. C-=Q • . . " - • c..a-=.!E--mc..=..c..E-=a>CcaE--um /'.c. - iU " _-. -c. . -m- aC=Lca'lW_-u.ac.c>­um Q) -c .1:: 0... e..:--= ca --Ic::::: -c .c Cl) . Q) ~)~ ' , P Pion Limited, 207 Brondesbury Park, London NW2 SJN Contents ,.", Authors v © 1971 Pion Limited Preface vii ·,1,·, [ All rights reserved. No part of this book may be reproduced in any form by ( >1 The laws of thermodynamics (P. T.Landsberg) I photostat microfilm or any other means without written permission from the v Mathematical preliminaries I publishers. + " Quasistatic changes 4 SBN 85086 023 7 • The First Law 6 [J The Second Law 9 "oj J 3 "Simple ideal fluids II Joule -Thomson effect 21 v 0 v Thermodynamic cycles 23 K v Chemical thermodynamics 28 'V The Third Law 33 "' Phase changes 34 " Thermal and mechanical stability 37 2 Statistical theory of information and of ensembles (P. T.Landsberg) 44 VEntropy maximisation: ensembles 44 v Partition functions in general 48 vEnt ropy maximisation: probability distributions 51 "Most probable distribution method 53 .Some general principles 60 3 Statistical mechanics of ideal systems (P. T.Landsberg) 63 Maxwell distribution 63 v ~ Classical statistical mechanics 67 "' Virial theorem 72 Oscillators and phonons 74 v or The ideal quantum gas 80 ",Constant pressure ensembles 92 v Radiative emission and absorption 93 ( 4 Ideal classical gases of polyatomic molecules (C.l. Wormald) 98 -The translational partition function 98 "Thermodynamic properties and the theory of fluctuations 100 The classical rotational partition function 103 v \ "The quantum mechanical rotational partition function 105 A convenient formula for the high temperature rotational partition function 108 Thermodynamic properties arising from simple harmonic mode" of vibration 110 ,((3> ( Corrections to the rigid rotator-harmonic oscillator model Contributions to the therm9dynamic properties arising from low lying ... -..~ electronic energy levels 118 Calculation of the thermodynamic properties of HCI from spectroscopic data 121 Set on IBM 72 Composers by Pion Limited, London. Thermodynamic properties of ethane 127 Printed in Great Britain by J.W.Arrowsmith Limited, Bristol. ii Contents Contents iii 23 Time dependence of fluctuations: correlation functions, power spectra, 5 Ideal relativistic classical and quantum gases (P. T.Landsberg) 132 Wiener-Khintchine relations (C W.McCombie) 472 6 Non-electrolyte liquids and solutions (A.J.E.Cruickshank) 140 Cell theories of the liquid state 142 24 Nyquist's theorem and its generalisations (C W.McCombie) 489 Equation of state treatment of liquids 154 .. 25 Onsager relations (C W.McCombie) 502 Binary solutions 168 ~ .../ 26 Stochastic methods: master equation and Fokker-Planck equation 7 Phase stability, co-existence, and criticality (A .i.E. Cruickshank) 193 (I. Oppenheim, K.E.Shuler, G." Weiss) 511 Singulary systems 193 Binary systems 214 27 Ergodic theory, H-theorems, recurrence problems (D. ter Haar) 531 8 Surfaces (/.M.Haynes) 230 28 Variational principles and minimum entropy production (S.Simons) 549 The Gibbs model of a surface 230 Macroscopic principles 549 Electron flow problems 557 9 The imperfect classical gas (P.CHemmer) 246 The equation of state 246 Author index 563 The virial expansion 254 Subject index 565 Pair distribution function. Virial theorem 261 10 The imperfect quantum gas (D. fer Haar) 270 The equation of state 270 Second quantisation formalism 278 ~ v 11 Phase transitions (D. fer Haar) 282 Einstein condensation of a perfect boson gas 282 Vapour condensation 289 Hard sphere gas 293 12 Cooperative phenomena (D. fer Haar) 303 13 Green function methods (D. fer Haar) 319 Mathematical preliminaries 319 General formalism 320 \ The Kubo formula 323 The Heisenberg ferromagnet 324 14 The plasma (D.ter Haar) 333 15 Negative temperatures and population inversion (U.M. Titulaer) 341 Dynamic polarization 346 A model of laser action 348 16 Recombination rate theory in semiconductors (/.S.Blakemore) 350 17 Transport in gases (D.J.Gri{{iths) 378 18 Transport in metals (/.M.Honig) 401 19 Transport in semiconductors (/.M.Honig) 432 \(20 Fluctuations of energy and number of particles (C W.McCombie) 448 v21 Fluctuations of general classical mechanical variables (C W.McCombie) 457 oJ 22 Fluctuations of thennodynamic variables: constant pressure systems, isolated systems (C W.McCombie) 465 r I .) Authors J .S.Blakemore Department of Physics, Florida Atlantic University, Boca Raton, Florida A.J .B.Cruickshank School ofChemistry, University ofBristol, Bristol D.J .G riffiths Department ofPhysics, University of Exeter, Exeter J.M.Haynes School ofChemistry, University ofBristol, Bristol P.C.Hemmer Institutt for Teoretisk fysikk, NTH, Trondheim J.M.Honig Department ofChemistry, Purdue University, Lafayette, Indiana P.T.Landsberg Department ofApplied Mathematics and Mathematical Physics, University College, Cardiff C.W.McCombie J.J. Thompson Physical Laboratory, University ofReading, Reading I.Oppenheim Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts K.E.shuler Department ofChemistry, University of California, San Diego, California S.Simons Queen Mary College, London D.ter Haar Department of Theoretical Physics, University ofOxford, Oxford U.M.Titulaer Institut voor Theoretische FYsica der Rijksuniversiteit, Utrecht G.H.Weiss National Institute ofHealth, Bethesda. Maryland C.J.Wormald • School of Chemistry, University ofBristol• Bristol v I Preface Problems and solutions! Their production is an annual ritual-feared and unpopular-for the university teacher. Nor are examination questions particularly liked by students. Yet, take away the examination aura, and the technique appears in a new light. The problem-and-solution style of writing presents to the author new difficulties and constraints, and thus offers him a novel challenge. Perhaps it is like teasing a sculptor with a new and promising type of stone, or a poet with a new rhythm. Viewed in this light, the potential author has the opportunities which go with a new medium: for example, a novel way of arranging important results and of setting them up to be seen more clearly than is possible in 1 a uniformly flowing and elegant exposition. As to the reader, he finds before him a series of hurdles. They are easy enough at first to bewitch him, beguile him and persuade him to join into the fun-until he gradually participates with the author in a neo-Socratic dialogue. The existence of these opportunities must surely be one of the considerations which explain how it was possible to find so select and experienced a group of persons to help in the construction of this book. Each author contributed in his own special area of research with the result that the book presents a penetrating view of statistical physics and its uses which, as regards the width of its sweep is, I suspect, beyond any one of today's experts. Thus, in spite of the age of our subject and the love and care which has been bestowed on it by successive generations, this book presents in some sense a new unit. I have planned its outline and have discussed various points with the authors, but have imposed only a limited uniformity of style. I have also attempted to ensure that overlap of material is not extensive, that cross references are adequate, and that the early problems in each chapter are reasonably easy. In this way I wanted the reader of each chapter to feel drawn into a fascinating world which opens up for him once he realises he can actually derive results for himself. Here then is a book for teachers, undergraduates or graduates who want to know what can be done with reasonably simple models in thermodynamics and statistical physics. It is also suitable for self-study. It should appeal to a wide range of numerate readers: mathematicians, physicists, chemists, engi­ neers and perhaps even economists and biologists. The foundation of the general theory of statistical physics is involved and difficult, and its discussion takes a great deal of time in a lecture course. One can turn to this book for ideas and for inspiration vii viii Preface if one wishes to pass on to areas of application while condensing the time spent on the foundations. This procedure will appeal to those who feel, as I do, that the foundations of the subject are best discussed briefly at first, and then again later from time to time as the effectiveness , and the power of the methods are being appreciated. It will come as no surprise to the experienced teachers that the need for care in the setting of a problem can stimulate original work. There are new presentations in various places (e.g. in Problems 1.24 to 1.26, 1.29-1.31, and 3.19 to 3.22), and in several sections of this book previously unpublished ideas will be found. For example, in Section 6 some looseness of logic inherent in previous work has been eliminated, through discussions between the author and the editor, and there are new approaches to equations of state, to the law of corresponding states and to the relationship between reduced equations of state (a) for chain molecules and (b) for their components. Problems 17.7 to 17.9 have also some novelty in the use made there of the random walk approach. There are several other places where recent research work has here been incorporated in book form for the first time (for example, in Problem 5.5). In conclusion, I wish to thank all contributors and the publishers for the cooperation which made this venture possible. P. T.Landsberg ., r 1 The laws of thermodynamics P.T.LANDSBERG (University College, Cardiff) MAJHEMATICAL PRELIMINARIES VI.I If each of the three variables A, B, C is a differentiable function of the other two, regarded as independent, prove that (~;) (~~t G~t (a) C = -1, (~~)B I/G~t· = Solution Let the functional dependence of A on Band C be expressed by f(A, B, C) = 0, then (a~) B,C dA + (:;) A,Cd B + (:~) A.B d C = O. If A is constant this becomes (:;)A'CG~)A (:~)A,B' = - i.e. G~)A -(:~t.B/(:;)A,C' = Similarly (:~t (a~)B,cl(:~t'B' = (~~)C A,C I(a~)B,C' = -(:;) Multiplication of these three equations yields (~~)C(~~t (:~t =-1. 2 Chapter 1 1.1 1.2 The laws of thermodynamics 3 Interchange of A and C in the second of these equations yields Solution (a) We have . i 1/ (~~) G~t = (dx+dy) (X2 -X )+(Y2 -YI), B = 1 (I) i (dx+dy) = (Y2 -Yl)+(X2 -xd· 1.2 (a) Integrations over the following two paths in a plane are to be (ii) performed: These two line integrals have the same value, namely (i) the straight lines (XI, Yl) ~ (x2 , yd ~ (X2, Y2); (ii) the straight lines (Xl,Yl) ~ (X 1 ,Y2) ~ (X2,Y2)' u(Q)-u(P) = (X2+Y2)-(Xl +Yl)' On the other hand P(xl>yIl, Q(X2,Y2) are two points, andx 1 =1= X2,Yl =1= Y2' The differen­ r tial forms to be integrated are ~(x~ dv -Xi)+X2(Y2 -Yl), J(i) du == dx+dy, f = -yd+t(x~ dv X (Y2 -xi). dv == x(dx+dy). 1 J(ii) Show that f Since the two results are unequal, there does not exist any function f = = vex, y) of which the differential form dv can be considered as an exact du du u(Q)-u(P), differential, for otherwise both integrals would yield v(Q)-v(P). 0) (il) where u = X and f i.e.I na ftuhnec tpiorens ge(nxt, yca)s eex iasvts, walhthicohu gcohn ivneerxtsa cdtv, thoa sa na enx ainctte dgirfafteinregn tfiaacl tor, f dv =1= dv, dF = gdv . (I) (ii) and discuss the result. If one integrating factor exists, then there exists an infinity of them, and [We shall denote differential forms with this property by av instead of a simple example is furnished in the present case by dv, and call them inexact, while du is an exact differentiaL In relations of the type du(x,Y, ... ) g(x,Y, ...)crv(x,y, ... ), g(x,y, ... ) will be called g = 1/x, dv du . integrating factors.] (b) For an exact differential (b)If dF = X(x,y)dx+ Vex, dF = ( aaxF )y dx+ (aaFy )x dy == Xdx+ Ydy, so that is an exact differential, show that a (ax) 2 F (ay) (~~t (~~t· axay = ay x ax y' = (c) Let crf = Xdx+ Ydy , (c) Pfaffian forms have the general form where X, Yare continuous differentiable and single-valued functions of dv or av = i L=n I A/(x1> x 2, ..· xn)dxi' twhiet hi nYd =e1=p e0n mdeenatn sv tahriaatb tlhese exq, uya. tioTnh is restriction on X and Y together dy X (dv may be exact or inexact). -=-­ dx Y Show that for n = 2, if the Xj are single-valued, continuous and dif­ has a solution of the form ferentiable functions, av has always an integrating factor provided X2 is C;t non-zero in the domain of variation considered. (d) Verify that a Pfaffian form with n 3 need not have an integrat­ F(x, = C, i.e. dF = (~~)ydx+ dy 0, ing factor by considering dv = X dy + k dz, where k is a non-zero constant. where C is a constant. 1.4 The laws of thermodynamics 5 4 Chapter 1 1.2 Comparing coefficients in df = 0 and dF = 0, Solution (a) From the first two expressions for dQ an equation for dt is ±(~~t ~(~~)x found: = == g(x,y). (Cp -Cv)dt = Iv dv -Ip dp . Hence Substitution for dt in the second expression for oQ = = gdf gXdx+gYdy dF _ IvCp ( IpCp ) and dfhas the integrating factor g(x, y). dQ - C -C dv + lp C -C dp. p v p v (d) Suppose dv(x, y, z) == xdy+kdz = g(x, y, z)dF. Then Comparison with the last form for oQ yields the required result. k ( aaxF ) y,z = 0, (aaFy ) x,Z xg ' (aazF ) x,y g ((cb)) CTvh ei sf irtshte ehqeuaatt iorenq uunirdeedr p(ae)r yuienlidts rtihseis orefs eumlt paitr iocnacl et.e mperature at It follows from part (b) that constant volume, also called the heat capacity at constant volume; Ip is a (a the heat required per unit rise of pressure at constant temperature, some­ 2F 1 1 g) times called the latent heat of pressure increase. The other coefficients ax y,; aXay=0=g-g2 can be described analogously. aax2aFz ° k (aaxg ) 1.4 Let = = -g2 y,z' ~(~~ a~ x(a~z) k(~) cxp == \ ayaz = -g2 x,y = -g2 ay x,; be the coefficient of volume expansion at constant pressure, and let These equations cannot be satisfied by a finite function g(x, y, z). _.l(av) K == ap t V t QUASISTATIC CHANGES(I) be the isothermal compressibility of a fluid. The notation of Problem 1.3 1.3 For a fluid and other simple materials any three of pressure p, will be used. volume v, and empirical temperature t are possible variables. They are (a) Show that the Griineisen ratio r == cxpv/KTCv of the material satis­ connected by an equation of state so that only two of the three variables fies are independent. An increment of heat added quasistatically may then v(~~)jcv. be expressed in the alternative ways r = oQ = Cvdt+lvdv = Cpdt+lpdp = mvdv+mpdp, (b) Show that the ratio of heat capacities 'Y == Cp/Cv satisfies where the coefficients are themselves functions and are characteristic of (~~))(~~)t' the fluid. The term empirical temperature refers to an arbitrary scale 'Y = and is used to distinguish it from the absolute temperature, denoted by T. Prove that the following relations hold: -0 = G~)jG~)p' _ IvCp __ IpCv mv '!!.E.. _ . (a) mv - Cp -Cv ' mp - Cp -Cv ' 1v + Ip - 1, 'Y~l (~~))(~~)v' = a (av) ( p) = _ Cp -Cv = Cp -Cv (b) where ( )a denotes a quantity evaluated under quasisfatic adiabatic con­ at v lp' at p Iv· ditions, i.e. for dQ = 0. (c) Express in words the physical meanings of the coefficients in the (c) If Ka is the adiabatic compressibility, prove that expressions for dQ. K K t = 'Y. a (I) Changes consisting of a continuum of equilibrium states. \ 6 Chapter 1 1.4 1.6 The laws of 7 (b) Prove that the Griineisen ratio of Problem 1.4 is (d) Show from a consideration of d(lnv) that _(aK v (~ap) = att) . r v I(~~)o mp t p (c) Find the most general equation of state of a fluid whose Griineisen Solution ratio is independent of pressure. (a) We have Solution r v(~)v (a) This is obvious since dU = oQ -ow. If dW were an exact -vG;)p = I differential, one could integrate to find Q = U+ W+constant, in contra­ where Problem 1.1 (a) has been used. diction with the inexact nature of oQ. Alternatively, note that we have (b) From the first equation in Problem 1.3: two independent variables, so that one could write v (aapv ) _ Cpl (~~)t Iv oW = pdv+xdy a mp - Cvl I ' p p where x = 0 y Tor p. If oW were exact we would have [from Problem 1.3(a) 1 (ax) G;)a 'G~)p = (aapy ) = av = 0 Iv ' mv v y • (~~)a (ap) _ This yields 1 0 if y is chosen to be p, and hence oW is inexact. lp , at v ­ (b) We start with We now take the ratios of terms in the first column to corresponding oQ = dU+pdv = Cvdt+l"dv terms in the second column. The first pair yields "I at once, The second whence .' pair yields, in conjunction with Problem l.3(a), Co Ca~)v (~~)v(~)v' = = _Cvmv __Cv~ Cp Iv - CpCp-Cv "I-I' Substitute in Problem 1.4(a) to find I v v The third pair yields "1/("1-1). v(~~)) (~~)v' r = (c) This follows from (b). C = (d) We observe that t and p are the independent variables needed. (c) Integrate the result of (b) to find The form of the equation to be proved suggests that the procedure of 1 Problem 1.2(b) may be involved. Fortunately we also have the hint to U = r(v)[pv+f(v)], consider d(lnv). Hence where rand f can be functions of volume, and fjr is a constant of the ~dV ~ [(~~)p dOnv) = = dt+(:;)tdP]. integration with respect to p. The equation of state of a solid is some­ times taken in this form: This is OI.p dt-Ktdp, and we can indeed apply Problem 1.2(b). pv = r(v)U(v, T)­ THE FIRST LAW 1.6 Show that 1.5 The first law states that an internal energy function U exists such (~~)v' (a) that for a fluid or similar material we have, in addition to the first equa­ mp = tion of Problem 1.3, (~~)p oQ = dU+pdv, mv = +p, where p is the pressure, and p dv is the mechanical work done by the (am system. (~av) = ap v) 1. fa) Show that the increment of work dW pdv is inexact p v