J . Phys. Chem. 1993,97, 214Q-2744
2140
Solute-Solute and Solute-Solvent Correlations in Dilute Near-Critical Ternary Mixtures: Mixed-Solute and Entrainer Effects Ariel A. Chialvo Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22903-2442 Received: September 8, 1992
The asymptotic expressions for the solute-solute, cosolute-cosolute (cosolvent-solvent), solute-cosolute (solutecosolvent), solute-solvent, and cosolute-solvent (cosolvent-solvent) total correlation function integrals of infinitely dilute ternary mixtures approaching the solvent's critical point are derived. All these integrals scale as the solvent's isothermal compressibility and diverge a t the solvent's critical point. The sign of those diverging quantities depends on the behavior of the short-ranged infinite dilution solute-solvent and cosolute-solvent direct correlation function integrals. The mixed-solute and entrainer effects are rigorously interpreted in terms of those correlation function integrals. Some important implications of the peculiar microscopic behavior of these mixtures are discussed, such as (a) the irreconcilable incompatibility between the microstructure of dilute near-critical mixtures and the physical basis underlying the van der Waals one-fluid conformal solution mixing rules, (b) the prediction of mixed-solute and/or entrainer effects for ternary systems whose individual solutesolvent or solute-cosolvent binaries behave as either weakly attractive or repulsive mixtures, (c) the independence of the mixed-solute effect on the solute-solute, solute-cosolute, and cosolute-cosolute interactions, and (d) the independence of the entrainer effect on the solute-solute, solute-cosolvent, and cosolvent-cosolvent interactions.
Introduction Because most supercritical systems of practical interest are highly dilute, researchers have devoted considerable attention to the study of solute-solvent interaction^,'-^ in search of a microscopic-based understanding of the behavior of supercritical mixtures aimed at predicting the solubility enhancement in systems close to their lower critical end points. In fact, systems with solute mole fractions of the order of 10-2-10-3 have been traditionally considered dilute, and consequently, their solutesolute interactions were invariably neglected.8-13 However, we currently have significant experiqental,'3-17 simulation,'* and t h e ~ r e t i c a l ' evidence ~ - ~ ~ which suggests that solute-solute interactions might be very important in near-critical mixtures, even at high dilution. Furthermore, recent experimentals~9~'3-'s~2' and t h e ~ r e t i c a l ~results ~ - ~ ~indicate that the presence of either a second dilute solute (mixed-solute system) or a dilutecosolvent (entrainer system) may have profound effects on the solubility of both solutes (selectivity). In spite of the increasing number of e ~ p e r i m e n t a l , ' ~t h. ~e o~ r. e~t~i ~ a l , ' ~and .~~ simulation'8 studies on the formation of solute-solute aggregates in dilute near-critical mixtures, little insight has been gained into the true role played by the solute-solute interactions, especially in mixed-solute and entrainer systems. As a necessary first step in that direction, in this work we investigate the behavior of the solute-solute (solute-solute, solutecosolute, and cosolute-cosolute) and solute-solvent (solute-solvent and cosolute-solvent) total correlation function integrals (TCFI) of infinitely dilute ternary systems near the solvent's critical point. We show that all these TCFI's scale as the solvent's isothermal compressibility, with prefactors which depend on solvent density, temperature, and short-ranged (local) effects measured by the corresponding direct correlation function integrals (DCFI). For mixed-solute systems, the likesolutesolute TCFI's diverge to plus infinity, while the unlike solute-solute and the solutesolvent TCFI's diverge to either plus or minus infinity depending on the behavior of the corresponding solute-solvent DCFI's. In terms of the mixture's microstructure, the above behavior implies that, close to the solvent's critical point, (a) the like solute-solute pair distribution functions [g22(r),g ~ j ( r )decay ] slowly to unity from above and (b) the unlike solute-solute pair distribution function [g23(r)]can decay to unity from either above or below.
For entrainer systems, the behavior of the TCFI's would be the same as for the mixed-solute case provided we rename the third component as the cosolvent. In what follows we develop a general derivation of the asymptotic behavior for the solute-solute, cosolutesosolute, solute-cosolute, solute-solvent, and cosolute-solvent total correlation function integrals (and its entrainer counterpart) of infinitely dilute nonelectrolyte ternary mixtures approaching the solvent's critical point. Then, we discuss the implications of our results in relation to the modeling of mixed-solute and entrainer effects in dilute near-critical mixtures. Specifically, we show that (a) the hypothesis underlying the van der Waals one-fluid (vdW 1) conformal solution mixing rules is incompatibility with the microstructure of dilute near-critical mixtures, (b) mixedsolute and/or entrainer effects might occur, in principle, not only in ternary systems with attractive solute-solvent or solutecosolvent binaries (under the Debenedetti and Mohamed definition'), but also in ternary mixtures with either weakly attractive or repulsive binaries, and (c) the solute-solute, solute-cosolute, and cosolute-cosolute interactions are not involved in the mixedsolute effect. Likewise, the solute-solute, solute-cosolvent, and cosolvent I and C,,"> I 6?< 0 and 6, C 0 (attractive) inconsistent with the behavior of the pair distribution functions 0 -< C,:-5 1 and 0 5 C, S I 6! > 0 and 6, > 0 (weakly attractive) of dilute mixtures close to the solvent's critical point. Although C,?" < 0 and C,1m 0 and 6, > 0 (repulsive) the scaling approximation is certainly a poor assumption, which so that the mixed-solute effect in dilute near-critical mixtures becomes poorer as the mixture approaches infinite dilution,3' the (solute synergism) depends on the strength (affinity) of both vdW 1 mixing rules appear to be somewhat successful due to a solute-solvent and cosolute-solvent interactions. The solutefortuitous but systematic cancellation of errors.38 This conclusion cosolute, solute-solute, and cosolutexosolute interactions play is also consistent with recent r e ~ u l t s which ~ ~ J ~suggest that the no role in the mixed-solute effect, contrary to what has been introduction of local density effectsshort-ranged effects on the suggested in t h e l i t e r a t ~ r e . ' ~ )This ~ ~ Jconclusion ~*~ is independent system's microstructureinto mean field approaches could greatly of the type of truncation used in expressions 41 and 42, because improve the ability of current models to describe mixed-solute K,J(i,j = 2, 3) does not depend on C,Jm (i, j = 2, 3), but Cl,"(i and entrainer systems. = 2, 3). At this point a fundamental question arises naturally: How The mixed-solute effect has been found for binary attractive does the described behavior for the mixture's microstructure systems;i.e.,K23>OwithC12m>1 andCI3->1 ineq 15. However, translate into the fluid-phase equilibria of mixed-solute and/or we are not aware of any experimental evidence of that effect for entrainer systems? Using a first-order MacLaurin expansion of dilute ternary systems whose corresponding binaries are either the fluid-phase fugacity coefficients J,, Jonah and C ~ h r a n ~ ~weakly attractive (i.e., K23 > 0 with 0 ICIZ"I1 and 0 IC I ~ " have recently shown that the first-order coefficient K23 plays a I1) or repulsive (i.e., K23 > 0 with C12-< 0 and C I3- 0 both dilute binaries-solvent (1)-solute (2) and solvent (l)-cosolute (3)-must show either where { x ) denotes constant values of (xz,...,x c ) . attractive (including weakly attractive) or repulsive behavior;'
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2744
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
After taking the derivative of eq A1 with respect to N,, at constant V and T, we obtain
+
-vK(aP/aNi)v,T,N,#, bi = 0
(A41
Chialvo (6) Cummings, P. T.; Cochran, H. D.; Simonson, J . M.; Mesner, R. E.; Karaborni, S. J. Phys. Chem. 1991, 94, 5606. (7) Cochran, H. D.; Cummings, P. T.; Karaborni, S.Fluid Phase Equilib. 1992, 71, I .
(8) Dobbs, J. M.; Wong, J. M.; Lahiere, R. J.; Johnston, K. P. Ind. Eng. Chem. Res. 1987, 26, 56.
where K is the isothermal compressibility, so that
(9) Johnston, K. P.; Eckert, C. A. AICHE J. 1981, 27, 773 ( 1 0) Johnston, K. P.; Ziger, D. H.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1982, 21, 191.
where p I o = N I / V is the number density of the pure solvent. Equation A5 is the first line of eq (7). Likewise, by taking the derivative of eq A3 with respect to xi, a t constant v and T, we have
( 1 1 ) Schmitt, W. J.; Reid, R. C. J. Chem. Eng. Data 1986, 31, 204. (12) Schmitt, W. J.; Reid, R. C. Fluid Phase Equilib. 1986, 32, 77. ( I 3) Ziger, D. H.; Eckert, C. A. Ind. Eng. Chem. Process Des. Deo. 1983, 22, 582.
(14) Kurnik, R. T.; Reid, R. C. Fluid Phase Equilib. 1982, 8, 93.
Then, by recalling that v = Cf~,xp,,and X I = 1 - x2 - x,, we have (au/axl)T,fJ,,,
= [a(~xJ'J)/axll
T,P.X,,,
... - x, ...
(16) Brennecke, J. F.; Eckert, C. A. In Supercritical Fluid Science and Technology; ACS Symposium Series 406; Johnston, Penninger, Eds.; American Chemical Society: Washington, DC, 1989.
=
c ' 1
- '1 + ~ x ) ( a b J / a x l ) T , p , X J # ,
(15) Kwiatkowski, J.; Lisicki, Z . ; Majewski, W. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 865.
(A7)
J=I
where the second term on the right-hand side of eq A7 is zero (Gibbs-Duhem). Finally, by invoking the following limiting condition
(17) Brennecke, J. F.;Tomasko,D. L.;Peshkin,J.;Eckert,C.A.Ind. Eng. Chem. Res. 1990, 29, 1682. (18) Chialvo, A. A.; Debenedetti, P. G. Ind. Eng. Chem. Res. 1992, 31, 1391. (19) Debenedetti, P. G.; Chialvo, A. A. J . Chem. Phys. 1992, 97, 504. (20) Wu, R. S.;Lee, L. L.; Cochran, H.D. Ind. Eng. Chem. Res. 1990, 29, 977.
lim O, = lim u = (plo)-' XI-I
x,-l
(21) Dobbs, J. M.; Johnston, K. P. Ind. Eng. Chem. Res. 1987,26, 1476.
we obtain from eq A6
( 2 2 ) Cochran, H. D.; Johnson, E.; Lee, L. L. J. Supercrit. Fluids 1990, 3, 157.
which is the second line of eq 7. Change of Variables in the Derivatives of the Activity Coefficients. Given the Gibbs free energy G = G(P,T,Nl,N2,...,N,), we can make the following change of variables,
(23) Pfund, D. M.; Cochran, H. D. In Supercritical EngineeringScience. Fundamentals and Applications; ACS Symposium Series; Brennecke, J. F., Kiram, E., Eds.; American Chemical Society: Washington, DC; in press. (24) Jonah, D. A.; Cochran, H. D. Submitted for publication to Fluid Phase Equilib. (25) Brennecke, J . F.;Tomasko, D. L.; Eckert,C. A . J . Phys. Chem. 1990,
G(P,T,Nl,Np.-.,N,)---cG ( P , T , N J ~ . J , ) (A91 so that the Kirkwood-Buff derivatives of the activity coefficients can be recast as follows: (a(zi))
94, 7692.
(26) Zagrobelny, J.; Bright, F. V. In Recent Advances in Supercritical Fluid Technology. Applicationsand FundamentalStudies. ACSSymposium Series 488; Bright, MacNally, Eds.; American Chemical Society: Washington, DC, 1992. (27) Kirkwood, J. G.; Buff, F. P. J . Chem. Phys. 1951, 19, 774. (28) O'Connell, J. P. Mol. Phys. 1971, 20, 27.
P,T.N,,,
(A101 with
(29) O'Connell, J . P. In Fluctuation Theory of Mixtures. Matteoli, E., Mansoori, G. A., Eds.; Taylor and Franics: New York, 1990. (30) O'Connell, J. P. Proceedings 6th ICFPPECPD (Cortina D'Ampezzo, 1992). Fluid Phase Equilib. in press.
so that
(31) Van Ness, H . C.; Abbott, M. Classical Thermodynamics of Non Elecrrolyte Solutions; McGraw-Hill: New York, 1982. (32) Reed, T. M.; Gubbins, K. E. Applied Statistical Mechanics; McGraw-Hill: New York, 1973; Chapter 8. (33) Levelt Sengers, J. M. H. J . Supercrit. Fluids 1991, 4, 215. (34) Levelt Sengers, J. M. H. In Supercritical Fluid Technology; Ely, Bruno, Eds.; CRC Press: Boca Raton, FL, 1991; p 14. (35) Chialvo, A. A. Unpublished results, 1992.
References and Notes ( I ) Debenedetti, P. G.; Mohamed, R. S. J. Chem. Phys. 1989,90,4528. (2) Pestche, I. B.; Debenedetti. P. G. J . Chem. Phys. 1989, 91, 7075. (3) Pestche, 1. B.; Debenedetti, P. G . J . Phys. Chem. 1991, 95, 386. (4) McGuigan, D. B.; Monson, P. A. Fluidfhase Equilib. 1990,57,227. (5) Lee, L. L.; Debenedetti, P. G.; Cochran, H. D. In Supercriricol Fluid Technology; Bruno, Ely, Eds.; C R C Press: Boca Raton, FL. 1991; Chapter 4.
(36) Lee, L. L. Molecular Thermodynamics of Nonideal Fluids; Butterworth Publishers: Stoneham, 1988. (37) Shing, K . S.;Chung, S.-T. AICHE J. 1988, 34, 1973. (38) Kahl, G.; Hansen, J.-P. Mol. Phys. 1989, 67, 367. (39) Kim, S.;Johnston, K . P. AICHE J . 1987. 33, 1603 (40) Brennecke, J. F.; Eckert, C. A. AICHE J. 1989, 35, 1409.
(41) Lupis, C. H. P. Acra Metall. 1977, 25, 751.
