Applications of Near-Critical Dilute-Solution ... - ACS Publications

Successful correlations have been produced for Henry's constants and K-factors. Application to apparent molar properties is discussed, and an explanat...
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Ind. Eng. Chem. Res. 1998, 37, 3080-3088

Applications of Near-Critical Dilute-Solution Thermodynamics Allan H. Harvey† Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Boulder, Colorado 80303

Much of the thermodynamics of dilute solutions near the solvent’s critical point is governed by a single thermodynamic derivative known as the Krichevskii parameter. This parameter manifests itself in some linear asymptotic relationships that describe the behavior of solute Henry’s constants and infinite-dilution K-factors in the vicinity of the solvent’s critical point. This paper reviews the application of these asymptotic relationships to practical thermodynamic calculations. Successful correlations have been produced for Henry’s constants and K-factors. Application to apparent molar properties is discussed, and an explanation is given for its quantitative inadequacy. The Henry’s constant relationship has been extended to describe the solubility of solids in supercritical fluids; current limitations and future prospects are discussed for this application. Finally, a new linear relationship is derived for the capacity factor in supercritical fluid chromatography and applied to some data from the literature. Introduction Many problems of interest in chemical engineering involve dilute solutions. Examples include the solubilities of organics in water, of gases in liquids at low and moderate pressures, and of solids in supercritical fluids. When the solute mole fraction is sufficiently small, the description of these systems can be simplified by ignoring solute-solute interactions and considering only the limit of infinite dilution. An important tool in engineering thermodynamics is the use of limiting cases (and asymptotic approaches to such limits), such as the ideal gas or the ideal solution. For infinite-dilution properties, a useful limiting case is the infinitely dilute solution in the vicinity of the solvent’s critical point. This work describes some relationships that have been derived for this case and reviews applications of these relationships for a variety of problems. Theoretical Background The important factor in the thermodynamics of dilute solutions near the solvent (component 1) critical point is the effect on the system of the addition of a small amount of solute (component 2). This is conveniently expressed by the Krichevskii parameter, which we write as

-AcVx

( )

∂p ) ∂x2

c

(1)

V,T,x2)0

where p is the pressure, V is the volume, T is the temperature, xi is the mole fraction of species i, and superscript c indicates evaluation at the solvent critical point. The shorthand notation -AcVx indicates a second differentiation of the molar Helmholtz energy A with respect to molar volume and solute mole fraction in a formulation where T, V, and x are the independent variables. Because of the path dependence of this †

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derivative (Chang et al., 1984), it is important that the limit x2 f 0 be taken before the limit of the critical point. Physically, the Krichevskii parameter corresponds to the change in pressure at the solvent’s critical point when an infinitesimal amount of solvent is transformed into solute at constant temperature and volume. It represents solute-solvent interactions; separating these slowly varying and well-behaved interactions from the divergent properties of the pure solvent is the key to understanding and modeling near-critical dilute mixtures. The fundamental importance of this parameter was recognized many years ago (Krichevskii, 1967; Wheeler, 1972; Rozen, 1976). In addition to the properties to be discussed below, it determines the initial departure of the mixture critical line from the solvent critical point and the initial slopes on p-x coordinates of the isothermal dew-bubble curves of a binary mixture near the solvent critical point. Further discussion of this ubiquitous parameter has been given by Levelt Sengers (1991a). For phase-equilibrium calculations in dilute solutions, an important quantity is Henry’s constant kH, defined by

kH ) lim (f2/x2) x2f0

(2)

where f2 is the fugacity of the solute. While it is common to restrict discussion of this quantity to solutes in the liquid along the solvent coexistence curve, the definition applies to any solvent state. The correct asymptotic behavior of kH near the solvent critical point was first derived by Japas and Levelt Sengers (1989) for departures from the critical point along the solvent coexistence curve. The derivation was extended by Harvey et al. (1991) to the general case of departure along any path. The result is

T ln(kH/f1) ) A + B(F1 - Fc,1)

(3)

where F is the molar density and Fc,1 is the solvent’s critical density. While the constants A and B in eq 3 can be fitted to data, they are also meaningful thermo-

This article not subject to U.S. Copyright. Published 1998 by the American Chemical Society Published on Web 04/17/1998

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dynamic quantities:

A ) Tc,1 ln(φ∞2,c1 pc,1/fc,1) B(RFc,12) )

( ) ∂p ∂x2

c

(4a) (4b)

v,T,x2)0

R is the molar gas constant and φ∞2,c1 is the infinitedilution fugacity coefficient of component 2 evaluated at the solvent critical point. Note that B, which is the slope in eq 3, is proportional to the Krichevskii parameter. A similar analysis can be performed for the vaporliquid partition coefficient (or K-factor) of a solute at infinite dilution, defined along the solvent’s coexistence curve by

K∞ ) lim (y2/x2) x2f0

(5)

For this quantity, the limiting relationship is (Japas and Levelt Sengers, 1989)

T ln K∞ ) 2B(FsL,1 - Fc,1)

(6)

where FsL,1 is the saturated liquid density of the solvent. B is the constant defined in eq 4b. This is the only parameter needed in the linear relationship for K∞, because K∞ must be 1 at the solvent’s critical point where the vapor and liquid become identical. Finally, the density is not the only possible linear variable in eqs 3 and 6. The asymptotic conditions would be satisfied equally well by choosing the volume, or even the logarithm of the density or volume. For properties on the solvent’s saturation curve, one also has the choice of using saturated vapor or liquid densities (or volumes) or the density (or volume) difference between the two phases. The choice of the density (and the liquid density for properties on the saturation boundary) in eqs 3 and 6 is empirical; that variable seems to produce linear relationships that work over the widest range of conditions.

Figure 1. Linear relationship (eq 3) for Henry’s constants for nitrogen in water.

Application to Henry’s Constants Equation 3 is only an asymptotic relationship; there is no a priori way to know whether the linearity will hold far enough away from the critical point to be of any practical use. Harvey and Levelt Sengers (1990) discovered that plotting the data for Henry’s constant of nonpolar gases in water according to eq 3 (T ln(kH/f1) versus FsL,1) gave straight lines from conditions near the critical point all the way to approximately the normal boiling point. Figure 1 shows this behavior for nitrogen in water. By adding an empirical correction term at low temperatures, they produced the following correlation:

T ln(kH/fsL,1) ) A + B(FsL,1 - Fc,1) + CTFsL,1 exp[(273.15 - T)/τ] (7) where τ was set to 50 K for all solutes. Equation 7, with only three adjustable parameters, was successful in fitting the Henry’s constants for a number of gases in water; a typical result (for nitrogen in water) is shown in Figure 2.

Figure 2. Henry’s constants for nitrogen in water as correlated by eq 7.

For engineering use, however, eq 7 has some disadvantages. First, the empirical correction term is specific to water as the solvent. This could be repaired fairly simply. The larger disadvantage is the presence of the fugacity and density of the saturated liquid. These quantities, particularly the fugacity, are not easily available for many solvents. Even for those solvents for which comprehensive equations of state exist, the calculations would be time-consuming. It would be preferable for engineering use to capture the qualities of eq 7 by a relatively simple function of temperature only.

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Figure 3. Henry’s constants for nitrogen in water. Extrapolation of eq 9 from the fit to data below 450 K.

Harvey (1996) did this by noting that the saturated liquid density is fairly well described by the scaling relationship

|FsL,1 - Fc,1| ∼ |T - Tc,1|β

(8)

β is a critical exponent with an asymptotic value of 0.326, but data over a larger range typically manifest an “effective” β of approximately 0.355. The fugacity of the pure solvent in eq 7 can be approximated by its vapor pressure ps1, which is typically available as a simple function of temperature. Finally, a general form was chosen for the empirical term that contributed far from the critical point. The result is

ln kH ) ln ps1 + A/T* + B(1 - T*)0.355/T* + C exp(1 - T*)(T*)-0.41 (9) where we have used the reduced temperature T* ) T/Tc,1 (T* e 1). A and B are no longer the quantities defined by eq 4a,b, but B is still proportional to the Krichevskii parameter. Equation 9 is better suited for practical use than eq 7, yet it fits data for solutes in water equally well. It can also be used for other solvents and fits data well for 12 solutes in n-hexadecane (Harvey, 1996). One important advantage of both eqs 7 and 9 (arising from their foundation in eq 3) is their extrapolation capability. Often, Henry’s-constant data are available only at low and moderate temperatures. A strictly empirical expression might be able to fit the data, but extrapolation to higher temperatures would be unwarranted. Since the equations presented here are constrained in a theoretically correct way as the critical temperature is approached, they can be extrapolated with more confidence. Figure 3 shows the extrapolation of eq 9 for the Henry’s constant of nitrogen in water from parameters obtained by fitting only the data below 450 K.

Figure 4. Infinite-dilution K-factor for nitrogen in water.

Application to Infinite-Dilution K-Factors The quantity K∞ defined by eq 5 is often of interest because it relates directly to the compositions of coexisting phases. As with the Henry’s constant, the linear relationship (eq 6) seems to hold to a greater distance from the critical point than one would have a right to expect. Figure 4 shows this relationship for nitrogen in water. Equation 6 has been extended with lowertemperature correction terms and applied to a number of systems by Alvarez et al. (1994). Japas et al. (1995) have shown that the asymptotic linear region should be smaller when the solute and solvent have similar volatilities, because the Krichevskii parameter in that case is small and higher-order terms become relatively more important. This was the case in their analysis of the fractionation of isotopes of water between vapor and liquid. The constant B, proportional to the Krichevskii parameter, appears in the slopes of both eqs 3 and 6. It was therefore surprising when it was discovered that, for several systems, the values of B obtained from these two methods disagreed, with the values obtained from Henry’s constants being higher by a factor of about 1.4. This can be seen by comparing Figures 1 and 4, where the slopes according to eqs 3 and 6 should differ by a factor of 2. Further analysis (Harvey et al., 1990) revealed that it was the relationship for K∞ that was giving the correct value of B. This conclusion was reached because the known endpoint (K∞ ) 1) at the solvent’s critical point agreed with extrapolations of eq 6 and because values of the Krichevskii parameter obtained from K-factors were in agreement with those obtained independently from critical-line slopes. This means that the linear Henry’s-constant relationship (eq 3), despite the fact that it gives a straight line over a large range of conditions, does not have the true asymptotic slope in this range. The extraordinary linearity displayed by eq 3 must therefore be to some extent fortuitous. By extension, eqs 7 and 9 are not as

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well-founded theoretically as one would hope. It appears that the asymptotic slope in eq 3 is obtained only within a range of perhaps 10-25 K (or, in terms of the saturated liquid density, to about 1.5-1.8 times the critical density) from the solvent’s critical point. For most systems, reliable data do not exist this close to the critical point, although such a crossover appears to be visible in the nitrogen-water and hydrogen-water data reported by Alvarez and Ferna´ndez-Prini (1991). The crossover in the nitrogen-water data was further analyzed by Gallagher et al. (1993). The reasons why eq 6 yields a correct value of the Krichevskii parameter while eq 3 does not may now be partly understood. The K-factor represents a difference between two phases, apparently causing the cancellation of some nonasymptotic effects that do not cancel in the case of the Henry’s constant, which is a singlephase quantity. This explanation has been phrased in terms of solvation effects by Chialvo et al. (1996). These slopes were also examined in the context of computer simulation of noble-gas-like solutes in water by Guillot and Guissani (1993). They found linear regions that were somewhat greater in extent for the Henry’s constant than for the K-factor (but extending less than 100 K below the critical temperature in both cases), and the slopes of these regions differed by approximately the theoretical factor of 2. The qualitative difference between the ratio of the slopes in their simulation results and the behavior of real fluids has not yet been explained. Application to Apparent and Partial Molar Properties It is common in solution chemistry to make use of standard states for solute volumes, enthalpies, etc., which are the infinite-dilution limit of the partial molar properties of the solute. While there are dangers and subtleties associated with this approach (Levelt Sengers et al., 1992), it can be useful if carefully applied, especially if the critical point is not approached too closely. These infinite-dilution properties can be derived from the standard-state Gibbs energy of hydration (or solvation if water is not the solvent) ∆hG0, corresponding to the transfer of the solute from the pure ideal-gas state to an infinite-dilution state at the same T and p. ∆hG0 is related to the Henry’s constant by

∆hG0 ) RT ln kH

(10)

The standard-state volume, entropy, and heat capacity follow from differentiation of eq 10:

( ) ( ) ( )

∂∆hG0 ∆h V ) ∂p 0

T

∆hS0 ) -

∂∆hG0 ∂T

∆hCp0 )

∂∆hS0 ∂T

p

p

(11a)

(11b)

(11c)

Other properties such as standard-state enthalpies may be obtained from appropriate combinations of these quantities.

Figure 5. Apparent molar volumes for carbon dioxide in water at high temperatures and pressures.

Harvey et al. (1991) substituted eq 7 for kH into eq 10, which was then differentiated to produce standardstate properties of hydration. [The statement on p 934 of that paper that the expressions for these properties are in agreement with ideal-gas results in the limit of zero density is incorrect. The derivative properties agree with ideal-gas results, but the Gibbs energy does not because of the A term in eq 7.] In each of eqs 11ac, the dominant term in the near-critical region is given by a thermodynamic derivative of pure water (for example, the isothermal compressibility for ∆hV0) multiplied by a constant proportional to the Krichevskii parameter. This simple procedure produced qualitatively correct derivative properties; quantitative evaluation was limited because of the data available at the time. New measurements have since been made (Hnedkovsky et al., 1996) for apparent molar volumes of aqueous solutes at high temperatures and pressures. In Figure 5, we compare measured apparent molar volumes for CO2 as a function of temperature at 28 and 35 MPa to the predictions of eq 11a, where the parameters in eq 7 were fit to Henry’s-constant data (Crovetto, 1991) for CO2 in water. This is not a comparison of identical quantities, since eq 11a gives infinite-dilution values and the apparent molar volume data are at CO2 molalities near 0.15. Hnedkovsky et al. (1996) indicated that, based on experiments with aqueous ammonia at different concentrations, the differences between their finite-concentration values and the infinite-dilution values were on the order of a few percent. Levelt Sengers et al. (1992) have pointed out that such extrapolations to infinite dilution can be tricky, but their calculations specifically for aqueous CO2 (see their Figure 8) indicate that the difference between apparent molar volumes at 0.15 m (0.003 mole fraction) and infinite dilution should not be more than 10% at Hnedkovsky’s experimental conditions. Also shown in Figure 5 is the correlation of O’Connell et al. (1996), to

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be discussed below. In producing Figure 5, properties of water were obtained from the new international standard formulation (Pruss and Wagner, 1998; Harvey et al., 1997) rather than the equations used in the original work, but the differences in water properties are negligible for this purpose. Figure 5 shows that eq 11a overestimates the solute partial molar volumes at high temperatures. Similar results are obtained with the other gaseous solutes studied by Hnedkovsky et al. (1996). The overestimation is consistently by a factor near 1.4. This is approximately the factor by which the fit of eq 7 to Henry’s-constant data overestimates the Krichevskii parameter, as mentioned in the previous section. This is not surprising, since the amplitude of these derivative properties is proportional to the Krichevskii parameter; any error in determining that factor will translate into errors in the derivative properties. One might think of using a “rescaled” value of B (perhaps divided by 1.4) to calculate these properties, or a value obtained from infinite-dilution K-factors, the initial slope of the mixture critical line, or some other method of determining the Krichevskii parameter. Unfortunately, although doing this (while keeping the other parameters determined in eq 7) matches the hightemperature data, the resulting partial molar volumes in low-temperature liquid water are too low. We therefore conclude that while direct differentiation of Henry’s constants correlated to eq 7 gives the right qualitative behavior, such a simple procedure will not work quantitatively. For quantitative calculations, it seems necessary to recognize that the asymptotic term does not work for all states. One may define a “generalized Krichevskii parameter”, which is the quantity defined by eq 1 except that the limit to the critical point is not taken. This can be related directly (via the isothermal compressibility of the solvent) to the solute’s infinite-dilution partial molar volume. It is a useful variable for correlations because, in contrast to the partial molar volume itself, it is a smoothly varying property without a critical divergence. O’Connell et al. (1996) used this approach, correlating the generalized Krichevskii parameter with a simple functional form. They were able to correlate quantitatively experimental partial molar volumes (Hnedkovsky et al., 1996) for several gaseous solutes and H3BO3 in water at high temperatures and pressures; the results for aqueous CO2 are shown in Figure 5. It remains to be seen whether this correlation can be expanded into a framework that can be used for all infinite-dilution properties. Application to Supercritical Solubility of Solids Thus far we have discussed applications where the solute is a gas dissolved in a less volatile solvent. However, the same relationships apply regardless of the relative volatilities. For instance, eq 3 also applies to the Henry’s constant of naphthalene in CO2. Therefore, these relationships may be applied to solid solubility in supercritical fluids. Harvey (1990) worked with an effective Henry’s constant obtained by relaxing the infinite-dilution condition in eq 2:

keff H ) f2/y2

(12)

Figure 6. Linear relationship of T ln(keff H /f1) versus F1 for naphthalene in carbon dioxide. The dashed line in this and subsequent figures is only a guide to the eye and is not the result of any optimized fit.

where we have switched from x2 to y2 for the solute mole fraction in keeping with common practice for these systems. This finite-concentration definition is necessary because, in a supercritical solubility problem, the solute mole fraction and fugacity never reach zero but rather are fixed by the fugacity of the equilibrium solid phase. This fixed fugacity produces (assuming that the solid is incompressible and that the pure-solute vapor pressure is low enough for its vapor to be ideal) the following expression for the effective Henry’s constant: s s keff H ) p2 exp[v2(p - p2)/RT]/y2

(13)

where v2 is the molar volume of the solid solute and ps2 is its vapor pressure. It was then hypothesized that, at least for solutes that were not too soluble so that the solution was near infinite dilution, eq 3 would apply for the effective Henry’s constant:

T ln(keff H /f1) ) A + B(F1 - Fc,1)

(14)

This assumption was later shown (Levelt Sengers, 1991b) to be reasonable as long as the mixture critical endpoint is close to the solvent critical point (which will usually be true if the solubility is small). When, as suggested by eq 14, T ln(keff H /f1) was plotted versus solvent density, data for supercritical solubility at various temperatures tended to collapse onto a single curve. Furthermore, this curve was linear over a substantial range of solvent densities. Figure 6 shows a typical case for naphthalene in CO2; the data are from Tsekhanskaya et al. (1964). Two features of Figure 6 merit comment. First, a few points deviate from the line at conditions very close to the critical point of CO2 (approximately 31 °C and 10.6 mol/L). This is not unexpected, since it is in this region that the finite concentration of the solute (the solubility of naphthalene at these conditions is on the order of 1 mol %) would induce a relatively large change in the density of the mixture. This means that a plot using the pure-solvent density such as Figure 6 would not represent the true state of the mixture. Small errors

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in experimental measurements, particularly the pressure, could also produce large deviations in this region. The second feature is the upturn from the straight line at a density approximately 1.5 times the critical value. While this may partly be due to the growing inadequacy of the infinite-dilution assumption at higher solvent densities, it is more likely due to departure from the asymptotic behavior; we recall that a density of 1.5Fc,1 is also approximately where the crossover to nonasymptotic behavior takes place for gaseous solutes in liquid water. However, even in the nonasymptotic regime, the data generally fall on a single curve when plotted on these coordinates. Unfortunately, eq 14 is not adequate for practical use in supercritical solubility applications. The most important range for application is the high-density regime (typically at least twice the solvent’s critical density) that would characterize the high-pressure side of an extraction process. The linear relationship fails in this range. However, the fact that the data still collapse onto a single curve suggests that the addition of an empirical high-density correction to eq 14 might produce a useful correlating equation. Another area for improvement might be the lowdensity behavior for supercritical solubilities; eq 14 does not go to the correct limit as the solvent density approaches zero. In practice, however, this part of the phase diagram is less important because the solubilities are so much smaller than they are on the high-pressure side that they can be considered negligible. Kumar and Johnston (1988) derived an equation similar to eq 14 in which ln(y2) was linear in ln(F1). Their derivation was limited to the solvent’s critical isotherm. As mentioned previously, linearity in the logarithm of the density is asymptotically equivalent to linearity in the density itself, so their result (which also differs from eq 14 in some details, such as the Poynting correction, that are relatively unimportant in the near-critical limit) is asymptotically consistent with eq 14. Application to Supercritical Fluid Chromatography

(ti - t0) t0

(15)

where ti is the retention time of solute i and t0 is the retention time of a solute that is not retained in the stationary phase. Under certain plausible assumptions, the capacity factor can be related to the solute’s solubility yi in the supercritical mobile phase:

k′i )

C(T) yiF1

small, that the critical point is not approached too closely within the column, and that the solute’s activity in the stationary phase is not affected (except for a Poynting correction which is assumed to be the same as that for the pure solid) by the pressure of the mobile phase. It is then straightforward to substitute yi from eq 16 into eq 13 (defining an effective Henry’s constant) and write an analogue of eq 14:

ln

The analysis of supercritical fluid chromatography (SFC) of solid solutes is related to that for the supercritical solubility of solids. The central quantity in SFC is the capacity factor k′, defined by

k′i )

Figure 7. Capacity factor relationship (eq 17) for naphthalene in supercritical CO2 at 318.2 K according to the data of Bartle et al. (1990b).

(16)

where C(T) is a constant depending only on temperature for a particular experimental arrangement. The conditions under which eq 16 is valid have been discussed elsewhere (see Bartle et al., 1990a). The most important assumptions are that phase equilibrium is attained throughout the column, that the solute is sufficiently near infinite dilution, that the column pressure drop is

(

)

k′2F1 exp[v2(p - ps2)/RT] ) A′(T) + B′(T)(F1 - Fc,1) f1 (17)

Here we have written the slope and intercept of the linear relationship as A′(T) and B′(T) to emphasize that they (in contrast to the constants in eq 14) contain temperature dependence both through the apparatusdependent factor C(T) and the solute vapor pressure ps2. This makes the relationship less general than eq 14, but it can still be used for analyzing SFC data on individual isotherms. Figure 7 shows a test of eq 17 for capacity factors reported by Bartle et al. (1990b) for naphthalene in CO2 at 318.2 K. The properties of CO2 were obtained from the equation of Span and Wagner (1996). While we might expect linearity over a range comparable to that in Figure 6 for the solubility, the data in Figure 7 show no discernible linear region. Similar results were obtained for other systems. This is somewhat surprising because eq 17 was produced directly from eq 14, which did give linear behavior for the solubility in these systems. We must remember, however, that the deviation in linearity from these asymptotic expressions is a function of the relative contributions of higher-order terms which will vary from problem to problem and system to system. Equally important, the choice of the linear variable (density,

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Figure 8. Capacity factor relationship (eq 18) for naphthalene in supercritical CO2 at 318.2 K according to the data of Bartle et al. (1990b).

Figure 9. Capacity factor relationship (eq 18) for phenanthrene and naphthalene in supercritical CO2 at 35 °C according to the data of Shim and Johnston (1991).

volume, logarithm of density, etc.) can have a major impact on the range in which the asymptotic behavior is manifested. This last factor leads us to consider a different, but asymptotically equivalent, choice for the linear variable, namely, the logarithm of the density. If we use the logarithm of the density, the resulting equation is

(

ln

)

k′2 exp[v2(p - ps2)/RT] ) A′(T) + B′(T) ln(Fr,1) f1 (18)

where the factor of ln(F1) on the left side of eq 17 has been absorbed into the right side of eq 18 and we have used the reduced density Fr,1 ) F1/Fc,1. Before evaluating the performance of eq 18, we note that a linear variation of ln(k′) with ln(F1) was also proposed by Chimowitz and Kelley (1989). Their expression did not have the pure-solvent fugacity and Poynting correction that appear in eq 18; the latter difference appears to be due to a term that was dropped partway through their derivation. However, the capacity factor is the dominant term (varying fastest with density) on the left side of eq 18 at most conditions, so they captured the most important part of the behavior. Their expression was based on that of Kumar and Johnston (1988) for the supercritical solubility and was therefore valid only for the critical isotherm. Figure 8 shows a plot according to eq 18 of the same data used for Figure 7 (the dashed lines on this and subsequent figures are included only for visual assessment of the linearity of the data and are not the result of any optimized fit). The linearity is striking. Figure 9 shows the capacity-factor data of Shim and Johnston (1991) for naphthalene and phenanthrene in CO2 at 35 °C. Figure 10 shows the data of Kautz (1997) for biphenyl in CO2 at temperatures near 329.6 K. In all

Figure 10. Capacity factor relationship (eq 18) for biphenyl in supercritical CO2 at 329.6 K according to the data of Kautz (1997).

of these cases, the data fall onto a line according to eq 18, with an upturn from the line at the highest densities. This upturn begins at densities near 1.5Fc,1, which is similar to the behavior observed for supercritical solubilities. We emphasize that there is no fundamental reason eq 18 should work better than eq 17; they are equivalent in the asymptotic limit at the solvent’s critical point. Somehow, probably fortuitously, the logarithm of the

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density provides a much better linear correlating variable for this particular property. For practical application, we can accept that good fortune and use eq 18 to correlate capacity factors. One further caveat is necessary with regard to SFC. The asymptotic correctness of eqs 17 and 18 depends on the correctness of the step used to go from eq 14 to eq 17, namely, eq 16 relating the capacity factor to the solubility. While most of the assumptions that went into eq 16 are reasonable, one can raise serious questions about assuming that the solute fugacity in the stationary phase is not affected (apart from a Poynting correction for the pressure) by the mobile phase. If increasing pressure causes the mobile phase to dissolve significantly in the stationary phase, that will affect the ability of the stationary phase to dissolve the solid solute. The stationary phases in these experiments often contain long hydrocarbon chains, and the solubility of CO2 in hydrocarbons is significant. Swelling of the stationary phase may also be important. These issues have been explored in more detail by Roth (1992). Discussion Much of dilute-solution thermodynamics near the solvent’s critical point is governed by a single thermodynamic derivative, the Krichevskii parameter. Simple relationships have been derived in which this parameter determines the behavior of solute properties in the nearcritical region. These relationships have proved to be useful for describing Henry’s constants and K-factors over a wide range of conditions. These relationships have also been extended and applied to describe apparent molar properties at infinite dilution for solutes in water and to describe the solubility of solids in supercritical fluids. While the application to apparent molar properties of solutes was not quantitatively successful, using the Krichevskii parameter (in generalized form) to separate solute-solvent effects from the diverging solvent properties is a fundamentally sound and promising approach. In the case of supercritical solubilities of solids, the method is capable of collapsing all the data for a system onto a single curve. Since, however, the linearity of this curve is not maintained in the high-density region, more work is required before the method can be put to direct engineering use. An extension of the supercritical solubility relationship to capacity factors in supercritical fluid chromatography has been derived. With the proper choice of variable for the linear expression, data for capacity factors can also be successfully described. Acknowledgment Dr. J. M. H. Levelt Sengers played an important role in much of the research reviewed here and also provided helpful comments on the manuscript. Prof. G. M. Schneider forwarded some SFC data taken in his laboratory. Literature Cited Alvarez, J.; Ferna´ndez-Prini, R. A semiempirical procedure to describe the thermodynamics of dissolution of nonpolar gases in water. Fluid Phase Equilib. 1991, 66, 309-326. Alvarez, J.; Corti, H. R.; Ferna´ndez-Prini, R.; Japas, M. L. Distribution of solutes between coexisting steam and water. Geochim. Cosmochim. Acta 1994, 58, 2789-2798.

Bartle, K. D.; Clifford, A. A.; Jafar, S. A. Relationship between Retention of a Solid Solute in Liquid and Supercritical Fluid Chromatography and its Solubility in the Mobile Phase. J. Chem. Soc., Faraday Trans. 1990a, 86, 855-860. Bartle, K. D.; Clifford, A. A.; Jafar, S. A. Measurement of Solubility in Supercritical Fluids Using Chromatographic Retention: the Solubility of Fluorene, Phenanthrene, and Pyrene in Carbon Dioxide. J. Chem. Eng. Data 1990b, 35, 355-360. Chang, R. F.; Morrison, G.; Levelt Sengers, J. M. H. The Critical Dilemma of Dilute Mixtures. J. Phys. Chem. 1984, 88, 33893391. Chialvo, A. A.; Kalyuzhnyi, Yu. V.; Cummings, P. T. Solvation Thermodynamics of Gas Solubility at Sub- and Near-Critical Conditions. AIChE J. 1996, 42, 571-584. Chimowitz, E. H.; Kelley, F. D. A New Representation for Retention Time in Supercritical Fluid Chromatography. J. Supercrit. Fluids 1989, 2, 106-110. Crovetto, R. Evaluation of Solubility Data of the System CO2H2O from 273 K to the Critical Point of Water. J. Phys. Chem. Ref. Data 1991, 20, 575-589. Ferna´ndez Prini, R.; Crovetto, R. Evaluation of Data on Solubility of Simple Apolar Gases in Light and Heavy Water at High Temperature. J. Phys. Chem. Ref. Data 1989, 18, 1231-1243. Gallagher, J. S.; Levelt Sengers, J. M. H.; Abdulagatov, I. M.; Watson, J. T. R.; Fenghour, A. Thermodynamic Properties of Homogeneous Mixtures of Nitrogen and Water from 440 to 1000 K, up to 100 MPa and 0.8 mole fraction N2. NIST Technical Note 1404; U.S. Government Printing Office: Washington, DC, 1993. Guillot, B.; Guissani, Y. Temperature dependence of the solubility of non-polar gases in liquids. Mol. Phys. 1993, 79, 53-75. Harvey, A. H. Supercritical Solubility of Solids from Near-Critical Dilute-Mixture Theory. J. Phys. Chem. 1990, 94, 8403-8406. Harvey, A. H. Semiempirical Correlation for Henry’s Constants over Large Temperature Ranges. AIChE J. 1996, 42, 14911494. Harvey, A. H.; Levelt Sengers, J. M. H. Correlation of Aqueous Henry’s Constants from 0 °C to the Critical Point. AIChE J. 1990, 36, 539-546. Harvey, A. H.; Crovetto, R.; Levelt Sengers, J. M. H. Limiting vs. Apparent Critical Behavior of Henry’s Constants and K Factors. AIChE J. 1990, 36, 1901-1904. Harvey, A. H.; Levelt Sengers, J. M. H.; Tanger, J. C., IV. Unified Description of Infinite-Dilution Thermodynamic Properties for Aqueous Solutes. J. Phys. Chem. 1991, 95, 932-937. Harvey, A. H.; Peskin, A. P.; Klein, S. A. NIST/ASME Steam Properties; NIST Standard Reference Database 10, Version 2.1; Standard Reference Data Program; NIST: Gaithersburg, MD, 1997. Hnedkovsky, L.; Wood, R. H.; Majer, V. Volumes of aqueous solutions of CH4, CO2, H2S, and NH3 at temperatures from 298.15 to 705 K and pressures to 35 MPa. J. Chem. Thermodyn. 1996, 28, 125-142. Japas, M. L.; Levelt Sengers, J. M. H. Gas Solubility and Henry’s Law Near the Solvent’s Critical Point. AIChE J. 1989, 35, 705713. Japas, M. L.; Fernandez-Prini, R.; Horita, J.; Wesolowski, D. J. Fractioning of Isotopic Species between Coexisting Liquid and Vapor Water: Complete Temperature Range, Including the Asymptotic Critical Behavior. J. Phys. Chem. 1995, 99, 51715175. Kautz, C. B. Fluidchromatographische Untersuchungen (SFC) an organischen Substanzen mit u¨ berkritischem Kohlendioxid; Cuvillier Verlag: Go¨ttingen, Germany, 1997. Krichevskii, I. R. Thermodynamics of Critical Phenomena in Infinitely Dilute Binary Solutions. Russ. J. Phys. Chem. 1967, 41, 1332-1338. Kumar, S. K.; Johnston, K. P. Modelling the Solubility of Solids in Supercritical Fluids with Density as the Independent Variable. J. Supercrit. Fluids 1988, 1, 15-22. Levelt Sengers, J. M. H. Thermodynamics of Solutions Near the Solvent’s Critical Point. In Supercritical Fluid Technology: Reviews in Modern Theory and Applications; Bruno, T. J., Ely, J. F., Eds.; CRC Press: Boca Raton, FL, 1991a; Chapter 1. Levelt Sengers, J. M. H. Solubility Near the Solvent’s Critical Point. J. Supercrit. Fluids 1991b, 4, 215-222. Levelt Sengers, J. M. H.; Harvey, A. H.; Crovetto, R.; Gallagher, J. S. Standard States, Reference States and Finite-Concentra-

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Received for review November 13, 1997 Revised manuscript received March 2, 1998 Accepted March 2, 1998 IE970800R