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Proc. Nat. Acad. Sci. USA Vol. 69, No. 11, pp. 3436-3439, November 1972

Some Thermodynamic Relations at the Critical Point in - Systems (analyticity/symmetry/specific heats/critical exponents/coexistence curve) 0. K. RICE AND DO REN CHANG Department of , 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 Pc to the analyticity of the chemical potential above the a.01aP)T = (6916PY('aPlbP)T critical 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- 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- 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 -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 and p is the density of particles. pCv, then, is the heat capacity per THE ONE-PHASE REGION unit . 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- 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 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 , ;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 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 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 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). Suppose that CvII diverges as (T, - T) - a ' with a"l > 0, the equality meaning logarithmic divergence. If CvII > CvI, we must then have all > a'i, where a' is the exponent for CvI. In the case of equality of the exponents there are two possibil- ities: either the coefficient for CvIi must be greater than that of Cv1 or the leading terms of CvI and CvI" are exactly equal and the difference remains in terms that do not become infinite or else become infinite more slowly, so that CV-I CvI c (T,, - T)-a [9] with

a <

FIG. 1. Sketch of (b&/Ab/T2)p (curve A) and (62P/PT')p t B. Widom. (1967) paper presented at the Western Reserve (curve B) in arbitrary units as functions of p for T > T,. University Conference on Critical Phenomena. Downloaded by guest on September 24, 2021 3438 Chemistry: Rice and Chang Proc. Nat. Acad. Sci. USA 69 (1972)

On the coexistence curve, (aP/bV)T behaveslike (PT-T)-, Inasmuch as a' is found to be less than (3, we see that a' is this being the definition of -y', while IV - V, behaves as (T. negative. It is thus seen that the observation that (b2p/bT2)v - T)". Then from Eq. 8 and 9 we obtain is not infinite leads to a rather remarkable conclusion concern- ing the similarity of Cv,"II and CvII, showing that CV,'II -aI.< yI + 23- 2 [10] - Cv, and Cv, II - Cv, approach each other in very nearly the condition for the equality being aii = a. Inasmuch as 1 - the same way. C I/CvII oc (T, - T) -a+cI it is seen that the condition for We can also use Eq. 17 to make some deductions concerning the equality reverts to the symmetry of the coexistence curve. Since Cv is linear in V we could just as well have writ-ten Eq. 17 lim Cvii/Cv'> 1 [11] T--Tic ( C jCv, - Cv Cvc [21] equivalent to the condition mentioned by Stanley (10) for the ZaV/T v,-Vc VVi-VC magnetic case. We now need to relate aII to a'. The specific heat at constant total volume in the two-phase region will be which shows that any lack of symmetry between V0, and VI is related to the specific heats just inside the coexistence curve exactly reflected in that between Cv,"Ii and Cv II". Thus if (3g by = #Il = 3, then by 19 both Cv,"I- Cvc and Cv",- -CvC behave as (T, - T)-a' or vice versa. CV"II = xzCV II + XoCvoII [12] Some further conclusions can be drawn by noting that along the coexistence curve (8) where x1 and x0, denote the mole fractions of liquid and vapor, respectively (VI, V., and V are molal ): dV/dT = (dV/dT)p + (d V/lP)TdP/dT Xi = (Vg - V)/(V - VI) and x9 = [dP/dT - (bP/aT)v]/(1P/aV)T [22] = (V - VI)/(V, - VI) [13] where the partial derivatives refer to the one-phase region just For the critical volume V may be replaced by V,~.Then, if (31 outside the coexistence curve. Since dP2/dT2 = b,(T, - T) - and (3 are equal, xl and xS will approach constant values as the where bi is a constant, critical point is reached; thus, we will have dP/dT = b,(l - a')-'(Te - T)-a' + (dP/dT), [23] a' = amI [14] Now since dV/dT changes sign at the critical point and (ZP/ where amII is the larger of a'I" and aI1, and, if -y'm is the larger V)T does not, it is clear that the numerator of the right-hand of 'y'z and 'y',, then 10 will become side of Eq. 22 must change sign there, and while there is a pos- sibility of a discontinuity it is still necessary that the numera- -a' < ym' + 23- 2 [15] tor vanish there. Thus, it is necessary that at the critical point If 'Y'I = 7y'0 = y'm then 15 takes the usual form (OP/1T)v = (dP/dT),, confirming a generally accepted re- lationship. -a' <. ' + 2(3-2 [16] Let us then set as has been demonstrated by Widomt. dP/dT - (bP/dT)v = +4b2,(TC-T)1-a [24] Since (62P/OT2)V = d2P/dT2 is independent of V in the two-phase region, where b2 and a" are constants, the positive sign holding for the vapor side, the negative for the liquid side. From Eq. 22, then, ( Cv\ - CV II Cv,"II [17] we conclude 6V /T V0- VI -a" = y' +(3-2 [25] is not as there is In real systems there complete symmetry, Since 16 is a near equality, we can conclude that 1 - a' > in the lattice gas, and and pCv, are probably not equal. pCv,, 1 - a", so the first term on the side of Eq. 23 is = = of the right-hand If agi' a1ii a', but the coefficients leading to the of 24, we see 2 and small compared right-hand side Eq. and terms of Cv,,Ii and Cv,III are different, then, by Eqs. 17, (11) that along the coexistence curve near T, d2P/dT2 behaves like (T. - T)- -a . Since Cv diverges only as (T, - T) -a', it is seen by Eq. 1 that d2M/dT' would also (bP/IT)v = (dP/dT), F b2(T - T)" +-1. [26] have to become infinite as (T, - T) -a -a in order to cancel the leading term of Vd2P/dT2. The evidence that d'2/dTl There is an inequality (12) that states that one then to that the does not become infinite leads conclude - [27] leading terms of CvIi and Cv II" must cancel, so that 7' 2 (3(6 1) This has often been treated as an equality (1, 13), and, we shall CV,9II - Cv II" = (CV,"II - Cv ) give a further argument supporting this view. Assuming the - (Cva1II - Cvc) o (Tc - T)-a' [18] equality, then, we have from Eq. 26 where a' < a'. Then (cP/Z8T)v = (dP/dT)c w b,(T -T)6- (OCV/C V)T OC (Tc- T) -a/-# [19] or where by Eqs. 2 and 3 (aS/aV)T = ('S/aV)Tc ± bslV - Vcj''-11 [28] a' = a' + ( [20] where b, is yet another constant. Downloaded by guest on September 24, 2021 Proc. Nat. Acad. Sci. USA 69 (1972) Thermodynamic Relations at the Critical Point 3439

In earlier (8) one of us used the equivalent of Eq. 22, the critical point and the departure from the curve together with the Maxwell and Clapeyron equations and the becomes appreciable) assumption of the expansibility of S about the critical point, aTe Tc to obtain a relation between 6 and (. The exponent y' in Eq. 27 _ (C-2P/C)VbT')dT1 > - (b2P/C-VbT')dT' was 1, which made 8 = (3-1 + 1. It is seen that we may invert this procedure to get an estimate of the behavior of the en- except at the critical point itself, and (bP/bV)T, cannot ap- tropy if a and (3 are known, though, strictly speaking, only an proach the integral of 29. So, under these circumstances, y' = inequality is certain. 0(3 - 1). If (3' > (3, this conclusion is reinforced. ,3' < (3 is We shall conclude with our argument that 27 is an equality, clearly impossible. Thus the equality must hold, unless the Liberman (12) has pointed out that, at a fixed volume, unlikely condition described in the parentheses above is fullfiled. We are indebted to Drs. B. Widom and J. M. H. Levelt Sengers (aP) _ Tc dT' 82P_ for discussion and for pointing out certain errors. This work was \dcV8 T, aVbT' supported in part by the Army Research Office. Durham, and in part by the UNC Materials Research Center under Contract No. where (bP/aV)T, is evaluated on the critical isotherm and DAH C 15-67-C-02334 with the Advanced Research Projects Agency. (bP/aV)T, at the same volume on the coexistence curve. Empirically, the right-hand side of this equation is greater 1. Widom, B. (1965) J. Chem. Phys. 43, 3898-3905. than or equal to zero 2. Widom, B. & Rowlinson, J. S. (1970) J. Chem. Phys. 52, (the latter, presumably, only at the crit- 1670-1684. ical point). Since- (OP/aV)T, C (T, - T)Y( - 1) and is posi- 3. Mermin, N. D. (1971) Phys. Rev. Lett. 26, 169-172, 957-959. tive, while (OP/IV)T, a: (T, - T)Y and is negative, this 4. Yang, C. N. & Yang, C. P. (1964) Phys. Rev. Lett. 13, 303- makes y' > (( 1). But if y' > ,B(( - 1), then -(P/CV)T, 305. approaches the integral at the critical point. 5. Rice, 0. K. (1967) in Statistical , Thermodynam- ics and Kinetics (Freeman, San Francisco), (a) pp. 308-309, Now let T, be the temperature of the spinodal line at the (b) p. 173. same volume as T, and Te. Since (6P/aV)T. = 0, we see that 6. Vicentini-Missoni, M., Levelt Sengers, J. M. H. & Green, M. S. (1969) J. Res. Nat. Bur. Stand. Sect. A 73, 563-583. rTc 7. Brown, G. R. & Meyer, H. (1972) Phys. Rev. A6, 364-377. (PP/)V)T.= (62P/bVZT')dT' [30] 8. Rice, 0. K. (1955) J. Chem. Phys. 23, 169-173. 9. Stephenson, J. (1971) J. Chem. Phys. 54, 890-894. Stephen- son has obtained a number of the results reported here, but the Now let us assume that the spinodal line also approaches the general point of view is somewhat different. 10. Stanley, H. E. (1971) in Introduction to Phase Transitions - critical point as (Tc T)O' with (3' = (. Then (unless the and Critical Phenomena (Oxford, University Press, New spinodal line has the same proportionality constant as the York and Oxford), p. 54. coexistence curve, so that the departure of the one curve 11. Stephenson, J. (1971) J. Chem. Phys. 54, 895-897. Stephen- son from the other arises only from terms, seems very has obtained equations equivalent to the following ones higher which from a scaling theory approach. unlikely since the dependence of the coexistence curve, 12. Liberman, D. A. (1966) J. Chem. Phys. 44, 419-420. (T, - T)I, remains rather accurate even fairly distant from 13. Widom, B. (1964) J. Chem. Phys. 41, 1633-1634. Downloaded by guest on September 24, 2021