Thermodynamics, Statistical Physics, and Kinetics

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Thermodynamics, Statistical Physics, and Kinetics Yu. B. Rumer, M. Sh. Ryvkin

MIR PUBLIHSERS MOSCOW

PyMep K). B., P ubkhr M. III. TEPMOflHHAMHKA, CTATHCTHH ECKAH (DH3HKA H KHHETHKA HanaTejibCTBo «Hayica» Mockbb

Yu. B. Burner, M. Sh. Ryvkin

Thermodynamics, Statistical Physics, and Kinetics Translated from the Russian by S. Semyonov

Mir Publishers Moscow

First published 1980 Revised from the 1977 Russian edition

Ha amAudeKOM Jisune

©

rnaBHBH peABKUHH $H3HKO-MaTeMaTHVeCKOH jiHTepaTypu HSAaTejibCTaa «HayRa», 1977 © English translation, Mir Publishers, 1980

PREFACE

This book is intended for readers taking up the study of thermo­ dynamics, statistical physics and kinetics. Accordingly, readers are assumed to have a knowledge of elementary physics, higher mathematics and quantum mechanics. The sections marked by asterisks require a deeper knowledge and can be omitted on the first reading. The book aims to gradually familiarize readers with methods used in thermodynamics, statistical physics and kinetics, to show how the concrete problems should be solved and to bring readers as soon as possible to a level which enables them to tackle more specialized books and papers. It has therefore proved Jnecessary to include some matter, perhaps too simple for some, as well as problems that may be too difficult for others. It was the desire to cover as quickly as possible the introductory stages and pass to concrete problems that dictated to a great extent the methods followed by the authors in introducing basic concepts. For instance, in the part of the book devoted to phenomenological thermodynamics the concepts of entropy and temperature are intro­ duced simultaneously early on, an extensive use of Jacobians then allowing us to offer an integrated approach to solving problems for a wide range of monovariant (and later on poly variant) thermodynam­ ic systems. For the same reason we begin the treatment of the fundamentals of statistical physics by outlining the “cells and boxes” method, which although only suitable for ideal gases, does provide a simple solution to a wide range of problems. The more general Gibbs ensemble method is introduced later. We would like to place special emphasis on the following. The treatment of the fundamentals of statistical physics begins with n discussion of the hypothesis of the indistinguishability of particles, without this being linked to any quantum-mechanical ideas. We wished to underline, thereby, the fact that along with the two meth­ ods historically used to uncover the wave nature of matter—matrix mechanics, based on the correspondence principle (Heisenberg-Bohr) and wave mechanics, based on the optical-mechanical analogy (Schrodinger-Hamilton)—a third way, obviously that used by A. Einstein, is possible, based on statistical mechanics. Our belief In the existence of this method derives from the fact that In the statistics of indistinguishable objects the volume ofj an elementary cell in the phase volume is not arbitrary, but is predeter­ mined unambiguously by natural laws. This is why, in principle.

6

Preface

this method should be experimentally discoverable, and Planck’s constant could appear in physics not as a result of an analysis of radiation laws (M. Planck) or photoeffect laws (A. Einstein), but from, for instance, the measurement of the heat capacity of an elec­ tron gas at low temperatures. For this reason we use Planck’s constant h, appearing naturally in statistical physics as a measure of the phase volume, and not the constant h, more suitable in quantum mechanics. Most sections of the book contain problems. We found it inexpe­ dient to give solutions to the problems, not simply for reasons of space, but also in the belief that the independent solution of prob­ lems is far more beneficial than the study of provided solutions. Thus usually only answers are given to the problems, with brief hints given in some cases. In conclusion, we wish to extend thanks to G. L. Kotkin, G. G. Mikhailichenko, V. G. Zelevinskii and V. S. L’vov for their great assistance in writing this book. Yu. B. Rumer, M. Sh. Ryvkin

CONTENTS

Preface Introduction

5 11

PART ONE THERMODYNAMICS

13

Chapter I. General Laws of Thermodynamics

13

1. 2. 3. 4. 5. 0. 7. 8. 9. 10.

14 16 18 20 23 25 28 31 33

Equilibrium states and equilibrium processes Temperature. The temperature principle Entropy. The entropy principle Absolute temperature and absolute entropy Work Adiabatic and isothermal potentials The energy principle. Supply and removal of heat Heat capacity of gases Cyclic processes. The Carnot cycle Axiomatics of thermodynamics. Generalization of the concept of entropy to arbitrary thermodynamic systems. Nernst's heat theorem Thermodynamic coefficients. Polytropic processes Thermodynamics of the van der Waals gas Gas cooling methods. Gay-Lussac and Joule-Thomson processes Thermodynamics of rods Thermodynamics of magnetics Thermodynamics of dielectrics Thermodynamics of radiation Thermodynamics of water The thermodynamic potential. Method of thermodynamic functions Thermodynamics of plasma Poly variant systems. Magnetostriction and the piezomagnetic effect

38 43 52 61 67 72 80 85 91 96 102 104

r.lmpter II. Systems with Variable Amount of Matter. Phase Transitions

110

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 20. 27. 2M. 29. 30. 31.

Systems with variable amount of matter. The chemical potential The increase in entropy in equalization processes. The Gibbsparadox Extrema of thermodynamic functions Thermodynamic inequalities Phase equilibrium. First-order phase transitions Three-phase equilibrium. Superheating and supercooling Second-order phase transitions Thermodynamics of superconductors Multicomponent systems. Phase rule Chemical equilibrium in a homogeneous system. The mass action law

110 113 121 128 130 141 148 150 155 158

8

Contents

PART TWO STATISTICAL PHYSICS Chapter III. Statistical Distributions for Ideal Gases

165

32. 33. 34. 35. 36. 37.

165 170 173 180 186

Statistical regularities. Distributions, most probable distributions u-Space. Boxes and cells Bose-Einstein and Fermi-Dirac distributions The Boltzmann principle The Maxwell-Boltzmann distribution Transition to continuously varying energy. Degeneracy conditions for ideal gases 38. The Q-potential of Bose and Fermi gases 39. Energy quantization. The Nernst theorem

190 194 197

Chapter IV. The Maxwell-Boltzmann Gas

202

40. The Maxwell-Boltzmann monoatomic gas in the classical approxi­ mation. Phase volume of a cell and the zero point entropy 41. The' Maxwell distribution 42. Spatial distribution of molecules 43. Polyatomic gases (classical theory). The equipartition theorem 44. The Maxwell-Boltzmann gas with two energy levels 45. Quantization of translational motion 46. Diatomic gas. Rotational degrees of freedom 47. Molecules consisting of identical atoms. Ortho- and paramodifications 48. Vibrational degrees of freedom 49. Thermal ionization of atoms 50. Thermal dissociation of molecules 51. Paramagnetic gas in a magnetic field

202 205 209 210 214 216 219 225 230 234 239 242

Chapter V. Degenerate Gases

246

52. Equilibrium thermal radiation. Photon gas 53. Thermal motion in crystals. Phonon gas 54. A degenerate Bose gas in the absence of a field. The Bose-Einstein condensation 55. The Bose gas in an external field 56. An electron in a periodic field 57. Degenerate Fermi gas. Electron gas in metals 58. Electrons in a semiconductor 59. Magnetism of an electron gas

246 253 262 270 273 276 283 286

Chapter VI. Systems of Interacting Particles. The Gibba Method

297

60. 61. 62. 63.

297 302 306

r-space. Liouville’s theorem Microcanonical and canonical distributions T-V-u and T-P-N distributions Anotner derivation of the T-V-N, T-V-p and T-V-N distribution. Thermodynamic corollaries 64. Derivation of the Bose-Einstein and Fermi-Dirac distributions with the aid of a grand canonical ensemble 65. Nonideal gases

309 322 325

Contents

9

III). 117. iih. llli.

Plasma. Debye’s screening Extreme and negative temperatures Second quantization Superfluidity. Bogoliubov’s theory To. Superconductivity

335 337 345 359 366

Chapter VII. theory of Fluctuations

384

71. 73. 73. 74.

384 388 391

Fluctuations in energy, volume and number of particles Fluctuations in main thermodynamic quantities Fluctuations in occupation numbers in idfcal gases Fluctuation limit of sensitivity of measuring instruments. Nyquist’s theorem

393

Chapter VIII. Phase Transitions

398

7!j. 76. 77. 78.

398 403 405

The Lee-Yang theory Critical exponents and phenomenologicalinequalities for them Critical point for the van der Waals gas Phase transition in ferromagnetic materials. Molecular field method and the Bragg-Williams approximation 79. The Landau theory of second-order phase transitions 80. Review of results. Comparison with experiment. Models with exact solutions 81. Fluctuations and phase transitions. The Ornstein-Zernike theory. Similarity hypothesis

409 418 427 435

PART THREE ELEMENTS OF KINETICS AND NON-EQUILIBRIUM THERMODYNAMICS

442

Chapter IX. Kinetics

444

H2. H3. H4. H5. Hll. H7.

444 448 454 459 465

HH. H!i. mi. lit. li!!. H3. III.

US. HU. Hi.

The Smoluchowski equation. The principle of detailed balancing The Fokker-Planck equation. Brownian motion Kinetic balance equation. Einstein’s derivation of Planck’sformula The Boltzmann kinetic equation The Bogoliubov equations Evolution stages for a non-equilibrium system. Bogoliubov's deri­ vation of Boltzmann's equation Dimensionless form of Bogoliubov’s equations. Factorization and the correlation functions. Free-molecule flow Equation of a self-consistent field. Collisionless plasma Oscillations of electron plasma The laws of conservation and the entropy increase law Local equilibrium The kinetic equation for plasma Equations of gas dynamics Methods of solution for Boltzmann's equation Irreversibility of macroscopic processes Density matrix and its variation with time. The Cubo method

472 483 487 489 495 501 505 511 521 531 542

Contents

10

Chapter X. Elements of Non-Equilibrium Ihermodynamics

549

98. Balance equations for mass, momentum, energy and entropy 99. Small deviations from equilibrium. Onsager’s principle 100. Corollaries of Onsager's reciprocal relations. Theorem on the minimum production of entropy for stationary states. Examples 101. Far-from-equilibrium states

549 557 562 566

MATHEMATICAL APPENDIX I. Jacobians (functional determinants) II. Stirling’s formula III. Lagrange's method of finding the conditional extremum IV. Integrals J n V. Probability function erf (x)

570 573 574 575 576

VI. Properties of function 6 (x) =

576

oe

VII. VIII. IX. X. XI. XII. X III. XIV.

^

e~nn2x

Integrals K,< and A'„ " °° Dirac’s delta-function 6 (x) and step function a (x) Integrals Ln Integral M The Laplace transform Integrals for section 59 n-Dimensional sphere The Gauss distribution for one and two variables

577 579 581 581 583 585 586 587

References

589

Name Index

591

Sub’el Index

593

INTRODUCTION

Thermodynamics, statistical physics and kinetics are concerned with the physical processes occurring in macroscopic systems, i.e. in bodies containing an enormous number of microparticles (atoms, molecules, ions, electrons, photons, etc. depending on the kind of the system). There are two methods of studying the states of macroscopic sys­ tems—the thermodynamic and the statistical method. The thermodynam­ ic method is independent of any models of the atomic and molecu­ lar structure of matter and is essentially a phenomenological meth­ od, meaning that its object is to establish relationships between directly observed (that is measured in macroscopic tests) quanti­ ties, such as pressure, volume, temperature, concentration of a solu­ tion, intensity of an electric or magnetic held and luminous flux. Quantities associated with the atomic and molecular structure of mailer (dimensions of an atom or a molecule, their masses, number, etc.) are not used by the thermodynamic method in the solving of problems. In contrast to this, the statistical method of studying the proper­ ties of macroscopic bodies is based from the outset on atomic-molec­ ular models. The main object of statistical physics can thus be formulated as follows: using the laws governing the behaviour of the particles of a particular system (molecules, atoms, ions, quanta, otc.) to establish the laws governing the behaviour of macroscopic quantities of substance. It is implied in all this that both the thermodynamic and sta­ tistical approaches have their merits and shortcomings in studying various phenomena. The thermodynamic method, being independent of conceptual models, is characterized by a rather high degree of generality, while the results obtained from the use of statistical physics arc true only to the extent that the assumptions made concerning the behaviour of the smallest particles of a system are valid. The thermodynamic method is distinguished, as a rule, by great simplicity and it opens the way, after a number of simple mathe­ matical procedures, to the solution of a large number of specific prob­ lems, without any information on the properties of atoms or molec­ ules being required. This is an invaluable advantage of the thermo­ dynamic method, especially when solving practical problems of a lerlmical character (engineering thermodynamics, heat engineer­ ing). The thermodynamic method possesses, however, a fundamental slmrl coining, in that with this method the internal (atomic and mo-

12

Introduction

lecular) mechanism of the phenomena involved remains unknown. For this reason in thermodynamics the question “Why?” is mean­ ingless. If, for instance, the thermodynamic method is applied to establish that a copper wire cools upon rapid extension, while a rubber band becomes hot (this problem will be considered in Sec. 14), we must be satisfied with the simple fact, with the physical mechanism leading to this result remaining hidden from us. In contrast to this, the solution of particular a problem by the methods of statistical physics is based from the very beginning on atomic and molecular concepts and permits us to visualize the mechanism of the phenomenon involved. The statistical approach allows one in principle to solve problems that cannot be solved at all within the framework of the thermody­ namic method; the most important examples of this kind are the derivation of the equation of the state of macroscopic systems, the theory of heat capacity, some problems of the theory of thermal radiation. Finally,-the statistical method permits, on the one hand, a rigorous substantiation of the laws of thermodynamics, and on the other hand, makes it possible to set the limits of applicability of the laws, and also to predict violations of the laws of classical thermodynamics (fluctuations) and estimate their scale. It is clear from the above that neither thermodynamics nor statis­ tical physics are restricted to any particular field of physical pheno­ mena, in contrast to optics, mechanics, electrodynamics and other branches of physics, being rather methods of studying any macro­ scopic system in a state of equilibrium. The methods of statistical thermodynamics can be used in studying any systems consisting of a sufficiently large number of particles: gases, liquids, solids, plasma, electrolytes, light radiation and even heavy nuclei containing hundreds of nucleons. The branches of physics concerned with non-equilibrium states and processes, i.e. non-equilibrium thermodynamics and physical kinetics, have been elaborated to a considerably lesser degree than statistical thermodynamics, but are rapidly developing especially over the last few years. Physical kinetics, in particular, studies the processes of relaxation (transition from a non-equilibrium state to an equilibrium one) and steady-state transport processes. Part I of the present book deals with the fundamentals of thermodynamics, Part II is devoted to the statistical theory of equilibrium systems, and Part III is concerned with the fundamentals of physical kinet­ ics and non-equilibrium thermodynamics.

14

Part One. Thermodynamics

Finally, in Sections 22 through 31 the thermodynamics of systems with a variable amount of substance and especially the problem of phase equilibriums and phase transitions are considered. 1. Equilibrium States and Equilibrium Processes

By a thermodynamic system will be meant any macroscopic body in an equilibrium or close to equilibrium state. The states of any thermodynamic system con be defined by means of a number of parameters. Thus, for instance, the states of a gas or liquid (homogeneous systems) can be set by means of the following parameters: P (pressure), V (volume), T (temperature); the state of a film of a liquid—with the aid of a (surface tension coefficient), a (film area) and T (temperature). The slate of a rod can be specified with the aid of parameters I (length), a (cross-sectional area), / (ten­ sile force), E (Young’s modulus). It should be noted, however, that not all parameters of a system necessarily have a definite meaning in any particular state. Consider, for instance, a vessel divided into two halves by a partition with a valve, initially the left-hand half of the vessel contains a gas and the other half of the vessel is evacuated. If we open the valve, a stream of gas will be forced through it. It is clear that during the first moments of this process the volume of gas is indefinite, with the density of the gas contained in the right-hand half of the vessel changing from point to point according to some intricate law. It is thus impossible to indicate the boundaries of the volume of the gas. Also states are possible in which any parameter of a system differs at different points, so that no single value of this parameter exists for the system as a whole. It is possible to visualize a system, for instance, the temperature of which changes from point to point, or a gas whose pressure varies from point to point. Experiments show, however, that when a thermodynamic system is in such a state, there exist fluxes (heat flux, mass flux, etc.) and this state does not remain invariable, unless it is maintained artificially with the aid of heat­ proof (impervious) partitions, gas-tight walls, etc. With the pas­ sage of time a state is established in which the value of each param­ eter of the system is the same at all points and remains constant as long as desired, provided the external conditions do not change. Such states are called equilibrium states. If no equilibrium is estab­ lished and there exist in the system gradients of macroscopic param­ eters (pressure, density, temperature, etc.) such a state is referred to as a non-equilibrium state. The process during which a thermodynamic system passes from a non-equilibrium to an equilibrium state is referred to as a relaxation process. The time it takes each system parameter to become the same over the entire volume of the system is known as the relaxation time

Ch. I. General Laws of Thermodynamics

15

nl llid givon parameter. The total relaxation time of a system, clear­ ly, Is t ho longest of the individual relaxation times. The relaxation lluids for various processes cannot be calculated within the frame­ work of thermodynamics, since relaxation phenomena are essential­ ly llie processes of molecular (atomic, electron, etc.) transfer of diinrgy, mass, momentum and of similar physical quantities. The dviiliintion of relaxation time, therefore, is a problem of physical kiiidlics. I,el us consider a process operating in a thermodynamic system at a rule that is considerably less than the rate of relaxation; this means Hint at any stage of the process there will be enough time for the |in rumeters to equalize over the entire system and such a process will represent a continuous succession of equilibrium states infinitely rliiso to each other. Processes this slow are called equilibrium or qnnsi-static. It is clear that all real processes are non-equilibrium li nes, approaching equilibrium only to a greater or lesser exIrill.. It should also be noted that in the case of an equilibrium process Hie gradients of all parameters are equal to zero at any moment of lime. It follows from this that by virtue of symmetry the process limy operate in a system both in a forward and reverse direction.(that Is, in the direction in which any system parameter increases and decreases). For this reason an equilibrium process can be reversed in time, and during the reverse process the system will pass through Hid same succession of states as during the forward process, but in it reversed sequence. In this connection equilibrium processes are iiIso referred to as reversible processes. In thermodynamics graphical representation of states and proc­ esses is widely practised. So, for instance, when dealing with homo­ geneous systems (gas or liquid) the states of the system are depicted liv points and processes by lines on a P-V plane. It is clear that such graphical representation is possible only for equilibrium states and equilibrium (reversible) processes, since system parameters (for Instance, pressure) have definite values only in equilibrium states. I'p to Chapter IX of this book only equilibrium processes will be • onsidered, unless otherwise specified. In concluding this section we note that the parameters describing n slate of a system are not independent quantities, but are interre­ lat'd by one or several equations that are known as the equations of oinlc. For instance, for a gas pressure, temperature and volume are mldirelated as follows: / (P , V, T) = 0, while for a rod the magniimlc of the tensile force, temperature and the length of the rod nrn interrelated. The form of the equations of state cannot be deter­ mined by thermodynamic methods, and the knowledge permitting l lii'ir derivation comes from experiments or from other branches of Ilii'iirolical physics (usually from statistical physics). However,

14

Part One. Thermodynamics

Finally, in Sections 22 through 31 the thermodynamics of systems with a variable amount of substance and especially the problem of phase equilibriums and phase transitions are considered. 1. Equilibrium States and Equilibrium Processes

By a thermodynamic system will be meant any macroscopic body in an equilibrium or close to equilibrium state. The states of any thermodynamic system can be defined by means of a number of parameters. Thus, for instance, the states of a gas or liquid (homogeneous systems) can be set by means of the following parameters: P (pressure), V (volume), T (temperature); the state of a film of a liquid—with the aid of a (surface tension coefficient), a (film area) and T (temperature). The state of a rod can be specified with the aid of parameters I (length), a (cross-sectional area), / (ten­ sile force), E (Young’s modulus). It should be noted, however, that not all parameters of a system necessarily have a definite meaning in any particular state. Consider, for instance, a vessel divided into two halves by a partition with a valve, initially the left-hand half of the vessel contains a gas and the other half of the vessel is evacuated. If we open the valve, a stream of gas will be forced through it. It is clear that during the first moments of this process the volume of gas is indefinite, with the density of the gas contained in the right-hand half of the vessel changing from point to point according to some intricate law. It is thus impossible to indicate the boundaries of the volume of the gas. Also states are possible in which any parameter of a system differs at different points, so that no single value of this parameter exists for the system as a whole. It is possible to visualize a system, for instance, the temperature of which changes from point to point, or a gas whose pressure varies from point to point. Experiments show, however, that when a thermodynamic system is in such a state, there exist fluxes (heat flux, mass flux, etc.) and this state does not remain invariable, unless it is maintained artificially with the aid of heat­ proof (impervious) partitions, gas-tight walls, etc. With the pas­ sage of time a state is established in which the value of each param­ eter of the system is the same at all points and remains constant as long as desired, provided the external conditions do not change. Such states are called equilibrium states. If no equilibrium is estab­ lished and there exist in the system gradients of macroscopic param­ eters (pressure, density, temperature, etc.) such a state is referred to as a non-equilibrium state. The process during which a thermodynamic system passes from a non-equilibrium to an equilibrium state is referred to as a relaxation process. The time it takes each system parameter to become the same over the entire volume of the system is known as the relaxation time

Ch. I. General Laws of Thermodynamics

15

o| ilio givon parameter. The total relaxation time of a system, clear­ ly, is t ho longest of the individual relaxation times. The relaxation 11mrs for various processes cannot be calculated within the frame­ work of thermodynamics, since relaxation phenomena are essential­ ly Ilie processes of molecular (atomic, electron, etc.) transfer of onorgy, mass, momentum and of similar physical quantities. The ('valuation of relaxation time, therefore, is a problem of physical kinetics. I.ct us consider a process operating in a thermodynamic system at a rale that is considerably less than the rate of relaxation; this means Ihat at any stage of the process there-will be enough time for the parameters to equalize over the entire system and such a process will represent a continuous succession of equilibrium states infinitely dose to each other. Processes this slow are called equilibrium or i/u/isi-static. It is clear that all real processes are non-equilibrium ones, approaching equilibrium only to a greater or lesser exlent. It should also be noted that in the case of an equilibrium process Ilie gradients of all parameters are equal to zero at any moment of lime. It follows from this that by virtue of symmetry the process may operate in a system both in a forward and reverse direction.(that is, in the direction in which any system parameter increases and decreases). For this reason an equilibrium process can be reversed in time, and during the reverse process the system will pass through l lie same succession of states as during the forward process, but in a reversed sequence. In this connection equilibrium processes are also referred to as reversible processes. In thermodynamics graphical representation of states and proc­ esses is widely practised. So, for instance, when dealing with homo­ geneous systems (gas or liquid) the states of the system are depicted by points and processes by lines on a P-V plane. It is clear that such graphical representation is possible only for equilibrium states and equilibrium (reversible) processes, since system parameters (for instance, pressure) have definite values only in equilibrium states. I'p to Chapter IX of this book only equilibrium processes will be iunsidered, unless otherwise specified. In concluding this section we note that the parameters describing a slate of a system are not independent quantities, but are interre­ lated by one or several equations that are known as the equations of ohih'. For instance, for a gas pressure, temperature and volume are interrelated as follows: / (P , V, T) = 0, while for a rod the magni­ tude of the tensile force, temperature and the length of the rod ar e interrelated. The form of the equations of state cannot be deter­ mined by thermodynamic methods, and the knowledge permitting Hi'-ir derivation comes from experiments or from other branches of ..... rolical physics (usually from statistical physics). However,

16

Part One. Thermodynamics

the derivation of equations of state in statistical physics is also a very intricate problem that can be solved for no more than a small number of the simplest systems. 2. Temperature. The Temperature Principle

The concept of temperature has already been used in Sec. 1, an intuitive understanding of the concept being assumed. We will now adopt a stricter approach to its understanding. The concept of temperature is based on a rather subjective and diffuse term—the degree of hotness of a body. The term can, how­ ever, be given a more objective meaning through a whole number of easily measured physical parameters depending on the degree of hotness. Examples of such parameters are the height of the column of liquid mercury in a glass tube, the pressure of a gas in a constantvolume vessel, the resistance of a conductor, the emissive power of a red-hot body, etc. The data obtained by measuring any of these parameters can be used to create an empirical thermometer, with the measurement scale for this conditional or empirical temperature being chosen arbitrarily. When using, for instance, a mercury glass thermometer, the name conditional temperature can be given to the height of the column of mercury, measured in any unit, or any monotonically increasing function of that height. It should also be noted that the held of application of each empirical thermometer is restricted (if only from one side). Thus the lower limit of applica­ bility of a mercury thermometer is the mercury solidification point, the lower limit of applicability of a gas thermometer is the gas con­ densation point, the upper limit of applicability of a resistance thermometer is the fusion point (or the boiling point) of metal, etc. Due to the fact that these fields of application partly overlap, we can select one of the empirical thermometers as a basis to the deter­ mination of conditional temperature on some arbitrary scale for quite a wide temperature range. Let us now introduce the concept of the thermostat. By a thermo­ stat (thermorelay) will be meant a body whose heat capacity is great compared with that of any reference bodies with which the thermo­ stat will be brought into contact. This means that, on the one hand, when brought into contact with reference bodies the conditional temperature of the thermostat does not change and, on the other hand, after a short relaxation time any reference body brought into contact with the thermostat acquires its temperature. Let us place into a thermostat a vessel containing a gas and change its volume with the help of a piston, simultaneously measuring the pressure. Plotting the results in the form of points on a P-V plane, we obtain a curve (isotherm), the form of which depends essentially on the kind of a gas and on the conditional temperature of the ther-

Ch. I. General Laws of Thermodynamics

17

niosliil. Changing the conditional temperature of the thermostat mill repeating each time the procedure described above we obtain a system, or family, of isotherms corresponding to different condi­ tional temperatures x„ (Fig. 1). Let us write down the equations of the isotherms 1, 2, ... in the form (pi(/J, F) = 0, (p2(P, V) = 0, ..., labelling the isotherms in a wav that the number n increases with the rising conditional tem­ perature. Let us now turn to an infinitely dense network of isotherms, assum­ ing that when passing from one isotherm to the adjoining one the change in the condit ional temper­ ature of the thermostat and, con­ sequently, of the gas is infinite­ ly small. The number of the iso­ therm must then be replaced by a continuously varying parame­ ter x, which we can take as the conditional (empirical) tempera­ ture. Let the isotherm correspond­ ing to the conditional tempera­ FIG. 1 ture x be described by the equa­ tion

1 is the adiabatic exponent, constant for n given gas. The law described by Eq. (3.1) is valid for monoatomic giisi's at any temperature lower than the characteristic temperature of gas ionization, and for polyatomic gases over a wide temperature range. In Sec. 8 it will be shown that y is the ratio of the heat capaci­ ty at constant pressure, CP, to the heat capacity at constant volume, f'i . The relation a = a (P, 7) will be referred to as the caloric niinition of the state of a substance, and the imagined gas for which llm equation a = a (PV*) is valid and for which the adiabats obey Fi|. (3.1) as the calorically ideal, or perfect, gas. It should be noted Mint a perfect gas is necessarily a thermally ideal gas as well. Generully speaking, any statement to the contrary is incorrect, and the i nt io CplCv of a thermally ideal gas can depend on temperature nver a wide temperature range. Iiy virtue of the fact that y > 1, on the P-V plane the adiabats run steeper (3.1) in respect to the 7-axis than the isotherms (2.1). I'nr this reason the fundamental fact that each adiabat intersects 1'iir.li isotherm at one and only one point becomes insignificant for a lii'i fecl gas, since the system of equations PV = a, PVv = b has mm solution

20

Part One. Thermodynamics

Generalizing experimental data, we are now in a position to for­ mulate the following postulate of thermodynamics, the entropy principle: there exists (at least one) a single-valued function of state, remaining constant for any processes'operating in a thermallyinsulated vessel, called the conditional entropy. Between the pairs of variables, P, V and t , a there is a one-to-one correspondence. The conditional entropy of gases can be considered as a function of pressure and volume, a — a (P , K). For a calorically ideal gas a — — a (PVv); it depends only on the product PVv. 4. Absolute Temperature and Absolute Entropy

In the preceding sections conditional temperature and conditional entropy were defined accurate to the transformations x' = f l (x) and a' — / 2(o), where f l and / 2 are monolonically increasing and continuous functions of their arguments. Any arbitrariness in the selection of conditional temperature and conditional entropy can, however, be removed in the following way. The condition required for the one-to-one correspondence of pairs of variables P, V and x, o for two infinitely small regions on the P-V and x-a planes is known to be the difference of the Jaco­ bian transformation D = d ( /\ V)ld (x, a) from zero and from in­ finity: D =jfc0, C "1 ^ 0 (see ‘‘Mathematical Appendix”, Al). The geometrical meaning of the Jacobian lies in the fact that its module gives the variation factor for the elementary area when passing from the P-V plane to the x-a plane dPdV -+■ \D | dx da. If we demand complete equivalence for the P-V and x-a planes, it is natural to impose the requirement that the Jacobian D be equal to unity. Such normalization of the Jacobian leads to a de­ fined selection (accurate to the reference point and scale) of the tem­ perature and entropy scales. The temperature and entropy read off these scales will be referred to as absolute temperature T and absolute entropy S (in the following sections the word “absolute” will be omitted for the sake of brevity). It should be noted that there exist thermodynamic systems for which absolute temperature and absolute entropy cannot be cali­ brated over the entire P-V plane with the aid of the relationship d{P,V) d (T, S)

,

(4.1)

An example of such a system is water (see Sec. 18) and some other liquids and solids. This concerns, however, small regions of the P-V plane, while for the remaining parts of the plane calibration with the aid of Eq. (4.1) remains possible.

Ch. I. General Laws of Thermodynamics

21

Tlii> n'qiiirement (4.1) allows the determination of temperature mill i>nt ropy as a function of pressure and volume for any thermoilviiiuiiir system for which the equations of the isotherms and adiaIin is m e known. A perfect gas is used as a working medium for the i'iiIiliral.inn of absolute temperature and absolute entropy. Since IIn* Ino pcni l in e of a perfect gas depends only on PV, and the entro­ py mi /’Tv, then, denoting PV = x, and PV^ = y, we obtain T = '/' (.1), S S (y) and {!•' V)

1 S' (Y— 1)!/ *

= r S '( Y - l ) 0 = l , ‘ T'

(4.2)

Tlie symbols T and S' in Eqs. (4.2) represent the derivatives of /' ( 1) mill S (//) with respect to their arguments. Since the right11n1111 ■!1111• iif K(|. (4.2) depends only on y and the left-hand side only mi 1, i'ih Ii of ilm Iwo sides is equal to the same constant -iTl . Then in' gel Ihi' i'i|iinlions

r (*) = - - . S ’ (y) . (Y—1) y ’

(4.3)

Ini low 1ii|i I In* iiili'griition of which we find

r ,v

II V •

10

In// I (■

p v + 0, — H

Y-1 In (PVy)-tC ,

(4.4)

alii'ii' ll mol I mi' 1111i'u in I Imi ronsliinls. Since it is convenient to Ini*....... 11iiii'iisliiiih'ns ii 11inhi