3389
J . Phys. Chem. 1984, 88, 3389-3391
The Critical Dilemma of Dilute Mixtures R. F. Chang, G . Morrison, and J. M. H. Levelt Sengers* Thermophysics Division, National Engineering Laboratory, National Bureau of Standards, Gaithersburg, Maryland 20899 (Received: April 2, 1984) Remarkable anomalies in excess properties and partial molar quantities recently reported in dilute mixtures near the solvent's critical point are explained as due to a solute-induced phase transition. A more intriguing effect, the path dependence of partial molar properties near the solvent's critical point, is analyzed for a classical (analytic) and a nonclassical (scaled) model. Asymptotic expressions are presented for partial molar volumes, enthalpies, and specific heats along a variety of paths to the solvent's critical point. Our results contribute to the formulation of supercritical solubility and of impurity effects in near-critical fluids. The thermodynamic behavior of dilute mixtures near the solvent's critical point, a subject studied in the USSR in the 1960's , recently received much attention in this country. and ~ O ' Shas New experiments have revealed a strong divergence in the partial molar volume of the solute,14 a curious concentration dependence of the excess enthalpy5 not seen in ordinary liquid mixtures, large negative values of apparent molar heat capacities,6 for which several ingenious explanations have been p r o p o ~ e d ,and ~ steep increases in supercritical s o l ~ b i l i t y . ~ Many ~ ~ ~of ~ these effects are, as we will show, the result of a phase transition induced in the near-critical solvent by the solute (impurity). The truly remarkable effect in dilute near-critical mixtures, however, is the path dependence, in the limit of infinite dilution, of thermodynamic derivatives such as the partial molar volume. This path dependence is a consequence of the incompatibility of the conditions that the ) ~ ~ near the critical line but osmotic susceptibility ( d x / d ~diverges vanishes in the infinite-dilution limit; here x is the solute's concentration and p the chemical potential. The path dependence of partial molar volumes was noted experimentally by Krichevskii and co-workers,1° formulated for classical (analytic) critical behavior by Rozen," and worked out in a nonclassical (scaled) model for the decorated lattice gas by Wheeler.12 In this Letter we present the results of classical and nonclassical analyses of the critical behavior of dilute mixtures for partial molar volumes, enthalpies, compressibilities, and specific heats along a variety of paths of approach to the pure-host critical point. Our analysis permits interpretation of the experimental behavior of dilute near-critical mixtures and reveals that this behavior can be described in terms of the properties of the pure solvent and the slope of the critical line. As stated above, the experimental effects mentioned have their origin in the induction of a phase transition by the s01ute.l~ An isothermal V-x diagram of a dilute mixture of a nonvolatile solute in a near-critical solvent slightly above the latter's critical temperature T , is shown in Figure 1 along with the coexistence curve and some tie-lines. A critical point occurs where the length of the tie-line shrinks to zero. The critical isotherm-isobar is tangent to the phase boundary at that point. The slope of that tangent, (1) van Wasen, U.; Schneider, G.M. J . Phys. Chem. 1980, 84, 229-30. (2) Paulaitis, M. E.;Johnston, K. P.; Eckert, C. A. J . Phys. Chem. 1981, 85, 1770-1. (3) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167-75. (4) van Wasen, U.; Swaid, I.; Schneider, G . M. Angew. Chem., Int. Ed. Eng. 1980, 19, 575-87. ( 5 ) Schofield, R. S.;Post, M. E.;McFall, T. A.; Izatt, R. M.; Christensen, J. J. J. Chem. Thermodyn. 1983, 15, 217-24. Wormald, C. J. Ibid. 1977, 9, 643-9. (6) Smith-Magowan, D.; Wood, R. H. J . Chem. Thermodyn. 1981, 13, 1047-73. (7) Gates, J. A.; Wood, R. H.; Quint, J. R. J . Phys. Chem. 1982, 86, 4948-5 1. (8) McHugh, M.; Paulaitis, M. E. J . Chem. Eng. Data 1980, 25, 326-9. (9) Johnston, K. P.; Eckert, C. A. AIChE J . 1981, 27, 773. (IO) Krichevskii,I. R. Russ.J . Phys. Chem. 1967,41, 1332-8 (in English). ( 1 1) Rozen, A. M. Russ. J . Phys. Chem. 1976, 50, 837-45 (in English). (12) Wheeler, J. C. Ber. Bunsenges. Phys. Chem. 1972, 76, 308-18. (13) Levelt Sengers, J. M. H.; Morrison, G . ; Chang, R. F. Fluid Phase Equilib. 1983, 14, 19-44.
( d V / d ~ ) defines ~ ~ , the partial molar volumes of the solvent and solute, respectively, as = V-X(dV/dx)pT 72 = I/+ (1 -X)(dV/dX)pT (1)
TI and V2at the mixture's critical point are indicated in Figure 1 as the intercepts of the tangent with the axes x = 0 and x = 1, respectively. In the limit that the temperature approaches T,, the tangent approaches the vertical and therefore --m in the case shown in Figure 1. This divergence is the subject of ref 1-4 and 10-1 2. Since the specific heat C,, in contrast to the volume V , diverges strongly at the solvent's critical point, it is to be expected that a partial molar heat capacity (as measured by Wood et al.637)shows an even stronger divergence than a partial molar volume, and our results will confirm this. To show the behavior of excess properties we have also plotted, in Figure 1, some isotherm-isobars for P C P,. The excess volume is the difference between the volume on such a curve and the straight line interpolated between V(x=O) and V(x=l). In Figure 2, VE is displayed at P = P,; its unusual features are due to the occurrence of a phase transition when sufficient solute is added. Since enthalpy and volume are both derivatives of the Gibbs free energy G(P,T,x) with respect to intensive variables (fields), the behavior of the excess enthalpyS is as unusual but unsurprising as that of the excess volume. As to the path dependence of thermodynamic behavior in the limit x 0 we can present within the confines of a Letter only our principal results. For details we refer to a future publication. The behavior of the partial molar volumes (1) is deduced from the thermodynamic identity
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v2
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(av/ex)PT =
VKTx(dP/dx)VT
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(2)
where KTxis the isothermal compressibility at constant composition. If the limit x 0 is taken first, (dV/dx),, becomes proportional to the compressibility of the pure solvent,'*12 K T1 diverging as IT - TclY, y equals 1 for classical, and 1.24 for nonclassical critical behavior. The proportionality factor is the value of ( d P / d ~ in) ~the limit x 0 which, asymptotically near the critical point, is of the simple form
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(3) Here dP/dxlzRLand dT/dxJzRLrefer to the initial slope of the critical line while dP/dilzxc denotes the critical slope of the solvent's vapor pressure curve. Krichevskiilonoted the importance of this particular combination of critical-line and pure-fluid vapor pressure properties and Rozen" subsequently denoted it by the symbol A . Equation 3 is readily derived for classical behavior by expanding the pressure P( V,x,T)at the solvent's critical point; ~) path ~ ~independent. The here the limiting value of ( d P / a is same relation (3) holds for our scaled model only if the limit x 0 is taken first. If this limit is not taken first, matters become more subtle. We find that in the classical case there are two, in the nonclassical case three, modes of approach to the solvent's critical point. We have calculated the limiting behavior of partial molar volumes
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This article not subject to U S . Copyright. Published 1984 by the American Chemical Society
3390 The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
Letters
TABLE I: Limiting Path Dependence of Partial Molar Properties path
v7,
H,
K,,
cm
v,
lim-,,xV,
limr4( V7/KTJ
isotherm-isobar isotherm-isochore isobar-isochore critical line isotherm-isobar isotherm-isochore isobar-isochore critical line
terms.14 We denote derivatives of a with repeated subscripts. Thus aWx d 4 a / d 2 V d T d x . The superscript c of a derivative refers to its critical-point value. The coefficients a,' and acw are zero. The material stability of the mixture demands that the determinant of second derivatives, D = a@,, - a v 2 , be positive everywhere except at the mixture's critical line where D = 0. From our expansion of the free energy to leading order, we obtain for the elements of determinant a, = -(VKTx)-l e aCw(6T) a b x x ac&6v)2/2
+
aVx= -(dP/dx)VT axx
a;x
N
+
a&
-A
(4)
+ RTJx
From eq 2 and 4 we obtain
0
XC
X
Figure 1. The V-x diagram of a vapor-liquid mixture at a temperature TI slightly above the critical temperature Tc of the solvent ( x = 0) but far below the critical temperature of the solute. The coexistence curve, a few tie lines, and the critical point are shown. Of the four isothermisobars also displayed, one passes through the mixture critical point, one through the extremum in the x( V)' coexistence curve. Two others are on either side of the one passing through the extremum, and enter the two phase region with changes of slope of opposite sign.20 The partial molar volumes at the mixture critical point are indicated as the intercepts of the tangent to the isotherm-isobar with the x = 0 and x = 1 axes, respectively.
( d v / d x ) , , = A/[aCWxx + a b A 6 T ) + aC&6v)2/2] (5) which, with (l), yields the partial molar volumes Vl and V2. Equation 5 contains only two derivatives a$, and acWxcharacteristic of the mixture. They can be. expressed in terms of the initial slope of the critical line so that this slope and the solvent's critical completely determine the behavior of properties, aCw and a-,' the mixture at infinite dilution." A measurement of this slope can therefore substitute for the difficult measurement of partial molar volume^.'^^ In terms of derivatives of the free energy, the relevant expression for the slope of the critical line in T-x space, obtained by setting D = 0, with (4), is found to be
That for dP/dxI& then follows from (3). Both aVxand awx can be solved in terms of these two relations. In eq 5 , the numerator is finite and the denominator vanishes at the solvent's critical point, thus causing (dV/dx),, and V2to diverge in a way that depends on the path. On any path along approach zero at least as fast as x , (dV/dx),, which 6 T and and Vzwill diverge as l/x; the amplitude will be path dependent and VI will not approach Vc. These paths include the critical line and the critical isochore-isotherm. If 6 T varies as xp with p 2 1, but GVas xq with q < l/z,then the term in dominates and (d/dx), will diverge more slowly than l/x, namely, as l/x2q, while Vl approaches Vc. On the critical isotherm-isobar, for instance, (dV/dx),, diverges as .f2f3. W e expect the "partial molar specific heat", = Cpx+ (1 - x ) (dCpx/dx)pT,to behave as the similarly defined fictitious "partial molar compressibility", Kn,which is readily analyzed by means of the expansion of K,;' cc a, in (4). On the critical , on the critical isoline, for instance, (dKTX/dx),, = x - ~ while therm-isobar (dKTx/dx)PT0: a very strongly diverging quantity indeed. This explains why Wood et al. observed large absolute values of the apparent molar heat capacity of dilute salt solutions not even close to the critical point of A summary of our results for the path dependence of partial molar quantities is given in Table I. As an aside, we point out that the large change in supercritical s o l ~ b i l i t y ~can , ~ ,also ~ be understood by expressing the quantity
(6v2
(6v2 cp2
0
X
1
Figure 2. The excess volume along an isotherm-isobar that passes through the two-phase region at a pressure PI below that corresponding to the extremum in x(V) (Figure 1). The shape of this curve is similar to some of the excess enthalpy curves reported in ref 5 . More subtle effects (also seen in the excess enthalpy plots) occur on the other isotherm-isobars indicated in Figure 1, due to a reversal of the sign of the change of slope as the two-phase region is entered at the small-x side.20
and compressibilities for all these cases. Classical critical behavior is explored by assuming that the molar Helmholtz free energy a(V,x,T) and therefore the pressure P(V,x,T) can be expanded at the solvent's critical point in powers of 6 T = T - T,, dV = V - Vc,and x, except for the ideal-mixing
(14) Rowlinson, J. S.;Swinton, F. F. "Liquids and Liquids Mixtures"; Butterworths Scientific: London, 1982; 3rd ed.
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3391
Letters (dx/dP),,,, where the subscript u indicates that the solute is present in an additional phase, in terms of derivatives of a. Using eq 2.7 in ref 15 and eq 6.8 in ref 14, we derive
where V, is the molar volume of the solute in the additional phase. This expression, considerably simpler than eq 2.1 7 in ref 15, reveals that the solubility changes strongly with pressure when D approaches zero, at the mixture’s critical end point; D approaches zero “strongly times weakly”, and ( d ~ / d P ) ~diverges ,, “strongly” in the sense of ref 16. Let us now explore the nonclassical critical behavior. First of all, since the pure-fluid compressibility KT diverges as IT - TclY, with y = 1.24, the quantity acm in (4) is zero. Furthermore, according to the theory of Griffiths and Wheeler16properties such ) ~ two , “field” variables fixed at differentiation, as ( d ~ / d p ~with have y-type divergences. Moreover, quantities such as KTx, Cpx, and ( d P / d ~ ) ~ f lin, which one “density” is kept fixed, diverge weakly, d i k e (a O.l), at the critical line. Therefore, both aw and aVxin (4)are zero on this line; an expansion of the type used in (4) does not exist. Finally, quantities such as ( d P / d T ) , remain finite; this constant-x isochoric slope, however, is predicted to approach the slope of the critical line at the mixture’s critical point, in contrast to classical behavior where the isochore intersects the ~~, is expected to critical line. The quantity ( d V / d ~ ) however, remain finite at the critical line, contrary to eq 4.3 of ref 15. Rozen, introducing nonclassical exponents into the classical expansion, ignored both the tangency of isochore and critical line, and the weak anomaly of the compressibility.” Even though both conditions may elude experimental observation, ignoring them leads to formal contradictions. To study the nonclassical limiting behavior we have used the scaled Leung-Griffiths In this model, the anomalous part of the thermodynamic potential w = P/RT, a function of p,/RT, p2/RT, and 1/RT, is written as a scaled function of two scaling fields, one of which, 7,is parallel to the coexistence surface, while the other, h, intersects it. A third field, {, on which the parameters in the potential depend, is parallel to the critical line.
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(15) Gitterman, M.; Procaccia, I. J. Chem. Phys. 1983, 78, 2648-54. (16) Griffiths, R. B.; Wheeler, J. C. Phys. Reu. 1970, A2, 1047-64. (17) Leung, S. S.; Griffiths, R. B. Phys. Rev. 1973, A8, 2670-83. (18) Moldover, M. R.; Gallagher, J. S. AIChE J . 1978, 24, 267-78. (19) Chang, R. F.; Levelt Sengers, J. M. H.; Doiron, T.; Jones, J. J. Chem. Phys. 1983, 79, 3058-66. Chang, R. F.; Doiron, T. Int. J . Thermophys., accepted for publication. (20) We are grateful to the reviewer for pointing this out.
{is related to the chemical potentials p l , p2 and is proportional to x for small x . The various paths of approach to the pure-fluid critical point are explored by defining a distance variable r which is a function of 7,h, and {.”J9 A path determines T and h in terms of l a n d permits one to express r a: S,with E > 0, as {- 0 along the path. The value of e characterizes the path’s approach to the critical point. If E is small, a path is obtained that stays close to the pure-fluid axis. If e is large, the path will probe the weak anomalies near the critical line. Thus we find that there are the following three regimes. (1) Paths asymptotically tangent to the pure-fluid axis, E < l / y (-0.8). Along such paths limM(xV2) = 0, and both V2and KTx behave as l/xYE. On the critical isotherm-isobar, for instance, r a {l/@‘, with p = 0.325, 6 = 4.82, l/pS 0.64;both F2 and KTx behave as 1/x+f/@’= 1/#I/*) 1/XO.’~. The “partial molar compressibility” would behave like 1 along this same path. (2) Paths asymptotically tangent to the critical line, t > l/a. Along such paths, the weak anomaly of KTX(not present in the classical case) manifests itself; V2 behaves as 1/ x , KTx as 1/ x f a , and the limit of V2/KTxfor x 0 is zero. (3) Intermediate paths for which l / y < t < l / a . Here V2and KTx behave as 1/x just as in the classical case. Contrary to classical behavior, however, the amplitude is path independent, which is consistent with the fact that nonclassically aCwT = 0 (Table I). The nonclassical critical behavior of partial molar volumes and compressibility is summarized in Table I for various paths to the solvent’s critical point. In summary, we have shown that many anomalous effects recently reported in near-critical dilute mixtures can be simply explained as due to a solute-induced phase transition. A more subtle but unique feature, however, is the path dependence of partial molar volumes, enthalpies, and specific heats, for which we present here the results obtained for both a classical and a scaled model. The limiting value of the partial molar volume of the solvent VI is not necessarily equal to V, at the solvent’s critical point; the divergence of the partial molar volume of the solute, V2,is path dependent; on the critical isotherm-isobar, impurity effects on the density of a fluid are not linear in x , but rather proportional to x1l8;at least asymptotically near the solvent’s critical point, a knowledge of the solvent’s properties and of the initial slope of the critical line suffices to obtain V, and V2. Finally, without incurring inconsistencies it is not possible to combine a classical treatment of the mixture with a nonclassical model for the solvent, or even with a high-quality classical model that “mimics” the solvent’s nonclassical features. An accurate treatment of dilute supercritical fluids will require a consistent nonclassical description for the mixture.
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