it increases very
rapidly as the distance diminishes, so as to become enor-
mous if the distance is less by a very small quantity than
that at which, the force first begins to be repulsive, the
phenomena will be precisely the same as those of smooth
elastic spheres. 1
Van der Waals, to whom we owe these applications, has
shown, further, how to take into account the action of mutual
forces such as those treated by Laplace in his theory of
1 Maxwell, Nature^ vol. x. p. 477, 1874.
Law of Gay-Lussac. 325
capillarity. The range of these forces is supposed to be
very small in comparison with the dimensions of ordinary
bodies, but large in comparison with the molecular dis-
tances. The effect of such forces in the virial equation is
to cause the addition to p, the pressure exercised by the
walls of the containing vessel, of Laplace's intrinsic pressure
K, which prevails in the interior of the liquid in consequence
of these forces. R.]
LAW OF GAY-LUSSAC.
Let us next consider two different gases in thermal equi-
librium. We have already stated that if M t M 2 are the
masses of individual molecules of these gases, and v l v 2
their respective velocities of_ agitation,_ it is necessary for
thermal equilibrium that MJ v t 2 = M 2 V 2 2 by equation (i).
If the pressures of these gases are # l and / 2 , and the
number of molecules in unit of volume N t and N 2 , then, by
equation (2),
p l J MJ N! v t 2 and / 2 = M 2 N 2 V 2 2 .
If the pressures of the two gases are equal,
MI N! Vj 2 = M 2 N 2 v 2 2 .
If their temperatures are equal,
Mj V, 2 = M 2 V 2 2 .
Dividing the terms of the first of these equations by those
of the second, we find
Nj = N 2 (6)
or when two gases are at the same pressure and tempera-
ture^ the number of molecules in unit of volume is the same in
both gases.
If we put (0, = M! N! and p 2 = M 2 N 2 for the densities of
the two gases, then, since NJ = N 2 , we get
A>i : ,o 2 :: M! : M 2 (7)
326 Molecular Theory.
or the densities of two gases at the same temperature and
pressure are proportional to the masses of their individual
molecules.
These two equivalent propositions are the expression of
a very important law established by Gay-Lussac, that the
densities of gases are proportional to their molecular
weights.
[In a subsequent publication the author recognised the
insufficiency of this proof. 'If the system is a gas or a
mixture of gases not acted on by external forces, the theorem
that the average kinetic energy for a single molecule is
the same for molecules of different gases is not sufficient
to establish the condition of equilibrium of temperature
between gases of different kinds, such as oxygen and
nitrogen, because when the gases are mixed we have no
means of ascertaining the temperature of the oxygen and
of the nitrogen separately. We can only ascertain the
temperature of the mixture by putting a thermometer
into it.' 1
The law of the equality of kinetic energies was stated by
Waterston in a memoir communicated to the Royal Society
in 1845. R.]
The proportion by weight in which different substances
combine to form chemical compounds depends, according to
Dalton's atomic theory, on the weights of their molecules,
and it is one of the most important researches in chemistry
to determine the proportions of the weights of the molecules
from the proportions in which they enter into combination.
Gay-Lussac discovered that in the case of gases the volumes
of the combining quantities of different gases always stand
in a simple ratio to each other. This law of volumes has
now been raised from the rank of an empirical fact to that of
a deduction from our theory, and we may now assert, as a
dynamical proposition, that the weights of the molecules of
1 Camb. Trans. 1879.
Law of Charles. 3 2 7
gases (that is, those small portions which do not part com-
pany during their motion) are proportional to the densities
of these gases at standard temperature and pressure.
LAW OF CHARLES.
We must next consider the effect of changes of temperature
on different gases. Since at all temperatures, when there is
thermal equilibrium,
MiV = M 2 v 2 a ;
and since the absolute temperature, as measured by a gas*
thermometer, is proportional to Vj 2 when the gas is of the
first kind, and to v 2 2 when the gas is of the second kind; it
follows, since Vj 2 is itself proportional to V 2 2 , that the
absolute temperatures, as measured by the two thermometers,
are proportional, and if they agree at any one temperature
(as the freezing point), they agree throughout. This is
the law of the equal dilatation of gases discovered by
Charles.
KINETIC ENERGY OF A MOLECULE.
The mean kinetic energy of agitation of a molecule is the
product of its mass by half the mean square of its velocity, or
' MV 2 .
This is the energy due to the motion of the molecule as a
whole, but its parts may be in a state of relative motion. If
we assume, with Clausius, that the energy due to this
internal motion of the parts of the molecule tends towards a
value having a constant ratio to the energy of agitation, the
whole energy will be proportional to the energy of agitation,
and may be written
\ ft M V 2 ,
where /3 is a factor, always greater than unity, and probably
equal to 1*634 for a "* an( i several of the more perfect gases.
For steam it may be as much as 2*19, but this is very
uncertain.
328 Molecular TJieory.
To find the kinetic energy of the substance contained in
unit of volume, we have only to multiply by the number of
molecules, and we obtain
T = i/3M Nv 2 . . . . . . . . . (8)
Comparing this with the equation (2) which determines
the pressure, we get
T v -f /3/ (9)
or the energy in unit of volume is numerically equal to the
pressure on unit of area multiplied by f /3.
The energy in unit of mass is found by multiplying this
by v, the volume of unit of mass :
T m = |/3/z> . ... . . . . (10)
SPECIFIC HEAT AT CONSTANT VOLUME.
Since the product p v is proportional to the absolute tem-
perature, the energy is proportional to the temperature.
The specific heat is measured dynamically by the increase
of energy corresponding to a rise of one degree of temperature.
Hence
*,=*(** .(II)
To express the specific heat in ordinary thermal units, we
must divide this by j, the specific heat of water (Joule's
equivalent). It follows from this expression that for any
one gas the specific heat of unit of mass at constant volume
is the same for all pressures and temperatures, because ^-~-
a
remains constant. For different gases the specific heat at
constant volume is inversely proportional to the specific
gravity, and directly proportional to /3.
Since p is nearly the same for several gases, the specific
heat of these gases is inversely proportional to their specific
gravity referred to air, or, since the specific gravity is pro-
portional to their molecular weight, the specific heat multi-
plied by the molecular weight is the same for all these gases.
Law of Dulong and Petit. 3 29
This is the law of Dulong and Petit. It would be accu-
rate for all gases if the value of /3 were the same in every
case.
It has been shown at p. 183 that the difference of the two
specific heats is Jf. Hence their ratio, y, is
u
If u is the velocity of sound in a gas, we have, as at p. 228,
u a = y p v ......... (12)
The mean square of the velocity of agitation is
v 2 = 3 /.E ......... (13)
Hence u =^/- v, or, if y = 1-408, as in air and severaJ
O
other gases,
u = '6858 v or v = i -458 u . . (14)
These are the relations between the velocity of sound and
the velocity of mean square of agitation in any gas for which
y = 1-408.
The nature of this book admits only of a brief account of
some other results of the kinetic theory of gases. Two of
these are independent of the nature of the action between
the molecules during their encounters.
The first of these relates to the equilibrium of a mixture of
gases acted on by gravity. The result of our theory is that
the final distribution of any number of kinds of gas in a
vertical vessel is such that the density of each gas at a
given height is the same as if all the other gases had been
removed, leaving it alone in the vessel.
This is exactly the mode of distribution which Dalton
supposed to exist in a mixed atmosphere in equilibrium, the
law of diminution of density of each constituent gas being
the same as if no other gases were present.
In our atmosphere the continual disturbances caused by
winds carry portions of the mixed gases from one stratum
330 Molecular TJicory.
to another, so that the proportion of oxygen and nitrogen at
different heights is much more uniform than if these gases
had been allowed to take their places by diffusion during a
dead calm.
The second result of our theory relates to the thermal equi-
librium of a vertical column. We find that if a vertical
column of a gas were left to itself, till by the conduction
of heat it had attained a condition of thermal equilibrium,
the temperature would be the same throughout, or, in other
words, gravity produces no effect in making the bottom of
the column hotter or colder than the top.
This result is important in the tneory of thermodynamics,
for it proves that gravity has no influence in altering the
conditions of thermal equilibrium in any substance, whether
gaseous or not. For if two vertical columns of different
substances stand on the same perfectly conducting horizontal
plate, the temperature of the bottom of each column will be
the same ; and if each column is in thermal equilibrium of
itself, the temperatures at all equal heights must be the same.
In fact, if the temperatures of the tops of the two columns
were different, we might drive an engine with this difference of
temperature, and the refuse heat would pass down the colder
column, through the conducting plate, and up the warmer
column; and this would go on till all the heat was converted
into work, contrary to the second law of thermodynamics.
But we know that if one of the columns is gaseous, its
temperature is uniform. Hence that of the other must be
uniform, whatever its material.
This result is by no means applicable to the case of our
atmosphere. Setting aside the enormous direct effect of
the sun's radiation in disturbing thermal equilibrium, the
effect of winds in carrying large masses of air from one
height to another tends to produce a distribution of tem-
perature of a quite different kind, the temperature at any
height being such that a mass of air, brought from one height
to another without gaining or losing heat, would always nnd
Diffusion, Conduction, and Viscosity. 331
itself at the temperature of the surrounding air. In thig
condition of what Sir William Thomson has called the Con
vective equilibrium of heat, it is not the temperature which
is constant, but the quantity (f>, which determines the adia-
batic curves.
In the convective equilibrium of temperature, the abso-
lute temperature is proportional to the pressure raised to
the power ^-^ , or 0*29.
The extreme slowness of the conduction of heat in air,
compared with the rapidity with which large masses of air
are carried from one height to another by the winds, causes
the temperature of the different strata of the atmosphere to
depend far more on this condition of convective equilibrium
than on true thermal equilibrium.
We now proceed to those phenomena of gases which,
according to the kinetic theory, depend upon the particular
nature of the action which takes place when the molecules
encounter each other, and on the frequency of these
encounters.
There are three phenomena of this kind of which the
kinetic theory takes account the diffusion of gases, the
viscosity of gases, and the conduction of heat through a gas.
We have already described the known facts about the
interdiffusion of two different gases. It is only when the
gases are chemically different that we can trace the process
of diffusion, but on the molecular theory diffusion is always
going on, even in a single gas ; only it is impossible to trace
the progress of the molecules, because we cannot tell one
from another.
The relation between diffusion and viscosity may be
explained as follows : Consider the case of motion of a mass
of gas, which has already been described in Chapter XXI., in
which the different horizontal layers of the gas slide over
each other. In diffusion the molecules pass, some of them
upwards and some of them downwards, through any
332 Molecular Theory.
horizontal plane. If the medium has different properties of
any kind above and below this plane, then this interchange
of molecules will tend to assimilate the properties of the two
portions of the medium.
In the case of ordinary diffusion, the proportions of the
two diffusing substances are different above and below, and
vary in the different horizontal layers according to their
height In the case of internal friction, the mean horizontal
momentum is different in the different layers, and when the
molecules pass through the plane, carrying their momentum
with them, this exchange of momentum between the upper
and lower parts of the medium constitutes a force tending to
equalize their velocity, and this is the phenomenon actually
observed in the motion of viscous fluids.
The coefficient of viscosity, when measured in the kine-
matic way, represents the rate at which the equalization of
velocity goes on by the exchange of the momentum of the
molecules, just as the coefficient of diffusion represents the
rate at which the equalization of chemical composition goes
on bv the exchange of the molecules themselves.
It appears from the kinetic theory of gases that if D is
the coefficient of diffusion of the gas into itself, and v the
viscosity measured kinematically,
v = 0-6479 D ......... (15)
D = J'5435 " ......... (16)
The conduction of heat in a gas, according to the kinetic
theory, is simply the diffusion of the energy of the molecules
by their moving about in the medium and carrying their
energy with them till they encounter other molecules, when
the energy is redistributed. The relation of the conduc-
tivity K, measured thermometrically, to the viscosity v,
measured kinematically, is
It appears, therefore, that diffusion, viscosity, and conduc-
Evaporation and Condensation. 333
tivity in gases are related to each other in a very simple
way, being the rate of equalization of three properties of the
medium the proportion of its ingredients, its velocity, and
its temperature. The equalization is effected by the same
agency in each case namely, the agitation of the molecules.
In each case, if the density remains the same, the rate of
equalization is proportional to the absolute temperature;
and if the temperature remains the same, the rate of equal-
ization is inversely proportional to the density. Hence,
if we consider the temperature and the pressure as defining
the state of the gas, the quantities D, v, and K vary directly
as the square of the absolute temperature and inversely as
the pressure.
MOLECULAR THEORY OF EVAPORATION AND CONDENSATION.
The mathematical difficulties arising in the investigation
of the motions of molecules are so great that it is not to be
wondered at that most of the numerical results are confined
to the phenomena of gases. The general character, however,
of the explanation of many other phenomena by the mole-
cular theory has been pointed out by Clausius and others.
We have seen that in the case of a gas some of the mole-
cules have a much greater velocity than others, so that it is
only to the average velocity of all the molecules that we can
ascribe a definite value. It is probable that this is also true
of the motions of the molecules of a liquid, so that, though
the average velocity may be much smaller than in the vapour
of that liquid, some of the molecules in the liquid may have
velocities equal to or greater than the average velocity in
the vapour. If any of the molecules at the surface of the
liquid have such velocities, and if they are moving from the
liquid, they will escape from those forces which retain the
other molecules as constituents of the liquid, and will fly
about as vapour in the space outside the liquid. This is
the molecular theory of evaporation. At the same time, a
molecule of the vapour striking the liquid may become
334 Molecular Theory.
entangled among the molecules of the liquid, and may thus
become part of the liquid. This is the molecular explanation
of condensation. The number of molecules which pass from
the liquid to the vapour depends on the temperature of the
liquid. The number of molecules which pass from the
vapour to the liquid depends upon the density of the vapour
as well as its temperature. If the temperature of the vapour
is the same as that of the liquid, evaporation will take place
as long as more molecules are evaporated than condensed ;
but when the density of the vapour has increased to such a
value that as many molecules are condensed as evaporated,
then the vapour has attained its maximum density. It is
then said to be saturated, and it is commonly supposed that
evaporation ceases. According to the molecular theory,
however, evaporation is still going on as fast as ever ; only,
condensation is also going on at an equal rate, since the
proportions of liquid and of gas remain unchanged.
A similar explanation applies to cases in which the vapour
or gas is absorbed by a liquid of a different kind, as when
oxygen or carbonic acid is absorbed by water or alcohol. In
such cases a ' movable equilibrium ' is attained when the
liquid has absorbed a quantity of the gas whose volume at
the density of the unabsorbed gas is a certain multiple or
fraction of the volume of the liquid ; or, in other words, the
density of the gas in the liquid and outside the liquid stand
in a certain numerical ratio to each other. This subject is
treated very fully in Bunsen's ' Gasometry.'
The amount of vapour of a liquid diffused into a gas of a
different kind is generally independent of the nature of the
gas, except when the gas acts chemically on the vapour.
Dr. Andrews has shown ('Proc. R.S.' 1875) tnat by mix-
ing nitrogen with carbonic acid, the critical temperature is
lowered, and that Dalton's law of the density of mixed
vapours only holds at low pressures and at temperatures
greatly above their critical points,
Electrolysis. 335
MOLECULAR THEORY OF ELECTROLYSIS.
A very interesting part of molecular science which has not
been thoroughly worked out, but which hardly belongs to a
treatise on Heat, is the theory of electrolysis. Here an
electromotive force acting on a liquid electrolyte causes
the molecules of one of its components to be urged in one
direction, while those of the other component are urged in
the opposite direction. Now these components are joined
together in pairs by chemical forces of great power, so that
we might expect that no electrolytic effect could take place
unless the electromotive force were so strong as to be able
to tear these couples asunder. But, according to Clausius, in
the dance of molecules which is always going on, some of the
linked pairs of molecules acquire such velocities that when
they have an encounter with a pair also in violent motion
the molecules composing one or both of the pairs are torn
asunder, and wander about seeking new partners. If the
temperature is so high that the general agitation is so violent
that more pairs of molecules are torn asunder than can pair
again in an equal time, we have the phenomenon of
Dissociation, studied by M. Ste. -Claire Deville. If, on the
other hand, the separated molecules can always find partners
before they are ejected from the system, the composition of
the system remains apparently the same.
Now Professor Clausius considers that it is during these
temporary separations that the electromotive force comes
into play as a directing power, causing the molecules of
one component to move on the whole one way, and those
of the other the opposite way. Thus the component mole-
cules are always changing partners, even when no electro-
motive force is in action, and the only effect of this force is tc
give direction to those movements which are already going on.
Professor Wiedemann, who has also taken this view of
electrolysis, compares the phenomenon with that of diffusion,
and shows that the electric conductivity of an electrolyte aiaj
336 Molecular Theory.
be considered as depending on the coefficient of diffusion of
the components through each other.
MOLECULAR THEORY OF RADIATION.
The phenomena already described are explained on the
molecular theory by the motion of agitation of the molecules,
a motion which is exceedingly irregular, the intervals between
successive encounters and the velocities of a molecule
during successive free paths not being subject to any law
which we can express. The internal motion of a single
molecule is of a very different kind. If the parts of the
molecule are capable of relative motion without being
altogether torn asunder, this relative motion will be some
kind of vibration. The small vibrations of a connected sys-
tem may be resolved into a number of simple vibrations, the
law of each of which is similar to that of a pendulum. It is
probable that in gases the molecules may execute many of
such vibrations in the interval between successive encounters.
At each encounter the whole molecule is roughly shaken.
During its free path it vibrates according to its own laws,
the amplitudes of the different simple vibrations being deter-
mined by the nature of the collision, but their periods
depending only on the constitution of the molecule itself.
If the molecule is capable of communicating these vibrations
to the medium in which radiations are propagated, it will
send forth radiations of certain definite kinds, and if these
belong to the luminous part of the spectrum, they will be
visible as light of definite refrangibility. This, then, is the
explanation, on che molecular theory, of the bright lines
observed in the spectra of incandescent gases. They repre-
sent the disturbance communicated to the luminiferous
medium by molecules vibrating in a regular and periodic
manner during their free paths. If the free path is long,
the molecule, by communicating its vibrations to the ether,
will cease to vibrate till it encounters some other molecule.
By raising the temperature we increase the velocity of
Radiation. 337
the motion of agitation and the force of each encounter.
The higher the temperature the greater will be the ampli-
tude of the internal vibrations of all kinds, and the more
likelihood will there be that vibrations of short period will
be excited, as well as those fundamental vibrations which
are most easily produced. By increasing the density we
diminish the length of the free path of each molecule, and
thus allow less time for the vibrations excited at each
encounter to subside, and, since each fresh encounter dis-
turbs the regularity of the series of vibrations, the radiation
will no longer be capable of complete resolution into a
series of vibrations of regular periods, but will be analysed
into a spectrum showing the bright bands due to the regular
vibrations, along with a ground of diffused light, forming a
continuous spectrum due to the irregular motion introduced
at each encounter.
Hence when a gas is rare the bright lines of its spectrum
are narrow and distinct, and the spaces between them are
dark. As the density of the gas increases, the bright lines
become broader and the spaces between them more
luminous.
There is another reason for the broadening of the bright
lines and the luminosity of the whole spectrum in dense
gases, which we have already stated at p. 245. There is
this difference, however, between the effect there mentioned
and that described here. At p. 245 the light from a
certain stratum of incandescent gas was supposed to pene-
trate through other strata, which absorbed the brighter rays
faster than the less luminous ones. This effect depends
only on the total quantity of gas through which the rays
pass, and will be the same whether it is a mile of gas at
thirty inches pressure, or thirty miles at one inch pressure.
The effect which we are now considering depends on the
absolute density, so that it is by no means the same whether
a stratum containing a given quantity of gas is one mile or
thirty miles thick.
338 Molecular Theory.
When the gas is so far condensed that it assumes the
liquid or solid form, then, as the molecules have no free
path, they have no regular vibrations, and no bright lines
are commonly observed in incandescent liquids or solids.
Mr. Huggins, however, has observed bright lines in the
spectrum of incandescent erbia and lime, which appear to
be due to the solid matter, and not to its vapour.
LIMITATION OF THE SECOND LAW OF THERMODYNAMICS.
Before I conclude, I wish to direct attention to an aspect
of the molecular theory which deserves consideration.
One of the best established facts in thermodynamics is
that it is impossible in a system enclosed in an envelope
which permits neither change of volume nor passage of heat,
and in which both the temperature and the pressure are every-
where the same, to produce any inequality of temperature or
of pressure without the expenditure of work. This is the
second law of thermodynamics, and it is undoubtedly true
as long as we can deal with bodies only in mass, and have
no power of perceiving or handling the separate molecules
of which they are made up. But if we conceive a being
whose faculties are so sharpened that he can follow every
molecule in its course, such a being, whose attributes are still
as essentially finite as our own, would be able to do what is
at present impossible to us. For we have seen that the
molecules in a vessel full of air at uniform temperature are
moving with velocities by no means uniform, though the
mean velocity of any great number of them, arbitrarily
selected, is almost exactly uniform. Now let us suppose
that such a vessel is divided into two portions, A and B, by
a division in which there is a small hole, and that a being,
who can see the individual molecules, opens and closes this
hole, so as to allow only the swifter molecules to pass
from A to B, and only the slower ones to pass from B to A.
He will thus, without expenditure of work, raise the tern-
Statistical Knowledge of Bodies. 339
perature of B and lower that of A, in contradiction to the
second law of thermodynamics.
This is only one of the instances in which conclusions
which we have drawn from our experience of bodies con-
sisting of an immense number of molecules may be found
not to be applicable to the more delicate observations and
experiments which we may suppose made by one who can
perceive and handle the individual molecules which we deal
with only in large masses.
In dealing with masses of matter, while we do not perceive
the individual molecules, we are compelled to adopt what I
have described as the statistical method of calculation, and
to abandon the strict dynamical method, in which we follow
every motion by the calculus.
It would be interesting to enquire how far those ideas
about the nature and methods of science which have been
derived from examples of scientific investigation in which
the dynamical method is followed are applicable to our
actual knowledge of concrete things, which, as we have seen,
is of an essentially statistical nature, because no one has
yet discovered any practical method of tracing the path
of a molecule, or of identifying it at different times.
I do not think, however, that the perfect identity which
we observe between different portions of the same kind of
matter can be explained on the statistical principle of the
stability of the averages of large numbers of quantities
each of which may differ from the mean. For if of the
molecules of some substance such as hydrogen, some were
of sensibly greater mass than others, we have the means
of producing a separation between molecules of different
masses, and in this way we should be able to produce two
kinds of hydrogen, one of which would be somewhat denser
than the other. As this cannot be done, we must admit that
the equality which we assert to exist between the molecules
of hydrogen applies to each individual molecule, and not
merely to the average of groups of millions of molecules.
z 2
Molecular Theory.
NATURE AND ORIGIN OF MOLECULES.
We have thus been led by our study of visible things to a
theory that they are made up of a finite number of parts or
molecules, each of which has a definite mass, and possesses
other properties. The molecules of the same substance are
all exactly alike, but different from those of other substances.
There is not a regular gradation in the mass of molecules
fiom that of hydrogen, which is the least of those known to
us, to that of bismuth ; but they all fall into a limited
number of classes or species, the individuals of each
species being exactly similar to each other, and no inter-
mediate links are found to connect one species with
another by a uniform gradation.
We are here reminded of certain speculations concerning
the relations between the species of living things. We find
that in these also the individuals are naturally grouped into
specie?,, and that intermediate links between the species are
wanting. But in each species variations occur, and there is
a perpetual generation and destruction of the individuals of
which the species consist.
Hence it is possible to frame a theory to account for the
present state of things by means of generation, variation,
and discriminative destruction.
In the case of the molecules, however, each individual is
permanent ; there is no generation or destruction, and no
variation, or rather no difference, between the individuals of
each species.
Hence the kind of speculation with which we have
become so familiar under the name of theories of evolution
is quite inapplicable to the case of molecules.
It is true that Descartes, whose inventiveness knew no
bounds, has given a theory of the evolution of molecules.
He supposes that the molecules with which the heavens
are nearly filled have received a spherical form from the
long-continued grinding of their projecting parts, so that,
Equality and Permanence of Molecules. 341
like marbles in a mill, they have ' rubbed each other's angles
down.' The result of this attrition forms the finest kind of
molecules, with which the interstices between the globular
molecules are filled. But, besides these, he describes another
elongated kind of molecules, the particula striata, which
have received their form from their often threading the
interstices between three spheres in contact. They have thus
acquired three longitudinal ridges, and, since some of them
during their passage are rotating on their axes, these ridges
are not in general parallel to the axis, but are twisted like
the threads of a screw. By means of these little screws
he most ingeniously attempts to explain the phenomena of
magnetism.
But it is evident that his molecules are very different from
ours. His seem to be produced by some general break-up
of his solid space, and to be ground down in the course of
ages, and, though their relative magnitude is in some degree
determinate, there is nothing to determine the absolute
magnitude of any of them.
Our molecules, on the other hand, are unalterable by any
of the processes which go on in the present state of things,
and every individual of each species is of exactly the same
magnitude, as though they had all been cast in the same
mould, like bullets, and not merely selected and grouped
according to their size, like small shot.
The individuals of each species also agree in the nature of
the light which they emit that is, in their natural periods of
vibration. They are therefore like tuning-forks all tuned to
concert pitch, or like watches regulated to solar time.
In speculating on the cause of this equality we are debarred
from imagining any cause of equalization, on account of the
immutability of each individual molecule. It is difficult, on the
other hand, to conceive of selection and elimination of inter-
mediate varieties, for where can these eliminated molecules
have gone to if, as we have reason to believe, the hydrogen,
&c., of the fixed stars is composed of molecules identical in
34 2 Molecular Theory.
all respects with our own ? The time required to eliminate
from the whole of the visible universe every molecule whose
mass differs from that of some one of our so-called elements,
by processes similar to Graham's method of dialysis, which
is the only method we can conceive of at present, would
exceed the utmost limits ever demanded by evolutionists
as many times as these exceed the period of vibration of a
molecule.
But if we suppose the molecules to be made at all, or if
we suppose them to consist of something previously made,
why should we expect any irregularity to exist among them ?
If they are, as we believe, the only material things which
still remain in the precise condition in which they first
began to exist, why should we not rather look for some
indication of that spirit of order, our scientific confidence
in which is never shaken by the difficulty which we expe-
rience in tracing it in the complex arrangements of visible
things, and of which our moral estimation is shown in all
our attempts to think and speak the truth, and to ascertain
the exact principles of distributive justice ?
APPENDIX.
Table of the Coefficients of Interdiffusion of Gases, from the Memoir of
Professor Loschmidt (see /. 279), in square centimetres per second
D
Carbonic acid . . Air . r . '1423
Hydrogen -'*+ . '5614
Oxygen . ; ~ . '1409
Marsh gas . . '1586
Carbonic oxide . . '1406
Nitrous oxide . . "0982
Oxygen . . . Hydrogen . . 7214
Carbonic oxide . . '1802
Carbonic oxide . . Hydrogen . . -6422
Sulphurous acid . . Hydrogen . .
Appendix. 343
Professor J. Stefan, also of Vienna, has undertaken a series of very
delicate experiments to determine the thermal conductivity of air and other
gases. He finds the thermometric conductivity, /c, of air 0-256 square
centimetres per second. The rate of propagation of thermal effects in
still air is therefore intermediate between the rate in iron, for which
= 0-183, and in copper, for which K =1-077. Stefan finds it inter-
mediate between iron and zinc.
The calorimetric conductivity, k, is 0*00005 5 8 for air, or about 20,000
times less than that of copper, and 3, 360 times less than that of iron.
As calculated from the coefficient of viscosity by the writer
= 0-000054.
Stefan has also found that the calorimetric conductivity is inde-
pendent of the pressure, and that it is seven times greater for hy-
drogen than for air. Both these results had been predicted by the
molecular theory. See Maxwell ' On the Dynamical Theory of Gases,'
Phil. Trans. 1867, p. S8.
INDEX.
ABSOLUTE, temperature, 51, 159, Thom-
son's scale of, 155 ; zero, 215
Absorption of heat, 243
Aciiabatic lines, 129, 135
Air, thermometer, 46; velocity of sound
in, 182
Aitken on the formation of fogs, 293
Amiot's method of double weighing, 69
Andrews' experiments on gases, 118, 323
Athermanous bodies, 13
Atmosphere, height of, 220 ; homo-
geneous, 220, 229
Available energy, 187
BAROMETER, 97 ; determination of
heights by, 217
Bernoulli on the pressure of gases, 312
Bismuth, conductivity of, 270
Black's experiments on latent heat, 58
Boiling, 23 ; point of a thermometer, 33 ;
conditions of, 291
Borda's method of weighing, 69
Boscovich's theory, 86, 324
Boyle's law, 27, 30
Bumping of liquids on boiling, 126, 129,
291
Bunsen's calorimeter, 61 ; Gasometry,
334
CAGNIARD de la Tour, experiments of,
1 18, 204
Cailletet's experiments on the liquefac-
tion ot gases, 17
Caloric, 57
Calorie, 7, 9
Calorimeter, 7 ; ice, 58
Calorimetry, 7
Capacity of a body for heat, 65
Capillarity, 281 ; angl^ of, 288; coeffi-
cient of, 283 ; Laplace's theory of, 292 ;
its connection with latent heat, 297 ;
with thermodynamics, 290
Capillary attraction, 285
Carnot's function, 155, 162 ; principle,
153 ; reversible engine, 139
Celsius degrees, 37
Centigrade scale, 37
Chamber of uniform temperature, 272
Charles, law of, 29, 327
Clapeyron, introduction of indicator
diagram by, 102
Clausius' statement of Carnot's princi-
ple, 153 ; on entropy, 162 ; development
of the mechanical theory of gases, 312
Clouds, subsidence of, 307
Cohesion figures, 286
Colloids, 280
Condensation, theory of, 333
Conduction, 10, n, 253 ; in a solid, 257
Conductivity, dynamical measure of,
255 ; electrical, 271 ; influence oi
temperature on, 271
Conservation of force, 91 ; of energy, 92
Convection, 10, 12 ; currents, 250
Convective equilibrium of heat. 250
Cooling, method of, 58 ; rate of, 68
Critical state, 204
Crystalloids, 280
Currents, convection, 251
Curves, adiabatic, 135
Cycle, definition of, 142
DALTON'S law, 28
Delambre and Mechain's measurement
of the size of the earth, 77
Density of a body, 82 ; maximum, of
water, 252
Descartes' theory of the evolution of
molecules, 340
Diagram of the effects of heat on water,
137 ; indicator, 102
Dialysis, 342
346
Index.
Diathermanous bodies, 13
Dielectrics, electrical conductivity of,
271 ^
Diffusion, of heat, 10 ; by conduction,
253 ; of liquids, 273 ; of gases, 276
Dilatability, 167
Dissipation of energy, 192, 204
Distribution of temperature, harmonic,
263
Dufour's experiments on boiling, 291
Dulong and Pe tit's law, 329 ; formula, 246
Dynamical measure of conductivity, 255 ;
equivalent of heat, 206
Dyni, 83
EFFICIENCY of an engine, 157, 158
Elasticity, 301 ; of a fluid, 107 ; modes
of measuring 171 ; perfect, 302
Electrolysis, molecular theory of, 335
Electrolytes, electrical conductivity of,
271
Energy, 87, 91 ; available, 192 ; dissipa-
tion of, 193 ; potential, 308 ; super-
ficial, 283
Engine, efficiency of an, 157, 158 ; heat,
138
Entropy, 162, 187, 189
Exchanges, Prevost's theory of, 240, 311
Expansion, free, 209
External forces, 94
Evaporation, theory of, 333
FAHRENHEIT degrees, 38
Faraday, liquefaction of gases by, 119 ;
on mental inertia ? 86
Fictitious thermal lines, 176
Flaugergues, discovered the change in
zero of a thermometer, 42
Flow of heat, periodic, 265 ; steady, 257
Fluids, 16 ; conductivity of, 271 ; defi-
nition of, 95 ; diffusion of, 273 ; elasti-
city of, 107 ; perfect, 96
Fogs, Aitken on the formation of, 307
Foot-pound, 87 -
Forbes, polarisation of heat, 237 ; on
conductivity for heat, 269
Force, 83 ; conservation of, 91 ; gravita-
tion measure of, 84
Forces, external and internal, 94
Fourier's theory of heat, 259
F*ee expansion, 209
Freezing point of a thermometer, deter-
mination of, 33 ; variation in, 44
French standard of mass, 79
Function, Carnot's, 155 ; potential, 91
Fundamental units, 76
Fusion, 19
GADOLIN, on specific heat, 65
Galileo, inventor of the air thermo-
meter, 46
Gallon, a legal measure, 81
Gas, formation of, 22
Gaseous state, 27
Gases, 1 6 ; Faraday's liquefaction of,
29, 325 ; observations
119 ; perfect, in
Gay Lussac's law, :
on boiling, 35
Gibbs' thermodynamic model, 195
Graham, on dialysis, 342 ; on the laws of
the diffusion of gases, 276
Grammej 79
Gravitation measure, 87
Gravity, intensity of, 84
Greay, spc ts, removal of, 300
HARMONIC distribution of temperature,
263
Heat, capacity of bodies for, 65 ; engines,
138 ; its effects on liquids, 21 ; invi-
sible, 15 ; and light, 233 ; latent, 19,
173 ; periodic flow of, 265 ; _ as a
quantity, 7 ; radiant, true meaning of
the term, 15 ; rays, 14 ; not a sub-
stance, 57 ; specific, 65
Heights, determination of, by the
barometer, 217
Helmholtz on the conservation of force,
91
Hermetical sealing, 42
Hirn's experiments on the steam engine,
146
Homogeneous atmosphere, 220
Hydrostatic pressure, 30
ICE, calorimeter, 58 ; influence of pres-
sure on the melting point of, 176
Indicator diagram, 102
Inertia, Faraday on mental, 86
Instability, conditions of, 205
Interference of light, 235 ; of heat, 236
Internal forces, 94
Intrinsic energy of a system of bodies,
185
Iodide of silver, anomalous expansion
of, 1 8
Iron, conductivity of, 271
Isenergetic lines, 198
Isentropic lines, 164
Isopiestic lines, 198
Isothermal curves, no; lines, 108; for
steam and water, 114
Isotropic strains, 99
JOULE, mechanical equivalent of heat,
147, 213 ; determination of the maxi-
mum density of water, 252 ; explana-
tion of the pressure of gases, 315.
KEW standard thermometer, 44
Kiloeramme des Archives, 79
Index.
347
Kinetic energy, 87, 91 ; of a molecule,
317, 329; theory of gases, 314.
Kirchhoff s principle, 295
Kronig, explanation of the property of
gases, 312
LAPLACE, theory of capillarity, 292 ; and
Lavoisier's calorimeter, 59
Latent heat, 19, 73, 173 ; determination
of, 71 ; its connection with capillarity,
292
Law, of Boyle, 321 ; of Charles, 29 ; of
Dalton, 29 ; of Dulong and Petit, 329 ;
of Gay Lussac, 29, 325
Length, standard of, 76
Light, interference of, 235
Lines, adiabatic, 135 ; of equal tempera-
ture, 108
Liquefaction of gases, 17
Longitudinal stress, 99 ; displacement
waves of, 225
MASS, standard of, 78, 79
Maximum density of water, 253
Mayer's calculation of the dynamical
equivalent of heat, 216
Measurement of quantities, 75 ; of
heights by the barometer, 217
Melloni's discoveries in radiant heat,
234
Method of cooling, 74, 247 ; of mixture,
63
Metre, its origin, 77
Milligramme, 79
Mixture, method of, 63
Molecular motion, 279, 311 ; theory, 08
Molecules, nature and origin of, 340
Momentum, 82
Motion, molecular, 279, 311
NEWTON, Sir Isaac, determination of
the fixed points of a thermometer, 34 ;
on the relation between work and
kinetic energy, 91
Node-couple, 203
PARLIAMENTARY standard of length, 77
Particulastriata, 341
Perfect gas, in
Periodic flow of heat, 265
Pictet's 'experiments on the liquefaction
of gases, 17
Polarisation of heat, 237
Potential energy, 91, 308
Pound, standard, 78
Poundal, 83
Pressure, 94 ; longitudinal. 95 ; in a
fluid, 97
Prevost's theory of exchanges, 240, 311
Propagation of waves, 223
RADIANT heat, 13 ; true meaning of the
term, 15
Radiation, 10, 13, 230 ; as depending on
temperature, 245 ; its effects on ther-
mometers, 248 ; molecular theory of.
33
Radius of the earth, 85
Rankine, on entropy, 162 ; on the propa-
gation of waves, 223
Rarity the converse of density, 82
Rate of cooling, 68
Reading of a thermometer, 32
Reaumur scale, 39
Regelation, 176
Regnault, on vapours, 28 ; on latent
heat, 175; on the expansion of gases,
215
Reversible engine, Carnot's, 139
Reynolds, Prof. Osborne, effects of oil
en waves, 298
Richards's indicator, 104
Rigidity of a body affected by tempera-
ture, 18
SATURATED vapour, 23
Scale of thermometer, 37
Scientific terms defined, 5
Sensible heat, 20
Shearing strains, 100 ; stresses, 302
Siemens' electrical thermometer, 53
Simmering, 24
Soap bubbles, energy of, 203
Solids, 16
Sound, velocity of, in air, 182, 229
Specific heat, 65 ; modes of measuring,
169 ; at constant volume, 328 ; ther-
mal conductivity, 255
Spinpde curve, 204
Stability of a system, conditions of, 201
Standard thermometer, Kew, 44 ;
French, of mass, 79 ; of length, 77 ;
pound, 78
Statical energy, 91
Steady flow of heat, 257, 265
Steam engine, Him '9 experiments, 146
Strains, 99 ; isotropic, 49 ; shearing, 100,
3Qi
Stresses, 95, 301, 302
Subsidence of clouds, 307
Superficial energy of a soap bubble, 283 ;
tension, 283 ; table of, 295
Surface tension, 283 ; thermodynamic,
196
TACNODAL point, 204
Tait, Professor, on thermo-electromotive
force, 53 ; on thermal conductivity, 271
Tears of wine, 300
Temperature, 4 ; absolute, 51 ; Thom-
son's scale of, 155 ; chamber of uniform,
272 ; its effect on rigidity, 18 ; in-
fluence on conductivity, 271; on
348
Index.
radiation, 245 ; harmonic distribution
of, 263 ; measured by electricity, 50 ;
lines of equal, 108 ; underground,
267; uniform, 272
Tension, 95 ; surface, 283
Theorie de la Chaleur, Fourier's, 258
Theory, of exchanges, Prevost's, 280,
311 ; of heat, Fourier's, 258 ; of mo-
lecular radiation, 336
Thermal, use of the word, 9 ; conduc-
tivity, 255 ; determination of, 268 : and
electrical conductivity, order of, 271 ;
unit, 9
Thermodynamic surface, 195, 206
Thermodynamics, 9 ; first and second
laws of, 152 ; Thomson's application
of capillarity to, 296
Thermo-electric current, 52
Thermokinematics, 9
Thermometer, air, 46 ; change of zero
point, 42 ; Galileo inventor of the, 46 ;
mercurial, 5 ; Newton's determination
of fixed points, 34 ; reading of a, 32 ;
scales, 37 ; Siemens' electrical, 50 ;
standard, at Kew, 44
Thermometric measure of conductivity,
255
Thermometry, 30
Thermostatics, 9
Thermotics, 9
Thomson's absolute scale of temperature,
155 ; application of thermodynamics
to capillarity, 289 ; on viscosity, 304
Thomson, James, on the influence of
pressure on the freezing point of water,
176
Time, unit of, 80
Tomlinson, Charles, on cohesion figures,
296
Tonne, 79
Tourmaline, its action on light, 237
Transparent bodies, 233
Troy pound, 78
Tyndall, Professor, on conductivity, 292
UNDERGROUND temperature, 267
Undulating theory, 232
Uniform temperature, 272
Unit, thermal, 9 ; of time, 80
VAN DER WAALS, application of the
virial, 325
Vapour, true meaning of, 23
Velocity, uniform, 82 ; of sound, 182,
229 ; of a gas, definition of, 318 ; of
molecular motion, 390
Virial, 322
Vis viva, 91
Viscosity, 301 ; coefficient of, 304 ; of
metals, 303
Viscous bodies, 303
Voit's determination of diffusion, 274
WATERSTON on capillarity and latent
heat, 292
Watt's indicator diagram, 105
Waves, propagation of, 223 ; of perma-
nent type, 227
Wilcke, measurement of heat on bodies
cooling, 58
Willard Gibbs' representation of the pro-
perties of a body by a surface, 195
Wine, tears of, 300
Work, 87
Zero point of a thermometer, change in,
42 ; determination of, 33; absolute, i
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