Proc. Nat. Acad. Sci. USA Vol. 69, No. 11, pp. 3436-3439, November 1972 Some Thermodynamic Relations at the Critical Point in Liquid-Vapor Systems (analyticity/symmetry/specific heats/critical exponents/coexistence curve) 0. K. RICE AND DO REN CHANG Department of Chemistry, University of North Carolina, Chapel Hill, N.C. 27514 Contributed by 0. K. Rice, September 2Q, 1972 ABSTRACT The relation of the symmetry of the ther- which follows from modynamic functions with respect to the critical density Pc to the analyticity of the chemical potential above the a.01aP)T = (6916PY('aPlbP)T critical temperature is discussed with the aid of thermo- dynamic considerations. Special consideration is- given to =V((P(P/)P= P-'(bP/?P)T [5] the differential coefficients (624/aT2)p and (Z2P/)T2)p. The discussion is extended to the two-phase region, and some of Then the differentiation of Eq. 3 gives the relations between critical exponents are analyzed. A proof is given that the vapor-pressure curve joins smoothly [6(PCv)/-P]T = CV + p(OCV/1p)T = -T(a2M/rT2), [6] with the critical isochore. The behayior of the entropy along the coexistence curve is considered. Finally, an argu- Let us now apply these equations to the case of particle-hole ment is presented that the Griffiths-Liberman inequality symmetry. In this case pCv is symmetric about Pc, the critical 'y' > #(a - 1) is an-equality. density, since Cv is the heat capacity per mol of particles and p is the density of particles. pCv, then, is the heat capacity per THE ONE-PHASE REGION unit volume. pCv is expected to have a ridge of maxima along There has been some discussion recently about the question of P = Pc, so [P(pCv)/Op]T and, hence, (b2u/aT2)' vanish along the analyticity of the chemical potential/h in the neighborhood Pc- of a critical point. It has been stated that above the critical Actually a lattice-gas model is a somewhat awkward con- temperature, Tc, ,u is analytical along the critical isochore of a cept to use for a representation of a real system, especially if simulated liquid-vapor system if there is particle-hole sym- one wishes to discuss the pressure of the real system, al- metry, as in a lattice gas, and such analyticity is assumed in though this can be done (5a). The essential feature of such a Widom's scaling theory (1). On the other hand, certain models system is that pCv has a maximum at p = PC for any tempera- (2, 3) that lack the particle-hole symmetry do not show this ture. We shall describe such a situation as resulting from par- analyticity; this has been the subject of a detailed discussion ticle-hole symmetry, but shall try to represent the pressure by Widom and Stillinger*. This discussion involved statistical more realistically by supposing that the pressure may, at small mechanical treatment of a fairly complex model. p, be represented by a virial equation It seems possible that a discussion based on thermodynamics P = RTp(1 + Bp) might have advantages both in generality and simplicity. It will be the attempt of this paper to supply such a treatment, for which (b2P/bT2)P = O at p = 0 and and also to consider some of its applications. Our treatment dB d will be based principally upon two well-known thermody- ~p = 2pR 2 + T BI namic equations that involve the molal heat capacity at con- aT26p \ dT dT2J stant volume, Cv, namely (4), From Eq. 4, then Cv = -T(62M/?.T2)v + TV(62P/6T2)V [1] = 2R(2 dB + T dd2B\ and 6T2p\)P dT dT2/ (bCv/aV)7 = T(2P/bT2)V [2] At very low densities, ;z = ;LO + RT ln p, where 1AO, the stan- be the value of = Writing the density as the reciprocal of the molal volume, p = dard chemical potential, would iuat p 1, pro- 1/V, we can recast Eq. 1: vided the ideal gas laws held at that density. Thus, at small densities, and by use of Eq. 6 (noting that the specific heat pCv = -Tp(c2/A/8T2)p + T(C2P//T2)p [3] of an ideal gas is independent of density) We differentiate Eq. 3 with respect to p, holding T constant, (b2A/bT2)P = d2/A5/dT2= -CV0T taking note of the fact that Therefore, (a2IA/ T2)p starts out negative at low p, while PC T/26p = c P/aT2ap [4] (b2P/bT2)P obviously starts out as zero. By differentiation of an approximate but general expression (5b) for B, it appears that, while 2dB/dT is positive, Td2B/dT2 is negative and over- * Manuscript submitted to J. Chem. Phys. balances the former, so b3/a/zT2bp is at first negative. Thus 3436 Downloaded by guest on September 24, 2021 Proc. Nat. Acad. Sci. USA 69 (1972) Thermodynamic Relations at the Critical Point 3437 (62iA/6TP), will have the general character depicted in Fig. 1, and use of Eq. 4 will give a curve for (b2p/bT2), such as that shown. The maximum in Cv will occur at p < Pc, where (b2p/bT2)p = 0, from Eq. 2. At p =PC we may write PcCv=T(=PIOTI)pe [7] from which it may be seen that, as Cv approaches infinity when T above the critical point approaches TC, the behavior of (62p/aT2)", parallels that of Cv. As the critical point is approached, the maxima in Fig. 1 become sharper and sharper, assuming that Cv and, hence, (b2A/bT2)) diverge, until at T, the curves have the appearance shown in Fig. 2. Inasmuch as (b2,u/bT2)pc is zero at all temperatures above T, it is clear that A is an analytic function of T for any tem- perature above T, and continuable through T,. On the other FIG. 2. Same as Fig. 1, for T = T,. hand, (8',u/6T')p is obviously not an analytic function of p at T = T., but it might be possible for 1A to be, though with most scaling hypotheses this is not the case at PC. nated by I) and those just inside the two-phase region (desig- It is now fairly easy to see what happens when we do not nated by II), from which relations between the critical expo- have particle-hole symmetry. We may still suppose that pCv nents may be obtained and related to the behavior of the de- has a maximum at some p for temperatures near T,. The line rivatives mentioned. From the relation (8) of these maxima will not, however, coincide with p = p,. Along CV11 - Cvl = - T()P/lbV)T(dV/dT)2 [8] the critical isochore, then, (b2j/bT2) P will not remain con- stant, but will gradually approach the infinite value at T,, so it where (bP/aV)T is evaluated in the one-phase region just out- cannot be said to be an analytic function of T at T,. It ap- side the phase boundary, Widomt obtained the Rushbrook pears, however, that there is no very essential difference be- inequality for the liquid-vapor case. tween the symmetrical and the nonsymmetrical case. It is worth noting that Eq. 8 has exactly the form of Cp - Cv with dV/dT substituted for (bV/16T)p. CvI is just Cv out- THE TWO-PHASE REGION side the coexistence curve and CVr" takes the place of CP. All the thermodynamic equations that are given in the preced- It is of interest, in the magnetic case, where the magnetic field ing section are also applicable to the two-phase region. In the H and the magnetization 111 take the place of - P and V, two-phase region ,, P and their derivatives with respect to T respectively, that the coexistence curve represents an equilib- are independent of p. Thus, as is well known, Cv is linear in V rium between two phases that are identical except for the di- by Eq. 2 and pCv is linear in p by Eq. 6. If there is particle- rection of magnetization, and hence have the same energy. hole symmetry pCv is constant, which makes (621A/bT2)p zero Thus, CII is independent of how the magnetization changes in by Eq. 6, and there seems to be some empirical evidence that any heating process. Indeed, it is the same thing as CH, since H = 0 under the coexistence curve, and in particular it is (a2i/AT2) , remains finite in actual cases (6, 7). If this is so, and if Cv diverges along the critical isochore (62P/bT2)o will di- specific heat along the coexistence curve. Also, dAM/dT along verge in the same way according to Eq. 1. If they refer to the the coexistence curve is (bM/bT)H, since H is constant along conditions inside the two-phase region, such derivatives as the coexistence curve; thus, we see that Eq. 8 is the exact ('2P/6T2)p and (WA'/aT2)p, involving quantities that are con- analogy of the expression for CH - CM that was used in the stant in the two-phase region, can be written as d2P/dT2 and original analysis of this problem for the magnetic case by d2/M/dT2 where the total derivatives indicate changes along Rushbrook. the coexistence or phase-boundary curve separating the one- Inasmuch as (bP/aV)T is never positive, we may infer, as phase from the two-phase region. There are several relations noted by Stephenson (9), that CvI" is never less than CvI between the quantities just inside the one-phase region (desig- (just as Cp is always greater than Cv, and CH than CM).