Ind. Eng. Chem. Res. 2004, 43, 4967-4972
4967
Calculation of Hansen Solubility Parameter Values for a Range of Pressure and Temperature Conditions, Including the Supercritical Fluid Region Laurie L. Williams,*,† James B. Rubin,‡ and H. W. Edwards† Department of Mechanical Engineering, Colorado State University, Fort Collins, Colorado 80523, and Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Starting from the original definition of the cohesive energy density and the Hildebrand solubility parameter, an approach utilizing an equation of state of the form P ) f(F,T) is used for calculating Hildebrand solubility parameters for a range of temperatures and pressures, including the supercritical fluid region. In addition, an approach is presented to determine Hansen solubility parameters over the same range of temperatures and pressures. The pure fluid used in these calculations is carbon dioxide (CO2). CO2 has been extensively studied, and its use as a supercritical fluid is well established in high-pressure chromatography and fluid extraction applications. Introduction During the last 2 decades, the concept of supercritical fluids (SCFs) as a solvent has grown out of knowledge developed through supercritical extraction and purification processes. In the U.S., commercial SCF applications include coffee decaffeination, hops extraction, extraction of organic wastes from water, fats from foods, and essential oils and spices from plants, and purification processes in the petrochemical industry.1 The ability of SCFs to solubilize other organic compounds has recently extended the scope of applications to include commercial degreasing and precision cleaning of machined parts contaminated with oil and greases. These cleaning applications are similar to the selective extraction and purification applications in that the objective is to solubilize, and thereby separate, organic compounds from matrixes. In addition, the expansion of polymer for foams, textile dying, and material processing of polymers with SCFs, especially supercritical carbon dioxide (CO2), are being commercialized.2 All of these applications require accurate and reliable solubilities of the solute in the chosen SCF or solubility of the SCF in a particular matrix. The Hildebrand and Scott3 solubility parameter methodology based on cohesive energy density is a useful approach to the solubility issue and has been used widely. Solubility parameters provide one of the simplest methods of correlating and predicting the solubility behavior of two components, based on the knowledge of the individual components alone. This simplicity, not surprisingly, also results in limitations on the accuracy of the methodology. As a result of this, the Hildebrand one-component model has been expanded by Prausnitz and Blanks4 into two components, by Hansen5 into three components, and by others6-9 into multicomponent models. However, of all of the solubility parameter models, the most widely used has been the three* To whom correspondence should be addressed. Tel.: 5056673706. Fax: 5056676561. E-mail:
[email protected]. † Colorado State University. ‡ Los Alamos National Laboratory.
parameter approach proposed by Hansen.5 As a result, there is a large body of tabulated HSP data, both measured and calculated, for a large number of compounds. Traditionally, the solubility parameter concept has been almost exclusively confined to mixtures of ambientpressure liquids. There are, however, important exceptions such as the empirical correlation proposed by Giddings et al.10 in 1968. This relation, although a good starting point for predicting trends qualitatively in the right direction, was found to be poor for quantitative predictions.11 Other notable works include Allada’s12 proposed generalized solubility parameter (which uses analytical equations of state (EOSs), such as LeeKesler or modified Redlich-Kwong, for evaluation), the modified solubility parameter proposed by Ikushima et al.13 (where the solubility parameter is expressed in terms of reduced parameters), and the EOS model proposed by Panayiotou.14 This later work utilizes the lattice fluid theory and a lattice fluid hydrogen-bonding model to evaluate solubility parameters and two separate components: physical (or van der Waals) and chemical (or specific, e.g., hydrogen bonding). The work proposed here also examines the thermodynamic EOS approach, which utilizes a fluid-specific EOS (in the case of this work, the fluid is CO2) for expanding both the Hildebrand and Hansen solubility parameters over a wide range of temperatures and pressures (including the SCF region). One-Component Hildebrand Parameter Hildebrand’s solubility parameter theory was derived from an approximation of the internal pressure of a fluid, which was later termed the cohesive energy density, based on work conducted in 1928,15 1929,16 1932,17 and 1950,3 where the two terms, internal pressure and cohesive energy density, were found to be related by the quantity n, as shown in eq 1, where ∆E/V
(∂E ∂V)
) T
10.1021/ie0497543 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/03/2004
n∆E V
(1)
4968 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004
is defined by Hildebrand as the cohesive energy density and (∂E/∂V)T is the internal pressure. Hildebrand and co-workers15-17 found that for nonpolar/nonassociating liquids, where intermolecular interactions are weak, n is not far from unity and the equality of the cohesive energy density and internal pressure is a good approximation. This same work also demonstrates that n is near unity for nonpolar liquids and also for polar liquids, where the dipole moment is less than 2 D (1 D is equal to 1.0 × 10-18 electrostatic units or 3.336 × 10-30 C m) and where specific interactions (particularly hydrogen bonding) are largely absent. While no direct evaluation of the value of n has been found in the literature for CO2, a comparison of the values found by Hildebrand and others18-21 strongly suggests that the value of n for CO2 is expected to be near unity, and as a result, the internal pressure and cohesive energy density are approximately equal. Accordingly, Hildebrand’s solubility parameter, defined as the square root of the cohesive energy density,3 can also be approximated by the square root of the internal pressure for nonpolar/nonassociating fluids.
δ)
(∆EV)
1/2 T
≈
(∂E ∂V)
1/2
(2)
T
Also, the internal pressure can be calculated from the thermodynamic EOS so that
δ2 ≈
(∂E ∂V)
T
)T
∂P (∂T )
V
-P
(3)
Total (one-component) solubility parameters can therefore be calculated using an EOS of the form P ) f(F,T). This approach is used in this work to calculate the total solubility parameter for pure CO2, using the empirical EOS of Huang et al.22
[
P ) FRT 1 + b2F′ + b3F′2 + b4F′3 + b5F′4 + b6F′5 + b7F′2 exp(-c21F′2) + b8F′4 exp(-c21F′2) + ∆F c22F′2 exp[-c27(∆T)2] + c23 exp[-c25(∆F)2 F′ ∆F c27(∆T)2] + c24 exp[-c26(∆F)2 - c27(∆T)2] (4) F′
]
b7 ) b8 )
c15 T′
3
c18 T′
3
+
+
c14 T′ c16 T′
4
c19 T′
4
+
+
c17 T ′5 c20 T ′5
(6)
The state constants (ci) are as defined in the Huang reference. The authors note that there is a wide range of available EOSs for CO2, and a comparison of Huang’s and others can be found in a review by Span and Wagner.23 These equations and the appropriate derivatives are then used to calculate CO2 solubility parameters over the temperature and pressure range for which the EOS is stated to be valid (220 K e T e 420 K and 0.1 MPa e P e 60 MPa). Figure 1 is a plot of the resulting onecomponent solubility parameters. Values calculated with this methodology compare well with values (shown graphically) determined by Allada,12 but they do not compare well with CO2 solubility parameter values (shown graphically) calculated by Panayiotou.14 Further, Panayiotou states that it has been observed, under the appropriate conditions of temperature and pressure, that CO2 may exhibit a solubility capacity comparable to n-hexane, which has a solubility parameter value of 14.9 MPa1/2.25 The conditions of temperature and pressure exemplified by Panayiotou are about 60 °C and 40 MPa. Extrapolating from Figure 2, in the referenced Panayiotou paper, at the suggested temperature and pressure a CO2 solubility of approximately 3.4 MPa1/2 is determined, whereas at the same temperature and pressure, this work predicts a CO2 solubility parameter value of 13.6 MPa1/2 and the value extrapolated from the referenced Allada work is approximately 13.9 MPa1/2. This discrepancy with the Panayiotou paper cannot be explained by this author. Three-Component (Hansen) Solubility Parameters: Pure CO2
b4 ) c10 +
c11 T′
A shortcoming of the early solubility parameter work of Hildebrand is that the approach was limited to solutions of weakly interacting components, as defined by Hildebrand and Scott, and did not account for association between molecules, such as those which polar and hydrogen-bonding interactions would require. Therefore, while the thermodynamic EOS method gives an accurate representation of the total (one-component) solubility parameter for CO2 over a range of temperatures and pressures, the (Hildebrand) one-component model does not accurately predict the solubility behavior of a real fluid. The basis of Hansen solubility parameters (HSPs) is the assumption that the total cohesive energy (E) is made up of the additive contributions from nonpolar (dispersion) interactions (Ed), polar (dipoledipole and dipole-induced-dipole) interactions (Ep), and hydrogen-bonding or other specific association interactions (including Lewis acid-base interactions) (Eh):
b5 ) c12 +
c13 T′
E ) Ed + Ep + Eh
where
T ′ ) T/Tc; ∆T ) 1 - T ′; F′ ) F/Fc; ∆F ) 1 - 1/F′ (5) and
b 2 ) c1 +
b6 )
c2 c4 c5 c6 c3 + + + + T ′ T ′2 T ′3 T ′4 T ′5 b3 ) c7 +
c9 c8 + 2 T′ T′
(7)
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4969
Figure 1. Total (one-component) solubility parameter of pure CO2, calculated using eqs 3 and 4.
Dividing each contribution by the molar volume gives the square of the total solubility parameter as the sum of the squares of the Hansen dispersion (δd), polar (δp), and hydrogen-bonding (δh) components
E Ed Ep Eh ) + + V V V V
(8)
δT2 ) δd2 + δp2 + δh2
(9)
so that
where
δd2 )
Ed Ep Eh ; δp2 ) ; δh2 ) V V V
(10)
Hansen’s total solubility parameter, δT, should equal the Hildebrand solubility parameter, eq 2, although the two quantities may differ for materials with specific interactions when these specific interactions are determined by different methods.24 Extending the HSP methodology to SCFs would significantly enhance the understanding of their solvent properties; however, few such studies appear to have been done. For strictly nonpolar gases (gases with no permanent dipole or higher moments, such as argon), the values of δp and δh will be zero and we could equate δd with the total solubility parameter, δT. However, CO2, which possesses a large quadrupole moment and can display Lewis acid-base characteristics, has nonzero values for both δp and δh.25 Determination of HSPs for (ambient condition) gases is usually based on the room-temperature solubility of the gas in different liquids of known δd, δp, and δh. Those liquids that show the highest solubility for the gas are assumed to have HSPs closer to those of the gas than those liquids that have lower solubilities for CO2. The solubility parameter values for CO2 have been determined by Williams,26 based on this suggested methodology, at 25 °C as being δd, δp, and δh equal to 15.6, 5.2, and 5.8 MPa1/2, respectively. Hansen, who proposed the methodology for gas HSP values used here, determined a set of CO2 HSP values (δd, δp, and δh respectively) equal to 15.3, 6.9, and 4.1 MPa1/2, using a separate liquid solubility dataset.25 It is also necessary to establish a pressure corresponding to this T ) 25 °C set of HSP values in order to use them to determine HSPs at arbitrary conditions of T and P. From eq 9, the determined HSP values for
Figure 2. CO2 HSP values as a function of the temperature and pressure: (a) dispersion parameter; (b) polar parameter; (c) hydrogen-bonding parameter.
CO2 result in a total CO2 solubility parameter of 17.4 MPa1/2.
δT2 ) δd2 + δp2 + δh2 ) (15.6)2 + (5.2)2 + (5.8)2 ) 304.04 MPa ) 17.4 MPa1/2
(11)
The PVT EOS, which calculates the total (Hildebrand) CO2 solubility parameter value, eq 4, was used to determine the combination of pressure and molar volume corresponding to T ) 25 °C and δT ) [T (∂P/ ∂T)V - P]1/2 ) 17.4 MPa1/2, which gave
P ) 13 300 psi (905 atm) VCO2 ) 39.13 cm3/mol
(12)
4970 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004
( )
Table 1. Experimentally Determined Values of Eh and dEh/dT
functional group
hydrogen-bonding parameter, Eh (cal/mol)
dEh/dT (cal/mol‚K)
-OH (aliphatic) -NH2 (aliphatic) -CN (aliphatic) -COOH (aliphatic)
4650 ( 400 1350 ( 200 500 ( 200 2750 ( 250
-10 -4.5 -7.0 -2.9
∂δd ∂V
temperature increment
δd δp δh
( ) ( ) ( )
P
P
P
) -1.25δdR ) -δp
() ( R 2
) -δh 1.32 × 10-3 +
( ) ( ) )( )
R 2
∂δd ∂P ∂δp ∂P ∂δh ∂P
T
T
δdref δd
δp
δpref δp
δh
δhref δh
)
( ) ( ) [ Vref V
Vref ) V
n + 1 -(n+1)/2 -1 [V ][V ] 2 (15)
∂δd ∂V ) -1.25 δd V
(16)
Equation 16 can now be either differentiated for a change in temperature or pressure or integrated. Results of both derivations are shown in Tables 2 and 3.
) 1.25δdβ
Temperature and Pressure Effects on HSPs: δp
() (
) δp
β 2
) δh
1.32 × 10-3β β + R 2
)
The first values of δp were assigned by Hansen and Skaarup using the Bo¨ttcher equation, shown here as eq 17. A simplified equation was later developed by Hansen
δp2 )
Table 3. Equations (Integrated Form) for the Temperature and Pressure Effects on HSPs δd
( )
(n +2 1)(V1 )
pressure increment
T
) k1/2
) -δd
Table 2. Equations (Derivative Form) for the Temperature and Pressure Effects on HSPs ∂δd ∂T ∂δp ∂T ∂δh ∂T
T,P
12108 - 1 (nD2 + 2)µ2 V 2 2 + nD2
[cal/cm3]
(17)
and Beerbower27
-1.25
δp ) 37.4µ/V1/2
-0.5
0.5
HSP values at other pressures and temperatures will be calculated based on this reference set of HSP values, using derived pressure and temperature integral functions, as shown in Table 3. Both temperature and pressure will influence the total and HSP component solubility parameters. Generally, an increase in the pressure at constant temperature will increase the total solubility parameter through an increase in the solvent density. Similarly, an increase in the temperature at constant pressure will decrease the total solubility parameter, as seen in Figure 1. The temperature and pressure dependence of individual HSPs, as a function of the temperature and pressure, is outlined as follows, where the temperature derivatives, originally derived by Hansen and Beerbower,27 are verified. Pressure derivatives, not found in any literature search, are derived in a manner parallel to that of the temperature derivatives. In addition, integral forms are shown in Table 3. Temperature and Pressure Effects on HSPs: δd Hildebrand,3 in his 1950 work, considered the effect of the temperature on solubility parameters by recalling the expression for the dependence of E on the volume
E ) -k/Vn
(13)
where k is a constant dependent upon the nature of the particular liquid and n is about 1.5 for normal (nonassociating or van der Waal) liquids. Substituting eq 13 into Hansen’s definition for the dispersion solubility parameter 1/2
(n+1)/2
δd ) -k /V
(18)
where µ is the dipole moment (in D). This equation is utilized for determining the change in δp either with respect to temperature at constant pressure or with respect to pressure at constant temperature.
( )]
Vref ) exp - 1.32 × 10 (Tref - T) - ln V -3
[MPa1/2]
(14)
allows one to calculate the change in δd produced by a change in the volume by differentiating eq 14.
( ) ∂δp ∂V
()
)-
1 -3/2 V (37.4µ) 2
)-
δp 1 37.4µ )1/2 2V V 2V
T,P
( )
∂δp ∂V )δp 2V
(19)
(20)
Equation 20 can now be either differentiated for a change in temperature or pressure or integrated. Results of both derivations are shown in Tables 2 and 3. Temperature and Pressure Effects on HSPs: δh In Hansen’s early work, the hydrogen-bonding parameter was almost always found by subtracting the polar and dispersion energies of vaporization from the total energy of vaporization. This is still widely used where the required data are available and reliable. Hansen,25 however, while noting that “there is no rigorous way of arriving at values of the temperature dependence of the hydrogen bonding solubility parameter”, developed an empirical approach for the determination of the temperature dependence of δh that involves experimental heats of vaporization data for hydrogen-bonded substances, which, in turn, are taken from Bondi and Simkin.28 From eq 10, the hydrogen-bonding solubility parameter, δh, is defined as
δh2 ) Eh/V
(21)
Eh ) Vδh2
(22)
so that
Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4971
where Eh is the hydrogen-bonding contribution to the total cohesive energy. Differentiating eq 22 with respect to the temperature at constant pressure,
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
∂Eh ∂T
P
∂δh ∂T
) V(2δh) ∂δh ∂T
2Vδh
∂δh ∂T
∂Eh ∂T
)
P
∂Eh ∂T
)
P
P
P
+ δh2
∂V ∂T
- δh2
∂V ∂T
- δh2
∂V ∂T
P
2Vδh
P
P
P
(23)
Simplifying, rearranging terms, and substituting in the isobaric coefficient of thermal expansion, R,
( ) ∂δh ∂T
P
) δh
(
)
( )
∂Eh ∂T P R 2Eh 2
(24)
Bondi,28 through exploratory calculations, has shown that the difference between the heat of vaporization of a hydroxylic compound (a compound displaying strong hydrogen bonding) and that of its hydrocarbon (or other nonpolar) homomorph constitutes a good measure of the hydrogen-bonding strength. This work also discusses the decrease in the heat of formation of the hydrogen bond with increasing temperature. Reference curves of dEh/ dT were constructed29 for various functional groups and are shown in Table 1, along with experimentally derived values of Eh.28 Averaging the rate of change of the hydrogen-bonding heat of vaporization with temperature (dEh/dT) and dividing by the average excess heats of vaporization (heat of vaporization of the hydrogen-bonding compound minus the heat of vaporization of its nonpolar homomorph) result in the following form of eq 24:
( ) ∂δh ∂T
P
) -δh
(
)
2.64 × 10-3 R + 2 2
(
-3
) -δh 1.32 × 10
R + 2
)
(25)
The change in δh with respect to pressure at constant temperature is obtained by utilizing the relationship
∂δh ∂δh ∂T ) ∂P ∂T ∂P
(26)
∂T β )∂P R
(27)
where
so that
( ) ∂δh ∂P
T
) δh
(
)
1.32 × 10-3β β + R 2
(28)
Equation 28 can be rearranged to a form that can also be easily integrated (Table 3). The derivative forms are summarized in Table 2 and the integrated forms in Table 3. The total solubility parameter, incremented for small changes in the temperature and pressure, can be
Figure 3. Plot of the differences in the CO2 total solubility parameter calculated by eqs 3 and 4 and the total HSP calculated by eq 9, in MPa1/2.
calculated from equations (derivative form) in Table 2
[ ( ) ( ) ] [ ( ) ( ) ] [ ( ) ( ) ]
δ2 ) δd +
2 ∂δd ∂δd ∆T + ∆P + ∂T P ∂P T 2 ∂δp ∂δp δp + ∆T + ∆P + ∂T P ∂P T ∂δh ∂δh δh + ∆T + ∆P ∂T P ∂P T
2
(29)
or from the equations (integrated form) in Table 3
[( ) ] [( ) ] {[
δ2 )
δdref
Vref V
-1.25
2
+
δpref Vref -0.5 V δhref
2
+
}
( )]
Vref exp -1.32 × 10 (Tref - T) - ln V -3
0.5
2
(30)
where the reference values are as determined earlier; δdref ) 15.6 MPa1/2, δpref ) 5.2 MPa1/2, δhref ) 5.8 MPa1/2, Vref ) 39.13 cm3/mol, and Tref ) 298.15 K. CO2 HSP values calculated with the equations in Table 3, as a function of the temperature and pressure, are shown in the CO2 HSP surface diagrams illustrated in Figure 2. It is important to note that the solubility parameter, or rather the differences in the solubility parameters, for a given solvent-solute combination has been foremost in determining the mutual solubility of the system.25 An analogy to “like to dissolves like” is appropriate. Therefore, the accuracy to which the solubility parameters for a binary pair can be known will be valuable in predicting the system’s behavior. Figure 3 is a contour plot comparing the one-component Hildebrand solubility parameter calculated from the CO2 EOS (eqs 3 and 4) and the total three-component HSP calculated from eq 9. Values on the contour lines represent values of δT,Hildebrand - δT,Hansen in units of MPa1/2 (SI unit for the solubility parameter). It can be seen that, for the range of temperatures and pressures
4972 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004
evaluated, the differences in the total solubility parameters, calculated by the two methods, are small and never exceed 1 MPa1/2. Conclusions The Hildebrand (one-component) solubility parameter provides a broad, qualitative indication of the mutual solubility behavior for most solvent/solute (or cosolvent) systems and gives good quantitative results for a very small number of nonpolar systems. Consequently, Hildebrand’s approach has become the most widely used method for describing intermolecular interactions in condensed (liquid and solid) systems. It has been shown through numerous examples, however, that the threeparameter approach of Hansen represents a significant improvement in the description of the solubility behavior of real fluids. HSPs give a semiquantitative measure of the extent of interactions for nonpolar, as well as many polar, systems.24 In addition to polymer solubility (swelling) studies, HSPs have been applied to biological materials, barrier properties of polymers, and the characterization of surfaces, pigments, fillers, and fibers.5 The ability to characterize organic and inorganic solids, liquids, and gases with the same three-parameter model also enables interpretation of situations involving mutual solubility and material compatibility. As a result, a fairly large industrial following has been built up, including the publishing of numerous compilations of measured parameters as well as well-developed calculational methods, when measured values are not available. This work introduces a theoretical methodology for generating solubility parameter values, both one-component Hildebrand and three-component Hansen parameters, for a pure SCF, using CO2 as an example. The approach adopted in this work was to first determine a set of ambient-condition HSPs for the pure fluid and then to devise a method of extending this single point in solubility parameter space to an arbitrary temperature and pressure. This entailed the verification of previously published temperature derivative functions and the derivation of novel pressure derivative functions. The resulting HSP values appear to be internally consistent in terms of the comparison shown in Figure 3, and the ability to express molecular interactions, in terms of HSPs, for a pure fluid solvent in a way that unites the liquid, gas, and supercritical phases represents an advancement in the understanding of the role of solvents in both existing and new applications. Literature Cited (1) Rubin, J. B.; Davenhall, L. B.; Barton, J.; Taylor, C. M. V. CO2-Based Supercritical Fluids as Replacements for PhotoresistStripping Solvents. Semiconductor Industry Association Occupational Health Conference on Current and Future Health and Safety Challenges in the Semiconductor Industry, Dallas, TX, Nov 3-5, 1998. (2) Teja, A. S.; Eckert, C. A. Commentary of Supercritical Fluids: Research and Applications. Ind. Eng. Chem. Res. 2000, 39, 4442. (3) Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes, 3rd ed.; Reinhold: New York, 1950. (4) Prausnitz, J. M.; Blanks, R. F. Thermodynamics of Polymer Solubility in Polar and Nonpolar Systems. Ind. Eng. Chem. Fundam. 1964, 3, 1.
(5) Hansen, C. M. The Three-Dimensional Solubility ParametersKey to Paint Component Affinities. I. Solvents, Plasticizers, Polymers, and Resins. J. Paint Technol. 1967, 39 (505), 104. (6) Beerbower, A.; Wu, P. L.; Martin, A. Expanded Solubility Parameter Approach. I. Naphthalene and Benzoic Acid in Individual Solvents. J. Pharm. Sci. 1984, 73, 179. (7) Karger, B. L.; Snyder, L. R.; Horvath, C. An Introduction to Separation Science; Wiley: New York, 1973. (8) Ignat, A. V.; Melder, L. I. Determination of the Components of the Solubility Parameters of Alcohols. J. Appl. Chem. USSR 1987, 60, 1070. (9) Munafo, A.; Buchmann, M.; Ho, N.-T.; Kesselring, U. W. Determination of the Total and Partial Cohesion Parameters of Lipophilic Liquids by Gas-Liquid Chromatography and from Molecular Properties. J. Pharm. Sci. 1988, 77, 169. (10) Giddings, J. C.; Myers, M. N.; McLaren, L.; Keller, R. A. High-Pressure Gas Chromatography on Nonvolatile Species. Science 1968, 162, 67. (11) Stahl, E.; Schilz, W.; Schutz, E.; Willing, E. Quick Method For Microanalytical Evaluation of Dissolving Power of Supercritical Gases. Angew. Chem. 1978, 17 (10), 731. (12) Allada, S. R. Solubility Parameters of Supercritical Fluids. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 344. (13) Ikushima, Y.; Goto, T.; Arai, M. Modified Solubility Parameter as an Index to Correlate the Solubility in Supercritical Fluids. Bull. Chem. Soc. Jpn. 1987, 60, 4145. (14) Panayiotou, C. Solubility parameter revisited: an equation-of-state approach for its estimation. Fluid Phase Equilib. 1997, 131, 21. (15) Westwater, W.; Frantz, H. W.; Hildebrand, J. H. The Internal Pressure of Pure and Mixed Liquids. Phys. Rev. 1928, 31, 135. (16) Hildebrand, J. H. Intermolecular Forces in Liquids. Phys. Rev. 1929, 34, 984. (17) Hildebrand, J. H.; Carter, J. M. A Study of van der Waals Forces between Tetrahalide Molecules. J. Am. Chem. Soc. 1932, 54, 3592. (18) Dack, M. R. J. Solvent Structure: The Use of Internal Pressure and Cohesive Energy Density to Examine Contributions to Solvent-Solvent Interactions. Aust. J. Chem. 1975, 28, 1643. (19) Renuncio, J. A. R.; Breedveld, G. J. F.; Prausnitz, J. M. Internal Pressures and Solubility Parameters for Carbon Disulfide, Benzene, and Cyclohexane. J. Phys. Chem. 1977, 81 (4), 324. (20) Allen, G.; Gee, G.; Wilson, G. J. Intermolecular Forces and Chain Flexibilities in Polymers: I. Internal Pressures and Cohesive Energy Densities of Simple Liquids. Polymer 1960, 1 (4), 456. (21) MacDonald, D. D.; Hyne, J. B. The Thermal Pressure and Energy-Volume Coefficients of the Methyl Alcohol-Water and tertButyl Alcohol-Water Systems. Can. J. Chem. 1971, 49, 2636. (22) Huang, F.; Li, M.; Lee, L.; Starling, K. An Accurate Equation of State for Carbon Dioxide. J. Chem. Eng. Jpn. 1985, 18 (6), 490. (23) Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1996, 25 (6), 1509. (24) Barton, A. F. M. CRC Handbook of Solubility Parameters and Other Cohesion Parameters, 2nd ed.; CRC: Boca Raton, FL, 1991. (25) Hansen, C. M. Hansen Solubility Parameters: A User’s Handbook; CRC: Boca Raton, FL, 2000. (26) Williams, L. Ph.D. Thesis, Colorado State University, Fort Collins, CO, 2001. (27) Hansen, C.; Beerbower, A. Solubility Parameters. KirkOthmer Encyclopedia of Chemical Technology, 2nd ed.; Interscience: New York, 1971; Suppl. Vol. (28) Bondi, A.; Simkin, D. J. Heats of Vaporization of Hydrogen bonded Substances. AIChE J. 1957, 3 (4), 473. (29) Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Glasses; Wiley: New York, 1968.
Received for review March 28, 2004 Revised manuscript received June 11, 2004 Accepted June 22, 2004 IE0497543
