1506
Ind. Eng. Chem. Res. 1988, 27, 1506-1510
Adaptation of the Flory-Huggins Theory for Modeling Supercritical Solubilities of Solids Anatoly Kramer and George Thodos* Chemical Engineering Department, Northwestern University, Evanston, Illinois 60208
The Flory-Huggins equation for the activity coefficient of a solute in an infinitely dilute solution has been extended in application to model the solubility of solids in supercritical fluids. This equation utilizes for a system a single adjustable parameter which has been found to be strongly dependent on the solubility parameter of the supercritical solvent. The insufficient analysis for the resolution of the total solubility parameter of solids into their dispersion, polar, and hydrogen bonding contributions has so far hindered the correlation of this parameter in a generalized manner. However, the present approach for modeling this single adjustable parameter with the solubility parameter of the supercritical solvent overcomes this limitation and also the frequently unavailable data associated with the solid solute such as its vapor pressure and other related physical properties. Supercritical gas extraction continues to receive considerable attention for the separation and recovery of highly nonvolatile and sensitive organic compounds. This method of separation is based on the ability of compressed gases to dissolve within their matrix these chemical constituents beyond their expected volatility and thus realize their separation upon altering the temperature and pressure conditions of the system. From theoretical arguments, it is possible to show that the volatility enhancement of these heavy organic compounds becomes greatest when the temperature of extraction is carried out in the vicinity of the critical temperature of the solvent gas and the pressure is maintained above its critical pressure. Present day applications involve the use of readily accessible supercritical solvents such as carbon dioxide. For this particular solvent, the temperature must be kept slightly above its critical temperature (T, = 304.2 K) and the pressure must exceed its critical pressure (P, = 73.82 bar). In this context, it is worth noting that supercritical carbon dioxide is presently employed to selectively decaffeinate green coffee beans as pointed out by Zosel(1980) and to extract nicotine from tobacco leaves and to remove the active ingredients from such spices as black pepper, nutmeg, and chilies as reported by Hubert and Vitzthum (1980). The use of carbon dioxide is being currently investigated for the extraction of oils from soybeans, corn, and cotton seeds. Furthermore, the pharmaceutical industry is presently exploring the removal of alkaloids and essential oils from plants for the recovery of such medicinals as morphine, codeine, atropine, and others as pointed out by Stahl et al. (1980). In this connection, Krukonis et al. (1979) indicate that they were successful in extracting antineoplastic agents from plant materials. Many other important applications of supercritical extraction are discussed in recent publications by Squires and Paulaitis (1987) and also by McHugh and Krukonis (1986). The design and operation of a supercritical extraction process requires an estimation of the solubility of a solute in a supercritical solvent. In this regard, equation of state approaches prove to be useful to satisfy this need, but only in a qualitative manner. Following classical thermodynamic arguments, the solubility of the heavy component in a solvent existing at supercritical conditions can be calculated from the relationship
where C$2v represents the fugacity coefficient of the solute
present in the supercritical fluid. The calculation of &" requires the application of an equation of state consistent with the rigorous thermodynamic relationship
Haselow et al. (1986) investigate 9 generalized equations of state and evaluate their ability to describe supercritical solubilities for 31 different binary systems. From their study, it can be concluded that the Redlich-Kwong equation of state produces the best overall results with an average deviation of 34%. These calculations require the use of mixing rules which involve binary adjustable interaction parameters. These parameters cannot be predicted or interpreted from pure component values but can be calculated only from existing experimental data. In order to obtain a better fit, binary interaction parameters are calculated independently at different temperatures, despite the fact that the typical experimental temperature range associated with supercritical extraction is quite narrow. However, using an alternative method, by treating the supercritical fluid as an expanded liquid, Mackay and Paulaitis (1979) utilize an equation of state approach to calculate the partial molar volume of the heavy component present in the vapor phase at infinite dilution. Following this argument, they show that the binary interaction applied in mixing rule parameters becomes essentially independent of temperature. In spite of this advantage, these binary interaction coefficients cannot yet be predicted in a generalized manner. Using a form of regular solution-solubility parameter theory, Giddings et al. (1969) propose a universal relationship for solubility enhancement which was hypothesized to be independent of the nature of the solute, solvent, and temperature. A modified version of this approach was developed by Ziger and Eckert (1983) for the prediction of solid-supercritical fluid phase equilibria. In order to improve the modeling of liquid and solid solubilities in dense supercritical fluids, the present study was undertaken to investigate the possibility of applying the FloryHuggins theory for athermal solutions associated with supercritical fluids (Flory, 1941; Huggins, 1941). Thermodynamic Treatment Using basic equilibrium arguments, it is required that the fugacity of the solid solute and that existing within the supercritical fluid must be equal,
f2'
= fzs
0888-5885/88/2627-1506$01.50/0 0 1988 American Chemical Society
(3)
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1507 under the existing temperature and pressure conditions of the system. Using this argument and treating the supercompressed fluid as an expanded liquid, it can be shown that Y2 =
(5)
(4)
7 72
where T~ = 6: + dh2 and b is a correction factor as pointed out by Hanson (1967). For solvents in the proximity of their supercritical state, the polar and hydrogen-bonding contributions can be neglected and therefore r approaches zero. However, both the polar and hydrogen-bonding contributions for the solute should be retained, thus simplifying eq 9 to the expression
The development for calculating the ratio, f2"/fio1, can be found elsewhere (Prausnitz, 1969) and is given as follows: where p12 accounts for the binary molecular interaction between the solvent and solute. If we assume that 6,, = 6dl, the solubility parameter for the solvent can be calculated by using as a first order of approximation the relationship proposed by Giddings et al. (1968): where Ahzfus = the molar heat of fusion, J/mol; AC, = constant pressure molar heat capacity change, J/(mol K); and Av2 = molar volume change in the course of fusion, cm3/mol. The last three terms of eq 5 contribute only in a small way to the overall result and may be neglected. Therefore, the pure component fugacity ratio, f2"/fZo1, can be calculated from basic physical properties associated with the solute. Using a statistical mechanics approach, Flory (1941) and Huggins (1941) independently derived an expression for the entropy of mixing of athermal solutions in which the molecular dimensions between solute and solvent are considerably different. Solutions with zero enthalpy of mixing are termed athermal solutions because their components mix with no liberation or adsorptim of heat. This is a good first approximation for the dissolution of solids in dense gases. Under these conditions, the Flory-Huggins approach defines the activity coefficient of the solute in an infinitely dilute solution as u2
In 72m = -(titl
RT
-
u2 u2 + 1- + In u1
(6)
u1
where 6,, and 6, represent the total solubility parameter for the solvent and solute, respectively. Working with infinitely dilute solutions of nonpolar solutes in a large variety of polar solvents, Weimer and Prausnitz (1965) consider the total solubility parameter to consist of nonpolar and polar contributions. Under these conditions, their expression for eq 6 takes the form u2
In T~~ = -[(Al
RT
-
u2 u2 x2)2 + T~~ - 2a12]+ 1 - + In u1
01
(7) where r2 = 6,2 - X2 in which X is the nonpolar contribution to 6,, the total solubility parameter. In order to account for the dipole moment and hydrogen-bonding contributions, Hansen (1967) proposes that the total solubility parameter is three dimensional and defines it as follows: 6: = 6d2
+ 6p2 + 6h2
(8)
where 6d is the dispersion molar attraction and equal to X and 6, and 6h are the polar and hydrogen-bonding contributions, respectively. All these three contributions are consistent with the definition of the total solubility parameter given by eq 8. Thus, Hansen modifies eq 6 as follows:
where 6d1 is in ( c a l / ~ m ~ P, ) ~isf ~in, atmospheres, pR is the reduced density of the fluid, and pR1 is the reduced density of the liquid state which is normally taken as 2.66. In eq 11the "state effect" may be identified with p R / p m , while the factor 1.25P,'f2 is recognized as the "chemical effect". The experimental density measurements of Douslin and Harrison (1976) for ethylene were used to obtain the needed reduced densities for the three binary systems associated with this solvent. On the other hand, for the other three binary carbon dioxide systems, the comprehensive reduced density compilation of Kennedy and Thodos (1960) for carbon dioxide was consulted. The dispersion part of the solubility parameter, 6dz, for naphthalene and benzoic acid was taken from the tabulated work of Barton (1983), while that for phenanthrene was approximated by using the group contribution method of Koenhen and Smolders (1975) and the homomorph concept of Bondi and Simkin (1956, 1957). It is of interest to note that Thomas and Eckert (1984) also utilize regular solution theory and a modified separation of cohesive energy density model (MOSCED) to predict y -, the limiting activity coefficient at infinite dilution. In their treatment, they develop an expression analogous to eq 10 which is specific to liquid materials and do not extend its application to the solubility of solids in solvents existing at supercritical conditions. Treatment of Available Literature Data To test the validity of eq 10, data for the solubility of solid solutes in supercritical solvents available in the literature for six binary systems were examined. These systems contain naphthalene, phenanthrene, and benzoic acid as the solutes dissolved in the supercritical solvents, ethylene and carbon dioxide. In this analysis, both of these solvents were treated as nonpolar. Therefore, the binaries of naphthalene and phenanthrene represent heavy aromatic hydrocarbons which are essentially nonpolar, while the benzoic acid systems include, in addition, significant polar contributions. These six binary systems, the temperatures investigated, and the sources of data are given in Table I. The critical temperatures of ethylene and carbon dioxide are 282.35 and 304.20 K, respectively. The experimental data obtained from these sources for each of these six binary systems were used with eq 10 to obtain the interaction parameter, p12. Because of the intrinsic development associated with the solubility parameter, 6,, which is fundamentally a liquid state property, these solid solutes were treated as subcooled liquids and
1508 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 Naphthalene- Ethylene
Table I. Binary Systems Investigated in This Study svstem temD. K source of data naphthalene-ethylene 285.15-338.15 Diepen and Scheffer (1948, 1953); Tsekhanskaya et al. (1964); Masuoka and Yorizane (1982) naphthalene-carbon dioxide 308.15-328.15 Tsekhanskaya et al. (1964) phenanthrene-ethylene 318-338 Kurnik et al. (1981); Johnston (1981) phenanthrene-carbon dioxide 318-338 Kurnik et al. (1981) benzoic acid-ethylene 318-338 Kurnik et al. (1981) benzoic acid-carbon dioxide 318-338 Kurnik et al. (1981)
Diepen ond Scheffer 11948: V 285 15 K
O t
401
Table 11. Molar Volumes at 298.15 K (cms/mol) naphthalene phenanthrene benzoic acid
Barton (1983) 123 158 100
this study 126.1 184.9 97.0
the correspondingthermodynamic and physical properties needed in these calculations were extrapolated from the liquid state region. The molar volume for the subcooled liquid state of naphthalene at 25 "C was determined from the compilation of Kudchadkar et al. (19781, while those for phenanthrene and benzoic acid at 25 "C were calculated by using the group contribution approach suggested by Fedors (1974). In this regard, it is of interest to note that the molar volumes for subcooled liquids applied in these calculations were somewhat different from those reported by Barton (1983). These differences are given in Table 11. The temperature dependence of the dispersion portion of the solubility parameter for all these solutes was calculated as a first-order approximation by using the expression given by Hildebrand and Scott (1950) for the total solubility parameter of nonpolar liquids: d In 6JdT = - 1 . 2 5 ~ ~ (12) where a is the coefficient of thermal expansion. The use of eq 12 for the dispersion contribution is also recommended by Hansen and Beerbower (1971). As a first-order approximation, the coefficient of thermal expansion for liquid naphthalene was taken as a = 0.0007 K-l as deduced from the work of Vargaftik (1975). Because of the lack of direct measurements, this value has also been applied to both phenanthrene and benzoic acid. Values for the calculated binary interaction parameter, P12, resulting from eq 10 were found to be strongly dependent on the total solubility parameter of the solvent. For the naphthalene-ethylene and naphthalene-carbon dioxide systems, this dependence is presented in Figure 1. For the naphthalene-ethylene system, the data reported by Diepen and Scheffer (1948, 1953) and Tsekhanskaya et al. (1964) show that the binary interaction parameter, P12, is essentially independent of temperature in the high-pressure region. On the other hand, for the data of these investigators for densities corresponding to solubility parameters, 6 < 4.0, these parameters exhibit a significant temperature dependence. In this context, the data of Masuoka and Yorizane (1982) obtained in the low-pressure region exhibit an essentially temperatureindependent behavior for 308.15, 323.15, and 338.15 K. After a careful examination of this disparity, Masuoka and Yorizane (1982) conclude that the earlier data of Diepen and Scheffer (1948, 1953) and Tsekhanskaya et al. (1964) in this low-pressure region are too high. It is worth noting that the experimental approach of Masuoka and Yorizane (1982) was designed for pressures ranging from 8.0 to 62.9 bar, and as such their procedure and resulting data should be sensitive in this pressure range. On the other hand, the
Naphthalene-Carbon Dioxide Tiehhonshwo, Iomlev ond Mushhino 119641 030815K 031815K O 3 2 8 1 5 K
Biz
0
1
2
3
4
6 , , Solubility
5
6
7
8
9
l
C
Parameter of Salvent
Figure 1. Dependence of interaction parameter, PI*, on solubility parameter of solvent for the binary systems naphthalene-ethylene and naphthalene-carbon dioxide.
works of Diepen and Scheffer (1948,1953) and Tsekhanskaya et al. (1964) were specific to higher pressures ranging from 38.5 to 304 bar. In view of these arguments, the data of Masuoka and Yorizane (1982) are more representative in the low-pressure region and exhibit a temperature-independent behavior consistent with that found to exist in the higher pressure region resulting from the data of Diepen and Scheffer (1948,1953)and Tsekhanskaya et al. (1964). In this context, it is worth noting that the amount of solute dissolved in the same volume of solvent in the static type of systems used by these earlier investigators can be several orders of magnitude lower than is presented in the higher-pressure region. These measurements require a higher sensitivity that was not always attainable with this earlier work. In addition, Johnston (1981) points out a weakness in the static-type measurements used by Diepen and Scheffer (1948,1953) and Tsekhanskaya et al. (1964) because of the nonequilibrium conditions that exist during the depressurization of the vessel which contains the sample, and therefore this condition could significantly change the final measurements. On the other hand, the measurements taken by Masuoka and Yorizane (1982), who utilized a continuous flow type system, were always carried out under equilibrium conditions. The naphthalene-carbon dioxide data of Tsekhanskaya et al. (1964) follow essentially the same temperature-independent patern that was exhibited with the naphthalene-ethylene system. Their data at 328.15 K in the high-pressure region begin to deviate. This may be due to the liquefaction of naphthalene encountered at these temperature and pressure conditions. Values of pl2 resulting from the data of Kurnik et al. (1981) and Johnston (1981) for the phenanthrene-ethylene system are related to the solvent solubility parameter in Figure 2. This figure also includes the phenanthrenecarbon dioxide system. The data of Kurnik et al. (1981) for the benzoic acid-ethylene and benzoic acid-carbon dioxide binaries, shown in Figure 3, also produce temperature-independent relationships. This pattern represents an interesting behavior where the interaction pa-
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1509 solubility of solid solutes in supercritical gases, if experimental data are available for a single isotherm. The accuracy for calculating the dispersion, polar, and hydrogen-bonding contributions of the solubility parameters of solids, which have been treated as subcooled liquids, presents difficulties for the development of a generalized approach for predicting Pl2, the binary interaction parameter. Therefore, additional studies should be made in this direction to resolve these individual contributions of pure solids. Several techniques suggested by Hansen and Beerbower (1971) can be applied to achieve this objective.
Phenanthrene-Ethylene Kurnik.Hellc ond Reid 119811 0 318 K 0328 K 0 3 3 8 K Johnston 119811 +31a15K
0
20
Phenanthrene- Corbon Dioxide Kumik, Holk and Reid (19811 0 3 1 8 ~0 x 8 K 0 3 3 8 ~
0
B. 0
t
0
-I0 -20
1
0
1
I
1
2
1
3
1
4
8, , Solubility
1
5
1
6
1
7
1
8
1
9
1
10
Parameter of Solvent
Figure 2. Dependence of interaction parameter, &, on solubility parameter of solvent for the binary systems phenanthrene-ethylene and phenanthrene-carbon dioxide. Benzoic Acid-Ethylene Kurnik, Hello end Reid 119811 0 318 K 0 3 2 8 K 0 3 3 8 K
Benzoic Acid-Corbon Dioxide 2o
t
Kurnlk, Holio ond Reid 11981) 0318K o 3 2 8 K 0 3 3 8 K
8, , Solubility Parameter of Solvent Figure 3. Dependence of interaction parameter, Plz, on solubility parameter of solvent for the binary systems benzoic acid-ethylene and benzoic acid-carbon dioxide.
rameter does not change significantly with the solubility parameter of the solvent. Conclusions The results of this study point to the existence of a single binary interaction parameter which when incorporated into the Flory-Huggins equation produces the activity coefficient of the solute at infinite dilution. This parameter has been found to be essentially temperature independent and has been shown to be related to the solubility parameter of the solvent of the binary system. This pattern of behavior was observed to exist for the six binary systems examined in this study at supercritical state conditions of the solvents. A comparable equation of state approach to calculate the solubility of solid solutes in dense supercritical fluids requires adjustable binary interaction parameters that are strongly temperature dependent and which cannot be correlated with any pure component physical properties. In addition, the approach presented in this study does not require the use of solute vapor pressure data which quite often are not accessible. The present approach can serve as a fiist-order approximation for calculating the
Nomenclature b = correction factor, eq 10 C = heat capacity, cal/(mol K) f = fugacity, bar Ahfus= molar heat of fusion, cal/mol n = number of moles P = pressure, bar R = gas constant, 1.9872 cal/(mol K) and 83.144 (cm3 bar)/(mol K) T = temperature, K y = mole fraction of component in vapor phase u = molar volume, cm3/mol z = compressibility factor Greek Symbols a = coefficient of thermal expansion, K-’ fl = interaction parameter, ( c a l / ~ m ~ ) ’ / ~ y = activity coefficient 6 = solubility parameter, ( c a l / ~ m ~ ) ’ / ~ A = nonpolar contribution, ( c a l / ~ m ~ ) ’ / ~ p = density, g/cm3 7 = (6,2 + 6h2)’/* 4 = fugacity coefficient = interaction coefficient Subscripts 1 = solvent 2 = solute c = critical d = dispersion h = hydrogen bonding i, j = component designation 1 = liquid phase m = triple point p = constant pressure p = polar R = reduced t = total Superscripts f = fluid o = pure component s = solid sat = saturated state v = vapor phase m = infinite dilution
Literature Cited Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC: Boca Raton, FL, 1983. Bondi, A.; Simkin, D. J. “The Hydrogen-Bond Contribution to the Heat of Vaporization of Aliphatic Alcohols”. J. Chem. Phys. 1956, 25, 1073-1074. Bondi, A.; Simkin, D. J. “Heats of Vaporization of Hydrogen-Bonded Substances”. AIChE J . 1957, 3, 473-479. Diepen, G. A. M.; Scheffer, F. E. C . “The Solubility of Naphthalene in Supercritical Ethylene”. J . Am. Chem. SOC. 1948, 70, 4085-4089. Diepen, G. A. M.; Scheffer, F. E. C. “The Solubility of Naphthalene in Supercritical Ethylene. 11”. J. Phys. Chem. 1953,57, 575-577.
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Douslin, D. R.; Harrison, R. H. “Pressure, Volume, Temperature Relations of Ethylene”. J . Chem. Thermodyn. 1976,8, 301-330. Fedors, R. F. “A Method for Estimating Both the Solubility Parameters and Molar Volumes of Liquids”. Polym. Eng. Sci. 1974, 14, 472. Flory, P. J. ”Thermodynamics of High-Polymer Solutions”. J. Chem. Phys. 1941, 9, 66C-661. Giddings, J. C.; Myers, M. N.; McLaren, L.; Keller, R. A. “High Pressure Gas Chromatography of Nonvolatile Species”. Science (Washington, D.C.) 1968, 162, 67-73. Giddings, J. C.; Myers, M. N.; King, J. W. “Dense Gas Chromatography at Pressures to 2000 Atmospheres”. J . Chromatogr. Sci. 1969, 7, 276-283. Hansen, C. M. “Three Dimensional Solubility Parameter and Solvent Diffusion Coefficient Importance in Surface Coating Formulation”. Doctoral Dissertation, Danish Technical Press, Copenhagen, 1967. Hansen, C. M.; Beerbower, A. “Solubility Parameters”. In “KirkOthmer Encyclopedia of Chemical Technology”, Suppl. Vol. 2nd ed.; Interscience: New York, 1971; pp 889-910. Haselow, J. S.; Han, S. J.; Greenkorn, R. A.; Chao, K. C. “Equation of State for Supercritical Extraction”. In “Equations of State: Theories and Applications”; Chao, K. C., Robinson, R. L., Jr., Eds.; American Chemical Society: Washington, DC, 1986. Hildebrand, J. H.; Scott, R. T. “The Solubility of Nonelectrolytes”; Reinhold: New York, 1950. Hubert, P.; Vitzthum, 0. G. “Fluid Extraction of Hops, Spices and Tobacco with Supercritical Gases”. In “Extraction with Supercritical Gases”; Schneider, G. M., Stahl, E., Wilke, G., Eds.; Verlag Chemie: Deerfield Beach, FL, 1980. Huggins, M. L. “Solutions of Long Chain Compounds”. J . Chem. Phys. 1941, 9, 440. Johnston, K. P . “Analytical Perturbed Hard Sphere Models Based on Solubility and Volumetric Studies of Organic Solids Interacting with Supercritical Fluids”. Ph.D. Dissertation, University of 11linois Champaign-Urbana, Urbana, IL, 1981. Kennedy, J. T., Jr.; Thodos, G. “Reduced Density Correlation for Carbon Dioxide: Gaseous and Liquid States”. J . Chem. Eng. Data 1960, 5, 293-297. Koenhen, D. M.; Smolders, C. A. “The Determination of Solubility Parameters of Solvents and Polymers by Means of Correlations with Other Physical Quantities”. J . Appl. Polym. Sci. 1975, 19, 1163-1179. Krukonis, V. J.; Branfman, A. R.; Broome, M. G. “Supercritical Fluid Extraction of Plant Materials Containing Chemotherapeutic Drugs”. Paper presented at the American Institute of Chemical
Engineers Meeting, Boston, MA, 1979. Kudchadkar, A. P.; Kudchadkar, S. A.; Wilhoit, R. C. ”Naphthalene”. American Petroleum Institute Publication 707, Oct 1978; American Petroleum Institute, New York. Kurnik, R. T.; Holla, S. J.; Reid, R. C. “Solubility of Solids in Supercritical Carbon Dioxide and Ethylene”. J . Chem. Eng. Data 1981,26, 47-51. Mackay, M. E.; Paulaitis, M. E.
