Statistical Physics E-Book

Franz Mandl

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: The Manchester Physics Series
• Sprache: Englisch

Seitenzahl: 569

Veröffentlichungsjahr: 2013

Beschreibung

The Manchester Physics Series General Editors: D. J. Sandiford; F. Mandl; A. C. Phillips Department of Physics and Astronomy, University of Manchester Properties of Matter B. H. Flowers and E. Mendoza Optics Second Edition F. G. Smith and J. H. Thomson Statistical Physics Second Edition E. Mandl Electromagnetism Second Edition I. S. Grant and W. R. Phillips Statistics R. J. Barlow Solid State Physics Second Edition J. R. Hook and H. E. Hall Quantum Mechanics F. Mandl Particle Physics Second Edition B. R. Martin and G. Shaw The Physics of Stars Second Edition A. C. Phillips Computing for Scientists R. J. Barlow and A. R. Barnett

Statistical Physics, Second Edition develops a unified treatment of statistical mechanics and thermodynamics, which emphasises the statistical nature of the laws of thermodynamics and the atomic nature of matter. Prominence is given to the Gibbs distribution, leading to a simple treatment of quantum statistics and of chemical reactions. Undergraduate students of physics and related sciences will find this a stimulating account of the basic physics and its applications. Only an elementary knowledge of kinetic theory and atomic physics, as well as the rudiments of quantum theory, are presupposed for an understanding of this book. Statistical Physics, Second Edition features:

• A fully integrated treatment of thermodynamics and statistical mechanics.
• A flow diagram allowing topics to be studied in different orders or omitted altogether.
• Optional "starred" and highlighted sections containing more advanced and specialised material for the more ambitious reader.
• Sets of problems at the end of each chapter to help student understanding. Hints for solving the problems are given in an Appendix.

Leseprobe

Contents

Editors’ preface to the Manchester Physics Series

Preface to the Second Edition

Preface to First Edition

Flow diagram

1 THE FIRST LAW OF THERMODYNAMICS

1.1 MACROSCOPIC PHYSICS

1.2 SOME THERMAL CONCEPTS

1.3 THE FIRST LAW

1.4 MAGNETIC WORK

2 THE SECOND LAW OF THERMODYNAMICS I

2.1 THE DIRECTION OF NATURAL PROCESSES

2.2 THE STATISTICAL WEIGHT OF A MACROSTATE

2.3 EQUILIBRIUM OF AN ISOLATED SYSTEM

2.4 THE SCHOTTKY DEFECT*

2.5 EQUILIBRIUM OF A SYSTEM IN A HEAT BATH

3 PARAMAGNETISM

3.1 A PARAMAGNETIC SOLID IN A HEAT BATH

3.2 THE HEAT CAPACITY AND THE ENTROPY

3.3 AN ISOLATED PARAMAGNETIC SOLID

3.4 NEGATIVE TEMPERATURE

4 THE SECOND LAW OF THERMODYNAMICS II

4.1 THE SECOND LAW FOR INFINITESIMAL CHANGES

4.2 THE CLAUSIUS INEQUALITY

4.3 SIMPLE APPLICATIONS

4.4 THE HELMHOLTZ FREE ENERGY

4.5 OTHER THERMODYNAMIC POTENTIALS

4.6 MAXIMUM WORK

4.7 THE THIRD LAW OF THERMODYNAMICS

4.8 THE THIRD LAW (CONTINUED)

5 SIMPLE THERMODYNAMIC SYSTEMS

5.1 OTHER FORMS OF THE SECOND LAW

5.2 HEAT ENGINES AND REFRIGERATORS

5.3 THE DIFFERENCE OF HEAT CAPACITIES

5.4 SOME PROPERTIES OF PERFECT GASES

5.5 SOME PROPERTIES OF REAL GASES

5.6 ADIABATIC COOLING

6 THE HEAT CAPACITY OF SOLIDS

6.1 INTRODUCTORY REMARKS

6.2 EINSTEIN’S THEORY

6.3 DEBYE’S THEORY

7 THE PERFECT CLASSICAL GAS

7.1 THE DEFINITION OF THE PERFECT CLASSICAL GAS

7.2 THE PARTITION FUNCTION

7.3 VALIDITY CRITERION FOR THE CLASSICAL REGIME

7.4 THE EQUATION OF STATE

7.5 THE HEAT CAPACITY

7.6 THE ENTROPY

7.7 THE MAXWELL VELOCITY DISTRIBUTION

7.8 REAL GASES

7.9 CLASSICAL STATISTICAL MECHANICS

8 PHASE EQUILIBRIA

8.1 EQUILIBRIUM CONDITIONS

8.2 ALTERNATIVE DERIVATION OF THE EQUILIBRIUM CONDITIONS

8.3 DISCUSSION OF THE EQUILIBRIUM CONDITIONS

8.4 THE CLAUSIUS-CLAPEYRON EQUATION

8.5 APPLICATIONS OF THE CLAUSIUS–CLAPEYRON EQUATION

8.6 THE CRITICAL POINT

9 THE PERFECT QUANTAL GAS

9.1 INTRODUCTORY REMARKS

9.2 QUANTUM STATISTICS

9.3 THE PARTITION FUNCTION

10 BLACK-BODY RADIATION

10.1 INTRODUCTORY REMARKS

10.2 THE PARTITION FUNCTION FOR PHOTONS

10.3 PLANCK’S LAW: DERIVATION

10.4 THE PROPERTIES OF BLACK-BODY RADIATION

10.5 THE THERMODYNAMICS OF BLACK-BODY RADIATION

11 SYSTEMS WITH VARIABLE PARTICLE NUMBERS

11.1 THE GIBBS DISTRIBUTION

11.2 THE FD AND BE DISTRIBUTIONS

11.3 THE FD AND BE DISTRIBUTIONS: ALTERNATIVE APPROACH

11.4 THE CLASSICAL LIMIT

11.5 THE FREE ELECTRON MODEL OF METALS

11.6 BOSE–EINSTEIN CONDENSATION

11.7 THERMODYNAMICS OF THE GIBBS DISTRIBUTION

11.8 THE PERFECT CLASSICAL GAS

11.9 CHEMICAL REACTIONS

A MATHEMATICAL RESULTS

A.1 STIRLING’S FORMULA

A.2 EVALUATION OF

A.3 SOME KINETIC THEORY INTEGRALS

B THE DENSITY OF STATES

B.l THE GENERAL CASE

B.2 THE SCHRÖDINGER EQUATION

B.3 ELECTROMAGNETIC WAVES

B.4 ELASTIC WAVES IN A CONTINUOUS SOLID

APPENDIX C Magnetic systems*

APPENDIX D Hints for solving problems

Bibliography

Index

Conversion Factors

Copyright © 1971, 1988 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,

West Sussex PO19 8SQ, England

Telephone [phone]

Email (for orders and customer service enquiries): [email protected]

Visit our Home Page on www.wileyeurope.com or www.wiley.co.uk

Second Edition 1988

Reprinted November 1989, November 1991, October 1993, April and December 1994, July 1995, March and November 1996, November 1997, October 1998, September 1999, October 2000, September 2002, December 2003, November 2004, October 2005, October 2006, July 2007, June 2008, February 2009, August 2010

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to ([phone].

This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Other Wiley Editorial Offices

John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA

Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA

Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany

John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia

John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809

John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0-471-91532-4 (HB)

ISBN 978-0-471-91533-1 (PB)

THE MANCHESTER PHYSICS SERIES

General Editors: D. J. SANDIFORD; F. MANDL; A. C. PHILLIPS

Department of Physics and Astronomy, University of Manchester

PROPERTIES OF MATTER

B. H. FLOWERS and E. MENDOZA

OPTICS

Second Edition

F. G. SMITH and J. H. THOMSON

STATISTICAL PHYSICS

Second Edition

F. MANDL

ELECTROMAGNETISM

Second Edition

I. S. GRANT and W. R. PHILLIPS

STATISTICS

R. J. BARLOW

SOLID STATE PHYSICS

Second Edition

J. R. HOOK and H. E. HALL

QUANTUM MECHANICS

F. MANDL

PARTICLE PHYSICS

Second Edition

B. R. MARTIN and G. SHAW

THE PHYSICS OF STARS

Second Edition

A. C. PHILLIPS

COMPUTING FOR SCIENTISTS

R. J. BARLOW and A. R. BARNETT

Editors’ preface to the Manchester Physics Series

The Manchester Physics Series is a series of textbooks at first degree level. It grew out of our experience at the Department of Physics and Astronomy at Manchester University, widely shared elsewhere, that many textbooks contain much more material than can be accommodated in a typical undergraduate course; and that this material is only rarely so arranged as to allow the definition of a shorter self-contained course. In planning these books we have had two objectives. One was to produce short books: so that lecturers should find them attractive for undergraduate courses; so that students should not be frightened off by their encyclopaedic size or their price. To achieve this, we have been very selective in the choice of topics, with the emphasis on the basic physics together with some instructive, stimulating and useful applications. Our second objective was to produce books which allow courses of different lengths and difficulty to be selected, with emphasis on different applications. To achieve such flexibility we have encouraged authors to use flow diagrams showing the logical connections between different chapters and to put some topics in starred sections. These cover more advanced and alternative material which is not required for the understanding of latter parts of each volume.

Although these books were conceived as a series, each of them is self-contained and can be used independently of the others. Several of them are suitable for wider use in other sciences. Each Author’s Preface gives details about the level, prerequisites, etc., of his volume.

The Manchester Physics Series has been very successful with total sales of more than a quarter of a million copies. We are extremely grateful to the many students and colleagues, at Manchester and elsewhere, for helpful criticisms and stimulating comments. Our particular thanks go to the authors for all the work they have done, for the many new ideas they have contributed, and for discussing patiently, and often accepting, the suggestions of the editors.

Finally, we would like to thank our publishers, John Wiley & Sons Ltd, for their enthusiastic and continued commitment to the Manchester Physics Series.

D. J. SandifordF. MandlA. C. PhillipsFebruary 1997

Preface to the Second Edition

My motivation for producing this second edition is to introduce two changes which, I believe, are substantial improvements.

First, I have decided to give much greater prominence to the Gibbs distribution. The importance of this formulation of statistical mechanics is due to its generality, allowing applications to a wide range of systems. Furthermore, the introduction of the Gibbs distribution as the natural generalization of the Boltzmann distribution to systems with variable particle numbers brings out the simplicity of its interpretation and leads directly to the chemical potential and its significance. In spite of its generality, the mathematics of the Gibbs approach is often much simpler than that of other approaches. In the first edition, I avoided the Gibbs distribution as far as possible. Fermi–Dirac and Bose–Einstein statistics were derived within the framework of the Boltzmann distribution. In this second edition, they are obtained much more simply taking the Gibbs distribution as the starting point. (For readers solely interested in the Fermi–Dirac and Bose–Einstein distributions, an alternative derivation, which does not depend on the Gibbs distribution, is given in section 11.3.) The shift in emphasis to the Gibbs approach has also led me to expand the section on chemical reactions, both as regards details and scope.

Secondly, I have completely revised the treatment of magnetic work in section 1.4, with some of the subtler points discussed in the new Appendix C. As is well known, the thermodynamic discussion of magnetic systems easily leads to misleading or even wrong statements, and I fear the first edition was not free from these. My new account is based on the work of two colleagues of mine, Albert Hillel and Pat Buttle, and represents, I believe, an enlightening addition to the many existing treatments.

I have taken this opportunity to make some other minor changes: clarifying some arguments, updating some information and the bibliography, etc. Many of these points were brought to my attention by students, colleagues, correspondents and reviewers, and I would like to thank them all—too many to mention by name—for their help.

I would like to thank Henry Hall, Albert Hillel and Peter Lucas for reading the material on the Gibbs distribution, etc., and suggesting various improvements. I am most grateful to Albert Hillel and Pat Buttle for introducing me to their work on magnetic systems and for allowing me to use it, for many discussions and for helpful comments on my revised account. It is a pleasure to thank David Sandiford for his help throughout this revision, particularly for critically reading all new material, and Sandy Donnachie for encouraging me to carry out this work.

January 1987

FRANZ MANDL

Preface to First Edition

This book is intended for an undergraduate course in statistical physics. The laws of statistical mechanics and thermodynamics form one of the most fascinating branches of physics. This book will, I hope, impart some of this fascination to the reader. I have discarded the historical approach of treating thermodynamics and statistical mechanics as separate disciplines in favour of an integrated development. This has some decisive advantages. Firstly, it leads more directly to a deeper understanding since the statistical nature of the thermodynamic laws, which is their true explanation, is put in the forefront from the very beginning. Secondly, this approach emphasizes the atomic nature of matter which makes it more stimulating and, being the mode of thought of most working physicists, is a more useful training. Thirdly, this approach is more economical on time, an important factor in view of the rapid expansion of science.

It is a consequence of this growth in scientific knowledge that an undergraduate physics course can no longer teach the whole of physics. There are many ways of selecting material. I have tried to produce a book which allows maximum flexibility in its use: to enable readers to proceed by the quickest route to a particular topic; to enable teachers to select courses differing in length, difficulty and choice of applications. This flexibility is achieved by means of the flow diagram (on the inside front cover) which shows the logical connections of the chapters. In addition, some sections are marked with a star and some material, insufficient to justify a separate section, is printed on a tinted background. Material distinguished in either of these ways may be omitted. It is not needed later except very occasionally in similarly marked parts, where explicit cross-references are always given.

My aim has been to explain critically the basic laws of statistical physics and to apply them to a wide range of interesting problems. A reader who has mastered this book should have no difficulties with one of the more advanced treatises or with tackling quite realistic problems. I have limited myself to systems in equilibrium, omitting irreversible thermodynamics, fluctuation phenomena and transport theory. This was partly for reasons of time and space, but largely because these topics are hardly appropriate for a fairly elementary account. For this reason also, I have not discussed the foundations of statistical physics but have based the theory on some simple intuitively plausible axioms. The ultimate justification of this approach lies in its success.

The development of statistical physics which I have given is self-contained, but the level of sophistication presupposes some previous acquaintance with the kinetic theory of gases, with the elementary descriptive ideas of atomic physics and with the rudiments of quantum theory. Fortunately, very little of the latter is required.

Over the past ten years I have given various undergraduate and postgraduate courses on statistical physics at Manchester University. In its present form, this book developed out of a course given to second-year undergraduates in physics, chemical physics and electronic engineering. This course of 26 lectures, of 50 minutes each, approximately covered the unstarred sections of the book, as well as the material of chapter 5 and of sections 7.5, 7.7, 11.4 and 11.5,* omitting all material printed on tinted background. In addition, students were expected to solve about 20 problems. The answers were corrected, returned together with sample solutions, and discussed in class.

The problems and hints for solving them form an important part of the book. Attempting the problems and then studying the hints will deepen the reader’s understanding and develop his skill and self-confidence in facing new situations. The problems contain much interesting physics which might well have found its way into the main body of the text.

This book was preceded by a preliminary edition which was used in my lecture course and was also distributed fairly widely outside Manchester University. I have received many comments and suggestions for improvements and additions from readers. I also had many stimulating discussions with students and colleagues at Manchester. As a result the original text has been greatly improved. I would like to thank all these people most warmly; there are too many of them to thank them all by name. However, I would like to express my appreciation for their help to Professor Henry Hall and to Dr David Sandiford who read the whole manuscript and with whom I discussed difficult and obscure points until—temporarily at least—they seemed clear. Not only was this intellectual pursuit of great benefit to this book, but to me it was one of the joys of writing it.

May, 1970

F. MANDL

*In the second edition, the numbers of these sections have become 11.5 and 11.6.

CHAPTER 1

The first law of thermodynamics

1.1 MACROSCOPIC PHYSICS

In spite of the enormous complexity of macroscopic bodies when viewed from an atomistic viewpoint, one knows from everyday experience as well as from precision experiments that macroscopic bodies obey quite definite laws. Thus when a hot and a cold body are put into thermal contact temperature equalization occurs; water at standard atmospheric pressure always boils at the same temperature (by definition called 100 °C); the pressure exerted by a dilute gas on a containing wall is given by the ideal gas laws. These examples illustrate that the laws of macroscopic bodies are quite different from those of mechanics or electromagnetic theory. They do not afford a complete microscopic description of a system (e.g. the position of each molecule of a gas at each instant of time). They provide certain macroscopic observable quantities, such as pressure or temperature. These represent averages over microscopic properties. Thus the macroscopic laws are of a statistical nature. But because of the enormous number of particles involved, the fluctuations which are an essential feature of a statistical theory turn out to be extremely small. In practice they can only be observed under very special conditions. In general they will be utterly negligible, and the statistical laws will in practice lead to statements of complete certainty.

Fig. 1.1. Gas exerting pressure on movable piston, balanced by external applied force F.

In contrast to this macroscopic determination of pressure consider how the pressure actually comes about.* According to the kinetic theory the molecules of the gas are undergoing elastic collisions with the walls. The pressure due to these collisions is certainly not a strictly constant time-independent quantity. On the contrary the instantaneous force acting on the piston is a rapidly fluctuating quantity. By the pressure of the gas we mean the average of this fluctuating force over a time interval sufficiently long for many collisions to have occurred in this time. We may then use the steady-state velocity distribution of the molecules to calculate the momentum transfer per unit area per unit time from the molecules to the wall, i.e. the pressure. The applied force F acting on the piston can of course only approximately balance these irregular impulses due to molecular collisions. On average the piston is at rest but it will perform small irregular vibrations about its equilibrium position as a consequence of the individual molecular collisions. These small irregular movements are known as Brownian motion (Flowers and Mendoza,26 section 4.4.2). In the case of our piston, and generally, these minute movements are totally unobservable. It is only with very small macroscopic bodies (such as tiny particles suspended in a liquid) or very sensitive apparatus (such as the very delicate suspension of a galvanometer — see section 7.9.1) that Brownian motion can be observed. It represents one of the ultimate limitations on the accuracy of measurements that can be achieved.