MUSLIM PHYSICS DEPARTMENT GADJAH MAOA UNIVf~)iiY Francis W. Sears Professor Emeritus, Dartmouth College Gerhard L. Salinger Associate Professor of Physics, Rensselaer Polytechnic Institute Thermodynamics, Kinetic Theory, and Statistical Thermodynamics THIRD EDITION Addison-Wesley Publishing Company Reading, Massachusetts Amsterdam • London • Manila • Singapore • Sydney • Tokyo I WORlD SlliDENf SERIES EDmON SIXTH PRINTING 1982 Copyri&flt 0 191.5 by Addison-Wesley P\lblishing Company, Inc. Philippines copyri&bt 197.5 by Addisoo-Wcslcy Publishing Company, Inc. All righu rosavtd. No part of this publiution may be ~produced, stored in a retrieval system, or transmitted, in any form or by any means, elcetronic, mec:hank:al, photoc:opying. recording, or otherwise, without chc prior written permission or the publisher. Printed in the United States of America. Published simultaneously in Canada. Ubrary of Congress Catalog Card No. 74-28.51. Preface This text is a major revision of An Introduction to Thermodynamics, Kinetic Theory, and Statistical Mechanics by Francis W. Sears. The general approach has been unaltered and the level remains much the same, perhaps being increased somewhat by greater coverage. The text is still considered useful for advanced undergraduatCJ in physics and engineering who have some familiarity with calculus. The first eight chapters are devoted to a presentation of classical thermo dynamics without recourse to either kinetic theory or statistical mechanics. We feel it is important for the student to understand that if certain macroscopic properties of a system are determined experimentally, all the properties of the system can be specified without knowing anything about the microscopic properties of the system. In the later chapters we show how the microscopic properties of the system can be determined by using the methods of kinetic theory and statistical mechanics to calculate the dependence of the macroscopic properties of a system on thermodynamic variables. The presentation of many topics differs from the earlier text. Non·PVT systems are introduced in the second chapter and are discussed throughout the text. The first la)V is developed as a definition of the difference in the internal energy of a system between two equilibrium states as the work in an adiabatic process between the states and in which the kinetic and potential energy of the system do not change. The heat flow is then the difference between the work in any process between two equilibrium states and the work in an adiabatic process between the same states. Care is taken to explain the effects of changes in kinetic and potential energy as well. After the discussion of the first law, various examples are presented to show which properties of the system can be determined on the basis of this law alone. The statement that "in every process taking place in an isolated system the entropy of the system either increases or remains constant" is used as the second law. It is made plausible by a series of examples and shown to be equivalent to the "engine" statements and the Caratneodory treatment. Thermodynamic potentials are presented in greater detail than in the earlier text. A new potential F" is introduced to make consistent the thermodynamic and statistical treatments of processes in which the potential energy of a system changes. The discussion of open systems, added in Chapter 8, is necessary for the new derivation of statistics. Ill lv PREFACE Kinetic theory of gases is treated in Chapters 9 and 10. Although the coverage appears to be reduced from the previous edition, the remaining material is dis cussed from the point of view of statistics in Chapter 12. The derivation of the distribution functions for the various types of statistics is completely different from previous editions. Discrete energy levels are assumed from the outset. The number of microstates belonging to each macrostate is calculated in the conventional manner for Bose-Einstein, Fermi-Dirac and Ma.well- . Boltzmann statistics. The entropy is shown to be proportional to the natural logarithm of the total number of microstates available to the system and not to the number of microstates in the most probable macrostate. The distribution of particles among energy levels is determined without the use of Lagrange multiplier> and Stirling's approximation, by calculating the change in the total number ol microstates when a particle in a particular energy level is removed from the system. The logarithm of this change is proportional to the change of entropy of the system. Only the single-particle partition function is introduced and it is used to derive the thermodynamic properties of systems. The coverage is much the same as the earlier text except that it is based entirely on discrete levels. The chapter on/ fluctuations has l)een omitted. The number of problems at the end of each chapter has been expanded. Some of the problems would become tedious if one did not have access to a small calcu lator. The International System (SI) bas been adopted throughout. Thus the units are those of the MKS system and are written, for example, as 1 kilomoJe-• K-• for specific heat capacity. The section on classical thermodynamics can be used for a course lasting one quarter. For a one-semester course it can be used with either the chapters on kinetic theory or statistical thermodynamics, but probably not both, unless only classical statistics are discussed, which can be done by using the development given in the sections on Bose-Einstein statistics and taking the limit that g1 » N1• We appreciate the he Ip fu l comments oft he reviewers of the manuscript, especially L. S. Lerner and C. F. Hooper, who also gave part of the manuscript a field test. One of us (GLS) wishes to thank his colleagues at Rensselaer for many helpful discussions. J. Aitken worked all the problems and checked the answers. Phyllis Kallenburg patiently retyped many parts of the manuscript with great accuracy and good humor. The encouragement of our wives and tolerance of our children helped considerably in this undertaking. Criticisms from teachers and students will be welcomed. Norwich, Vermont F.W.S. Troy, New York G.L.S. October 1974 MUSLIM Contents PHYSICS O~t'ARit.IENT GADJAH MADA u;-.;:VERSITY Fundamental concepts 1-1 Scope of thermodynamics 2 1-2 Thermodynamic systems 3 1-3 State of a system. Properties 3 1-4 Pressure 4 l~s Thermal equilibrium and temperature. The Zeroth law s 1-6 Empirical and thermodynamic temperature 7 1-7 The International Practical Temperature Scale IS 1-8 Thermodynamic equilibrium . 16 1-9 Processes 17 2 Equations of state 2-1 Equations of state 24 2-2 Equation of state of an ideal gas 24 2-3 P-v-T surface for an ideal gas 26 2-4 Equations of state of real gases 28 2-5 P-u-T surfaces for real substances 30 2-6 Equations of state of other than P-u-T systems 40 2-7 Partial derivatives. Expansivity and compressibility. 42 2-8 Critical constants of a van der Waals gas 49 2-9 Relations between partial derivatives 51 2-10 Exact differentials 53 3 The first law of thermodynamics 3-1 Introduction 62 3-2 Work in a volume change 62 3-3 Oi~er forms of work . 65 3-4 W rk depends on the path 69 3-5 Configuration work and dissipative work 70 3-6 The first law of thermodynamics 72 3-7 Internal energy 73 3-8 Heat flow 74 3-9 Heat flow depends on the path 77 3-10 The mechanical equivalent of heat 77 3-11 Heat capacity 80 3-12 Heats of transformation. Enthalpy 83 3-13 General form of the first law. 86 3-14 Energy equation of steady flow 87 vi CONTENTS 4 Some consequences of the first law 4-1 The energy equation 98 4-2 T and v independent 98 4-3 T and P independent 100 4-4 P and v independent. . . . , . . . . , 101 4-5 The Gay·Lussac-Joule experiment and the Joule-Thomson experiment 102 4-{; 1 Reversible adiabatic processes . . . . . . . 108 4-7 The Carnot cycle . . . . . . . . . Ill 4-8 The heat engine and the refrigerator 113 5 Entropy and the second law of thermodynamics S-1 The second law of thermodynamics 122 S-2 Thermodynamic temperature. 124 5-J Entropy . . . . . 127 5-4 Calculations of entropy changes in reversible processes 130 5-S Temperature. .n tropy diagrams 132 S-6 Entropy changes in irreversible processes 133 5-7 The principle of increase of entropy . . . . 135 S-8 The Clausius and Kelvin· Planck statements or the second law 138 6 Combined fir.st and second laws 6-1 Introduction 148 6-2 T and o independent 149 6-3 T and P independent IS3 6-4 P and v independent IS4 6-5 The T ds equations ISS 6-{i Properties of a pure substance IS7 6-7 Properties of an ideal gas IS9 6-8 Properties of a van der Waals gas . 160 6-9 Properties of a liquid or solid under hydrostatic pressure . 163 6-10 The Joule and Joule-Thomson experiments 164 6-ll Empirical and thermodynamic temperature 166 6-12 Multivariable systems. CarathCodory principle 168 7 Thermodynamic potentials 7-1 The Helmholtz function and the Gibbs function 178 7-2 Thermodynamic potentials . 181 7-3 The Maxwell relations . · 18S ~-4 Stable and unstable equilibrium 186 -5 Phase transitions . . 190 7-{i The Clausius·Ciapeyron equation 193 7-7 The third law of thermodynamics 196 8 Applications of thermodynamics to simple systems 8-1 Chemical ~tential . . . 206 8-2 Phase equtlibrium and the phase rule 210 8-3 Dependence or vapor pressure on total pressure 216 CONTENTS vii 8-4 Surface tension . . . 218 8-S Vapor pressure or a liquid drop 221 8-6 The reversible voltaic cell . 223 8-7 Blackbody radiation . 22S 8-8 ThermodJnamics or magnetism 228 8-9 Engineenng applications • 233 9 Kinetic theory 9-1 Introduction . 2SO 9-2 Basic assumptions 2SI 9-3 Molecular fluK • 2S4 9-4 Equation or state or an ideal gas 2S8 9-S Collisions with a moving wall . . 262 9-6 The principle or equipartition or energy . 264 9-1 Classical theory of specific heat capacity 267 9-8 Specific heat capacity or a solid 271 10 Intermolecular forces. Transport phenomena 10-1 Intermolecular forces 276 10-2 The van der Waals equation of state 276 10-3 Collision cross seclion. Mean free path 279 10-4 Coefficient or viscosity 286 10-S Thermal conductivity . 292 10-6 Diffusion . 294 10-7 Summary . 296 11 Statistical thermodynamics I 1-1 Introduction 302 11-2 Energy states and energy levels 302 11-3 Macrostatn and microstates 307 11-4 Thermodynamic probability 310 11-S The Bose-Einstein Slatistics. 312 11-6 The Fermi-Dirac statistics . 317 I 1-7 The Maxwell-Boltzmann statistics 320 11-8 The statistical interpretation or entropy 323 11-9 The Bose-Einstein distribution function 327 I 1-10 The Fermi-Dirac distribution function 331 11-11 The classical distribution function . . • . . 333 11-12 Comparison or distribution functions for indistinguishable particles 333 11-13 The Maxwell-Boltzmann distribution function 334 11-14 The partition function • . . 336 Il-lS Thermodynamic properties of a system 337 12 Applications of statistics to gases ".1 2-1 The monatomic ideal gas . 3SO rq-2 The distribution or molecular velocities. 3S4 12-3 EKperimental verification or the Maxwell-Boltzmann speed distribu· tion. Molecular beams 362 12-4 Ideal gas in a gravitational field . 366 viii CONTENTS 12-5 The principle or equipattition or energy 370 ·~ The quantozed linear oscillator . . 372 12-7 Specific beat capacity or a diatomic gas 376 13 Applications of quantum statistics to other systems 13-1 The Einstein theory or the specific heat capacity or a solid 386 13-2 The Debye theory or the specific heat capacity or a solid 387 13-3 Blackbody radiation • • • • • • • 395 13-4 Paramagnetism . 399 13-5 Negative temperatures 405 13-6 The electron gas. 407 APPENDIX A Selected differentials from e condensed collection of thermodynamic formulas by P. W. Bridgman 419 B The Lagrange method of undetermined multipliers 421 C Properties of factorials 424 D An alternative derivation of distribution functions 427 E Magnetic potential energy 432 Answers to problems 43S Index 445 1 Fundamental concepts 1-1 SCOPE OF THERMODYNAMICS 1-2 THERMODYNAMIC SYSTEMS 1-3 STATE OF A SYSTEM. PROPERTIES 1-4 PRESSURE 1-5 THERMAL EQUILIBRIUM AND TEMPERATURE. THE ZEROTH LAW 1-6 EMPIRICAL AND THERMODYNAMIC TEMPERATURE 1-7 THE INTERNATIONAL PRACTICAL TEMPERATURE SCALE 1-6 THERMODYNAMIC EQUILIBRIUM 1-9 PROCESSES 2 FUNDAMENTAL CONCEPTS 1-1 1-1 SCQPE OF THERMODYNAMICS Thermodynamics is an experimental science based on a small number of principles that are generalizations made from experience. It is concerned only with macro· scopic or large-scale properties of matter and it makes no hypotheses about the small-scale or microscopic structure of matter. From the principles of thermo dynamics one can derive general relations between such quantities as coefficients of expansion, compressibilities, specifi" heat capacities, heats of transformation, and magnetic and dielectric coefficients·, especially as these are affected by tem· perature. The principles of thermodynamics also tell us which few of these rela tions must be determined experimentally in order to completely specify all the properties of the system. The actual magnitudes of quantities like those above can be calculated only on the basis of a molecular model. The kinetic theory of matter applies the Jaws of mechanics to the individual molecules of a system and enables one to calculate, for example, the numerical value of the specific heat capacity of a gas and to understand the properties of gases in terms of the law of force between individual molecules. The approach of statistical thermodynamics ignores the detailed consideration of molecules as individuals and applies statistical considerations to find the distri bution of the very large number of molecules that ll)ake up a macroscopic piece of matter over the energy states of the system. For those systems whose energy states can be calculated by the methods of either quantum or classical physics, both the magnitudes of the quantities mentioned above.and the relations between them can be determined by quite general means. The methods of statistics also give further insight into the concepts of entropy and the principle of the increase of entropy. Thermodynamics is complementary to kinetic theory and statistical thermo dynamics. Thermodynamics provides relationships between physical properties of any system once certain measurements are made. Kinetic theory and statistical thermodynamics enable one to calculate the magnitudes of these properties for those systems whose energy states can be determined. The science of thermodynamics had its start in the early part of the nineteenth century, primarily as a result of attempts to improve the efficiencies of steam engines, devices into which there is an input in the form of heat, and whose output is mechanical work. Thus as the name implies, thermodynamics was concerned with both thermal and mechanical, or dynamical, concepts. As the subject developed and its basic laws were more fully understood, its scope became broader. The principles of thermodynamics are now used by engineers in the design of internal combustion engines, conventional and nuclear power stations, refrigeration and air-conditioning systems, and propulsion systems for rockets, missiles, aircraft, ships, and land vehifles. The sciences of physical chemistry and chemical physics consist in large part of the applications of thermodynamics to chemistry and chemical equilibria. The production of extremely low temperatures, in the neigh borhood of absolute zero, involves the application of thermodynamic principles 0