J . Phys. Chem. 1990, 94, 8403-8406 TABLE I: Comparison of Hydration Free Energies" of a Model Dipolar Solute Obtained from Continuum and Molecular Methods
diwle momentb
continuum'
2.2
-9.5
4.4 8.8
-38.1
moleculaf
-1 52.5
-7.8 -37.2 -1 58.4
diuole fieldd x 109 x 109 9.6 x 109 2.4 4.8
Energies in kcal/mol. bDipole parameters: p is measured in debyes. 'Solvation free energies calculated from continuum solvent model using the Onsager formula;Ic calculated from molecular solvent model using free energy perturbation simulations." Electric field in units of volts/meter calculated at a distance of 2.3 A from the dipole origin. shell. The orientational dielectric saturation of waters in the first hydration shell of a spherical ion ( R = 2.2 A) is plotted in Figure 1 as a function of the field strength, as the charge on the ion is varied between and -2 in a series of simulations. Similar results are obtained for the dielectric saturation in the first hydration shell of a model dipolar solute as the dipole moment is varied between 2.2 and 8.8 D. The small rate of change of the orientational polarization with increasing charge is indicative of strong dielectric saturation effects at the ion surface. That the maximum value of the polarization is -0.6 instead of 1.0 reflects the competition between the orienting electric field of the ion and the specific anisotropic interactions between the first-shell waters and its nearest neighbors. A recent analysis of the orientational response at the surface of a small cation ( R = 0.95 A) to increasing charge reported by Jayram et al.98is qualitatively consistent with the present results, although the maximum polarization of the hydration layer is greater for the smaller cation. The results presented in Figure 1 imply that continuum treatments of solute-solvent interactions that use a bulk water dielectric constant to characterize the dielectric response will fail for solutes that generate electric fields in the solvent region which exceed lo9 V/m. Recent studies have shown, however, that calculations of molecular solvation free energies based on con-
-
8403
tinuum and microscopic treatments of water are in close agreement for many s o l ~ t e s . ~Indeed, J~ we have now verified this result for neutral dipolar solutes with very large dipole moments (up to 8.8 D); the solvation free energies are listed in Table I. The apparent contradiction is resolved by noting that the continuum formulas for solvation free energies in high dielectric solvents are quite insensitive to dielectric saturation.Ic For spherical solutes the dielectric constant enters in the expression ( l / c - I), so that a IO-fold decrease in the dielectric constant of water would only change the hydration free energy by 12%. Other dynamic and thermodynamic properties will be more sensitive to the dielectric saturation of water by solute electric f i e l d ~ . ~ ~The ' ~ Jexistence ~ of dielectrically saturated hydration shells around ions can significantly decrease the extent of the screening of the ionic interactions between ions in water. Limited information concerning these effects is available from recent computer simulations of the potentials of mean force (pmfs) between ions in water at relatively small ( R C -6 A) ion separation^.'^ At these separations, the pmfs are highly structured and the minima and maxima are associated with specific solute-solvent configurations. Further work is required to clarify the spatial extent of dielectric saturation in the vicinity of solvated ion pairs. Acknowledgment. This work has been supported by a grant from the National Institutes of Health (GM-30580). H.E.A. has been supported in part by a supercomputer postdoctoral fellowship from the New Jersey Commission on Science and Technology. (17) (a) Hirata, F.; Redfern, P.; Levy, R. M. Int. J. Quantum Chem. 1988, 15, 179. (b) Roux, B.; Hsiang-Ai, Y.; Karplus, M . J . Phys. Chem. 1990,94, 4683. ( 1 8) Levine, S.;Rozenthal,
D. K. In Chemical Physics of Ionic Solutions; Conway, J., Barradas, P., Wiley: New York, 1976, Chapter 8. (19) (a) Berkowitz, M.; Karim, 0.; McCammon, J. A.; Rossky, P. J. Chem. Phys. Lerr. 1984, 105, 577. (b) Dang, L. X.;Pettitt, B. M . J . Am. Chem. SOC.1987, 109, 5531. (c) Jorgensen, W. L.; Buckner, J.; Huston, S.; Rossky, P. J. J . Am. Chem. Soc. 1987,109, 1891. (d) Huston, S.E.; Rossky, P. J. J . Phys. Chem. 1989, 93, 7888.
Supercritical Solubility of Solids from Near-Critical Dilute-Mixture Theory Allan H. Harvey Thermophysics Dioision, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 (Received: August 27, 1990)
A recently derived asymptotic theory for Henry's constant near the critical point of a solvent is adapted to the description of the solubility of solids in supercritical fluids. When solubility data for several systems are examined in the manner suggested by the theory, all the experimental isotherms for a given system (with the exception of a few near-critical points where the deviations are attributed to finite-concentration effects) collapse onto a single curve that is linear over a substantial range of solvent densities. This work provides a theoreticaljustification for some often-noted empirical relationships between solubilities and enhancement factors and the density of the supercritical solvent.
Introduction
The use of supercritical fluids as solvents for extraction and other processes continues to attract intere~t.l-~ Design and evaluation of supercritical fluid technology is often hampered by the difficulty of predicting and modeling the phase equilibria in these highly asymmetric systems. A variety of semitheoretical ( I ) McHugh, M. A.; Krukonis, V. J. Supercritical Fluid Exrrocrion; Butterworths: Boston, 1986. (2) Johnston, K. P., Penninger, J. M. L., Eds. Supercrirical Science and Technology; ACS Symposium Series No. 406; American Chemical Society: Washington, DC, 1989. (3) Brennecke, J . F.; Eckert, C. A. AIChE J 1989, 35, 1409.
approaches have been used (with mixed success) to model supercritical mixtures; a review is given by Johnston et ala4 For systems where the supercritical fluid is equilibrated with a solid, many workers5-10 have noticed a striking regularity in the (4) Johnston, K. P.; Peck, D. G.; Kim, S.Ind. Eng. Chem. Res. 1989, 28, 1115. (5) van Wasen, U.; Swaid, I.; Schneider, G. M. Angew. Chem., Inr. Ed. Engl. 1980, 19, 575. (6) Johnston, K. P.; Eckert, C. A. AIChE J . 1981. 27, 773. ( 7 ) Zieer. D. H.; Eckert, C. A. Ind. Ena. Chem. Process Des. Deu. 1983,
22,'582. (8) Paulaitis, M. E.; Kander, Phys. Chem. 1984, 88, 869.
R. G.; DiAndreth, J. R. Ber. Bunsen-Ges.
--his article not subject to US. Copyright. Published 1990 by the American Chemical Society
8404 The Journal of Physical Chemistry, Vol. 94, No. 22, 1990
experimental data. When the logarithm of the solid's solubility (usually expressed as mole fraction y 2 ) is plotted versus solvent density p I for a given temperature, a straight line is obtained over a wide range of densities. Although at low densities (where a second-virial-coefficientdescription of the mixture is adequate) such linearity is expected from theory,"*12 explanations for this behavior at higher densities are for the most part lacking. Some justification at near-critical densities does come from the work of Kumar and Johnston,lo who derived a linear relationship between In y 2 and In ( ~ ~ / p ~where , ~ ) pc,l , is the solvent's critical density. In the limit p1 pc,l, the linear variation in In (pl/pc,l) is asymptotically equivalent to linearity in ( p i - pc,,). Their derivation was, however, limited to the critical isotherm. Meanwhile, the theory of dilute mixtures near the critical point of a solvent has been used to derive asymptotic expressions for the variation of Henry's ~ o n s t a n tand ~ ~ other ~ ' ~ infinite-dilution propertiesI5 of solutes in the neighborhood of the solvent's critical point. These limiting relationships were applied successfully to a variety of systems, with an emphasis on aqueous solutions of nonpolar gases. Here this work is extended and applied to the solubility of solids in supercritical fluids. It is demonstrated that the resulting relationship provides an explanation for the observed linear behavior noted above. Comparisons are made with literature data for solubilities and partial molar volumes for several systems, with good agreement for the solubilities and mixed results for the volumes.
-
Theory The quantity of central interest to our development is Henry's constant k H , defined as
wheref2 is the fugacity of the solute. In the neighborhood of the solvent's critical point, the asymptotic variation of kH is given by the following h e a r expression:
where T i s the absolute temperature and f l is the fugacity of the pure solvent. A and B are constants that can be written in terms of thermodynamic derivative~.'~Though eq 2 was originally derived'j for states along the solvent's saturation curve, a more general derivationt5has shown that it is also asymptotically correct in the one-phase region. While eq 2 has thus far only been applied for volatile solutes in a near-critical solvent (where the criterion for a solute to act as a "volatile" is that (dp/dy2)& the infinite-dilution derivative of the pressure with respect to solute mole fraction at constant volume and temperature, be positive at the solvent's critical point), there is no such restriction in the derivation. It is as valid for a "nonvolatile" solute such as naphthalene near the critical point of C 0 2 as it is for C 0 2 near the critical point of water. This suggests that it might be fruitful to reformulate supercritical solubility as a Henry's constant problem and apply eq 2. The chief obstacle to this approach is the inapplicability of eq I . In a supercritical solubility problem, the infinite-dilution limit cannot be reached because the equilibrium solute fugacity f2 is fixed at a nonzero value by the presence of the pure solid. We therefore define an "effective" Henry's constant by relaxing the infinite-dilution condition in eq 1 : kHCff = f d y z
(3)
(9) Schmitt, W . J.; Reid, R. C. In Supercritical Fluid Technology; Penninger, J. M. L., Radosz, M., McHugh, M. A,, Krukonis, V. J., Eds.; Elsevier: Amsterdam, 1985; p 123. (IO) Kumar, S. K.; Johnston, K. P. J . Supercrit. Fluids 1988, I , 15. ( I I ) Robin, S.; Vodar, B. Discuss. Faraday SOC.1953, I S , 233. ( I 2) Ewald, A. H.; Jepson. W. B.; Rowlinson, J. S. Discuss. Faraday SOC. 1953. I S , 238. (13) Japas, M. L.; Levelt Sengers, J . M. H. AlChE J . 1989. 35, 705. (14) Harvey, A. H.; Levelt Sengers, J. M. H. AIChE J. 1990, 36, 539. ( 1 5 ) Harvey, A. H.; Levelt Sengers, J . M. H.; Tanger, J. C. J . Phys. Chem., in press.
Letters -1000 Bo'
-1 250
I-
1
1
1
1
1
1
Tsekhanskaya et ai., 1964
"0
'P
0
=35"C
0 = 45" A = 55"
&+
c c
O'A
-1500I
c %x
5 -1750-
-E
+
Ob O
-2000-
%
O
B
0
%
i
-2250-
Figure 1. Linear relationship of T In (kHcN/h) versus pI for naphthalene in C 0 2 . T h e dashed line in this and subsequent figures is only a guide to the eye and is not the result of any optimized fit.
Here f2 and y 2 have their equilibrium values. The finite-concentration analogue of eq 2 is then (4)
Unlike eq 2, eq 4 is by no means rigorous. The theory of finite-concentration mixtures near the solvent's critical point (and near the mixture's critical endpoint) is fraught with intricacies; the reader is referred to a recent reviewI6 for a careful consideration of such matters. Since the solubilities of solids in supercritical fluids seldom exceed a few mole percent, it is not unreasonable to expect that the errors introduced by replacing kH with kHeffwill be small under most circumstances; larger errors would be expected at finite concentrations very near the solvent's critical point. Finally, we note that because Henry's constant is directly related to the Gibbs energy of solvation, eq 2 can be differentiated to yield the infinite-dilution partial molar volume of the solute ij;. The resulting expression ists 0; = BR(ap,/ap)T
+ UI
(5)
where R is the molar gas constant and ul is the solvent's molar volume. The value of B used in eq 5 should be the asymptotic infinite-dilution value (from eq 2); as a necessary approximation we use values of B from eq 4 to compare with experimental data for partial molar volumes of solids in supercritical fluids. Comparison with Data. To convert an experimental solubility measurement to an effective Henry's constant via eq 3, the fugacity of the pure solid at the experimental T and p is required. This is given by"
where p; is the vapor pressure of the solid at T and (p; is its fugacity coefficient at saturation. Because the vapor pressures dealt with here are very low, we may assume (ps = 1 . The exponential factor is the Poynting correction, which accounts for the effect of pressure on the fugacity of a pure solid. Since the molar volume of a solid is effectively independent of pressure, uz may be moved outside the integral. Equation 3 then becomes (7)
For this work, vapor pressures for solid naphthaleneI8 and hexa(16) Levelt Sengers, J. M. H. In Supercritical Fluid Technology: Bruno, T. J.. Ely, J. F., Eds.; C R C Press: Boca Raton, FL, in press. (17) See for example: Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986; Chapter 3. (IS) de Kruif, C. G.;Kuipers, T.; van Miltenburg, J . C.; Schaake, R. C. F.: Stevens, G .J . Chem. Thermodyn. 1981, 13, 1081.
Letters
The Journal of Physical Chemistry, Vol. 94, No. 22, 1990 8405 -1000
r
1
-""p
I
I
I
I
I
/
I
-1500
. d c
-6
-
-
-
-
-8
-
aI>N -2250
-10 -
-2500t
'
'\
-3000 2
I
'
'
4
6
I
8
I
10
'
12
14
16
= 35.23" C
-
70
18
0 0 Eckerl et al., 1986
- Eq.5
80
CpH4 DENSITY (mol I L)
Figure 2.
T -
-12 -
-2750
\
..
Linear relationship of T In ( k ~ ~ ~versus / f )p , for naphthalene
in C2H4.
90 100 110 120 PRESSURE (bars)
130
140
Figure 4. Comparison of experimental infinite-dilution partial molar volumes for naphthalene in C02 with predictions from eq 5 .
van Gunst, 1950 = 16.5' C 0 = 20.0'
-800
c
A = 23.5" c
.
-
a N -15 I>
.T=12"C
t
-2wo -24001 2
-20 0
I
4
I
6
1
8
I
I
10
12
I
14
16
50
CpH4 DENSITY (mol / L)
Eckerl et al., 1986
- Eq.5
-25 55
l
l
I
60 65 70 75 PRESSURE (bars)
I
80
-I 85
Figure 3. Linear relationship of T In ( k ~ ~ ~ versus / f , ) p , for hexachloroethane in C2H,.
Figure 5. Comparison of experimental infinite-dilution partial molar volumes for naphthalene in C2H4with predictions from eq 5.
chloroethanei9were taken from the literature. Solid molar volumes are from ref 20. Use of eq 4 also requires values forfi and pi, the fugacity and density of the pure solvent. These were calculated from recent comprehensive equations of state for carbon dioxide2I and ethylene,22the solvents of interest here. Tsekhanskaya et aL2' measured the solubility of naphthalene in supercritical carbon dioxide at three different temperatures. Figure 1 is a plot from their data of T In (kHCfl/J)versus solvent density p , . Not only do all the data (with the exception of a few points at the lowest temperature; these will be discussed later) collapse (with a scatter comparable to that in the data) onto a single curve, but the curve appears to be linear (in accordance with eq 4) at densities up to approximately 15 mol/L. (For C02, pc = 10.63 mol/L). Figure 2 shows a similar plot for naphthalene in supercritical ethylene, also measured by Tsekhanskaya et aL2' Solubilities for this system were also measured by Diepen and S ~ h e f f e r ;their ~~.~~
measurements are omitted from the graph for clarity but are in good agreement with those shown. Figure 3 is the same plot for hexachloroethane in ethylene, as measured by van Gunst.26 Both solutes in ethylene display the same behavior observed in Figure 1: a lengthy linear region followed by an upturn in the graph at a density of approximately 1.5pC,,. (For C2H4,pc = 7.63 mol/L.) Figure 2 also shows a few deviant points near the critical density at the lowest (closest to critical) temperature studied. From the linear regions of Figures 1-3, values of B may be estimated and then used in eq 5 to predict infinite-dilution partial molar volumes. For the two naphthalene systems, near-critical partial molar volumes have been mea~ured,~' and the predictions can be tested. There is some uncertainty (perhaps 10%) in determining B from eq 4; we used values of -1 05 and -1 35 K L/mol for naphthalene in carbon dioxide and ethylene, respectively. These slopes are consistent with the dashed lines in Figures 1 and 2. It is assumed here that these are the asymptotic slopes. While work on Henry's constants with eq 2 has shown28that it is possible to be misled by apparent linearity in a nonasymptotic regime, the existence here of data both above and below the critical density makes such an error less likely. In Figures 4 and 5, predicted infinite-dilution partial molar volumes are compared with data for naphthalene in carbon dioxide and in ethylene. The error bars are the experimenters' estimates
(19) Ivin, K. J.; Dainton, F. S . Trans. Faraday SOC.1947, 43, 32. (20) CRC Handbook of Chemistry and Physics, 59th ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1978. (21 ) Ely, J. F.Proceedings of the 65th Annual Gas Processors Convention; Gas Processors Association: Tulsa, OK, 1986; p 185. (22) Jacobsen, R. T.; Jahangiri, M.; Stewart, R. B.; McCarty, R. D.; Levelt Sengers, J. M. H.; White, H. J.; Sengers, J. V.; Olchowy, G. A.
Ethylene (Ethene), International Thermodynamic Tables of the Fluid State-lO; Blackwell Scientific Publications: Oxford, U.K., 1988. (23) Tsekhanskaya, Y.V.; lomtev, M. 8.;Mushinka, E.V. Russ. J . Phys. Chem. 1964, 38, 1173. (24) Diepen, G. A. M.;Scheffer, F. E. C. J . Am. Chem. Sor. 1948, 70, 4085. (25) Diepen, G. A. M.; Scheffer, F. E. C. J . Phys. Chem. 1953, 57, 575.
(26) van Gunst, C. A. Ph.D. Thesis, Technical University of Delft, Netherlands. 1950. ---(27) Eckert, C . A.; Ziger, D. H.; Johnston, K. P.; Kim, S. J . Phys. Chem. 1986, 90, 2738. (28) Harvey, A. H.; Crovetto, R.; Levelt Sengers, J. M. H. AIChE J . , in
press.
8406 The Journal of Physical Chemistry, Vol. 94, No. 22, 1990
of uncertainty; none are shown where they would lie within the symbol representing the data point. For both systems, q 5 appears to underpredict 0; (in the sense of predicting too large a negative number). In Figure 4 one might claim a reasonable prediction within the experimental uncertainties, but in Figure 5 the predictions are clearly too low by approximately 50%.
Discussion The empirically observed linearity of In y 2 with pI along isotherms can be recovered from this work by substituting eq 3 into eq 4 and rearranging: In y 2 = - [ A
+ B ( p l - P,J)I/T+ 1n.h - i n f l
(8)
At a fixed temperature, f2is only a weak function of pressure through the Poynting correction. l n f l varies less weakly cfl is roughly linear in the pressure), but the dominant variation (because p 1 changes so rapidly near the critical point) is the term linear in density. A similar relationship can be derived for the enhancement factor E , defined as E = yzp/pi. The result is T In E = - [ A
+ B(pl - P ~ , I ) I+ u2(P - P S ) / R - T In
PI
(9)
where cpI is the fugacity coefficient of the pure solvent. Again the density variation dominates the other terms near the solvent’s critical point. Equation 9 provides an explanation for the observation of Schmitt and Reidg that plots of In E versus p , separated into finely spaced (and nearly parallel) lines at different temperatures. Had they plotted T In E, the lines would have very nearly coincided. This also suggests that a good “shortcut” approximation to eq 4 would be to correlate T In E versus solvent density; this might be desirable in cases where the calculation of solvent fugacity f1is impractical. As mentioned in the Introduction, near-critical linearity in p I is asymptotically equivalent to linearity in In ( p I / p c , ] ) . Since it has been foundlo that, for some systems, plots of In y 2 are more linear versus In ( ~ , / p ~ it, ~might ) , also be worthwhile to try the logarithmic quantity in the context given by this work. This can be accomplished by replacing (pI - p , , ) with pC,]In ( p ] / ~ ~in, ~ eqs 2 , 4, 8, and 9. Figures 1 and 2 both contain several points at densities near pC,]that deviate from the linear trend. This is most likely an error introduced by replacing k , with the finite-concentration kHCK.The deviant points are those closest in temperature and density to the solvent’s critical point: they therefore represent highly compressible states where ii2will be large and (for less-volatile solutes) negative. A finite amount of solute will produce a finite contraction in the solvent; as a result the density of the mixture will be substantially higher than the pure-solvent value pI. It is physically reasonable that i t is this final density at equilibrium that determines kHeff. If reported mixture densities23 are used instead of p , in Figures 1 and 2, the near-critical points move into much closer agreement with the linear trend. Most often the mixture densities will not be known, but it might be possible to make an approximate correction to eq 4 for the effect of finite concentration on the density by using eq 5 to estimate 02. This would have to be done
)
Letters carefully and as self-consistently as possible (probably as an integration beginning at x2 = 0), as O2 itself is a strong function of density and composition near the critical point. The source of the deviations from linear behavior at high densities is less clear; we have identified two possible causes. The first is the effect of finite concentration. Since solubilities are greater at higher solvent densities, it may be that solute-solute interactions are becoming important and Henry’s law is no longer applicable. If this were the case, however, one would expect to see the higher temperature points departing from the linear trend sooner, since their solute concentrations are higher. The fact that the isotherms show little separation beyond the linear region in Figures 1-3 suggests the solute-solute interactions are not the primary cause of the deviation. The alternative explanation is that, at higher densities, the asymptotic behavior of eq 2 no longer prevails. This would be consistent with recent findings2* on the application of eq 2 to Henry’s constants for gases in liquid solvents along the saturation curve. It was found that (despite the presence of a long, nonasymptotic linear region) the true asymptotic behavior of eq 2 was manifested only within a few degrees of the critical point. While the near-critical data were too sparse to determine the location of the crossover to nonasymptotic behavior, a transition at approximately 1.5 times the critical density would be consistent with these findings. It is worth noting that, as exemplified here, it is easier to study densities near when the solvent is supercritical rather than subcritical. The results for infinite-dilution partial molar volumes (Figures 4 and 5 ) , while qualitatively correct, are disappointing from a quantitative standpoint. This is especially so given the success of similar predictions (from a fit of Henry’s constants) for nonpolar gases in water.lS Deviations in the high-pressure region are not unexpected because at these conditions the density exceeds the asymptotic regime. The overprediction of the peak sizes, however, remains unexplained. Impurities in the solvent could reduce the size of the experimentally observed peaks; however, rough calculations indicate that if the solvents were of the purity the impurity effect is not large enough to explain the observed deviation. Conclusions I t has been demonstrated that a near-critical limiting relationship for Henry’s constants can be adapted to the solubility of solids in supercritical fluids. The result is the collapse of all solubility data for a given system onto a single curve that is linear at densities up to approximately 1.5 times the critical. This promises to be a useful tool for correlating and predicting the solubilities of solids in supercritical fluids. .4cknowledgment. This work was supported by the Office of Standard Reference Data of the National Institute of Standards and Technology. I am grateful to Dr. J. F. Ely and to the Center for Applied Thermodynamic Studies at the University of Idaho for making available computer programs for calculating the properties of C 0 2 and C2H4,respectively, and to Dr. J. M. H. Levelt Sengers and Prof. K. P. Johnston for helpful discussions.
