Thermodynamics, Kinetic Theory, and Statistical Thermodynamics

462 Pages·1975·48.2 MB·English

by  Francis W. Sears

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