Thermodynamics

Citation preview

I.P. Bazarov

THERMO– DYNAMICS

T

THERMODYNAMICS

Thermodynamics I. P. BAZAROV P H O F E S S O H O F S T A T IS T IC A L P H Y S IC S , M O SC O W S T A T E U N IV E H S IT Y

Translated by

F. IMMIRZI Translation edited by

A. E. J. HAYES

A Pergamon Press Book

THE MACMILLAN COMPANY NEW YORK

1964

T H E M ACM ILLAN COMPANY 60 Fifth Avenue New York 11, N.Y.

T his book is d istrib u ted b y T H E M A C M IL L A N C O M P A N Y pursuant to a special arrangem ent w ith P E R G A M O N P R E S S L IM IT E D O xford, E ngland

C opyright © 1964 P e ro a m o n P r e s s L td .

Library o f Congress C atalog Card Num ber 63— 16495

T h is is a translation o f th e original R ussian volum e TepMcduHCLMUKa (Termodinamika), published in 1961 b y F izm atgiz, Moscow

CONTENTS Preface

ix

Introduction

x

Chapter I: Basic concepts and initial propositions of thermodynamics

1

§ 1 Thermodynamic system and thermodynamic parameters. Thermo­ dynamic equilibrium from the molecular viewpoint 1 § 2 Initial propositions of thermodynamics and theirdiscussion 4 § 3 Quasi-static and non-static processes 9 § 4 The internal energy of a system. Work and heat 10 § 5 Thermic and caloric equations of state 16 Problems 20 Chapter II: Fundamental laws and equations of thermodynamics. The first law of thermodynamics 22 § 6 The equation of the first law of thermodynamics § 7 Thermic and caloric properties. Thermal capacities and latent heats. Connexion between thermal capacities § 8 Fundamental thermodynamic processes and their equations § 9 The connexion between elasticity coefficients and thermal capa­ cities § 10 Hess’s rule. Thermo chemical equations Problems

33 35 36

Chapter HI: The second law of thermodynamics

38

§11 General features and initial formulation of the second law § 12 Reversible and irreversible processes § 13 The second law of thermodynamics for quasi-static processes. Entropy and absolute temperature § 14 Mathematical justification for the existence of entropy and abso­ lute temperature § 15 The fundamental equation of thermodynamics for quasi-static processes. The evaluation of entropy § 16 The second law of thermodynamics for non-staticprocesses § 17 Connexion between the thermic and caloric equationsof stato

38 43

V

23 26 30

45 47 58 62 65

VI

CONTENTS

§ 18 The efficiency of heat-engines. Carnot’s cycle and Carnot’s theo­ rem. Types of heat-engines § 19 Spontaneous flow of heat § 20 Limits of applicability of the second law. The “theory” of the thermal death of the universe Problems

69 73 74 78

Chapter IV: Methods of thermodynamics

84

§ 21 The method of cyclical processes § 22 The method of thermodynamic potentials Problems

84 86 99

Chapter V: Certain applications of thermodynamics

100

§ 23 The Gibbs-Helmholtz equation and its application to a galvanic cell 100 § 24 The cooling of gas in reversible and irreversible adiabatic expan­ sion. The Joule-Thomson effect. Kapitsa’s turbo-cooler 103 § 25 The thermodynamics of dielectrics and magnetic substances 108 §26 Radiation thermodynamics 117 § 27 Thermodynamics of plasma 127 § 28 The thermodynamics of systems with a variable number of par­ ticles 131 Problems 134 Chapter VI: Conditions of thermodynamic equilibrium and their appli* cation 136

§29 § 30 § 31 § 32 § 33

Homogeneous and heterogeneous systems. Phases and components General conditions of thermodynamic equilibrium Stability conditions for the equilibrium of a homogeneous system Equilibrium in a homogeneous system Equilibrium in a heterogeneous system Problems

136 138 145 149 153 158

Chapter VII: Surface phenomena

161

§ 34 Surface tension and surface pressure § 35 Equilibrium form of a crystal. Wulf’s theorem § 36 The role of surface tension in the formation of a new phase. Initial stages § 37 Gibbs’ equation for adsorption. Surface-active substances Problems

162 164 166 169 171

CONTENTS

Chapter VIIIj Phase transformations and critical phenomena

§ 38 § 39 § 40 §41

Phase transitions of the first kind. Clapeyron-Clausiusequation Phase transitions of the second kind. Ehrenfest’s equation The theory of phase transitions of the second kind Critical phenomena Problems

VU

172

174 178 182 185 192

Chapter IX s Nernst’s heat theorem

194

§ 42 Nernst’s theorem §43 Corollaries of Nernst’s theorem Problems

194 197 201

Chapter X: Negative absolute temperatures

202

§ 44 The possibility of the existence of states with a negative absolute temperature 202 § 45 A system with negative absolute temperature 204 § 46 The thermodynamics of systems at negative absolute tempera­ tures 208 § 47 Stability of the state of a system at negative absolute temperature 214 Chapter XT: Fundamentals of the thermodynamics of irreversible pro­ cesses 216

§ 48 The fundamental propositions of the thermodynamics of irrever­ sible processes 216 § 49 Thermoelectrical phenomena 222 Problem 227 Solutions to problems

228

Index

283

PREFACE This book is the first part of a course on “Thermodynamics and

Statistical Physics” held at present in physical and physicomathematical faculties of our universities. In comparison with existing courses of thermodynamics the book has the following distinctive features: 1. Greater attention than in other books is devoted to a dis­ cussion of the initial propositions of thermodynamics, which en­ ables us to establish both the limitations of thermodynamics and its organic connexion with statistical physics. 2. The content of the second law of thermodynamics is analysed logically and with greater consistency, by taking into account the existence of negative absolute temperatures. 3. The methods of thermodynamics are considered and discussed in detail. 4. The theory of critical phenomena is expounded according to Gibbs, which enables us to establish the connexion of these pheno­ mena with phase transitions of the second kind. 5. The thermodynamics of systems with negative absolute tem­ peratures is treated. 6. The fundamentals of the thermodynamics of irreversible pro­ cesses are given. 7. The large number of problems included in the book form an in­ tegral part of the course. Many are devoted to certain additional questions that are not always treated in the course and do not enter in the main text of the book. For these reasons the solutions of the problems are set out in as much detail as possible. The author considers it his duty to express here his gratitude to Academician N. N. Bogolyubov and Prof. A. A. Vlasov for their com­ ments on various questions of thermodynamics and statistical physics, as well as to the members of the methodological seminar of the phy­ sical faculty of Moscow State University for discussing at this seminar methodological problems arising in connection with the book. I. P. B azarov

INTRODUCTION of the various courses of both general and theoretica physics is determined by the gradual transition to the study of more and more complex forms of motion of matter. Mechanics studies the laws of displacement, the simplest form of motion of matter. Thermodynamics and statistical physics consider phenomena caused by the combined action of a very large number of continually moving molecules or other particles, of which the bodies surrounding us consist. Owing to the large number of par­ ticles their disorderly motion acquires new qualities. We have here an example of dialectic transition of quantity into quality when the increase of the number of mechanically moving particles in a body gives rise to a qualitatively new type of motion, namely thermal motion.-)Thermodynamics and statistical physics study the thermal form of motion of matter with its own specific laws. This does not mean that either thermodynamics or statistical physics are sciences of the thermal phenomena; they study not only thermal phenomena but also electric, magnetic and other phenomena in bodies. The study of these phenomena, however, is carried out from the view­ point of the specific properties of the thermal motion in them. The main contents of both thermodynamics and statistical physics con­ sist in the analysis of the laws of thermal motion in systems that are found in thermal equilibrium^ and in the passage of systems to a state of equilibrium. I t can be seen from this th at the object of the study of both thermodynamics and statistical physics is one and the The

seq uence

•f “M otion is n ot o n ly a change o f p o sition ; in super-m echanical regions it is also a change o f q u ality. T he discovery th a t h eat is a certain m olecular m otion opened up a n ew epoch in th e history o f scionce. H ow ever, if I h ad n oth in g else to say ab ou t h ea t e x cep t th a t it represents a certain displacem ent o f m olecules, I should b etter sa y n o th in g alto g eth er” (F. E n gels, The D ialectics of N ature, G ospolitizdat, 1952, page 201). T his statem en t o f E n gols p o in ts out tho fact th a t a q u a lita tiv ely n ew form o f m otion— therm al m otion— does n ot reduce [sim p ly] to th e m echanical m otion o f th e separate particles. J See § 1 for more d etails on therm al equilibrium .

INTRO DUCTION

XI

same. Their essential difference from each other consists in the meth­ ods of investigation. Whereas thermodynamics studies the general properties of phys­ ical systems in equilibrium by proceeding from two basic laws, called the laws of thermodynamics, together with a whole series of other experimental results, and does not explicitly use notions on the molecular structure of a substance, in the analysis of these pro­ perties statistical physics proceeds at its very beginning from mole­ cular models of the structure of physical systems, by making wide use of the methods of the mathematical theory of probabilities. The macroscopical approach of thermodynamics, i.e. its not being connected with the molecular-kinetic essence of the laws studied by it, leads on the one hand, to great and important results concerning the properties of physical systems, and, on the other hand, sets a limit to the depth to which these properties can be studied since it does not enable us to discover the nature of the phenomena studied. As a consequence of this, there has developed, alongside with thermo­ dynamics, the molecular-kinetic theory of the properties of physical systems, and all the investigators whose names are connected with the development of thermodynamics have devoted great attention to the molecular-kinetic foundation of the results of thermodynamics. Thermodynamics is a first mighty stride on the way to studying the laws in a large set of continuously moving and interacting partic­ les (the so-called statistical laws); a detailed and more complete anal­ ysis of these laws requires the use of statistical methods. Thermodynamics, however, must not be dismissed as part of the material of statistical physics. Though, in the end, all properties of physical systems are caused by the molecular motion in them, thermo­ dynamics enables us to establish many of these properties without having recourse to notions of the molecular structure of bodies. The methods of thermodynamics are sufficient for solving very many practically important problems. All this, on the one hand, limits the scope of thermodynamics and, on the other hand, gives it a definite advantage in comparison with molecular theories. There are no grounds today for drawing a sharp boundary between thermodynamics and statistical physics; nevertheless the definite advantage of thermodynamics and the specific nature of its methods make it important that a preliminary separate treatment of thermo­ dynamics be given, making use of such qualitative molecular repre­ sentations as are required.

XII

IN TRO DUCTIO N

The advantage of thermodynamics as compared with statistics consists in that it enables us by means of its principles to take into account easily laws observed experimentally and to obtain funda­ mental results from them. In just this manner was Nemst’s heat theorem established in its time, the degeneracy of gases at low tem­ peratures was predicted, the theory of phase transitions of the second kind was developed etc., while today the thermodynamic theory of kinetic phenomena in physical systems (the so-called thermodynamics of irreversible processes) is being successfully developed. The statis­ tical theory of macroscopic processes lags in this respect behind thermodynamics. I t can be seen from this that the treatment of thermodynamics as a separate, to a certain extent independent, subject is justified not only on pedagogical grounds. Historically, thermodynamics'}* arose from the needs of heat en­ gineering. The arising of thermodynamics was stimulated by the development of productive forces. The invention at the beginning of the nineteenth century of the steam-engine put before science the task of the theoretical investigation of the work of heat-engines with a view to increasing their efficiency. An investigation of this type was carried out in 1824 in the first work on thermodynamics by the French engineer and physicist Sadi Carnot and led him to establish theorems that determine the largest [possible] efficiency of heat-engines. These theorems later enabled one of the basic laws of thermodynamics— the second law, to be formulated. Somewhat later, in the forties of the nineteenth century, as a result of the investigations of Mayer, Joule, Helmholtz and others the mechanical equivalent of heat was established and, on this basis, the law of conservation and trans­ formation of energy, called the first law of thermodynamics, was discovered. As a formal scientific system proceeding from Carnot’s work and from the law of conservation and transformation of energy, thermo­ dynamics first occurred in the fifties of the past century in works of Clausius and Thomson-Kelvin, in which the modem formulations of the second law were given and the most important concepts of entropy and absolute temperature were introduced. The main method f T herm odynam ics m eans literally n o t th e stu d y o f h ea t m otion b u t th e science o f “ m o tiv e pow er” arising in therm al processes. T hus th e first work on therm odyna­ m ics w as called “ R eflections on th e m o tiv e pow er o f fire an d on th e m achines capable o f develop in g th is pow er” (Sadi Carnot, 1824), in w hich th e term “m o tiv e pow er” denotes th e useful action (work) th a t an en g in e can provide at th e expense o f heat.

INTRODUCTION

X lll

of investigation of thermodynamics in the nineteenth century was the method of closed-cycle processes. Of exceptionally great importance for thermodynamics were Gibbs’s works published at the end of the nineteenth century. In them a new method of thermodynamic investigations, the method of thermodynamic potentials, was originated, general conditions for thermodynamic equilibrium were established, and the theory of phases and capillarity was developed. In the twentieth century thermodynamics developed considerably beyond the initial requirements of heat engineering and is concerned, as has already been said, with the laws of the thermal form of motion of matter mainly in equilibrium systems and in the passage of systems to an equilibrium state. Thermodynamics is a deductive science that derives its main con­ tents from two fundamental laws—the laws of thermodynamics, and makes use a t the same time of a whole series of other experimental facts. The first law of thermodynamics expresses the quantitative aspect of the law of conservation and transformation of energy. Although the law of conservation and transformation of energy (just as the concept itself of energy as a measure of motion) is only applicable to physical forms of motion (see § 4) and is not applicable to higher forms of motion of matter (biological and social motions), it has nonetheless a universal value. This follows from the univer­ sality of physical forms of motion: each higher form of motion of matter contains in itself physical forms of motion though it does not reduce [simply] to them. And if in the transformation of a physical form of motion into another, one of them disappears (partially or completely) while the second one increases quantitatively (the trans­ formation of mechanical motion into thermal or electromagnetic motion and vice versa), then in the arising of a new higher form of motion of matter the various physical forms of motion generating it do not disappear but exist as “their higher unity” (see F. Engels, The Dialectics of Nature, page 199, 1952). The destruction of this unity leads to the vanishing of this higher form of motion and to the liberation and separation of the different physical forms of motion generating it and having their measure of energy. Hence it follows that the law of conservation and transformation of energy is directly applicable to only physical forms of motion of matter and establishes the indestructibility and inter-transformability of only these forms of material motion, but at the same time it also

XIV

INTRO DUCTION

expresses the fact that motion cannot be destroyed or created in general and therefore it is indissolubly bound with the dialecticmaterialistic ideology which recognizes the priority of matter with its imperishable attributes.! From the viewpoint of this unitary scientific ideology we can categorically deny the possibility of existence of phenomena (either at the scale of our planet or at a cosmical scale) in which the law of conservation and transformation of energy be invalidated. To admit the possibility of existence of such phenomena would mean not to see in the law of conservation and transformation of energy anything more important than in any other law (for example in Boyle-Mariotte law, known from the course of general physics, pV = const, which can also be expressed as a law of conservation of a sum of quantities, log p + log V = const, and which is verified in processes in a rare­ fied gas but is not valid in a denser gas). The second law of thermodynamics is the law of entropy. A con­ sequence of this law is, for example, the impossibility of processes the only result of which would be the transformation of heat into work, or the spontaneous passage of heat from a cold body to a hot one when these bodies come into contact etc. Just as the first law of thermodynamics, the second law has about ten different formulations. The majority of them are equivalent to each other and express the entire content of the law itself. However, the variety of formulations of the second principle is connected with the fact th at these laws become apparent in some or other concrete phenomena, so th at each of these formulations corresponds to a determinate more or less general phenomenon. The formulation that expresses the law of a phenomenon closer to our experience and prac­ tice can be taken as the initial one in establishing and analysing each of the laws. On the basis of the first and second laws, thermodynamics in­ vestigates the properties of real systems. Thermodynamics is applicable only to systems consisting of a large number of particles. This establishes a lower bound for the dimen­ sions of systems to which thermodynamics is applicable. But the applicability of thermodynamics has also an upper bound: it is int “ Modern n atu ral science has been com pelled to borrow from philosophy th e th esis o f th o in d estru ctib ility o f m otion ; w ith ou t th is th esis natural science could no longer e x ist n o w ” (F. E n g e ls, The Dialectics of N ature * p age 16, 1952).

INTRODUCTION

XV

applicable to systems of infinite dimensions such as the universe or an infinite part of it. The initial propositions of thermodynamics are established for finite systems but with a large number of particles and for finite intervals of time. In 1906 to the two laws of thermodynamics there was added one more experimental fact which is referred to as Nernst’s theorem. According to this theorem, at temperatures tending to the absolute zero equilibrium processes occur without entropy variation, while the entropy itself of any body tends to zero as the absolute zero is steadily approached. This theorem is of great importance for find­ ing the entropy constants and chemical constants of substances taking part in chemical transformations. Nernst’s theorem does not follow from the first and second laws of thermodynamics but expresses a new law of nature and therefore it is often referred to as the third law of thermodynamics. I t can be established by proceeding from the basic propositions of quantum statistical mechanics. Otherwise, as far as its value for thermodyna­ mics is concerned, Nernst’s theorem is less important than the first and second laws. A great contribution to the development of thermodynamics has been given by our scientists. At the end of the nineteenth century a professor of Kiev University, N. N. Shiller, gave a new formulation of the second law of thermodynamics which in 1909 was developed by the German mathematician Caratheodory. In 1928 T. A. Afanas’yeva-Ehrenfest, by analysing critically the work of Shiller and Caratheodory, showed for the first time that the second law of thermo­ dynamics consists of two independent propositions that are generaliza­ tions of experimental data and concern, on the one hand, states of equilibrium and, on the other hand, different processes. Of special importance has been the role of Russian scientists in the study of critical phenomena. The concept itself of critical tem­ perature first occurs in D. I. Mendeleyev’s work. Mendeleyev es­ tablished that in approaching a certain temperature, surface tension tends to zero, so th at the distinction between liquid and vapour disappears. Mendeleyev called this temperature the absolute boiling point. A. G. Stoletov, M. P. Avenarius and others devoted further study to critical phenomena. The Russian scientists V. A. Mikhel’son and B. B. Golytsin gave a considerable contribution to the thermo­ dynamics of radiation. Golytsin first introduced the concept of radi­ ation temperature which was accepted by scientists and has been

XVI

INTRODUCTION

retained to the present day. D. P. Konovalov, N. S. Kurnakovand others have studied the application of thermodynamics to physical chemistry. A great contribution to thermodynamics and statistical investiga­ tions are the works of L. D. Landau on the theory of phase tran ­ sitions, the works of M. A. Leontovich on the thermodynamical func­ tions of non-equilibrium states, N. N. Bogolyubov’s works on the theory of superfluidity and superconductivity, on kinetic equations and the theory of real gases, V. K. Semenchenko’s works on the theory of solutions and critical phenomena and others. This course of thermodynamics has been organised according to the following plan: at first the initial propositions of thermodyna­ mics are established and discussed, and the contents and main co­ rollaries of the first and second laws are expounded, then the methods of thermodynamics are considered, and next by means of these methods the most important problems of the behaviour of systems in a state of equilibrium and in various processes are investigated. The last chapter is devoted to the fundamentals of the theory of irreversible processes.

CHAPTER

]

BASIC CONCEPTS AND INITIAL PROPOSITIONS OF THERMODYNAMICS T hermodynamics deals with the study of the laws of thermal motion in equilibrium systems and in the transition of systems to equilibrium [states] and it also extends these laws to non-equilibrium system s.f

Before we pass to the study of properties of systems we shall firstly elucidate the content of the basic thermodynamic concepts (thermo­ dynamic system, thermodynamic equilibrium, process, heat, work etc.) and shall discuss the initial propositions of thermodynamics. This will enable us to assess the significance and limits of applicability of thermo­ dynamics. § 1. Thermodynamic system and thermodynamic parameters. Thermodynamic equilibrium from the molecular viewpoint

Any material object, any physical body, consisting of a large number of particles is called a macroscopic system. The dimensions of macro­ scopic systems are always considerably larger than the dimensions of atoms and molecules. All macroscopic attributes characterizing a system and its relation to the surrounding bodies are called macroscopic parameters. They comprise, for example, such quantities as density, volume, elasticity, concentration, polarization, magnetization etc. Macroscopic parameters are sub-divided into external and internal ones. Quantities that are determined by the position of external bodies not entering into our system are called the external parameters a{ (i = = 1,2, ...); for example, the volume of a system is one of its external parameters in that it is determined by the position of external bodies; the intensity of a field of force is also an external parameter since it depends on the position of the sources of the field, charges and cur­ rents, which are no part of our system, etc. t The la tter refers to th e therm odynam ics o f irreversible processes (see Chap. II). 2

1

2

THERMODYNAMICS

External parameters are, therefore, functions of the coordinates of external bodies. Other quantities, however, that are determined by the combined motion and distribution in space of particles that are part of our sys­ tem are called internal parameters bj (j = 1, 2, . . .); for example, dens­ ity, pressure, energy, polarization, magnetization etc. are internal parameters, since their value depends on the motion and position of particles of the system and of charges occurring in them. Since the relative position in space of particles entering into our system, atoms and molecules, depends itself on the relative position of external bodies, it follows that internal parameters are determined by the position and motion of these particles and by the value of external parameters. The set of the independent macroscopic parameters of a system de­ termines the state of the system. Quantities that are independent of the previous history of the system and are fully determined by its state at a given instant (i.e. by the set of the independent parameters) are called parameters of state. A state is called stationary, if the system parameters are constant with time. If, in addition, not only are all parameters in the system constant with time, but there are no stationary currents whatsoever owing to the action of some external sources, then such a state of the system is called an equilibrium state (a state of thermodynamic equilibrium). Thermodynamics studies mainly the properties of physical systems that are found in an equilibrium state. Usually, therefore, not all mac­ roscopic systems but only those of them that are found in thermodyna­ mic equilibrium are called thermodynamic systems. By analogy we call thermodynamic parameters those parameters that characterize a system in its thermodynamic equilibrium. What do thermodynamic internal equilibrium parameters represent from the molecular viewpoint? In order to clarify this, let us consider a most simple example—the density of a gas (the number of particles per unit volume of gas). If the gas is found in a non-equilibrium state the gas density will be dif­ ferent at different points. After'some time has elapsed the gas will reach an equilibrium statef and the density q = mn (m is the mass of a molecule and n is tho number of molecules per cm3) will have a cerf See § 2 on th is p o in t.

BASIC CONCEPTS OF THERMODYNAMICS

3

tain, macroscopically constant, equilibrium value (Fig. 1). This equi­ librium value of the density @0 can be defined as the mean value of the density, q, over a large interval of time T: T ~ . 1 r q0 = q = lim —- \ g(t) d t . (1. A) r - * “

TJ

o Similarly, the equilibrium value of any other internal parameter is the mean value, over a long interval of time, of the function of coordina­ tes and rates of change corresponding to this parameter. Statistical physics, pro­ ceeding from a determinate molecular model of the struc­ ture of matter, enables us to evaluate equilibrium values of internal parameters. How­ ever, even Avithout carryingout these calculations, Ave can bring to light laAvs of systems in equilibrium states, by bearing in mind that in practice equilibrium parameters can be measured in many cases directly from experiment. Thermodynamics provides just this first stage in the theory of equilibrium states. As we shall see below, systems are characterized in a state of thermo­ dynamic equilibrium both by determined equilibrium values of the macroscopic parameters indicated above (density, pressure, volume, magnetization, etc.) and by such typically thermodynamic parameters, that in the absence of equilibrium in the system are deprived of mean­ ing for all systems, as, for example, temperature, entropy, etc. Thermodynamic parameters that are independent of mass or of the number of particles in the system are called intensive parameters (for example, pressure, temperature etc.), Avhile those parameters that are proportional to mass and to the number of particles in the system are called additive or extensive parameters (for example, volume, energy, entropy, etc.). Determined notions, based on macroscopic experiment, on the pro­ perties of the thermodynamic equilibrium of finite systems are assumed in thermodynamics as postulates, on which basis and by means of the basic laws (principles) of thermodynamics the properties of systems in equilibrium states are studied.

4

THERMODYNAMICS

§ 2. Initial propositions of thermodynamics and their discussion

A system that does not exchange energy with external bodies is said to be isolated. External actions, by influencing the motion of particles of the system, also alter the thermal motion in it. I t is postulated in thermodynamics that, an isolated system always reaches, in the course of time, a state of thermodynamic equilibrium and can never depart from it spontaneously (the first postulate of thermodyna­ mics ). This first initial proposition of thermodynamics can be called the general principle of thermodynamics, since, similarly to the first and second laws, which establish the existence of determined functions of state, it leads to the existence of a whole series of functions of state of a system in thermodynamic equilibrium. From the viewpoint of statistical physics, which takes into account explicitly the motion of particles in a system, the meaning of the thesis on thermodynamic equilibrium (the first postulate of thermodynamics) consists in that in each isolated macroscopic system there exists a determined and unique state that is generated by the continuously moving particles most often of all (the most probable state); an isolated system reaches in the course of time just this most probable state. It can be seen from this that the postulate of the spontaneous passing of an isolated system to equilibrium and of its remaining there for an in­ definitely long time is not an absolute law of nature but merely expres­ ses the most probable behaviour of a system; the never ceasing motion of particles of the system leads to spontaneous deviations (fluctua­ tions) from the equilibrium state. Thus thermodynamics, by assuming the first postulate, restricts it­ self, in that it renounces the consideration of all phenomena connected with the spontaneous departure of a system from its equilibrium state. A justification for accepting the general principle of thermodynamics is the fact that, as statistical physics shows, relative spontaneous deviations of a system from equilibrium are the smaller the more particles there are in the system, and, since thermodynamic systems consist of an enormous number of particles (N g> 1), fluctuations are altogether neglected in thermodynamics. However, in those phenomena where fluctuations are essential, the thermodynamic approach to the study of these phenomena is no longer correct owing to the initial proposition assumed, and a statistical con­ sideration is necessary. In these cases the conclusions of thermodyna-

BASIC CONCEPTS OF THERM ODYNAM ICS

5

mics and statistical physics will be at variance, which is caused by the restricted scope and relativity of the first initial proposition of thermodynamics. The clarification of this fact shows that thermodyna­ mic and statistical considerations do not exclude, but are comple­ mentary to, each other. The first postulate on thermodynamic equilibrium restricts, on the other hand, the application of thermodynamics when applied to in­ finite systems (the universe or an infinite part of it), since in a system of an infinite number of particles (N -*■ °°) all states are equiprobable, and therefore there exists no equilibrium state (as the most probable state) to which the system should pass in the course of time.f The second initial proposition of thermodynamics is connected with other properties of thermodynamic equilibrium as a special form of thermal motion. Experiment shows that if two equilibrium systems A and B are brought into thermal contact,J then, independently of whether the external parameters in them are different or equal, they either remain as before in a state of thermodynamic equilibrium, or the equilibrium in them is destroyed and, after a certain time, as a result of thermal exchange (exchange of energy) both systems reach another equilibrium state. If then this complex system is brought into thermal contact with a third equilibrium system G, then, similarly as before, either equilibrium in the systems is not altered, or is des­ troyed and, after a certain time, they all reach a certain new equi­ librium state. If, later, thermal contact between the systems is inter­ rupted (either simultaneously or in sequence), then, both after this and after renewed establishing of thermal contact, equilibrium in each of the systems is not altered, and therefore equilibrium of the system C separately with the systems A and B involves equilibrium of the systems A and B with one another (the transitivity property of thermodynamic equilibrium). I t follows from all this that a state of thermodynamic equilibrium of a system is determined not only by its external parameters at but also by another quantity t characterizing its internal state. In the presence of thermal contact of various equilibrium systems, as a result of exchange of energy, the values of this quantity t are equalized and t See § 20 an d problom no. 64. J I t i3 said o f bodies th a t th e y are found in therm al con tact (or olso are brought into therm al c o n ta c t)if, b y somo m ethod or other (therm al co n d u ctivity and b y m oans o f radiation) therm al exchange is m ade possiblo for thorn, w ith ou t it being possible for a substanco th a t is p art o f one body to ponotrato insido th e other.

6

THERMODYNAMICS

remain the same for all of them both while thermal contact lasts and after it has been discontinued.I The transitivity property of states of thermodynamic equilibrium enables us to compare the values of the quantity t in different systems without bringing them into direct thermal contact with each other, but by making use of some other body. This quantity, expressing the state of interna 1motion of an equilibrium system, having one and the same value in all parts of a complex equilibrium system independently of the number of particles in them, and being determined by external parameters and by the energy corresponding to each part, is called temperature. Being an intensive parameter, temperature is in this sense a measure of the intensity of thermal motion. The proposition outlined stating the existence of temperature as a special function of the state of equilibrium of a system represents the second initial proposition of thermodynamics. I t is referred to some­ times as the “zero-th principle”, since, similarly to the first and second laws which establish the existence of certain functions of state, it establishes the existence of temperature in an equilibrium system. Temperature, as we see, is a parameter of thermodynamic equilib­ rium since it only exists in systems in thermodynamic equilibrium, and moreover only in such systems the (macroscopic) parts of which do not interact with each other, so that the energy of the system is equal to the sum of the energies of its parts.{ For a system close to equilibrium temperature can only be assumed as an approximate concept, and, for a system not in equilibrium, the concept of temperature is deprived of meaning altogether. The proposition stating the existence of temperature can also be formulated in the following manner. We have established in § 1 that the internal parameters of a system are determined by the position and motion of the molecules that are part of the system and by the values of external parameters, in which connexion, to characterize a state (an equilibrium state) of the system, thermodynamics uses not actual functions of the coordinates and momentum of molecules but mean values of functions over a sufficiently large interval of time. How­ ever, the proposition on the existence of temperature establishes that f In th is connection the energy o f different system s is, generally speaking, different. This follow s from the id en tity o f tho soparato binary equilibrium o f system s A + B , A + G and B + C and o f their ternary equilibrium state (A + B ) + C. J T his follow s from tho fact th a t in interrupting or establishing thormal con tact botwoon parts o f a system , their states, as wo havo obsorvod, do n o t vary.

BA SIC CONCEPTS OF THERM ODYNAM ICS

7

a state of thermodynamic equilibrium is determined by the set of external parameters and temperature.f Therefore, though internal parameters characterize the state of a system, they are not independent parameters of an equilibrium system. All equilibrium internal parameters of a system are functions of the external parameters and temperature (the second postulate of thermo­ dynamics ). Since the energy of a system is one of its internal parameters, then, in equilibrium, energy will be a function of external parameters and temperature. By expressing, from this equation, temperature in terms of energy and external parameters, we can formulate the second ini­ tial proposition of thermodynamics also in the following manner: under equilibrium conditions all internal parameters are functions of external parameters and energy.t The second initial proposition of thermodynamics enables us to determine the temperature variation of a body from the variation of any of its internal parameters, on which fact is based the construction of the various thermometers.fi In order to establish which of two temperatures is the larger, an additional condition is introduced: it is assumed that in communicat­ ing energy to a body under constant external parameters its tempera­ ture increases (although we could assume that it decreases) or else (which is found in accordance with the condition given but is not the | Since th e sta te o f a therm odynam ic system is determ ined by external param eters and tem perature w hile th e concept o f tem perature (as a measure o f the in te n sity o f therm al m otion) is extraneous to m echanics, then, consequently, m echanical system s are n o t therm odynam ic system s (they can be considered form ally as a particular case o f therm odynam ic system s in which th e in ten sity o f therm al m otion is equal to zero). I t is understandable, therefore, th a t certain conclusions o f therm odynam ics are in a p ­ plicable to m echanical system s. These are m ain ly those results th a t are connected w ith d istin ctive features o f therm al m otion in m olecular macroscopic system s (irrevers­ ib ility , uniqueness o f direction o f natural processes etc.). Thus, to avoid m isunder­ stan d in gs in th e definition o f som e or other therm odynam ic concepts, it is necessary from the very begin n in g n o t to consider m echanical system s on a par w ith therm o­ dynam ic ones and to apply the results o f therm odynam ics to m echanical system s each tim e after special consideration, since the passage from m echanical system s to therm o­ dynam ic (statistical) ones is connected w ith a qualitative saltus from one form o f m otion to another. X E quilibrium system s in which th e internal param eters are functions o f th e external param eters and th e energy are called ergodic. Therefore therm odynam ics studies ergodic system s. I t The fact th at in therm om eters, th e tem perature t is often determ ined from the volum e V which is n o t an internal b u t an external param eter is explained b y th e existen ce o f th e equation o f state (see § 5).

8

THERMODYNAMICS

samef) if in the presence of thermal contact of the bodies A and B energy passes from A to B it is assumed that the temperature of the body A is larger than the temperature of the body B (though we could also have assumed the contrary). Such an additional condition to spe­ cify further the concept of temperature enables us to choose for the internal energy of a system a monotonically increasing function of temperature, which is perfectly possible as a consequence of two experi­ mental facts: the uniqueness of the distribution of energy over the parts of the system and the simultaneous growth of the energy of parts when the total energy of the system increases (see problem 1). In the practical determination of temperature we have to use some determined scale connected with some substance or other. As the thermometric parameter we use most often of all the volume of this substance, and the scale chosen is Celsius’ scale: the difference of the volumes of the body when it is in thermal equilibrium with boiling water at atmospheric pressure and with melting ice at the same pres­ sure is divided in 100 parts; each division corresponds to one degree and the temperature of melting ice is taken equal to 0°C.t We call the empirical temperature of a body a measure, determined by experiment, of the deviation of the thermodynamic state of a body from the state of thermal equilibrium with melting ice found under the pressure of one physical atmosphere. The readings of two thermometers with different thermometric sub­ stances are never, generally speaking, coincident except at zero and 100°C, since such a determination of temperature as an objective measure of the intensity of thermal motion is arbitrary. This arbit­ rariness is partly removed if we use as the thermometric substance sufficiently rarefied (ideal) gases. Their coefficient of thermal expansion a does not depend either on temperature or on the nature of the gas. The scale of a gas thermometer is graduated in the same manner as the Celsius scale but the zero temperature is taken equal to —a-1 degrees Celsius (Kelvin’s scale). We shall denote temperature measured by means of an ideal gas f Concerning this see in §§ 19 and 30. t Therm om eters were in v en ted a few years after 1600 b y Galileo and, in dependently o f him , b y the D utchm an Drebbel. T he freezing p oin t o f water was introduced as a co n sta n t p o in t o f the therm om eter in 1664 b y H ooke and th e boiling p oin t in 1665 b y H uygens. Celsius in 1740 began d en otin g the m elting p o in t o f ice by 100° and the boiling p o in t o f water b y 0°. T hus he introduced th e centigrade scale, b u t the direction o f th is scale w as opposite to th e one used tod ay.

BASIC CONCEPTS OF THERMODYNAMICS

9

according to Kelvin’s scale by the letter T; it is evident that T = = a-1 + t where t is the temperature according to Celsius’ scale. The readings of all other thermometers are reduced to the gas thermo­ meter. As will be shown in the sequel, the second law of thermodynamics removes all arbitrariness in the definition of temperature by enabling us to establish an absolute scale of temperature (absolute temperature) independently of both the substance chosen and any thermometric parameter whatsoever. § 3. Quasi-static and non-static processes

Until now we have considered the properties of systems in a state of thermodynamic equilibrium when no parameter of the system varies with time, and inside the system there are no macroscopic motions. If some parameters of a system vary with time then we shall say that a process takes place in such system; for example, in a variation of volume a process of compression or expansion of the system takes place; in a variation of external field a process of magnetization or polarization of the system occurs, etc. If the system is moved away from a state of equilibrium and is left to itself, then, after a certain time, it will return to the initial equilibrium state. This process of transition of a system back to an equilibrium state is called relaxation, and the interval of time during which the system returns to a state of equilibrium is called the relaxation time r.f A process is called an equilibrium or quasi-static process if all para­ meters of the system vary physically indefinitely slowly, so that the system is found all the time in equilibrium states.} Physically an inf The relaxation tim e is different for different processes: it varies from 1 0 ~ 18 sec for the estab lish in g o f th e equilibrium pressure in a gas up to several years for th e equaliz­ in g o f concentration in hard a lloys. In therm odynam ics the largest relaxation tim e, during w hich equilibrium is reached for all param eters o f a given system s, is taken as th e relaxation tim e o f th e system . J The in d ication th a t in quasi-static processes all param eters (both in ten siv e and exten siv e ones) v a ry p h ysically in d efin itely slow ly elim inates th e need for introducing the concept, unnecessary for th e therm odynam ic in vestigation s, o f the so-called pseudoequilibrium processes (in which certain in ten siv e param eters are varied artificially b y a finite am ount). M oreover, as can bo seen from th e definition g iv en , quasi-static processes are n o t o n ly in d efin itely slow p h y sically b u t also alw ays begin from some equilibrium sta te. T his rem oves, as unnecessary in therm odynam ics, the need to underline th a t, although each equilibrium process is quasi-static, n o t every quasi-static process (defining it m erely as in d efin itely slow) is an equilibrium process (as an e x ­ am ple o f such non-equilibrium b u t in d efin itely slow process, one u su ally cites the

10

THERM ODYNAM ICS

definitely slow or quasi-static variation of some parameter a is called such a variation of this parameter with time that the rate of change dajdt be considerably smaller than its mean rate of variation in relaxa­ tion; thus, if the parameter a has varied by the amount Aa while the relaxation time is r, then in quasi-static processes we have da dt

Aa r

-----

(

1 . 1)

If a variation of some parameter a occurs during a time t smaller than or equal to the relaxation time x (t =s£ r) so that da dt

Aa r ’

(1.

2)

then such a process is called non-static. The concept of an equilibrium process and all considerations con­ nected with it prove possible only on the basis of the general principle of thermodynamics that an equilibrium state is not destroyed spon­ taneously. In fact, the direction of an equilibrium process will be fully determined by the character of the external actions if and only if spontaneous variations of the thermodynamic state of a system are excluded. The study of equilibrium or quasi-static processes is important since it is found (this will be verified later) that in these processes a whole series of practically important quantities (work, efliciency of machines, etc.) has limit, maximum possible, values. Therefore, results obtained in thermodynamics for quasi-static processes have in thermodynamics the role of some kind of limit theorems. § 4. The internal energy of a system. Work and heat

Physics studies the laws of various forms of motion of matter (me­ chanical motion, thermal motion, electromagnetic processes, atomic and nuclear processes and the motion of micro-particles). The common measure of material motion in its transformations from one type to another is called energy. Whatever processes occur in the world, process o f therm al exchange b etw een bodies at different tem peratures, m ade arbitrarily slow b y th e introduction b etw een them o f a therm al resistance). T he definition given in th e te x t identifies equilibrium an d quasi-static processes. A slowed-down therm al exch an ge is n o t an equilibrium process and therefore is non -static (though indefinitely slow) since a t th e in itia l in sta n t, in estab lish ing therm al con tact betw een th e bodies, equilibrium h as been destroyed.

BASIC CONCEPTS OF THERMODYNAMICS

11

whatever conversions of forms of motion are accomplished, the total quantity of energy always remains unaltered. The law of conservation and transformation of energy has a most important role in all natural sciences, in particular in thermodynamics, since it concerns an attribute, an inalienable property, of matter. Engels called it the “great fundamental law of motion,”f and consid­ ered the law itself the foundation stone of the principal theses of ma­ terialism. The law of conservation and transformation of energy has both a quantitative and a qualitative aspect. The quantitative aspect of the law of conservation and transformation of energy consists in the state­ ment that the energy of a sj'stem is a single-valued function of its state and is preserved in all processes in an isolated system, being only converted from one type to another according to a rigorously" deter­ mined quantitative equivalence relationship; the qualitative side of this law consists in the never-exhausted possibility of material motion to newer and newer transformations. Thermodynamics studies the laws of thermal motion. Every thermodynamic system consists of a huge number of particles. The energy of these continuously moving and interacting particles is called the energy of the system. The total energy of the system divides into internal and external energy. External energy comprises the energy of motion of the system as a whole and the potential energy of the system in a field of forces. All the remaining part of the energy of a sj^stem is called its internal energy. The motion of a system as a whole and the variation of i ts potential energy in such a motion is not considered in thermodynamics, and therefore in thermodynamics the energy of a system is its internal energy.J The internal energy of a system comprises the energy of all forms of motion and interaction of the particles that are part of the system: the energy of the translational and rotational movements of the molecules and of the oscillatory motion of atoms, the energy of molecular interaction, the inter-atomic energy of filled electron levels, the inter-nuclear energy etc. t F . E ngels, A n ti-D u h rin j, p. 13 (1957). J The p osition al en ergy o f a system in a field o f external forces is p art o f its external onergy on condition th a t th e therm odynam ic state o f th e system be n o t varied for a displacem ent in th e field o f forces. If, how over, th e therm odynam ic stoto o f a systom is varied in its d isplacem ent in a field o f forces, then a determ ined fraction o f the p oten tia l energy form s n o w a part o f th e internal onergy o f th e system .

12

THERMODYNAMICS

The internal energy U is an internal parameter and therefore, in equilibrium, it will depend on the external parameters a, and the tem­ perature T U = U { a 1, a2) . . . , a„, T ) . (l.B) The dependence of the internal energy U on the temperature T of nearly all systems encountered in the nature surrounding us is such that an unbounded increase of temperature is accompanied with an unbounded increase of the internal energy. This occurs since each molecule or any other element of an “ordinary” thermodynamic sys­ tem can have an arbitrarily large value of energy. I t has been established a few years ago that there also exist systems in which, as temperature increases, the internal energy tends asymp­ totically to a finite limit value, since each element of the system is limited in its maximum possible energy. Such “extraordinary” sys­ tems are the sets of nuclear spins of certain crystals, i.e. the sets of nuclear magnetic moments bound at nodes of the lattice and inter­ acting with each other when their energy of interaction with the lattice is extremely smallf in comparison with the energy of spin-to-spin interactions. In the interaction of a thermodynamic system with the surrounding medium, energy is exchanged. In this connexion two different methods of transmission of energy from the system to external bodies are pos­ sible: with variation of the external parameters of the system and without a variation of these parameters. The first method of transmission of energy, the one connected with variation of the external parameters, is called work, the second method, the one without variation of the external parameters, is called heat, while the transmission process itself is called thermal exchange. The quantity of energy transmitted by a system with variation of its external parameters, is called work W (and not quantity of work), while the quantity of energy transmitted by the system without varia­ tion of its external parameters is called quantity of heat Q. As can be seen from the definition of heat and work, these two different methods of transmission of energy considered in thermodynamics are not equi­ valent. In fact, whereas expended work can directly pass into an in­ crease of any form of energy (electric, magnetic, elastic, potential energy of gravitational force etc.), heat can only pass directly (i.e. without being preliminarily converted into work) into an increase of | On certain paradoxical properties o f such system s see § 46.

BASIC CONCEPTS OF THERM ODYNAM ICS

13

the internal energy of a system. This leads to the fact that, in the trans­ formation of work into heat, it is possible to restrict ourselves to two bodies only, the first of which (for a variation of its external para­ meters) transmits by thermal contact energy to the other body (with­ out a variation of its external parameters); on the contrary in the trans­ formation of heat into work there must be at least three bodies: a first one that gives up energy in the form of heat (thermal source), a second one th at receives energy in the form of heat and yields energy in the form of work (it is called the working body), and a third one that receives energy in the form of work from the working body. If a system does not exchange energy with surrounding bodies it is called, as has already been said, an isolated or closed system; if, how­ ever, it does not exchange energy with other bodies in the form of heat only, it is called an adiabatically isolated or adiabatic system. The work W and the quantity of heat Q have the dimension of energy but work and heat are not forms of energy: they represent the two dif­ ferent methods of transmission of energy considered in thermodynamics and therefore characterize a process. The work W and the quantity of heat Q are only different from zero in a process undergone by the system; to a state of the system, however, there corresponds no value of W or Q whatsoever. I t is assumed to consider the work W as positive if it is accomplished by a system on external bodies, while the quantity of heat Q is assumed positive if energy is transmitted to the system without variation of its external parameters.f For an indefinitely small equilibrium variation of the parameter a the work accomplished by the system is equal to 6W = A d a ,

(1.3)

where A is a generalized force associated with the external parameter a and which, under equilibrium conditions, is a function of the external parameters a%and the temperature T. For a non-static indefinitely small variation of the parameter a the work 6Wn accomplished by the system is also equal to

Al

4jz

«3 = A > (l.C )

— E v) A S= - ~ E Z. 471 y 3 471 z

’ A ~~

In the case of an isotropic dielectric when D is parallel to E this work will be S W = - — E dD

(a = D ,

471

4 71

A= - —

e

\ .(l.D )

The polarization work proper (or polarization work in the true sense) (3Wp is the work d W less the work —d(E2j 871 ) of excitation of the field in a vacuum dWv= d W + d

E2

= - EdP

(a = P,

A = — E).

(1.9)

(d) In a similar manner the elementary work for a variation in a magnet with induction B of a magnetic field H is equal to oiffy, (1-30) which is very important in practice for the determination of y in solid and liquid bodies, since it is impossible to raise the temperature of a bod}7, without a variation of its volume (or of the envelope in which it is contained). PROBLEMS 1. Show tlin t the tw o follow ing experim ental facts: the uniqueness o f the energydistribution o f an equilibrium system am ong its parts and the sim ultaneous increase o f th e energy o f these parts for an increase o f the total energy o f the system , enable us to choose for th e internal energy a m onotonically increasing function of tem ­ perature. 2. Show th a t the differential expression for the elom ontary work 511’ — ^ , A , da,

^

w is n o t an ex a ct differential. 3. E valu ate the work o f evaporation o f one m ole o f water when this is converted into vapour nt 100°C and under norm al pressure. D eterm ine also the q u a n tity o f h eat com m unicated nt th e sam e tim e. 4. E valu ate the work accom plished b y th e u n it volum e of the core of a long solenoid when its m agnetic p olarity is reversed tw ice, i f it is known th a t the area o f the loop o f th e hysteresis curve for the core in th e coordinates H and A1 is equal to S. 5. Show th a t the elem entary work o f polarization per u n it volum e o f an isotropic dielectric is equal to SW-= —

1 -E(1D,

4 7t

(l.H)

w hile th e elem entary work o f polarization proper (see p. 25) is equal to 8 ]Vp = — E d P .

(l.J)

6. Transverse w aves o f frequency v and am plitude a are propagated from loft to right along a string. The tension o f the string is equal to T. D eterm ine the work accom plished per period b y th e portion o f string situated on the left o f a certain p oin t o f the string on tho portion o f the string situated on the right o f this p oin t. 7. E stablish th a t for an y sim ple system subject to the action o f a generalized force A (associated w ith tho extornal param eter a) the follow ing id en tity is valid

8. E stab lish th e connexion betw een tho therm ic coefficients a, /S and y. 9. Considering an ideal g a s as a set o f non-intoracting continouslv m oving particles, find for it tho therm ic and clnoric equations o f state. 10. A t a certain tem perature T = T c and a certain pressure p = p c tho difference betw een tho specific volum es of liquid V, and gas Vff van ish es (F / = Vg = Vc). Such a stato of a substnneo is called critical, w hile tho values of the param eters T c, p c and Fc for which th is occurs are called critical param eters.

BASIC CONCEPTS OF THERMODYNAMICS

>

i

21

E xpress the critical param eters Ve, p c and T c o f a van der W aals’ gas in term s o f th e con stan ts a and b o f th is gas and ovalutao the critical coefficient

s—

RTC

(l.L)

Vcvi

11. F ind expressions o f tho critical param eters Vc, p c and T e b y proceeding from D iotcrici’s equation

p{V— b) = RTe~a,nrV .

(!-M)

E valu ate tho critical coefficient a = R l ' ej pcVc for th is equation and com pare it w ith tho experim ental value and tho value ob tain ed from van der W auls’ equation. Show th a t for largo volum es D ietorici’s equation reduces to van der W aals’ equation. 12. E valu ate tho critical coefficient a for D iotorici’s second equation

\v+ ^ Y v - b ) = RT

(l.N)

and com pare it with tho experim ental value and the value obtained from van dor W ea ls’ equation. 13. I f tho critical param eters aro used as units for tho m easurem ent o f pressure, volum e and tem perature, wo obtain tho reduced variables

7t=

V , Pc

V

Gv, finds a relation between these thermal capacities and the elasticity moduli, defines quantities the variations of which indicate the direction of natural processes, obtains equilibrium conditions etc. Thermodynamics cannot, however, determine the functional dependence of thermal capacity on tempera­ ture, cannot obtain the equation of state, leaves the so-called entropic and chemical constants undetermined etc. When approaching the study of phenomena from a purely thermodynamical viewpoint, all these data must be taken from experiment. Such a study cannot of course be considered definitive and complete; this raises the need for a molecularstatistical approach to the study of phenomena, which is just the con­ tent of statistical physics. The results of thermodynamics have the advantage of being based on the most general empirical laws without being tied to any particular conception of the properties of the particles that make up a system (a system of classical particles, a system of Bose-Einstein’s quantum particles, a system of Fermi-Dirac’s quantum particles etc.); the rela­ tions of thermodynamics are indeed true for all statistical systems. The laws of thermodynamics, expressed quantitatively in the form of determined equations are the basic equations of thermodynamics. We shall proceed now to discuss the contents of these basic laws and the basic equations of thermodynamics corresponding to them. T h e r m o d y n a m ic s

23

THE FIR ST LAW OF THERM ODYNAM ICS

§ 6. The equation of the first law of thermodynamics T

t h

o f

h e

e

q

f i r s t

u

a n

o r k ,

e

t h

e r m

a s p e c t

h a s

s t a g e

i . e .

t h

w

o

a n

a n

d

i n t o

o n e

I t

o f

b e e n

r k

t h

e

o f

e

w

o

o f

i n

h i c h

o f

s a m

w

e

o

y

n

a m

t h e

t h e

w

a s

d i s c o v e r y

d

d

i c s

l a w

r k

d

t h

o f

o

e

m

a

m

a

r e s u l t

o f

f a c t

e a t

c o n

e m

a l w

s t a n

o

f

p h y s i c s

t h

i s

a t h

a t i c a l

c o n s e r v a t i o n

d i s c o v e r y

h

r i g o r o u s l y

f

a s

a i n

t h e

i n t o

i s

o

e s t a b l i s h e d

i n v e s t i g a t i o n s

c l u d i n g

w

t i t a t i v

e n e r g y .

r e t i c a l

l a w

t

q

o f

a

t

a y s

u

a n

t h e

t h

e x

a n

p

d

t r a n

e r i m

d

e n

c h e m

e

t r a n

t i t a t i v

s f o

l a t i o

s f o

r m

a n

i s t r y ,

r m

o f

a t i o

p l i s h e d

e

u

t a l

e q u i v a l e n c e

a c c o m

a n

f o r m

o f

a t i o

n

d

t h

h

n

t h

e

e o

c o n ­

e a t

n

o f

a n

h

n

d

e a t

a c c o r d i n g

r e l a t i o

­

t o

. f

the internal energy of a system is a single-valued function of its state and varies only under the influence of external actions. T

h

T

w

e

f i r s t

o

t y

a c t i o n s

s y s t e m

w

i t h

n a l

e s

p

( t h e

a r a m

t o

o f

o f

t h

i n t e g r a l

w

i t h

s y s t e m

n

e t e r s

t h e

e r m

o

d

y

e x t e r n a l

c o n n e c t e d

v a r i a t i o

c a t e d

I n

p

l a w

o f

o r

s y s t e m

f o r m

,

p

i c s

e s t a b l i s h e s

a r e

v a r i a t i o n

a c c o m

t e m

a m

a c t i o n s

a

e x t e r n

n

a l

p l i s h e s

p

e r a t u

a r a m

r e

( a

c o n s i d e r e d

o

w

t h

f

t h

e

b

u

c e r t a i n

a n

t

q

t

i n

e x t e r n a l

o r k ) ,

e t e r s

a

d

p

e r m

a r a m

a c t i o n s

c a u s i n g

a

u

o f

a n

t h

t i t y

n

o

o d y n a m

e t e r s

t

v a r i a t i o

h

e a t

i s

i c s :

o f

t h e

c o n n e c t e d

n

c o

o f

m

i n

m

t e r ­

u

n

i ­

) .

i . e .

f o r

a

f i n i t e

p r o c e s s ,

t h

e

Ui - U 1 = Q - W ,

f i r s t

l a w

i s

d e s c r i b e d

t h

u

s

( 2 . A

)

o r

Q = U 3 - U 1-\-W t

( 2 . 1 )

| The im possibility o f a m echanical perpetual m otor, i.e. o f such a d evice as w ould enable us by its m eans to obtain m echanical work w ithout th e expenditure o f any energy, had already been discovered in th e eigh teen th century. In 1748 M. L. Lom onosov in a letter to E uler, sta tin g his thoughts on the law o f conservation o f m atter and exten d in g it to m aterial m otion, wrote: “ a body th a t by its im petus stim ulates another body into m otion loses as m uch m otion as it com m unicates to th e other” . In 1755 the F rench A cadem y o f Sciences declared ‘‘once and for ev er” th a t it w ould no longer accept any projects o f perpetual m otors. In 1840 H . H . H ess form ulated th e law of the indep en d en ce o f th e h e a t effect o f chem ical rections o f interm ediate reactions. In 1842-1850 a w hole series o f in vestigators (Mayer, Joule, H elm holtz an d others) arrived at th e discovery o f th e equivalence o f heat and work, and th e m echanical equivalen t o f h ea t w as determ ined: 1 kcal = 427 kg. The establishing o f th e eq u iv a ­ lence principle w as th e last stage in th e form ulation o f th e q u an titative aspect o f th e law o f conservation and transform ation o f energy, and therefore th e date o f the esta b ­ lishin g o f th is principle is u su ally identified w ith th e date o f th e discovery o f th e first law o f therm odynam ics. I t can be seen from th e historical inform ation above th a t several decades were required in order th a t science could m ove all the w ay from th e mere conviction o f the im possibility o f a perpetual m otor to th e modern form o f the law o f conservation and transform ation o f energy.

24

THERMODYNAMICS

where U2— U1 is the variation of internal energy of the system for the transition from a first to a second state, Q is the quantity of heat received in this connection by the system and W is the work accom­ plished by the system. According to the first law the variation of internal energy U2 — U1 will be one and the same independently of the path taken by the system for passing from the state 1 to the state 2 (Fig. 2), whether the path denoted conventionally by a or the path denoted conventionally by b is used; however, Q and IF will be dif­ ferent. This means that Q and W are not functions of state but characterize a process and, if there is no process, the system will have neither Q nor W while the internal energy always exists. The dependence of Q and W on the path can be seen from the elemen­ tary example of the expansion of a gas. The work for the passage of the system from 1 to 2 (Fig. 3) along the path a 2

W .= fp (V ,T )d V i

(2.B)

(a)

is represented by the area bounded by the contour A \a 2 B A ) while the work for the passage along the path b is represented by the area bounded by the contour A \b2BA: 2

Wb= \ p ( V , T ) d V . 'i (6)

(2.C)

The difference of Wa and Wb is explained by the fact that, since the pressure depends not only on the volume but also on the temperature, then in the presence of different temperature variations along the paths a and b in passing from one and the same initial state (pv Vx) to one and the same final state (p 2, V2) a different work is obtained. I t can be seen from this that in the closed process (cycle) la26l the system accomplishes work that is not equal to zero. On this is based the work of all heat engines. I t

f o l l o w

a c c o m

a

t

t h e

s

p l i s h e d

f r o m

t h

e i t h e r

e x p e n s e

o f

e

a

f i r s t

t

c o m

t h

m

e

u

l a w

o f

e x p e n s e

n

i c a t i n

g

t h

e r m

o f

t o

a

t h

o

d

y

n

a m

v a r i a t i o n

e

s y s t e m

i c s

o f

a

t h

a

t

w

o r k

i n t e r n a l

q

u

a n

t i t y

c a n

e n e r g y

o f

b e

o r

h e a t .

THE FIRST LAW OF THERMODYNAMICS

25

In the case when the process is cyclical the initial and final stages coincide, U2 — Ux = 0 and W = Q, i.e. in a cj^clical process, work can be accomplished only at the expense of heat received by the system from external bodies. For this reason the first law is often formulated as a proposition stating the impossibilit}^ of a per­ petual motor of the first kind} i.e. the impossibilitj^ of such peri­ odically acting device as would accomplish work without receiv­ ing energy from without. The proposition of the impos­ sibility of a perpetual motor of the first kind admits of another formulation: work cannot be created from nothing or be reduced to nothing. The equation that expresses the first law for an elementary process will be 6Q = d U + d W , (2.2) or 6Q = d U + Z Aidai(2 . 3 ) X,) = dx2 6 6

x

3

A

.

(nX,)

(rt) = J r W

,

.

(3.K)

6_ (fix,) = - - - ( fiX ,) . 6x1 By eliminating from these equations the function /u(x1 , x2 , x2), we shall multiply the first equation by X 1 the second by X 2 and the third by X 3 and after this we shall add them up. We shall then obtain the necessary condition of holonomicity of the Pfaffian form (3.11) X t K - s- - dM

+ x M ^ - dM

+ X 3f ^ 2 - dM

= 0 , (3.12)

or, more briefly, (Z, rot Z) = 0,

(3.13)

where Z is a vector of components Xj_ , X 2 and X 3. This condition is not only necessary but also sufficient: when it is satisfied the Pfaffian form (3.11) is holonomic. In the general case of an arbitrary number of variables (to > 3) for the Pfaffian form (3.10) to be holonomic it is necessary that the following identity (Z, rot Z) = 0, (3.L) be satisfied for any three of the functions Xi, Z being a vector of com­ ponents equal to three of these functions. By equating a Pfaffian form to zero we shall obtain Pfaff’s equation. SITa = 0 (3.14)

50

THERMODYNAMICS

for a holonomic form (3.11), i.e. for /i 8IT3 = d &2> • • ■> fyn) >

) |

(3-24)

A ~ • • • >fr/rt) • ) Since Aj does not depend on and A2 does not depend on a,-, then it followsfrom (3.22) and (3.23) that A does not depend on ai and bk, Aj does not depend on a%, and A2 does not depend on bk. Thus we have from (3.24) that Ai =

(ffi),

T 2. f See Y . S. M artynovskii, Teplovyye nasosy (H eat P um ps), G osenergoizdat, 1953.

74

THERMODYNAMICS

The unilateral character of heat transmission can serve as the basis for defining the concept of larger or smaller temperature. Generally speaking, this concept can be defined both within the limits of quasi­ static processes (in communicating heat to a body its temperature increases) and by an analysis of non-static processes (if of two bodies A and B in thermal contact heat flows A to B, then the temperature of the body A is larger than the temperature of body B). The second law shows that the two definitions are not independent of each other so that the definitions given in the text do not contradict each other (see § 31). In the scientific and pedagogical literature the statement is often encountered that the law of the increase of entropy and the fact of the existence of entropy are independent theses, since the latter is perfectly compatible with the opposite thesis “the law of decreasing entropy” or any other. However we must observe the following. Ju st as we have seen, if we take the absolute temperature (for ordinary systems) to be positive, then only the law of the increase of entropy is compatible with the fact of the existence of entropy and the law of the decrease of entropy is incompatible with this fact; if, however, a negative absolute temperature is taken the contrary will be true. On the other hand, the result that the absolute temperature cannot vary its sign (in quasi-static processes) is a consequence of the fact of the existence of entropy in every equilibrium system, and the choice of this sign depends on the definition of the concept of higher or lower temperature. The second law of thermodynamics effectively consists of two inde­ pendent theses which are expressed either in the form Q> W

and

W = Q,

(3.CC)

or, in the language of entropy, in the fact of the existence of entropy and its unilateral variation in non-static processes in adiabatic closed systems. § 20. Limits of applicability of the second law. The “ theory” of the thermal death of the universe T he second law o f therm odynam ics is an expression o f th e natural features proper o f phenom ena connected w ith therm al m otion. I t establishes a difference o f principle o f th e m icrophysical form o f transm ission o f energy, h eat, from th e m acrophysical form which is connected with a variation o f external param eters, work. This differ­ ence which leads to th e existen ce an d to th e increase o f entropy is v a lid both in large

THE SECOND LAW OF THERMODYNAMICS

75

and in sm all bodies. H ow ever, in th e case o f sy stem s th e dim ensions o f which are com ­ parable w ith th e dim ensions o f m olecules, th e difference betw een the concept o f heat an d work van ish es, a n d therefore th e therm odynam ic param eters, entropy, tem pera­ ture etc., h a v e no m eaning for such m icrosystem s, f T his leads to th e sta te m e n t th a t th e second law o f therm odynam ics is inapplicableto m icrosystem s. The la tter doos n o t m ean th a t in such system s th e second law is v io la t­ ed— w e can n ot im p lem en t a p erpetual m otor o f th e second kind w hatever th e system s w e use*— it o n ly m ean s th a t to speak o f a perpetual m otor o f th e second k in d as a d evice capable o f con vertin g h ea t in to work w ith o u t com pensation w ould be w ith ou t a m ean in g w hen applied to m icrosystem s, since for them th e distinction betw een h ea t an d w ork does n o t exist. Thus a lower boundary o f ap p licab ility ex ists for th e second law a n d therefore for all therm odynam ics: th e second law is in ap p licab le to m icrosystem s. An upper boundary o f ap p licab ility o f therm odynam ics can also bo show n. The in itia l th esis o f th erm odynam ics o f th e therm al equilibrium , o f th e stead y transition o f an iso la ted sy stem to an equilibrium sta te and the self-indestructibility o f this state, and th e law o f th e increase o f en trop y in such system s are the result o f generalizations o f exp erim en tal d ata in system s o f fin ite dim ensions. T he uncritical exten sion o f these law s to system s o f in fin ite dim ensions, w ith ou t an a n a lysis o f the profound q u alita tiv e v ariation s th a t can be in v o lv ed in such a transition from finite to infinite system s, can lead and h as a ctu a lly led certain scien tists to anti-scientific conclusions on th e so-called therm al death o f th e universe. The “ th eo ry ” o f th e therm al d eath o f th e universe w as ex p licitly form ulated about one hundred years ago b y C la u siu sff w ho, b y ex ten d in g th e law o f therm odynam ics t Therefore such system s are n o t th erm odynam ic system s. t I t m ig h t appear a t first sig h t th a t th e presence o f fluctuations g iv es u s in p rin ­ cip le th e p o ssib ility o f constru ctin g a p erpetual m otor o f th e second kind. H ow ever, th is is n o t so. L e t us consider, for exam p le th e fluctuation o f d en sity in a gas. I t m ight appear possible to “ ca tch ” th e pressure differences arising b y m eans o f special va lv es an d apparatus capable o f dealing w ith single m olecules (such d evices were called by M axw ell, dem ons (consciousness, in telligen ce)) an d to use them for accom plishing work or for sep aratin g a m ixture o f gases. T his is how ever, im possible n o t o n ly p ra cti­ cally b u t also th eoretically. A ll our apparatus, v a lv es etc. th em selves con sist o f m ole­ cules a n d th em selves e x h ib it oscillation s a b ou t th e position o f equilibrium , th ese being com p letely in d ep en d en t o f th e oscillation s o f gas d en sity. T he desired result could be ob ta in ed a t a certain in sta n t o f tim e, b u t w ould be com pensated a t th e n e x t in sta n t b y th e oscillation s o f th e apparatus an d th e gas. t f Since th en , th is pseudo-scientific theory has ob tain ed fairly w ide acceptance in bourgeois scientific circles. A n ti-scien tific fab rications on th e “ therm al d ea th ” o f the universe are b ein g diffused also in our tim es b y certain bourgeois scien tists. Thus, for exam p le, in J e a n s’ The Universe A round Us (1930, p . 328) w e read: “The universe can n o t e x ist for ever; sooner or later a tim e m u st com e w hen its la st erg o f en ergy w ill reach th e h ig h est step on th e ladder o f decreasing usefulness, and, a t th is in sta n t, the a ctiv e life o f th e universe m u st cease.” E d d in gton in The Nature of the Physical World (1935, p . 90) wrote: “The en tire universe w ill reach therm al equilibrium in th e future a t a tim e th a t is n o t in d efin itely rem ote. There ex ists a trend o f th o u g h t th a t finds th e id ea o f th e w ear o f th e w orld fra n k ly repellent. Various rejuvenation theories are favoured b y th is school. B u t I confess th a t I h ave personally no great desire th a t the final stop p in g o f th e un iverse b e a verted . . . I w ould feel more gratified if th e universe accom plished som e great schem e o f evolu tion and, h a v in g accom plished all th a t could be accom plished, fell back to chaotic im m u tab ility, than if its purpose were debased b y u n w earyin g rep etitio n ” .

7G

THERMODYNAMICS

to th e entire in fin ite universe wrote: “T he energy o f th e world rem ains con stan t, the entropy o f th e world tends to a m axim um ” . This m eans th a t th e universe w ill sooner or later arrive at a sta te o f therm odynam ic equilibrium; then all processes w ill cease and settle down in a sta te o f “ therm al d eath ” : th e tem perature a t all places o f the universe w ill be one an d th e sam e, all other in ten siv e factors w ill be equalized and there w ill be no more causes capable o f g iv in g rise to an y process w hatsoever. This “ th eory” o f th e therm al d eath leads directly to religious superstition— to the existen ce o f God. In fact, since according to Clausius th e universe m oves continuously towards therm odynam ic equilibrium , w hile it is n o t in an equilibrium state a t the present m om ent, th en it follow s th a t eithor th e universe has n o t alw ays existed and has been som ehow created, or som e sort o f external force has a t som e tim e rem oved it from its equilibrium sta te and w e are now livin g the epoch o f th e returning o f the universe to a sta te o f equilibrium . This m oans th a t God m ust e x is t.f The reactionary view s o f Clausius h a v e been subject to E n gels’ crushing criticism . From th e positions o f dialectic m aterialism E ngels proved tho com plete incon sisten cy o f th e “ th eory” o f th e therm al death. In his Dialectics of N ature he wrote: “In w hatever form the second thesis of Clausius etc. is p u t forwrad, in any case according to him energy is lo st q u alita tiv ely if n o t q u a n titatively. Entropy cannot disappear by natural means but instead can be created. T he world at th e b eginning m ust h ave been w ound up and th en it w ill go u n til it reaches a sta te o f equilibrium , and on ly a miracle can m ove it aw ay from th is sta te and start it back into m otion. The energy expended to w ind up th e clock has disappeared, a t least q u alitatively, and can o n ly be restored by m eans o f a stim ulus from without. Thus a stim ulus from w ithout was also needed at th e beginning; th u s the q u an tity o f m otion or energy available in th e universe is n o t alw ays th e same; th u s energy m ust h ave been created: th u s energy can be created; thus energy can be destroyed” (F. E n gels, The Dialectics of Nature, G ospolitizdat, 1 9 5 2 , p . 2 2 9 ).

E ngels show s in th is m anner th a t th e “th eory” o f th e therm al death o f th e universe contradicts th e theory o f conservation an d transform ation o f energy, since th is law , as w e have already ind icated , affirms n o t o n ly th e q u a n titative indestructibility o f m otion o f m atter b u t also its never-exhausted ab ility for qualitative transform ations o f various form s o f m otion in to one another. “The in d estru ctib ility o f m otion — wrote E ngels— is to be understood n o t on ly in a q u an titative b u t also in a q u alitative sense. M atter, th e purely m echanical d isp lace­ m en t o f which does con tain in itse lf tho p ossib ility o f transform ation under favourable conditions in to h eat, electricity, chem ical action, life, b u t which is n o t in a sta te o f generating b y itse lf these conditions, such matter would suffer a certain deterioration in its motion. M otion th a t had lost its cap acity for transform ation in to d i f f e r e n t forms peculiar to it, though it still possesses dynam is (possibility) it no longer possesses energeia (efficacy) an d th u s is p artially destroyed. B u t such m otion an d such m atter are u n th in k ab le” (ibid., p. 17). E n gels established th e fa lsity o f th e conception o f tho possibility itse lf o f therm al equilibrium in a universe infinite in space an d tim e, for in m oving m atter itself there lies th e cause o f its m otion. H e wrote: “ . . . m atter in all its transform ations alw ays re­ m ains one and th e sam e, n on e o f its attributes can ever be exhausted and therefore by f In 1952 in his address to a session o f th e “Pontifical A cadem y o f Sciences” th e P ope Pius X I I said: “The law o f entropy discovered b y R udolph Clausius has given us th e certain ty th a t spontaneous natural phenom ena are alw ays connected w ith a certain loss o f th e free energy capable o f utilization, w hence it follows th a t in a closed m aterial system those processes at th e m acroscopic scale w ill in th e end at som e tim e cease. This sad n e c e ssity . . . is an eloquent evidence o f th e existence o f a Necessary B ein g ” .

THE SECOND LAW OP THERMODYNAMICS

77

th e sam e iron n ecessity b y which som e d a y it w ill destroy on E arth its h igh est flower, in tellig en ce, it m u st again gen erate it som ew here in another p lace a t anoth er tim e (ibid., p . 19). L eading scien tists o f th e la st cen tu ry h ave also com e forward a g ain st C lausius’ con cep tion o f th e th erm al d eath o f th e universe. O f great progressive v a lu e are, in th is co n n exion , th e works, first o f all, o f th e lea d ­ in g p h y sicist-m a teria list B oltzm an n an d the works o f th e w ell-know n P olish p h y si­ cist-m a teria list Sm oluchow ski. To cou n teract th e “ th eo ry ” o f th e th erm al death o f th e universe B oltzm an n a d ­ van ced th e so-called “fluctuation h y p o th esis” . B o ltzm n a n w as th e first to establish the sta tistica l naturo o f th e second law o f th erm odynam ics. A ccording to B oltzm an n , a sta te o f therm odynam ic equilibrium is o n ly on e th a t occurs m ost o ften , a m ost probable state; in an equilibrium sy stem there can alw ays sp on tan eou sly arise arbitrarily large fluctu ation s. B y ex ten d in g th ese d ed u ction s to th e w hole universe, B oltzm an n arrives at the conclusion th a t th e un iverse is found, gen erally speaking, ina sta te o f therm odynam ic equilibrium ; in it, how ever, th ere in e v ita b ly arise arbitrarily large fluctuations. Such an enorm ous fluctuation is th e p art o f th e u niverse in w hich w e are found. E ach fluc­ tu atio n m u st v a n ish b u t eq u ally in e v ita b ly there w ill arise fluctuations o f a sim ilar k in d in other p la ces o f th e universe. T hus, according to B oltzm an n , som e w o ild s perish an d others com e in to bein g. I n h is Lectures on the Theory of Oases (§ 90) B oltzm an n w ro tef “ W e can think o f th e universe as a m echanical system con sistin g o f an enorm ous num ber o f com ponent parts a n d w ith an enorm ous duration o f existen ce, so th a t th e dim ensions o f our s y s ­ tem o f fixed stars are n egligib le in com parison w ith th e exten sion o f th e universe and the duration o f its existen ce. T hen in th e universe w hich, in general, is found ev ery ­ where in a sta te o f therm al equilibrium , i.e. d eath , here an d there com p aratively sm all regions o f th e exten sion o f our stellar sy stem m ust ex ist (we shall call th em solitary w orlds), th a t in th e course o f a com p aratively b rief tim e era d eviate considerably from therm al equilibrium , there being equally p arts in w hich th e p rob ab ility o f sta te in ­ creases an d parts in which it decreases. Therefore, for th e universe, th e tw o directions o f tim e are in d istin gu ish ab le, ju st as in space there ex ist n o up an d dow n. B u t, ju st as a t a determ ined p o in t o f th e E a rth ’s surface th e direction tow ards th e centre o f th e E arth is defined as th e dow n direction, liv in g substance th a t is found a t a determ ined p h ase o f tim e on such a solitary w orld w ill determ ine th e direction o f tim e tow ards less probable sta tes rather th a n th e op p osite direction (the first as th e p a st, th e origin, and th e second as th e future, th e end), an d , in accordance w ith such designation, th e sm all “ b eg in n in g ” regions, isolated from th e universe, w ill alw ays be found in a sm all p ro­ b a b ility sta te . T his m eth od seem s to m e th e o n ly m ethod b y w hich w e can conceive th e second law , th e therm al death o f each solitary world, w ith ou t a unilateral v a ria ­ tion o f th e w hole universe from a determ ined origin to an ultim ate final sta te .” Sm oluchovski, w ho had b rillia n tly proved th e existen ce o f flu ctu ation s, w as a p ar­ tisa n o f B o ltzm a n n ’s view . In o n e o f h is declarations h e said: “Clausius m ain tain ed , on th e basis o f em pirical th erm od yn am ics, th a t th e entrop y o f th e universe co n tin u a lly increases, a n d th a t, therefore, th e universe m u st p ass, in tim e, to a stage o f num bness, th e notorious W aerm etod, in w hich all p o te n tia l en ergy w ill be converted in to h ea t and all differences in tem perature w ill b e equalized. T he k in etic th eory, on th e contrary, m ain ta in s th a t after a sta g e o f n u m b n ess n ew life w ill again arise, sin ce a ll sta tes return w ith tim e in p erp etu al ro ta tio n .” B oltzm a n n , Sm oluchovski and, in our tim e, m an y others consider th a t th e fa llacy o f th e dedu ction o f th e therm al d eath o f th e universe lies in con ceivin g th e law s o f t L. B o ltzm a n n , L ektsii po teorii gazov (transl.), G ostekhizdat (1956).

78

THERM ODYNAMICS

therm odynam ics as absolute law s, i.e. in n eglectin g th e statistical nature o f the second law , since th e fluctuation h yp oth esis supported b y them rejects th e “th eo ry ” o f the therm al doath on th e basis o f th e sta tistica l nature o f th e second law . As a m atter of fact th e “th eory” o f th e therm al death o f th e universe has been falsely developed n o t as a consequonce o f ignoring th e sta tistica l nature o f the second law but, on th e contrary, ow ing to th e in ap p licab ility o f th e second la w to infinite system s as a consequenco o f its statistica l nature. T his has been show n recently b y I. R . Plotkin who in his paper “On th e increase o f en trop y in an in fin ite un iverse” (Zh. eksp. teor. fiz. 20, 1051 (1950)) has established th a t in an in fin ite universe there exist no more probable states (the con cep t o f therm odynam ic equilibrium as th e m ost probable sta te has, therefore, no m eaning w hen applied to the universe) and therefore the law o f the increase o f entropy, which is valid in finite isolated system s, is ineffective in th e w hole universe or in an in ­ finite part o f it (see problem no. 64). In passing from finite to infinite system s a quali­ ta tiv e saltu s occurs in tho tem poral evolution o f such system s. This establishes an upper bound o f applicab ility o f th e second law and reveals tho fallacy o f th e form ulation itself o f th e question o f th e therm al death o f th e universe. T he m etaphysical character o f B oltzm an n ’s hyp oth esis can also be seen from the fact th a t, according to B oltzm an n , tho d evelopm ent o f the W orld has th e character o f therm al fluctuations on a general background o f “therm al d eath ” and n o t as a m o­ tion along an ascen d in g lin e, n o t as th e transition from an old q u alitative sta te to a new q u alitative sta te, n o t as a develop m en t from sim ple to com plex, from low er to higher, as is considered b y dialectic m aterialism . B o ltzm a n n ’s deduction according to which a state o f therm odynam ic equilibrium prevails in th e universe, is contradicted m ore and more by th e growing experim ental d ata o f astronom y. A t th e sam e tim e th is astronom ic m aterial confirms th e view s o f dialectic m aterialism on th e d evelop m en t in nature. T he recent observations and dis­ coveries o f astrophysicists on th e arising a n d d evelopm ent o f stellar associations show th a t m atter never exh ib its an exhaustion o f its capacity for energy concentration and for transform ation o f form s o f m otion in to one another. A t the sam e tim e th ey establish th a t th e process o f form ation o f stars from dispersed m atter obeys determ ined laws, goes through determ ined stages a n d in no case can be reduced to random fluctuations o f th e distribution o f energy in th e universe. PROBLEM S 28. I s a process possible, in which h eat taken from a tnerm al source is entirely co n ­ verted in to work? 29. A ttem p ts were repeatedly m ade in th e la st century to find such processes as would prove a contradiction to th e second law . A more accurate analysis o f such proces­ ses has shown each tim e th a t their contradiction to th e second law is on ly apparent. W . W ien ’s lig h t-v a lv e f paradox (1900) belongs to this group. B etw een tw o bodies o f equal tem perature there is situated a lig h t va lv e (Fig. 10). L igh t from th e bod y 1 is in cid en t on th e prism I. One h a lf o f the in cid en t energy J , i.e. J /2 , is transm itted through th e prism I , th en through th e prism I I and is absorbed

j" A lig h t v a lv e is a device as follows: two N icol prism s are situ ated on the path o f a ray o f lig h t in such a m anner th a t their principal planes are rotated b y 45° with respect to each other; betw een th em there is found a layer o f a substance in which a rotation o f th e plane o f polarization is caused by a sta tic m agnetic field directed along the ray, for exam ple from th e first N icol prism to the second (Faraday’s phenom enon). In contrast to a natural rotation o f th e plane o f polarization (w ithout a m agnetic-field, for exam ple a sugar solution) th e direction o f rotation o f the plane o f polarization is

THE SECOND LAW OP THERMODYNAMICS

79

by th e b o d y 2, w hile th e other h a lf, after to ta l internal reflection in th e prism I is returned b y th e mirror S 1 to th e b o d y 1. On th e other hand, th e lig h t sen t b y the body 2 is n o t tran sm itted b y th e lig h t v a lv e. H a lf o f th e en ergy J radiated b y th is body, equal to Jj'2, after to ta l in tern al reflection in th e prism H , w ill return, b y reflection

S

S2 '////////////// a

from th e mirror S t to th e b o d y 2. T he second h a lf o f th e energy, on passing through th e prism I I , w ill traverse the rotatin g m edium undergoing a rotation o f the plane o f polarization b y 45°, w ill n o t be tran sm itted through th e prism I b u t, b y to ta l reflection in it, w ill b e reflected back b y th e mirror S 3, and, after to ta l internal reflection in th e prism I w ill return to th e b o d y 2. Thus th e b od y 1 receives th e en ergy J j 2 and th e body 2 th e en ergy 3J/2 and w ill be h eated in contradiction to the second law o f therm o­ dynam ics. W ien reckoned th a t com pensation for heatin g th e body 2 w ill be obtained on accou n t o f som e unknow n processes in th e m agneto-rotating substance. R ayleigh p oin ted o u t an elem en tary error in th e above reason­ in g b y W ien. In w h a t does th is error consist? 30. On a h o t san d -b ath there is p laced a high hardglass te st tu b e on th e b o tto m o f w hich a layer o f a n i­ lin e is poured an d ab ove it a su fficien tly large q u a n tity o f w ater. A fter a certain tim e a drop o f an ilin e rises to th e surface o f th e w ater b y accom plishing work again st th e gravitation al forces an d th en falls again to th e b ottom . This process w ill be repeated as long as th e bath is heated . H ow can w e exp lain such a m otion o f the drop ? D oes it n o t contradict th e second law on the im possib ility o f periodically accom plishing work a t the expen se o f h ea t from a single therm al source ? 31. A beau tifu l transparent illustration o f th e sec­ ond law o f therm odynam ics is provided b y th e Chi­ nese “d u ck lin g” to y , which should becom e a standard device for physical dem onstration in every p hysical laboratory. The “d u ck lin g” (Fig. 11) is a h erm etically-

n o t determ ined b y th e direction o f th e ray b u t b y th e direction o f th e m agnetic field. Therefore if th e in te n sity o f th e m agnetic field has been so chosen th a t in th e propaga­ tion o f lig h t from th e prism I to th e prism I I th e plane o f polarization is rotated by 45° and lig h t passes through th e prism I I , th en in a ray o f opposite direction, in the propagation o f lig h t from prism I I to prism I , the plane o f polarization w ill be rotated b y 45° in th e sam e direction as in th e first case and lig h t does n o t pass through the prism I.

80

THERMODYNAMICS

sealed shaped am pule o f glass on a m etal base. The am pule is filled w ith a volatile liquid. In equilibrium th e stem o f th e duckling is in clin ed by several degrees w ith respect to th e vertical. T he h ead an d th e beak are covered w ith a th in layer of cotton w ool. I f th e head is som ew hat m oistened, for exam ple, b y lowering, th e beak in a little glass con tain in g w ater, th e d uckling itse lf w ill after this con tin u ally “drink” water from th e little glass. E xp lain such a behaviour o f the duckling. 32. T he process o f diffusion o f different gases is irreversible. H ow can w e accom plish the m ixin g o f gases reversibly ? 33. Show th a t en trop y increases in th e follow ing processes: (a) H o t w ater cedes h ea t to an equal m ass o f cold w ater and their tem peratures aro equalized. (b) Two equal vessels ad iab atically isolated from th e surrounding m edium and con tain in g equal m asses o f ideal gases a t different pressures are connected b y a tube w ith a stop-cock. T he stop-cock is opened an d th e states o f th e gas in both vessels b e­ com e th e sam e. 34. T he h olon om icity o f th e elem en t o f h ea t 8Q occurs on ly for therm ally hom o­ geneous system s. Show th a t a th erm ally non-hom ogeneous system w ill be non-holonom ic. 35. The co n ten t o f th e second law o f therm odynam ics for quasi-static processes is, according to Caratheodory, th e holon om icity o f th e elem ent o f h eat 8Q. In his book “ T herm odynam ics” P lan ck presents th is rem arkable fa ct as som ething trivial, express­ in g n o special properties o f bodies: b y th e exam ple o f an ideal gas he evaluates directly th e expression 8Q/T an d satisfies h im self th a t it is a to ta l differential, w hile he reckons it possible to p rove th a t th is expression w ill b e a to ta l differential also for an y other system b y considering th e com plex system consisting o f an ideal gas O an d th e given system S . H e lets th is com plex system O +

The free energy, proper, for unit volume of dielectric, connected with the presence of a field is evidently equal to Tp2 p _ 1 ^proper

(T > D )

= F ( T

, D) -

^

E 2.

(5.31)

THERMODYNAMICS

112

This expression can also be obtained by integrating the equation for dFproper &om (5.28) at constant values of T and V i> 6— 1 (5.V) F proper(T,P) = E dP — E 2, 871 since P = (e — l)Ej4ji. The variation of internal energy of a dielectric during the time of its polarization at constant temperature and volume can be found from the Gibbs-Helmholtz equation (4.45) with the external parameter a = D: (5. W) U (T, D )= F (T , D) — T By using (5.30) we obtain 7 )2

U ( T ,D ) =

77 7 )2

-+ -----871 87I E 2

(5.32)

— = — (£ + 2 ’— dT 87rl dT

The internal energy, proper, per unit volume of dielectric ^proper (T , D) is evidently equal to 7^2

Jp2 (

\

U»roper( T ,D ) = U ( T ,D ) ------= — £ - 1 + 2 i — L 871 871 v dT!

(5.33)

which also follows directly from the Gibbs-Helmholtz equation E W , N .

P)

=

(T

,P)-T

M

.

(5 . X )

I t can be seen from (5.32) that the internal energy of a dielectric U(T , D) connected with the presence of a field is not equal to the energy of the electric field in the dielectric eE 2I8tt. This is explained by the fact that, in electro-dynamics, by the field energy eE 2/87i is meant all the energy that must be expended for generating the field in the dielectric at constant temperature (and not constant entropy!), whereas the ex­ pression U(T , D) determines the variation of internal energy of a dielectric for its polarization, also at a constant temperature, but al­ lowing for the energy ceded to the thermostat if the polarization causes a variation of temperature of the dielectric. As a consequence of this, the internal energy proper connected with polarization, Uproper( T , D), of a dielectric can prove to be exactly equal to zero. For example, in the particular case of an ideal dipolar gas, for which according to Curie’s law 6 = 1 + (CjT) (C is Curie’s constant), the internal energy ■U(T , D) is equal, according to (5.33), to E2/8tz, i.e. to the energy of

CERTAIN APPLICATIONS OF THERMODYNAMICS

113

the field in a vacuum. Therefore the internal energy proper UpTaper(T , D) of such a gas is evidently equal exactly to zero: UVTOveT(T , D) = = U(T , D) — (E2/8ti) = 0. In the light of what has been expounded above this result is not unexpected. I t can be seen from the electrodynamical definition of field energy that the quantity £E2j8ji is not the energy but the free energy of the field in the dielectric. As (5.30) shows, it is exactly coincident with the free energy of a polarized dielectric; on the other hand, the internal energy of a field in the thermodynamic sense coincides with the internal energy of a dielectric in the field, (5.32). I t is easy to see that the integration of the equation (5.27) for dU for assigned entropy and volume does not give for the variation of energy of a dielectric, that satisfies the linear relationship D = eE, the quantity D2j8szs (5.Y) since in the adiabatic variation of the induction D the temperature of the dielectric does not in general remain constant and e can no longer be considered as a constant. Only in the particular case when e does not depend on temperature, we have F (T , D)

F0= U ( S , D ) - U 0=

eE2 ~8n’

(5.Z)

and the energy of the field coincides with its free energy. From the fundamental equation of thermodynamics for dielectrics (5.2G) in terms of the independent (electric) variable E we obtain d F ' = — S d T — p d V - } D dE.

(5.34)

471

By integrating (5.34) at constant temperature and volume for dielec­ trics that satisfy the linear relationship D = eE, we shall have (by neglecting quantities independent of the field): eE

F ' ( T , E ) = — 8-

2

(5.35)

and (T, E) = F ' ( T , E ) - { - g ) = - ~

E\

(5.36)

114

THERMODYNAMICS

By comparing (5.30) with (5.35) and (5.31) with(5.3G) we observe that the potentials F{T D) and F'(T E) and also Fproper(T, Z))and F'ProPer{T , D) differ from each other only in their sign ,

,

F { T , D) = — F' { T , E ) = Fp,n„ T , D ) = (

-

J

’; r o p e ,

sE2 871 3*5

(5.37)

(T ,E )=

E\

which is similar to the relation, known from electrodynamics, between the variation of field energy (i.e. of the free energy of the field, as has been established above) of conductors in a vacuum d9U (occurring for constant potentials q>of the conductors) and its variation deU (arising for constant charges e of the conductors): d9U = —deU > 0. Such a connexion between 69U and deU (and not their equality) is due, as is known, to the fact that if for constant potentials of the conductors and for a variation of their charges the work of the forces of the field is accomplished at the expense of the energy of the external e.m.f.’s (which maintain these potentials constant), then for constant charges and varying potential of the conductors the work of the forces of the field is accomplished at the expense of the energy of the field. Similarly to this, the relations (5.37) arise from the fact that whereas F(T D) determines the variation of free energy of the dielectric (or the variation of free energy of the field in the dielectric) at the expense of positive work of the external sources displacing the charges in the field, the expression F'(T E) determines the variation of free energy of the dielectric (or the variation of free energy of the field in a dielectric) for generating the field in the dielectric by allowing for the work against the external sources. d9U deU ,

,

T

h e

a b o v e

F(T D) ,

a n

m

u l a t i o n

o f

v a r i a b l e

D

a n a l o g y

F(T E)

d

,

t h e

a n

d

b e t w o n

p r o b l e m

w

h i c h

e e n

t h e

a n

o t h e r ,

c o r r e s p o n d s

t o

t h e

c h o i c e

d

o n

e n a b l e s

t o

o

t h e

f

t h

u s

t h e

t o

c h o i c e

e

i n

o n e

c l a r i f y

o f

t h e

d e p e n d e n t

h

w

i n d

a n

d

a n

h i c h

e p

e n

d

f o r ­

d

v a r i a b l e

e n

t

E.

3. Magnetostriction, eledrostriction and piezoelectric ef.ee t The expressions (5.27) and (5.28) for the differentials of the thermo­ dynamic potentials of dielectrics (and the analogous expressions for magnetic materials) enable us to establish a number of relations be­ tween various properties of dielectrics and magnetic materials.

CERTAIN APPLICATIONS OF THERMODYNAMICS

115

Thus, from the expression for the thermodynamic potential per unit volume of a magnetic material dZ = - S d T + V d p - M d H

(5.38)

we find f

a

n

_

\d H J p ,T

(dM\

(5.39)

\ d p/ j T t H

Here (dVldH)pT is the variation of volume of magnetic material caused by the magnetic field and is called volume magnetostriction; the quan­ tity (dMldp)r,n determines the variation of magnetization for a pres­ sure variation, called the piezomagnetic effect. The relation (5.38) con­ nects these two magnetothermic phenomena. From the relation dZ = — S d T + V d p - P d E

(5.40)

we find in a similar manner for dielectrics the connexion between electrostriction {dV{dE)vj and the piezoelectric effect (dP/dp)B,T, he. (5.41) The formula (5.41), just as (5.39), also corresponds to volume piezo­ effect, though piezoelectric phenomena are usually observed in crystals in determined crystallographic directions. A slab cut from a piezo­ electric crystal and provided with a pair of electrodes undergoes a deformation under the action of an external electric field, which fact causes in it elastic oscillations. Vice versa, a deformation induced mechanically causes electric charges on the electrodes of the slab. Piezoelectric crystals find wide application in radio engineering, electro-acoustics and ultrasonic acoustics and in many other branches of science and engineering, connected with the transformation of peri­ odic electrical processes into mechanical processes and vice versa.4 4. Magnetic and nuclear cooling The cooling of a body can be caused not only by adiabatic expansion but also by an arbitrary adiabatic work of the system. Thus, according to a suggestion by Debye (1926) the adiabatic demagnetization of para­ magnetic crystals is employed as one of the basic methods for obtaining ultra-low temperatures {T The phenomenon of temperature variation in adiabatic demagneti­ zation is called the magnetocaloric effect. The quantitative value of
, s

I n

x

b

p

y

a r a m

C

a g

n

u r i e ’s

e t i c

l a w

m

i s

M = xH,

a t e r i a l s

i n v e r s e l y

a n

d

t h e

p r o p o r t i o n a l

(5. AD)

m

t o

a g n e t i c

t e m

p

s u s c e p t i b i l i t y

e r a t u

r e

G T

x=

(5.42)

—

.(G is Curie’s constant, G > 0). From (5.42) we have (dM\

_

(

HG

_

(5.AE)

and therefore (d T \

H

e n c e

i s

l o w

i t

c a n

(dT

e r e d

p e r a t u r e s

b e

i s

s e e n


v' at melting and, therefore in such bodies, just as in the case of boiling, dTjdp > 0. However, in water, cast iron and bis­ muth the volume decreases with the melting of the solid phase, so that dTjdp < 0 for them, i.e. the melting point decreases with an increase of pressure. In secondary-school physics an efficacious experiment is demonstrat­ ed by means of ice and a wire with a weight, which confirms this behaviour of the melting point in ice. In nearly all textbooks this property of ice is used for explaining in the following manner the slip­ periness of ice, i.e. the known fact that skating on ice is easy in winter: as a consequence of the large pressure exerted by the blades of skates the ice melts at a temperature below 0°C; this provides a watery lubri­ cant which is responsible for the slipperiness of ice. Such an explanation of the slipperiness of ice was given about 100 years ago by the English physicists Tyndall and Reynolds, and has been widely accepted although it does nofi at all correspond to the facts. I t follows, in fact, from the Clapeyron-Clausius equation that in order to lower the melting point of ice by only a few degrees, such a high pressure is needed that the ice would not be capable of sustaining it. In fact the specific volume of ice at 0°Cis v' = 1-091 cm3/g, and that of water is v" = 1 cm3/g. The heat of melting is A = 80 cal/g. Therefore dp dT

------------- = — 80 *-41-3- atm/°C = — 134 atm/°C T{v" — v’) 273 X0-O91 (1 cal = 41-3 atm • cm3),

(8.E)

i.e. for lowering the melting temperature of ice by 1°G, the pressure must be increased by 134 atmospheres. And for ice to begin to melt, for example at —10°C the pressure must be increased on the average to 1300 atm; the ice cannot sustain such a pressure!

P H A S E T R A N S F O R M A T IO N S A N D C R IT IC A L P H E N O M E N A

177

As has now been conclusively shown by experiment,f the slipperiness of ice is caused by the formation in the sliding plane of a watery lubri­ cant due to the conversion into heat of the work of the motive forces overcoming friction. 3. Thermodynamics considers phase transitions as point phenomena (at a temperature T 0 and pressure p 0). In reality, however, as a con­ sequence of heterophasic fluctuations the temperature point of a phase transition is blurred into a certain small interval.

F

ig

. 30.

In the thermodynamic approach to a phase transition of the first kind we establish that the thermal capacity at the transition point is equal to infinity: Gp = f-^-1 , vdT J p and therefore

but

6Q ^0,

and

dT = 0,

Cp = °o.

(8.F) (8.6)

It has been observed experimentally th at in approaching the melting point (both on the side of the solid phase T ' < T 0 and on the side of f S. S. B u d n evich a n d B . V. D eryagin; Zh. tekh. jiz. 22, page 1907 (1962).

178

T H E R M O D Y N A M IC S

the liquid phase T" > T 0) the curves that represent the temperature behaviour of the thermal capacity bend rapidly upwards (see Fig. 30 on which is shown the temperature dependence of the thermal capacity of paraffin in the neighbourhood of the melting point). Such a behaviour of the thermal capacity in phase transitions of the first kind agrees with that which the theory of hetcrophasic fluctuations predicts. In phase transitions of the first kind a system, being dispersive, has a highly developed surface. The taking into account of the surface leads, as we have seen in Chapter VII to the fact that the system can be found in the vapour state at pressures exceeding the saturation pressure at the given temperature. This possibility of the existence of supercooled or superheated phases (supercooled vapour, superheated liquid) is the typical distinctive feature of phase transitions of the first kind. § 39. Phase transitions of the second kind. Ehrenfest’s equation

In phase transitions of the second kind no heat is absorbed and the specific volume does not vary (s' = s", v' = v") but the thermal capa­ city cp, the coefficient of thermal expansion a and the compressibility /? undergo jumps. The connexion between these jumps and the slope of the equilibrium curve at the transition point is determined by Ehrenfest’s equations, who first introduced (1933) the concept itself of phase transitions of the second kind in an analysis of the H e-I to H e-II transition.f Let us derive these equations. The right-hand side of Clapeyron-Clausius equation dp _ s" — s' d,T ~~ v" — v

(8.3)

at the point of a phase transition of the second kind assumes the indeterminate form 0/0. In order to resolve this indeterminacy we shall use L ’Hopital’s rule. By differentiating the numerator and denominator | The form o f th e curve o f th e tem perature variation o f the therm al capacity in the v ic in ity o f the H e -I to H e -I I transition p o in t recalls the specular im age o f the Greek letter A (lambda) and therefore this curve is called a A-curve, and the transition p oin t the A-point. The presence o f a A-point on the curve o f the thermal capacity has com e to be considered ns th e distinguishing feature o f a phase transition o f the second kind.

179

P H A SE TRANSFORMATIONS A N D CRITICAL PHENOM ENA

of the right-hand side of (8.3) with respect to either T or p we shall obtain ds" ds' AcP dp dT dT _ (8.7) dT dv" dv' f dv \ ~dT ~dT [dT ) p and (W _ c W dp _ dp dp (8.G) d T ~ dv^_dv' dp dp or dv A dT dp (8 . 8) dv j dT A dp)T since from d/j, = — sdT + vdp it follows that — (dsjdp)T = (dvjdT)p. From (8.7) and (8.8) we obtain Ehrenfest’s equations (8.9) (8 . 10)

A

If not only the force of pressure p but also some other generalized force A corresponding to the external parameter a acts on the system, then Ehrenfest’s equation will have the form Aca = - T dT)

dT

U

A

(8 .

11 )

(8 .

12 )

t

\ , (dA)T

Let us apply these equations to the case of the transition of a con­ ductor from normal (n) to superconducting (s) state. As is known, such transitions occur in some conductors at a determined temperature T e. Super-conductivity can be destroyed if a sufficiently intense magnetic field H c is superimposed. The temperature dependence of the critical field H c is analogous to the temperature dependence of pressure p =

THERMODYNAMICS

180

— p[T) in liquid-vapour equi­ librium and is represented on the II , T diagram by the curve shown in Fig. 31 (a sufficiently accurate analytical represen­ tation of this curve is the para­ bola HC(T) = Hc[ l~ { T ! T cY]). If a conductor is placed in a magnetic field its transition to superconducting state is ac­ companied by a heat effect, and therefore it is a phase tran­ F ig . 31. sition of the first kind. It has been shown by Keesom that the transition is determined in this case by the Clapeyron-Clausius equation. In the absence of a magnetic field the heat of transition is equal to zero and the n-to-s transition is a phase transition of the second kindf. By putting in the equation (8.11) A — H and a — — M we shall obtain for the jump of the thermal capacity! =

f

l

j

Q

-

(S-.3)

For a normal conductor Mn = (fi — l)H/47i, and for a superconductor fi = 0 and M s ~ —HjArc. Thus dMn 1

ns

1 1

ro_ U H)

1 4ji

H —1 471

( 8 . H)

V471

( 8. J)

and 4 d* \ -

dM, dH

dMn d ll

However, fi = 1 -f- 4nx, and x for paramagnetic and diamagnetic sub­ stances is of the order of 10~6 to 10“6, and therefore I dH I

4n

(S.K)

t Tho modern microscopic theory o f superconductivity duo to B ardeen-C ooperSchriffor and B ogolyubov loads to a finite jum p o f therm al capacity in the transition o f superconducting m etals from normal to superconducting state. J The m inus sign in (8 .1 3 ) is retained in such a su b stitu tion since in (8.S ), from which (8.11) is obtainod, thore m ust bo a plus sign if th e variables are H and M .

PH ASE TRANSFORMATIONS A N D CRITICAL PHENOM ENA

181

and _ T fdHY 471 \d T )h= o

(8.14)

This formula is called Rutgers's formula. I t agrees to a high degree of accuracy with experimental data, as can be seen from the following table. Substance

Experimental value

| I

cal

cal d e g re e

x

Calculated value

m o le

|

d e g re e

x

m o le

\ Tin .............. T h alliu m . . . In d iu m . . . .

0-00290 0-0014S 0-00202

0-00261 0-00146 0-00201

The temperature behaviour of the thermal capacity of tin in its transition to superconducting state (in the absence of a magnetic field) is represented by the curve shown in Fig. 32 according to Kcesom’s data. This curve has little in common with the A-curve. Thus in this phase transition of the second kind there is no overshooting of the

curve of the thermal capacity at the transition point, and therefore the presence of A-curves in phase transitions is no proof of the fact th at these transitions are transitions of the second kind. This conclusion is also shown convincingly by the example of the phase transition of ammonium bromide NH4Br. At low temperatures anomalies are observed (A-curves; see Fig. 33) in the temperature be­ haviour of NH4Br (as well as in that of NH4C1 and NH4I). These pro-

182

THERM ODYNAMICS

cesses are usually attributed to phase transitions of the second kind on the ground that no release of latent heat is observed. Therefore Elirenfest’s equation (8.7) ought to be valid in this case. On the basis, however, of very accurate measurements of the temperature behaviour of the thermal capacity of NH4Br it has been shown-)' that, in reality, Cp

F

ig

. 33.

the Clapeyron-Clausius equation (8.4) is valid for such transitions, which fact indicates that these transitions belong to the class of phase transitions of the first kind. This is also indicated by the variation of the crystal lattice in such transitions. Though in the s-to-n transitions of NH4Br, NH4C1 and NH4I no (or, more correctly, almost no) heat effect is observed, its presence is revealed by the peak of the thermal capacity at the transition point (/l-curvo). § 40. The theory of phase transitions of the second kind

After Ehrenfest had introduced, in 1933, the concept of phase tran­ sitions of the second kind, their thermodynamic theory was developed in 1937 by Landau. This theory concerns first of all those transitions which are connected with a variation of the symmetry of a body or of the pattern of atoms. If a transition is accomplished at the temperature Tc, then the phase | V. Y u. Urbakh; Zhurnal jizicheskoi kh im ii 30, p . 217 (1956).

PH A SE TRANSFORMATIONS A N D CRITICAL PHENOM ENA

183

that is stable for T < T c is called ordered, and the phase stable for T > T e is called non-ordered. For a mathematical description of a phase transition of the second kind a certain essentially positive inter­ nal parameter! rj is introduced, which is called the ordering parameter. The thermodynamic potential Z of a body is therefore a function of T, p and rj. In the non-ordered state the parameter r] is taken equal to zero. The continuity of the variation of Z in a phase transition of the second kind means that in the vicinity of the transition point rj assumes an arbitrarily small value; let us therefore expand Z(T, p, rj) in the ordered phase in the vicinity of the transition point in a power series with respect to the ordering parameter rj %{T, P , V ) = Zo{T,P) +

V+

1 - P ( P , T ) v 2+ ■ ■■

• (8.L)

The equilibrium state corresponds to th at value of rj for which Z has an extremum (dZ/dr) = 0), which gives a + ^ = 0,

(8.M)

whence (8.15) This state will be stable if rj corresponds to a minimum of Z ----> 0, drj2

whence

/? > 0.

(8.N)

It follows from (8.15) that, to accord theory with experiment, we need to postulate that for each assigned pressure p there exists a certain temperature T c such that for T > T e the condition a(p , T) = 0 is verified, while for T < T ethen a(p , T) ^ 0. In other words, the pres­ ence of ordering for T < T cand its absence for T ~>Te cannot be derived from the theory but must be introduced in it by proceeding from experimental data. In the vicinity of the transition point, a(p , T) can be expanded in a power series with respect to (T — Te) a {p ,T )

=

cL{p,Tc) + { T - T c) l d£ \

,

(8.0)

\ y 1 J T=T„

f A n in tern al param eter can also be introduced for the characteristic o f phase transitions o f th e first kind. Therefore from the fact th a t L andau’s m ethod can be applied successfully to th e analysis o f phase transitions o f the second kind it does not follow th a t all processes where th is m ethod is applicable are phase transitions o f th e second kind.

184

THERM ODYNAMICS

whence doc)

«(2>, T) = (T — T c)

(8.P)

d T ) r . T;

since x ( p , T c)

=

0

(8. Q)

.

Thus in the vicinity of the transition point the thermodynamic poten­ tial Z and the entropy in the ordered state (T < T c) are equal to 3 = £0 S= -

_

2/3

( T - T J * fdx

y

(8.16)

2/5

dZ

= S„ + i ( T - T . ) _d«Y dT dT)T=Tc P

(8.17)

where Z0 and S 0 arc the thermodynamic potential Z and the entropy in the non-ordered state. We obtain from (8.17) that the thermal capacity Gp = T{dSjdT)v undergoes a jump at the transition point ACn = C„ ~ C (v =

Tcfdccy /5 \d T )T=Te

(8.18)

Since /5 > 0, it can be seen from the formula obtained that the thermal capacity of the ordered state proves to be larger than the thermal capacity of the non-ordered state. Such a conclusion of the theory is found to be in agreement with experiment, which can be seen from the data shown for the transition of a conductor to the super-conduct­ ing state. As the functions a and /5 are single-valued, the solution (8.15) for the equilibrium value of rj is also single-valued. This leads to the im­ possibility of the existence of metastable states: for T T c there exists only one ordered phase and the non-ordered phase is absent (unstable); for T T c, on the contrary, only the non-ordered phase is possible but the ordered phase is unstable. In other words, in phase transitions of the second kind neither supercooling nor superheating are possible. By knowing the jump of thermal capacity, we can easily find also the jumps of other second derivatives of the thermodynamic potential. Landau’s theory is a phenomenological theory, since it cannot deter­ mine the form of the functions a{p , T ) and /?(p, T). By comparing the formula (8.18) of Landau’s theory with Ehrenfest’s formula (S. 11) for the jump of the thermal capacity ACa, we can put a = A and /5 = — A(dajdA)T(. Though Landau’s theory uses


P H A SE TRANSFORMATIONS A N D CRITICAL PHENOM ENA

185

certain conceptions of a model of matter (ordering, non-ordering etc.), the formula (8.18) of this theory is sometimes even less valuable than Ehrenfest’s purely thermodynamic formula (8.11). Thus, in the case of the phase transition of a conductor to the superconducting state we obtain, from Ehrenfest’s formula, Rutgers’ formula T dH (8.14) zlc = dT u=o’ which, on the basis of an experimental measurement of (dII{dT)Tc gives a quantitative value for the jump of the thermal capacity AG. The formula (8.18) cannot give anything similar. Moreover, Rutgers’ thermodynamic formula enables us to establish for a given transition the values of a and in the formula (8.18): a = H and = 4n. On the whole, however, in the analysis of phase transitions of the second kind Landau’s theory is superior to Ehrenfest’s purely thermodynamic ap­ proach. Thus previous to Landau’s work the question cf the possibility of supercooling and superheating in phase transitions of the second kind had been incorrectly answered, as a consequence of which contra­ dictions had arisen (see problem no. 14). § 41. Critical phenomena In 1860 D. I. Mendeleyev, studying the temperature dependence of surface tension, established that at a certain temperature, called by him the absolute boiling temperature, the difference in the properties of the two coexisting phases, liquid and vapour, vanishes. This state, charac­ terized by determined values of tem­ perature, T ct pressure, p CT, and volume, VCT, was later called the critical state. The equilibrium curve of liquid and vapour on the T, p diagram ends at the critical point. In 1869 the critical phenomenon was F ig . 34. investigated by Andrews and from 1873 onwards, by a group of Kiev physi­ cists under the leadership of M. P. Avenarius. In 1878 Gibbs developed a general theory of critical phenomena on the basis of his work on the equilibrium and stability of heterogeneous systems. In our exposition of the theory of critical phenomena we shall follow Gibbs.

186

THERMODYNAMICS

1. The theory of critical phenomena according to Gibbs Under the influence of external actions the state of a phase can vary both continuously (without the arising of a new phase) and discontinuously (with the arising of a new phase). Accordingly there exist two types of boundaries of the region of stability of a phase. These are boundaries of phase stability with respect to (1) continuous and (2) dis­ continuous variations of state. A boundary of the first type is the set of states of the phase taken by itself, which correspond to the points C of Fig. 24 for various temperatures (the curve I in Fig. 34). A boun­ dary of the second type is the equilibrium curve of phases (Fig. 25 or the curve I I in Fig. 34). In the passage through a boundary of stability with respect to discontinuous phase variations, one of the phases be­ comes relatively more stable than the other, which fact causes a phase transition of the first kind. At a boundary of stability with respect to continuous phase variations the thermodynamic inequalities (6.26), determining the stability of the existence of a phase, assume limit values, namely zero or infinity. As can be seen from Fig. 34, when the stability of a phase with respect to the arising of a new phase fails the stability of the phase with respect to its continuous variations is either retained or fails. Stable states that lie simultaneously on a boundary of stability both with respect to discontinuous and with respect to continuous phase variations are defined by Gibbs as critical states (the point K in Fig. 34). A critical phase is therefore the limit case of the equilibrium of two phases for which the stability conditions are satisfied both with respect to discontinuous and with respect to continuous variations. Mathematically the boundary of the region of stability of a phase with respect to continuous variations is determined by reducing to equalities the thermodynamic inequalities (6.26), i.e. by the equations

m

dSjp

A{,

=o.

|) d/V|

P )

=o,

d V j T . A i , in

=

(8.19)

= 0 .

o a i J p , T ,m

dN i)'T ,p, A(

or p )

(8 .

20 )

Avhere are thermodynamic forces (T , p, II, E, p, . ..) and are the coordinates associated with them (S , V, M, P, N, . ..). This defini-

187

PH A SE TRANSFORMATIONS A N D CRITICAL PHENOM ENA

tion of the boundary of stability with respect to continuous variations of phase can be written in the following more general form. When thermodynamic coordinates are taken as independent vari­ ables, the differentials of the thermodynamic forces are equal to dX { = (— *]

dxx +

V d x ^ J x 2,X3, ,...X n

dxz + . . .

(8.R)

V^*^3 ) x l>x 3t ••• >x n

ATT\ ...

+

dxn) xlt..., xn_i

dx„

{i = 1 ,2 , . . . ,n ). The determinant of this system will be dX x (dXx dxxJx2, ... f xn dx2/Xlt x3, ..., xn

(dXi dx„n) &lr >xn—l

( 8. S)

D = dX„ \

(dXn

x 3, ..., x n v, dx2)X \, x 3, .... x n

dX, dxnj x j, x 3, ..,, x n_ x _d{Xx,X 2, . . . , X n) d{xx, x2, . . . , xn)

In the case of the two independent variables S and V we have (8 .T )

'dZM fd T D=

d{T,p) _ d{T,p) d (S ,p ) = (d T \ D d(S,V) d(S,p) d(S,V ) 115

d S J A W ja d p i fd p )

dSJrldVjs where DX1 = (dp/dV)s is the minor of the determinant D corresponding to the element of the first column and first row. In the general case we have, by analogy, 'dXA D= I Du dXi y . r i , X 2 , . . . , x J- i , x i + i ,

(8.U)

and dX { v d X { J X i , X 2 , X j —j X i + i , ...

D

(8 .

21)

D #

Therefore, according to (8.20), the boundary of stability of a phase with respect to continuous variations will be written in the form D=

2 -• • ’ X nL = 0. d(xx,x 2, . . . , x n)

(3.22)

THERMODYNAMICS

188

Since critical states are stable also with respect to discontinuous variations, then small fluctuation-type variations of the coordinates of a phase for constant thermodynamic forces must not remove the system from the stability boundary (8.22), neither must they lead to the formation of a new phase, i.e. the condition D = 0 must be main­ tained for such fluctuations. This is equivalent to the equality to zero of the derivatives (dD/dXi) x2, • • •> xi—v j i +1, • • • and to the in­ equality (d2Dldxi2)xk < 0, which is expressed in the most general form, in the first place, by the equality r __ d(-D,-AT2, . . . ,X n) __ q (8.23) d(x1,x 2, . . .,x n) i.e. by equating to zero the determinant of the system (8.20) in which any one of the rows is replaced by a row formed from the coefficients of the linear form dD = I'-?-') \ d x 1 J x 2, . . . , x n