24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr, w laborat on th idas prsntd by T. H. Hill [1] with th dirct purpos of giving a molcular ustification to quations of stat and xcss Gibbs nrgy modls usd in practical work. CANONICAL ENSEMBLE AND PROBABILITY For this introduction, w considr a closd systm having rigid and diathrmal walls, immrsd in a thrmostatic bath at a tmpratur T. Th systm has a fixd numbr of molculs insid and a fixd volum, and it can only xchang nrgy in th form of hat with its surroundings. Th molculs in th systm ar moving around, hitting ach othr and also hitting th rigid walls. A dtaild dscription of th intrnal nrgy of this systm would rquir th spcification of th position and momntum coordinats, that is, th xtrnal dgrs of frdom, of ach molcul at any givn instant. In addition, it would b ncssary to spcify th rotational, vibrational, and lctronic coordinats, that is, th intrnal dgrs of frdom, of ach molcul at any givn instant. Bcaus pr ach mol insid th systm thr ar 10 23 molculs, this dtaild dscription bcoms an impossibl task. On th othr hand, macroscopically th systm is charactrizd by thr paramtrs: th tmpratur T, th total volum V, and th numbr of mols n or, mor proprly, th numbr of molculs N that it contains. Each st of valus of th intrnal and xtrnal coordinats of th molculs compatibl with th valus of th macroscopic paramtrs T, V, and N rprsnts a possibl quantum stat of th systm. Thr ar vry many of such quantum stats, ach with a valu E i for th intrnal nrgy of th macroscopic systm. Th intrnal nrgy U of th systm is thn a macroscopic avrag of th intrnal nrgis of ths vry many possibl quantum stats. Th probability P i of having a particular quantum stat i with an intrnal nrgy E i will b qual to th ratio btwn th numbr of quantum stats N (E i ) having that particular valu E i of th intrnal nrgy and th total numbr of quantum stats compatibl with th macroscopic paramtrs T, V, and N: with N ( Ei) Pi = N ( E ) (24.1) P = 1 (24.2) 273
274 Classical Thrmodynamics of Fluid Systms INTERNAL ENERGY AND THE PARTITION FUNCTION Hnc, according to Equation 24.1, as th intrnal nrgy U(T, V, N) of th systm is a wightd avrag of th intrnal nrgy E i of ths quantum stats, it taks th form U =PE (24.3) Considring th larg numbr of possibl quantum stats, th probability of finding on particular stat with intrnal nrgy E i is wll rprsntd by a normalizd Boltzmann distribution function, namly, P = i Ei E (24.4) whr k is th Boltzmann constant, givn by th ratio of th gas constant ovr Avogadro s numbr. In this quation, it is convntional to dfin th partition function Q as Q E (24.5) With th purpos of conncting ths xprssions with classical thrmodynamics, w diffrntiat Equation 24.3 and writ du = P de + EdP (24.6) Th nxt stp is to xprss th right-hand sid of this quation in trms of th macroscopic thrmodynamic variabls. In classical thrmodynamics, w hav or So, by analogy, w dfin U = P V n E V n p de = p dv Multiplying this quation by P and adding ovr all, w hav ( P ) P de = p dv = PdV (24.7)
Elmnts of Statistical Thrmodynamics 275 In this stp, w hav idntifid th prssur of th systm with th wightd avrag of valus of p. For th scond trm of th right-hand sid of Equation 24.6, from Equation 24.4 and th dfinition of th partition function, Equation 24.5, w writ for a quantum stat E = ln P ln Q (24.8) Multiplying this xprssion by dp and adding ovr all, EdP = lnpdp lnq dp But according to Equation 24.2, d P = d P = 0 so th scond trm on th lft-hand sid vanishs, and w writ EdP = ln PdP This xprssion can b transformd furthr considring that d P ln P = ln P dp + P d ln P but 1 P dln P = P dp = dp = 0 P so d = P ln P ln P dp Thus, ( ln ) EdP = d P ln P = Td k P P (24.9) and rplacing Equations 24.7 and 24.9 in Equation 24.6, w obtain du = PdV + Td( k P ln P ) (24.10)
276 Classical Thrmodynamics of Fluid Systms ENTROPY, PROBABILITY AND THE THIRD LAW OF THERMODYNAMICS From th combination of th first and scond laws of thrmodynamics, w know that du = TdS P dv So, by comparison with Equation 24.10, w conclud that th ntropy is rlatd to th probability by th Boltzmann quation, S = k P ln P (24.11) Equation 24.11 is of importanc in information thory. A prfct crystal at 0 K is in its ground stat. Thus, for this singl quantum stat th valu of P is unity and S = 0. This is a statmnt of th Third Law of Thrmodynamics. EQUATION OF STATE, CHEMICAL POTENTIAL, AND PARTITION FUNCTION Aftr obtaining this first bridg btwn statistical and classical thrmodynamics, w multiply Equation 24.8 by P, and add ovr all quantum stats to obtain or P E = T P P P k ln ln Q Thus, rplacing in this xprssion Equations 24.2, 24.3, and 24.11, U = TS ln Q U TS = ln Q But from thrmodynamics w know that th Hlmholtz function is dfind by Equation 4.1 as Thus, w conclud that A = U TS A = ln Q (24.12) Equation 24.12 is on of th most important links btwn statistical thrmodynamics and classical thrmodynamics, as it prmits us to obtain xprssions for th quation of stat and for th chmical potntial basd on th xact rlations.
Elmnts of Statistical Thrmodynamics 277 From Equation 4.15, And from Equation 4.16, A lnq P = V = V (24.13) Tn, Tn, A lnq µ i = n = n i TVn,, i i TVn,, i (24.14) Th nxt stp, thn, is to say somthing about th partition function for gass and liquids. This is discussd in th chaptrs that follow. REFERENCE 1. Hill, T. L. 1960. An Introduction to Statistical Thrmodynamics. Rading, MA: Addison-Wsly.