Thermal Physics Energy and Entropy Written by distinguished physics educator, this fresh introduction to thermodynamics, statistical mechanics and the study of matter is ideal for undergraduate courses. The textbook looks at the behavior of thermodynamic variables and examines partial derivatives the essential language of thermodynamics. It also explores states of matter and the phase transitions between them, the ideal gas equation and the behavior of the atmosphere. The origin and meaning of the laws of thermodynamics are then discussed, together with Carnot engines and refrigerators and the notion of reversibility. Later chapters cover the partition function, the density of states and energy functions, as well as more advanced topics such as the interactions between particles and equations for the states of gases of varying densities. Favoring intuitive and qualitative descriptions over exhaustive mathematical derivations, the textbook uses numerous problems and worked examples to help readers get to grips with the subject. david goodstein is the Frank J. Gilloon Distinguished Teaching and Service Professor Emeritus at the California Institute of Technology. He has extensive research experience in condensed matter physics, and his book States of Matter (Prentice Hall, 1975) was hailed as launching this important field. He directed and hosted the popular television series The Mechanical Universe, which was based on his lectures at Caltech.
Thermal Physics Energy and Entropy DAVID GOODSTEIN California Institute of Technology
University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: /9781107080119 D. Goodstein 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Goodstein, David L., 1939 author. Thermal physics : energy and entropy / by. pages cm Includes index. ISBN 978-1-107-08011-9 (Hardback) 1. Thermodynamics. 2. Entropy. 3. Statistical mechanics. I. Title. QC311.G585 2015 536.7 dc23 2014021242 ISBN 978-1-107-08011-9 Hardback ISBN 978-1-107-46549-7 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents Preface Acknowledgments page vii viii 1 The basic ideas of thermodynamics and statistical mechanics 1 1.1 From atoms to thermodynamics 1 1.2 Counting quantum states 5 1.3 Equilibrium, equal probabilities and temperature 14 1.4 The way entropy works 17 1.5 Some difficulties and their resolutions 21 2 The care and feeding of thermodynamic variables 28 2.1 The thermodynamic variables 28 2.2 Partial derivatives 29 2.3 Some special partial derivatives 33 2.4 Thermodynamic operations 37 3 Gases and other matters 45 3.1 The states of matter 45 3.2 Ideal and non-ideal gases 59 3.3 The atmosphere 66 4 The laws of thermodynamics 69 4.1 The origin and meaning of the first and second laws 69 4.2 Ideal engines 73 4.3 Tools of the trade 81 5 The Boltzmann factor and the partition function 91 5.1 The Boltzmann factor 91 5.2 What happened? 96 v
vi Contents 5.3 The density of states 97 5.4 Energy and entropy 100 5.5 The entropy of an ideal gas 105 6 Thermodynamic functions 108 6.1 Free energies 108 6.2 The chemical potential 113 6.3 Variational properties 116 6.4 Phase equilibrium in systems of more than one constituent 117 7 Statistical mechanics for fixed and variable N 119 7.1 Statistical mechanics for fixed N 119 7.2 The Gibbs distribution and the partition function 121 7.3 Statistical mechanics for variable N 124 8 More advanced topics 127 8.1 Interacting particles 127 8.2 Doing sums as integrals 128 8.3 The partition function for interacting particles 130 8.4 What happened? 131 8.5 The statistical mechanics of real molecules 133 8.6 Two atoms 133 8.7 More atoms 135 8.8 More about equations of state 139 8.9 The van der Waals equation of state 143 Solutions to the problems 147 Index 164
Preface This is a book for students who have some familiarity with general physics and the calculus to learn what thermodynamics and statistical mechanics is all about, starting with the fact that you don t have to know what a system is doing, just how many things it could be doing, to get precise values of its thermodynamic variables, particularly its temperature and pressure. That insight is discussed and explained in the first chapter, along with some simple observations about the peculiar behavior of very large numbers. It uses the perfect gas to illustrate its points. Chapter 2 looks into the behavior of thermodynamic variables and gives a lesson in partial derivatives, the essential language of thermodynamics. It also considers the various ways of expanding an ideal gas. Chapter 3 deals with the states of matter and the phase transitions between them, and also covers temperature scales, the ideal gas equation of state and the behavior of the atmosphere. Chapter 4 considers the origin and meaning of the first and second laws of thermodynamics and goes on to discuss Carnot engines and refrigerators and the notion of reversibility. Chapter 5 addresses the probability of finding a system in a particular state and goes on to deal with the partition function and the density of states. Chapter 6 concerns itself with the various energy functions and includes a neat mnemonic device for remembering them all. Chapter 7 deals with statistical mechanics for fixed and variable N and the grand partition function. Chapter 8 treats some more advanced topics, particularly interactions among the particles and equations of state of gases of varying densities. The book is designed to offer a good rigorous introduction to thermodynamics, statistical mechanics and the study of matter in general. vii
Acknowledgments I am very happy to acknowledge the generous assistance given me by my wife, Dr. Judith Goodstein. It was she who among many other things found all of the photographs in the places given below. Photographs of James Prescott Joule, James Clerk Maxwell, Rudolf Clausius, Max Planck, Albert Einstein, Michael Faraday, Josiah Willard Gibbs, William Thomson (Lord Kelvin), Nicolas Le onard, Sadi Carnot, Courtesy of the Archives, California Institute of Technology. Photographs of Ludwig Boltzmann and Johannes Van der Waals, Emilio Segre`, Visual Archives, American Institute of Physics. Photograph of Joseph and Maria Goeppert Mayer, Mandeville Special Collections, UC San Diego Library. viii