7288
J. Phys. Chem. B 2007, 111, 7288-7290
Salting Out near the Critical Point Robert M. Mazo Institute of Theoretical Science and Department of Chemistry, UniVersity of Oregon, Eugene, Oregon 97403 ReceiVed: February 6, 2007
The salting out of solid solutes near the critical point of the solvent is investigated using the results of a previous paper (J. Phys. Chem. B 2006, 110, 24077) on the fluctuation theory of salting out. It is found that the salting out coefficient at infinite dilution of cosolvent is approximately proportional to the compressibility of the solvent and is consequently quite large near the critical point. No estimate is given for the range of cosolvent concentrations over which the infinite dilution slope might be a good approximation. Far from the critical point, it is known to be a good approximation over a considerable cosolvent concentration range.
Introduction paper1
In the course of submitting an earlier on the fluctuation theory2 of the salting out effect, a referee asked about salting out near the critical point of the solvent. I could not answer this question at the time of review but resolved to return to this subject at the earliest opportunity. This paper is an attempt to answer the referee’s question. It turns out that much of the machinery needed is already in the literature in papers of Chialvo,3 Kusalik and Patey,4 and Chialvo et al.5 In this paper, we consider a three-component solution consisting of a solvent, denoted as component 1, a solute, denoted as component 2, and a cosolvent, denoted as component 3. The cosolvent may be a salt of the general formula Mν+Xν-. In this case, the ionic constituents are denoted by subscripts + and -. A phase containing all three components is in equilibrium with a phase containing only component 2 (or sometimes both components 1 and 2, especially if it is a gas phase). The salting out coefficient, km, is defined by1
( ( ))
m2 ∂ km ) - lim log 0 c3f0 ∂m3 m2
is a surface in the phase diagram giving the saturation value of x as a function of P and T, that is, the solubility surface of the solute. The critical line will terminate on the solubility surface, since x2 cannot exceed its saturation value. This is the critical point of the solution in equilibrium with the pure solute. There is a similar critical line emanating from the critical point of the solute that will also intersect the solubility surface. We shall be concerned here with solutes whose critical point is far above that of the solvent, so this latter critical point will not concern us in this paper. This is type III behavior in the classification of van Koneynenburg and Scott.7 For a three-component solution, the phase diagram space is four-dimensional. The critical line becomes a critical twodimensional surface, and the solubility “surface” is threedimensional. Salting out deals with properties of the solubility surface. Results The result of ref 1 for km is
(1)
P,T, µ2
where m2 is the solubility (molality units) of component 2 in the presence of 3 and m02 is its solubility in the absence of 3. c3 is the molarity of 3. Empirically at ambient temperatures, it is observed that log(m02/m2) is a linear function of c3 over a considerable range of c3 (Sechenow’s law), so the solubility curve is a straight line. I am not aware of any experimental studies of the shape of the solubility curve as a function of c3 near the critical point of the solvent. Consequently, eq 1 must be regarded as merely giving the limiting slope of the curve; it does not predict the range over which the linear term alone is a valid approximation. The phase diagram near the critical point of one component is more easily visualized in a two-component system than in a three-component one. The phase diagram space is threedimensional: pressure, P, temperature, T, and solute mole fraction, x. In this three-dimensional space, a curve emerges from the critical point of the pure solvent (x ) 0).6 Each point on this line represents the critical point of a mixture with composition x. Thus, there is a critical line in the phase diagram emanating from the critical point of pure component 1. There
(
2c01M1 1 0 0 km ) + G011 + G2+ - G1+ - G012 2303λ c0 1
)
(2)
where the subscripts refer to species and the superscript “0” means infinite dilution of the cosolvent. c01 is the concentration of the solvent, M1 is its molecular weight, and λ is the factor that converts from molality to ionic strength for a single electrolyte. We shall shortly use symbols with two superscript zeros, “00”. These will denote infinite dilution in both solute and cosolvent. The Gij’s are so-called Kirkwood-Buff integrals
Gij ) 4π
∫0∞[gij(r) - 1]r2 dr
(3)
where gij(r) is the radial distribution function of the pair of species i and j. We shall now evaluate the terms in eq 2 seriatim, assuming that the solution is infinitely dilute in both solute and cosolvent. We assume infinite dilution of cosolvent because we are calculating the limiting slope in cosolvent concentration. The solute is assumed to be at infinite dilution as an approximation to make calculations feasible. We shall discuss this assumption later.
10.1021/jp071046i CCC: $37.00 © 2007 American Chemical Society Published on Web 05/31/2007
Salting Out near the Critical Point
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7289
What we shall do, following the idea of ref 3, is isolate in km those parts that get large near the critical point of the solvent from the parts that presumably remain bounded in this neighborhood. (1) 1/c01 + G00 11. This quantity is just the well-known compressibility relation and is
1 0 + G00 11 ) kTκ1 0 c1
(4)
where κ01 is the compressibility of the pure solvent. 00 (2) G1+ . In this paper, we are considering the cosolvent to be a salt, Aν+Bν-. We set ν ) ν+ + ν-. According to Chialvo,3 modified to account for the dissociation of the cosolvent,1 0 ) G1+
0 0 (ν+C1+ + ν-C1) c3G+ 0 1 - c1C11
(5)
where Cij denotes the integral of the direct correlation function c(r)
Cij ) 4π
∫0∞cij(r)r2 dr
(6)
Note that the C integrals defined in ref 3 have an extra factor of c (the total concentration) and are consequently dimensionless. The C’s that we use here have the dimension of volume, the same as the G’s. Now, as c3 f 0, (1 - c1C011)-1 f c01κ01kT and c3G0+ - f 1/ν.4 Hence, 00 G1+ ) c01κ01kT
(
)
00 00 ν+C1+ + ν-C1ν
(7)
(3) G00 12. According to ref 3, 0 0 00 G00 12 ) kTκ1c1C12
(4) This quantity can be evaluated using a method similar to that of ref 4. From the Ornstein-Zernike equation, we have
G2+ ) C2+ + C21c1G1+ + C2+c+G++ + C2-c-G- + + C22c2G2+ (9) and a similar equation for G2-. Adding these two equations and using the fact that G2+ ) G2- yields after some algebra
1 [C c G + c3(C2+ν+ + C2-ν-)G+ -] 1 - c2C22 21 1 1+ (10)
Since c2C22 f 0 as c2 f 0 and c3G+ - f 1/ν as c3 f 0, we finally have 00 G2+
)
00 00 00 00 (ν+C1+ + ν-C1) (ν+C2+ + ν-C2) + ν ν (11)
c01κ01kT
Putting all of these results together, we have for our final expression for km
km ) -
[
Discussion Equation 12 contains two terms. The first contains the compressibility of the pure solvent. This diverges at the critical point of the solvent and is very large in its vicinity. However, it must be remembered that this property of the pure solvent enters only because we have approximated the G’s pertaining to the solute at its saturation concentration by those at infinite dilution. The saturation G’s will doubtlessly diverge at the critical point of the solution but not at the critical point of the solVent. The main conclusion of this paper is that the salting out, or Sechenow, coefficient near the critical point of the solvent is very large, varying approximately as the compressibility of the solvent. To estimate how this varies with temperature, one should use the classical or mean field value of the variation of the compressibility with temperature, κ ∼ |T - Tc|-γ′, along the critical isobar. Here, γ′ is the susceptibility exponent for T < Tc, or 1.0. For T > Tc, the susceptibility exponent γ is also 1.0. The classical exponents rather than the true critical exponents should be used because the fact that κ01 enters our equations is a result of our approximations and the singularity at the pure solvent critical point is therefore an artifact. One should not get too close to (T00, p00) when applying eq 12. (Again, the double zero superscripts refer to infinite dilution with respect to both solute and cosolvent.) The integrands of the direct correlation function integrals are generally believed to behave asymptotically like the intermolecular potentials between the species concerned, that is,
(8)
00 G2+ .
G2+ )
It is important to realize that eq 12 is approximate, for the G integrals in eq 2 refer to the solution at infinite dilution of cosolvent but the concentration of solute is the saturation value. That is G0’s rather than G00’s are required. In passing to the form of eq 12, we have assumed that the solute saturation concentration is small enough that G0 can be replaced by G00 to good approximation.
]
00 00 c01M1 2 (ν+C2+ + ν-C2) kTκ01(1 - c01C00 ) + 12 1000 2.303λ ν (12)
cij(r) ) c/ij(r) + uij(r)/kT
(13)
where c/ij is a short range, integrable function and uij is the intermolecular potential. When uij is short ranged, the first term in eq 12 is large only because of the κ factor. The second term, however, contains C’s that may arise from ion-dipole potentials and perhaps other long range electrostatic interactions involving the ionic charges. These could potentially be divergent. However, all such electrostatic interactions arising from the ionic charges are linear in the charges. Since ν+z+ + ν-z- ) 0, the potentially divergent parts of the second term in eq 12 cancel and this term will remain finite. One must realize that our considerations here have been for the situation where a pure phase, the solute, is in equilibrium with its saturated solution. From an experimental point of view, this practically restricts the solute to be a pure solid. The solubility and salting out behavior of supercritical solvents have been much studied, both for their intrinsic interest and for their immense commercial importance. Also, the effect of an added component on liquid-liquid critical points (consolute points) has been much studied. However, I am not aware of any experimental studies of solid solubility and salting out near the critical point of the solvent. Thus, this paper is making a prediction that I hope will be tested in future experiments. Gitterman and Procaccia8 have presented a thermodynamic theory of solubility in supercritical fluids. Chang et al.9 have studied the divergence of partial molar volumes of the solute
7290 J. Phys. Chem. B, Vol. 111, No. 25, 2007 at the solvent critical point. These phenomena were first studied by Krichevskii10 and analyzed on the basis of mean field theory by Rozen.11 It turns out that the partial molar volumes as well as other partial molar quantities are singular in a nonuniform way near the critical point; the nature of the divergence depends on the path in state space by which the critical point is approached. Since the G integrals are closely related to partial molar volumes,2 it is not surprising that the G integrals themselves should show anomalous behavior. A useful review of the near critical behavior of solutions has been published by Morrison.12 However, none of these articles considered salting out phenomena. Our calculations, culminating in eq 12, are not restricted to the neighborhood of the critical point. Away from the critical point, there is no advantage to eq 12 over eq 2. In eq 12, the “dangerous” compressibility has been separated from the nondivergent terms. Away from the critical neighborhood, there is no point to this. It is true that, in the supercritical region, the solvent is much more compressible and generally has an appreciably lower density than fluids at conventional temperatures and pressures (a factor of roughly 1/3 for water and roughly 1/2 for CO2). Nevertheless, eq 12 is expressed in terms of the direct correlation function integrals which have no direct intuitive physical meaning. They cannot be measured directly, as can the G’s (in principle) by radiation scattering, nor can they be calculated except through the G’s and the OrnsteinZernike relation. Thus, the inaccessibility of the Cij integrals renders eq 12 useful only when the compressibility is large and is changing rapidly. Then, one may test the theory by plotting the Sechenow factor versus κ01. If the theory is correct, the result should be a straight line. It should be emphasized that what has been computed here is an estimate of the initial slope of the logarithm of the
Mazo solubility of the solute as a function of cosolvent concentration near the critical point of the solvent. This function is essentially the activity of the solute based on a standard state of a saturated solution with no cosolvent present. It is an empirical fact (Sechenow’s law) that this function is linear, or extremely close to linear, over a wide range of cosolvent concentrations at normal laboratory conditions of temperature and pressure. It is not at all clear at the present time whether the initial slope remains the actual slope as the cosolvent concentration increases. That is, as one adds cosolvent, the system moves further and further from the solvent critical point. Consequently, the singularity at the critical point will become less and less important, but the magnitude of this effect is difficult to estimate. Acknowledgment. I would like to thank the anonymous referee of ref 1 for raising the question addressed here and Professor John Schellman for valuable discussions. References and Notes (1) Mazo, R. M. J. Phys. Chem. B 2006, 110, 24077 (2) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774. (3) Chialvo, A. A. J. Phys. Chem. 1993, 97, 2740. (4) Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1987, 86, 5110. (5) Chialvo, A. A.; Cummings, P. T.; Simonson, J. M.; Mesmer, R. E. J. Chem. Phys. 1999, 110, 1075. (6) Rowlinson, J. S. Liquids and liquid mixtures, 2nd ed.; Butterworths: London, 1969. See particularly Chapter 6. (7) van Konynenburg, P. H.; Scott, R. L. Philos. Trans. R. Soc. London, Ser. A 1980, 298, 495. (8) Gitterman, M.; Procaccia, I. J. Chem. Phys. 1983, 78, 2648. (9) Chang, R. F.; Morrison, G.; Levelt Semgers, J. M. H. J. Phys. Chem. 1984, 88, 3389. (10) Krichevskii, I. R. Russ. J. Phys. Chem. 1967, 41, 1332. (11) Rozen, A. M. Russ. J. Phys. Chem. 1976, 50, 837. (12) Morrison, G. J. Solution Chem. 1988, 17, 887.
