Elements of Classical Thermodynamics by a.B.pippard

CAMBRIDGE

ELEMENTS OF CLASSICALTHERMODYNAMICS

A.B.PIPPARD

ELEMENTS OF CLASSICAL

THERMODYNAMICS

ELEMENTS OF CLASSICAL

THERMODYNAMICSFOR

ADVANCED STUDENTS OF

PHYSICS

BY

A. B. PIPPARDM.A., Ph.D., F.R.S.

J. H. Plum'I'IUr ProJessor oj Physics in the University oj Cambridgeand Fellow oj Clare College, Cambridge

CAMBRIDGEAT THE UNIVERSITY PRESS

1966

PUBLISHED BY

THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS

Bentley House, 200 Euston Road, London, N.W.IAmerican Branch: 32 East 57th Street, New York, N.Y. 10022

West African Office: P.M.B. 5181 Ibadan, Nigeria

@(~AMBRIDGE UNIVERSITY PRESS

1957

Fir8t printed 1957Reprinted

with corrections 1960Reprinted 1961Reprinted

with corrections 1964Reprinted 1966

Fir. printed in Great Britain at the Univer.nty Press, OambridgeReprinted by o.fJ8et-Utho by John Dickens Ltd., Northampton

T

CONTENTS

PREFAOE

1. INTBODUOTION

2. THE ZEROTH AND FIRST LAws OF

THERM:ODYN AMICS

Fundamental definitionsTemperatureInternal energy and heat

3. REVERSIBLE CHANGES

Reversibility and irreversibilityDifferent types of work

4. THE SECOND LAW OF THERMODYNAMICS

page vii

1

5

57

13

191923

29

5. A MISCELLANY OF USEFUL IDEAS 43

Dimensions and related topics 43Maxwell's thermodynamic relations 45Identity of the absolute and perfect gas scales oftem~mtme 46

Absolute zero, negative temperatures and the thirdlaw of thermodynamics 48

An elementary graphical method for solving thermo-dynamic problems 53

The functions U, H, F and G 55

6. ApPLICATIONS OF THERMODYN AMICS TO SIMPLE

SYSTEMS 57

Introduction 57Relations between the specific heats 60The adiabatic equation 61Magnetic analogues of the foregoing results 63The Joule and Joule-Kelvin effects 68Radiation 77Surface tension and surface energy 84Establishment of the absolute scale of temperature 87

vi Oontents

7. THE THERMODYNAMIC INEQUALITIES

The increase of entropyThe decrease of availabilityThe conditions for equilibrium

8. PHASE EQUILIBRIUM

The phase diagram of a simple substanceClapeyron's equationLiquid-vapour equilibrium and the critical pointSolid-liquid equilibriumThe phase diagram of heliumThe superconducting phase transition

9. HIGHER-ORDER TRANSITIONS

Classification of transitionsAnalogues of Clapeyron's equationCritique of the theory of higher-order transitions

EXERCISES

INDEX

page 94

94100104

112

112115116122124129

136

136140146

160

163

vii

PREFACE

The first treatment of thermodynamics which a student of physicsgets nowadays is likely to be given in an early year at the university,and to be concerned more with the descriptive aspect of the subjectthan with its systematic application to a variety of problems. If ina later year he attends a more thorough course it will probably bequite brief, on account of the pressure of other subjects which nowadays rightly rank as of greater urgency, and he cannot normally beexpected to supplement his meagre ration of lectures by protractedstudy of the standard monographs. It is this situation which I havekept in mind in writing a short account of the fundamental ideas ofthermodynamics, and to keep it short I have deliberately excludeddetails of experimental methods and multiplicity of illustrativeexamples. In consequence this is probably not a suitable text-bookfor the beginner, but I hope the more advanced student will find herea statement of the aims and techniques, which will illuminate anyspecialized applications he m'ay meet later. I hope too that formathematical students whose ambitions point towards theoreticalphysics it may serve as a concise introduction to the theory of heat.It may be objected by some that I have concentrated too much onthe dry bones, and too little on the flesh which clothes them, but Iwould ask such critics to concede at least that the bones have anaustere beauty of their own.

The problems which are provided as an aid to learning are of twokinds. From time to time in the text I have suggested extensionsto the arguments which may be profitably developed, and at the endI have placed a small number of exercises, most of which are ratherdifficult. I am indebted to Mr V. M. Morton for checking these andsuggesting improvements.

The relief from teaching which was afforded by an invitation tospend a year as visiting Professor at the Institute for the Study ofMetals, in the University of Chicago, provided the stimulus andopportunity for writing this book. I should like to express mygratitude to all at the Institute who made my visit so enjoyable.My warmest thanks are also due to Professor J. W. Stout, Dr D.Shoenberg, F.R.S., and Dr T. E. Faber for their critical comments,and to Mrs Ruth Patterson and Mrs Iris Ross for typing themanuscript.

A.B.P.May 1967

viii

PREFACE TO FOURTH IMPRESSION

A number oferrors and blemishes in the first edition have been pointedout by reviewers and friendly critics, and some of the more seriouswere corrected in the second printing ofthat edition. I have taken theopportunity afforded by a further printing to bring some parts of thelater chapters up to date, but have otherwise made no substantialchanges.

A.B.P.August 1963

1

CHAPTER 1

INTRODUCTION

The science of thermodynamics, in the widest sense in which the wordis used nowadays, may be said to be concerned with the understandingand interpretation of the properties of matter in so far as they areaffected by changes of temperature. In this sense thermodynamicsranks as one ofthe majorsubdivisions ofphysical science, and a varietyof mathematical and experimental techniques may be invoked to aidits advancement, with the ultimate aim of providing an explanationof the observed properties of matter at all temperatures in terms of itsatomic constitution and the forces exerted by atoms upon one another.This statement covers perhaps a wider field of investigation than canlegitimately be called thermodynamics. For example, it can hardly beclaimed that the theory of the chemical forces which bind together theatoms of stable chemical compounds is a branch of thermodynamics;rather is it a branch of quantum mechanics in which the concept oftemperature plays no part. On the other hand, as soon as we becomeinterested in the excitation or dissociation of molecules as a consequence of heating, the matter becomes truly one in which thermodynamical considerations are involved. In the same way the existenceof solids and a great many properties of solids which are only toa minordegree affected by temperature maybeexplainedsatisfactorilyas purely mechanical consequences of the forces between atoms;thermodynamics strictly enters only when we attempt to account forthe temperature-dependent properties, such as heat capacity and (incertain solids) magnetic susceptibility. And, of course, the study ofphase transitions (solid to liquid, liquid to gas, or changes ofcrystallinestructure) is in its essence thermodynamical, and provides indeed someof the most interesting problems of present-day thermodynamics.

This is the wide use of the term, but there is a narrower sense inwhich it is used, and this is what may conveniently be distinguished ascla88ica.l thermodynamics, the subject ofthis book. Here the method ofapproach takes no account of the atomic constitution of matter, butseeks rather to derive from certain basic postulates, the laws ofthermodynamics, relations bet\\"een the observed properties of substances. In contrast to the atomic theory of thermal phenomena,classical thermodynamics makes no attempt to provide a mechanisticexplanation of why a given substance has the properties observedexperimentally; its function is to link together the many observableproperties so that they can all be seen to be a consequence ofa few. For

2 Classical thermodynamics

example, if the equation of state of a gas (the relation between itspressure, volume and temperature) be known, and a determinationmade of its specific heat at constant pressure over a range of temperatures, then by thermodynamical arguments the specific heat at constant volume may be found, as well as the dependence of both specificheats on pressure or volume. In addition, it may be predicted whetherthe gas will be heated or cooled when it is expanded through a throttlefrom a high to a low pressure, and the magnitude of the temperaturechange may be calculated precisely. This is a comparativelyelementaryexample of the application of classical thermodynamics to a physicalproblem. The applications discussed in this book should give some ideaof the power of thermodynamics to deal with a considerable variety ofphenomena, but to appreciate the full scope of the method, and to seehow much can be achieved by the use of only simple mathematics,reference should be made to more detailed treatises on the subject.There is none which encompasses the whole field, for the applicationsof thermodynamics range over many branches of physics, chemistryand engineering, and in each are so extensive as to demand separatetreatment if anything like completeness is to be attained. Nevertheless, in spite of a great diversity of methods of presentation, the ideasinvolved are exactly the same in principle.

The two approaches to thermodynamical theory, the classical orphenomenological approach on the one hand and the statisticalapproach through molecular dynamics on the other, are so differentthat it is worth discussing the relationship between them, especially aswe shall have no more concern with the latter in what follows. Thelaws of thermodynamics were arrived at as a consequence of observation and generalization of experience; continued application of themethods of classical thermodynamics to practical problems showedthese laws to predict the correct answer in all cases. This is theempirical justification for regarding the laws as having a very widerange of validity. But classical thermodynamics makes no attemptto explain why the laws have their particular form, that is, to exhibitthe laws as a necessary consequence ofother laws ofphysics which maybe regarded as even more fundamental. This is one of the problemswhich is treated by statistical thermodynamics. From a considerationof the behaviour of a large assembly of atoms, molecules or otherphysical entities, it may be shown, with a fair degree of rigour (enoughto satisfy most physicists but few pure mathematicians), that thoseproperties of the assembly which are observable by macroscopicmeasurements are related in obedience with the laws of thermodynamics. This result, which is not derived by considering any veryspecific model, but which has as wide a range of validity as the laws ofmechanics themselves, has perhaps tended to encourage an under-

Introduction 3

valuation of classical thermodynamics. For from the point of view ofthe physicist who aims to penetrate as far as possible into the deepestmysteries ofthe physical world, and to find the fundamental principlesfrom which all physical laws derive, thermodynamics has ceased to bean interesting study, since it is wholly contained within the laws ofdynamics.

Not all practitioners, however, of the physical sciences (in whichterm we may include without prejudice chemists and engineers) havethis particular ambition to probe the ultimate mysteries of their craft,and many who have are forced by circumstances to forgo their desire.For these, the great majority, thermodynamics is not so obviouslya trivial pursuit; indeed, in many branches it is an almost indispensabletool. For often enough in the pure sciences, and still more in the appliedsciences, it is more important to know the relations between theproperties of substances than to have a clear understanding of theorigin of these properties in terms of the molecular constitution. Andeven the theoretical physicist who is concerned with a detailedexplanation of these properties may find classical thermodynamicsa valuable aid, since it reduces the number of problems which requireseparate statistical treatment-once certain results have been derivedthe rest follow thermodynamically, as will become clearer in laterchapters.

These are some of the reasons which make a study of classicalthermodynamics a valuable part of the education of a physical scientist, but there is another, less purely practical reason. The development of the ideas in thermodynamics has a formal elegance which isexceedingly satisfying aesthetically. It has not perhaps quite therigour of a perfect mathematical proof, but it approaches nearer thatlogical ideal than almost any other branch ofnatural science. For thisreason alone it may be regarded as an important part of the educationof a scientist.

Because classical thermodynamics is capable ofso rigorous a formulation it is desirable, in the author's opinion, to present at least theearly steps of the argument in a way which brings out the logicaldevelopment clearly. At the same time a wholly mathematical approach may prove either repellent to the student, or, what is worse,formally intelligible and yet meaningless in terms of physical reality.The early chapters which follow therefore represent some sort ofa compromise in which the ideas are expressed in as unmathematicala form as is consistent with exactitude. By the use of more mathematics the arguments could be shortened on paper; it is doubtfulwhether such a treatment would lead to a speedier assimilation of theideas by any but the most mathematically minded students.

The pursuit of rigour involves almost inevitably abandoning the

4: Olassical thermodynamics

historical approach. Great ideas are more often arrived at by a combination of intuition and a judicious disregard of niceties than bya systematic and logical development of explicitly formulated premisses, and certainly the history of thermodynamics bears out thisview. But once the goal has been attained it is possible to go back overthe road and see how the same end could have been reached morelogically. While it is fascinating for the historian of science to see howCarnot, in his astonishing memoir Sur la puissance motrice du feu(1824), arrived at so many correct conclusions after having startedwith the incorrect caloric theory of heat, only confusion would resultfrom trying to base a modern treatment on this work. Carnot's mainresults were reproduced and extended, principally by James andWilliam Thomson, Clausius and Rankine, after the experiments ofJoule (about 1843-9) had provided convincing evidence for interpreting heat as a form of energy and had thus extended the law ofconservation of energy to include thermal processes. Even this work,however, suffers from a number of defects from the point of view oflogical presentation. For example, it is undesirable in a purelyphenomenological development to have recourse to an unobservableatomic interpretation of the nature of heat, if it can be avoided, as itcan; and secondly, there is little explicit discussion in this early workof the meaning of that all-important term temperature. To be sure, ourbodily senses allow us to comprehend with ease the idea of temperature, and without such direct apprehension the development of thermodynamics would surely have been considerably retarded. But thatis no reason for continuing to regard the idea of temperature asessentially intuitive, if a satisfactory definition of the term can begiven which does not rely on our qualitative sensory impressions only.

We therefore begin our formal treatment of the subject by showinghow the ideas of temperature and heat may be systematically formulated on the basis of experiment, so that the subsequent development may be as free as possible from the suspicion that it is based onintuitive concepts or atomistic interpretations.

5

CHAPTER 2

THE ZEROTH AND FIRST LAWSOF THERMODYNAMICS

Fundamental definitions

Thermodynamics is concerned with real physical systems, whichmay be solid or fluid, or mixtures of both, or even an evacuated spacecontaining nothing but electromagnetic radiation. Usually the systemconsidered must be contained within a vessel of some kind with whichit does not react chemically.

Now the walls ofdifferent vessels differ considerably in the ease withwhich influences from without may be transmitted to the systemwithin. Water within a thin-walled glass flask may have its propertiesreadily changed by holding the flask over a flame or by putting it ina refrigerator; or the change brought about by the flame may besimulated (though not so easily) by directing an intense beam ofradiation on to the flask. If, on the other hand, the water is containedwithin a double-walled vacuum flask with silvered walls (Dewarvessel), the effects of flame or refrigerator or radiation are reducedalmost to nothing. The degree ofisolation ofthe contents from externalinfluences can be varied continuously over such a wide range, that it isnot a very daring extrapolation to imagine the existence of a vesselhaving perfectly isolating walls, so that the substance containedwithin is totally unaffected by any external agency.t Such an idealwall is termed an adiabatic wall, and a substance wholly containedwithin adiabatic walls is said to be isolated. Walls which are notadiabatic are diathermal walls. Two physical systems separatedfrom each other only by diathermal walls are said to be in thermalcontact.

An adiabatic wall may be so nearly realized in practice that itmay be claimed to be a matter of experience that when a physicalsystem is entirely enclosed within adiabatic walls it tends towards,

t It is perhaps too much of an extrapolation to imagine the walls imperviousto gravitational fields. Rather than postulate the existence of such a wall weshall for the moment avoid problems involving gravitation. It is hoped thatwhen, later in the book, we make occasional reference to gravitational fieldsthe reader will feel enough confidence in his physical understanding of thermodynamics to be able to ignore lacunae in the basic formulation. If he cannotovercome his scruples he must work out a better treatment for himself.


and eventually reaches, a state in which no further change is perceptible, no matter how long one waits. The system is then said tobe in equilibrium.

Mechanical systems exhibit a number of different types of staticequilibrium, which may be exemplified by the behaviour ofa sphericalball resting on curves of different shapes and acted upon by gravity.Thermodynamic systems show analogies to some of these, and it isconvenient to point them out at this stage, although it is not strictlypertinent to the argument and, indeed, necessitates the use of concepts which have not yet been defined. The following discussionshould therefore be regarded as an explanatory parenthesis only.Stable equilibrium may be represented by a ball resting at the bottom ofa valley; the equilibrium ofa pure gas at rest at a uniform temperatureis analogous to this. There is no realizable thermodynamic analogueto unstable equilibrium,t as of a ball poised at the top of a hill. A ballresting on a flat plane is in neutral equilibrium; so is a mixture of waterand water vapour enclosed in a cylinder and subjected to a constantpressure by means of a frictionless piston, the whole being maintainedat such a temperature that the vapour pressure of the water is exactlyequal to the pressure exerted by the piston. For just as the ball mayremain at rest at any point on the plane, so the proportion of liquidand vapour may be adjusted at will by movement of the piston.Finally, there is meta8table equilibrium, represented by a ball restingin a local depression at the top of a hill, and stable with respect tosmall displacements while unstable with respect to large displacements. It is difficult to find a strict thermodynamic analogue to thistype ofmetastability, but perhaps the nearest approach is exemplifiedby a supercooled vapour or by a mixture of hydrogen and oxygen.Both the systems have the appearance of stability and may be subjected to small variations of pressure and temperature as if they weretruly stable; yet the effect of a condensation nucleus on the former ora spark on the latter shows clearly that they have not the stability of,say, helium gas in equilibrium. The analogy with the ball in metastableequilibrium is not perfect, for these thermodynamic systems are neverstrictly in equilibrium. Given long enough a supercooled vapour willeventually condense of its own accord, and given long enough a mixture ofhydrogen and oxygen will transform itself into water. The timeinvolved may be so enormous, however, perhaps 10100 years or more,that the process is not perceptible. For most purposes, provided therapid change is not artificially stimulated, the systems may beregarded as being in equilibrium.

Although we may discover analogies between thermodynamic andsimple mechanical systems it is well to bear in mind an important

t See p. III for f~ther discussion of this point.

Zeroth and first laws 7

difference. The equilibrium of a thermodynamic system is neverstatic; the' matter ofexperience' mentioned above, that systems tendto a state from which they subsequently do not change, is not strictlyan experimental truth. A microscopic examination of minute particles suspended in a fluid reveals them to be in a state of continuousagitation (Brownian movement), and in the same way delicatemeasurements on a gas would reveal that the density at a given pointis subject to incessant minute fluctuations about its mean value. Thisis, of course, a consequence of the rapid motions of the moleculescomposing the system, and is an intrinsic property of the system. Ifone were prepared to wait long enough, and in most cases long enoughmeans a time enormously longer than the age of the universe, onemight observe really sizeable departures from the average state of thesystem. For example, there is no reason why I c.c. of a gas, in a stateof complete thermal isolation, should not spontaneously contract tohalf its volume, leaving the other half of the vessel evacuated, and justas suddenly revert to its average state of virtually uniform density.But the whole fluctuation would take only about 10-4 sec. to beaccomplished, and might be expected to occur once in about 10(1019

)

years, so that the possibility of making such an observation need notbe seriously contemplated. For most purposes it is quite satisfactoryto imagine an isolated system to tend to a definite and invariant stateofequilibrium, and classical thermodynamics assumes the equilibriumstate to have this static property. In so far as this assumption is notstrictly true we must expect' to have to revise any results we mayderive before applying them to problems in which fluctuations playa significant part. We shall return to this point later (Chapter 7).

Temperature

At this stage it is convenient to consider an especially simple type ofthermodynamic system in order to arrive at an idea of the meaning oftemperature, and we shall for the present confine our attention tohomogeneous fluids, either liquids or gases. A gas, of course, of itsvery nature fills its containing vessel; we imagine any liquid under consideration to be contained exactly by its vessel, leaving no free spacefor vapour. The especial simplicity of a fluid derives from the fact thatits shape is of no consequence thermodynamically; deformation of thecontaining vessel, if unaccompanied by any change in volume, may inprinciple be accomplished without the performance of work, and doesnot alter the thermal properties of the fluid within. By contrast,a solid body can only be altered in shape by the application of considerable stresses, and the thermal properties are in general affected inthe process.

8 Olassical thermodynamics

It is a fact of experience that a given mass of fluid in equilibrium iscompletely specified (apart from its shape which is, as just pointed out,ofno significance) by a prescription of its volume, V, and pressure, P.tIf we take a certain quantity of gas, enclosed in a cylinder witha movable piston, we may fix the volume at some predetermined valueand then, with the help of such well-known auxiliary devices as anoven and a refrigerator, set about altering the pressure to any requiredvalue. It may readily be verified by experiment that whatever theprocess by which the pressure and volume are adjusted, the final stateof the gas is always the same, no matter what property is examined(e.g. colour, smell, sensation of warmth, thermal conductivity, viscosity, etc.). That is to say, any property capable of quantitativemeasurement may be expressed as a function of the two variables,Pand V.

Let us consider now the behaviour of two systems which are notthermally isolated from one another. If we take two isolated systemsand allow them to come into equilibrium separately, and then bringthem into thermal contact by replacing the adiabatic wall whichdivides them by a diathermal wall, we shall find in general that changestake place in both, until eventually the composite system attains a newstate of equilibrium, in which the two separate systems are said to bein equilibrium with one another. As a matter of sensory experience weknow that this is because two systems chosen independently will notin general have the same temperature, and the changes which occurwhen they are brought into thermal contact result in their eventuallyattaining the same temperature. But there is no need at this stage toemploy the, as yet, meaningless word temperature to describe thisparticular fact ofexperience. It is sufficient to realize that two systemsmay be separately in equilibrium and yet not in equilibrium with oneanother. In particular, two given masses offluid are not in equilibriumwith one another if their pressures and volumes (the para1neters ofstate) are chosen arbitrarily. Of the four variables of the compositesystem, PI and VI for the first fluid, and P2 and V2 for the second, threemay be fixed arbitrarily, but for the two fluids to be in equilibriumwith one another the fourth variable is then determined by the otherthree. One may, for example, adjust both PI and VI by placing thefirst fluid in an oven until the required values are reached; if then V2 isfixed it will be found necessary to adjust P2 by placing the second fluidin the oven before the two fluids are in equilibrium with one another.This may be expressed in a formal manner by saying that for two givenmasses of fluid there exists a function of the variables of state,

t We leave out of consideration for the moment the possible influences ofelectric and magnetic fields on the properties of the fluid, assuming such fieldsto be absent.

(2-1)


F(P1, VI' P2,V2) such that when the fluids are in equilibrium with oneanother,

The form of the function will depend of course on the fluids considered,and may be determined, if required, by a sufficient number of experiments which measure the conditions under which the fluids are inequilibrium_

In order to establish the existence of the important propertytemperature it is necessary to demonstrate that (2-1) may alwaysbe rewritten in the form

rpl(P1, VI) = rp2(P2,V2), (2-2)

in which the variables describing the two systems are separated. Thiswe shall do first by a formal mathematical argument, and then (sinceto many students the mathematical argument appears too abstract tobe altogether meaningful) by an equivalent argument in terms ofhypothetical experiments. Both arguments depend on a further factof experience which has come to be regarded as sufficiently importantto be designated the zeroth law of thermodynamics:

If, of three bodies, A, Band 0, A and B are separately in equilibriumwith 0, then A and B are in equilibrium with one another.

It is also desirable, for reasons which will appear later, to state what isessentially the converse of the zeroth law:

If three or more bodies are in thermal contact, each to each, by means ofdiathermal walls, and are all in equilibrium together, then any two takenseparately are in equilibrium with one another_

The sort ofsimple experiment upon which the zeroth law is based maybe illustrated by the following example. Let 0 be a mercury-in-glassthermometer, in which the mercury is a fluid at roughly zero pressure(if the thermometer is evacuated) and with a volume determined byits height in the tube; the height ofthe mercury in a given thermometeris sufficient to determine its state. Then according to the zeroth law ifthe reading of the thermometer is the same when it is immersed in twodifferent liquids, A and B, nothing will happen when A and Bareplaced in thermal contact. It is easy enough to multiply examples.ofthe application ofthis law, which expresses so elementary and commonan experience that it was not formulated until long after the first andsecond laws had been thoroughly established.

Consider now three fluids, A, Band O. The conditions under whichA and 0 are in equilibrium may be expressed by the equation

F1(PA , VA' Po, Vo)=O,


which may be solved for Pc to give an equation of the form

PC=fl(PA,VA' Vc)· (2·3)

Similarly, the conditions under which Band 0 are in equilibrium maybe expressed by the equation

F2(PB , VB, Pc, Ve )=0,

or, again by solving for Pc,

Pe =f2(PB,VB, Ve ). (2·4)

Hence the conditions under '\\Thich A and B are separately in equilibrium with C may be expressed, from (2·3) and (2·4), by the equation

(2·5)

But if A and B are separately in equilibrium with 0, then according tothe zeroth la,v they are in equilibrium with one another, so that (2·5)must be equivalent to an equation of the form

(2·6)

It will be seen now that while (2·5) contains the variable Ve , (2·6) doesnot. If the two equations are to be equivalent, it can only mean thatthe functions 11 and f2 contain Ve in such a form that it cancels out onthe t\\ro sides of (2·5) [e.g. fl(PA,VA, Ve) might take the form

¢>l(PA, T'A) ~( Vc) +1J( Ve)].

When this cancellation is performed, (2·5) will have the form of (2·2),

¢>l(PA, VA) =¢>2(PB, VB),

and by an obvious extension of the argument

91(PA, VA) =¢>2(PB, VB) =¢>3(Pe, Jc),and so on for any number offluids in equilibrium with one another. Wehave thus demonstrated that for every fluid it is possible to finda function ¢(P, V) of its parameters of state (different of course foreach fluid) which has the property that the numerical value of¢ (=(), say) is the same for all fluids in equilibrium with one another.The quantity () is called the empirical temperature, and the equation

¢>(P, V) =() (2·7)

is called the equation of state of the fluid. In this way we have shown,by means of the zeroth law, that there exists a function of the state ofa fluid, the temperature, which has the property of taking the same

Zeroth an.d first laws 11

value for fluids in equilibrium with one another. Since 0 is uniquelydetermined by P and V, t it is possible to regard the state of the fluid asspecified by any two of the three variables P, V and O.

In order to make the physical meaning of this result somev~'hat

clearer, we shall now show how it may be derived from a considerationof certain simple experirnents. Suppose we have two nlasses of fluid,a standard massS and a test mass T. Keeping S in a fixed state (PsandVs constant) we may vary PT and VT of the test mass in such a way asto keep Sand T always in equilibrium ,,,ith one another. Since of thefour variables only one is independent (cf. (2·3)), ,ve shall find a relation between PT and VT which may be represented by a line such as Lin fig. I. Such a line is termed an isotherm. It may now be readily

VT

Fig. 1. Isotherms of a simplo fluid.

seen that the form of the isotherm is independent of the nature of thestandard mass. :For suppose we had choHen instead of J~ anotherstandard mass S' which was in equilibrium \vith S. Then for any stateof T corresponding to a point on L, ,ve should have S in equilibriulnwith bothS' and T; hence, according to the zeroth law, S' and T wouldalso be in equilibrium, and the isotherm of T determined by the usc of8' rather than S would pass through this point.

We have shown then how to construct the isotherms ofa fluid on thep- V diagram (indicator diagram), and also that the isotherlns so constructed are determined by the nature ofthe fluid and not by the choiceof the subsidiary standard body. By continued experiment a whole

t Apart from occa.sional ambiguities arising from multiplo solutions of (2'7);e.g. at a pressure of 1 atmosphere water has tho same volume at 2:) C. and a.t6°C.

¢J(P, V)=O.

PV =f(O),


family of isotherms may be plotted out, as shown schematically infig. 1. Now let us label each isotherm with a number, 0, chosen at will,which we call the empirical temperature corresponding to the givenisotherm. Then provided there is some system, however arbitrary, inthe labelling of the isotherms, there will exist a relationship (notnecessarily analytic) between P, V and 0 which may be written in thesame form as (2·7),

Once this labelling of isotherms has been carried out for one particular mass of fluid, however, there exists no latitude of choice so faras other fluids are concerned, if consistency is to be achieved. For theisotherm ofa second fluid in equilibrium with the first must be labelledwith the same O. If, and only if, this is done can we say that all fluidshaving the same value of 0 are in equilibrium with one another. Thisbrings us to the same result as was derived before; the two argumentsare equivalent.

It is because of the element ofchoice in the labelling ofthe isothermsofthe first fluid to be selected (the thermometric body) that the quantity 0is referred to as the empirical temperature. It is usual to choose as thethermometric body a fluid whose properties make a rational choice ofo particularly simple. For example, in a mercury-in-glass thermometer there is effectively only one variable, the volume ofthe mercury,and 0 is taken to be a linear function of the volume. The particularstraight line selected depends on the choice of scale; according to theCelsius scale, 0 is put equal to 0 at the temperature ofmelting ice, and100 at the temperature of water boiling at standard atmosphericpressure. Two fixed points are sufficient to determine the linear relation. Consider now the perfect gas scale of temperature. This iscapable of simple definition because of the analytical simplicity of theisotherms, which for perfect gases follow Boyle's law, PV = constant.Thus the equation ofstate of a perfect gas on any empirical scale musttake the form

and the nature of the empirical scale determines the form of thefunction f(O). It happens that if the empirical scale is fixed by amercury-in-glass thermometer, f(O) is very nearly a linear functionover a wide range of temperature. This experimental result makes itconvenient to establish an empirical scale in terms of a perfect gasby adopting as a definition of 0 the equation

PV=RO.The constant R is chosen for any particular mass of gas in ~uch a waythat the value of 0 shall change by 100 between the melting-point ofice and the boiling-point of water.

Zeroth and fir8t laW8 13

For the purpose of the foregoing analysis we have considered onlythe simplest type of system, a fluid, whose state is definable by twoparameters. The argument may easily be extended to more complicated systems, such as solids under the influence of more thansimple hydrostatic stresses, or bodies acted upon by electric or magnetic fields, in which cases more than two parameters must be specifiedin order to determine the state uniquely. The only change involved bythis extension is that instead of isothermal lines the body possessesisothermal surfaces in three or more dimensions, and the equation ofstate may be formally expressed

!(x1, X 2, ••• , xn ) = (J,

in which Xl ••• X n represent the parameters needed to define the state.The existence of temperature may be proved in exactly the same wayas before.

It should be noted that our knowledge of temperature at this stageis insufficient to correlate the empirical temperature of a body with itshotness or coldness. There is no reason why a body having a high valueof (J should necessarily be hotter (in the subjective sense, or any other)than one having a low value, since the choice of a temperature scale isentirely arbitrary. It is in fact possible, as in the perfect gas scale, toarrange that the' degree of hotness' of a body is a monotonic functionof its temperature, but we cannot demonstrate this without firstinquiring into the meaning of hotness and coldness, and findinga definition which is based on something less subjective than physiological sensation. This involves an investigation of the significance ofthe term heat; only when we have placed this concept on a secureexperimental basis can we resolve objectively the relationship betweentemperature and hotness.

Internal energy and heat

In pursuance ofour plan ofdeveloping thermodynamics as a phenomenological science, we shall pass over any consideration of themolecular interpretation of heat, and the historical controversiesbetween the followers of the caloric and the kinetic theories. The workof Rumford, Joule and innulnerable other experimenters has in truthfirmly established the kinetic theory (so firmly, indeed, that it is hardto realize that the matter was extremely controversial little more thana century ago), but although the work itself is of fundamental importance to the phenomenological aspect of thermodynamics, its molecularinterpretation is entirely irrelevant.

Let us consider the way in which an experiment, designed to measurethe mechanical equivalent ofheat, is conducted, taking Joule's paddle-


wheel experiment as typical. A mass of water is enclosed in a calorimeter whose walls are made as nearly adiabatic as possible, and throughthese walls is inserted a spindle carrying paddles, so that mechanicalwork can be performed on the system consisting of water, calorimeter,paddles and spindle. A measured amount of ,vork, Ga, is done byapplying a known couple, G, to the spindle and rotating it througha known angle, a. As a result of this work the temperature of the wateris found to change. The experiment is repeated with a differentamount of water in the same calorimeter, and it can then be deduced,after corrections have been applied to allow for the walls beingimperfectly adiabatic, what change of temperature a given isolatedmass of water would suffer if a given amount of mechanical work wereperformed on it. An alternative experiment of the same nature maybe carried out by replacing the paddle-wheels and spindle by a resistivecoil of wire. Work is then performed by passing a measured current,i, through the wire for a measured time, t. If the potential differenceacross the resistance is E, the work done is Eit (in Joules if practicalunits are used, in ergs if absolute electromagnetic or electrostaticunits). The observed outcome of this experiment is that the sametemperature change is produced as in the paddle-wheel experiment bythe performance of the same amount of work. In many other similarexperiments, using different kinds ofmechanical work, the same resultis obtained.

It should be most particularly noted that in none of these experiments is any process carried out which can be legitimately called, adding heat to the system'. All are experiments in which the state ofan otherwise isolated mass of water (or other fluid) is cha~ged by theperformance ofmechanical work. It would be a purely infei"ential, andphenomenologically quite unjustifiable, interpretation of the experiments to regard the mechanical work as transformed into heat, whichthen raises the temperature of the water. So long as we take accountonly of what is observed, the deduction to be drawn from the experiments is one which may be stated in the following generalized form:

If the state ofan otherwise isolated system is changed by the performanceof work, the amount of work needed depends solely on the change accomplished, and not on the means by which the work is performed, nor on theintermediate stages through which the system passes between its initial andfinal states.

This statement contains rather more than can be justified by theexperimental evidence presented, particularly in its last clause. Inorder to verify its validity more thoroughly, a different type ofexperiment is required, in which a given change of state is effected bytwo different processes, involving markedly different intermediate


states of the system. One might, for instance, change the state of anisolated mass of gas from that represented by the point A in fig. 2 tothat represented by B, by the two processes indicated, ACB and ADB.In the former the gas is expanded from A to C, and caused to doexternal work in the process, andis then changed to B at constantvolume by means ofwork supplied,say, electrically. In the latter theelectrical work is performed first, tobring the state to D, and the workof expansion second. According to Athe statement made above the total pwork performed in each processshould be the same. Unfortunately,it does not seem that experiments Bof this kind have ever been carriedout carefully. This is historically Cmerely a consequence of the rapidand universal acceptance of the V

first law of thernl0dynamics, and Fig. 2. Different ways of achievingof the kinetic theory of heat, which the same change of state.follo",·ed the work of Joule. Wemust therefore admit that the statement which we have enunciatedhere, and ",·hich is equivalent to the first law of thermodynamics, isnot \vell founded on direct experimental evidence. Its manifold consequences, ho,,"ever, are so ,veIl verified in practice that it may beregarded as being established beyond any reasonable doubt.

As a consequence of the first law we may define an importantproperty of a. thermodynamic system, its internal energy, U. If anisolated systenl is brought from one state to another by the performance upon it of an amount of work W, the internal energy shall besaid to have increased by an amount ~U which is equal to W. The firstlaw, in stating that W is independent of the path between the initialand final Rtates, ensures that ~U is determined solely by these states.This nleans that. once [] has been fixed arbitrarily at some value Uo foranyone particular state of the system, the value of U for every otherstate is uniquely deterlnincd. It Inay not be an easy matter in practiceto 11leasure the difference in internal energy bet\veen any two givenstates, since the experiluental processes involved in get.ting fronloneto the other solely by the performance of work on the isolated systemInay be difficult to acconlplish. Nevertheless, ho","ever roundaboutthe journey, a suitable path can ahvays be devised in principle forgetting froln one given state to another, or vice versa. It will be seenlater to be a consequence of the second law of thermodynamics that


the path may not necessarily be traced out in both directions under thespecified conditions, but for the purpose of determining aU it is onlynecessary to achieve one or the other, not both..

We have therefore shown the existence of a function ofstate, U, tnatis to say, a function which is determined (apart from an additiveconstant Uo) by the parameters defining the state of the system. Totake a simple fluid as an example,

U = U(P, V) or U = U(P,O) or U = U(V,(J).

For any system, in an adiatkermal process, i.e. one performed by meansof work on an otherwise isolated system,

aU=W. (2·8)

(2·9)

Up to this point we have considered only systems contained withinadiabatic walls, and it is only to these that (2·8) applies. If the systemis not so contained, it is found possible to effect a given change ofstate in different ways which involve different quantities of work.Abeaker ofwatermay be broughtfrom 20° to 100°C. by electrical workperformed on a resistance immersed in it, or alternatively by lightinga Bunsen burner under it, the latter process involving no work at all.This does not mean t~t any fault is to be found with the concept ofinternal energy, but that the equating of 6.U with W is only correctunder adiathermal conditions. For any change between given endstates aU can always be uniquely determined by carrying out anadiathermal experiment, for which 6.U = W, but if the conditions arenot so specialized then in general6.U =t= W. Instead we may write theequation

and thus define the quantity Q which is a measure of the extent towhich the conditions are not adiathermal. The quantity Qso defined iscalled heat, and with the sign convention adopted in (2·9) it is definedto be the heat transferred to the system during the change, just as W isthe work done on, rather than by, the system.

Such a manner of introducing and defining heat may appear somewhat arbitrary, and in justification it is necessary to show that thequantity Q exhibits those properties that are habitually associatedwith heat. In point of fact the properties concerned are not many innumber, and may be summarized as follows:

(I) The addition of heat to a body changes its state.(2) Heat may be conveyed from one body to another by conduction,

convection or radiation.{3) In a calorimetric experiment by the method ofmixtures, or any

equivalent experiment, heat is conserved if the experimental bodiesare adiabatically enclosed.

(2·10)

L\U=O,

L\U1 =Ql' L\U.=Q2 •

Ql+QI=O.Hence

and


It hardly requires proof that the quantity Q exhibits properties(1) and (2); from the known existence of diathermal walls it followsthat changes of state may be produced not solely by the performanceofwork, and that the change ofstate is not necessarily inhibited by theintervention of solid or fluid barriers, or even of evacuated spaces.That Q possesses property (3) is readily demonstrated by consideration of a typical calorimetric experiment, in which two bodies atdifferent temperatures are brought into thermal contact within a vesselcomposed of adiabatic walls. It is clear, from the definition of U andthe fact that the work separately performed on two distinct bodies maybe summed to give the total work performed, that U is an additivequantity; if Uland U2 are the internal energies of the two bodiesseparately, the internal energy of the whole enclosed system, U, maybe written as U1 +U2•

Therefore 6.U =L\U1 +L\U2 in any change in which the bodiesmaintain their identities. When the bodies are brought into thermalcontact, no work is done on either,t and since they are sUITounded byadiabatic walls

IfQis interpreted as heat, (2·10) expresses the conservation of heat inthe experiment, so that property (3) is demonstrated. We see thenthat Qpossesses all the properties habitually associated with heat, andthe use of the term is justified. Moreover, we may now understand thesignificance, as expressed by (2·9), of the common statement of thefirBf, law of thermodynamics:

Energy is conserved if heat is taken into account.

Let us now return to the question of what is meant by the termshotter and colder, which, as we have seen earlier, do not necessarily bearany relation to higher and lower temperatures on an empirical scale.In the experiment just analysed, we found that the gain ofheat by onebody equalled the loss bythe other, and this behaviour, although it doesnot necessarily imply that heat is a physical entity whose movementcan be followed from one body to another, is called, for the sake ofconvenience, the transfer ofheat from one body to the other. In generalif any two bodies are brought into thermal contact under such conditions that no work is performed on either, there will be a transfer ofheat with accompanying change ofstate of both, unless they are at the

t \Ve are neglecting the work done by the surrounding atmosphere if thevolumes of the bodies alter during the experiment. The argument may readilybe extended to include this effect, without modification of the conclusion.


same temperature and consequently in equilibrium "ith one another.The rate at which the transfer occurs may usually be varied over a widerange by altering the nature of the diathermal wall separating thebodies. The rate is a measure of the thermal conductance of the wall.The body which loses heat (Q negative) is said to be hotter than thatwhich gains heat (Q positive), and the latter is said to be colder.

It now remains to demonstrate that the scale of hotness, so defined,may be consistently linked with a scale of temperature, in the sensethat all bodies at a temperature (Jl shall be hotter than all bodies ata temperature ()2' if ()l is greater than ()2. To prove that this is possiblewe consider the consequences of presuming it to be impossible. Giventhree bodies, A , Band C, ofwhich A is at a temperature ()1 and Band Cat a temperature ()2' let us suppose that A is hotter than B, and thatC is hotter than A. We may now vary the temperature of B slightly sothat it is hotter than C while still colder than A. If the three bodies arethen placed in thermal contact, heat will be transferred from A to B,from B to C, and from C to A; by adjustment of the thermal conductances the rate of transfer may be made the same at each contact, anda state of (dynamic) equilibrium will be established. But if any two ofthe bodies are taken away they will be found not to be in equilibrium,since all are at different temperatures, and this is in conflict with theconverse to the zeroth law (p. 9). We conclude then that if a body at()1 is hotter than anyone body at ()2 it is hotter than all bodies at ()2.

Therefore if we take a suitable thermometric substance and label itsisotherms in such a way that successively hotter isotherms are ascribedsuccessively higher empirical temperatures, we have achieved a correlation between temperature and hotness which is equally valid lfor allsubstances. Henceforth we shall take the term' higher tempetature'to imply' of greater degree of hotness'.

In the above argument it has been assumed that no performanceof work accompanied the transfer of heat from one body to another.This is unnecessarily restrictive; all that is necessary is that suchparameters as are needed, together with the temperature, to definethe states of the bodies shall remain constant when the thermal contact is made. The correlation of temperature and hotness thenfollows exactly as above. This has the important consequence thataddition of heat to a body whose independent parameters of state,apart from the temperature, are maintained constant always causesa rise of temperature, and therefore the principal specific heatst ofa body are always positive.

t Here and elsewhere we use the tenn 8pec~'fic heat to mean the thermalcapacity of the thermodynamic system. regardless of its size or nature. If thespecific heat is me88ured under such conditions that the independent parametersof state (other than the temperature) are mantained constant, the me88uredproperty is one of the principal specific heats (e.g. Cp and C v for a fluid).

19

CHAPTER 3

REVERSIBLE CHANGES

Reversibility and irreversibility

In this chapter we shall discuss the significance and some applications of the first law of thermodynamics:

L\U = Q+ W in any change.

However satisfactory this equation may be as an expression ofa physical law, from an analytical point of view it leaves much to bedesired. For although U is a well-behaved function ofstate in the sensethat L\U depends only on the initial and final states of the system andnot on the path by which the change is effected, the same does notapply to either Qor W. There do not exist functions of state, of whichQ and Ware finite differences, as may be illustrated by a simpleexample. The temperature of a vessel of water may be raised either byperforming work (as in Joule's experiments) or by supplying heat;thus for any given change the values ofQand W may be altered at will,only their sum remaining constant. And this is true not only for finite,but also for infinitesimal changes, which we may represent by theequation

dU=q+w. (3·1)

It is quite impermissible to write the increments q and w in the formd~ and d1f/, and to regard them as differential coefficients of hypothetical functions of state, !2 and if/", since there is no sense in whicha given body can be said to contain a certain total quantity of heat, ~,ora certain total quantity of work, if/". All that can be said along theselines is that the internal energy U is well defined (apart from a certainarbitrariness in fixing the zero of energy), and that U may be alteredby means of work or of heat, the two contributions being usuallysubject to some measure of arbitrary variation, according to themethod by which the change is effected.

Nevertheless, if certain restrictions are imposed on the way inwhich the change occurs, it is possible to treat qand w as well-behaveddifferential coefficients, dQ And d W, without becoming involved inmathematical absurdities. For if the conditions of an infinitesimalchange are such that either qor w depends only on the initial and finalstates and not on intermediate states, it follows, since dU has this

20 Classical tkermodynamiC8

property, that the third term in the equation must also be independentof path. Then (3·1) may legitimately be written in the form

dU=dQ+dW, (3·2)

and each term will have a unique value for any given infinitesimalchange.

We must now inquire what are the restrictive conditions whichvalidate this procedure. Clearly one simple possibility is to carry outthe change under such conditions that either qor w vanishes; then (3·1)takes either of the forms dU =w or dU =q, and clearly wand q areuniquely defined and may be treated as differential coefficients, d Wand dQ. A more interesting state of affairs comes about, however, ifneither q nor w vanishes, but w is expressible as a function of theparameters ofstate. This may conveniently be illustrated by referenceto a simple fluid. If the fluid is contained within a cylinder, andvolume changes are produced by movement ofa piston, the work doneon the system (fluid +cylinder and piston) in an infinitesimal volumechange d V is - P'd V, where P' is the pressure exerted by an outsideagency on the piston. In particular, if the piston moves without friction the pressure P' must be the same as the pressure P·in the fluid ifthe system is to remain in equilibrium, and the work w in a smallchange of volume is just -PdV. Now for any infinitesimal changebetweengiven states of the fluid, the value of P is uniquely defined bythe state, as is also the value of d V by the change considered, so thatw takes a definite value, and we may write (3·2) in the form

dU=dQ-PdV. (3·3)

Note, however, that care must still be exercised when finite changes

are considered, for in such a change the work done is - IPdv, and the

value of this integral, unlike that of IdU, depends on the path of

integration. So that although dQ is a well-defined quantity for anyinfinitesimal change, it is not an exact differential, that is, thedifferential coefficient of a function of state.

In order that we may equate w to - P d V in this case, two conditionsmust be satisfied, first that the change must be performed slowly, andsecondly that there must be no friction. It is easy to understand thereason for these conditions if we imagine experiments in which theyare violated. Suppose the fluid to be a gas, and the piston to be withdrawn a small distance suddenly; then a rarefaction of the gas will beproduced just inside the piston, and the work done by the gas willbe less than if the pressure were uniform. An extreme example of thisbehaviour occurs in Joule's experiment, shown schematically iIi fig. 3.

Reversible changes 21

Vacuum

Fig. 3. Expansion of gasinto a vacuum.

Gas is contained at uniform pressure within a vessel, of which one wallseparates it from an evacuated chamber. If this wall is pierced the gasexpands to fill the whole space available, but it does so withoutperforming any external work; w is zero even though PdV does notvanish. If we are to equate w to - P d V it is essential to perform theexpansion so slowly that the term -PdV has a meaning. During thecourse of a rapid expansion the gas becomes non-uniform, so thatthere is no unique pressure P. In fact the expansion must take placesufficiently slowly that the gas is changed from its initial to its finalstate by the process of passing through all intermediate states ofequilibrium, so that at any time its statemay be represented by a point on the indicator diagram. Such a change is calledquasistatic. In the experiment shown infig. 3 only the initial and final states fall onthe indicator diagram; any intermediatestate would need for its adequate representation a specification ofthe distributionof pressure and velocity at all points in thevessel-requiring obviously many morethan the two parameters of state which aresufficient for a fluid inequilibrium. Strietly,of course, if the fluid is to remain inequilibrium at all times during a changeof state the change must be performed infinitely slowly. Usually in practice, however, a fluid requires verylittletime to equalize its pressure, and the change can 'be carried out quitequickly without provoking any significantdeparture from equilibrium.

Consider now the effect of friction in the expansion or compressionof a gas. When the gas is being compressed the pressure, P', to beexerted by the external agency must exceed the pressure, P, of the gasitselfif there is friction between the cylinder and piston, and when thegas is being expanded P' must be less than P. In a cycle ofcompressionand expansion, then, the relation between P' and V is as illustrated infig. 4, while the broken line shows the relation between P and V. Nowin any infinitesimal change of volume the work done by the externalagency is - P' d V, and clearly for a change between given initial andfinal states, the value of this quantity depends on the magnitude of thefrictional forces and on the direction in which the change is carried out.Indeed, during the course of the cycle, at the end of which the gas hasbeen returned to its initial state, the work done by the external

agency is - fP'd V, which is positive and equal to the area of the cycle

in fig. 4. Since the internal energy of the gas has been returned to its


initial value, the cycle shown can only be performed if an amount of

heat - fp' d V is extracted from the system during the cycle. We may

say then that in this process work has been converted into heatthrough the effect of the frictional forces. By this we do not intend toimply any particular microscopic mechanism, although an analysis ofthe frictional process from an atomic point of vie"r would surelyreveal the degradation of ordered mechanical energy into disorderlymotion of the atoms composing the system. From the point of view ofclassical thermodynamics this is merely a special example of a verycommon type of process which we call for convenience conversion of

p'

VFig. 4. Effect of friction on P -V relation.

work into heat, and which may be generally defined as the use ofmechanical work to perform a process which could be equally wellperformed by means of heat. All determinations of the mechanicalequivalent of heat are examples of the conversion of work into heat inthis sense.

This, however, is somewhat of a digression from our main argument,which is that friction prevents the equating of w to - P d V and henceprevents both wand q in (3'1) from being treated as well-behaveddifferential coefficients. This analysis of a special example shouldenable us to see the importance of that particular class of idealizedprocesses which are designated reversible. A reversible process is definedas one which may b~ exactly reversed by an infinitesimal change in theexternal conditions. If then a fluid is to be compressed reversibly theremust be no friction between piston and cylinder, otherwise the pressureapplied externally to the piston ""ould need to be reduced by a finite


dU==dQ-PdV.

amount before the gas could be expanded. In the same way it may beseen that compressions and expansions must be performed slowly, inorder to avoid non-uniformities of pressure in the fluid which wouldprevent the process from being reversible in the sense defined above.In fact, the condition of reversibility in a change is such as to ensurefirst that the system passes from one state to another without appreciably deviating from a state of equilibrium, and secondly that theexternal forces are uniquely related to the internal forces. It shouldalso be noted, though the significance of this will not appear untillater, that in order to transfer heat reversibly from one body toanother the hotter of the two bodies should be only infinitesimallyhotter, so that by an infinitesimal reduction of its temperature it maybecome the cooler, and the direction of heat transfer may be reversed.

To sum up, if a fluid be caused to undergo an infinitesimal changereversibly, it is legitimate to apply the first law to the change in theform (3'3),

(3-4)dU=dQ+!dl_

Different types of work

Sofar we have only considered in any detail the reversible performance of work by means of pressure acting on a fluid. Let us now seehow (3'3) may be extended so as to apply to situations in which othertypes of work are involved. There are a number of simple extensionswhich may be written down immediately, such as the work done bya tensile force in stretching a wire, or the work done in enlarginga film against the forces of surface tension. In the former case, ifa wire is held in tension by a force! and increases in length by dl, thework done by the external agency exerting the force isfdl. Providedthat the wire is deformed elastically, and not plastically, the changeof length will be reversible, and we may apply the first law to the wirein the form

It should be noted that the validity of this equation does not dependon the 'wire obeying Hooke's law, but only on the absence of elastichysteresis so that at a constant temperature stress and strain areuniquely related. The corresponding result for a film having surfacetension y and area.91 t.akes a similar form

dU=dQ+ydd, (3-5)

subject again to the condition that y shall be a unique function ofd atconstant temperature. This condition is normally satisfied by y beingpractically independent ofd.

Somewhat more careful consideration is needed for the importantextension of the argument to situations where a body is influenced by


electric and magnetic fields, and we shall consider the latter case insome detail. There has been in the past a certain amount of confusionconcerning the correct formulation of the expression for work performed by means of a magnetic field, but no difficulty need arise if wespecify exactly what the experimental arrangement is which we areseeking to analyse. This is, in any case, a highly proper attitude, forthermodynamics is above all a systematic formulation ofthe results ofexperiment, and we need feel no shame at keeping in mind a concretephysical example ofthe problem under consideration. Let us thereforesuppose that we produce a magnetic field by means ofa solenoid whichis excited by a battery, as in fig. 5. In practice the solenoid wouldnormally possess electrical resistance, but as the result we shall obtainis found by more detailed analysis to be independent of the resistance,we may simplify the argument by taking it to be zero. Then the functionof the battery is to provide the work needed to create or change thefield in the solenoid. It may be regarded as being capable of giving ane.m.f. which is variable at will; and so long as conditions in the solenoidremain steady its e.m.f. may be reduced to zero. The thermodynamicsystem is now taken to consist of everything within the enclosuremarked by broken lines in fig. 6, the solenoid and magnetizable body.Heat may be transmitted through the walls of the enclosure direct tothe body (we may neglect any thermal effects in the solenoid itself),and work may be done on the system by the battery.

Consider first the empty solenoid (fig. 5). If at any point the field is~ the magnetic energy is known from elementary electromagnetic

considerations to be given by 8~f.Jf'2d V.t This is the work which the

all space

battery must do in creating the field. We shall not assume .Jf' to beconstant within the solenoid, but shall suppose that at every point it isproportional to the current, i, through the battery,

~=hi, (3·6)

where h is a vector function of position. Now consider the effect produced by the body. Ifits magnetization changes for any reason, a backe.m.f. will be induced in the solenoid, and the battery will have toprovide an e.m.f. of the same magnitude in order to maintain thecurrent constant; similarly, a back e.m.f. will be produced if .Jf'isvaried by a change of i. The rate at which work is being done by thebattery on the system is just this e.m.f. times i, and, for example, if thefield of the empty solenoid is changed,

dW =~{~f~2dV} (3.7)dt dt 817' •

t All magnetic terms are expressed in electromaWletic units.


In order to calculate the rate of working due to changes in the magnetization ofthe body, we make use oftwo simple principles: first, thatthe law of superposition holds, so that each element of the body maybe treated independently of the rest, and the total effect found byintegration over the whole body; and secondly, that the back e.m.f.induced by the creation of an elementary magnetic dipole does notdepend on the nature of the dipole, but only on the rate of change ofthe dipole moment and a geometric factor depending on the solenoid.We may therefore calculate the work done in creating an elementarydipole dm by use of any simple model dipole, and for our purposea small current loop, as in fig. 7, is as convenient as any. Suppose then

r:----~

I II II II II II IL

r----~

I IIIII II II II I~

r:----~

I II a I

I ~. :I II I~

Fig. 5. Empty Fig. 6. Solenoid and Fig. 7. Solenoid andsolenoid. magnetizable body. current loop.

Figs. 5-7. Calculation of work in reversible magnetization.

(3·8)dW = 1M) dmdt en. dt •

we have a loop ofarea a (the vector being directed normal to the planeof the loop) carrying a current i'. Since unit current in the solenoidproduces a field h at the loop, the flux linkage due to unit current ish •a, and this is the mutual inductance between the loop and thesolenoid. If then i' changes, the e.m.f. generated in the solenoid ish.adi'/dt,so that the battery must work at a rate ih. adi'/dt. But byAmpere's theorem ai' is the magnetic moment of the loop, and by (3·6)the solenoid field at the loop is hi, so that we may write for thiselementary process,

Now, removing the time derivatives in (3 0 7) and (3·8), and integrating over all space, we have the equation

dW=d{8~f;K2dV}+f(:K.d..¢")dV, (3'9)


(3-10)

in which § is the intensity of magnetization (the magnetic momentdm of an element d V is §dV). This may also be written in the form

dW= 4~f(.7e.dgj')dV,

in which the new vector&l' is defined as.7e + 411§. It should be particularly noted that this is not the same as the conventional vector~(induction), for the field variable.7e which enters intoBiJ' is not thelocal value of the magnetic field within the body, but the field due tothe solenoid in the absence of the body; it is in fact what is usuallycalled the external or applied field. t

If the body is situated in a uniform external field, (3-9) takes theform,

(3-11)

in which ;;f( is written for f-' d V, the magnetic moment of the whole

body. This is the form which is applicable to most problems ofinterest_The results which we have derived express quite generally the work

done in the particular type of experiment envisaged. We may, however, only substitute any of these expressions for dW in (3·2) andregard d W as a well-behaved differential coefficient if the change isreversible. For this reason we must exclude ferromagnetic and othersubstances which exhibit hysteresis from our future discussion, justas we had to exclude plastic deformation in considering the stretchingof a wire. Considerable attention has been paid to the question of howto include hysteresis effects in the framework of classical thermodynamics, but it is doubtful whether any really satisfactory treatmentof such problems can be given without a more detailed microscopictreatment, which falls outside our scope. We shall therefore onlyconsider applications to substances for which § is uniquely (notnecessarily linearly) related to the local field strength, and to whichthe first law can be written in either of the forms

d U= dQ + d{~f.7e2d V} + .7e•d.;/(+other reversible work terms,811 (3-12)

ord U' =dQ +.7e •dJf+other reversible work terms. (3-13)

t It is also possible to express d W as .!-j(.Tt'z.dBl1)d V, in which .Tt'z and BIz4~

are the local values of the magnetic field and induction, but the form (3-10) isusually more convenient in practice. A proof that these fonus are equivalent isgiven by ·V. Heine, Proc. Calnb. Phil. Soc. 52,546 (1956).

(3·14)


In (3-13) the term expressing the field energy of the empty solenoid has

been incorporated in U', which equals U - ~JJt'2d V.817

The corresponding case ofelectric fields and dielectric bodies may betreated in a quite analogous manner, taking, for example, a condenseras the source of the electric field. Then it is found that, expressed inthe electrostatic system of units,

dW =d{Sl1TJ82dV} +J(8.d9')dV,

in which 8 is the electric field in the empty condenser, and 9J is thedipole moment per unit volume. Equation (3-14) is analogous to(3-9), and the analogues of (3-10-3·13) may easily be constructed.

The results we have derived are typical of the form which d W takesin a reversible change, and can be represented in general by the productof a generalized force X and a generalized infinitesimal displacementdx, so that the first law may be written for reversible changes:

dU=dQ+ ~ Xidx i i

(3-15)

In the examples considered, Xdx takes the forms -PdV,jdl, ydA,

4~J(Jt'.d8i")dVtand 4~J(8'~')dV;similar forms may be con

structed for changes of state which involve other physical variables.In many practical applications of thermodynamics there is only onework term of importance_ We shall, for the sake of illustration, usuallyconsider the simple fluid as typical of these applications and write d Was - PdV, but it should be remembered that all the results so derivedmay be transformed to meet different situations by changing P and Vinto different forms ofX and x (care being taken, of course, to alter thesigns and supply numerical coefficients where necessary).

By restricting ourselves to reversible changes we have been enabledto write the first law in the form of a differential equation (3·2) inwhich, however, only one term is an exact differential. But our analysisofvarious physical systems has shown how the inexact differential d Wmay be written in the form ~ Xidxi , in which all the Xi and Xi are

~

functions ofstate, and we have thereby advanced considerably towardscasting the law into a form in which it may be manipulated mathematically with ease. Nevertheless, we have the inexact differentialdQ still not expressed in terms of functions of state. The problem of

t The reader should consider carefully what are the analogues of X and x inthis expression_


finding the appropriate form for dQ is one which cannot be solved bymeans of the first law alone; new experimental evidence must beinvoked, and this evidence is summed up in the second law of thermodynamics. With its aid we may do for dQ what we have in this chapterdone for dW, and at the same time remove the restriction on theapplication of our results solely to reversible changes.

29

CHAPTER 4

THE SECOND LAW OF THERl\IODYNAMICS

The first law of thermodynamics expresses a generalization of the lawof conservation of energy to include heat, and thereby imposesa formidable restriction on the changes of state which a system mayundergo, only those being permitted which conserve energy. However,out of all conceivable changes ,vhich satisfy this law there are manywhich do not occur in practice. We have already implicitly noted thefact that there is a certain tendency for changes to occur preferentiallyin one direction rather than for either direction to be equally probable.For example, we have taken it as a basic assumption, in accord withobservation, that systems left to themselves tend towards a welldefined state of equilibrium. It is not observed that a reversion to theoriginal non-equilibrium state occurs; indeed, ifit did it would be verydoubtful whether the term equilibrium would have any meaning.Again we have found it valuable to make a distinction betweenreversible and irreversible changes; the former are as readily accomplished in the backward as in the forward direction, the latter have notthis property. It is the irreversible change which should be regarded asthe normal type of behaviour. In order that a change shall occurreversibly very stringent conditions must be imposed which clearlymake it a limiting case of an irreversible change. Strictly speaking thereversible change is an abstract idealization-all changes which occurin nature are more or less irreversible, and exhibit therefore a preferential tendency. However, the idea that there is a preferred direction fora given change has been perhaps most clearly expressed in our discussion of the terms hotter and colder. There is an unmistakable tendencyfor heat to flow from a body of higher temperature to one of lowertemperature rather than for either direction of flow to occur spontaneously. The second law of thermodynamics is little more thana generalization of these elementary observations. In essence it statesthat there is no process devisable whereby the natural tendency ofheat to flow from higher to lower temperatures may be systclnatically reversed. Of the many statements of the law \vhich havebeen proposed, that of Clausius approaches most clearly thi8 pointof view:

It is impossible to devise an engine which, working in a cycle, shallproduce no effect other than the transfer of heat froln a colder to a hotterbody.


The clause 'working in a cycle' should be noted. Only by the use ofa cyclical process can it be guaranteed that the process is exactlyrepeatable so that the amount of heat which can be transferred isunlimited. It is easy to devise non-cyclical processes which transferheat from a colder to a hotter body. Consider, for instance, a quantityofgas contained in a cylinder and in thermal contact with a cold body.The gas may be expanded to extract heat from the body; if it is thenisolated and compressed it will become hotter, and may be broughtinto equilibrium with a hot body; further compression will enable heatto be transferred to the hot body. But at the end of this process the gasis not in its originalstate, and no violation ofthe second law has occurredyet, even though the total amount of work done by or on the gas mayhave been made to vanish. Only if the gas can be brought back to itsoriginal state without undoing the heat transfer already effected canany violation ofthe second law be claimed. In fact, no violation can bebrought about in this case, nor with any of the ingenious and oftensubtle engines which have been devised with the object of circumventing the law. Moreover, the consequences of the law are sounfailingly verified by experiment that it has come to be regarded asamong the most firmly established of all the laws of nature.

Kelvin's formulation of the second law is very similar to that ofClausius, with'its emphasis rather more on the practical engineeringaspect of heat engines:

It is impossible to devise an engine which, working in a cycle, shallproduce no effect other than the extraction ofheat from a reservoir and theperformance of an equal amount of mechanical work.

Kelvin's law denies the possibility of constructing, for example, anengine which takes heat from the atmosphere and does useful work,while at the same time giving out liquid air as a 'waste product'. It isnot difficult to show that a violation ofClausius's law enables Kelvin'slaw to be violated, and vice versa, so that the two laws are entirelyequivalent; a proof of this assertion is left as an exercise for the reader.

A third formulation, due to Caratheodory, is not so clearly related:

In the neighbourhood of any equilibrium state of a system there arestates which are inaccessible by an adiathermal process.

Each formulation has its enthusiastic supporters. For the engineer orthe practically minded physicist Clausius's or Kelvin's formulationsare more directly meaningful, and, moreover, the derivation fromthem of the important consequences of the law may be made withouta great deal of mathematics. On the other hand, Caratheodory'sformulation is undoubtedly more economical, in that it demands theimpossibility of a rather simpler type of process than that considered

Second law 31

in the other formulations. A typically impossible process is the coolingof the water in Joule's paddle-wheel experiment; this is a very simpleexample since there is only one means whereby work may be performed, and the impossibility of cooling arises from the obvious factthat the paddle-wheel can only do work on the water and not extractenergy. But Caratheodory asserts that however complex the system,however many the different forms of work involved, which may beboth positive and negative, it is still true that not all changes may beaccomplished adiathermally. Caratheodory's law appears to differ inoutlook from the others. The average physicist is prepared to takeClausius's and Kelvin's laws as reasonable generalizations of commonexperience, but Caratheodory's law (at any rate in the author'sopinion) is not immediately acceptable except in the trivial cases, ofwhich Joule's experiment is one; it is neither intuitively obvious norsupported by a mass of experimental evidence. It may be arguedtherefore that the further development of thermodynamics should notbe made to rest on this basis, but that Caratheodory's law should beregarded, in view of the fact that it leads to the same conclusions asthe others, as a statement of the minimal postulate which is needed inorder to achieve the desired end. It bears somewhat the same relationto the other statements as Hamilton's principle bears to Newton'slaws of motion.

In view of the wealth of discussion which has centred on thedevelopment of the consequences of the second law, no harm will arisefrom giving several different approaches, not always in completedetail. The reader may then select which he prefers, or, if none is tohis taste, consult other texts for further varieties of what is basicallythe same argument. In everything that follows we shall be closelyconcerned with adiabatic changes, that is, reversible adiathermalchangest which may be analytically represented by putting dQ equalto zero in (3·15):

dU - L""~idxi=O for an adiabatic change. (4'1)i

If U is treated as a function of the generalized position coordinates Xi

and of one other coordinate X r then

_~8U . audU - ~ ~ dxa+ ~X dXr ,

t uXi u r

80 that an adiabatic ~hangemay be represented by the equation

(4·2)

t Some writers prefer to use adiabatic in the sense of our adiatherm.al, and tocall our adiabatic change a reversible adin.batic or '£8entropic change.


which for convenience we may rewrite in the form

L YidYi=O, (4-3)

in which all the coordinates Xi and X r are relabelled Yi' and eachcoefficient Yi is a function of state.

For a simple fluid, (4-3) takes the form

{(:~t+P}dV+(~~tdP=O

or

where

dP/dV=F(P, V), }

F(P, V)=-{(~~t+p}/(~~t·(4-4)

(4'8)

At every point on the indicator diagram of a fluid, the gradient of theadiabatic is uniquely defined by (4'4), and a step-by-step integrationofthe equation, starting from any point, will lead to a unique adiabaticline. Thus (4-4) represents a family of adiabatic lines covering thewhole indicator diagram.

As soon, however, as we go to more elaborate systems, havingmore than two independent parameters of state, the situation alters.A three-parameter adiabatic equation,

Y1dYl+ Y2dY2+ YadYa=O, (4'5)

does not necessarily represent a family ofsurfaces in three-dimensionalspace. Only, in fact, if the coefficients in (4·5) satisfy the equation

Yl(OY2 _ OYa) + y:2(OYa_ OY1) + y:a(OY1 _ OY2) =0,

0Ya 0Y2 0Yl oYa 0Y2 0Yl (4'6)

can (4-5) be integrated as a family of surfaces. For example, theequation

y1dYl+Y2dY2+Yadys=0 (4-7)

may be integrated immediately as the family of spheres

y~+Y:+ Y: = constant,

but no such integration is possible for the equation

Y2 dYl +dY2+ dYa=0,

for which the coefficients do not satisfy (4·6). This does not mean thatan adiabatic change governed by an equation such as (4'8) is necessarilyimpossible-clearly one can always carry out a step-by-step integration of (4·S)-but that what results is not a unique adiabatic surface_It is easy to see in fact that one can connect any two points by means

Second law 33

of a line which everywhere satisfies (4·8) (whereas of course ifwe moveso as to satisfy (4·7) we can only connect points which lie on the samesphere). To see this property of (4·8) let us start from the origin ofcoordinates and move first in the plane Y2 = constant = 0; then dYa = 0and the path follows the Yl-axis. Next, starting from any point on theYl-axis let us move in the plane Yl =constant; then dY2+dYa =0, andwe move in the plane Y2 + Ys = O. Thus by a combination of these twoprogressions we can reach any point in the plane Y2 + Ya = o. But fromany point in this plane we may move so as again to keep Y2 constant,tracing out the straight line dYa/dYl = - Y2 = constant. The family ofsuch lines starting from every point in the plane YI+Ya=O fills thewhole of space, so that we have found a way of connecting all pointswhile still obeying (4·S).

This example will serve to show that we must not, in what follows,take for granted the existence of adiabatic surfaces, except in thespecial case of simple two-parameter fluids for which the existence ofadiabatic lines has been proved. As CarathOOdory pointed out, theassumption that adiabatic surfaces exist enables one of the mostimportant consequences of the second law to be deduced purelymathematically. We shall therefore allow ourselves to consideradiabatic changes, but shall not assume that they constitute lines onwell-defined adiabatic surfaces until, by means of the second law, wehave proved the existence of such surfaces.

For the first development of the consequences of the second law weshall start from Kelvin's formulation, and proceed along the conventionallines, making use of Oarnot cycle8. Consider a system of anydegree of complexity, and in particular consider two isotherms of thesystem which correspond to temperatures (Jl and (J2. These isothermswill be lines for simple fluids or 'surfaces' of two or more dimensionsfor more complex systems. Draw two adiabatic lines which cut the(Jl-surface in A and B, and the (J2-surface in D and O. The performanceof a Carnot cycle then ~onsists of changing the state of the systemreversibly from A to B isothermally at (Jl' from B to 0 adiabatically,from 0 to D isothermally at (J2 and finally from D to A adiabatically.Let us suppose that in the isothermal change AB the system takes inan amount of heat Q1 from a reservoir at (Jl' and along OD an amountQ2 from another reservoir at (J2. Then since the system returns to itsoriginal state A at the conclusion of the cycle there is no net change inits internal energy, and in consequence it must have performed workequal to Ql + Q2"

We may now use Kelvin's law to show that for given values of(Jl and(J2 the ratio - Ql/Q2 is the same for all Carnot cycles, irrespective of thenature of the system concerned. We use a negative sign because Ql andQ2 are necessarily ofopposite sign, as follows from Kelvin's law. For if


Ql and Q2 were of the same sign it would be possible to accomplish thecycle in such a sense that both were positive; of the work done,Ql +Q2' an amount Ql could be put back into the reservoir at 01 bysome irreversible process, such as friction or Joule heat, and all thatwould have happened in the cycle would have been the extraction ofQ2 from the reservoir at O2and the performance of work, in contradiction of Kelvin's law.

To show that - Ql/Q2 takes a constant value for given °1 and °2,

consider two systems performing Carnot cycles. Let the first absorbheats Ql and Q2 in a single cycle, and the second Q~ and Q~, and letn IQl I= m IQ~ I, where nand m are integers. Then consider the composite system of the two systems taken together, and let a completecycle of this system consist ofn Carnot cycles of the first and m Carnotcycles of the second, the sense of the cycles being chosen that Q~ andQ1are of opposite sign. In this complete cycle the heat absorbed fromthe reservoir at°1, nQl +mQ~, has been made to vanish, while the heatabsorbed from the reservoir at O2 and converted into work in accordance with the first law, is equal to nQ2+mQ;. By Kelvin's law thiscannot be positive, Q Q' 0 (4.9)

n 2+ m 2~ •

But since everything is reversible we may carry out the whole cyclebackwards, and thus prove that

-nQ2-mQ~~0,

from which it follows thatnQ2= -mQ~.

(4·10)

But by definition

Hence

nQ1= -mQ~.

which proves the proposition.We may therefore infer the existence of a universal function!(°1, 0i)

having the property that

-Ql/Q2=!(01' (2) for a Camot cycle. (4·11)

Further, we may readily show that !(Ol' ( 2 ) is decomposible into thequotient ¢J(01)/¢J(02)' where ¢J(O) is some function of empirical temperature. Since!(Ol' ( 2 ) is a universal function it is only necessary toprove the decomposition in anyone special case in order to establishits generality, and the special case which we shall choose is a simplefluid, for which isotherms and unique adiabatics may be dra""n, as infig.8.

Here AB, FO, ED are portions of isotherms corresponding to thetemperatures 0t, (}2 and 0a respectively, while AFE and BOD are

Second law 35

adiabatics. Let the heat absorbed in going from left to right along thethree isotherms be Ql' Q2and Qa respectively. Then according to (4·11),

Ql/Q2 =f(Ol' ( 2 ),

Q2/Qa = f(02' Oa)

and Ql/Qa = f(Ol' Oa)·

Hence f(Ol' 0a) = f(Ol' (2)f((}2' 0a)· (4·12)

Since°2 is an independent variable which appears only on the righthand side of (4·12), the functionf(Ol' ( 2)must take the form ¢J(Ol)/r/J(02)so that°2 may vanish from the equation.

p

VFig. 8. Isotherms and adiabatics of a simple fluid.

This argument justifies the introduction of the absolute temperaturescale by the definition ofT as proportional to the new function r/J(0), or,from (4·11), by writing

-Ql/Q2=T1/T2 for any Carnot cycle. (4·13)

This equation determines the absolute scale (so called because it doesnot depend upon the properties of any particular substance, but onlyon a general property of Carnot cycles) except for an arbitrary constant of proportionality. This constant is normally fixed by thesubsidiary requireInent that there shall be 1000 bet\,.recn the meltingpoint of ice and the boiling-point of water.t It would in principle be

t It has recently boon agreed intornationally to abandon the boiling-point 88

a fixed point and to define the absolute degree in such a way that the ice-pointis exactly 273'15° K. This means that the boiling-point is very nearly, butnot precisely, 373·15° K. The symbol K (for Kelvin) is used to designate theabsolute scale of temperature.


possible to determine the form of the function ¢J(O) for any givenempirical scale by the performance of Carnot cycles under as nearlyideal conditions as possible and the determination of - Ql/Q2' Since,however, it is possible to show that T =0 if the perfect gas scale is usedto define 0, this tedious process is unnecessary. We shall defer a proof ofthis result until the analytical development of the second law hasproceeded far enough to make it a very straightforward matter.

We have shown that ~Q/T=O in a Carnot cycle, and may nowproceed to establish that a similar result holds for any reversible cycle.Consider a system, u, which executes a cyclical process of any degreeof complexity. In the course of traversing one element of this cyclework will be done in general on or by the system, and it will also benecessary to transfer heat to, or abstract heat from, the system. It isconvenient to imagine that each element of heat q is transferred to thesystem from a subsidiary body, u', at temperature T, which may becaused to execute Camot cycles between the temperature T and thefixed temperature, To, of a heat reservoir. The traversal of the elementof the main cycle may then be considered as involving the followingprocedure:

(I) u is in its initial state; u' is at temperature To.(2) (1" is brought adiabatically to T.(3) (I' moves to its final state, absorbing heat q from (1", which

proceeds along an isothermal at temperature T.(4) (1" is returned adiabatically to To and compressed or expanded

isothermally until it regains its initial state exactly.Since in this infinitesimal process (1" loses heat q at temperature Tin

stage 3, it must gain from the reservoir in stage 4 heat amounting toqTo/T, by (4'13). In the course of the whole cycle of (1', then, the heat

lost by the reservoir isequal to TOfq/T, the integral being taken around

the cycle of (1'. Now the intemal energies of both (I' and (1" are the sameat the end as at the beginning of the cycle, so that this heat, if positive,is equal to the work done by (I' and (1" during the course of the cycle. Itfollows, then, from Kelvin's law that (since To is constant),

(4'14)

If the cycle of u is reversible we may imagine the whole process carnedout backwards, when for every element q in the original cycle we havenow -q. To avoid violation of Kelvin's law it must now be true that

Second law 37

Hence for a reversible cycle

!q/T=fB q/T+f.d q/T=O,J .4.pat.hl Bpathi

which proves the proposition. If then we introduce a function S, theentropy, by means of the definition

SB-SA= f;q/T for a reversible change from A to B, (4·16)

or, differentially,

fq/T=O. (4·15)

It should be noted that T here is strictly the temperature of the bodyu' which supplies heat q; but for the cycle to be reversible (J' and (J"

must have the same temperature when any heat transfer occurs, sothat T may be also interpreted as the temperature of the system uwhich receives heat q. If, however, the cycle of (J' is not reversible thenwe can only establish the inequality (4'14), which is known asOlausius's inequality. And in these circumstances we must be carefulto interpret T as the temperature of the body which supplies the heat.This distinction will be of importance when we come to considercyclical changes in which the system (J' may not be in equilibrium atall times, and may not have a single definable temperature.

We shall not at present discuss the consequences of Clausius'sinequality, but consider first the important consequences arising from(4'15), which is applicable to reversible processes only. It is animmediate corollary of (4'15) that for all reversible paths between two

equilibrium states (A and B) of a system, the integral f~q/T takes the

same value. For if we consider two such paths we may traverse onein the opposite sense from the other and thus construct a reversiblecycle. Hence

d8=q/T for an infinitesimal reversible change, (4·17)

it follows that the difference SB - S.4. is independent ofthe path chosento connect A and B, and that S is therefore a function of state, completely determinable once its value has been arbitrarily fixed for oneparticular state of the system. This result enables us to infer immediately that adiabatic surfaces exist for all systems, however complex.For in an adiabatic (reversible) change Q=O and hence S is constant;the family of surfaces S =constant constitutes the adiabatic surfacesof the system.

In deriving the existence of adiabatic surfaces and of the entropyfunction, we have used the second law liberally in order to avoid


mathematical argument. Let us now see how we can arrive at thesame result by a more economical use of physical principles anda more lavish use of mathematics. We shall still employ Kelvin'sformulation of the second law, but shall first establish the existence ofadiabatic surfaces, without employing Carnot cycles, and thencededuce the existence of entropy and of an absolute scale of temperature. The argument is more readily visualized if we consider a systemdefined by three parameters, for which the isothermals are twodimensional surfaces. Let us construct an adiabatic surface by thefollowing procedure. Draw any adiabatic line, Le. one which at allpoints satisfies the equation (which is the same as (4·5))

YldYl + Y2 dY2+ YadY2=O. (4·18)

Starting from each point on this line construct an adiabatic line whichlies on the isothermal surface through that point. In this way we haveconstructed an adiabatic surface, and we must no,"' show, by use ofKelvin's law, that our procedure has led to a unique surface satisfying(4·18) at all points. To do this we first consider two points, P and P',lying in the surface and infinitesimally close to one another. It ispossible to join these by means of a curve fornled from parts of theadiabatic lines whereby the surface was constructed, so that along this

JP 1

curve p dQ=O. Hence the infinitesimal element PP' must satisfy

(4·18). For if it did not it would satisfy an equation of the form

Y1 dYl + Y2dY2+ YadYa=dQ

with a non-zero value of dQ; this would imply that a reversible cyclecould be constructed in which heat dQ was absorbed from a reservoirat the temperature corresponding to the point P, and an equal amountof work done, in violation of Kelvin's law. Thus all line elements in thesurface are Rolutions of (4·18), so that all curves lying wholly in thesurface are adiabatics. Moreover, since the tangent plane to the surfaceat P has now been shown to be a solution of (4·18), it follows that aninfinitesimal line element PP", joining P to any neighbouring pointpIt not lying in the surface, is not a solution of (4·18) and hence PP"is not an adiabatic line; by the ~ame argulnent as before we concludethat there can be no adiabatic line, however roundabout, connectingP and J.J". The surface we have constructed is thus entirely surroundedby adiabatically inaccessible points, and 80 is a unique adiabaticsurface. The argument may readily be extended to systems of anynumber of parameters.

Kno,ving that adiabatic surfaces exist, ,ve may now prove withoutfurther reference to the second law, that in a reversible change dQ lnaybe \\'Titten as T d8', in which T is a function of empirical tenlperature

Second law 39

(4·21)

only, and S is a function of state. To do so we imagine each separateadiabatic surface of a system having n independent parameters to bearbitrarily labelled with a number ¢J, and thus introduce a new function of state ¢J(Yi) which has the property that ¢J =constant for anadiabatic change, or o¢J

~-dYi=O. (4·19)i 0Yi

In addition, for an adiabatic change, (4·3) holds:

~ YidYi=O.,Therefore, if A(Yi) is any function of state, it follows from (4·3) and(4·19) that on an adiabatic surface

1(Yi-"-::)dYi=O. (4·20)

Now of all the n coordinates Y" only (n-I) are independent, since thedYi must satisfy (4·19). Let us regard Yl as the dependent variable, andchoose A such that

The first term in (4·20) now vanishes and we are left with an equationinvolving only independent variables. If then (4·20) is to be generallytrue, each coefficient of dYi must vanish separately, and

Yi= "- ~t/J for all i.UYi

Hence in a reversible change which is not adiabatic,

(4·22)

It remains to demonstrate that A is a function of () and ¢J only.For this purpose consideracornposite system consisting oftwo separatesystems in thermal equilibrium, at a common empirical temperature(J, the first system having n coordinates Yl ... Yn' and the second mcoordinates Zl ... Zm. For each system we may introduce the newfunctions, ¢J(Yi) and A(Yi) for the first, ¢J'(Zi) and A'(Zi) for the second,and we have that in a reversible change

dQ=Ad¢J for the first system,

dQ' = A' d¢J' for the second system.

For the combined system the same arguments may be used to demonstrate the existence of functions <D(Yi' Zi) and A(Yi' Zi), such that for

40 Classical tkermodynamiC8

or

any reversible change,

Ad<1' = dQ+dQ' = Ad¢J+A'd¢J',

A A'd<1'=i\ d¢J+i\ d¢J'. (4-23)

Now since 0, ¢J and ¢J' are all functions of state, we may carry outa change ofvariables and describe the first system by (), ¢J and n - 2 ofthe original y-coordinates, and the second system by (), ¢J' and m - 2 ofthe original z-coordinates_ Then although <1' is in principle a functionof all these variables, nevertheless (4-23) shows that there are only twoterms in the total differential of <1'; thus ell is in fact a function of only¢J and ¢J' and is independent ofall the other coordinates, inparticular of0_ Hence o<1'/o¢J, Le_ A/A, and 8<1'/o¢J', Le_ A'/A, are independent of allcoordinates except ¢J and ¢J'. Therefore since Ais independent of thez-coordinates, A also must be independent of the z-coordinates; andsince A' is independent of the y-coordinates, A also must be independent of the lI-coordinates. Hence

A=A(¢J, ¢J', 0), A=A(¢J,O), A' =A'(¢J', 0).

But, further, since AIA and A'IA are functions only of ~ and ¢J',

:0 (~) =~(~)=0.

Le_ :0 (log A) =:0 (logA') =:0 (log A)_ (4-24)

In (4-24) we have the first term a function of ¢J and 0 only, the seconda function of ¢J' and 0 only, ¢J and ¢J' being independent variables_Hence in reality each term in (4-24) is a function only of O. We havetherefore proved that for any system

880 (log A) =g(O), (4·25)

in which (/(0) is a universal function of 0, or, integrating (4-25),

A=F(,p) exp Ug(O) dO], (4-26)

the form of the function F(¢J) being determined by the way in whichthe adiabatic surfaces were labelled, Le_ by the form chosen for thefunction ¢J. If now we define

T(O):=oexpUg(O)dO]. (4-27)

S(,p):=~fF(,p)d,p, (4-28)

Second law 41

then (4·22) takes the form dQ= T dB, in which Tis a universal functionof (), and S is constant in an adiabatic change. We have thus arrived atthe result expressed in (4·17) by a quite different route.

Finally, let us see in outline how Caratheodory's law may be usedinstead of Kelvin's to establish the existence of adiabatic surfaces.According to this law there are in the neighbourhood of any stateother states which are inaccessible by an adiathermal, and a fortiori byan adiabatic, process. By the word neighbourhood we need not implythat the inaccessible states are infinitesimally close, although that isthe conclusion we shall reach eventually. All we mean is that theinaccessible states are always close at hand, and do not consist of a fewsingular points or surfaces which are inaccessible from everywhereelse. Ifthen the nearest adiabatically inaccessible point Qfrom a givenpoint P is a finite distance from it, we may assume that this holds ingeneral wherever P may be situated. But this is clearly impossible.For if we join P and Qby a line L, there will be on L a point P' which,being nearer than Q, must be accessible from P but inaccessible fromQ; since P' may be made as close as we choose to Q it follows that thenearest inaccessible point to P is at a finite distance from P, ,vhile thenearest inaccessible point to Q is infinitesimally close. Hence weconclude that Caratheodory's law can be obeyed only if there areadiabatically inaccessible points infinitesimally close to any givenpoint. We may now proceed as in the earlier analysis and, startingfrom any point P, draw an adiabatic line through P, an adiabaticsurface through this line, and so on, according to the number ofparameters involved. If we apply the same procedure to two points,Q and Q', infinitesimally close to P on opposite sides ofthe surface, andinaccessible from P, we may cordon off the surface through P by twoinaccessible surfaces as close as we please, and thus demonstrate theuniqueness of the adiabatic surface constructed. From this pointonward the rest of the argument proceeds as before.t

We have thus by a variety of means demonstrated that for a reversible change dQ may always be written as TdS; further, in an irre-

versible cycle fqJT < 0, where T is the temperature of the body

supplying the element of heat q. We have only proved Clausius'sinequality by one means; to construct a proof without invokingcyclical processes may be left as a not very easy exercise for thereader. We shall not at this stage discuss Clausius's inequality further,

t This argument does not do justice to tho precise mathE'matif'al roasoningof Caratheodory's development, but is intended to show in a simplo way howCaratheodory's principle leads to a proof of the existence of adiabatic: sur-fa(·ns.For more careful discussions consult A. H. Wilson, Thermodynamics and StatisticalMechanic8 (Carnbridge, 1956), R. Eisenschitz, Sci. Progr. 170, 246 (19;);')).


but first confine our attention to the equality applicable to reversiblechanges. Since in a reversible change, according to the first law,

dU=dQ+ ~Xidxi'i

we have that (4·29)

(4·30)

(4-31)

dU=q+w

dU=TdS-PdV.and

This result, derived by a consideration of reversible changes, statesa relationship between functions which are all functions of state. Wemay therefore make a most important extension of the range ofvalidity of (4·29), and declare that it is applicable to any differentialchange, reversible or irreversible. For example, for any change ina fluid,

It is only for a reversible change, however, that q=TdS andw = - P d V; neither oftheseequalities hold for an irreversible changeq=l=TdS and w=+= -PdV-but (4·31) is still valid; if q=TdS-E thenW = - PdV+ E. This point is illustrated by the experiment pictured infig. 3, the expansion of gas into a vacuum under isolated conditions.Here PdV has a definite, non-zero value, but w=O; similarlyq=O butT dS has a non-zero value, which could be determined by compressingthe gas reversibly to its initial state and determining how much heatmust be extracted during the compression. We should find that it wasjust equal to PdV, and that in consequence the entropy increaseduring the irreversible expansion was PdVIT.

We have now solved the problem which we set ourselves, of findingexpressions for q and w in terms of functions of state, so that we shouldhave only exact differentials to handle. Equation (4·29) is the solution,valid for all changes, and it is from this equation that the analyticalapplication of thermodynamics to phYSIcal problems stems. Thefollowing chapters will be concerne~with such applications.

43

CHAPTER 5

A MISCELLANY OF USEFUL IDEAS

Dimensions and related topics

In the last chapter we derived the important result that for anydifferential change

dU=TdS+ LXidxii

In particular, for a fluiddU=TdS-PdV,

(5·1)

(5·2)

and corresponding results hold for other two-parameter systems inwhich P and V are replaced by the appropriate X and x; for instance,a solid magnetizable body may often be r~garded as unaffected bypressure changes and under this assumption it obeys the equation

dU'=Td8+K. d.;/(. (5·3)

We shall consider the consequences of (5·2); they can obviously beapplied, mutatis mutandis, to (5·3) or similar modifications.

If any thermodynamic variable is a function of two parameters, itis possible to consider any two functions of state as independentvariables, and we have available for a fluid the variables, ll, P, V, T, S,or combinations of these such as H the enthalpy, defined as U + P V;F the Helmholtz free energy, defined as U -TS; G thc Gibbs function,defined as U -TS+PV; or any other suitable combinations. It isdesirable of course that the combination chosen should be dimensionally homogeneous, and clearly this condition is satisfied by H, p" andG, which all have the dimensions of energy. In this conncxion it isworth remarking on three points. First, there is nothing in theway temperature and entropy are introduced which ena bles usto determine their dimensions separately; all we can say is thatthe product TS has the same dimensions as U. We must expecttherefore that S will never appear in any equation without T to renderit dimensionally meaningful, unless the equation is in itself hOlllOgeneous in S so that its dilnensions are unimportant. The secondpoint concerns the variation of the thermodynamic paramcters \\'it.hthe size of the body considered. Unless surface phenonlclla areconcerned the behaviour under equilibrium conditions of silnilarbodies in given circumstances is not dependent on their size, and thework required to cause a given change oftcmperature, say, to a bodyof given constitution under conditions of thermal isolation is simply


proportional to the size of the body. It follows that U is proportionalto the size of the body, and we shall therefore expect TSand P V (orany XiXi) to be proportional to the size, otherwise (5·1) would predictsize-dependent behaviour. It is clear that this proportionality isusually achieved by having only one member of each pair of variablessize-dependent. There are variables such as T, P or £ (intensivevariables) which determine the conditions to which the body issubjected, and which are size-independent, and there are the corresponding variables S, V and vii (extensive variables) ,vhich undergiven conditions are proportional to the size of the body. In theseexamples the intensive variables have happened to be the coefficientsof the differentials in (5·1), but this is not a general rule. For example,if we consider a wire in tension we cannot ascribe the terms intensiveor extensive uniquely, since the behaviour of the wire depends notsimply on its volume but on its shape as well. Ifwe are concerned onlywith wires of a given cross-sectional area, the work term if writtenfdl has f as an intensive and l as an extensive variable, but if it iswrittenflde (de:=dl/l), then according to our definitionfl is extensiveand € intensive. We need not concern ourselves with such hairsplitting, however; in any practical application no ambiguity arises.

Finally, we may note that if we have a composite system, consistingof several subsystems, each in equilibrium within itself, though, ifthermally isolated, not necessarily in equilibrium with the others, thevalues of the extensive variables V, U and S for the whole system maybe taken without inconsistency to be simply the sum of the contributions from all the subsystems. For V this is obvious; proof for theother variables, based on the definitions of LlU and LlS, is left to thereader. The proviso that each subsystem shall be in equilibrium is ofcourse essential if the entropy is to have any meaning. This additivityrule applies also to the free energy F if all the subsystems have thesame temperature, since

the summations being taken over all subsystems. But if the subsystems are not at the same temperature the free energy of the system asa whole is not a meaningful concept (since the temperature of thesystem as a whole is undefined); it may, however, be arbitrarilydefined as 'LFi • Similarly, the additivity rule applies to the Gibbsfunction G only if T and P are constant throughout the system,though again, if required, G may be defined as ~Gi even in nonuniform systems.

Useful ideas 45

Maxwell's thermodynamic relations

Returning now to (5-2), we may regard U and all other parametersas functions of two independent variables, say y and z, and in general

(5-4)(~~t= T(~Z)'-pe:).(aU) _T(OS) _p(8V)

OZlI- oZ'JI OZ'JI-

In particular, we may choose either or both ofS and Vas our independent variables_ If Sand z are chosen,

and

ifS and V are chosen,

and

(5-5)

(5-6)

(5-7)

From (5-6) and (5-7) an important result is obtained by a seconddifferentiation_ From (5-6)

82U (8T)oVoS= 8V S'

82U (8P)and from (5'7) oSoV= - oS v'

Since in the double differentiation the order of differentiation isimmaterial, we have that

(M_I)

This equation is the first of Maxwell's four thermodynamic relationsIn all that follows these relations will be referred to as M_I, 2, 3 and 4.The other three lnay be obtained by use of the functions H, F and G_For example, since H = U +P V, in general

dH=dU+PdV+ VdP

=TdS+ VdP from (5-2)_

Hence by double differentiation, taking Sand P as independentvariables, we have that

(~t= (~~t· (M.2)

46 OlaBBical tkermodynamiCB

The reader may easily verify in a similar manner the last two:

and

(M.3)

(M.4)

(5·9)

(5·8)

These four equations should not be regarded as independent deductions from (5·2); given one, the others may be deduced by manipulation alone, by means of the mathematical identities for functions oftwo variables:

and

To show this, let us start from M. 1, which may be rewritten by use of(5·8),

and by use of (5·9),

which is the same as M.2. Repetition of the procedure yields M.3 andM.4·t

Identity of the absolute and perfect ~as scales of temperature

The chiefvalue ofMaxwell's relations is that they allow a rearrangement of differential coefficients so that the relationship betweendifferent observable phenomena is exhibited. This is particularly true

t In view of the wide application of Maxwell's relations it is worth whilecommitting them to memory, even though their derivation is 80 simple. Thefollowing remarks may help in remembering them. First we may note that theyare dimensionally homogeneous, in that cross-multiplication yields each timethe pairs TS and PV (the operator ais of course dimensionless). Secondly, theequations may always be written 80 as to exhibit the independent variables inthe denominator. Maxwell's relations represent the four possible equationswhich satisfy these requirements, so that one can write them down immediately,complete except for the signs, of which two are positive and two negative. Thesigns can easily be found by inspection if we consider the meaning of the equations when they are applied to a perfect gas. To take M. 1 as an example, theleft-hand differential coefficient describes the increase in temperature when thegas is expanded adiabatically, and is therefore negative, since a gas cools onexpansion; the right-hand differential coefficient describes the increase inpressure when a gas is heated at constant volume, and is therefore positive. Thenecessity for the negative sign is now clear. The reader should satisfy himselfthat he understands the meaning of the other three relations in the same way.

dU=TdS-PdV,

Useful ideas 47

of M.3 and M.4, which contain on the right-hand side quantitieswhich are difficult to measure in practice, as they involve calorimetry,while the quantities on the left-hand side may be computed from theequation ofstate, which may usually be found experimentally withoutexcessive difficulty. This point should become clear as we proceed, inlater chapters, to the applications of thermodynamics. A simpleexample of the use of Maxwell's relations is, however, convenientlyintroduced here, to justify an assertion made earlier that the perfectgas scale and the thermodynamic scale of temperature are identical.We must first define what we mean by a perfect gas without referenceto a scale of temperature, and for this purpose we make use of thefollowing experimental facts:

(I) As the pressure of a real gas tends to zero, the product PV atconstant temperature tends to a finite limit (Boyle's law).

(2) As the pressure of a real gas tends to zero, the internal energyat constant temperature tends to a finite limit (Joule's law).

The experimental evidence for the second statement is, first, Joule'sobservation that a gas expanding into a vacuum under isolated conditions experiences no change of temperature, and, secondly, morerecent experimentst in which more precise observation has revealeda dependence of U on P which becomes negligible as the pressure islowered. We take the laws of Boyle and Joule to define a perfect gas, inconfidence that a good approximation to a perfect gas can be found inpractice, and that the behaviour of perfect gases may in principle befound experimentally by extrapolating the behaviour of real gases tozero pressure (see p. 89). The two laws are sufficient to enable us todeduce the equation of state of a perfect gas thermodynamically.

For any gas,

(5·10)

80 that(8U) =T(BS) _pBV T BV T

=T(OP) _p from M.4.BT y

In particular, for a perfect gas (BUI8V)T=0 from Joule's law, so that

(:;t=~.or P=TF(V), where F(V) is an as yet unknown function of V. Butaccording to Boyle's law, at constant temperature PocIIV. ThereforeF( V) ocll V, and PV = kT, where k is a constant. Now the perfect gasscale of temperature is defined by the equation P V = RfJ, so that fJIT is

t F. D. Rossini and M. Frandsen, J. Ru. Nat. B1W. Stand. 9, 733 (1932).


a constant. By choosing the same fixed points the constant can bemade to equal unity, and the scales of temperature coincide. Thisargument does not of course solve the practical problem ofcalibratinga thermometer according to the absolute scale; it indicates a methodwhereby it can be done. The experimental problems of constructinga gas thermometer and extrapolating its behaviour to zero pressureare very small ifonly moderate accuracy is desired, but ifuncertaintiesof 1 ~ 0 0 or less are sought very elaborate precautions are needed. Weshall not enter upon this problem at all here, but refer the interestedreader to other treatises for more detailed accounts.t Some of thethermodynamical problems involved in establishing the absolutescale will be discussed more fully in the next chapter.

Absolute zero, ne~ative temperatures and the third law ofthermodynamics

Measurements with gas thermometers, and by other means, haveestablished that the melting-point of ice on the absolute Centigradescale of temperature is 273.150 K. By the use of gas liquefiers lowertemperatures than this are readily attained, the normal boiling-pointof liquid helium, for example, being 4.2 0 K. A bath of liquid helium,contained in a Dewar vessel to minimize the leakage of heat from outside, may be cooled further by pumping away the vapour and causingthe liquid to boil at a reduced pressure. In this way temperatures aslo"\\P as 0.80 K. are attainable, but it is not feasible to go much lowerthan this since the vapour pressure ofhelium falls very rapidly towardszero below 0.80 K., and no pump will maintain a low enough pressureagainst the large volume ofgas which evaporates from the liquid underthose conditions. To reach still lower temperatures the technique ofadiabatic demagnetization was devised (p. 67), and the lowest temperature yet reached by this means is about 0.001 0 K. These experiments in themselves suggest that, however low the temperature maybe brought, there may be some limitation to all cooling methods whichprevents the absolute zero, 00 K., from ever being reached.

It was once a rather widely held belief that the second law providedarguments against the possibility of attaining the absolute zero, andwe shall now indicate the lines of thought involved, although as weshall see later they cannot be regarded as convincing. The most efficientway of lowering temperature is to employ an adiabatic process, suchas expanding an isolated mass of gas reversibly, lowering the pressureover an isolated bath of liquid, or reducing the magnetic field appliedto an isolated block of paramagnetic salt. All these methods depend

t Temperature, Its Measurement and Control in Science and Indu8try (Reinhold, 1941). F. Henning, Temperaturmu8Ung (Barth, 1951).

Useful ideas 49

for their operation on the use of a substance or system in which theentropy may be varied at constant temperature by the change of someconvenient parameter {J (e.g. V, P and l/e in the three examplesabove). For if (OS/O(J)T does not vanish, then by (5·S) neither does(oS/oT)p (oT/o{J)s; therefore so long as (oS/oT)p is finite,t an adiabaticchange of {J will alter the temperature. For a given change of fJ thegreatest cooling is achieved by making the change reversibly, as thereader should be able to prove from the second law.

Now suppose that by such an adiabatic process the temperature ofthe system may be reduced to zero. It might appear that this providesthe possibility of operating a Carnot engine between zero and a nonzero temperature, and that this would enable the second law to beviolated, since such an engine need discharge no heat at the lo,veroperating temperature and can still do useful work. But we must becareful in jumping to this conclusion, since a Carnot cycle involves theperformance ofan isothermal change ofentropy at the lower operatingtemperature, and if this is zero the isothermal change is also adiabatic,in the sense that no heat is needed to alter the entropy at 0° K. This is,however, to some extent only an ambiguity of terms, for the essentialdistinction to be made is between a reversible adiathermal change andan isentropic change, which are of course identical except at 0° K.If a reversible change in the value offJ at 0° K. alters the entropy, thenthe two adiabatic portions of the Carnot cycle are performed atdifferent values of the entropy, and this enables heat to be taken in atthe upper operating temperature, and the second law to be violated.On the other hand, if the entropy is not dependent upon fJ at 0° K. thetwo adiabatic portions necessarily coincide, the cycle is inoperative,and the validity of the second law is preserved.

It has been argued in this way that the second law requires that anyreversible change taking place at the absolute zero shall involve nochange in entropy; and this requirement is readily seen to precludethe attainment of the absolute zero. For if we construct adiabaticsurfaces in the coordinate system of T and any relevant parameters(X, x), the best hope of reaching 0° K. is to move on one of these

t If real physical systems were governed by the laws of classical mechanicsthe problem of trying to reach 0° K. would never arise. For Boltzmann's law ofequipartition of energy would ensure that the specific heat remained at a nonvanishing value at all temperatures. Thus as T tended to zero (oSjoT)/3 wouldbecome infinite as T-l. In other words, the entropy of all substances wouldtend to - 00 as T tended to zero, and no isentropic process could roduco T tozero. In fact, however, it is a consequence of the quantal behaviour of matterthat all specific heats tend to zero at 0° K. at least as fast as T (for metalsOy oc T, for non-metals and liquid helium C v oc T3 at the lowest temperatures).Thus the entropy tends to a finite limit and the question of the attainability ofabsolute zero is one which must be considered.

50 Ola881,coJ, tkermodynamiC8

surfaces from some starting temperature To and finish eventually atOOK. But if the above argument is valid, a single adiabatic surfacecovers the whole variation ofthe parameters (X, x) at 0° K., and therefore the only chance of reaching 0° K. is for this surface (on which wemay for convenience put S=O) to have a branch into the region ofhigher temperatures, as is indicated for one parameter p in fig. 9. Itwill now be seen that this automatically involves (as/aT), and therefore 0, becoming negative in some region of the diagram. However,op being a principal specific heat is always positive, as pointed outon p. 18, and therefore the zero entropy contour cannot branch asin fig. 9 and the absolute zero is unattainable.

o

o

TFig. 9. Hypothetical branched adiabatic.

Three criticisms may be directed against the foregoing argument.The first is that we have considered an idealized experiment, in thatwe have postulated perfect reversibility in the Camot cycle, andperfect isolation in the adiabatic parts. Although this may be legitimate at non-zero temperatures, since, for example, there is a meaningto be attached to the limit, as 8Q goes to zero, of the entropy change8Q/T resulting from a leakage of heat, this limit is meaningless whenT = o. Any heat leak, however small, from the surroundings to thesystem at 0° K., raises its temperature above 0° K. by an amountwhich is never negligible~ Secondly, it may be argued that it is takingthe second law too far beyond the range of its experimental founda-

U8eful ideas 51

tions to apply it in the vicinity of 0° K. ; it might be that the Carnotengine considered constitutes the one exception to its general validity.But the third criticism is the most cogent. We have assumed that ifthe system is such that (88/8P)T does not vanish at 0° K., it is possibleto operate a Camot cycle without a reservoir at the lower operatingtemperature, 0°K. Certainly this is true in the sense alreadydiscussed,that an isothermal change at zero temperature involves no transfer ofheat. But this very circumstance makes the cycle inoperative; forthere is no practical way of compelling the isolated system to performthe isothermal change rather than an adiabatic (isentropic) change.The ambiguity of adiabatic and isothermal changes at 0° K., out ofwhich we attempted to talk our way on p. 49, is thus a crucial matterwhich invalidates any attempt along these lines to demonstrate thatthe unattainability of the absolute zero is a consequence of the secondlaw.

In view of this failure we must, if we are to incorporate the idea inthe development of thermodynamics, introduce it as a new postulate,the third law of thermodynamics:

By no finite aeries ofprOCU8es i8 the absolute zero attainable.

By reversing the argument given above it is now readily seen that noisentropic surface connects any point at 0° K. with any other point ata higher temperature, and therefore, since S remains finite down to0° K., all points at 0° K.lie on a single isolated isentropic surface. Thisenables us to give an alternative statement to the third law:

A8 the temperature tentlB to zero, tAe magnitude ofthe e'IIJropy change inany reversible proces8 tentlB to zero.

We shall meet, in the following chapters, a few examples whichdemonstrate the application of the third law, but it has not the sameimportance in physics as it has in chemistry, where it plays a mostvaluable role in enabling the equilibrium constants of chemical reactions to be calculated from the thermal properties of the reactants.The success which attends its application in this field leaves no roomfor doubt of its correctness. In view, however, of the limited use weshall make of the law, and the need for a full development of chemicalthermodynamics in order to appreciate its application, we shall notdiscuss it any further.

From the unattainability of the absolute zero and the experimentalfact (which is a consequence of the validity of classical mechanics indescribing the atomic behaviour of matter at high temperatures) thatthe specific heat ofa complete system does not tend to zero as T tendsto infinity, it follows that the temperatures ofall bodies have the sameIign, which is positive by definition. Recently an experiment was


reported in which part of a system (the atomic nuclei of a solid) wasbrought by magnetic means into what may be described with somejustification as a state of negative temperature.t A full discussion ofthis experiment lies outside the realm of classical thermodynamics,since it involves a microscopic viewpoint to see the assembly ofatomic nuclei as forming a subsystem which can be considered isolatedfrom the rest of the solid lattice. The specific heat of such a subsystemtends to zero at high temperatures sufficiently rapidly for only a finiteamount ofenergy to be needed to raise the temperature to infinity. Thethermal conductance between the nuclear subsystem and the rest ofthe lattice is so low that this may be achieved while the lattice ismaintained at ordinary temperatures. It is even possible to add stillmore energy to the subsystem, and this is equivalent to forcing itstemperature into the negative region. A microscopic analysis, bystatistical means, shows that +00 and - 00 are indistinguishable astemperatures. This is because from the point of view of statisticalthermodynamics it is liT, rather than T, which is the physicallysignificant parameter, and there is no energetic barrier in these particular experiments preventing passage through the origin of liT.Thus negative temperatures are hotter than any positive temperature,and the hottest possible temperature is -0, while the coldest is +0.This experiment shows very clearly the ~ssentiallysingular nature ofthe absolute zero, which is quite impassable by any means. It islegitimate to inquire whether the second law may be violated byworking a Carnot engine between two temperature baths ofwhich oneis negative and the other positive. It may be shown that no violationis possible, since no isentropic surfaces connect positive and negativetemperatures, and therefore no reversible cycle may be constructed.In the experiment the passage from positive to negative temperaturewas effected by an ingenious trick which does not correspond to anyordinary reversible process.

This experiment has been mentioned solely in order to point outthat there are exceptions to the rule that temperatures are positive.But these exceptions never occur with complete systems in equilibrium, only for very special isolable subsystems, and for normalpurposes we take T to be always a positive quantity.

t The reader cannot be expected to understand the nature of this experimentwithout a rather detailed knowledge of the field of spin resonance. The experiment is described and discussed by N. F. Ramsey, Phys. Rev. 103, 20 (1956).

Useful ideas 53

An elementary graphical method of solving thermodynamicproblems

We shall now examine briefly a graphical method which is sometimes used to solve elementary thermodynamic problems. The substance considered is imagined taken around a Carnot cycle betweentwo neighbouring temperatures T and T - oT. If the area of the cyclein the indicator diagram is 0-,,4, and the heat absorbed at T is Q, thenthe heat given out at T - aT is Q- 0-,,4, since 0-,,4 is the work done in thecycle. Hence, by (4·13),

or

Q-o-,,4 T-oT-Q---T-'

oA/Q=oT/T. (5·11)

Evaluation of the quantities occurring in (5·11) leads to the desiredresult. We may illustrate the method by deriving Clapeyron's equation for the variation of vapour pressure ofa liquid with temperature.In fig. 10 are drav~'1l t\\ro neighbouring isotherms of a liquid-vapour

pD

T B

T-oT

v~'ig. 10. Camot cycle for deriving Clapeyron's equation.

system, the horizontal portions corresponding to the two phasescoexisting in equilibrium. The Carnot cycle consists of two longisotherms AB and CD and two short adiabatics BC and DA. Wesuppose the variation from A to B to correspond to the evaporation atconstant pressure P ofunit mass ofliquid, necessitating the absorptionof heat equal to the latent heat per unit mass l. To find the area of thecycle we note that the length AB is vv-v" the difference in volumebetween unit mass ofvapour and liquid, while the vertical \vidth ofthecycle is (dP/dT) oT, P being the equilibrium vapour pressure. Thus

(5·12)


8A=(vv -vz)(dP/dT)aT, any difference in slope of AD and BObecoming negligible as aT~o. Hence from (5·11),

~(Vtl-Vl)(~) oT=oTjT,

dP ldT=T(V1)- vz) ,or

which is Clapeyron's equation, expressing the variation of vapourpressure with temperature in terms of other measurable quantities.

Let us now see what this method amounts to, by applying it to aninfinitesimal Carnot cycle of a simple fluid. In fig. 11, AB and OD are

p

VFig. 11. Infinitesimal Carnot cycle for simple fluid.

infinitesimal portions of isotherms at temperatures T and T - aT, andBO and AD are adiabatics. Let the volume change between A and Bbe aV; the vertical distance between the two isotherms is (8P/8T)y aT,

80 that (8P)OA = aT v oV oT. (5·13)

Now in the course of the reversible isothermal expansion AB the heatabsorbed is T(8S/8V)TaV. Therefore from (5·11) and (5·13),

(8P/8T)yaVaT aTT(8S/8V)TaV T'

i.e.

dU=TdS-PdV,

dH=TdS+ VdP,

dF= -SdT-PdV,

dG= -SdT+ VdP,

Useful idea8 55

which is just M.4. We should not therefore expect this graphicalmethod to yield any result which could not be derived, probably withless labour, by the direct application of a Maxwell relation. In fact,the Clapeyron equation may be derived immediately from M.4 if it isapplied to a mixture of a vapour and liquid in equilibrium. For so longas both phases are present (oP/oT)v is independent of V and equal tothe variation of vapour pressure with temperature. In consequence(OS/OV)T is also independent of V. Let us therefore make a finiteexpansion in which unit mass ofliquid is evaporated; then L\ V = V v - vz,and L\S=l/T. Substituting L\S/L\V=(OS/OV)T in M.4 we reachClapeyron's equation at once.

It is possible that the reader may feel some uneasiness in applyingMaxwell's relations, which were derived for a simple fluid, to a problemin which liquid and vapour coexist in an inhomogeneous mixture. Ifso,let him imagine the liquid and vapour to be enclosed in an opaquecylinder fitted with a piston, so that he may think himself to be dealingwith a peculiar fluid whose isotherms have the unusual form of fig. 10.In fact, Maxwell's equations apply, not simply to a fluid, but to anysystem which is determined by two parameters, r' and T say, and themixture of liquid and vapour belongs to this class.

The functions U, H, F and G

We have introduced in this chapter the four functions, internalenergy U, enthalpy H, free energy F and Gibbs function G, which arerelated to one another and to P, V, Sand T by the definitions

H:=U+PV,

F:=U-TS,

G= U-TS+PV=F+PV=H-TS.

The differential coefficients of these functions may be expressed in thefollowing forms:

and it was by use of these forms that we derived Maxwell's equations.Since the functions play important roles in the application of thermodynamics to specific problems we shall outline some of their propertieshere.

First it will be observed that a specification of each as a function ofthe appropriate parameters provides a complete specification of the


(5·14)

thermodynamic properties ofthe system. The appropriate parametersfor each function are as follows: U(8, V), H(8,P), F(T, V), G(T,P).For example, if U(8, V) is a known function, the temperature andpressure corresponding to any particular state (8, V) may be foundimmediately, since (oU/oS)y=T and (oU/oV)s= -Po Hence anyother of the thermodynamic functions may be constructed and inaddition the equation ofstate is found. It is necessary for this purposethat each function shall be known in terms of the appropriate parameters and not any two parameters. If, for instance, we know U(T, V)we cannot derive therefrom the equation of state. It is this circumstance which gives the free energy its particular importance, sincethe appropriate parameters are V and T. In many problems involvingthe calculation of the properties of a system by the use of statisticalmechanics it is most convenient to consider the system as defined byits volume and temperature, and therefore it is only necessary tocalculate the free energy in order to derive all the equilibrium properties, which are related to F by the equations

(oF/oT)y= -8,

(OF/OV)T= -P,

U=F-T(oF/oT)y= -T2(~~) .oTT y

Equation (5·14) is known as the Gibbs-Helmholtz equation. Analogous results may be similarly derived:

In any reversible change taking place at constant temperature,

6.F=6.U-T6.8

=6.U-Q

= W, from the first law.

From this it follows that the work done by a system in an isothermalreversible change is equal to its decrease in free energy. This is theorigin of the name ascribed to F. It will be seen from the arguments tobe presented in Chapter 7 that the decrease in F of a system representsthe m,aximum work which can be performed as a consequenceof a givenisothermal change, and that this maximum is only achieved bycarrying out the change reversibly.

57

CHAPTER 6

APPLICATIONS OF THERMODYNAMICSTO SIMPLE SYSTEMS

Introduction

We shall see in this chapter how thermodynamics may be used tocorrelate the thermal properties of simple systems, such as fluids, inthe equilibrium state. Taking fluids (or solids subjected only to hydrostatic pressure) as typical of two-parameter systems we may tabulatethe thermal propertiest as follows:

(I) The equation of state. This includes, for liquids and solids, suchproperties as isothermal compressibility and expansion coefficient.

(2) The adiabatic equation, the relation between any two of P, Vand T in an adiabatic change.

(3) The specific heats, particularly the two principal specific heatsOp, the specific heat at constant pressure and Oy, the specific heat atconstant volume. In general these are not constant but are functionsof the parameters of state.

(4) The Joule coefficient, the relation between temperature andvolume when a fluid (especially a gas) is expanded from one equilibriumstate to another in an isolated enclosure without doing work.

(5) The Joule-Kelvin coefficient, the relation between temperatureand pressure when a fluid (especially a gas) is expanded from oneequilibrium state to another through a throttle valve.

If all these properties had to be measured individually for all statesof the fluid the amount of experimental effort required would beenormous, but fortunately they may be so closely related by thermodynamic reasoning that the number of independent properties whichmust be measured in order to give complete information is not tooformidable. The properties which may be determined most easily varyaccording to the system studied, but usually they are the equation ofstate and one of the principal specific heats, for solids and liquids Opand for gases either Op or Oy. Specific heats are most accuratelydetermined over a wide range of temperature by means ofan adiabatic(Nernst) calorimeter, and this will provide values ofOp at zero pressurefor solids fairly readily and values of Oy at moderate pressures for

t Non.equilibrium thermal properties, such as thermal condu('tivity, aro ofcourso beyond the range of classical thermodynamics, and the Joule and J ouleKelvin effects t.ahulated bolow ((4) and (5» are amenable to t,hArn10dynarnicmethods only because the initial and final states are equilibrium states.


f(P, V,T)=O.

gases with somewhat more trouble; the specific heat which is measuredfor a liquid will depend on whether the liquid is given room to expandor not, but there is not usually much difficulty in deciding what isactually measured. Flow methods for determining Cp for a gas havebeen developed to give results of high accuracy. The other properties,except perhaps the Joule-Kelvin coefficient, present considerabledifficulties in their accurate determination. There are occasions, aswill be mentioned in due course, when measurements of the JouleKelvin coefficient may usefully supplement studies of the equation ofstate.

It will be seen, then, that the amount of experimental informationwhich may be obtained by standard means is rather limited, butfortunately it is sufficient, as we may easily show. All the propertieslisted above maybeformulated in termsofthermodynamicparameters:

(I) The equation of state expresses the relation between P, V andT in equilibrium,

(6·1)

(6·2)and

therefore

(2) An adiabatic change is one which takes place at constantentropy, so that complete knowledge of the differential coefficient(oP/o V)s for all states is sufficient to determine the relation betweenP and V in an adiabatic change; similarly, knowledge of (oP/oT)sand (oV/oT)s determines the relations between P and T, and V and T,respectively.

(3) The specific heat is defined as dQ/dT in a reversible changeunder specified conditions. According to the first law,

dQ=dU+PdV;

°v=(oU/oT)y

Cp= (OU/oT)p+ P(oV/8T)p.

and

Alternatively, since in a reversible change dQ= TdS,

Ov=T(oS/oT)y

Op=T(oS/oT)pe

(6·3)

(6·4)

Yet another convenient expression for Op may be obtained in termsof the enthalpy H. For dB = dU+PdV + V dP in general, so thatdB = dQ+ V dP for a reversible change, and

Op=(oH/oT)p. (6·5)

(4) In the expansion ofan isolated fluid without the performance ofwork, the internal energy stays constant, since Q= W =0. The Joulecoefficient is therefore, as we shall discuss in more detail later,(oT/oV)u·

Applications to simple systems 59

(5) We shall see that in a throttling process the enthalpy is conserved, and the Joule-Kelvin coefficient is given by (OT/OP)H.

If, therefore, ,ve know all the thermodynamic functions for allstates ofthe fluid we can derive complete information. But the thermodynamic functions themselves are not independent; if we know theequation of state and the value of any of U, S, H (or For G for thatmatter) for all states of the fluid, the values of the others may bederived. If, for example, we have a complete specification of S(T, V)we may use the fundamental equation

dU=TdS-PdV

to find dU for any infinitesimal change (dT, dV); for the equation ofstate gives the value ofP at any (T, V) and the rest ofthe quantities onthe right-hand side are known or defined by the change considered.The function U may therefore in principle be constructed by step-bystep integration. A similar result may easily be shown to hold for anyother thermodynamic function.

Lastly, we may note that it is not necessary to know by experimentthe value, say, of S(T, V) for all values of T and V, since a littleexperimental information may be largely extended by use of thermodynamics and the equation of state. Since S is here treated as a function of T and V, we have that

dS= (~~LdT+ (~~LdV

Ov (OP)="T dT+ aT v dV, byuseofMA.

JTb YoOv JTb YI(OP)

Hence S(TI ,VI)-S(To, Vo)= 'F dT+ oT dV. (6·6)TOt Yo Tit Vo v

The integrations are supposed here to be performed in two stages, thefirst integral being evaluated at constant volume from (To, yo) to(TI , Yo), and then the second evaluated at constant temperature from(TI , Vo) to (TI , Vi). Since knowledge of the first integrand demandsknowledge only of Cvasa function of temperature at a certain standard volume Vo, and the second integrand depends on the equation ofstate, it is clear that this is all the information needed to determineS completely, and hence to find all the listed properties for all states ofthe fluid. We have taken a special case to illustrate this point, but thereader may verify that similar results hold for the other thermodynamic variables, and that a complete specification is given by thefollowing data:

(1) The equation of state.(2) One of the specific heats along any line on the (P, T) or (V, T)

or


diagram, provided that the line runs through all temperatures forwhich the information is needed.

Having thus demonstrated how little experimental information isneeded, we now turn our attention to the explicit evaluation of thelisted properties in terms ofthe measurable quantit.ies, beginning withthe specific heats.

Relations between the specific heats

Since we have seen that there is no need to measure the specific heatsin all states ofthe fluid, but only to have knowledge ofone specific heatalong one line on the (P, T) or (V, T) diagram, we expect to be able todeduce Op from 0v or vice versa, and also to determine either ata given temperature for all pressures or volumes, once a single value isfound experimentally. For this purpose it is convenient to expressOp - 0v, (oCP/OP)T and (oCv/o V)T in terms of quantities which can becalculated from the equation of state.

Let us first consider (oOP/OP)T and (oOV/OV)T. Since from (6·4)Op=T(oS/oT)p, we have that

ea;L=T(a~L (~~L =T(8~L (~~L = -T(~~t from M.3.(6·7)

Similarly, (~;L=T(~:::L by use ofMA. (6'8)

The quantities on the right-hand sides of (6·7) and (6·S) may beevaluated from the equation of state. The reader should verify thatthey both vanish for a perfect gas (as is to be expected since U and Hare functions of temperature only) and that while (oCV/OV)T=O fora van der Waals gas, (oC}P/OP)T=t=O.

In order to calculate Op - 0v it is convenient to regard S as a function of T and V, so that

dS= (~~)v dT+ (~~Ld V,

(~JiL = (~~,) v + (~~L (~~L·Hence, from (6'3) and (6'4),

Cp-Cv=T(:~)l' (~) p

=T(~~L (~;L from MA, (6'9)


This expression for Cp - 0v is convenient if the equation of state iskno,vn explicitly, but for solids and liquids it is usually more convenient to express the result in terms of the expansion coefficient fJ

( 1 (0 II") ). ( 1 (0 ll) )== V aT p and the Isothermal compressibility kT == - V 8P 7' I

which is the reciprocal of the isothermal bulk modulus. Since, by (5'S),

(:~) y = - (~~L (~~)p'

(6'9) can be put in the form

Op-Cv = _T(OV)2 (OP)aT p oV T

= VTfJ2/kT • (6'10)

(6,11 )

This form of the result sho","s that Op can never be less than 0v, sincekT is always positive; Op= Ov when fJ=O, as, for example, in \\·ater at40 C. or liquid helium at about 10 K.

Equations (6'7), (6'S) and (6'9) or (6'10) provide all the thermodynamic information needed to extend the bare minimum of dataconcerning the specific heat into complete knowledge for all states ofthe fluid. In addition, the ratio of the principal specific heats'Y (== OplOv) is of course determined by either (6'9) or (6'10),

T (8P) (OV) VTfJ2y-l = Gy aT y aT p = Gyk

T•

In general 'Y varies with the state of the fluid, even for a perfect gas if,as is quite possible, Ov depends on temperature (as in hydrogen). Thephysical importance of'Y is twofold: in the first place its value for a gasprovides evidence concerning the number of degrees of freedom of themolecules constituting the gas, a matter which has no place in ourpresent discussion, and in the second place the adiabatic equation ofa gas is conveniently expressed in terms of 'Y.

The adiabatic equation

If we ,,,ish to find the relationship between P and V in an adiabaticchange, we naturally seek to find an expression for (aPia l!}s in ternlS ofmeasured properties of the fluid (or solid if only hydrostatic stressesare applied), in which category we are now able to include y. This isvery simply done, for

(oP/oV)s (oS/oV)p(oPjoS)v (8S18T)p_(aPia V)r = (aT/a Jl)p-(§P/oT)~ = (oSloTh~ - y,

(6']2)

62 Oln,ssical thermodynamics

by use of the mathematical identities (5'S) and (5'9). Thus the adiabatic bulk modulus is 'Y times the isothermal bulk modulus for allfluids. Equation (6'12) expresses the general solution of the problemof determining the adiabatic relationship between P and V, for(OP/OV)T may be calculated from the equation of state and theadiabatic curve is found by integrating the equation

(6-13)

(6-15)

if necessary step-by-step. For example, if we consider a perfect gas,PV =RT and (OP/OV)T= -PlY, so that (6'13) takes the form

(:~t= -yP/V,

with solution PVY = constant, if'Y is constant.The fact that we have derived (6'12) by using only mathematical

identities, and without employing Maxwell's relations, suggests thatthe result does not depend on the second law, and this is in fact so. Weshall now show how the first law alone leads to the same result. Forthis purpose consider an infinitesimal portion ofthe indicator diagram,as in fig. 12, so that the two neighbouring isotherms and the adiabaticshown may be represented by straight lines; moreover, the internalenergy may be considered to vary in a linear manner over the rangeshown; if the point A is (PO) Yo),

U(Po+oP, Vo+oV)=U(Po,Vo)+(~~v oP+ (~~toV,

higher order terms in the expansion being neglected. We shall nowcalculate the difference in U between ",,4 and C, first along the pathABC and then round the path ",,4DO. Equating these two gives thedesired re~ult. For the path ABC we have that

UB-U,A=CyoT,

OAand therefore UC-U,A=BAOvoT=gOvoT, (6'14)

where g is the ratio of the gradients of CD and BD, i.e.

(OP/0 V)adtabattc/(OP/0 Vhsothermal.

For the path ADO we have that

Un-U,A=OpoT-PooV, where oV=AD,

and Uc - Un =PooV, since DC is an adiabatic. Therefore

UC-U,A=OpoT.

A pplications to simple systems 63

Hence from (6-14) and (6'15),

gOy=Op or g=/"

which is equivalent to (6'12).By retaining/, in the expression for (oPjoV)s we avoid recourse to

the second law; but if we do not know /" but only Cp or Cv, we must

c

p

D

VFig. 12. Indicator diagram for simple fluid.

invoke the second law through (6'11). So too for the other adiabaticcoefficients; thus

Similarly,

(OT) = _ (OT) (OS) from (5-8)oP 8 oS P oP T

T(OV)=Op aT p from M.3. (6'16)

(6-17)

Ma~netic analo~ues of the fore~oin~ results

Let us now apply to magnetizable bodies the principles which wehave used in deriving the properties of fluids, taking as the fundamental equation

dU'=TdS-PdV +:K.d';/{_ (6·18)

To simplify the notationwe shall replace the scalar product :K _d.;/{ by£ <LA, in which vii is to be interpreted as the component of .;/{ in thedirection of:K_ We shall also, as (6'18) implies, only consider cases in

84 Olassical therrnodynamica

therefore

which the external field is uniform. Under many circumstances it ispermissible to neglect the term PdV in comparison with £<LL, andwe shall first examine the relative magnitudes of these quantitiesunder different conditions_ It is convenient to note that a largenumber ofanalogues ofMaxwell's relations may be derived from (6-18),of which the particular one we require for the present purpose isobtained by considering the differential properties of the Gibbsfunction 0' (= U' -TS+PV):

dO'= -SdT+ VdP+£<LL;

(00') (00')- =V and ovll =£,oP T,~ T,P

whence (6-19)

The remaining twenty-three analogues may be derived by similarmethods. Now let us Imagine a differential change taking place atconstant temperature and pressure, in which a variation of£ leads toa change in both V and vii; clearly the ratio, fT,p, of the two termsin (6-18), PdV and £<LL, is given by the expression

P(OV)fT,p= Je oJ(

T,P

P (aYe) from (6-19)_ (6-20)= Je oP T,~

For the purpose of estimating the magnitude of fT,p let us assume thatthe body is linearly magnetizable, Le. J( =a(T, P) Ye, where a is thesusceptibility of the body itself. Then from (6'20),

rT

•p = _~ (oa) =PkT! (oa) ,

a oP T a oV 7'

where kT is the isothermal compressibility. For many magnetic substances (particularly paramagnetic substances), the property of magnetizability owes its origin to the internal structure ofone of the atomscomposing the substance, and under these conditions a is to allintents and purposes independent of the pressure or volume, sinceeach atom acts nearly independently. For a paramagnetic gas, such as

oxygen, PkT =1, but~ (:;) T ~ 1, so that rT.p~ 1;forasolid~ (:;) T

may not be so small as for a gas, but it is unlikely to exceed the orderof magnitude of unity, while PkT~ 10-5 to 10-8 for most solids atatmospheric pressure, so that fT,p is only likely to be comparable with


unity at extremely high pressures or with exceptional substances for

which ~ (~~) T ~ 1. On the other hand", if conditions are such that

T and~, or P and .11: are kept constant it is not so clear that theterm PdV is negligible; indeed, if in the former case a is independentof pressure, or in the latter ca.se a is independent of temperature, theneYe <LA' = 0 and the only work term is that due t.o pressure. It wouldtherefore be necessary in applying thermodynamics under these conditions to examine the relative magnitudes of the terms carefullybefore discarding any. For present purposes, however, ","e shall beconcerned only with phenomena in solids, which may be examinedexperimentally at very low pressures, and no significant error 'Yillresult from assuming for the fundamental equation

dU'=TdS+eYe<LA'. (6,21 )

All relations between the principal specific heats which werederived for fluids may now be taken over without further thought byreplacing P and V by:K and -vii. Thus

(6'22)

in which (x, the susceptibility vII/:K of the body: need not be independent of£;

(6'23)

(6'24)

Remembering that by definition vii = a(£, T) £, and defining thedifferential isothermal susceptibility a~ as (Ovll/O.Yt')T,t we can cast

t If the magnetization of the body is proportional to the field, as is comnlonlythe case except at low t.emperatures and high field strengt.hs (we excludo forromagnetics from this discussion for tho reasons explained in Chapter 3), thf'n thegenerally defined susceptibility IX ( == .A/~), which may be a function both of Tand Jf', becomes simply a function of T and identical with the isothermaldifferential susceptibility a~.

It should be remembered in all this section that IX is defined for a particularbody, and is in general dependent upon the shape 01 the body. If wo considf}ra long rod set parallel to~, the field .Yt'i \vithin the body is simply~, and IX isthe same 80S VX, where X is the volume susceptibility,J/~,of tho material. Forany other shape, ho\vo\"er, ~i and jIf? are not the same. The only shapes whichmay bo handled easily are ellipsoids, for which Jt7i is unifonn. If"one of theaxes of the ellipsoids coin~ides in direction with~ then ~ is also parallel to

Vx


(6·24) into the convenient form

T:Ye2(Oa)201- 0.,((= lX~ aT · (6·25)

From this expression we see that when dF=0, 01 =OJ(, as would beexpected, and that for a paramagnetic material having a~ positive0l~OJ(, while for a diamagnetic material having a~ negative°1~0J(.

By analogy with a; we may also define a differential adiabaticsusceptibility a~ as (oJ(/oeJ't')s, the relationship between a~ and a~

being analogous to that between the compressibilities kT and ks of

a fluid, ' r' · hi h r 0 /0 (6·26)(XT= (Xs, m w c = 'I 'JI.

The adiabatic susceptibility of a magnetic material is what is usuallymeasured by any method which involves alternating fields (as, forinstance, measuring in an a.c. bridge the inductance ofa coil containingthe sample as core), since there is usually no time for the sample toexchange heat with its surroundings. From (6·25) it will be seen thatr = 1 when~= 0, and departs from unity as £2, so that when smallmeasuring fields are used no error results from the assumption that it isa~ which is measured. But if a large steady field be applied while thea.c. measurements are made there may be a significant differencebetween a~ and a~. The implication of such a difference is that whena body is magnetized adiabatically it may change its temperature, andthis is seen to be so by writing down the other adiabatic equations, theanalogues of (6·16) and (6·17),

(OT) ~T(O(x) (6.27)oJt's=-0IoT 1 '

(oT ) £T (oa) (6.28)oJ( s = - aOJl oT JI .

.JIe, and may conveniently be connected with Jf' by introducing the idea ofa demagnetizing eoeffteient, n, such that

~= Jf'-41TnJ.

The values of n range from zero (long rod parallel to .Jf') to unity (flat slabnonnal to Jf'), with such typical intennediate values 80S t for a sphere and 1 fora long circular cylinder nonna! to Jf'. The relation between a and X is nowreadily derived:

a 1+41Tnx .

If, 88 in many practical applications, X< 10-4., the demagnetizing correction isvery small and a may be taken 80S Vx without serious error, whatever the shapeof the body. But in some C80Ses of interest, particularly paramagnetic salts atlow temperatures for which X may approach unity, and superconductors forwhich 41TX is - 1, the shape of the body may play an important role in determining its behaviour.


From (6·27) it is seen that if a is temperature-dependent, there is indeeda change in temperature when a body is magnetized adiabatically.This is the magneto-caloric effect; it is of extremely small magnitude atroom temperature, but may under favourable circumstances becomelarge at low temperatures, where it is used for producing temperatureswell below 10 K. by adiabatic demagnetization. The magnitude of theeffect may be estimated by considering an idealized paramagneticbody which obeys Curie's law at all temperatures,

x=a/T, where a is a constant. (6·29)

(6·30)

(OS/O£)T= V£dX/dT= -aV£/T2,

S(~ T) =8(0, T) - fa V£2/T2,so that

For such a body the entropy S( £; T) in a field £ may be seen, byapplication of the analogue of M. 3, to vary with £ according to theequation

where 8(0, T) is the entropy in zero field. If a process of adiabaticdemagnetization is carried out, starting at temperature T 1 in a field £?,and ending at temperature T2 in zero field, then according to (6·30) T1

and T 2 are related by the equation

8(0, T1 ) -laY£2/T~=S(0,T2). (6·31 )

Now at room temperature, on account of the presence of T~ in itsdenominator, the second term in (6·31) is small for any achievablefield £; moreover, the specific heat in zero field is large, so that8(0, T) is a steep function of T. This means that T1 - T2is small, andapproximately, from (6·31),

aV£2T1 -T2= 2TC

o' (6·32)

(6·33)0.Jt'=00+aV~2/T2,

where 0 0 is the value of 0.Jt' in zero field. Equation (6·32) can of coursebe deduced immediately from (6·27) on the assumption that CJF isindependent of~ For a typical paramagnetic salt at room temperature, with £ equal to 104 gauss, T 1 - T 2 is less than 10-4 degrees. Butas the temperature is lowered the denominator in (6·32) falls morerapidly than T, since, except over a certain range at a very lowtemperature, Co normally decreases steadily with lowering of thetemperature, and the cooling effect becomes more pronounced. Attemperatures of a few degrees absolute the approximations made inderiving (6·32) are invalid and a different approximation will serve toexhibit the sort of behaviour to be expected. As may be seen from(6·30)

and at low temperatures quite a moderate value of £ will cause the


second term to dominate the first. If then we neglect 0 0 , and combine(6·33) with (6·27), putting a= VX, we arrive at the equation

(aT) TaYe s = ;YtJ'

or Tj£=constant.

Thus the temperature falls in proportion to the magnetic field duringadiabatic demagnetization, and a very considerable reduction intemperature is possible. It should be realized that this argument isbased on an idealized model which neglects both saturation effects anddepartures from Curie's law (6·29); in addition, it is applicable only atsuch field strengths that 0 0 in (6·33) may be neglected. All these effects,as might be expected, serve to prevent the absolute zero from beingattained by simply reducing the field to zero. Nevertheless, by choosinga paramagnetic salt which obeys Curie's law to as low as possiblea temperature, such as chrome alum or copper potassium sulphate, andby demagnetizing from a field of perhaps 20,000 gauss and an initialtemperature of 10 K., temperatures as low as one or two thousandthsof a degree may be reached, far below the limit of any other coolingmethod. To enter further into the rich field of thermodynamics andlow-temperature research afforded by adiabatic demagnetizationwould take us too far beyond the scope of this book, and the reader isreferred elsewhere for more detailed information.t We shall, however,return to the topic at the end of the present chapter in discussingmethods of establishing the absolute scale of temperature.

The Joule and Joule-Kelvin effects

We now revert to the problem of expressing the thermal propertiesof fluids in terms of measurable quantities. So far we have consideredonly reversible changes between equilibrium states; the Joule andJoule-Kelvin effects are by contrast concerned with changes betweenequilibrium states effected by irreversible processes. In the Jouleeffect a fluid (particularly a gas) is allowed to expand into a vacuum, sothat it increases its volume irreversibly without performing any work;in addition, the system is thermally isolated so that no heat enters orleaves. Under these conditions it clearly changes irreversibly from oneequilibrium state to another having the same value of U. Nowalthough the change certainly does not proceed by way ofintermediateequilibrium states, we may assume for the purpose of calculating theeffect that it does so. For the final state is specified precisely by thevalues of V and U, and it does not matter how the final state is

t C. G. B. Garrett, Magnetic Oooling (Harvard University Press, 1954).

(6·34)


attained since we are concerned solely with functions of state. Thusthe temperature change accompanying an infinitesimal Joule expansion (the Joule coefficient of the fluid) may be ascertained by expressing(oT /0 V)u in terms of known quantities, in the following way:

(8T) (8U/8V)ToV U = - (ou/fir)v by (5·S)

= _~{T(8S) _p) from (5.2)Oy oV T

= _~{T(8P) _p} from M.4,Ov 8T v

and this expresses the Joule coefficient in terms of Ov and quantitiesdeterminable from the equation of state. It is easily verified that theJoule coefficient vanishes for a perfect gas, as may be seen directlyfrom the definition of a perfect gas as one for which U is a function oftemperature only. The reader should evaluate the Joule coefficient fora van der Waals gas and attempt to understand from an atomic standpoint why it takes the form it does. For a finite volume change theresultant temperature change may of course be calculated in principleby integration of (6·34). There may, however, be a considerable gulfbetween principle and practice, for it is only with especially simpleequations ofstate and variations ofOv with V and T that (6·34) can becast into the form of a differential equation in V and T which may besolved with any ease. In general, it is probably less troublesome toreturn to first principles, and use available data to construct lines ofconstant U on a V-T diagram by processes analogous to thoseexplained on p. 59 in connexion with the calculation of S. From sucha diagram the magnitude of the cooling in a given Joule expansion isof course immediately plain.

The irreversibility of the Joule effect gives rise to a change in theentropy of the fluid even though the system is adiabatically enclosed.The change is readily calculated, since it follows immediately from(5·2) that (oS/oV)u=P/T, as already remarked on p. 42. Therefore inthe expansion the entropy always increases. ~"'or a perfect gasP/T=RjV and the entropy change in a Joule expansion froln VI toV2 is R log (V2/V1 ).

The practical itnportance of the Joule effect is very small, but thesame is not true of the related Joule-Kelvin effcct, the temperaturechange which accompanies the expansion of a gas from high to lowpressure through a throttle (in the early experinlents on this phenomenon a porous plug ,vas used instead of a rrlcchallical valve, and theexperiment is sometilnes referred to as the porous-plug experiment).


The experimental arrangement is shown schematically in fig. 13; gasenters at a pressure Po and temperature To and leaves at a lowerpressure ~ and, in general, a different temperature 1l. On each sideofthe throttle the gas is moving but is otherwise in thermal equilibrium.In the idealized form of the experiment it is assumed that there is noheat flow through the walls of the tubes.

This experimental arrangement is one example of a large class of8tationary flow phenomena, which are characterized by a non-equilibrium state of a fluid system which maintains the same configurationfrom time to time. In its simplest form a system in stationary flow mayconsist simply of a tube of varying cross-section, as in fig. 13, throughwhich a steady flow is maintained from external sources, and in whicha constant pressure and temperature distribut.ion is set up. But thetube need not be a real tube-in the streamline flow of a fluid it is

---"__P_O_'_T._

O__....~ - PI' T

1

Fig. 13. Joule-Kelvin expansion through a throttle.

permissible to isolate for consideration any part of the fluid whosemotion is bounded by streamlines. It is now easy to show from thefirst law that if there is no transport ofheat by conduction either alongor across the lines of flow the enthalpy H of a given portion of the fluidis conserved as it passes through the system. Consider the tube of flowin fig. 14, and let theelementary volumes described ateach end containthe same mass, m, of fluid. Then as a quantity of fluid having mass menters the shaded region at one end, an equal mass leaves at the other,and since the flow is stationary the internal energy of the fluidcontained in the shaded region is unaltered. The factors tending tochange this internal energy are:

the energy muo transported in by the fluid entering (here and elsewhere we use small letters to denote the values of extensive thermodynamic quantities per unit mass);

the energy mUl transported out by the fluid leaving;the work mPovodone by the fluid outside the shaded region on the

fluid inside;the work mJi VI done by the fluid inside on the fluid outside.If there is no heat contribution to the internal energy of the shaded

region, we may therefore write

UO-UI+POVO-PIVI=O, }(6·35)

or ho=hl , where h is the enthalpy per unit mass.

ApplicationB to simple systems 71

Since the length of the shaded region was chosen arbitrarily, it followsthat h is constant along a tube of flow. The result may obviously begeneralized to cover more complicated stationary flows in which more

Fig. 14. Tube of flow, to illustrate Bernoulli's theorem.

Fig. 15. Schematic diagram of stationary flow.

than a single stream is involved, as represented schematically in fig. 15.If the mass entering the system in unit time is mi for the ith channel,and the enthalpy per unit mass is hi' then Lmi=O and, provided

ithere is no heat exchange between the system and its surroundings,

72 OlasBical tkermodynamiCB

(6·36)

h=u(O)+gx+ ~W2+P/p,

and no heat flow along the channels at the points where the hi aremeasured,

It should be noted that there need be no restriction on heat exchangewithin the system.

The constancy of h along a streamline is a result more familiarlyknown as Bernoulli's theorem when applied to an incompressiblefluid. The internal energy density u may be considered as composed ofa number of terms:

u(O), the internal energy of unit mass at rest at a standard height,under the same conditions of temperature and pressure that obtain inthe stream;

gx, the potential energy, the work required to raise the mass toa height x above the standard;

!w2, the kinetic energy, the work required to accelerate the massfrom rest to a velocity w.

Thus

where p is the dEmsity. In particular, for an incompressible fluid thetemperature and u(O) remain constant along the stream if there is noheat transport, and therefore gx+ !w2+P/p is a constant of the flow,which is the well-known form of Bernoulli's theorem.

Let us now return to the Joule-Kelvin effect. If there is no heat flowat the output and input the enthalpy is conservedt during the irreversible pressure drop at the throttle, and if the flow is slow enough thekinetic energy may be neglected in calculating the enthalpy. We mayuse the same argument as with the Joule effect to replace this irreversible process by a hypothetical reversible process in which h isconserved, and therefore calculate the temperature change

1;-To=f:'l(~~tdP. (6·37)

We now express the Joule-Kelvin coefficient (oT/oP)", in terms ofmeasurable quantities, as follows:

(OT) (Oh/OP)ToP h = - (ohloT)p

since

=_~{T(OS) +v),Cp oP T

dh=Tds+vdP and (ohjoT)=cp.

t Since this is an irreversible process the entropy need not be conserved. Thereader should show that there is always an increase of entropy at the throttle.

Applications to 8imple 8y8temlJ

Therefore, by use of M. 3,

73

(6·38)

which is the desired form, involving cp and quantities calculable fromthe equation of state. The use of (6·38) in (6·37) to calculate the temperature change resulting from a finite pressure drop involves thesame difficulties in general as were remarked on in connexion with theJoule effect (p. 69). Here also the practical treatment of the problemis to use experimental data to construct lines of constant enthalpy ona P-T diagram, or, as is obviously equivalent, isotherms on an H-Pdiagram, of which fig. 18 is an example.

It follows from (6·38) that the Joule-Kelvin effect vanishes fora perfect gas; this may be seen also from the fact that the definition ofa perfect gas implies that h is a function of temperature only. It mayalso vanish for an imperfect gas under certain conditions, as may beseen by applying (6·38) to the equation of state expressed as a power

series, Pv=rT+BP+Op2+ ... , (6.39)

in which r is the gas constant per unit mass and the virial coefficientstB, 0, etc., are functions oftemperature only. Combining this equationwith (6·38), we find that

(6·40)

For all gases the behaviour ofB is qualitatively similar, as indicated infig. 16. At not too high pressures the gas may be adequately represented by terminating the virial series after B. One can then see fromfig. 16 that there is a temperature, the Boyle temperature TB , at whichB=O and the gas approximates closely to a perfect gas, and thatat a higher temperature, the Inversion ;emperature TIt TdB/d'l'-Bvanishes and with it the Joule-Kelvin coefficient. Above the inver8ion temperature dB/dT is less than BIT and the gas is warmed by theexpansion; below PI it is cooled. In showing inversion the JouleKelvin effect contrasts with the Joule effect, for real gases (in consequence of the fact that their intermolecular forces are always attractive at distances larger than the molecular diameter) always coolwhen expanded into a vacuum.

t Strictly this term should be reserved for the coefficients of the power seriesPv =A (1 + BIv + CIv"+...) introduced by Kamerlingh Onnes as an empiricalequation of state; but we shall use it for similar equations in the absence ofany altemative designation.

74 Clas8ical thermodynamics

In order to examine the inversion phenomenon at higher pressures,further virial coefficients must be considered, or the equation of statemust be known in a closed form. It is clear from (6·38) that the vanishing of the Joule-Kelvin coefficient occurs whenever (ov/oT)p=v/T,and the substitution of this condition in the equation of state yieldsthe inversion curve. We shall calculate the form of the inversion curvefor a gas obeying Dieterici's equation,

P(v-b)=rTe-a/(rTv), (6-41)

since the result is particularly simple to derive for this equation ofstate. To facilitate the interpretation of the result it is convenient to

B

01'-----4-----------

Fig_ 16. Second virial coefficient, showing Boyle temperature (TB )

and inversion temperature (TI)-

use the equation in its reduced form (IT=P/Pc, 0=T/Tc, <P=v/vc'

where ~, Vc and Tc are the critical pressure, volume and temperature, which may readily be shown to equal a/(4e2b2

), 2b and a/(4br)respectively), ( 2 )

IT(2<P-I)=0exp 2-e<p • (6-42)

We now differentiate with respect to 0, keeping IT constant, andreplace (o<P/o0)n everywhere by <P/0_ Then after a little rearrangement, we find that the inversion curve on the 0-<1> diagram takes thesimple form 4

<P=S_0. (6-43)

(6·44)

20


In order to find the form of the inversion curve on the 0·n diagram wesubstitute (6·43) in (6·42), to yield the result, plotted in fig. 17,

II=(8-0)exp (~-~).

Comparison of this theoretical curve with a typical experimental curve,as given in fig. 17, sho\\--s that Dieterici's equation predicts the shapefairly closely, but is not so satisfactory as regards the scale; it is, however, quite as good as could be expected in view of the approximations

10Reduced pressure (I I)

Fig. 17. Inversion curves: (a) according to Dieterici's equation; (b) experimental curve for nitrogen. The ringed point marks the critical point.

which are made in the derivation of Dieterici's equation.t Within theinversion curve a gas is cooled on expan~ion,outsidc it is warmed. r.rhemagnitude of the cooling which is obtainable may be seen fronl fig. 18,which shows the isotherms of helium on an H-P diagraln. JoulcKelvin expansion corresponds on this diagram to a horizontal movement to the left, so that the dotted locus of minima is the inversioncurve. If helium at 100 K. and 20 atmospheres pressure is eXJ>anded toI atmosphere the process is that represented by the arrow, and the finaltemperature is about 6.5 0 K., or little over one-half of t.he initialtemperature.

t The reader may find it instructivo to dorive tho oquation corrosponding to(6·44) frolll vall der Waals's equHt.ion, (P+a/v2)(v-b)=r~Jl; he will find (asu,pplios to llCHrly all eu,lcuhttions based on these equat ions) tbat the derivation 1M

not ~o straightforward, and the result when obtained not u,ppreciably bettor.


There are anumber of applications ofJoule-Kelvin expansionwhichmake it an important process, and the basic reason for its importancelies in the ease with which it may be carried out. Since it is a continuousprocess it is possible to make accurate determinations of the pressqredrop and temperature change, and it is not difficult to arrange the gascircuits and throttle so that very nearly ideal conditions are realized as

30

20o K.

18

1620,-....-eIbO 14.;~~.e..2a~

10

oL..L ooAo-__

1 10 100

Pressure (atmospheres) (logarithmic scale)

Fig. 18. Isotherms of helium on an H-P diagram (W. H. Keesom,Helium, p. 252, Elsevier, 1942).

regards heat insulation and gentle flow (to minimize the kinetic energyterm in h).t It is therefore a convenient method for determining theisenthalps ofa gas. Expansionfrom various pressuresandtemperaturesdeterminesthe form of the isenthalps; measurement ofcp at a standardpressure as a function of temperature enables numerical values of h to

t See, for example, N. Eumorfopoulos and J. Rai, Phil. Mag. 2, 961 (1926).

Applications to simple Bystem,s 77

be ascribed to each isenthalp. The engineering importance ofsuch datawill be readily understood when it is realized how many useful devices,particularlyturbines and other heatengines,andgas liquefiers,t dependfor their working on the flow of gases. The work of Callendar on thethermodynamic properties ofsteamt may be consultedto illustrate thispoint. In this connexion it is worth remarking that although we set outin deriving (6·38) to express the Joule-Kelvin effect in terms ofwhatwehave called measurable properties, in fact this equation has proved farmore useful in the reverse sense. Since the effect only occurs asa result of departures from the perfect gas law it may be, and has been,used to derive better forms of the equation of state for gases whoseequation is not easily determined directly, such as steam. Similarly,(6·38) may be employed to estimate corrections to the perfect gas lawfor nearly perfect gases, and hence to correct gas thermometers to theabsolute scale of temperature. We shall discuss this application morefully later in the chapter.

Radiation

As another example ofthe application ofthermodynamics to simplesystems we shall consider the properties of the electromagneticradiation which is present inside a cavity bounded by opaque wallsmaintained at a uniform temperature (cavity radiation). It is not at allobvious that we are justified in applying thermodynamics to sucha problem, since the formulation of the laws of thermodynamics wasbased on experience of material bodies, and we need feel no a prioriconfidence that they are of sufficiently wide validity to embracemechanical processes in which radiation provides the motive force.The success of thermodynamics in these circumstances is perhaps thestrongest evidence we possess for regarding the laws as valid in allphysical situations to which they can be applied.

By means of electromagnetic radiation energy may be transportedfrom one place to another; it is on this fact that the application ofthermodynamics rests. Anything in the nature of a general thermodynamic theory of radiation is rendered difficult by virtue of the factthat the term equilibrium is not normally applicable. The light emittedby a glowing filament is no more in equilibrium than is a stream ofmolecules escaping from a gas-filled vessel into a vacuum. This is thereason for the peculiar importance of cavity radiation, since it represents an equilibrium situation. Moreover, as Kirchhoff showed, theproperties of cavity radiation are such as to make it, if anything, aneven simpler thermodynanlic system than a fluid. Kirchhoff used the

t For a detailed account of gas liquefaction consult M. and B. Ruhemann,Low Te"lnperatuTe Physics (Cambridge, 1937).

t H. L. Callondo.r, Propertie.s of Stearn (Arnold, 1920).


second law to prove that, provided the walls of the cavity are opaque,the quality of the radiation in equilibrium inside is independent of thenature ofthe walls and depends only on their temperature. The proofisvery simple. Imagine two cavities, A and B, maintained at the sametemperature T and connected by a narrow tube through which radiation may pass from one to the other. If the energy transported fromA to B exceeds that transported from B to A, it would be possible toraise the temperature of B to T +~T and for B still to receive moreenergy than it emitted; there would then be a steady flow of energyfrom a colder to a hotter body and the second law would be violated.It follows, then, that whatever the nature of the walls of A and B theenergy flow must be the same from both, and hence the energy densitythe same in both. By imagining colour filters or polarizers inserted inthe connecting tube it can be proved that the radiation is unpolarizedand that the spectral distribution ofcavity radiation is independent ofthe walls; it is also easily seen that the radiation must be isotropic,Le. at no point within the cavity is there any preferred direction eitherof polarization or of energy flux. t

Thus the quality of cavity radiation depends on the temperaturealone, and in particular all the extensive thermodynamic variables ofthe radiation in a cavity are proportional to the volume, i.e. U =uV,8 =8V, etc., where u, 8, etc., are functions of temperature only.

We may now treat the cavity containing the radiation exactly as ifit were a vessel containing a gas, and apply the fundamental equationto the system. For this purpose we must know what pressure is exertedon the walls of the cavity by the radiation, and this may be derived byelectromagnetic theory.t A useful alternative trick which avoidsdetailed calculations is to employ the photon theory of radiation inconjunction with the elementary kinetic theory of gases. If we think

t The assumption is inherent in this proof that a hole can be cut in thecavity wall which is large enough to allow radiation to escape freely, but not solarge as to disturb the radiation in the cavity appreciably. This assumption isno longer valid if the dimensions of the cavity become comparable with thewavelength of the radiation considered. For example at a temperature of 10 K.the energy of cavity radiation is largely concentrated in wavelengths aroundI cm., and in order to allow such energy to escape freely the hole cut must beabout I cm. or more in diameter. Such a hole would interfere considerablywith the radiation in a cavity of only a few cubic centimetres, so that the~onditions postulated in Kirchhoff's proof cannot be met. The quality of theradiation in such a cavity is indeed different from that in a very large cavity atthe same temperature, and the Stefan-Boltzmann law (6·48) does not apply,nor does Planck!s law in the fonn (6·50). A case of this sort needs specialtreatment, which, however, is not difficult by the statistical methods used toderive Planck's law.

t G. P. Hamwell, Principles of Electricity and Electromagnetism (McGrawHill, 1938), p. 537.


of the radiation as a gas whose particles all move with the velocity oflight, c, and have different energies according to the spectral distribution in the radiation, we may write for the' pressure

P=tpc2,

in which P is the density (mass per unit volume) of the photons. Butaccording to Einstein's mass-energy relation, pc2 is simply the densityof energy in the gas, so that

p=tu, (6·45)

dU=TdS-PdV,

which is the required answer, and the same as derived by electromagnetic arguments. It follows then that the pressure of cavityradiation is also a function of temperature only. The fundamentalequation,

may now be rewritten, by putting U=uV, S=sV, P=!u, in the form

Idu=Tds+ y (Ts-tu) d V.

(6·48)

(6·49)

where a is a constant,u=aT4,

and

But u and s are functions of temperature only, whence

du=Tds (6·46)

and Ts=tu. (6·47)

From these equations we see that du/ds=tu/s, so that uoc st; hence,from (6·47),

The result expressed in (6·48) is the Stefan-Boltzmann fourth-powerlaw for radiation, which is completely confirmed byexperiment.t

To obtain more information about the nature of cavity radiation,such as the spectral distribution of energy, is not possible by purethermodynamic means. An interesting extension of the thermodynamic result was achieved by Wien, who considered the modifications suffered by the radiation during an adiabatic expansion,tbut the complete solution of the problem requires the methods ofquantum statistics, and leads to Planck's celebrated formula for the

t A more concise proof of this result may be obtained by use of theGibbs-Helmholtz equation (5·14). Since dF= -SdT-PdV, we have that(oF/aV)T= -Po But the expansion of radiation at constant temperature onlycreates new radiation of the same quality, so that (aF/aV)T=j, the free energyper unit volume. Hencej= -P= -iu; substituting this in (5·14) we find that4u=Tdu/dT, or 'l.t oc T.a.

t G. P. Hamwell and J. J. Livingood, Experimental Atomic Phy8ic8(McGraw-Hill, 1933), p. 58.

80 OlasBical tkermodynamie8

energy density of radiation, u.,dv, between the frequencies v and

v+dv, 811hv8dv/e3

u.,dv= 1&.,/kT l' (6·50)e -

which we introduce here solely in order to facilitate the followingdiscussion. It will be seen by integrating over all frequencies that thisformula is in accordance with the Stefan-Boltzmann law.

So far we have considered only the radiation in a cavity; we nowextend the argument to the radiation emitted by a heated surface.First let us consider an opaque black surface, one which completelyabsorbs all radiation falling on it from any angle. We have Seen thataccording to Kirchhoff's law a cavity at uniform temperature lined

dS

Fig. 19. Calculation of radiation flux.

with such a surface would contain cavity radiation having the spectraldistribution given to Planck's formula (6·50). Such a situation canonly occur if the surface is emitting spontaneously the same radiationas falls on it,t which may readily be calculated in terms of u." theenergy density. Consider an element ofthe surface dB (fig. 19); in timedt the radiation propagated within a solid angle dw which will fall ondB from angles around 0 to the normal will be that contained ina cylinder ofside edt and volume edtdB cos O. The density of radiationin the solid angle dw, and frequency range dV,is u.,dvdw/(41T), since the

t Here and in what follows we take for granted the truth of Prevost's theoryof exchanges, according to which the processes of absorption and scattering onthe one hand, and spontaneous radiation on the other, are independent. Suchan assumption implies a mechanism not immediately deducible from theobservations, so that arguments based on it are, strictly speaking, outside therealm of classical thennodynamics.


radiation is isotropic, so that the radiation falling on the surface isCUI' cos OdvdwdtdS/(41T), or cu.,cos Odv/(41T) per unit solid angle, perunit area of surface, per unit time_ We conclude therefore that theradiation emitted by unit area of a black body per unit time, per unitsolid angle, is given by the fermula

cE., wdv=-4 u., cos fJdv_

t 1T (6-51)

(6-53)

By integrating over all angles between 0 and l1T, we arrive at anexpression for the total radiation of the surface in the frequencyrange dv, E d -L- d

I' v-~ul' v, (6-52)

and, by integrating over all frequencies,

E=icu

=!caT' from (6'48).

Ifthe surfa.ce is not black, but absorbs only a portion (X ofthe incidentradiation, then the emission will be reduced by a factor (X from theformulae (6-52) and (6-53) derived for a black body_ The emittedradiation will not necessarily have an angular distribution as cos 0,which necessarily holds for a black body; what is emitted must havesuch a distribution as will make up for the absorbed radiation andcombine with the unabsorbed radiation to produce an isotropic distribution of radiation leaving the surface_ For example, a rough surfacescatters such radiation as it does not absorb in all directions, moreor less with a co~ 0 angular variation, since the projected area of thesurface varies as cos 0; the radiation emitted from the surface willthen have approximately a cos 0 variation also_ But it is possible toproduce smooth reflecting surfaces which absorb strongly in certaindirections only; these will then emit strongly only in ~hose directions_Many different situations are possible with partially absorbing surfaces, but the general rule holds good, that they emit in such a way asto preserve the normal distribution of radiation when they form partof the wall of a cavity_ By application of this rule it is possiblein principle to determine exactly what will be emitted by any givensurface at any temperature if its absorbing and scattering propertiesare known_

The same method ofreasoning may be applied to surfaces which arenot perfectly opaque, and we shall illustrate it by considering a layerwhich does not reflect or refract radiation but which attenuates it sothat a fraction f is absorbed, the rest being transmitted- If weplace this layer, at temperature T, in front of a black surface at thesame temperature, the two together form a black surface at uniform


temperature, since all radiation incident is absorbed either by the layeror by the black surface. The composite body thus emits in accordancewith (6·51), as does the black surface. But of the radiation emitted bythe black surface only a fraction (I-f) penetrates the absorbinglayer; it follows that the radiation by the absorbing layer is f timesthat ofa black body. A transparent body is thus a poor radiator, a factwhich is well illustrated in laboratory practice when fused silica isworked in a gas flame. The softening temperature ofsilica is very muchhigher than can usually be reached with the gas flames used, but silicafortunately has the property of maintaining its transparency overa wide band ofoptical and infra-red wavelengths up to a high temperature. It is thus heated by molecular bombardment in the flame and hasno adequate radiative means of losing its heat. In consequence itstemperature rises much higher than that of glass in the same flame.One has only to contaminate the surface of the silica with a littlecarbon, which is a good radiator, to make it quite impossible for therequired working temperature to be reached.

The sun provides a more interesting illustration of this same principle. If, instead of its visible radiation, the radiation which it emitsat radio-frequencies corresponding to wavelengths about 5 m. isexamined, the sun appears to be several times larger than its visiblediameter. Moreover, the intensity of radiation suggests that theemitter is a black body with a temperature as high as one milliondegrees, enormously greater than the temperature of the visiblesurface, about 6000° C. If shorter radio waves are studied in the sameway there is observed a marked fall in apparent size and temperatureof the sun. These observations are explicable by the hypothesis, towhich other phenomena give strong support, that the visible 8phere ofthe sun is surrounded to a distance of many times its diameter bya highly rarefied atmosphere, consisting mainly of protons and electrons, at an extremely high temperature. Such an ionized gas has anabsorbing power for radiation which varies as the square of the wavelength. It may therefore be almost completely transparent to visiblelight, and opaque and 'black' to radio waves. Correspondingly it maybe a powerful emitter ofradio waves and yet so weak an optical emitteras only to be visible, as the corona, at a time of total solar eclipse.

Finally, before leaving the topic of radiation, it is of interest to notean application of the foregoing results at such low frequencies thattransmission lines and lumped circuit components become practicable.We may imagine two large cavities, maintained at a uniform temperature T, which are connected by means of radio aerials terminatinga transmission line, as shown schematically in fig. 20. The aerials,which may take any form or size, are supposed to be matched to thetransmission line, i.e. ifpower is sent along the line towards one of the

Applications to Bimple systems 83

aerials, it is all radiated and none is reflected. Then Kirchhoff's argument may be repeated to show that the power picked up by each aerial,and hence by any matched aerial, is the same. If the aerial is imperfectly matched, and radiates only a fraction! of the power sent alongthe line, reflecting the rest, then it is easily seen that its capacity topick up power from the radiation field in the cavity is also reduced by afactor!. By considering in detail a suitably simple aerial it is possibleto show that when matched to its line it will pick up in the range of

I I II I I

Fig. 20. Temperature enclosures connected by transmission line.

frequency dv power Wdv equal to c3uy dv/(81Tv2), which from (6·50) isseen to be given by the expression

hvdvWdv= ehYlkT-I

~ kTdv when hv~ kT. (6·54)

The approximation (6·54), which is normally valid for radio-frequencyphenomena, is equivalent to the use of the Rayleigh-Jeans radiationformula (uvdv = 81TkTv2dv/c3) rather than that of Planck.

Now let us replace one of the aerials in fig. 20 by a resistor whichmatches the transmission line and which is maintained at temperature T. Then the theorem of Kirchhoff shows that the resistor mustgenerate power at the rate given by (6·54). This result is the celebratedNyquist formula for the' Johnson noise' spontaneously generated bya resistor. It will be seen that this noise is the exact analogue of theradiation emitted by a black body. But what is more important, andthe main reason for our introducing the topic, we may see from thisexample how a typical 'fluctuation' phenomenon, Johnson noise,which may be thought of as originating in the Brownian motion of theelectrical carriers, may be related to an 'equilibrium' phenomenon,the radiation in equilibrium in a cavity. This example shows clearly,what we have stressed before, that fluctuations are not to be regarded

84 OlasBical thermodynamics

as spontaneous departures from the equilibrium configuration ofa system, but are manifestations of the dynamic character of thermalequilibrium itself, and quite inseparable from the equilibrium state.

Surface tension and surface ener~y

The elementary facts of surface tension are well known and we shallbase the following discussion on what is probably the clearest-cut fact

Fig. 21. Fonnation of liquid drop.

of all, that the pressure inside a spherical surface of a liquid is greaterthan that outside by an amount 2u/r, where u is the surface tensionand r the radius of the surface. We shall concern ourselves only withthe surface behaviour of liquids and say nothing of the properties ofsoap films.

Let us now consider a hypothetical experiment, illustrated in fig. 21,in which a spherical drop is formed from the bulk liquid at the end ofa very narrow pipette. We assume the liquid to be incompressible, sothat its free energy per unit volume is independent of pressure (since(OF/OP)T= -P(OV/OP)T=O). The reader may readily extend theargument to apply to a compressible liquid. If the atmospheric


pressure is ~, the pressure required on the plane surface of the reservoir is Po+ IlP, where tlP=2CT/r. Let us now increase the radius of thedrop reversibly by an amount dr, keeping the temperature constant.A volume 41Tr2dr of liquid must be forced through the pipette, and thework required is 41Tr2dr tlP, i.e. 81TrCTdr. This is equal (see p. 56) to theincrease of free energy of the system. But the free energy of thereservoir has decreased by an amount 41Tr2fodr, where fo is the freeenergy per unit volume of the bulk liquid. Therefore

Le.

eFo~op)T = 41TT2fo +81TTO' ,

417' 31 2FdroP=3r Jo+41Tr U

= Vfo+dCT, (6·55)

where V is the volume of the drop and d its surface area. The freeenergy of the drop is thus made up of two terms, one proportional tothe volume and one to the area, and the surface tension may be interpreted as the free energy per unit area of surface.

The molecular reason for the appearance of surface tension is madeclear by this result. The forces between the molecules of a liquidextend only over a short range, and at a distance within the surfacegreater than this range the molecular behaviour is unaffected by theproximity of the surface, and the free-energy density of an elementof volume is in consequence independent of the position of thatelement. Thus in a mass of liquid whose dimensions are much greaterthan the range of molecular forces the major contribution to thefree energy is a term proportional to the volume, the first term of(6·55). Close to the surface, however, the molecular behaviour isaffected by the surface, and the second term of (6·55) represents thecorrection needed for this effect. From this argument it will be seenthat the surface contribution to the free energy is not solely a propertyof the liquid state of matter, but may be expected to exhibit itself insolids and even gases, though only in liquids does it give rise to thestraightforward observable behaviour characteristic ofsurface tension.As will be shown in Chapter 7 (p. 107), the equilibrium state of a system maintained at constant volume and temperature is that of minimum free energy, and the tendency for a liquid drop to assumea spherical shape is a simple manifestation of this rule, since a spherehas the minimum surface area for a given volume. In a crystallinesolid, the surface contribution to the free energy may vary widely fordifferent orientations of the surface with respect to the crystallineaxes, and the equilibrium shape of a small crystal is not now spherical,


but of such a form as exposes most fully the crystal faces of smallestfree energy. This accounts in general terms for the well-marked, andoften complicated, shapes into which solids crystallize naturally fromthe vapour or from solution.t

So long as the dimensions of the body considered are much large.rthan the range ofmolecular forces, and provided there are no surfaceswhose radii of curvature are smaller than this range, the result (6·55)may be expected to be valid. But it cannot be applied when twosurfaces are so close together that there is direct molecular interactionbetween them; the distinction between volume and surface contributions becomes inapplicable. Similarly, at a sharp edge it may beexpected that additional corrections to the free energy will be needed,by addition of a term proportional to the length of the edge, and theremay be yet another term to be applied where three or more surfacesmeet in a point. It must be remembered, however, that these termswill depend on the angles between the surfaces. There are few phenomena in which they play any significant part and we shall considerthem no further.

The surface contribution to the free energy finds of course itsanalogues with the other thermodynamic functions, and there will be,in general, surface contributions to the internal energy, entropy J etc.It is easily seen that these quantities are related by expressionsanalogous to those relating the thermodynamic functions of completesystems, e.g. if A is the internal energy per unit area of surface and1/ the entropy per unit area,

and

1/= -du/dT

dCTA=u+T7J=u-TdT'

(6·56)

(6·57)

which is the analogue of the Gibbs-Helmholtz equation (5·14). It is ofinterest to note the consequences of Eotvos's rule when it is applied tothese results. Over a very wide temperature range in a liquid, accordingto this empirical rule, the surface tension of a liquid in contact with itsvapour is proportional to ~-T, where ~ is the critical temperature.This implies the rather remarkable fact that 7J and Aare independentof temperature. The rule cannot, however, be valid either near~ or atvery low temperatures. For at the critical point the liquid and vapourphases become indistinguishable, so that the phase boundary ceasesto exist and there can be no entropy contribution from it. We therefore

t This is a considerable over-simplification of the real state of affairs, fora crystal is normally not in its state of minimum free energy, and its shape maybe determined more by the detailed processes of its growth than by thermodynamic conditions. See F. C. Frank, Advanc. Phys. 1, 91 (1952).

Applications to simple systema 87

expect that the surface tension eventually falls to zero with a vanishinggradient, and this seems to be confirmed by experiment; at any rate,Ramsey and Shields have suggested that the critical temperature inEotvos's rule should be replaced by a temperature about 6° lower fora great many liquids to improve the agreement with experiment.A direct measurement of (]' within a few degrees of the critical temperature is very hard to carry out. If the liquid phase can persist totemperatures approaching the absolute zero (as with liquid helium)we may expect that (]' will tend to become independent oftemperature,since according to the third law the surface contribution to the entropymust vanish at 0° K. This behaviour is indeed shown by liquid helium.

The surface contributions to the thermodynamic functions differfrom the functions of the majority of complete systems in that it ispossible to ascribe absolute values to them without the normalambiguity of additive constants. This is because the surface area maybe varied at will without alteration of the volume. In principle therefore one may determine experimentally the work or heat required toincrease the area of surface by a known amount and thus determineunequivocally the free energy, entropy, etc., per unit area. It will benoted that the same property is possessed by cavity radiation.

Establishment of the absolute scale of temperature

The establishment of the absolute scale of temperature involves, inprinciple, the calibration of a convenient empirical thermometer interms of absolute temperature, and the systems which we haveanalysed in this chapter provide four different methods of doing thisover limited ranges of temperature. We shal,l not concern ourselveshere with any technical details, which are often of considerable complexity on account of the great refinement needed to ensure highaccuracy. The details may be found by reference to standard worksconcerned with the problems ofthermometry.t It is, however, worthmentioning that the aim of establishing the absolute scale is notsimply to possess one empirical thermometer and a calibration curvefor it so that the ownermay be able to measure temperature absolutely.I t is also desirable that replicas may be constructed inotherlaboratoriesand calibrated without a complete repetition of the arduous experiments involved in calibrating the first standard thermometer. Forthis reason much work has gone into finding suitable materials forsubstandard thermometers, whose properties shall be sufficientlyreproducible and sufficiently smooth functions of temperature thatcalibration at a few fixed points will be adequate to enable thecomplete calibration curve to be inferred. An example of such a

t See references on p. 48.

88 Classical thermodynamiCB

thermometer is the platinum resistance thermometer. No two speci.mens of platinum have resistances which vary in exactly the same waywith temperature, but it is known from comparison ofmany specimensthat allowance may be made for variation by calibration at a fewquite widely spaced fixed points, since the resistance over a wide rangeis very nearly a linear function of absolute temperature. Once suitablethermometric substances have been developed there remains the taskof discovering accurately reproducible fixed points and determiningtheir absolute temperatures as well as possible. The melting-point ofice (273·15° K.) is fixed by definition, and the boiling-point of water(373·15° K.) by accurate measurement; other melting-points (or, preferably, triple points) and boiling-points have been added to extendthe range of temperature both upwards (b.p. of sulphur, 717·8°K.;m.p. of gold, 1336°K.) and downwards (sublimation point of COs,194·7° K.; b.p. of oxygen, 90·2° K.; triple point of hydrogen 14.00 K.).

We shall discuss briefly the thermodynamic principles involved infour methods:

(I) correction of the gas scale by extrapolation to zero pressure;(2) correction of the gas scale by means of the Joule-Kelvin effect;(3) radiation pyrometry;(4) calibration of the magnetic scale below 10 K.Of these we shall dismiss the third in a few words. The Stefan

Boltzmann fourth-power law enables the temperature within a cavityto be determined from the total radiation emitted from a small holecut in the cavity wall. The thermodynamic principles have alreadybeen discussed sufficiently fully; the technical details, and formidableexperimental difficulties which must be overcome, will not be enteredinto. No great accuracy (better than about 1°) has been attained inradiation pyrometry, but it is the only method of temperaturemeasurement which can be carried into the region of very hightemperatures (over 2000° K.) and still pretend to be related to theabsolute scale.

The gas thermometer, on the other hand, is capable of very highaccuracy (about 0.01 0 or better) in the measurement of empiricaltemperature, so that the methods of making the small corrections toreduce its empirical scale to the absolute scale are ofgreat importancein thermometry. As we saw in Chapter 5, a perfect gas, obeyingBoyle's and Joule's laws, needs no calibration, since, if used in a constant-volume gas thermometer, the pressure is exactly proportionalto the absolute temperature. No real gas is perfect, of course, and weshall now extend the analysis of Chapter 5 to show that the extrapolation to zero pressure of the properties of a real gas is indeed justified,in that it yields a measure of the absolute temperature. For thispurpose we assume, as is found experimentally, that the departures of

ApplicatiO'1UJ to Bimple systemIJ 89

a real gas from Boyle's and Joule's laws may be expressed as powerseries in the pressure

and

PV=A +BP+Op2+ .

U =cx,+{JP+yP2+ ,

(6·58)

(6·59)

in which the coefficients A, B, cx" {J, etc., are functions of temperatureonly. Then from the fundamental equation and M.3 we may write

or, by use of (6·58) and (6·59),

fJ+2yP+ ... =(A-T:;) ~-T:;- (O+T:~P- ....Hence, comparing coefficients, we see that

or

A -TdA/dT=O,

AocT.

Thus the value of PV does indeed tend, as P tends to zero, to avalue which is proportional to T, and the extrapolation of a realgas to zero pressure yields correct values of the absolute temperature.

The process of extrapolation, however, involves in practice precisemeasurements with a gas thermometer at a number of differentpressures, and this procedure may be largely avoided by use ofsubsidiary determinations of the Joule-Kelvin coefficient of thethermometric gas. It will be recalled that the Joule-Kelvin effectvanishes for a perfect gas, so that it gives directly a measure of theimperfection of the gas. It is not in itself sufficient, as will be understood by recalling the inversion phenomenon; the absence of a JouleKelvin effect does not guarantee the perfection of the gas. We musttherefore look more closely into the matter to see what use can bemade of the Joule-Kelvin effect and what additional information isneeded. According to (6·38), the Joule-Kelvin coefficient (oT/oP)1&(which we shall denote by p), is expressible in the form,

pCp=T(ov/oT)p-v. (6·60)

Here all temperatures are absolute, but in any experiment on theeffect they are measured in terms of an empirical scale. Let us definethe empirical temperature in terms of a given constant-volume gasthermometer, putting ()=aP, where P is the pressure in the thermometer and a is a constant, which may be chosen so that there are 100°


(6-64)

between the melting-point of ice and the boiling-point of water. Thento express (6·60) in terms of 0 rather than T, we note that

, CO) dO (6·61)P == fJP ,,=PdT'

, en) dT (6·62)Cp == 00 P = cp dO

and (8V) ep) CV) dO (6·63)

fJT p = - fJO v oP 0 dT ·

If the Joule-Kelvin coefficient is measured with the same gas andunder the same conditions as exist in the gas thermometer, the term(oP/fJ{})v in (6·63) is by definition PIO. Hence, combining equations(6·60)-(6·63), we have

, , {PT (fJV) dO }P cp = -v -;;0 fJP odT+ I ·

If c~ is the specific heat per unit mass, so that V= lip, P being thedensity of the gas,

T dO = _ 0(1 + ' c' ) (O(log P)) =~(O).dT PP p fJ(log v) (J

The function on the right-hand side may be regarded as a function ofoonly, since all the quantities entering therein are prescribed to bemeasured under the conditions which prevail in the gas thermometer.It will be observed that everything that occurs in~(O) is a measurablequantity, and that the information needed besides the value of p'consists of the density and specific heat of the gas, and the quantity

- (~~II:~~;) fJ which is the isothermal bulk modulus of the gas divided

by its pressure. If the gas is perfect the last is equal to unity andP' vanishes, so that~(O)=0 and O=T. If the gas is not quite perfect,as is the practical situation, pp,'c~ is much smaller than unity, so thatthe precision of measurement required for these quantities is not sohigh as the precision desired in correcting the gas scale. On the otherhand, the bulk modulus must be measured very accurately, butthis is fortunately exactly what can be done in a well-designed gasthermometer.

Equation (6·64) may be integrated in the form

(T1) fOl dO

log To = fJ, $'(0)" (6·65)


If §"(O) is measured between 0 and 1000 C. this integration may beperformed numerically to give the ratio of the absolute temperaturesat the boiling-point and melting-point, and thus to fix the value of theabsolute zero on the Celsius scale (or to determine the :qlelting-point ofice on the absolute Centigrade scale). The relation between any otherT and () is then found by integrating from the melting-point to (), andhence the gas scale is calibrated in terms of the absolute scale.

Lastly we tum to the problem of establishing the temperature scalein the range of temperatures below 10 K. which are reached by adiabatic demagnetization of paramagnetic salts. Gas thermometry isimpossible in this range since even helium has so Iowa vapour pressure(e.g. 10-32 em. of mercury at 0.1 0 K.) that no useful measurements onthe gaseous state can be made. The helium gas thermometer has beenused to establish the absolute scale with an accuracy of about 0.01 0 orbetter down to 20 K., and the scale has been extended to 10 K. bymeans of paramagnetic salts which are known, from their behaviourin demagnetization experiments, to obey Curie's law accurately downto 10 K., so thatothe reciprocal of the susceptibility may be taken asa measure of T. In this range it is convenient to use as an empiricalthermometer the vapour pressure of helium, and it is this which isdetermined as a function of T by means of a gas thermometer ora paramagnetic salt. We shall therefore ~ssume that any temperatureof 10 K. or higher who~e value is required in establishing the absolutescale below 10 K. may be determined absolutely by reference to a tableof helium vapour pressures. We shall not expect to achieve greaterrelative accuracy below 10 K. than has been achieved above 10 K.,since the calibrations are interdependent.

As an empirical thermometer below 10 K. the susceptibility ofa paramagnetic salt serves admirably, and the salts which have beenused are those which obey Curie's law down to very lo\v temperaturesand which, by virtue of this fact, attain a very low temperature ina demagnetization experimento The empirical ('Curie') temperature,which is customarily denoted by '1Y*, is defined by the equation

T* == constant/x, (6·66)the constant being chosen so that T* tends to equal T as the temperature is raised to 10 K. or over. The problem now is to determine therelation between T* and T at lower temperatures. A number ofdifferent methods have been devised, of which we shall discuss onlyone, which has been used successfully to calibrate several differentsalts. This method starts from the fact that (see (6·21)), in the absenceof a magnetic field, T = d UIdS

= (;.) / (d~*)' (6'67)


The experiments carried out are aimed at determining the two differential coefficients of (6·67) independently over a range of Curietemperatures.

The coefficient (dU/dT*) is clearly the specific heat of the salt inzero magnetic field measured on the Curie scale, and is determinable inprinciple by demagnetizing the salt to a low temperature andmeasuring the change of susceptibility when a known amount of heatis supplied. The chief difficulty in practice is to ascertain the rate ofsupply of heat which, it may be mentioned, should be supplieduniformly to the salt on account of the extremely poor thermalconductivity of the salt at these low temperatures. In the earliestwork the heat was provided by mixing a little y-radioactive materialwith the salt, and determining the amount ofy-ray energy absorbed ina separate experiment at higher temperatures. More recently analternative means of heating has been devised which makes use of thehysteresis losses which the salt suffers when subjected to an alternatingmagnetic field; any heat production in the salt will show up asa resistive component in the inductance of the coil which is used tomeasure X, and the heat supplied may be determined from the magnitude of this resistance and the current in the coil. There are otherdifficulties in determining (dU/dT*), mainly concerned with unavoidable heat leaks to the salt and desorption ofhelium gas from its surfaceon warming, but these only limit the accuracy attainable withoutinvalidating the method, and we shall discuss them no further.

A quite different procedure is used to measure (dB/dT*) as a functionof T*. The first stage involves the determination ofS as a function ofJI'at some convenient temperature To in the range above 10 K., wherethe absolute value of To may be found. This is carried out by makinguse of the analogue of M. 3,

By studying the magnetization of the salt as a function of £ overa range of temperatures around To the differential coefficient on theright may be determined as a function of £ at temperature To, andhence by numerical integration the curve of S against~ is plotted.The second stage consists of a series of adiabatic demagnetizations ofthe salt, starting always from temperature To but varying the startingvalue of the field. From the first set ofmeasurements the starting valueof the entropy is known, and this is also (ideally) the final value, sincethe demagnetization is adiabatic. It is therefore possible, by measuringthe value of T* when the field reaches zero, to find how S varies withT* and hence to construct a curve of the variation of (dS/dT*) withT*. From measurements such as these a curve showing the relation

Applications to simple systema 93

between T* and T may be constructed, of \vhich fig. 22 is an example,the salt here being chrome alum. The inaccuracies ofthe measurementsare mainly due to imperfect realization of adiabatic conditions, but itwill also be appreciated that there is a considerable amount of

0·5

r*

o 0·5reK.)

Fig. 22. T-T* relation for chrome alum (E. Ambler and R. P. Hudson,Rep. Progr. Phys. 18, 251, 1955).

numerical manipulation of the data and smoothing out of experimental scatter involved, and that the errors in these procedures areliable to be cumulative. It is thus not surprising that very highaccuracy has not been attained, and that different workers do notalways agree about the detailed shape of the curve, particularly at thelower end of the temperature range. Nevertheless, the measure ofconsistency achieved at such an extreme of temperature is quitea triumph of applied thermodynamics.

94

CHAPTER 7

THE THERMODYNAMIC INEQUALITIES

(7·3)

(7-2)

(7·1)

fq/T~~S,or, for a differential change,

TdS~q.

The increase of entropy

We have so far discussed the consequences of the second law only inso far as they are concerned with the existence ofentropy and absolutetemperature. In this chapter we shall begin the discussion ofClausius'sinequality (4·14), which holds the clue to the difference betweenreversible and irreversible processes and provides a criterion by whichit may be determined whether a given process is physically possible.

According to Clausius's inequality, for any closed cycle of a system,during which heat may be supplied or extracted by an external bodyat temperature T (not necessarily constant during the change),

fq/T~O.The equality sign expresses the situatian which obtains when the cycletakes place in a reversible manner. If in any change the equality signis found to hold, there is no thermodynamic reason why the cycleshould not be exactly reversed, although there may be practicaldifficulties in the way of accomplishing the reversal, and we shall referto all such cycles as reversible without considering the matter ofpracticability. Let us consider a cycle which may be decomposed intotwo parts, the first an irreversible change from one equilibrium state Aof the system to another equilibrium state B, and the second a rever-

sible return to the state A. In this cycle f q/T <0, so that

fB q/T+fA q/T<O....4 irrev. B rev.

Now for the reversible change represented by the second integral we

may replace fA q/T by S,A-SB' since this is how an entropyBrev.

change is defined. Hence

f B q/T<SB-S,A.A irrev.

We may simplify the notation and include reversible changes bywriting (7-I) in the form

(7·4)t!S~O,

Thermodynamic inequalities 95

The inequality (7·2) is the fundamental thermodynamic inequality;once again we stress that the temperature T which occurs in (7·2) and(7·3) is the temperature of the body which supplies the heat, and notthe temperature of the system undergoing the change.

If the system is thermally isolated, so that q= 0, (7·2) takes a particularly simple form,

which expresses the law of increase of entropy:

The entropy of an isolated system can never diminish.

This law, which we shall refer to as the entropy law, provides thethermodynamic criterion for deciding which processes that couldconceivably occur in an isolated system (without violating the firstlaw) can actually be effected without violating the second law. Anexamination of simple examples will show the operation of thislaw. Consider first two bodies at different temperatures, TI and T2

(TI > T2 ), and let them be brought momentarily into thermal contactso that heat q flows from the hotter to the colder; the entropy of thehotter body decreases by q/~, while that of the colder increases bya greater amount q/T2, and the resultant total effect is that the entropyof the system increases. The reverse process, of heat flowing from thecolder to the hotter body, is excluded by the law.t Secondly, considera quantity of gas contained within a vessel which is surrounded by anevacuated space. If the vessel is pierced the gas expands irreversiblyto fill the whole space, and its entropy is thereby increased (see p. 69).As a third example consider a moving body coming to rest under theinfluence of frictional forces. The decrease of kinetic energy is accompanied by an increase in temperature, and consequently in entropy, ofthe body and its s~oundings.

t It will be seen that the second law provides a proof of what we havehitherto regarded 88 a consequence of the converse of the zeroth law, that it ispossible to construct a temperature scale (of which the absolute scale is anexample) such that there is a monotonic correspondence between hotness andtemperature. I t is in fact possible to develop thennodynamics without introducing the converse of the zeroth law, if the second law is formulated in sucha way that the concepts of hoUer and colde" do not enter; Kelvin's and Caratheodory's fonnulations fulfil this condition, but it would be harder to avoidimplicit 88sumption of the converse of the zeroth law if Clausius's formulationwere taken as a basis. Although the treatments of the second law given inChapter 4 avoided the use of Clausius's fonnulation, it has been felt to bedesirable to state the converse to the zeroth law explicitly at an early stage,since the ideas of hotness and coldness are so much part of our intuitive perception that they should be placed on a more rigorous basis as early as possible,and not left to follow 88 a trivial consequence of the much mot'6 sophisticatedsecond law.

96 Olas8ical thermodynamics

In the first two examples the changes treated involved the transitionfrom one equilibrium state to another, and were effected by alteringthe constraints imposed upon the systems, in the first by removalof anadiabatic wall and in the second by altering the volume to which thegas was confined. The third example is slightly different, since theinitial state was not an equilibrium state, but in this particular caseno difficulty arises if we define the entropy of a moving body, which isin equilibrium in a coordinate frame moving with its centre ofmass, tobe the same as if it were at rest (at any rate so long as the velocity issmall in comparison with the velocity of light; relativistic thermodynamics requires special treatment which will not be attemptedhere). In all the examples, then, the initial and final entropies are welldefined, and the operation ofthe entropy law is clear. It should not benecessary to point out that the entropy law cannot be applied to anytransition in which the entropies of the states concerned are notdefinable, and this means that, except for trivial exceptions like thethird example, it is not applicable except to transitions betweenequilibrium states. Now for any given set of constraints a thermodynamic sy@tem has only one true equilibrium state,t and we maytherefore formulate the entropy law in a slightly different way:

It is not possible to vary the constraintB of an isolated system in such away a8 to decrease tht, entropy.

This formulation focuses attention on the constraints to whicha system is subjected, and it is instructive to follow up this line ofargument in connexion with the fluctuations which are an essentialfeature of the equilibrium state. To make the meaning clear we shallconsider a specific example, the second of the three mentioned above,in which a gas is released from a smaller into a larger volume. Whenthe gas is in equilibrium in the larger volume its density is very nearlyuniform, but is subject to continual minute fluctuations. Veryoccasionally larger fluctuations will occur, and there is a continuousspectrum ofpossible fluctuations ranging, with decreasing probability,from the very small to the very large; so that it is a theoreticalpossibility (though it is overwhelmingly improbable of observationeven on a cosmic time scale) that the gas may spontaneously collapseinto the smaller volume from which it originally escaped at the piercingof the wall. It will subsequently expand again to fill the full volume atjust the same rate as at the first escape. We may now inquire what

t The existence of 'metastable' states of the sort discussed in Chapter 2,e.g. a mixture of hy--lrogen and oxygen, does not necessarily invalidate theargument, since we have elected to treat these exactly as if they were stablestates, that i~ to ignore the possibility that a chemical reaction could occur.Soo, however, the second paragraph of the next page.


happens to the entropy of the gas during this large-scale fluctuation,and to this question the only satisfactory answer is the perhapssurprising one-nothing. For the continuous spectrum offluctuationsof all magnitudes is, as stressed before, part of the nature of thermodynamic equilibrium; the huge fluctuation just envisaged does notrepresent a departure from equilibrium-it is simply an extremelyrare configuration of the gas molecules, but still just one of theenormous number of different configurations through which the gaspasses in its state of equilibrium subject to given constraints. If weascribe a definite value to the entropy of the gas in equilibrium wemust ascribe it not to any particular, most probable, set of configurations, but to the totality of configurations of which it is capable. Thuswe see that the entropy (and ofcourse other thermodynamic functions)must be regarded as a property ofthe system and ofits constraints, andthat once these are fixed the entropy also is fixed. Only in this sensecan any meaning be attached to the statement that the entropy of anisolated mass of gas, confined to a given volume, is a function of itsinternal energy and volume, S=S(U, V). It follows from this thatwhen the gas is confined to the smaller volume it has one value of theentropy, when the wall is pierced it has another value, and that it isthe act of piercing the wall and not the subsequent expansion thatincreases the entropy. In the same way when two bodies at differenttemperatures are placed in thermal contact by removal ofan adiabaticwall, it is the act of removing the wall and not the subsequent flow ofheat which increases the entropy. It will be seen then that our secondstatement of the entropy law has much to recommend it in that itconcentrates upon the essential feature of a thermodynamic change,the variation of the constraints to which a system is subjected.

To take this argument to its ultimate logical conclusion leads toa rather curious situation. Since no walls are absolutely impervious tomatter or to heat we may consider that no constraints are perfect; notwo bodies in the universe are absolutely incapable of interaction withone another. Therefore the entropy of the universe is fixed once andfor all, and the present state of the universe either is, or for thermodynamic purposes simulates, an enormous fluctuation from the meanstate of more or less uniform density and temperature. But, apartaltogether from the entirely unjustifiable assumption that the universecan be treated as a closed thermodynamic system, this point ofview isnot very useful, since it makes it difficult, if not impossible, to applythe entropy law in any situation. It is better by far to make a reasonable compromise, of ~he same nature as those which we made inChapter 2. Although truly adiabatic walls do not exist, we imagine forthe sake of argument that they do, so that small portions of theuniverse may be considered in isolation. A similar compromise was

98 Ola88ical thermodynamics

involved in our discussion of metastability, in which we concludedthat no harm would arise from assuming that reactions which proceedimmeasurably slowly are not proceeding at all. We are then enabledto define the entropy of physically interesting systems, and apply theentropy law to them without difficulty.

The point of view that it is the constraints, rather than the immediate configuration, which determine the entropy is satisfactory in thatit enables fluctuations to take their natural place in the thermodynamicscheme, but it carries one slightly unfortunate consequence with it,that the entropy law is no longer universally valid. A typical violationof the law is exemplified by the following experiment. Let two bodiesat different temperatures be initially isolated from one another andthen be brought into thermal contact; before they have had time toreach the same temperature isolate them again. In the first stage theentropy is increased by removing the adiabatic wall, in the second it isreduced once more, perhaps to nearly its original value, and thesecond stage can be thought ofas a violation ofthe entropy law, whichcan be made as great a violation as we please by choosing the initialtemperatures to be as far apart as desired. It will be observed, however, that in the complete experiment the entropy is increased, sincewe cannot increase the temperature difference between the bodies.Thus no useful decrease in entropy is achieved, and no violation of thesecond law can be effected by means ofsuch a violation of the entropylaw. We conclude therefore that our deduction ofthe entropy law fromthe second law is not logically flawless.

The reason for this is to be found in an inconsistency in our point ofview concerning large-scale fluctuations. In the development of thelaws Qf thermodynamics we took it for granted that no fluctuationswould lead to any observable temperature difference between twobodies in equilibrium, so that we were able to say that bodies inequilibrium were characterized by the same temperature. Now, however, we have committed ourselves to the view that the entropy isdetermined by the constraints, so that bodies in thermal contact butnot at the same temperature are treated as if they were indulging ina huge fluctuation from the average configuration, ofsuch a magnitudeas never to have the remotest chance of being observed as a spontaneous fluctuation. If we were to be consistent we should have tomake a clear distinction between the temperature of the combinedsystem of two bodies and the temperatures of the bodies separately.The former would be, like the entropy, a function of the constraintsand invariant in even the largest fluctuations; the latter would bevariable, and only rather imperfectly defined, just as the entropy ofone of the bodies is not well defined so long as it is in thermal contactwith the other. This discussion is, however, leading us into rather deep


waters, and it does not seem useful to carry it any further within theframework of classical thermodynamics. It is by no means easy toincorporate fluctuations consistently, and a full treatment is onlypossible with the aid of a microscopic picture and the techniques ofstatistical thermodynamics.

The difficulties which we have just been discussing need not force usto the conclusion that we are wrong to regard the entropy as det~r

mined by the constraints. Apart from the type of useless violation ofthe entropy law mentioned above, the law, in both formulationsgiven, is entirely valid in practice, and we need not hesitate to useeither formulation as the basis for determining what changes arepermitted by the second law. As a matter of fact, the entropy law hasa much wider validity than might be supposed from the foregoingdiscussion, from which it might have appeared likely that fluctuationslay outside the realm of the entropy law. But it seems most probablethat spontaneous fluctuations cannot be used in any way to violateeither the entropy law or the second law. One ofthe earliest discussionsof this topic was by Maxwell, who hypothesized a 'demon' having thenecessary endowments to enable him to violate the second law.Maxwell imagined two vessels of gas at the same temperatureseparated by an adiabatic wall, in which was a small hole of little morethan molecular dimensions. The demon controlled a trapdoor in sucha way as to let through the hole from left to right only such moleculesas had more than the average velocity and from right to left only suchas had less than the average. In this way he was eventually able toraise the temperature of the gas on the right and lower that of the gason the left, and so decrease the entropy of the gas as a whole. Thishypothetical experiment is essentially one which systematicallyemploys fluctuations (in this case fluctuations of the energy of the gasmolecules in a given region) to violate the second law. Now it isimplicit iIi Maxwell's discussion that the entropy of the demon neednot enter the problem, and recently Brillouint has pointed out howunjustified this assumption is. On either side of the adiabatic wall thetemperature is uniform, and in consequence the radiation in the vesselis isotropic. Therefore the demon cannot distinguish the form or position of any object in the vessel, and cannot tell when to open or closehis trap door. He must be provided with a small flash-lamp to illuminate the oncoming molecules, and this flash-lamp, since it must giveout radiation different in character from that in the vessel, necessarilyoperates irreversibly. Brillouin shows that however well designed theflash-lamp may be, the entropy it generates always exceeds thedecrease due to any segregation of molecules achieved with itsaid. Thus there is no net decrease of entropy. Although very few

t L. Brillouin, J. Appl. Phys. 22, 334 (1951).


hypothetical experiments employing fluctuations have been analysedin such detail, it appears most probable that they all fail to violatethe second law on account of the necessary entropy generation by theobserver who controls the process. There is thus no justification forthe view, often glibly repeated, that the second law ofthermodynamicsis only statistically true, in the sense that microscopic violationsrepeatedly occur but never violations of any serious magnitude. Onthe contrary, no evidence has ever been presented that the second lawbreaks down under any circumstances, and even the entropy lawappears to have an almost universal validity, except in such futileexperiments as we have discussed above, the removal and reapplication of constraints.

The decrease of availability

So far we have considered the entropy law only in relation to anisolated system. Let us now extend the argument to include systemswhich are immersed in a bath at constant temperature To and subjectedto a constant external pressure Po. We shall not assume that there isnecessarily thermal or mechanical contact between the system and itssurroundings, and in the event that the system is enclosed withinrigid adiabatic walls the results we shall derive will be identical withthose derived for an isolated system. But they will now include asother special cases of importance systems which are in either thermalor mechanical contact, or both, with their surroundings.

In any change of the system from one equilibrium state to another,during which the bath is the sole source ofheat supplied to the system,the heat entering from the bath must satisfy (7·2), or, since To isconstant,

We may therefore write for the change in internal energy ofthe system

dU=Q+ W~TodS+w.Also if the only source of work is the external pressure,

W=-~dV

and ~U~TodS-PodV,

or dA ~O, where A=lJ-ToS+~V. (7·5)

The new quantity, A, is called the availability of the system. It willbe observed that since To and Po, rather than T and P, enter in itsdefinition, A is not a property of the system alone, but of the systemin given surroundings. The availability is a function which has beenmore commonly employed by engineers than by physicists, and the


(7'6)

name expresses its technically important property of measuring themaximum amount ofuseful work which can be extracted from a system(e.g. a boiler) during a given change in given surroundings. To provethis we need consider only reversible changes, since it is readilyshown from the second law that these enable the greatest amount ofwork to be performed in a given change. For any infinitesimal change

dA=dU-TodS+~dV

=(T-To)dB- (P-Po)dV,

in which T and P are the temperature and pressure within the system.tIf the change is reversible, - T dB is the heat extracted from thesystem, which may be most usefully employed by transferring it to anideal Carnot engine working between the temperatures T and Too Thework done by the engine will then be - (T - To) dS. Also if the systembe allowed to expand reversibly by an amount d V, an additionalpressure (P-~) must be applied externally, and the useful work ofexpansion is that performed against the additional pressure; theremaining work done by the system is not useful, being expended inpushing back the atmosphere (or any other passive source of theexternal pressure ~). Hence (P-~)dV is the useful work, and thetwo work terms add to give - dA. Thus the decrease in A is a measureof the maximum useful work available.

This interpretation of A leads us to expect that A takes a minimumvalue when T = To and P =~, for then the system is in equilibriumwith its surroundings and cannot act as a source of useful work. Inorder to examine this idea we may expand A as a Taylor series inS and V about the point where T=To and P=~, on the assumptionthat such an expansion is valid:

A =Ao+~ (:~L(~S)2+ (:rl8~S~V-~ (:~t(~V)2+ ,..,

the first-order terms vanishing (from (7'6)) at the origin of the expansion. Thus A takes a minimum value if the second-order terms areessentially positive for all ~S, L\ V, Le. if

and

(OT/OShr> 0,

(oP/oV)s<O

(oT/o V)§ < - (oT/oS)v (oP/o V)s'

(7'7)

(7'S)

(7'9)

t By pressure within the system we mean that pressure which would have tobe applied externally to maintain the system in the same state if it wereenclosed in completely flexible and extensible walls. We assume in what followsthat P is the same for all parts of the system, but the argument is readilygeneralized to apply to more complex systems.


The first inequality (7·7) is satisfied if 0v is positive but not infinite.That Oy is positive we saw to be a consequence of the zeroth law; weshall return in a moment to the situation which might arise if Ov wereinfinite. The second inequality (7·S) is satisfied if the adiabatic com-

pressibility ks ( ;: -~ (~~)Jis positive but not infinite. That ks

should be positive is a mechanical requirement for stability; if it werenegative the system would spontaneously collapse, since by doing so itwould lessen the pressure needed to keep it in equilibrium. The thirdinequality (7·9) may be transformed, by means of (6·17) and the factthat (aT/oS) v is positive, into the form

(oP/oT)v (aT/a V)s > (oP/oV)s·

Now (oP/o V)s= (OP/OV)T+ (oP/oT)v (oT/oV)s'

and therefore (7·9) implies that

(OP/OV)T< o.Thus (7·9) is satisfied if the isothermal compressibility k T is positive,and this is also a mechanical requirement for stability. Under normalcircumstances then, when 0v' ks and kT are finite, A takes a minimumvalue when T = To and P = ~, as expected. We may, however, conceiveof less usual situations in which some or all of the inequalities becomeequalities. Two cases may be considered, the first being that in which(7·9) is an equality (kT infinite) while Ovand ks are both finite. In thiscase the second-order terms may be either positive or zero, and thehigher order terms in the expansion must take such values as to giveA an absolute minimum. This is the situation which arises at thecritical point of a liquid-vapour system (see Chapter 8), and it may beshown by a more detailed analysis that (82Pj8V2 )T must vanish and(03P/OV3)T must be negative. The second case to be considered is thatall the inequalities become equalities. Now the third-order terms inthe expansion must all vanish and the fourth-order terms must beessentially positive. It may be shown that this imposes more conditions on the form of A than can be simultaneously satisfied, and thattherefore Ovand ks can never become infinite.t

This analysis is valid, however, only on the assumption that a seriesexpansion of A is possible, and we must inquire into the justificationfor this assumption. Undoubtedly the conclusions reached are correctin the majority of cases, for it is only under exceptional circumstancesthat Op and kT become infinite, and it is still more exceptional to findoy or ks infinite. Nevertheless, quite simple exampIes may be adducedto show the danger oftaking for granted the validity of the expansion,

t L. Landau and E. Lifshitz, Statistical Physics (Oxford, 1938), p. 102.

Thermodynamic in,equalities 103

and the most elementary of these is a system consisting of two phasesin equilibrium, a liquid and solid for instance, and open to the surroundings so that P = Po, T = To. If 16 and To are such that both phasesmay coexist with arbitrary proportions of the material in each phase,it is clear that k T is infinite, since isothermal reduction ofvolume merelyalters the proportions (e.g. causes more solid to be formed if the solid is

s

VFig. 23. Contours of availability on the S- V diagram of a two-phase system,when (Po, To) is a point on the phase-equilibrium line. The system is in equilibrium, with varying proportions of the two phases, along the central line.

denser than the liquid) without involving any change of pressure.Such a compression changes S and V without changing the value ofA;thus the surface representing A as a function of S and V takes theform shown diagrammatically in fig. 23. So long as both phases arepresent the surface has a perfectly flat valley, and it is only when theextent of the compression or expansion is such as to eliminate one orother of the phases that the bottom of the valley begins to rise and toreveal that in equilibrium A does indeed take a minimum value. Infact, as remarked in Chapter 2, this is a situation ofneutral equilibrium,and no series expansion of A can demonstrate that at the ends of therange ofneutrality the surface rises rather than falls. Itwill be observedthat the direction of the bottom of the valley is such that

dS/d V = (8z-8s)/(Vz-vs)'

which by Clapeyron's equation (5·12) is just the rate of variation ofthe melting pressure with temperature, (dP/dT)meltlng. For most

104 Ola88ical thermodynamics

substances (dP/dT)melttnl is neither zero nor infinite, since usually(81-88) and (V,-f),) do not vanish. Under these circumstances both(8IA/8SS)y and (8IA/8VI)S are non-vanishing, &8 may be seen fromfig. 23, and Oy and lesremain finite even though Op and ieI' are infinite.There is no reason, however, why under very special conditions(dP/dT)meltlnlshould not become either zero or infinite, as illustratedby the points X and Y in fig. 24. It is believed that a point such as Xexists on the melting curve of the light isotope of helium, IHes (seep. 124), and undoubtedly a point such as Y would exist on the melting

p

~X

p

T T(a) (6)

Fig. 24. Special forms of melting curve.

curve of ice were it not for the transformation ofice into another solidmodification before this point is reached. At X the valley in fig. 23 hasswung round parallel to the V-axis, so that (82A/8V2)s is zero, or ksinfinite, while at Y the valley lies parallel to the S-axis, and 0,. isinfinite. These examples should make it clear that the finiteness of Oyand les is not an absolute thermodynamic requirement, but at thesame time exceptions are very rare indeed.t

The conditions for equilibrium

We are now in a position to apply the result expressed by (7·5) to seewhat it is that characterizes the equilibrium state of a system undergiven constraints. What we wish to do is to compare various states ofthe system, ofwhich only one is truly the equilibrium state, and to seehow this is distinguished from the rest. As a result ofthis investigationwe may expect to derive a rule for calculating in any given situationthe configuration of the equilibrium state. At first sight it mightappear a hopeless task, since only in the equilibrium state are wejustified in ascribing values to the entropy of the system and to the

t More subtle cases where series expansion of thennodynamic functionsmust be treated with great caution are examined by L. Tisza, Pha8e Transformations in Solids, ed. Smoluchowski, Mayer and Weyl (Wiley, 1951), ch. I.

Thermodynamic inequalitie8 105

other thermodynamic functions which depend on the entropy. However, by the judicious use ofadditional constraints we may in imagination convert a variety of non-equilibrium states into equilibriumstates, and then inquire into the consequences of removing theseconstraints. If we are to employ the result expressed by (7·5), theadditional constraints which we use must be of such a nature that nosources of work or heat, apart from the surroundings, are involved intheir removal. We may, for example, interpose an adiabatic wallbetween two bodies, or separate two masses ofgas by means ofa partition; but we must be""are of dividing a body into such small piecesthat surface effects are introduced or modified. It is difficult to formulate any firm rules for what additional constraints are permissible, butit should become clear in wha t follows that most examples of physicalinterest are so simple as to raise few doubts as to their validity.

Let us express the result of (7·5) in language similar to the secondformulation of the entropy law:

If the constraint.! of a system, whose surroundings are maintained atconstant temperature and pre88Ure, are altered in any manner whichinvolves no work or heat other than that provided or absorbed by thesurroundings, the availability of the system is not thereby increa8ed.

This tells us that when we remove any additional constraints, andallow the systenl if it wishes to take up a new configuration, theavailability ofthe new configuration cannot be greater than that oftheold. Therefore, if we calculate the availability of all conceivableconstrained states of a given system, that which has the smallestavailability is the equilibrium state of the unconstrained system. Forif we remove the additional constraints from this state there is noother configuration to which it can change without making possible,by reapplication of the constraints, an increase of the availability.

A few simple illustrations should make the argument clear. First,two bodies at different temperatures in an enclosure formed of adiabatic walls. If we interpose an adiabatic wall between them so thateach is in equilibrium we may calculate the total availability as thesum of the two separate availabilities, whatever the temperatures ofthe bodies may be. The argument given above now implies that fora given value of the total internal energy the equilibrium state for thebQdies in thermal contact is that for which the cOlnbined availabilityof the separated bodies is smallest, which in this case is the same aswhen the combined entropy is greatest, and this of course is ,vhen thetemperatures are equal. }'or if we have the bodies separated by anadiabatic wall when they are in this state, and then remove the walland replace it again after an interval, however long, the final stateofthe separated bodies cannot have changed from the original since


there is no state of lower availability to which transition is possible.We have thus discovered the equilibrium state ofthe bodies in contact.In a similar way we may show that the density of a gas in equilibriumis uniform, by separating the gas into regions of varying density bymeans of removable partitions, and calculating the availability of thenon-uniform gas thus artificially produced.

These two examples are trivial, since the results derived eitherfollow from the zeroth law or are implicitly assumed in the development of the subject. But there are many applications of the samemethod of reasoning where the answer obtained is not trivial but ofconsiderable value. Consider, for instance, a closed vessel containingonly a liquid and its vapour, maintained at a constant temperature To.What proportion of the material will be in the liquid phase1 To solvethis problem we clearly wish to apply an additional constraint so thatthe proportion of liquid, (x, may be varied at will without departurefrom a state of equilibrium, and obviously the way to do this is tointerpose a barrier at the surface of the liquid, separating the twophases. Now we can write the combined availability as a function of(x, and find for what value of (X it takes a minimum value; this is therequired equilibrium condition.

In this example we have considered the constraint to take the formof a mechanical barrier, but we might just as well have imagined theevaporation rate of the liquid and condensation rate of the vapour tobe reduced to zero, to enable the system to be in metastable equilibriumfor any value of (X. The real justification for this procedure is perhapsonly to be found in a microscopic picture of the processes occurring atthe interface between liquid and vapour, from which it is clear that theequilibrium state is only affected to a quite negligible degree by therate at which it is established. But there is experimental evidence onthis point as well; for instance, the rate of evaporation of liquidmercury is highly dependent on the cleanness ofthe surface, while theequilibrium vapour pressure is not. This device of imagining the rateof progression of a change of state to be reduced to zero is a veryversatile way of applying additional constraints, and finds its mostimportant application in chemical thermodynamics. If we wish tofind the equilibrium concentrations of the substances taking part ina given chemical reaction, it is convenient to suppose that the systemcan be 'frozen' into an unreactive state with any values of the concentrations. It may then be treated as an equilibrium state for thepurpose of calculating the availability.t Again the fundamental justification for this procedure is to be found in the microscopic viewpoint;in a mixture of reacting gases, for example, at any instant, the number

t The detailed procedure for carrying put such calculations may be found inany text-book of chemical thermodynamics.


ofmolecules which are actively engaged in interacting with others is sosmall, even in a fast reaction, that they do not appreciably affect thethermodynamic parameters of the mixture. But there is also powerfula posteriori justification in the success which attends the application ofthermodynamical reasoning to problems of chemical equilibrium. Itwill be seen that this argument is only an extension, into the realm ofmeasurably fast chemical reactions, of our early assumption (Chapter 2) that' metastable' equilibrium may be treated exactly as stableequilibrium, Le. immeasurably slow reactions may be ignored.

It is now possible to give in one general statement (which must beinterpreted in the light of the foregoing discussion) the rule whichdetermines the equilibrium configuration of a system:

For a system, whose sole external sources of heat and work are itssurroundings maintained at constant temperature and pressure, of allconceivable configurations that one is stable for which the availabilitytakes a minimum value.

This statement is not the most usual statement of the criteria forequilibrium. We may derive the conventional forms by consideringthree special types of system:

(1) An isolated system. Whether its volume can alter or not,U+Po V stays constant, and the condition that A shall take a minimumvalue is the same as the condition that the entropy shall take a maximum value.

(2) A system of constant volume in thermal contact with a constanttemperature bath. Here P is not necessarily equal to Po when A isminimized (in (7·6) d V =0 so that P-Pois indeterminate). But Po Visconstant, and when A is minimized T = To. Thus the condition forequilibrium is that the free energy F (== U - TS) shall be minimized,subject to V being constant and T being equal to the externaltemperature.

(3) A system acted upon directly by a co~tant external pressure, inthermal contact with a constant-temperature bath. If all parts of thesystem have the same pressure, then, as we have already shown,A takes a minimum value when T = T and P = Po. Thus the conditionfor equilibrium is that the Gibbs function G (== U - TS +PV) shall beminimized, subject to P and T being equal to the external pressure andtemperature respectively.

I t will be observed that in each of these cases the particular functionwhich takes an external value may be calculated for the completesystem by summation over all constituent parts, as pointed out onp.44. In particular, in case (3), the Gibbs function is defined for thesystem as a whole because P and T are constant over the system. Nowalthough the temperature takes the same value at all points ofa system


in equilibrium, however complicated the system may be, it is notnecessary that the pressure shall be uniform. For example, the pressure within a liquid drop is greater than that without, on account ofsurface tension, but a system consisting ofthe drop and its vapour maystill be in equilibrium. In a case like this the Gibbs function of thewhole system may be defined, if it is so wished, as the sum of contributions from the different parts; but there is little point in making theeffort to define G under these conditions, since it has now no relevanceto the criteria for equilibrium. If the system is open to the surroundings, and P takes different values in different parts, the conditions are not those of case (3) above, and we must revert to our originalstatement of the criterion for equilibrium, that the availability Atakes a minimum value. Since the pressure and temperature, Po and To,which enter into A are by definition constant, A is always an additivequantity for a system in equilibrium. We shall return at the end of thischapter to a more detailed analysis of the equilibrium of a liquid dropwith its vapour.

From this discussion it will be clear that the choice of criterion fordistinguishing the equilibrium state of a system depends on thenature of the constraints to which the system is subjected, and it is ofinterest to demonstrate the relation between the different criteria bya specific calculation. We shall consider the equilibrium betweena liquid and its vapour, under two different constraints: first, withthe vessel immersed in a constant-temperature bath, and open toa constant external pressure; and secondly, with the vessel closed andthermally isolated. In the first case, let us suppose that in equilibriumthere are masses a of liquid and (I-a) of vapour. The Gibbs functionof the whole system may be written in the form

(7·10)

in which Ul and Uv are the Gibbs functions per unit mass of liquid andvapour. In equilibl:ium,

(7·11)

Since the pressure and temperature remain constant, Ul and Uv areindependent of a. We see therefore that the condition for equilibriumis that the Gibbs functions per unit mass shall be the same in bothphases.

Turning now to the isolated vessel, we may no longer assume Ul andUv to be independent ofa, since varying a changes the temperature andpressure in the vessel. There are, however, certain invariants of thesystem, its total volume V, and its internal energy U; moreover, in


(7 ·14)

(7·12)

(7·13)

or

or

or

equilibrium the entropy is maximized and therefore stationary withrespect to small variations of a. We may thus write

V =XVl + (I-a) vv '

d V =advl+(I-a)dvv+(vl-vv)da=O;

U =aul+(I-a)uv,

dU =adul+(I-a)duv+(uz-uv)da=O;

8=asl+(I-a)sv,

dS=adsl+(I-a)dsv+(sl-sv)da=O.

Now multiply (7·12) by P, the internal pressure, and (7·14) by T, theinternal temperature, both of which will have the same value in bothphases. Then

dU+PdV-TdS

=a(dul+Pdvz- Tdsl)+(I-a) (duv+Pdvv-Tdsv)

+ (u,+ Pv,- Tsl)da - (uv+Pvv- Tsv)da =o.But from the fundamental equation of the second law the coefficientsof a and (I-a) vanish, while the remaining terms are ulda and Uvda.We arrive therefore at the same equilibrium condition as before, thatUz=Uv. The reader may verify by analogous arguments that this resultis also obtained with a thermally isolated vessel open to the externalpressure (8 maximized with P and H constant) and with a closedvessel in thermal contact with a constant-temperature bath (F minimized with V and T constant). The liquid and vapour cannot be inequilibrium unless the Gibbs function per unit mass takes the samevalue in both phases. This result forms the basis of the theory ofphase equilibrium, to which the following chapters are devoted.

We now return to the problem raised on p. 108, and discuss inmore detail the equilibrium between a small liquid drop and itsvapour, when the latter is maintained at constant temperature To andpressure Po; on account ofsurface tension the pressure within the dropis Po + 2a/r, where a is the surface tension and r the radius of the drop.The equilibrium state is characterized as that for which the availabilityof the system is stationary with respect to small variations of r. It isconvenient to express the availability of the whole system as a sum ofthree terms, contributed by the vapour, the bulk of the liquid, and thesurface of the drop. So far as the vapour is concerned its availabilityAl may be equated to its Gibbs function

A1(To' Po) =mv(uv- Tosv+Povv)=mvgv(To, ~),


since To and Po are the temperature and pressure, both external to thesystem and in the vapour itself. On the other hand, the equating ofavailability and Gibbs function is not strictly possible for the liquidwithin the drop. For its availability per unit mass is (ul- To8z+Povz),in which uz, 8 zand Vzare to be calculated not at a pressure Po but ata pressure Po+2u/r. If, however, we make the reasonable assumptionthat the liquid is in effect incompressible, it follows that uz- To 8 zand Vzare independent of pressure, and we may then write for the liquid

A 2(To' Po) =mzUz(To, ~)·t

Finally, we have for the surface term

Aa(To, Po) = 41Tr2(A - To'fJ) = 41Tr2u(To) from (6·57)

Thus for the whole system of mass m

A(To, PO)=A 1 +A 2 +Aa= (m - !1Trapl) Uv(To, Po) +t1TrSpz Uz( To, Po) +41Tr2u(To),

where pz is the density of the liquid. Then the equilibrium condition(oA/or)T"p, = 0 leads to the equation

2u(To)Uv(To' Po) - Uz(To, Po) =-- . (7 ·15)

rpz

From this equation it may be seen that if the drop is large, so that theright-hand side is small, ~ ana To must be adjusted in equilibrium sothat Uv=Uz, and this is the result we obtained before. If r is small theequilibrium relation between Po and To is altered, but the new vapourpressure at temperature To is readily expressed in terms of the vapourpressure Pbofthe bulk liquid (i.e. for the case r=oo). For we have that

Uv(To' Pb ) =Uz(To' Pb ) =Uo, say;

also (oUV/OP)To =vv=RTo/(MP),

where M is the molecular weight of the vapour, and R is the gasconstant per mole; and

Hence

and

t It will be instructive for the reader to work out how this expression andthe final result are to be modified if the compressibility of the liquid is notnegligible.

Thermodynamic inequalitie8 III

80 that (7 ·15) may be written

pzRTo (Po) 200~ log P

b-(P.J-Pb)= -;:. (7·16)

If the second term on the left.hand side be neglected, we arrive atKelvin's formula for the vapour pressure of a drop,

Po=Pbexp~~:~J (7·17)

The neglect of (PO-Pb) in (7·16) may be justified by expanding theexponential in (7 ·17) as far as its second term, whence it may bededuced that (~-Pb) is smaller than 200fr by a factor p,,)Pz, p'fJ beingthe density of the vapour. Thus (7·17) is a good approximation undermost circumstances.

It may be noted that the equilibrium of a drop with its vapour isunstable, for if the drop grows a little its equilibrium vapour pressuredecreases, and if it is kept in contact with vapour at constant pressureit will continue to grow. Similarly, if it starts to evaporate it willcontinue evaporating until it disappears. This means, as may beverified by direct calculation, that under isothermal conditions theavailability of the system (drop +vapour) has a maximum, not a mini·mum, when the pressure is that given by (7-17). In practice, therefore,the continued coexistence ofdroplets and vapour will not be observedifthe vapour pressure is maintained constant. On the other hand, ifthesystem is enclosed so that the total volume remains constant, it isfound, as the reader may verify, that (7·17) gives the condition for Ftotake an extremal value, and if the drop is not too small the extremummay be a minimum. It is possible in an enclosed system for dropletsto exist in stable equilibrium, but only ifthey are larger than a certaincritical size. We shall make use of this result in Chapter 9.

112

CHAPTER 8

PHASE EQUILIBRIUM

In this chapter we shall consider applications of the result derived inthe last chapter that (provided surface effects are negligible) twophases of the same substance can only coexist in equilibrium if theypossess the same Gibbs function per unit mass. The examples we shalltreat are the equilibrium of the solid, liquid and vapour phases ofa substance at various pressures and temperatures, the phase diagrams of the two isotopes of helium, and the equilibrium between thenormal and superconducting phases of a metal, for which we mustintroduce the magnetic field as an additional parameter. Theseexamples should be sufficient to illustrate the methods involved in thetreatment ofphase equilibrium. No essentially new ideas are involvedin the extension of these methods to more complex systems, such assolids which may exhibit many allotropic forms (ice, sulphur). Further,a full understanding of these examples should enable the reader, ifhewishes, to grasp the thermodynamic theory of chemical reactions andof the phase diagrams of alloys with little difficulty, as the methodsthere employed are fundamentally of the same nature.

(S·l)therefore

The phase diagram of a simple substance

We consider a simple substance ~hich under suitable conditionsexhibits three modifications, solid, liquid and vapour; by 'simple' weimply no more than that only these three modifications exist, at anyrate under the conditions of temperature and pressure with which weare concerned. In order to determine the circumstances under whichany of these phases can coexist it is convenient to study the variationwith temperature and pressure of the Gibbs function g (in whatfollows it will be taken for granted that we are considering unit massof each phase). Since for any change

dg= -sdT+vdP,

(:t= -8, (~t=V.

If then we exhibit gas a function of P and T, the surface representing g(g-surface) will always slope upwards in the direction of increasingpressure, the gradient being small and nearly constant for a liquid orsolid, and steep for a gas, decreasing as P increases. The slope in the

PluJ,se equilibrium 113

direction of increasing temperature will depend on the choice of thearbitrary constant in 8, but if, as is usual, this is chosen so that 8 ispositive at all temperatures the slope of the surface will always benegative, increasingly so as T increases since cp and therefore (o8/oT)pare always positive. Thus the surface has everywhere a negativecurvature,t but the detailed shape may be expected to vary considerably for different phases of the same substance.

Let us suppose that we can construct one ofthese g-surfaces for eachphase, irrespective of the question of whether the phase is observableat all pressures and temperatures. We then have three surfaces ofwhich any two in general intersect along a line, the three lines soconstructed meeting at the point at which all three surfaces intersect

s

p

Pc

o TFig. 25. Phase diagram for a simple substance.

one another. The projection of these three lines onto the P-T plane willgive a diagram of the type shown in fig. 25. Along these lines the twophases corresponding to the intersecting surfaces may coexist inequilibrium, since they have the same value of g, and at Pt all threephasesmaycoexist. Thepointpt is thus the triple pointofthesubstance,and our experimental knowledge ofthe behaviour ofnormal substancesenables us to interpret OPt as the sublimation curve, along which solidand vapour are in equilibrium, PtPc as the vapour pressure curve,along which liquid and vapour are in equilibrium, and PtS as themelting curve, along which liquid and solid are in equilibrium. Atpoints removed from these lines one of the g-surfaces lies lower than

t Strictly this statement is true only if the set of quadratic tenns in theexpansion of g 88 a power series in P and T is essentially negative. It may beleft as an exercise for the reader to demonstrate that this is so if Cy>O.


the others and this determines which phase is stable, since the equilibrium state of the system at given P and T is that for which the Gibbsfunction is minimized. It is clear that in the region below 0PtPc thevapour phase is stable, for it possesses the largest volume and thereforethe Gibbs function falls most steeply as the pressure is reduced. It isnot so obvious that the region to the left of OPtS corresponds to thestable existence of the solid and the remaining region to the stableexistence ofthe liquid, and here we must rely on experimental 0 bservation to ascribe the regions correctly. As the curves are drawn in thefigure it will be seen that in crossing the line PtS vertically upwards wemove from a region of liquid stability to one of solid stability; thisimplies that (ogz!OP)T > (ogB/OP)T' i.e. that Vz> VB' or that the substanceexpands on melting. For substances such as water or bismuth whichcontract on melting, PtS will have the opposite slope. If we moveacross an equilibrium line horizontally (P constant, T increasing) it isobvious that the phase which is stable on the high-temperature sidemust have the greater negative slope - (og/oT)p, that is~ by (S·l), thehigher entropy. Therefore in a reversible transition from the lowtemperature to the high-temperature phase heat must always beabsorbed, and the latent heat is in consequence always positive.

It might be argued that since only one phase is stable except alongthe equilibrium lines it is not justifiable to imagine the g-surfaces toextend over the whole plane for each phase, and indeed it is quiteunnecessary to make this supposition. We might equally well havestarted constructing the surface from three points, one in the middleof each region, .making use of experimental information to decidewhich phase to consider in each region; the intersections could then befound without supposing the surfaces to be capable of crossing oneanother. In fact, for many substances, particularly if they are highlypurified, it is possible to observe the metastable existence of a phasein a region where it should have made a transition to another phase.The most striking examples of this behaviour are the supercooling ofvapour8, which may stay uncondensed at pressures four times or morehigher than the equilibrium vapour pressure (the Wilson cloud chamber owes its operation to this fact), and the supercooling of liquidswithout solidification. For instance, small drops of water in a cloudmay be cooled to - 40° C. before they freeze. The converse effects mayalso be demonstrated, as, for example, the superheating of liquids,which is often manifested as 'bumping', and which is the basis of theoperation of the bubble chamber used to detect high energy particles.There is therefore no real obstacle to imagining each surface to continue beyond its intersection with another, and the foregoing argument could be dismissed as trivial except for the fact that muchconfusion has been generated in discussions ofhigher-order transitions

Phase equilibrium 115

(see Chapter 9) by a too-facile assumption that u-surfaces are alwayscontinuable beyond an equilibrium line. Before leaving the questionof the metastable persistence of phases, it may be pointed out that theequilibrium lines may be imagined prolonged through the triple point,asPtA,pt B andptC in fig. 24. Along PtA , for example, gz=us, but bothare greater than Uv. There is no reason in principle why a superheatedliquid should not be made to solidify by crossing the curve Pt A, thoughthe author is not aware ofany observation of this phenomenon. On theother hand, the coexistence of a supercooled liquid with its vapouralong the line Pt C is a commonplace of meteorology; the fact that thevapour pressure of the supercooled liquid is higher than that of thesolid at the same temperature (as shown in fig. 25) means that any icecrystal in a cloud containing supercooled water drops tends to growrapidly, since the latter maintain a high vapour pressure.

It is a result of general validity that the vapour pressure of a supercooled liquid is higher than that of the solid at the same temperature.For if this were not so the line Pt C would lie in the vapour region below0PtPc; but in this region Uf) is lower than either Ul or U..p while at thepoint C, Uv = Ul. It follows then that around a tripIe point there is only onemetastable line in each region, and that therefore the angles betweenneighbouring equilibrium lines are never greater than lS0°.

Clapeyron'8 equation

Along any equilibrium line separating two phases in the P-Tdiagram the Gibbs functions are equal, Ul = U2. The suffix l'denotesthat phase which is stable on the low-temperature side of the equilibrium line. For small variations ofpressure and temperature, OP and6T, which alter the state of the system to a neighbouring state still onthe equilibrium line, the variation ofUl and U2 must therefore be equal,so that

or

(aU1) oT + (aU1

) oP= (aU2) oT + (aU2) oP,

aT p ap T 'aT p ap T

(S2- S1)oT=(V2 -V1)OP from (S·l).

(S-2)

Now the ratio oP/oT tends, as oT ~ 0, to the slope dP/dT of the equilibrium line, so that

dP 8 2 -81 ldT= V2 -V1 = T(V2 -V

1)'

where l is the latent heat of the transition, T(S2- S1)' per unit mass.This is Clapeyron's equation which we derived (5·12) in Chapter 5by a different argument. From this point of view it is clear that

116 Classical thermodynamic8

Clapeyron's equation is simply an expression of the fact that along atransition line the g-surface has a sharp crease. If one describes theslope of the g-surface by the two-dimensional vector grad g, at thetransition line there is a discontinuity L\(grad g) in the slope, andClapeyron's equation states that the vector L\(grad g) is directednormal to the transition line.

Since l cannot be negative it may be seen from equation (8·2), as wededuced by a geometrical argument in the last section, that a solidwhich expands on melting (v2 > VI) has a positive pressure coefficientof the melting temperature, and one which contracts has a negativecoefficient. The comparison, by James and William Thomson in1849-50, of the actual pressure variation of the melting-point of icewith that predicted by Clapeyron's equation is of historical interest asbeing perhaps the first successful application of thermodynamics toa physical problem, and the success of this simple test undoubtedlycontributed largely to the spirit of confidence which underlay andencouraged the rapid development of the subject.

Liquid-vapour equilibrium and the critical point

Along the curvePtPc in fig. 25 the g-surfaces for the liquid and vapourintersect and the two phases are in equilibrium. This line does not,however, continue indefinitely, for at the critical point Pc the liquidand vapour become indistinguishable. The critical phenomenon isclearly illustrated by the ~othermsof a typical liquid-vapour systemshown in fig. 26. If the system is in the vapour phase at a temperaturebelow the critical temperature Tc' and is compressed isothermally,a stage is reached, as at 0, when liquid begins to form and the systembecomes inhomogeneous. Subsequent compression, at constant pressure, eventually leads to the state 0' where the vessel is wholly filledwith liquid, which is far less compressible than the gas phase. In theinhomogeneous region 00' the isothermal compressibility, defined as

- ~ (:;)T' is of course infinite. At the critical temperature the

compressibility decreases at first as the volume is reduced, and thenrises to infinity at the horizontal point of inflexion Pc, which corresponds to Pc in fig. 25; further compression reduces the compressibilitysteadily. At no point is there a separation into two phases. Attemperatures well above Tc the compressibility decreases monotonically as the volume is reduced, so that although at large volumes thesubstance may be regarded as vapour-like, and at low volumes asliquid-like, there is no point at which any transition from one phase toanother may be said to have occurred. It is therefore possible, bytravelling around the point Pc in fig. 25, to make a transition from


what is undoubtedly the vapour phase to the equally unduubtedliquid phase, without any abrupt change in the properties of thesystem. This means that the g-surfaces for the liquid and the vapourmust both be parts of the same surface, which, however, is not ofa simple form, but intersects itself along the line PtPc, though notbeyond Pc. In order to obtain a picture of the form of this surface it is

57·8

57·7,.....e~0.S·§~

Q,.

57·6

0·901·30 1·10Density (g. em. -3)

Fig. 26. Isotherms of xenon noar the critical point (H. W. Habgood andw. G. Schneider, Canad. J. Chetn. 32, 98, 1954). The broken linA marks theregion of coexistent phases, and the dotted line is the critical isothermaccording to the van der Wsals equation.

convenient to make use of the equation of state proposed by van clerWaals (p. 75 footnote) to express the continuity of the liquidand vapour phases. rrhe critical isotherm of a van cler Waals gasand a typical isotherm for a telnperature below 'Fc are shown infig. 27, together with the conjectured horizontal line CIC' whichcorresponds to the mixture of phases. We shall for the moment ignorethis line and suppose that the whole of the curve for a temperature less


than Tc is experimentally realizable, in spite of the fact that in thecentral region the compressibility is negative and the hypotheticalhomogeneous phase is intrinsically unstable.

We now use this isotherm to calculate the variation of g withpressure at constant temperature, since (OgjOP)T=V, and therefore

g(P, T) =g(Po, T) +fPvdP,P.

the integral being taken along the isotherm from Po to P. Let us take~he point D to correspond to the initial pressure Po. The first stage of

2 G

F

O~ Iooo- '-"' ---IL-

2Reduced volume, <J)

Fig. 27. Van der Waals isothenns: (a) T=To, (b) T=O·9To•

3

the integration, from D to E, is straightforward, and leads to thecurve DE in fig. 28. Between E and F the contribution of the integralis negative, and g falls, but at the same time P also falls, so that(ogjoP)T is still positive (as it must be, being equal to v). The curvefor g shows a cusp at E, the tangents to the two curves becomingcoincident at E. Similarly, there is another cusp at F, and then thecurve proceeds steadily upwards towards and beyond G. Clearly theportion DE corresponds to the vapour phase and the portion FG to theliquid phase, and equilibrium of the two phases is possible at thepressure Pe where they intersect. The difference in gradient between


DE and FG at this point represents the volume difference between thetwo phases. Since g takes the same value at 0 and 0', it is clear that the

fe'

lineOO'mustbedrawnatsllchapressureP.that c vdP=O, that is, the

areas O'FO' and OEO' must be equal.t Now this argument, and thatgiven in the footnote, both assume that the whole of the van der Waalsisotherm is physically realizable as a succession of equilibrium statesof the system. As this is not true it is very doubtful what validity thearguments possess. From a logical point of view they are worthless,but still it is probable that they lead to the correct conclusion. Itwould require a much more detailed investigation of the model ~o

justify this result completely.

g

G

Po P, P

Fig. 28. Gibbs function obtained by integrating thevan der Waals equation.

Returning now to fig. 28 we can see that as the temperature is raisedand v l1 - vzdiminishes, the two branches DE and FG intersect more andmore nearly tangentially, and the cusped region becomes steadilysmaller, until at the critical temperature the curve degenerates intoa single continuous curve. Just at the critical temperature the gradientof the curve, which is equal to tJ, is everywhere continuous, but thecurvature (8v/8P)T becomes momentarily infinite at the critical pressure, since at this point the van der Waals gas is infinitely compressible.Above T c the curves for g are everywhere continuous in all their

t An alternative argument leading to the same conclusion runs 88 follows:imagine the substance to be taken through the cycle CEO"FO'O'O reversibly.This cycle is completely isothennal, so that its efficiency must be zero, i.e. nowork must be done in the cycle. Hence the area of the cycle must vanish,a condition satisfied by making the two areas 0'FO" and OEO' equal.


derivatives. A sketch of the surface for g is shown in fig. 29. The topportion, between the two lines of cusps, is the completely unrealizableregion; a part of the regions between the line of intersection and thelines of cusps may be realized by superheating of the liquid or supercooling of the vapour.

T

(8·3)

PFig. 29. Gibbs surface for the van der Waals gas in vicinity

of the critical point.

Along with anomalous compressibility in the neighbourhood of thecritical point the specific heat also shows certain anomalies, which wemay exhibit qualitatively with the help of van der Waals's equation.It will be recalled (see p. 60) that according to this equation(oCyjov)T=O, so that if, as is a good approximation for many gases,(ocyjoT)y is effectively zero, cy is the same for all states of the gas orliquid. This is certainly not true in practice, and illustrates one of thelimitations of van der Waals's equation, but, nevertheless, it serves toshow that we need not expect any markedly unusual behaviour in cynear the critical point. On the other hand, as the isothermal compressibility k T goes to infinity at the critical point, so does cp - cy, since,as follows easily from (6·9),

(OP) 2

cp-cy=vTkT aT ,,'

A quantitative calculation from van der Waals's equation is straightforward. If we take the critical point as our origin of coordinates, andwork in reduced coordinates, writing

PluJ,se equilibrium 121

(8·4~3x3 +(2 +3x) y- Sz=O.

the equation takes ·the form (when all powers of x above x3 areneglected),

Equation (6·9) may be rewritten in the form

_ _ PevcT (Oy) (OX)cp cv--- - -Tl oz~azll'

(S·5)Sr

C -c =------p v (2 +3x) (y+3x2 )·

and near the critical point, where T may be put equal to ~, andPcvc=ir~,

'Y -+ (2+3x) (y+3x2)'

while the isothermal compressibility tends to the form ~ 3(2 +:X 2r c Y+ x)

Therefore k -k / ~ (2 3)2S- T Y-+24rPe + X ,

which remains finite when X = o.It is important to remember, however, that although the van der

Waals model illustrates qualitatively the thermodynamic beh8lviournear the critical point, it is very far from being quantitatively exact.The isotherms of a real substance in the vicinity of the critical pointtend to be much flatter than those predicted by van der Waals'sequation, as may be seen in fig. 26. It seems that not only (02Pjo V2)T'but (03P/OV3)T and (04P/OV4)T also vanish at the critical point, and itis likely that the behaviour cannot be adequately represented, as invan der Waals's and similar equations, by functions ,vhich are capableof expansion as a Taylor series involving values of the derivatives atthe critical point itself. But this is too complex a problem to beentered into here.

From this result it will be seen that if the pressure is maintained at thecritical value (y = 0) and the volume altered by changing the temperature, cP-cf , and hence cp, approach infinity as const./x2• Since, from(S·4), zocx3 wheny=O, we see that at the critical pressure cp approachesinfinity as const./(~-T)f. If the volume is maintained at Vc (x=O),cp approaches infinity as const./(Tc-T).

We may further use this model to demonstrate that although cp andkT become infinite at the critical point, the adiabatic compressibilityremains finite as asserted in Chapter 7. For, from (8·5), as the criticalpoint is approached 'Y (:=cp/cy) tends to infinity,

8r/cy


Solid-liquid equilibrium

Let us now examine the question whether the transition linebetween the solid and liquid phases (PtS in fig. 25) continues indefinitely, or whether it also terminates abruptly in a critical point. It maybe said at the outset that there are weighty theoretical argumentsagainst the possibility of a critical point. For ifsuch a point existed itwould be possible, by going around it, to make a continuous transitionfrom the liquid to the solid phase. Now the properties of normalliquids are strictly isotropic; they possess no crystalline structurewhich singles out anyone direction as different from another, whiletrue solids (excluding glasses and similar amorphous phases) possessnon-spherical symmetries which are characteristic of the regulararrangement of their molecules in a crystalline lattice. In order to gofrom the liquid to the crystalline phase, therefore, it is necessary tomake a change of the symmetry properties, and this is of necessitya discontinuous process. The symmetry properties of a lattice aredescribable in terms ofcertain geometrical operations, such as translation or reflexion, which displace every atom on to another identicalatom and so leave the lattice unaltered. A given phase either possessesor does not possess any given symmetry property, and thus no continuous transition is possible and no critical point can exist.

This is the theoretical argument, which has appeared to some to bea little too straightforward to be absolutely convincing. There is littledoubt, however, that the experimental evidence all points strongly tothe truth of its conclusion. Of this evidence the most complete is thatof Simon and his coworkers on solid and liquid helium. The choice ofhelium as a suitable substance for extensive investigation wasgoverned by the following considerations. According to the law ofcorre8ponding state8, the phase diagrams of most simple substances arevery similar in form and scale ifthey are plotted in reduced coordinates,P/~, V/Yc and T/Tc. There are, of course, differences in detail, but onthe whole the law is well obeyed. It is therefore desirable to studya substance for which the highest values ofPfPc andTfTc are attainable,and this is achieved by using helium, whose critical pressure andtemperature are 2·26 atmospheres and 5.20 K. respectively. Since themelting curve of helium can be followed up to pressures of severalthousand atmospheres, it follows that values of P/Pc ofmore than onethousand may be attained, far more than is possible with any othersubstance.

Apart from the lowest temperature region, in which helium behavesunlike any other substance (this will be discussed more fully in thenext section), the melting curve takes a form which closely resemblesthat ofmany other substances, in that the melting pressure varies with

PMse equilibrium 123

temperature according to the law P/a=(T/To)C-l, in which a and Toare constants for a given substance and c is an exponent which variessomewhat from one substance to another, but usually lies between1·5 and 2. The melting curve for helium is shown in fig. 30. There isthus no indication ofa critical point for the liquid-solid transition evenat temperatures eight times as high as that ofthe liquid-vapour criticalpoint. It is interesting to note that if the same law for the melting

8000

4020oT(°l{.)

Fig. 30. Melting curve of helium (F. A. Holland, J. A. W. Huggilland G. O. Jones, Proc. Roy. Soc. A, 207, 268, 1951).

pressure holds beyond the range of existing data, helium could besolidified at room temperature by application of a pressure of 110,000atmospheres, not so much higher than the highest pressures producedby Bridgman. There are, however, technical reasons limiting thepressure which can be applied to helium at a much lower value.

From the point ofview ofdetermining whether a liquid-solid criticalpoint exists, an even more instructive set of measurements is that ofthe entropies of the liquid and solid phases, which have been computed from specific heat measurements at high pressures. It is found


that the entropy difference between liquid and solid helium increasesslightly as the melting pressure is increased. Since at a critical point thetwo phases, being identical, must have the same entropy, the experiments make it highly probable that no critical point will be found.t

The phase diagram of helium

Naturally occurring helium consists almost entirely of 2He4, theisotope of mass 4 (two protons and two neutrons comprising thenucleus), with an admixture of one part in 105-106 of 2He3, the lightisotope of mass 3 (two protons and one neutron). Until recently littlewas known of the properties of the latter, but with the development ofmethods of separating the two isotopes and, still more, with theproduction on a comparatively large scale of2He3 in nuclearreactors,tit has become possible to carry out experiments on the thermodynamicand other properties ofthis very interesting substance, and even to useit as a liquid bath in cryostats. Since it behaves in a rather more simplemanner than 2He4 we shall discuss it first.

Both isotopes have the property, unlike any other substance, thatthey may remain liquid down to the lowest temperatures. The reasonfor this, which is essentially quantum-mechanical, need not concern ushere,§ since we shall limit our discussion to thermodynamical aspectsof the phase diagram. The diagram for 2He3 is shown in fig. 31, fromwhich it will be seen that there is no triple point.

The form of the melting curve is unique among liquid-solid transitions. At all temperatures the solid is the denser phase, but whereasabove 0·32° K. the-entropy of the liquid exceeds that of the solid, as isusual, below 0·32° K. the reverse is true. There is a range of temperatures and pressures within which the liquid is the low temperaturephase and the solid the high temperature phase. If a vessel containing2He3 is maintained at a pressure of 30 atmospheres and warmed upfrom a very low temperature, it starts as.a liquid, freezes at 0·18° K.,and melts once more at 0'49° K. It is worth noting in passing that solidiron behaves somewhat analogously, as may be seen in fig. 41; onheating it makes a transition from the a-modification to the y-modification, and at a higher temperature reverts to the a-modification.The melting curve of 2He3 has been studied down to 0'06° K.II and is

t For details of these experiments see J. S. Dugdale and F. E. Simon, Proc.Roy. Soc. A, 218, 291 (1953).

t The reaction aLie+ont~ sHe&+ tHa, creates tritium tHa, which decays byp-emission with a half-life of twelve years to 2Hea.

§ See W. H. Keesom, Helium (Elsevier, 1942), p. 332.II D. O. Edwards, J. L. Baum, D. F. Brewer, J. G. Daunt and A. S.

McWilliams, Proc. 7th Int. Can!. on Low Temperature Physic8 (Toronto, 1961),p.610.


still rising steadily. Ultimately it must become level, since accordingto the third law Ss-Sz must vanish at the absolute zero, but thetemperature at which this occurs may prove to be very low indeed.

The melting curve of 2He4 is more normal, though again there is notriple point, and a pressure of at least 25 atmospheres is needed toproduce the solid. Of particular interest in the phase diagram (fig. 32)is the line separating two different forms of the liquid phase.

40

30

10

o

Liquid

2

Vapour

3

Pc

Fig. 31. Phase diagram for sHe3 •

The higher-temperature form, He I, sho,,·s no markedly unusualproperties; but as the temperature is lowered a sharp transition to anew form, He II, occurs, which possesses such remarkable and uniqueproperties of high heat conductivity and low viscosity as have earnedit the title' superfluid'. Undoubtedly, the great fascination of heliumis due to its transport properties and hydrodynamic peculiarities, butits thermodynamic behaviour is not devoid of interest, since it istypical of a large class of transitions which hitherto \\ye have notconsidered.

126 Classical tkermodynamies

The specific heat of liquid 2He4 in contact with its vapour, that is,very nearly at constant pressure, varies with temperature in themanner shown in fig. 33. The sharp rise at 2·172° K. corresponds to thetransition from He IT to He I, and is called, on account of its shape, theA-point. The most careful measurements have been unable to determine with certainty how high the peak really is, and it is quiteprobable that an ideal experiment would reveal that it is virtuallyunbounded. If Op does in fact tend to infinity at the A-point it must

50

40Solid

i 30u.:Q..<I)

0

120~

Liquid He II Liquid He I

10

Pc

00 2 3

T (OK.)

Fig. 32. Phase diagram for sHe&.

do so more slowly than as (T).. -T)-l, since the energy change associ

ated with the transition,fCpdT, must remain finite. In fact the rise

is much slower than this, being represented rather closely by a function of the form A +B In ITA - TI on both sides of the transition, withA taking different values on the two sides;t the specific heat has beendetermined within a few microdegrees of T A' There is no latent heatassociated with this transition; if an isolated vessel of He II is suppliedwith heat at a constant rate, the rate of temperature rise steadilydecreases as T A is approached, and appears to become momentarilyzero at T A before increasing suddenly once more, but there is no

t M. J. Buckingham and W. M. Fairbank in Progress in Low TemperaturePhysics, ed. C. J. Gorter (Amsterdam: North Holland, 1961), Vol. III, p. 80.


halting of the temperature for a measurable time at T A as would occurif there were a latent heat. The entropy is thus a continuous functionof temperature, though there may be a discontinuity in (oS/oT)p. Itis easy to see that the volume is also continuous by applying M. 3 tothe liquid. In crossing the A-line (AB in fig. 32) at constant tempera-

4

........I

#i 2

bO

~tJi

01 2 3 4

Fig. 33. Specific heat of liquid 2He4 in contact with its vapour.

100'05

lOO·OO ......---......a.--.......-...L..----........-----'--2·14 2·16 TA, 2·18 2·20 2·22

TeK.)Fig. 34. Variation of volume of liquid tHet near the A-point. Note the

vertical tangent at TA•

ture the entropy is a continuous function of pressure, although(OS/OP)T may become infinite on the line; correspondingly the volumeis a continuous function of temperature, although (oV/oT)p maybecome infinite on the line, with the opposite sign to that of (OS/OP)T'This is illustrated in fig. 34. We have then to deal with a new type oftransition which in some respects resembles a smeared-out phase


transition-the latent heat is, as it were, absorbed over an interval oftemperature instead of at one fixed temperature. It also resembles,and perhaps more closely, the specific heat behaviour at the criticalpoint. Here, however, instead of a single point on the phase diagramat which Cp becomes infinite, we have a line ofsuch points. Clapeyron'sequation cannot be applied as it stands to relate the slope of this lineto discontinuities of entropy and volume at the transition, since thereis no discontinuity ofeither. We shall consider in the next chapter howto derive thermodynamic relationships analogous to Clapeyron'sequation for this type of transition and others.

The Gibbs function for the liquid phase exhibits singular behaviouralong the A-line, but of course in a less striking fashion than thespecific heat or entropy. IfCp rises to infinity, the corresponding curvefor 8 shows momentarily a vertical tangent, and the curve for g a pointat which the curvature becomes momentarily infinite. The surfaceg(P, T) is therefore not folded along the A-line, as it would be if therewere a latent heat at the transition, but shows, as it were, an 'incipientfold', the gradient changing rapidly but continuously. The intersectionof the g-surface for the liquid with that for the solid, which is quiteregular and free from kinks, should in principle reflect the singularityof the liquid in the shape of the liquid-solid equilibrium line. Just asat a triple point, where a regular surface meets a folded surface, thereis a sharp change in the gradient of the equilibrium line, so here, wherea regular surface meets an almost-folded surface, the equilibrium lineexhibits a region of rapidly changing gradient, with one point at whichthe curvature becomes infinite. The same behaviourshould occur at theliquid-vapour equilibrium line. Unfortunately, neither for the liquidsolid nor for the liquid-vapour transition are the experimental datasufficient to show this clearly. Certainly as far as the former is concerned the region of rapidly changing gradient on the melting-curveought to be readily apparent,t but it would be very hard to recognizeon the vapour-pressure curve. The reason for this is easily understood.The volume ofunit mass ofvapour is a rapidly varying function oftemperature, so that the curvature of the vapour-pressure curve is high,as follows from Clapeyron's equation, and the change in entropy ofthe liquid resulting from the specific heat maximum is small comparedwith the difference in entropy between the liquid and vapour phases.In consequence, the anomalous variations in curvature resulting fromthe existence of a A-transition are comparable with the curvaturealready present only within a few thousandths of a degree of theA-point, and give rise to no perceptible deviation in the general trendof the vapour-pressure curve.

t See C. A. Swenson, Phys. Rev. 79, 626 (1950), where the available data areclearly displayed.


In concluding this section the properties of mixtures of the twohelium isotopes deserve mention, since they illustrate the third lawmost strikingly. We have not entered into the subject of chemicalthermodynamics, and have thus neglected to introduce the conceptof 'entropy of mixing' . We shall therefore not go into details butmerely remark that the entropy of a mixture ofisotopes is greater thanthat of the separated isotopes, unless the isotopes can achieve anordered arrangement, analogous to the ordered state of an alloy shownin fig. 49. The third law demands that the isotopes either order themselves as zero temperature is approached, something that is hard toconceive in a liquid, or else that the two liquids become mutuallyinsoluble. For all that they are so similar it is in fact the latter thatoccurs. For instance, if a solution containing 3 2He3 atoms to 2 2He4atoms is cooled, phase separation begins just below 0·9° K. At firsta solution richer in 2He3 floats on a solution richer in 2He4, but withfurther cooling each phase expels its minority constituent so that aszero temperature is approached the system tends towards the idealstate of zero entropy, pure 2He3 floating on pure 2He4.

The investigation of liquefied helium isotopes, both pure and mixed,has revealed many new phenomena and contributed notably towardsthe understanding of the quantum physics of condensed systems, butthese are matters beyond our scope and the reader is referred tospecialized texts for further study.t

The superconducting phase transition

Certain metals, for example, lead, mercury, tin, aluminium, niobiumand tantalum, have the remarkable property of losing all trace ofelectrical resistance when cooled sufficiently. The disappearance ofresistance may take place gradually, over a temperature interval ofio to 1°, but it is found that if sufficient care is taken to ensurechemical and physical purity, by using single crystals of highly refinedmetal, the transition usually becomes very sudden. With a goodsample of tin, between the point at which the resistance is firstobserved to diminish and that at which it has entirely vanished, theremay lie a temperature interval of only one or two thousandths ofa degree. For tin the transition occurs at 3·73° K.; for other elementarymetals it varies between 9°K. for niobium and 0·14°K. for iridium.Metal-like compounds have been found which have transition temperatures as high as 18° K., but there seems little doubt that thephenomenon is one which only occurs at temperatures very far belowroom temperature. As with liquid helium, the chieffascination of thestudy of superconductors lies in properties which are not strictly

t K. R. Atkins, Liquid Helium (Cambridge, 1959).


thermodynamic, and which it is beyond our scope to discuss here.We shall only use superconductivity as an example of a phase transition which involves three independent variables, in this case P, T and.Tt', and because it exhibits under certain circumstances a type ofphase transition which is different from any we have encounteredhitherto·t

The most important single property of superconductors is not theirabsence of resistance but their perfect diamagnetism. A magneticfield is unable to enter a Buperconducting sample, being restrained

Fig. 35. Magnetic field around superconducting sphere.

from entry by shielding currents which flow in a very thin surfacelayer. The property ofperfect diamagnetism implies that the magneticfield at the surface of a superconductor is everywhere parallel to thesurface (a normal magnetic field cannot be prevented from enteringand is therefore prohibited by the property of perfect diamagnetism),and if the strength of the field is Jt7 outside and zero inside there mustbe a surface current density of £/411 (since curl e7l'=411J, J being thevolume density of current). A typical example of the magnetic fielddistribution around a superconductor is shown in fig. 35, where thesample is a sphere. If the mean direction of the field is regarded asdefining an axis of the sphere (vertical in fig. 35), the surfacecurrents flow along lines of latitude. Since a magnetic field may bemaintained indefinitely by means of a permanent magnet, withoutany external source ofpower, it follows that the screening currents are

t For a detailed account of superconductivity see D. Shoenberg, SupeJOconductivity (Cambridge University Press, 1952).

P1uue equilibrium 131

entirely non-dissipative, and in this case the property of perfectconductivity is an immediate corollary of the perfect diamagnetism.This is not so obviously true for multiply-connected superconductors,or superconducting wires fed by an external circuit, but it is ratherprobable that, although no rigorous deduction ofperfect conductivityfrom perfect diamagnetism has been given, it is nevertheless not farfrom physical reality to regard the latter as the primary property towhich the former owes its existence. The present discussion will beconfined to simply-connected bodies where this element of doubt isabsent. We shall also consider only massive superconductors forwhich the thickness ofthe surface layer carrying the screening currentsis negligibly small, so that the description of the body as perfectlydiamagnetic (f!A=O) is entirely adequate.

If a long cylindrical superconductor is placed in a magnetic fieldparallel to its length, the field does not enter until a certain criticalfield, ~, is reached. When the critical field is exceeded the metalreverts to its normal, resistive, state and the external field penetrates,so that within the metal f!A =;e. In reality the susceptibility of thenormal metal is not exactly zero, but it is so small that the errorinvolved in puttingf!A and £ equal is negligible in practice. There is nodifficulty in modifying the following arguments to include the normalsusceptibility. Ifthe field is slowly reduced from its value greater than~ the metal reverts to its perfectly diamagnetic state at the fieldstrength ~, with expulsion of all the magnetic flux from its interior,a striking phenomenon known, after its discoverers, as the MeissnerOchsenfeld effect.t Thus the magnetic transition between the superconducting and the normal states is reversible in the thermodynamicsense, and the curve (fig. 36) showing how~ depends on temperature

t Or often, with inequitable brevity, simply the Meissner effect. It should bepointed out that this description of the effect is an idealization on two counts.First, it is very commonly observed, particularly with pure metals, that thefield must be reduced below ~ before the superconducting state is re-established. This behaviour is analogous to the supercooling of vapours, and does notoccur if a nucleus of superconducting material is fonned either in regions ofimpurity or physical strain, or by artificial means. The phenomenon of supercooling is attributed, 88 in the vapour-liquid system, to the influence of a highinterphase surface energy inhibiting the formation of small nuclei from whichthe transition may proceed. A second idealization is that the magnetic flux isentirely expelled at the transition. In fact a little is always trapped within thespecimen, but in a well-prepared specimen it need not be more than -h % of theflux present before the transition started. This trapped flux is not uniformlydistributed over the specimen, but is confined to small channels which do notbecome superconducting. In the superconducting regions 11= o. Neither ofthese effects invalidates our assumption in what follows that the magneticbehaviour may be idealized 88 in the text without essentially falsifying thephysical picture.


is a transition curve between two phases in the same sense as the termwas used in discussing the phase diagram of simple substances.

An extra degree of freedom enters the discussion \\-Then it is realizedthat the transition temperature and the value of~ at a given temperature are affected, if only to a small extent, by pressure. If weinclude the influence of pressure we must regard the state of the metalas determined by three parameters of state, for which T, :Ye and Parethe most convenient choice, and the transition line now becomesextended into a transition surface in (T, ~P) space. Since the effectof pressure is small (the transition temperature of tin is changed from

:Itc

Superconducting

T Tc

Fig. 36. Variation with temperature of critical magneticfield of a superconductor.

(8·6)g':=u' -Ts+Pv-£m,

3·73 to 3·63° K. by a pressure of about 1700 atmospheres), sections ofthe transition surface at different attainable pressures do not differgreatly from the form of fig. 36.

Let us now consider the analogues of Clapeyron's equation for thisthree-parameter system. We shall confine our attention to a specimenofunit mass in the form ofa long cylinder, for which the demagnetizingfactor is zero. t It may easily be seen, by extension of the argumentsdeveloped in the last section, that the two phases are in equilibrium atsuch points that their 'magnetic Gibbs functions', g', are equal, g'being defined by the equation (see (3·13) for the significance of theprimes),

where m is the magnetic moment per unit mass. For the superconducting phase m,= -Vs:Ye/41T, since the volume susceptibility of

t See Shoenberg, loco cit., for a full discussion of shape effects, and a morecomplete account of the thermodynamic properties of superconductors.


a perfect diamagnetic is -1/417', while for the normal phase mn =0and 9~=g~. From (8·6), since du' =Tds-Pdv+£dm,

dg'= -sdT+vdP-mdJft: (8·7)so that

(09') (09') (89' )oT = -s, 8P =V and 8£ = -me (8·8)P,I T,JF T,P

The analogues of Clapeyron's equation now follow immediately by thesame argument as before. By considering a section of the transitionsurface at a constant pressure we have

(8·12)

(8·11)

(8·10)

(89~) (89;)(O~) = _ IYJ'. P, Jt' - aT, P, Jt' = _ 8 n -s. =_~ (Sn -s.).

8T P (8gn ) _ (898 ) mn-ms vs~o£ P,T oYe P,T (8·9)

(o~) 417'ap T = v.~ (vn-v.)

(OP) sn-ssoT .Jf'.c = vn-vs '

the last equation being identical with Clapeyron's. The values ofSn' ss' V n and Vs to be used in these equations are of course the valuestaken on the transition surface, although in fact their variation withmagnetic field is extremely small. Since, by the magnetic analogue ofM.3, (os/oYe)T,p=(om/oT).1f',p, we see that neither Sn nor S8 aresensibly field-dependent, the former because mn = 0 and the latterbecause m,= -v~/417', independent of temperature, except fora minute effect caused by thermal expansion. And from (8·7) itfollows that (ov/OJe)P,T= -(om/oP)JF,T' so that Vn is field-independent and V s also, apart from an equally minute effect which resultsfrom the finite bulk modulus of the superconductor. Thus, exceptperhaps for the term (vn-vs) which is a small difference of largequantities, it is precise enough in practice to use in (8·9), (8·10) and(8·11) the values ofssand vsin zero magnetic field, and to regard Sn andV n as the values which the normal state would possess if it could existin zero field.

From (8·9) it is seen that the change of entropy at the transition isgiven by the equation

vs£; (oYec)Sn-Ss= -417 oT p.

Fig. 36 shows that Sn - Ss vanishes at 0° K. in accordance with thethird law and also at the critical temperature where ~=O and

and

Similarly


(()~/()T)p is finite. The transition in zero field at ~ is thereforeaccomplished without any latent heat. This introduces us to a type oftransition different from either the simple phase transition or theA-transition, for here there is a finite discontinuity in specific heat, notinfinite as in the A-tr&llSition. This may be seen from (8·12), since

_ vsT ( ()2 1DJ 2)Cp" -CP.- - 811' aT2 (.ne ) p. (8·13)

Near Tcthe critical field is approximately proportional to Tc-T, sothat the second derivative of£~ is a well-defined finite quantity; since

0·008

Oa...- ~ ....... ____

432T (OK.)

Fig. 37. Specific heat ofnonnal and superconducting tin (W. H. Keesomand P. H. van Laer, PhyBica, 5, 193, 1938).

it is positive, cP, > CPn at the transition temperature. The specific heatof tin is shown in fig. 37 for both the normal and superconductingphases. From this experimental curve and (8·13) the variation of~with temperature may be calculated, and is found to agree well withwhat is measured directly.

Analogous results to (8·13) may be obtained by differentiating(8·10) with respect to pressure and temperature. In this operationonly a negligible error will result from taking VB in the denominatoras a constant and equal to V; the results of interest arise from the


variations of the smalldifference (vn - VB) caused bychangesof pressureand temperature_ Thus

and

1 (02 .re. 2) _1 {(OVn) (Ovs)}811 opt ( c) 2'-fj oP 2' - oP 2' = -(k..-k.)

1 (02 .re. 2) _1 {(OVn) (OVs)}_811 oPoT( c) -fj aT p - aT p -P,,-P.,

(8-14)

(8-15)

(8-18)

(8-17)

(S-16)

since

where k is written for the isothermal compressibility, and P for thevolume expansion coefficient_ The results (8-13)-(8-15) hold at allpoints on the transition surface_ When~= 0, i_e_ along the transitionline in zero field, they can be cast into simpler forms:

Cll-C~ = _ VTo(8.Tt:)1

,. , 411' aT p'

1 (8.re.)1k.. -k.= - 411 op

c'/

P R 1 (0.Tt:) (8.Tt:)"-11.= 417' oP 2' aT p'

from which it follows, by combining these equations in pairs, that

(aTe) "=vT. P..-P. = k.. -k. (8.19)oP Ic-O c CPn -Cp, Pn-P.'

eo:::tl(o;;t=- (~i)JF;The experimental verification of these results is not easy, for althoughthe change in specific heat is measurable with considerable accuracy,the expansion coefficients in both states are very small indeed, whileon account of the small value ofaPc/oP (itselfnot easily measured) thecompressibility changes by only a few parts in a million_ Such data ashave been obtained, however, agree satisfactorily with (8-19)_ Theseequations, which we have derived by considering the special case ofa superconductor, are in fact the analogues to Clapeyron's equationapplicable, as we shall see in the next chapter, to any transition inwhich there is no latent heat and a finite discontinuity in Op_

136

CHAPTER 9

HIGHER-ORDER TRANSITIONS

Classification of transitions

In the last chapter we noted three distinct types of thermalbehaviour occuning along lines separating different phases or modifications of a substance, the normal transition with latent heat, theA-transition without latent heat but a very high (perhaps infinite)peak of specific heat, and the so-called second-order transition inwhich there is no latent heat and a finite discontinuity in specific heat.The first and last are members of a classification introduced byEhrenfest, in which the 'order' of a transition is determined by thelowest order of differential coefficient of the Gibbs function whichshows a discontinuity on the transition line. Thus in a phase transitionwhich involves latent heat, g is continuous across the line, but itsderivatives (og/oT)p and (og/OP)T' -8 and v respectively, are discontinuous; such a transition is said to be of the first order. In the transition of a superconductor in zero magnetic field there is no latent heatand no volume change, so that the first derivatives ofg are continuous,but the second derivatives, representing specific heat, expansioncoefficient and compressibility are discontinuous, so that this isa transition of the second order. The classification may be extendedindefinitely, though as the order of the transition increases it becomesless and less clear that it is appropriate to think of the process asa change from one phase to another since the discontinuity in properties becomes progressively less significant. For instance, in a thirdorder transition the specific heat is a continuous function of temperature, only its gradient showing a sharp break, while in a fourth-ordertransition the Op-T curve is distinguished merely by possessing a discontinuity of curvature. Thus in practice it is only the first- andsecond-order transitions which usually arouse interest, and we shallmainly confine our discussion to these.

Now this classification ofEhrenfest's, while having served a valuablepurpose in pointing to a distinction between different types of transition, is of only limited application, since true second-order transitionsare exceedingly unusual. It is probably true to say that of physicallyinteresting systems (that is, excluding ad hoc models, of which weshall discuss one in detail) there is only one class, the superconductingtransition, which bears any resemblance to an ideal second-ordertransition. On the other hand, there are many transitions known to

Higher-order transiti0n8 137

occur in widely different varieties of substance which do not comfortably conform to Ehrenfest's scheme. The transition in liquidhelium is an example, and others will be mentioned later. A few conceivable specific heat curves are drawn diagrammatically in fig. 38,

2 2a

T

TFig. 38. Classification of transitions.

and arranged so as to present the variants of Ehrenfest's classification.Type 1 is the simple first-order transition, with latent heat. Types 2,2a, band c show no latent heat but a discontinuity in Op; in 2a aOp/aTtends to infinity as the transition is approached from one side, in 2bit tends to infinity from both sides; in 2c the discontinuity in Op isinfinite. The third-order transitions are analogous variants of thestandard Ehrenfest third-order transition (type 3).


Not all these types have been observed in practice, but thereappears to be no thermodynamical reason why they should not occur.A few examples of those which have been observed are given below,ttogether with some idealized theoretical models which have beencalculated exactly and found to exhibit interesting transitions. Thelatter are placed in square brackets.

0·2".......

IbOlU

"0

IbO

tU~

{..;)~

0·1

Lo 200 400 600

T CO C.)

Fig. 39. Specific heat of p-br88s.

(1) First-order transitions: Solid-liquid-vapour transitions. Manyallotropic transitions in solids, e.g. grey to white tin.

(2) Second-order transitions: Superconducting transition in zerofield (fig. 37) . [Weiss model offerromagnetism, t Bragg-Williams modelof the order-disorder transformation in ,B-brass.§]

(2c) A-transitions: Order-disorder transformation in ,B-brassll(fig. 39), ammonium salts,~ crystalline quartzll (fig. (0), solid hydrogentt (fig. (1) and many other solids.

t For an account of the phenomena of ferromagnetism, antiferromagnetismand order-disorder transitions, consult C. Kittel, I ntroduction to Solid Stat~

Physic8 (Wiley, 1953).t F. Seitz, Modern Theory oj Solida (McGraw-Hill, 1940), p. 608.§ Ibid. p. 507. II H. Moser, Phys. Z. 37,737 (1936).~ F. E. Simon, C. von Simson and M. Ruhemann, Z. PhY8. Chem. A, 129,

339 (1927).tt R. W. Hill and B. W. A. Ricketson, Phil. Mag. 45, 277 (1954).

Higher-order transitions 139

600400T (0 C.)

Fig. 40. Specific heat of crystalline quartz.

0·32

0·24~....... -a- --I _

200

0·36

",.....-IbOu~-IbO

!CJQ.. 0·28

8

r--,6I

#

0 ......----~----...60-----.--.---1·0 1·5 2·0 2·5T (OK.)

Fig. 41. Specific heat of solid hydrogen containing 74 % orthohydrogen,26 % parahydrogen.


(3) Third-order transitions: The Curie points of many ferromagneticst (fig. 42).

(3c) Symmetrical A-transitions: The antiferromagnetic transitionin MnBr2t (fig. 43). Liquid 2He4 (fig. 33). [Two-dimensional Isingmodel of the order-disorder transformation. §]

One might also include in category (3c) the critical point of liquidvapour systems, as discussed in the last chapter. This is rather differentfrom the rest, however, as the infinity in Cp occurs only at one point,and not along a transition line.

It is well to remember that the distinctions made between differenttypes of transition in this catalogue may prove to have been too nice.I t is very easy for the character of a transition to be falsified throughimperfections in the samples or experimental inaccuracy, and it is notunlikely that almost every transition which is not of first order wouldbe found to be of type 3c under ideal conditions. The only exception tothis statement is the superconducting transition, which obstinatelypersists in exhibiting a finite discontinuity in Cp in the most refinedexperiments.

Analo~ue8 of Clapeyron'8 equation

For all transitions except those of the first order, not only the Gibbsfunction but the entropy is continuous across the transition line, sincethere is no latent heat. It follows at once from Clapeyron's equation,since in general the slope ofthe transition line, dP/dT, is finite and nonzero, that the volume also is continuous, and Clapeyron's equa.tiondegenerates to the form dP/dT=O/O. But now the continuity of sand venables arguments similar to those applied to g in derivingClapeyron's equation to be applied to sand v. We shall first considera second-order transition of type 2 (Ehrenfest's second-order transition). For this s and v are continuous, but (os/oT)p is not on accountof the jump in Cp • We therefore write immediately,

dP (~t-·(~tdT = - (OS2) _ (OSI)

ap T 8P T

(OV2) _ (av1)

aT p aT p

(OV2) (8V1) ,

8P T - ap '1'

(9·1)or, from M. 3, dP 1 cP.-CPt fJ2-fJldT= vT fJ2-fJl = k2-k1'

in which, as usual, fJ is the volume expansion coefficient and k theisothermal compressibility. Equations (9·1), which are of courseidentical with (8·19), are Ehrenfest's equations for a second-order

t H. Moser, loc. cit.; J. B. Austin, Industr. Engng Chern. 24, 1225 (1932).t W. P. Hadley, private comm. § L. Onsager, Phys. Rev. 65,117 (1944).


transition. We discussed in the last chapter the degree of success withwhich they have been verified for the superconducting transition, theonly type of transition to which they may strictly be applied.

0·3

a-iron

/

400 800 1200T CO C.)

1600

Fig. 42. Specific heat of iron.

6·0

5·0

IJ!~ 4,0

1210I I

4o 2

tf~ 3·0

~{..;)Q..

2·0 III

II

1·0 "II

I,o ' I

Fig. 43. Specific heat of anhydrous MnBrl·


For Ehrenfest's third-order transition (type 3), not only g, s and vare continuous, but also cp , fJ and k, while the derivatives of the latterare discontinuous. There are in consequence three analogues ofClapeyron's equation for a third-order transition, which the readerwill easily verify to have the form

(OCP2) (OCP1) (OP2) (OPI) (OP2) (OPI)dP 1 aTp-aT p Mp-aT p aPT-aPT

dT vT (OP2) _ (OPI) (Ok2) _ (Ok l ) (Ok2) _ (Ok l )

oT p oT p oT p aT p oP T oP T

(9·2)

Relations between higher derivatives similarly exist for transitions ofhigher order. No experimental verification has been attempted forany transitions of higher than second order.

T TA,

Fig. 44. Variation of entropy with temperature at a "'-point.

Since in transitions of the types (2a) and (2b) the discontinuities incpare finite, even though ocp/oT may become infinite at the transition,Ehrenfest's equations for the second-order transition are applicable tothem. On the other hand, the infinities in Cp for types (2c) and (3c) andthe infinities in ocp/oT for types (3a) and (3b) convert equations like(9·1) and (9·2) into indeterminacies of the form 00/00. Although thiscircumstance rules out Ehrenfest's method of approach, it enables analternative approach to be made, which may be illustrated by consideration oftypes (2 c) and (3 c). Here the specific heat rises to infinity,so that as the transition is approached from below (os/oT)p and(o2s/oT2)p tend to infinity, and the variation of entropy with temperature, at constant pressure, takes the form shown diagrammatically infig. 44. At the transition temperature TA, the curve reaches a point of


(9·3)8=S",(P) +/(P-aT).

inflexion, A, with vertical tangent, and either continues withoutdiscontinuity of gradient, for a transition of type (3c), or breakssharply at the inflexion, for a transition oftype 2Co The value, s"', of theentropy at T", will normally be a smoothly varying function ofpressure,so that the surface s(T, P) will have a regular fold along the transitionline, and at any point near this fold (on either side for (3 c), but only onthe left-hand side for (2 c)) there will be a very great second derivativein a plane normal to the transition line, and only a much smaller one ina plane tangential to the transition line. We may therefore hope toapproximate to the shape of the entropy surface by treating it, overa short range of pressure, and in the vicinity of T"" as a cylindricalsurface:

In this approximation it is taken to be adequate to treat the line ofinflexion points A as straight; this is equivalent to the assumptionthat the second derivative of this line is negligible in comparison withthe second derivative of the function I which describes the shape ofthe curve in fig. 44. Clearly a is equal to (dP/dT)"" the slope of thetransition line. Then, since a is taken as constant and s",(P) tocontain no quadratic or higher terms, (9·3) yields the results

(:;:L=a~r, (a;~p) = -ar and (:;:L=r, (9'4)

in which I" is the second derivative ofI with respect to its argument.Hence

(:;)A=a= - (:~)pi(a;~p) = - (a~;p)I(:~)T'

or, by use ofM.3,

a~(:;L=aa~(:;L and a~(:;L=aa~(:;L· (9'5)

The results (9·5) imply that in the vicinity of the transition line,(osjoT)p is a linear function of (ovjoT)p, or that

cp = avT,\fJ+const. (9·6)

It follows from (9 0 6) that as cp tends to infinity, so does fJ, and therefore that the surface representing v(T, P) also has a fold like that fors(T, P). The same argument may then be applied to v(T, P), to yield

the equation fJ k t (9·7)=a +cons .

Equations (9·6) and (9 0 7) are based upon a cylindrical approximationto the form of the entropy and volume surfaces, and may not apply at


any great distance from T).; they should, however, become increasinglymore exact as the A-point is approached.

It will be noted that there is a marked similarity between theseequations and Ehrenfest's equations (9·1), and indeed Ehrenfest'sequations are readily derived from (9·6) and (9·7). At a transition oftype 2 the entropy surface has a sharp fold where the gradient changesabruptly. Ifthis fold is smoothed out into a cylinder ofhigh curvature,

60

30

o,242·4°K.

o\~ I

" 10 15 20237·0° K. Expansion coefficient P(x 104, deg. -1)

Fig. 45. Cp v. Pfor NHtCl in vicinity of A-pOint. The temperaturesinvolved are shown along the line.

(9-6) holds on this cylinder, so that between one side ofthe cylinder andthe other ~Op=avT). ~fJ; when the radius of curvature of the cylinderis allowed to tend to zero, this result becomes identical with one ofEhrenfest's equations. The other is similarly derived from (9·7).

There is very little experimental material available to provide a testof these equations, but the A-transition in ammonium chloride hasbeen sufficiently studied to enable (9·6) to be applied to it. In fig. 45values of Op and p measured at the same temperature are plottedagainst one another, and it is clear that the linear relation predictedfrom (9·6) holds over an interval of several degrees below TA (the data


above T).. are insufficiently reliable to be useful). From the slope of theline ex may be found, and it is predicted that the A-temperature shouldbe raised by 10 by applying a pressure of 113 atmospheres. Fig. 46shows the only measurements of T).. as a function of pressure. In viewof the scanty data the agreement of predicted and measured slopesis satisfactory. Much more information is available concerning theA-transition in liquid helium, but it turns out that the slope deducedfor ex when Op is plotted against {J, or fJ against k, may be incorrect tothe extent of a factor of nearly two if the temperature scales used in

".......NI

8000

o240

<;>I,

II

II

II

II

/L.

300

Fig. 46. Variation of A-point of NHaCI with pressure.The line is calculated from (9·6).

the determination of these quantities (by different groups of workers)differ from one another by as little as 0.001 0 K. As an error of thismagnitude is only too likely in practice, we cannot regard liquidhelium as providing a satisfactory test. However, the validity of (9·6)and (9·7) is not open to serious doubt in the vicinity of a A-point, andit is possible that their chief application lies in testing critically theexperimental data for a transition such as that of helium, and ineliminating relative errors in the temperature scales of differentworkers·t

We have seen earlier that, although cp may become infinite, Cv must

t For further discussion, see the article by Buckingham and Fairbankreferred to on p. 126 (footnote). For the application of (9·6) and (9·7) to thea.-p transition in quartz, see A. J. Hughes and A. W. Lawson, J. Ohem. Phys.36, 2098 (1962).

146 OlaBBical thermodynamiCll

remain finite (except under the most unusual circumstances), and thisanalysis of the A-transition provides a convenient illustration.Writing (9·6) and (9-7) in the form

cp-co=avT)..p,

p-po=ak.

and making use of (6·10), we find that near the A-point

( Po)-lcp-cv=(cp-co) 1- P

= (cp-co) (I +~+~+ ...) I

or CJ1'=(Co-IXVTAPO)-IXVTA,BO(~+".)' (9·8)

Now since the area under the curve of cp must be finite, cp , and hencep, must go to infinity less rapidly than (T).. - T)-l, so that liP mustgo to zero with a vertical tangent at T).. _This means that as the A-pointis approached Cy rises (if a is positive) with a vertical tangent to thefinite value given by the first bracket in (9·8). The contrast betweenCp and Cy is thus very marked, and indeed in ammonium chloride ananalysis of the data leads to no evidence of any significant change incy as T).. is approached. This result has importance in connexion withattempts to analyse in detail the mechanisms responsible for A-points_In most calculations by statistical methods it is much easier to handlea system maintained at constant volume than one at constantpressure.But it must always be remembered that in the former the symptomsof a A-transition may be much less pronounced than in the latter, andit may even be possible to lose the transition altogether by employingapproximate means of calculation on a constant-volume system.

Critique of the theory of hi~her-ordertransitions

After Ehrenfest's thermodynamical treatment of second-ordertransitions was published it became a target for several critical attackswhich, taken at face value, appeared to be not without substance.Following these attacks the literature on the subject has becomerather unnecessarily confused, and we shall attempt by the followingdiscussion to remove some of the resulting obscurity. A completediscussion can bemade moreeasilywith the aid ofstatistical mechanics,but we shall avoid this elaboration by analysing artificial modelswhich may be regarded as analogues of the real physical systems ofinterest.


First let us look at the criticisms of the theory, of which the two ofimportance are essentially concerned with pointing out a contrastbetween transitions of the first and second orders. Fig. 47 shows howthe Gibbs function varies with temperature for the two types oftransition. For the first-order transition the curves for g in the twophases (at a given pressure) cut each other at the equilibrium temperature, so that phase 1 is stable below and phase 2 above the equilibrium temperature. In the second-order transition, since there is noentropy difference between the two phases at the equilibrium temperature, but a difference in specific heat, the two curves osculate, andthe difference in curvature at the point of contact ensures that there isno cross-over (in a third-order transition there would be a three-point

g

2

2

t

2

2

T T(a) 1st order (b) 2nd order

Fig. 47. Suggested variation of the Gibbs function in firstand second-order transitions.

contact at the equilibrium temperature and the curves would cross).It may now be pointed out in criticism of the conception of a secondorder transition that

(1) whereas two lines in general cross each other at some point, thechance of their meeting so as to osculate exactly is so small as never tobe observed in the world of physical phenomena;

(2) if the two lines osculate without crossing, the line correspondingto phase 1 remains below that for phase 2 both above and belowthe equilibrium temperature, and phase 1 is therefore stable at alltemperatures.

On both these counts the critics of Ehrenfest's theory have claimedthat the second-order transitions cannot occur in nature, in spite ofthefact that the superconducting tra.nsition appears to provide convincingrefutation of their view.

It is now convenient to analyse in some detail a simple physicalsystem which simulates a second-order transition, in order to see how


Vapour

Pressure=P

Pressure=Pi

the criticisms can be explained away. This model of a second-ordertransition wassuggested by Gorter, who did notgive a detailed analysis,no doubt regarding its behaviour as obvious to the instructed imagination. As however even this model has been denounced as imperfect itwill be as well, and in addition it will provide an instructive exampleof thermodynamical reasoning, to show how it leads to Ehrenfest'sequations. Gorter's model is depicted in fig. 48. A vessel containsa small amount of liquid, the remaining spacebeing filled with its vapour; the walls of thevessel have negligible thermal capacity and arenot quite inextensible, so that the volume of thevessel is altered by a variation of the difference,n, between the external pressure P and internalpressure ~. The vapour is assumed to behaveas a perfect gas, having Op and Ov independentof pressure and temperature. As the vessel iswarmed, P being constant, liquid evaporatesinto the vapour phase so as to keep ~ equal tothe vapour pressure, and the effective thermalcapacity of the vessel includes not only the Fig.48. Gorter's modelthermal capacities of liquid and vapour but of a second-order tran-also the latent heat of evaporation. At a sition.certain temperature, To, the last drop of liquidevaporates (the amount of liquid is adjusted so that this occurs belowthe critical point), and thereafter there is no latent heat contributionto the effective thermal capacity, which thus shows a sharp drop, asin a second-order transition. At the same time the internal pressure~, which up to To had been the vapour pressure, now begins to increasein proportion to T, following the perfect gas law. There is thus a discontinuity in (on/oT)p, and hence in (oV/oT)p, just as in a secondorder transition. It will also be seen that the compressibility showsa discontinuity at ~.

We now analyse the model in detail (we shall leave the reader toverify some of the intermediate steps of the calculation). When P=O,let To be the' transition' temperature at which the last drop of liquiddisappears, and correspondingly let the internal pressure be ~ andthe volume of the vessel Vo. Let the elasticityof the vessel be such that(dV/dn) = - a. First we calculate how the transition temperaturedepends on P. When P is increased from zero to oP, let the transitiontemperature change from To to To +oT, and the internal pressure at thetransition temperature from Po to ~+o~. Then, at the transitiontemperature ~ is equal to the vapour pressure, and from Clapeyron'sequation,

o~=AoT,

Awhere

Higher-order transitions

l

149

Corresponding to the changes OP and o~, V changes by oV, where

o¥=a(o~-oP)=a(AoT-oP).

Now at the transition all the material is in the vapour phase, to whichthe perfect gas law applies, so that

or

Thus the slope dP/dT of the transition line is given by the expression

a=dP/dT = A(Vo+~)-R.a 0

(9·9)

(9·11)

To find the expallsion coefficient {J, we note that expansion iscaused (P being constant) by the i~crease of ~ with temperature.Below To, by Clapeyron's equation dPi/dT is simply A; above To, bythe gas law, dPi/dT is R/(Vo+aPo). Hence the discontinuity in fl isgiven by the expression

Afl- a R-A(Vo+aPo)L.l - -- - (9·10)

-Yo ~+a~·

A similar analysis of the variation of V caused by changing P leads tothe discontinuity in isothermal compressibility,

a2P'8k= - ---- ---~-- .

Vo(Vo+aPo)

From (9·9), (9·10) and (9·11) it will be seen that one of Ehrenfest'sequations is satisfied, 8{J/8k=a.

To verify the other of Ehrenfest's equations we must calculate thediscontinuity in specific heat, 8Cp , and for this purpose it is convenientto write down expressions for the entropy of the material in the vessel.Let there be unit mass of material altogether, and let Sz and 8 v be theentropy if all the material were in the liquid and vapour phaserespectively at temperature To and pressure~.Then To(sv -Sl) = l. Ata ternperature To - oT the entropy ofthe liquid at its saturation vapourpressure would be sz-cpzoTjTo, while that of the vapour would be

150 Classical thermodynamiC8

8 11 - (Cp" -AVo) aT/To.t If then at To-aT an amount aq of liquid ispresent, the entropy of the system takes the form

l cR -AVo8=811 - T. aq- v T. aT.

o 0

Since it follows from the gas law and Clapeyron's equation that

o =A~ A (aPo+Vo)-R aTq 0 Rl '

therefore

{A2VO }aT

S=8~- R (aPo+ Yo)-2AVo+cp~ To'

and the effective thermal capacity just below To,

A2~

0PI = R 0 (aPo+ Vo)-2A~+cPt1·

The thermal capacity just above To is easily seen to take the form

RVoOP2=CPv--P,TT·

a 0+"'0

Hence !i.0 =0 -0 = To{A(a~+Vo)-R}2 (9'12)P - PI P2 Po a~+Vo ·

From (9·9), (9·10) and (9·12) it follows that ~Op/~fJ=~ Tocx, in agreement with Ehrenfest's equation.

This model serves to show that there is no intrinsic violation ofthermodynamic principles involved in the existence of a second-ordertransition, and we may now examine the criticisms mentioned abovein the light ofthe model. It is immediately clear that they are based onan erroneous analogy with first-order transitions. To take the secondcriticism first, at temperatures below that at which the lines 1 and 2(fig. 47 b) meet, the stable phase 1inGorter's model is that inwhich thevessel contains both liquid and vapour, while phase 2 corresponds tosupercool~dvapour only. As the transition temperature is approached,the amount of liquid steadily diminishes until at the point of contactof the lines it finally vanishes. The continuation of line 1 above this

t ds~=(~i)pdT+(~~)TdP

cp~ (Ovv)=pdT- aT pdP from M.3

=cp~-AtJ dT from the gas law and Clapeyron's equation.T


temperature is physically absurd-it could only mean a state in whichthe vessel contains a negative amount of liquid. Thus the true diagramshould rather consist of a single line at temperatures above the transition, which breaks into two branches below the transition temperature.By the same token the first criticism is also disposed of. The two linesdo not correspond to totally different phases, such as liquid and vapour,which can be considered independently of one another so that thephase transition results from an almost fortuitous meeting of the lines;rather, the two phases become more and more similar in constitutionas the transition temperature is approached, and at that temperaturethey are actually identical.

We may see from this analysis what is the characteristic differencebetween first-order transitions and transitions of the second andhigher orders. The two phases which are in equilibrium at a first-ordertransition are different in their physical constitution, greatly differentwhen they are solid and vapour, not so greatly when they are solid andliquid, but always sufficiently different that they possess differentenergies, entropies and volumes. In a second-order transition, on theother hand, the two phases (and at this stage we may begin to doubtthe wisdom of using this terminology to describe the two states oneither side of the transition line) are identical in constitution, energy,entropy and volume. Where they differ is in the rates of variationwith respect to temperature, pressure, etc., of these primary thermodynamical parameters. In fact whereas a first-order transition marksthe point at which a major change in properties occurs, a second-ordertransition only marks the point at which a change begins to occur.Thus in Gorter's model the jump in specific heat (as the temperature isdecreased) occurs when the liquid phase begins to condense. Similarlythe transition in p-brass, according to the simplified description ofBragg and Williams, marks the onset of an ordering process, not theestablishment of an ordered state. Above the transition temperaturecopper and zinc atoms, present in equal numbers, are arranged atrandom on a body-centred cubic lattice; at very low temperatures thetwo types of atom take up a completely ordered arrangement onalternate lattice sites, as shown in fig. 49. At the transition temperature itself the arrangement is still quite random, in the sense that ifwecount the number of copper and zinc atoms on one set of sites (thosedenoted by open circles for example) we shall find them to be equal.But as the temperature is lowered the tendency begins to show for thecopper atoms to favour one set of sites and the zinc atoms the other,and this tendency becomes more pronounced as the temperature fallsuntil at 0° K. there is complete order. It should be emphasized thatthis is a simplified theoretical model which leads, when worked out indetail, to a second-order transition of the Ehrenfest type, while the


reality conforms more nearly to a transition of type 2 c. This does not,however, detract from its value as an illustration of the way in whicha second-order transition marks the beginning, not the achievement,of a change in character of a physical system.

We have seen how the critics ofhigher-order transitions were misledby a false analogy with first-order transitions, so that they implicitlyassumed the existence of two g-surfaces, one for each' phase' and eachcapable, as it were, of existing independently of the other. The justification for this assumption in the theory of first-order transitions, aswas discussed in Chapter 8, essentially lies in the possibility of establishing superheated and supercooled states, and thus ofdemonstrating

Fig. 49. Ordered structure of P-brass.

the actual intersection of the surfaces. Let us conclude our examination of higher-order transitions by inquiring into the extent to whichany analogues ofsuperheating and supercooling can be expected in thevicinity of the transition line. We may deduce immediately fromGorter's model and the ensuing discussion that superheating isinconceivable, since there is no continuation above the transition lineof the surface corresponding to that phase which is stable at lowertemperatures. We might also conclude that supercooling is a theoretical possibility, since the vapour in the vessel may remain in a supercooled, uncondensed state below To, but this conclusion will turn outto be misleading, since it is in this one, almost trivial, respect thatGorter's system is an unsound model of a real second-order transition.

The possibility of superheating and supercooling at a first-ordertransition is due to the essential difference in the constitution of thetwo phases concerned. Just as the volume contributions to their

Higher..order transitionB 153

energies and entropies are different, so in general we must expect thesurface contributions to differ also, so that there will be a surfacetension at a boundary between the phases. This surface tensionimposes what may be thought of, in mechanical terms, as a potentialbarrier between the two phases. If we imagine the condensation ofa slightly supercooled vapour to proceed by the formation and subsequent growth of a drop of liquid, we may represent the availability ofthe whole system as a function of the radius of the drop, making thecmde approximation that the contribution of the drop is simply its

Fig. 50. Availability of vapour containing one liquid drop of radius ,..

mass times the availability per unit mass of liquid, plus the surfacetension times the area of the drop. If the vapour is supercooled,g" > Uz, and the dependence of the availability on the radius of the dropis expressed by the equation

A (r) - A (0) = 41Tr2u -t1Tr3pz(g" - gz),

in which A(r) is the availability of the system containing a drop ofradius r, A (0) is the availability ofthe system before the drop is formed,u is the surface tension and pz the density of the liquid. As on p. 110,U" and Uzmust both be calculated at the pressure and temperature ofthe vapour. This expression is shown in fig. 50, from which it will beseen that although the availability is lowered by the production of& large drop, the early stages offormation ofa drop involve a raising ofA, which is precluded by the arguments ofChapter 7. This is in essencethe reason why supercooling is possible, although the argument given


here is too superficial to be entirely correct; taken at its face value itwould imply that however great the supercooling there could never beany condensation to the liquid phase, except by the introduction ofanextraneous impurity on which condensation could readily occur.A more sophisticated approach to the problem, by fluctuation theoryor by treating the early stages of formation of the drop as a quasichemical reaction between gas molecules,t shows that there willalways be present a very small number ofdroplets, and that condensation may in principle always occur when the vapour is supercooled, bythe growth of these droplets. The supercooled vapour is thus not'metastable in the strict mechanical sense that it corresponds toa local, rather than an absolute, minimum of A; it is rather to beregarded as slowly transforming itself into the liquid phase, but at soslow a rate as to be inappreciable. The rate may be calculated, and it isfound that water vapour at room temperature and at a pressure whichis twice the equilibrium vapour pressure should condense, withoutexternal aid, in about 1()40 years. As the pressure is increased the timefor condensation diminishes, until at six times the vapour pressure thetime is calculated to be only a few seconds, and this accords fairly wellwith careful observations on pure water vapour. It is clear that to thepurist the supercooled state is not acceptable as an equilibrium state.Nevertheless if the supersaturation is not too great, the contributionto the Gibbs function of the system by the small number of dropletspresent after a reasonable time is quite immeasurably small, so thatthe g-surface apparently runs smoothly across the equilibrium linu.Under these circumstances, as discussed in Chapter 2, we can affordfor thermodynamical purposes to treat the supercooled system as if itwere genuinely metastable.

In Gorter's model the possibility ofsupercooling at the second-ordertransition arises from the possibility ofsupercooling ofa vapour which,we have seen, is the consequence of a non-zero surface tension. But ifwe take account of surface tension in the model the transition ceasesto be strictly of the second order. We may imagine the vessel justbelow the transition temperature to contain one droplet suspended inthe vapour. On account ofsurface tension the equilibrium pressure inthe vessel will be modified in accordance with Kelvin's vapour-pressurerelation, (7·16), and will be higher than the vapour pressure overa plane surface. Now although (see p.III) a drop surrounded byvapour maintained at constant pressure is at best in unstable equilibrium, if the vapour is confined to a fixed volume the equilibrium isstable for drops greater than a certain critical size; this is because theact of evaporating a little liquid from the drop increases the pressureof the vapour at a greater rate than is required by (7·16). But below

t J. Frenkel, Kinetic Theory oj Liquids (Oxford, 1946), p. 382.


this critical size the drop is unstable and collapses suddenly. Thus inGorter's model the introduction of surface tension implies that theamount of liquid does not fall smoothly to zero, and in fact the transition becomes one of the first order, albeit with a very minute latentheat. If we wish to simulate precisely a second-order transition wemust put the surface tension equal to zero, and then we automaticallyeliminate the prime cause of supercooling in the vapour.

By this analysis we have removed any justification for taking theGorter model as favouring the possibility of supercooling in a secondorder transition, and indeed there is no reason to expect supercoolingto occur. For, in contrast to a first-order transition, no meaning isattachable to the idea of a phase-boundary at a second-order transition, since the two' phases' are identical along the transition line.t Nodiscontinuity of properties occurs in crossing the transition line, andthere is therefore nothing to inhibit the transition. In fact no supercooling has even been observed at any transition of higher than firstorder.t The behaviour of some superconductors exemplifies clearlythe contrast between first- and second-order transitions. In a magneticfield the transition is ofthe first order and one may expect superheatingand supercooling to be observed. Both phenomena have in fact beenobserved, and the latter is particularly striking in aluminium, whichmay when very pure be maintained in the normal (non-superconducting) state in a magnetic field whose strength is only one-twentiethof the critical field,~. It has never, however, been found possible toreduce the field to zero at any temperature below ~ without thesuperconducting state being established. Diagrammatically the behaviour may be represented as in fig. 51; the central curve shows ~,the field at which g~=g;, while the upper and lower curves representthe limits of superheating and supercooling respectively. At anynon-zero value of :Ye both supercooling and, to a lesser extent, superheating are possible, but when ;TtJ = 0 the transition proceeds withouthysteresis at the transition temperature ~. At temperatures below~,with;TtJ= 0, the normal state cannot exist in any sort ofequilibrium,stable or metastable.

We must therefore conclude that from a thermodynamical point of

t This is true only for a homogeneous sample. In an inhomogeneous so.mplo,88 for instance a very tall cylinder in which, on account of gravity, the pressurevaries with height, it is in principle possible to establish a phaso boundary inequilibrium, and the stable position of the phase boundary will alter 8S thetemperature is changed.

t It is often observed with alloys such as p-brass, which exhibit the orderdisorder phenomenon, that the disordered state may be maintained at temperatures well below the transition temperature by sufficiently rapid cooling.The reason why this should not be regarded as analogous to supercooling willbe explained on pp. 157-8.


view (and this conclusion is strengthened by a detailed analysis of realphysical systems by statistical means) we are not justified in drawinga diagram like fig. 47 b to represent a second-order transition. Nophysical meaning can in general be attached either to the line representing phase 1 above ~ or to the line representing phase 2 below ~.

It is far better to regard the g-surface as a single-valued function ofP and T, which shows a discontinuity in its second derivatives alonga certain line, the transition line. This provides the simplest, andphysically most exact, picture of a second-order transition, and leads

T

Fig. 51. Superheating and supercooling of a superconductor.

immediately to Ehrenfest's equations without admitting any validityto the criticisms which were based on analogy with a first-ordertransition. In no useful sense can the second-order transition beregarded as the limiting form of a first-order transition in which thelatent heat has been allowed to tend to zero.

This conclusion is just as valid for other higher-order transitions, asmay be illustrated by two simple examples. The perfect Bose-Einsteingas shows a third-order transition at its point of condensation (fig. 42).The theory of this phenomenon allows one and only one state of thegas for any given values of pressure and temperature, and reveals howthe g-surface possesses a line of discontinuities in the third differentialcoefficients. There is no question here ofany conceivable continuationof either branch of the surface across the transition line. As a secondexample let us consider a A-transition such as that of ,B-brass. Thetheory of Bragg and Williams, mentioned above, led to a second-ordertransition at the point where the ordering process began as the temperature was lowered, but this is not true to the facts (fig. 41). In particular the rounded portion ofthe curve for Op just above T).. is evidence,as more refined theories show, that order is not completely destroyed


r-ww(R):=-.

r+w

at T)., but that there is a persistence of' local order' to higher temperatures, in the sense that there remain in the equilibrium state smallgroups of atoms in which there is a marked tendency for alternation ofcopper and zinc atoms. If we choose any copper atom at random weshall find on the average that its nearest neighbours are rather morelikely to be zinc that copper atoms, and the next nearest neighboursrather more likely to be copper than zinc. As we move away from thechosen atom, however, we shall find that this discrimination betweenlattice sites becomes less and less marked. It is possible to define therange of local order along the follo,ving lines. Starting from a givencopper atom we may label alternate sites as 0 (copper) or Z (zinc) sitesaccording to their occupants in a perfectly ordered arrangement. Then,selecting those sites which lie at a distance R from the chosen centralcopper atom we may count the number, r, of copper atoms on O-sitesand zinc atoms on Z-sites, and the number, w, of copper atoms onZ-sites and zinc atoms on O-sites, and define the degree of order w(R)by the equation

In a completely ordered lattice w= 1, since w=O; in a completelydisordered lattice w= 0, since there is random occupation of the sites,and r=w. Let us now plot was a function ofR; we may expect to find,after repeating this procedure many times for different central atoms,an average diagram such as fig. 52 and Ro, the width of the plot (toa value of R at which w(Ro)= !w(O), say) is a measure of the range oflocal order. As the temperature is lowered towards T).., both w(O) andR o increase until at T). the range of order beconles infinite; w(R) doesnot tend to zero as R tends to infinity. Further lowering increases bothw(O) and the value to which w(R) tends as R tends to infinity, which isa measure of the degree of long-range order. This picture of the transition fixes the transition point as the temperature at which long-rangeorder first appears on cooling. But it will be seen that now there is nosharp distinction between disordered and partially ordered states;there is no definable state at temperatures above T).. which can beextrapolated into metastable existence below T).. without being initself a state of long-range order.

In the preceding description of the ordering process it has been assumed that at every temperature there is an equilibrium state of order(either of short or long range), and that any experiment is performedsufficiently slowly that at each temperature the equilibrium state isestablished. It may now be argued that of necessity this precludes theoccurrence of supercooling, just as a sufficiently slow cooling of avapour through its condensation temperature would avoid the establishment of the supersaturated vapour phase. If the alloy is cooled


rapidly from a temperature well above T). to one well below T). it mayretain its disordered state virtually indefinitely on account of theveryslow rate of migration of the atoms at low temperatures. There is,however, an important distinction to be made between these two cases.On the one hand a vapour just above its condensation temperaturereaches equilibrium in a very short time, and on cooling to the supersaturated state just below the condensation temperature still retainsthis property of reaching what is apparently equilibrium extremelyrapidly. As we have said before, the state is not strictly one of equilibrium since over"), very long period condensation will occur. Stillthere is a sufficiently large margin between the short time for apparentequilibrium of the vapour (10-8 sec., say) and the long time of conden-

R

Fig. 52. Illustrating degree and range of local order.

sation (perhaps> 10100 years for small supercooling of a pure vapour)that a meaning may be attached to the concept ofa supercooled vapourin equilibrium. On the other hand the time taken for an alloy justabove TA to reach its equilibrium state of partial short-range order isnot significantly different from the time taken to establish long-rangeorder just below T).., since the same migration mechanism is involvedin both cases. When a disordered alloy is quenched by rapid cooling toa temperature well belowT)., it is frozen into a particular configurationwhich is not analogous to the state of a supercooled vapour. For thestate of the vapour is thermodynamically well defined, in that it doesnot depend on how fast the vapour was cooled or from what temperature, whereas the state of the quenched alloy shows more or less orderaccording to the rate of cooling and the initial temperature. The onlyway in which the state of the whole alloy below T).. may be madeindependent of the manner in which it was achieved is by allowingsufficient time for migration to occur, and this necessarily involves, aspointed out above, the production of the long-range ordered state, as


being the only well·defined state at temperatures below TA- Theexistence of the quenching phenomenon does not therefore invalidatethe view that true supercooling is never observed, and is indeedunobservable, in any but first.order transitions.

This last example has taken our reasoning well beyond the confinesof classical thermodynamics, but the conclusion we arrive at is notdependent on the illustrations given of its application. The relationswhich may be derived for the thermodynamic behaviour in the vicinityof a transition line, of first or higher orders, in no case depend on theassumption that such a line marks the intersection or meeting ofdifferent surfaces; only for first.order transitions can such an assump·tion lead to a clarification rather than an actual misinterpretation ofthe physical situation.

160

EXERCISES

I. According to some textbooks, a knowledge of the Joule-Kelvincoefficient of a gas, and its specific heat at constant pressure, asfunctions of temperature and pressure is sufficient to enable theequation of state to be determined. Show that this is not so, and thatfurther information, e.g. the form of one isotherm, is required.

2. A saturated vapour is expanded adiabatically; if L is the latentheat of vaporization, show that it becomes supersaturated or un-

saturated according as whether - T d~ (~) is greater or less than the

specific heat of the liquid under its own vapour pressure.

3. A simple liquefier is constructed so that compressed gas entersat room temperature To and a high pressure P, and passes througha heat exchanger to a throttle, where it is expanded to a low pressure;part condenses and the rest returns through the heat exchanger,leaving the liquefier at room temperature and pressure. Show thatthe fraction of gas liquefied is greatest when the pressure of the gasentering is adjusted so that (P, To) is a point on the inversion curveof the gas.

4. A Simon helium liquefier consists essentially of a vessel intowhich helium gas is compressed to a high pressure P at lOoK. (abovethe critical point ofhelium). The vessel is then thermally isolated, andthe gas is allowed to escape slowly through a capillary tube until thepressure within the vessel is I atmosphere, and the temperature 4.20 K.,the normal boiling-point of helium. Assuming that the thermal isolation is perfect, that the heat capacity of the vessel is negligible incomparison with that of the gas, and that the gas obeys the perfect gaslaw, calculate what value of P must be chosen for the vessel tobe entirely filled with liquid. [Latent heat of liquid helium at4.20 K. = 20 cal. mole-I; Or of gaseous helium = 3 cal. mole-I deg.-1.]

5. Show that in a small Joule expansion of a fluid having negativeexpansion coefficient the pressure changes more than in the corresponding adiabatic expansion, the ratio of the two changes beingI-PVP/Op.

6. In a certain compressor gas at room temperature To andatmospheric pressure Po is compressed adiabatically, and is thenpassed through water-cooled tubes until eventually it emerges atpressure PI and temperature To. Find an expression for the workrequired for this process, compared with what would be needed for areversible isothermal compression leading to the same result, and

Exercises 161

show that the ratio is not less than unity. Discuss also the changesof entropy occurring in the two processes.

7. An ellipsoid made of a magnetically isotropic substance is free torotate about a vertical axis in a uniform horizontal magnetic field,with two unequal axes horizontal. The susceptibility is independentof field strength.

(a) Show both by direct calculation of couples and by the thermodynamic conditions for equilibrium that the ellipsoid tends to setitself with the longer horizontal axis (Le. that having the smallerdemagnetizing coefficient) parallel to the field, whether the substanceis paramagnetic or diamagnetic.

(b) If the ellipsoid is thermally isolated, and constructed of a paramagnetic material which obeys Curie's law, derive an expression forthe variation of its temperature when it is rotated in a constant field,in terms of the demagnetizing coefficients along the principal axes.For the sake of simplicity assume that the susceptibility is muchsmaller than unity.

8. For a system consisting of two phases of a substance in equilibrium, the specific heat at constant volume and the adiabatic compressibility are related by the equation

Ov/ks= VT(dP/dT)2,

where dP/dT is the slope of the equilibrium line on the phase diagram;show that this result holds if the transition between the phases iseither of the first order or a A-transition.

9. On the basis of the following information, which is partly hypothesis and partly somewhat simplified experimental data, calculatethe melting pressure of 2He3 at OOK.:

(a) Between 0 and 10-50 K. the specific heat of the solid is very high,but between 10-5 and 10 K. it is much less than that of the liquid.

(b) The specific heat of the liquid is proportional to T below 10 K.(c) The expansion coefficient of both phases may be assumed to be

zero.(d) At 0.320 K. the melting pressure Pm is 29·4 atmospheres and

dPm/dT=O; at 0·7°K. Pm is 33 atmospheres.

10. According to experimental measurements shown in fig. 42, airon transforms into y-iron at 9060 C. and back to a-iron at 14000 c.Between these temperatures the specific heat of y-iron rises linearlyfrom 0·160 cal. g.-1 deg.-1to 0·169 cal. g.-1 deg.-1. On the assumptionthat a-iron, if it were stable between 906 and 14000 C., would havea specific heat constant at the value 0·185 cal. g.-1 deg.-1 that it has atboth these temperatures, calculate the latent heat at each transition


and comment on the experimental value for the 9060 C. transition,3·86 cal. g.-I.

II. The transition point of the Sa~SP transition is 95·5° C., and themelting point of SP is 119·3°C., at atmospheric pressure. The latentheat of the transition Sa ~SP is 2·78 cal. g.-I, and the latent heat offusion of SP is 13·2 cal. g.-I. The densities of Sa, SP and liquid sulphurare 2·07,1·96, and 1·90g.cm.-3, respectively. Assuming the latentheats and densities to be independent of temperature and pressure,find the co-ordinates (P, T) of the triple point of Sa, SP, and liquidsulphur.

12. Discuss qualitatively the effects produced by gravity in experimental determinations of

(a) the specific heat of a substance exhibiting a second-order orA-transition;

(b) the critical isotherm of a simple fluid.

13. Show that for a substance obeying van der Waals's equation,the latent heat drops to zero with a vertical tangent as the criticalpoint is approached, Le. lim (dL/dT) = -00. What behaviour is to

T~Te

be expected for a real substance such as xenon, for which the iso-therms near Teare shown in fig. 261

14. The specific heat of unit volume of a metal may be veryapproximately represented by the formulae

O,=aT3 in the superconducting state,On=bT3+ yT in the normal state,

where a, band yare constants.Show that these formulae lead to the following results:

(a) the transition temperature in zero field, T c=(3Y/(a-b))1;(b) the critical magnetic field :Kc =:Ko(1-t2) where t = TITc and

Jt?0 = T c.J(21TY) ;(c) the difference between the internal energies of the two states, in

zero field, has a maximum when the temperature is Tcl.J3;(d) if a magnetic field applied to an isolated superconductor is

increased very slowly to a value above the critical, the transition tothe normal state is accompanied by a cooling of the metal. Find anexpression for the drop in temperature;

(e) if the field is applied suddenly, instead·of slowly, the metal isheated rather than cooled if the strength of the field exceeds

~o[(1+3t2) (l-t2)]1.

Furtherexercises, some more straightforward than those given here,\\'ill be found in Oavendi8h Problems in 0la88icalPhY8ics (C.U.P. 1963).

163

INDEX

Absolute temperaturedefinition, 35identical with gas scale, 47, 89establishment below 10 K., 91

Absolute zero, unattainabilitysecond law, 48equipartition law, 49third law, 51

Additivity of thermodynamic functions,44

Adiabatic change, 31demagnetization, 67equation, 61surface, 32wall, 5

Adiathermal process, 16Antiferromagnetism, 140Availability

definition, 100and useful work, 101of two-phase system, 103decrease in irreversible change, 105of liquid drop, 110, 153

Bernoulli's theorem, 72Boyle's law, 12, 47Boyle temperature, 73Brownian motion, 7, 83

Caratheodory's statement of secondlaw, 30

Camot cycle, 33Clapeyron's equation, 53, 115

analogues for superconductors, 133;second-order transitions, 140;third-order transitions, 142;lambda transitions, 143

Clausius's inequality, 37, 94statement of second law, 29

Compressibility, 61Critical point

liquid-vapour, 116liquid-solid, 122

Curie's law, 67Curie temperature, 91Cyclical method, 53

Demagnetization, adiabatic, 67Demagnetizing coefficient, 66

Diathennal wall, 5Dieterici's equation, 74Dimensions, 43

Ehrenfestclassification of transitions, 136equations for second-order transi

tions, 140Enthalpy

definition, 43conservation in stationary flow,

70diagram for helium, 76

Entropydefinition, 37increase in Joule effect, 69increase in Joule-Kelvin effect, 72law of increase, 95maximized in equilibrium, 107related to equation of state, etc.,

59of surface, 86

Eotvos's rule, 86Equation of state, 10Equilibrium

types of, 6conditions for, 104

Exercises, 160Expansion coefficient, 61Extensive variables, 44

Ferromagnetism, Weiss model, 138First law, statements, 14, 17Fluctuations, 7, 83, 96Free energy (Helmholtz)

definition, 43minimized in equilibrium, 107of surface, 85

Free expansion, see Joule's experiment

Friction, 21

Gas thermometer correction, 89Gibbs function

definition, 43g-surface, 112magnetic, 132minimized in equilibrium, 107and phase equilibrium, 108

164 Index

Gibbs-Helmholtz equation, 56

Heat, definition, 16Hotness and coldness, 18

Indicator diagram, 11Intensive variables, 44Internal energy

definition, 15of surface, 86

Inversion curve, 75temperature, 73

Isentropic change, definition, 31Isothenn

definition, 11critical, 117

Johnson noise, 83Joule coefficient, 57, 58, 69

effect, 68Joule's experiment, 21, 42

law, 47Joule-Kelvin coefficient, 57, 59, 72

effect, 69, 89

Kelvin's statement of second law,30

vapour-pressure relation, Ill, 154Kirchhoff's radiation law, 78

Lambda transitions, 138in' helium, 126, 143in ammonium chloride, 144

Latent heat positive, 114Liquid drop, stability, 153

Magneto-caloric effect, 67Maxwell's demon, 99

relations, 45ltIeissner-Ochsenfeld effect. 131Melting curve, 113, 122

of helium, 123

Noise, electrical, 83

Order-disorder transfonnationin p-brass, 138Bragg-Williams model, 138, 151Ising model, 140degree and range of order, 157

Perfect gasdefinition, 47temperature scale, 12, 47

Phase diagramsimple substance, 113aHa', 125iron, 124sHee, 126

Phase equilibrium, 112Planck's radiation law, 80Prevost's theory of exchanges, 80

Quasi-static change, definition, 21Quenching of disordered alloy, 158

Radiationblack-body or cavity, 77from SUD, 82pressure, 79

Rayleigh-Jeans radiation law, 83Reversible process, definition,

22

Second Jaw, statements, 29, 30Specific heat

helium, 127superconductor, 134fJ-brass, 138quartz, 139iron, 141manganous bromide, 141

Specific heats, principalalways positive, 18difference, 60ratio, 61variation with pressure, etc., 60magnetic, 65

Stationary flow, 70Stefan-Boltzmann law, 79, 88Sublimation curve, 113Superconductivity, 129Supercooling, 114

of superconductors, 131, 155of vapour, 154at second-order transition, 152

Surface tension, 84Susceptibility, 65

Temperaturedefinition, 7empirical, 10negative, 52

Temperature scaleabsolute, 35,47, 91perfect gas, 12, 47magnetic, 91fixed points, 88

Index 165Third law

statement, 51and surface tension, 87

Transitionsclassification, 137second-order, 136, 140third-order, 140lambda, 126, 142criticisms of theory, 147Gorter's model" 148

Triple-point, 113

Vapour-pressure curve, 113Virial coefficients, 73

Work, reversibleelastic, 23surface tension, 23magnetic, 24electric, 27general, 27

Zero, absolute, 48Zeroth lawand converse,statements, 9

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I _ _ _ __ __ -

I

THE ELEMENTS OF CLASSICALTHERMODYNAMICS

By A. B. PIPPARD

John Humphrey Plummer Professor ofPhysics in the University ofCambridge and Fellow ofClare College, Cambridge

There can be no hesitation in recommending this book to all undergraduates and postgraduates interested in thermodynamics, and manyusers ofmore advanced thermodynamics might well find pleasure ina study of this well-written account. NatureThis is a superb little book.••.It has something ofthe flavour ofanengrossing novel, or detective story, in which the author explores,with the aid of thermodynart!ic techniques, some of the mysteriesof the more exciting parts of physics; in particular those, such assuperconductivity, related to phase changes. Even when d~with old and familiar topics it frequendy imports a fresh and invigorating approach. Science ProgressThis little book is intended for advanced studenb of physics; but,with the possible exception ofthe 6nal chapter, itcould be read withprofit by any student of engineering who seeks a fairly rigorousbasis for his applied thermodynamics. It has the enormous advantagethat the author makes a real effort to pinpoint some ofthe difficultiesofhis subject and to clarify them.

Jou""l ~f tbe Royal Aeronautical SocietyDr Pippard's book, whilst paying adequate attention to technique,is partiCularly to be recommended for providing the reader withan understanding ofthermodynamics. Many awkWard poinb, whichare glossed over in other treatises, are discussed clearly and comprehensively...anyone with a rudimentary knowledge of thermodynamics cannot fail to derive benefit and stimulus from its pages.

Philosophical MagazinePippard attains the fine level ofexcellence one is accustomed to findin books from the Cambridge University Press. Succinct but notbrie~ thorough but not boring, instructive but not pedantic describethe general tenor. Journal ofthe American Chemical SocietyIt has freshness, brevity and elegance which will delight the physicistwho is (or who is prepared to become) enthusiastic about thermodynamics. Proceedings ofthe Physical SodetyIf there possibly exist studenb who have at one time felt thermodynamics to be a somewhat dry and uninspiring subject, this bookis to be recommended to them for refreslmient.

American Institute ofPhysics

CAMBRIDGE UNIVERSITY PRESSBentley House, 200 Euston Road, London, N.W. I

American Branch: 32 East 57th Street, New York, KY. 10022

o 521 09101 2

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Elements of Classical Thermodynamics by a.B.pippard

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