Revision of MOSCED Parameters and Extension to Solid Solubility

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Ind. Eng. Chem. Res. 2005, 44, 4075-4083

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Revision of MOSCED Parameters and Extension to Solid Solubility Calculations Michael J. Lazzaroni,†,⊥ David Bush,*,† and Charles A. Eckert*,† School of Chemical & Biomolecular Engineering and Specialty Separations Center, Georgia Institute of Technology, Atlanta, Georgia 30332-0100

Timothy C. Frank,‡ Sumnesh Gupta,§ and James D. Olson| The Dow Chemical Company, 1319 Building, Michigan Operations, Midland, Michigan 48667, The Dow Chemical Company, 1400 Building, Michigan Operations, Midland, Michigan 48667, and The Dow Chemical Company, 740 Building, 3200 Kanawha Turnpike Technical Park, South Charleston, West Virginia 25303

New pure component parameters are presented for the MOSCED limiting activity coefficient model for 133 solvents with an absolute average deviation of 10.6% to experimental literature data. The MOSCED model has been applied to solid-liquid equilibria correlation and compared with the experimental data available in the literature. The correlation of solubility of 26 solids in organic solvents has an average absolute deviation of 25%. This compares favorably to the prediction of the modified UNIFAC model. Introduction Quantitative estimation of multicomponent phase equilibria is important for the design of many chemical processes.1,2 Improved methods can facilitate a more comprehensive survey of solvents and other materials for use in chemical manufacturing and product formulations. They can also provide for greater productivity in terms of reducing the experimental effort and time required for commercialization of new products and processes. Limiting activity coefficients are most useful in characterizing phase equilibria3 because they truly represent unlike-pair intermolecular interactions and are readily applied to the estimation of multicomponent phase equilibria by combination with some general gE expression, such as Wilson, NRTL, or UNIQUAC.4 Often, there is little mixture data available for a given system to correlate the necessary binary interaction parameters, and some type of prediction is necessary to facilitate the process design. In particular, for the design of crystallization processes, the necessary solidliquid equilibrium data for a wide range of solvents is not available, and a predictive method for solubility in pure and mixed solvents would be beneficial for optimum solvent selection. At present, “state-of-the art” methods for the estimation of the infinite dilution activity coefficients are based mainly upon the group contribution methods, such as UNIFAC5 and ASOG,6,7 or based upon the modifications of the solubility parameter approach, such as Hansen * To whom correspondence should be addressed. Tel.: (404) 894-7070. Fax: (404) 894-9085. E-mail: david.bush@ che.gatech.edu, [email protected]. † School of Chemical & Biomolecular Engineering and Specialty Separations Center, Georgia Institute of Technology. ‡ The Dow Chemical Company, 1319 Building, Michigan. § The Dow Chemical Company, 1400 Building, Michigan. | The Dow Chemical Company, 740 Building, 3200 Kanawha Turnpike Technical Park, South Charleston. ⊥ Present address: Albemarle Corp., Baton Rouge, Louisiana 70805.

solubility parameter8,9 and MOSCED.10 The possibility of using molecular simulation methodology may also exist for the future, and we refer to several reviews that provide some idea of the advantages and difficulties of using this methodology.2,11-14 Modified UNIFAC15 is currently the most widely used in the chemical industry, but it is often limited in that it does not have any explicit representation of specific interactions, such as hydrogen bonds, and often performs less well for multifunctional molecules. We present here a reevaluation of parameters for the MOSCED model (MOdified Separation of Cohesive Energy Density), which specifically characterizes strong, specific interactions, and show its application to purification of pharmaceuticals and pharmaceutical precursors by crystallization. The classic estimation technique for γ∞ is regular solution theory (RST).16 RST is limited to nonpolar, nonassociating solvent systems; it performs poorly for associated and solvating systems, and it cannot predict negative deviations from ideality (γ < 1). The extension of RST most widely used in the chemical industry is the Hansen model8,9 which divides the regular solution solubility parameter into three parameters accounting for dispersion, polarity, and hydrogen bonding. The use of three adjustable parameters helps the correlation, but the Hansen model may also perform poorly for associated and solvating systems, and it too cannot predict negative deviations. This is a serious flaw, because in choosing solvents for separations, one frequently seeks specific solvation, or negative deviations. The UNIFAC17 and mod. UNIFAC15 models have been used to predict all types of phase equilibria to some degree of success. The model assumes that each functional group has a specific interaction energy with each other functional group; to parametrize the interactions, experimental data must be available for every functional group pair. The UNIFAC model has been used to predict solubility data for monofunctional solid compounds with success.18-20 However, many predictions of solid compounds of interest to the pharmaceutical and fine

10.1021/ie049122g CCC: $30.25 © 2005 American Chemical Society Published on Web 04/20/2005

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chemical industry are not possible because of missing interaction parameters or missing functional groups. Compared to UNIFAC-type models, MOSCED offers advantages in certain applications because knowledge of group interaction parameters is not required. This allows the use of MOSCED even when chemical structure and/or group interaction information is lacking. And, compared to methods based on molecular modeling approaches to estimating properties,21 the calculation effort required to use MOSCED is relatively low, thus facilitating greater use by practicing chemists and engineers. Several versions of MOSCED have been proposed in the literature.10,22,23 In the most recent model by Hait et al., all adjustable parameters per compound are predicted from empirical equations for each functional family that relate solvatochromic parameters to model parameters. This limits predictions for multifunctional compounds, common to many solids, which do not fit into a distinct family and where the solvatochromic parameters are generally unavailable. For this reason, we have focused our efforts on the original formulation of MOSCED which is given in the equations below. ∞

ln γ2

[

q12q22(τ1T - τ2T)2 v2 2 ) (λ1 - λ2) + + RT ψ1

]

(R1T - R2T)(β1T - β2T) ξ1

d12 ) ln

() v2 v1

aa

+1-

() v2 v1

+ d12 (1)

aa

(2)

aa ) 0.953 - 0.002314((τ2T)2 + R2T β2T)

(293K T )

RT ) R

0.8

(293K T )

, βT ) β

0.8

, τT ) τ

(293K T )

0.4

ψ1 ) POL + 0.002629 R1T β1T

(3) (4) (5)

ξ1 ) 0.68(POL - 1) + 1.5 (293K/T)2

[3.24 - 2.4 exp (-0.002687(R1β1) )]

(6)

POL ) q14[1.15 - 1.15 exp( - 0.002337(τ1T)3)] + 1 (7) There are 5 adjustable parameters per molecule: λ, the dispersion parameter; q, the induction parameter; τ, the polarity parameter; R, the hydrogen-bond acidity parameter; and β, the hydrogen-bond basicity parameter. Equation 2 is the Flory-Huggins combinatorial expression with an empirically derived exponent calculated from eq 3. The temperature dependence of λ is the same as in RST; the temperature dependence of R, β, and τ is shown in eq 4. ψ1 and ξ1 are empirical asymmetry terms, calculated from eqs 5-7. Although the original MOSCED parameter set has been shown quite useful in a priori prediction,24 the number of parametrized solvents is limited. We have decided to refit the parameters of the original solvent list as well as add parameters for additional solvents. In addition, the ability of the MOSCED model to correlate parameters for solid compounds from solid solubility data is examined and compared to solubility predictions using the UNIFAC model.

MOSCED Model Reevaluation. The increase in the available literature data for γ∞ in the last two decades has prompted a reexamination and new regression of parameters for the MOSCED model. Several new techniques have been developed that allow for faster determination of activity coefficient data such as headspace gas chromatographic techniques,25-29 dew point techniques,30 and others of which there are several reviews.31,32 The original dataset was heavily weighted to nonpolar alkane systems. Since then, infinite dilution activity coefficient data for a larger range of organic compound structures and functionalities have been reported, including the large data sets measured by Gmehling and co-workers33-41 and Carr and co-workers,27,29 as well as the DECHEMA activity coefficient compilation.42 Since a minimum of 6 experimental data points are required to parametrize a solvent, we added, when available, infinite dilution activity coefficients extrapolated from evaluated vapor-liquid equilibrium data,43 which comprised about 15% of the database. A complete listing of the selected data can be found in the supplementary data and in the thesis of Lazzaroni.44 The dataset was vetted for suspect points by comparison with other existing data, either with the same system, if available, or with a homologous series. In addition, if data were available at multiple temperatures, a plot of the data versus inverse temperature was useful in identifying suspect data. When a preponderance of data from a single reference source were deemed suspect, the entire reference was removed from the database. In general, data measured using the liquid chromatography technique were removed from the database because of the disagreement with the other existing data and known experimental uncertainties in data measured by this technique.45 The database used to reevaluate/refit the MOSCED model was limited to 6441 data points for 132 organic solvents, water, and five permanent gases. The parameters were regressed by minimizing the objective function in eq 8 using the Powell algorithm.46

O.F. ) (lnγ∞exp - lnγ∞pred)2

(8)

The root-mean-squared error for ln γ∞ was 0.148. The overall average absolute deviation (AAD) for γ∞ was 10.6%, which is slightly larger than the original correlation where the dataset was roughly half the size but well within the general uncertainty of the infinite dilution activity coefficients data. The parameters, in units of (J/cm3)1/2 except for q which is dimensionless, are shown in Table 1. The model is able to correlate accurately data for the following classes: nonpolar, polar aprotic (dipolar and hydrogen bond accepting), aromatic/halogenated (large dispersion and slight hydrogen bond donating/accepting), and associated (dipolar and strong hydrogen bond donating/accepting). AAD’s are shown in Table 2. MOSCED performs best for nonpolar and aromatic/halogenated compounds with slightly higher errors for polar aprotic and associated compounds. Mod. UNIFAC is able to predict the dataset to an average error of 16.3% with the highest errors for systems with associated solvents and solutes. For the mod. UNIFAC model, the predictions for 2,6-dimethylpyridine, butanenitrile, and glutaronitrile are consistent outliers (studentized residuals larger than 3) from the average deviation. For the case of 2,6-dimethylpyridine and butanenitrile, mod. UNIFAC is unable to account properly for the chemical effect upon addition

Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4077 Table 1. Parameters for the MOSCED Modela compound propane butane pentane isopentane cyclopentane hexane cyclohexane methylcyclopentane 3-methylpentane 2-methylpentane 2,3-dimethylbutane 2,2-dimethylbutane heptane methylcyclohexane cycloheptane 3-methylhexane 2,2-dimethylpentane 2,4-dimethylpentane 2,3,4-trimethylpentane octane 2,2,4-trimethylpentane ethylcyclohexane cyclooctane 2,5-dimethylhexane nonane decane dodecane tetradecane hexadecane squalane 1-pentene 1-hexene 1-octene R-pinene benzene toluene p-xylene ethylbenzene isopropylbenzene butylbenzene methanol ethanol 1-propanol 2-propanol 1-butanol 2-butanol 2-methyl-2-propanol 2-methyl-1-propanol 1-pentanol 1-hexanol 1-octanol phenol benzyl alcohol m-cresol

ν 75.7 96.5 116.0 117.1 94.6 131.4 108.9 113.0 130.4 132.9 131.2 133.7 147.0 128.2 121.7 146.4 148.9 150.0 159.5 163.4 165.5 143.0 134.9 165.6 179.6 195.8 228.6 261.3 294.2 526.1 110.3 125.8 157.8 159.0 89.5 106.7 123.9 122.9 139.9 156.6 40.6 58.6 75.1 76.8 92.0 92.0 94.7 92.9 108.5 125.2 158.2 88.9 103.8 105.0

λ 13.10 13.70 14.40 13.87 16.55 14.90 16.74 16.10 14.68 14.40 14.30 13.77 15.20 16.06 17.20 14.95 14.26 14.29 14.94 15.40 14.08 16.34 17.41 14.74 15.60 15.70 16.00 16.10 16.20 14.49 14.64 15.23 15.39 17.32 16.71 16.61 16.06 16.78 17.09 17.10 14.43 14.37 14.93 13.95 14.82 14.50 14.47 14.19 15.25 15.02 15.08 16.66 16.56 17.86

τ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.22 0.44 0.15 3.95 3.22 2.70 2.98 3.23 2.51 3.77 2.53 1.39 1.95 1.86 1.56 2.55 1.85 1.46 1.27 1.31 4.50 5.03 4.16

θ 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 0.93 0.95 0.95 0.90 0.90 0.90 0.90 0.90 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 1.00 0.90

2-ethoxyethanol methyl acetate ethyl acetate propyl acetate butyl acetate benzyl acetate methyl formate ethyl benzoate diethyl phthalate acetone 2-butanone 2-pentanone cyclohexanone 4-methyl-2-pentanone 2-heptanone

97.3 79.8 98.6 115.8 132.0 142.9 62.1 144.1 199.7 73.8 90.2 107.3 104.1 125.8 140.7

15.12 13.59 14.51 13.98 15.22 16.17 18.79 16.48 16.33 13.71 14.74 15.07 15.80 15.27 14.72

7.39 7.54 5.74 5.45 4.16 6.84 8.29 4.97 6.14 8.30 6.64 5.49 6.40 4.71 4.20

1.00 1.00 1.00 1.00 1.00 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

a

R 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.63 0.57 0.27 0.23 0.20 0.10 17.43 12.58 11.97 9.23 8.44 8.03 5.80 8.30 8.10 7.56 4.22 25.14 15.01 27.15

β 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.24 0.29 0.51 1.30 2.24 2.23 1.87 1.83 2.57 1.83 14.49 13.29 10.35 11.86 11.01 10.21 11.93 10.52 9.51 9.20 9.35 5.35 6.69 2.17

3.77 16.84 0.00 8.38 0.00 7.25 0.00 7.53 0.00 6.40 0.54 5.53 0.37 8.62 0.28 2.40 1.07 7.81 0.00 11.14 0.00 9.70 0.00 8.09 0.00 10.71 0.00 6.34 0.00 6.08

compound 1-phenyl-1-butanone acetophenone -caprolactone dichloromethane chloroform carbon tetrachloride 1,1-dichloroethane 1,2-dichloroethane 1,1,1-trichloroethane trichloroethylene 1-chlorobutane chlorobenzene bromoethane bromobenzene iodomethane diiodomethane iodoethane acetonitrile propanenitrile butanenitrile benzonitrile glutaronitrile nitromethane nitroethane 1-nitropropane 2-nitropropane nitrobenzene DMF N,N-dibutylformamide N,N-dimethylacetamide N,N-diethylacetamide N-methylformamide N-methylacetamide N-Ethylacetamide aniline 2-pyrrolidone N-methylpyrrolidone 1-ethylpyrrolidin-2-one 1,5-dimethyl-2-pyrrolidinone N-formylmorpholine pyridine 2,6-dimethylpyridine quinoline sulfolane DMSO dioxane tetrahydrofuran diethyl ether dipropyl ether dibutyl ether diisopropyl ether methyl tert-butyl ether anisole tetraethylene glycol dimethyl ether acetic acid dimethyl carbonate acetaldehyde butanal carbon disulfide triethylamine tributyl phosphate water argon oxygen nitrogen carbon monoxide carbon dioxide [emin][(CF3SO2)2N] [emmin][(CF3SO2)2N]

τ 4.98 6.50 9.65 6.23 4.50 1.82 6.22 6.58 3.15 2.96 3.38 4.17 4.41 4.29 4.21 5.19 3.58 11.51 9.82 8.27 8.21 12.59 12.44 9.96 8.62 8.30 8.23 9.51 5.02 9.46 6.71 8.92 5.90 4.91 9.41 11.36 9.34 8.31 8.45 10.91 6.13 4.16 5.96 12.16 13.36 6.72 4.41 2.79 2.00 1.73 1.90 2.48 5.63 6.73

q 1.00 0.90 1.00 0.96 0.96 1.01 0.92 0.94 1.01 1.00 1.00 0.89 1.00 0.89 1.00 1.00 1.00 1.00 1.00 1.00 0.90 1.00 1.00 1.00 1.00 1.00 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 1.00 1.00 1.00 1.00 1.00 0.90 0.90 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 1.00

R 0.88 1.71 0.43 3.98 5.80 1.25 3.28 2.42 1.05 2.07 0.11 0.00 0.22 0.00 1.16 2.40 0.51 3.49 1.08 0.00 0.15 3.76 4.07 1.19 0.28 0.55 0.98 1.22 0.24 0.00 0.25 8.07 5.28 4.14 6.51 2.39 0.00 0.00 0.00 2.42 1.61 0.73 2.17 1.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.75 0.00

ν 145.2 117.4 106.8 64.4 80.5 97.1 84.7 79.4 100.3 90.1 105.1 102.3 75.3 105.6 62.7 81.0 93.6 52.9 70.9 87.9 103.0 95.8 54.1 72.0 89.5 90.6 102.7 77.4 182.0 93.0 124.5 59.1 76.9 94.3 91.6 76.8 96.6 114.1 115.2 100.6 80.9 116.7 118.5 95.3 71.3 85.7 81.9 104.7 137.6 170.4 141.8 119.9 109.2 221.1

λ 16.46 16.16 16.42 15.94 15.61 16.54 16.77 16.60 16.54 17.19 15.49 16.72 15.72 17.10 19.13 21.90 17.39 13.78 14.95 14.95 15.43 15.12 13.48 14.68 15.17 14.60 16.06 15.95 15.99 15.86 15.66 15.55 16.22 16.07 16.51 16.72 17.64 16.74 16.50 16.10 16.39 15.95 16.84 16.49 16.12 16.96 15.78 13.96 15.20 15.13 14.72 15.17 16.54 16.08

β 6.54 7.12 13.06 0.92 0.12 0.64 1.56 1.34 0.85 0.21 1.17 2.50 1.56 3.13 0.83 2.08 1.96 8.98 6.83 8.57 7.41 9.11 4.01 4.72 5.83 3.43 3.29 22.65 14.07 21.00 18.67 22.01 23.58 22.45 6.34 27.59 24.22 20.75 22.66 19.29 14.93 13.12 12.10 13.52 26.17 10.39 10.43 6.61 5.25 5.29 6.39 7.40 3.93 13.53

57.6 84.7 56.5 90.4 60.6 139.7 345.0 36.0 57.1 52.9 50.0 49.0 42.2 258.6 275.9

14.96 3.23 1.00 24.03 7.50 17.81 8.05 1.00 0.00 7.32 13.76 8.48 1.00 0.00 6.50 15.11 5.97 1.00 0.00 5.27 19.67 1.04 1.00 0.59 0.33 14.49 1.02 1.00 0.00 7.70 15.05 4.87 1.00 0.00 14.06 10.58 10.48 1.00 52.78 15.86 9.84 0 1.0 0 0 8.84 0 1.0 0 0 7.48 0 1.0 0 0 8.15 0 1.0 0 0 8.72 5.68 1.0 1.87 0 15.18 10.72 0.9 9.79 4.75 15.25 10.83 0.9 7.20 5.11

Parameters λ, Τ, R, and β are in units of (J/cm3)1/2 and v in cm3/mol.

of methyl groups to pyridine and acetonitrile, which are not consistent outliers from the average error.

The form and parameters of the model was reexamined in light of the currently available data. The error

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Table 2. Absolute Average % Error in Regressed Activity Coefficients for Different Classes of Compounds: Nonpolar, Polar Aprotic, Aromatic and Halogenated, Polar Associated and Water solute nonpolar polar aprotic aroma./halogen associated nonpolar polar aprotic aroma./halogen associated nonpolar polar aprotic aroma./halogen associated nonpolar polar aprotic aroma./halogen associated nonpolar polar aprotic aroma./halogen associated

solvent nonpolar

polar aprotic

aroma/halogen

associated

water

MOSCED (%)

UNIFAC (%)

6.1 11.8 8.3 13.4 10.6 10.4 11.0 13.3 7.6 10.2 7.0 13.5 10.5 8.9 11.5 14.3 51.7 59.5 29.9 38.8

5.8 12.6 13.9 16.0 14.9 14.8 16.4 21.1 14.4 17.1 14.7 21.4 14.1 23.3 11.7 21.7 96.6 54.2 172.1 43.4

and scatter of data was found to be too large to make substantial changes to empirical constants of the model found in eqs 3-7. Therefore, only compound parameters were refit to the existing data with some changes to finding the best set of parameters. As with the original formulation of the MOSCED model, 5 parameters are used to characterize the energy of interaction of a solute in solution. Of these 5 parameters, chemical intuition is used to determine which parameters need to be fit for a given compound, e.g., the hydrogen-bond acidity parameter of 2-butanone was set to zero. We will discuss the significance of each parameter and the approach taken to arrive at the best set of compound parameters. Dispersion Parameter, λ. The initial formulation of the MOSCED model used two functions of the refractive index, one for nonaromatic and one for aromatic compounds, to calculate the dispersion parameter. The original linear correlations for the dispersion parameter were found to be insufficient to fit the data for very polar and basic compounds such as DMSO and NMP. No suitable correlation could be found that could represent the dispersion parameters for all classes of compounds. The original correlation is also not suitable for dispersion parameters of solid compounds, for which values of the refractive index of the liquid are not available. In this refitting of parameters, the dispersion parameters were fit for each compound, with the exception of alkanes and cycloalkanes, which were set to the value of the Hildebrand solubility parameter. Polarity Parameter, τ. The polarity parameter is a measure of the fixed dipole of a compound in solution. The original formulation used a homomorph method, but this approach was not used in the refit, as it was generally not applicable to aromatic or branched carbon backbones or to multi-functional solvents. The values found for polar compounds are consistent with the gasphase dipole moment data with the τ for DMSO (3.96 D) being the largest at 13.36, lower for nitromethane (3.46 D) at 12.44, and less for acetone (2.88 D) at 8.30. No sufficiently quantitative correlation could be found that relates the dipole moment to the regressed value of τ, although there is an approximate linear correlation with dipole moment divided by the molar volume. 1,4-

Dioxane is one example where the zero dipole moment in the gas phase is not in agreement with the expected more polar behavior in liquid solution. The polarity parameter value of 6.72 is similar to that of the moderately polar 2-butanone. One might speculate that this disparity is related to the chair-boat transitions in dioxane. Induction Parameter, q. The induction parameter accounts for the dipole-induced dipole and induced dipole-induced dipole interactions that can occur in compounds with large dispersion (polarizability) parameters. For compounds with large dispersion parameters, namely aromatic and halogenated compounds, the increased interaction of the dispersion forces tends to lessen the dipolar interactions, and thus the value of the induction parameter would be less than one. For aromatic compounds q is set to 0.9 and for halogenated compounds the polarity parameter is varied for best fit. Acidity and Basicity Parameters, r and β. The acidity and basicity parameters account for specific interactions due primarily to hydrogen bond formation through both association and solvation. As in the initial formulation, the R parameter was kept at a value of zero unless deemed chemically reasonable for that particular compound. The R parameter can also account for the Lewis acidity, as in the case of acetonitrile and nitromethane, where a nonnegligible value of R is necessary to correlate the data. In the case of alcohols the values of R and β were allowed to correlate independent of each other and were not forced to the same value. This resulted in better fits for the alcohols and a larger R parameter than β parameter for short chain alcohols with the R parameter decreasing rapidly with increasing carbon chain length so that at long chain lengths, e.g., 1-octanol, the β parameter is larger than the R parameter. The β parameter is also able to capture the strongly basic nature of compounds such as DMSO, DMF, and NMP which had the largest correlated β values. Parameters for Water. The magnitude and range of the infinite dilution activity coefficients for organics in water (10-1 to 1010) are much larger than the other organic data. In addition, the variability and discrepancies in experimental data are much larger for aqueous systems than most other organic solvent data due to experimental difficulties.45 Using the actual molar volume of water (18 mL/mol) in the model resulted in a poor fit of the data and gave unreasonably low values for the activity coefficient of water in organic solvents. Therefore, the molar volume of water was treated as an adjustable parameter with, and the optimum value was 36 cm3/mol. The extensive hydrogen bond network present in water could possibly cause water to act with a larger molar volume in solution. With this change, MOSCED is able to correlate the activity coefficients of the organics in water to 41.1% AAE, which is good considering that the uncertainty for nonpolar solutes in water (particularly for hydrocarbon-water mixtures) is close to 50% or even more in some cases. Parameters for Permanent Gases. MOSCED parameters for gaseous solutes were correlated with γ∞ calculated from Henry’s constant data at 1.103 bar and 298.15 K for oxygen,47 argon,48 nitrogen,49 carbon monoxide,50 and carbon dioxide51 and the hypothetical liquid fugacity from Prausnitz and Shair.52 In addition to the MOSCED parameters, the hypothetical-liquid molar volume was also fit. It was found that only the dispersion parameter was necessary to correlate accurately

Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4079

Figure 1. MOSCED prediction versus experimental Henry’s constants for Argon (b), Oxygen (0), Nitrogen (2), Carbon Monoxide (]), and Carbon Dioxide (3).

solubility data for oxygen, argon, nitrogen, and carbon monoxide. However for carbon dioxide, the polarity and acidity parameters along with the dispersion parameter were necessary for accurate correlation. The optimum values of the molar volume and parameters are found in Table 1, and experimental data and predictions are compared in Figure 1. The values for argon, oxygen, nitrogen, and carbon monoxide differ from the regular solution theory values4 because of the addition of the Flory-Huggins contribution with the MOSCED model. The optimum MOSCED parameters for carbon dioxide reveal interesting insights into the behavior of CO2 in solution. Although CO2 has no net dipole moment, it does have a quadrupole moment, and the nonzero polarity is necessary to explain the higher than expected solubility in polar solvents. The high solubility of CO2 in basic solvents may be explained by the Lewis acidity in solution of carbon dioxide. This acidity is accounted for in the small but significant R parameter allowing for accurate prediction of solubility. Parameters for Ionic Liquids. MOSCED parameters for nonelectrolyte solutes in two ionic liquids, 1-methyl-3-ethyl-imidazolium bis(trifluoromethyl-sulfonyl) amide and 1,2-dimethyl-3-ethyl-imidazolium bis(trifluoromethyl-sulfonyl) amide, correlated well with no change to the model. Application to Solid Solubility Prediction. The solubility of a pure crystalline solid, x, in a solvent is found from eq 9

[ (

)

(

)]

∆Cp Tm Tm ∆Hfus 1 1 1 +1 x) exp ln γ R Tm T R T T

(9)

where γ is the activity coefficient at a concentration of x, ∆Hfus is the enthalpy of fusion at the melting point temperature Tm, R is the universal gas constant, and ∆Cp is difference in heat capacity of the liquid and crystalline solute. Equation 9 makes the following valid assumptions: the effect of pressure is negligible,53 the difference between the heat capacity is insensitive to temperature changes, and the triple point temperature is the same as the melting point temperature. The exponential term in eq 9 is commonly named the ideal solubility. Although the heat capacity contribution to the overall solubility is small compared to the enthalpy of fusion term, it should not be neglected for compounds with melting points far from the temperature of interest. For example, if the temperature of interest is 298 K and the melting point of the solid is 498 K, a ∆Cp of 10 J/mol-K

will change the ideal solubility by 20%. Also, if there is any transformation in crystalline structure during dissolution of the solid, the enthalpy of that polymorphic transition must be added to correctly determine the ideal solubility. The binary interaction parameters in a 2-parameter gE expression (∆λ12 and ∆λ21 for Wilson, ∆u12 and ∆u21 for UNIQUAC) are fit to the two infinite dilution activity coefficients for solvent (1) and solid (2), γ1∞andγ2∞, which are calculated with MOSCED (Note: γ1∞ is for solvent in a hypothetical-liquid solid). The mole fraction concentration of the solute in the liquid phase, x, and the finite concentration activity coefficient, γ, calculated from the gE expression, are then found that satisfy the relationship in eq 9. The MOSCED parameters for the solid are found by minimizing the sum of squared error between experimental and calculated values for solubility in several different solvents. To validate the ability of the MOSCED model to describe accurately solid-liquid equilibria, solubility data for a solid solute in a variety of organic solvents are necessary. Solutes selected were limited to those with data in a variety of solvents to allow for accurate parametrization and demonstration of the capabilities of the model. We found literature data for 26 solids at ambient temperature that met this criterion. The regressed MOSCED parameters are shown in Table 3 along with the AAE% for both MOSCED and mod. UNIFAC. It should be noted that the deviations for MOSCED are for the correlation and the deviations for UNIFAC are for predicted values. For 10 of the 26 solutes studied, either necessary UNIFAC functional groups are missing or interaction parameters are not available, and no predictions for mod. UNIFAC were made. For all 26 solutes in this study, the MOSCED model with the Wilson gE model is able to correlate the 700 data points of solubility to an AAE of 24.9%. MOSCED performs similarly to the UNIFAC model for polyaromatic hydrocarbons and is superior in predicting solubility of polar and multi-functional solid compounds. From these dataset, five solid compounds that reflect a variety of structure and functionality were chosen to discuss in detail. The solubilities of phenanthrene (Tm ) 372.4 K, Hfus ) 3934.8 cal/mol, ∆Cp ) 3 cal/mol, Ttrans ) 339.2 K, Htrans ) 312.4 cal/mol) in 37 organic solvents54 were correlated with MOSCED and predicted with UNIFAC. The large dispersion term is expected for the polarizable π electrons in the poly-aromatic structure, which can act also as a weak base, reflected in the small β term. The UNIFAC model predicts the solubility to 37% absolute average error (AAE) and MOSCED is able to correlate the data to 23.8% AAE. A plot of the experimental mole fraction versus the predicted mole fraction is shown in Figure 2, where a perfect prediction of the data is represented by the solid line. The MOSCED model does not require extensive solubility data in order parametrize a given solute. Solubility data for a small but chemically diverse solvent set is sufficient to describe the possible interactions that a given solute can experience in solution. For example, from the 37 available data points of phenanthrene solubility, we select 6 solvents that cover a range of functionality: hexane (nonpolar), ethyl acetate (polar aprotic), 2-butanone (polar aprotic), acetonitrile (polar weak acid), ethanol (polar associated), and 2-propanol (polar associated). The best-fit parameters to this

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Table 3. Pure Component Parameters and Regressed MOSCED Parameters for Solid Solutesa solute

ν

λ

τ

q

R

β

2-hydroxybenzoic acid 2-nitro-5-methylphenol 4-nitro-5-methylphenol acenaphthene acetaminophen anthracene benzyl biphenyl diphenyl sulfone diuron fluoranthene hexachlorobenzene ibuprofen monuron naphthalene p-aminophenylacetic acid phenanthrene phenylacetic acid p-hydroxybenzoic acid p-hydroxyphenylacetic acid p-nitroaniline N,N-dimethyl-p-nitroaniline pyrene thianthrene trans-stilbene xanthene

119.6 123.5 123.5 137.8 105.4 183.3 183.0 149.4 161.5 164.8 184.4 164.9 214.2 152.8 131.0 137.9 167.1 110.9 95.7 123.9 131.6 140.8 184.9 156.0 188.6 150.0

14.72 17.10 17.60 18.26 18.45 18.02 18.90 17.28 16.75 16.99 19.96 19.64 15.20 16.44 17.78 16.08 18.48 14.29 15.16 14.45 18.15 18.02 18.63 19.54 17.98 19.07

4.77 7.76 7.06 4.31 2.67 4.73 6.25 5.09 9.74 4.12 4.59 2.13 5.02 5.48 4.53 7.23 5.31 1.95 3.68 4.18 8.94 8.25 5.81 5.14 4.79 4.84

0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

22.49 1.04 25.94 0.00 16.19 0.00 1.74 0.00 0.00 7.88 2.51 0.00 10.11 7.16 0.00 5.74 0.00 15.16 31.27 17.40 11.14 0.00 0.00 0.00 0.00 0.00

4.89 2.53 4.39 1.24 13.18 1.29 3.61 0.43 7.45 9.88 2.10 0.00 3.54 9.65 3.03 7.41 1.74 3.57 4.98 5.62 6.70 4.20 2.45 2.71 3.24 1.52

a

UNIFAC AAE% (n)

MOSCED AAE% (n)

ref

47.1% (14)

20.1% (14) 19.7% (17) 37.4% (19) 14.8% (27) 36.6% (19) 24.8% (36) 13.3% (28) 13.2% (32) 22.3% (32) 36.3% (37) 28.0% (32) 26.3% (30) 20.6% (38) 22.0% (32) 10.9% (33) 38.6% (25) 23.8% (37) 11.8% (9) 20.4% (9) 50.1% (32) 31.3% (39) 20.6% (33) 35.8% (27) 16.5% (18) 19.9% (16) 18.6% (27)

62 63 64 65 56 66 67 68 69 70 66 55 71 72 54 71 66 71 71 71 59 59 66 73 74 75

27.0% (36) 24.0% (43) 78.8% (36) 12.4% (42) 54.6% (45) 22.2% (38) 11.0% (36) 36.7% (37) 29.4% (9) 39.5% (9) 48.6% (30) 69.4% (17) 39.9% (30) 41.6% (16)

Average absolute error % (AAE%) and number of data points (n) for both UNIFAC and MOSCED predictions.

Figure 2. Predictions of the mole fraction solubility of phenanthrene with UNIFAC (b) and MOSCED (O).

smaller data set results in some small changes in the values, with the dispersion parameter increasing slightly to 18.93 from 18.48, the polarity parameters decreasing to 5.16 from 5.31, and the basicity parameter increasing to 2.38 from 1.74. These new parameters predict for the whole 37 point data set with a slight increase in absolute error to 26.3% and no increase in the number or magnitude of outliers. The solubilities of hexachlorobenzene (Tm ) 501.7 K, Hfus ) 6099.4 cal/mol) in 30 solvents55 for which MOSCED parameters were available were used to regress parameters. The best-fit parameters result in a 26% AAE (see Figure 3). The large dispersion term is a result of the number of free electrons in the benzene ring and attached chlorine atoms. The nonzero polarity parameter is consistent with that of other single ring aromatic solvents (benzene τ ) 3.95, toluene τ ) 3.22). There are no acidic moieties in the compound thus R ) 0 and the electron-withdrawing chlorines have eliminated the basicity of the aromatic ring. The results for the mod. UNIFAC model show a complete failure at predicting the solubility, with the possible exception of the solubility in 1,4-dioxane and less so in methanol. This may be because the model does not account for

Figure 3. Predictions of the mole fraction solubility of hexachlorobenzene with (b) and MOSCED (O).

neighboring-group effects and treats the six chlorine substituents as the sum of six single chlorine substituents, which where most likely fit to equilibria data with chlorobenzene. The capability of MOSCED to correlate a multifunctional molecule was tested with acetaminophen solubility data in 19 solvents.56 The large hydrogen bond donor value is expected because of the two acid protons in the molecule and the hydrogen bond acceptor value is reasonable because of the carbonyl and aromatic ring moieties. The smaller polarity term may be due to the para positioning of the two side groups-thus a small net dipole in solution. The MOSCED model is able to correlate the solubility data over 4 orders of magnitude in solubility with the results shown in Figure 4. A comparison of the solubility prediction with the mod. UNIFAC model is not possible because the molecule cannot be constructed with the available groups, due to a missing secondary amine attached to an aromatic carbon group. Rasmuson57 has suggested two approximations for building acetaminophen from the available mod. UNIFAC groups, but both approximations resulted in several very large deviations from experimental values.

Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4081

Figure 4. Predictions of the mole fraction solubility of acetaminophen with MOSCED (O).

Figure 5. Predictions of the mole fraction solubility of pnitroaniline with (b) and MOSCED (O) and N,N-dimethyl-pnitroaniline with MOSCED (4).

The solubilities of p-nitroaniline and N,N-dimethylp-nitroaniline were also considered since this pair of solvatochromic probes is used for Kamlet-Taft β scale.58 The scale is based upon the assumption that the only differences in interaction in solution are due to the change in the amine group from the acidic primary amine to the nonacidic tertiary amine. The solubilities of p-nitroaniline in 39 solvents and N,N-dimethyl-pnitroaniline in 33 solvents were used to regress the solute parameters.59 The results of the fit are shown in Figure 5. We can see from the regressed MOSCED parameters, as shown in Table 3, the dispersion and polarity terms are similar for the two compounds, and the difference in hydrogen bond acidity is expected, with a large term for p-nitroaniline (R ) 11.14) and zero for the dimethyl compound. We do see some more significant differences in the parameters for the hydrogen bond basicity term, which could be due to some differences in stability of the possible resonance structures of the two compounds. The mod. UNIFAC model does have an aromatic amine group available, but it is missing many of the interaction parameters for the solvents in this data set and for those available it generally under predicts the solubility. There is no aromatic tertiary amine group available in the mod. UNIFAC model and thus no predictions can be made for N,N-dimethyl-p-nitroaniline. The MOSCED model is readily extended to predict solid solubility in mixed solvents. Because the model predicts only the infinite dilution activity coefficients, the accurate prediction of solubility in mixed solvents is strongly dependent upon the ability of the gE model to predict the correct binary solvent behavior. There is often a solvent pair that will give a maximum in

Figure 6. Mole fraction solubility of 2-nitro-5-methylphenol in hexane + ethanol (b) Data from Buchowski et al. Solid line MOSCED prediction and Wilson model extrapolation.

Figure 7. Mole fraction solubility of acetaminophen in dioxane + water mixtures at 298.15 K. (b) Data of Bustamante et al.,. Solid line MOSCED with UNIQUAC extrapolation.

solubility. One example of a synergistic effect of a solvent mixture is the solubility of 2-nitro-5-methylphenol in a hexane/ethanol mixture.60 The prediction of the MOSCED model with the Wilson gE model is in good agreement with the experimental data as shown in Figure 6. One explanation for the existence of this maximum in solubility is the hexane interfering with the hydrogen bond network of the ethanol solvent sufficiently to allow some solvation of the 2-nitro-5methylphenol compound that possess both acidic and basic moieties. Another system that exhibits a maximum in solubility with a mixed solvent is the solubility of acetaminophen in a 1,4-dioxane + water mixture as measured by Bustamante.61 At a 50/50 mole ratio of solvent, acetaminophen has a solubility over four times greater than the solubility in pure 1,4-dioxane. This maximum at equal mole fraction implies a specific interaction of both solvents with the solute molecule. The acidic and basic moieties on the acetaminophen molecule are solvated by the basic ether and the acidic protons of the water molecule. As shown in Figure 7, the MOSCED model with the UNIQUAC gE model is able to predict the maximum in solubility at around a 50/50 mixture, however the magnitude of the maximum is underpredicted. Considering the challenge of predicting aqueous systems for many thermodynamic models, this result is promising. Conclusion New MOSCED parameters for 133 solvents and 5 gases were regressed to a large database of infinite

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dilution activity coefficients with an average absolute deviation of 10.6%. The MOSCED model is intuitively appealing and has the quantitative capabilities to aid in solvent selection for chemical processes. Because MOSCED predicts only the infinite dilution activity coefficients, any suitable activity coefficient model can be used to extrapolate to finite compositions. The model offers a useful companion to existing models such as the UNIFAC model for prediction of phase equilibria. Solid solubility can be correlated and predicted accurately using MOSCED to predict UNIQUAC or Wilson binary interaction parameters. For 26 solutes of various functionalities the solubility was predicted with an average error of 24.9%. Only limited solubility data (5 to 8 data points) in a chemically diverse solvent set are necessary to determine MOSCED parameters for a given solute, and then solubility can be predicted for other solvents. Acknowledgment The authors are grateful for financial support from The Dow Chemical Company, the National Science Foundation, and the donors of the J. Erskine Love, Jr., Institute Chair in Engineering. Supporting Information Available: Experimental infinite dilution activity coefficients used in the regression of the MOSCED parameters including absolute average deviation for both the MOSCED and UNIFAC models (Supporting Tables). This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Frank, T. C.; Downey, J. R.; Gupta, S. K. Quickly Screen Solvents for Organic Solids. Chem. Eng. Prog. 1999, 95, 41-61. (2) Gupta, S.; Olson, J. D. Industrial Needs in Physical Properties. Ind. Eng. Chem. Res. 2003, 42, 6359-6374. (3) Schreiber, L. B.; Eckert, C. A. Use of Infinite Dilution Activity Coefficients with Wilson’s Equation. Ind. Eng. Chem., Proc. Des. Dev. 1971, 10, 572. (4) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall: Englewood Cliffs, New Jersey, 1986. (5) Fredenslund, A.; Gmehling, J.; Michelsen, M. L.; Rasmussen, P.; Prausnitz, J. M. Computerized Design of Multicomponent Distillation Columns Using the UNIFAC Group Contribution Method for Calculation of Activity Coefficients. Ind. Eng. Chem. Proc. Des. Dev. 1977, 16, 450. (6) Tochigi, K.; Minami, S.; Kojima, K. Prediction of VaporLiquid Equilibria with Chemical Reaction by Analytical Solutions of Groups. J. Chem. Eng. Jpn. 1977, 10, 349-354. (7) Tochigi, K.; Tieges, D.; Gmehling, J.; Kojima, K. Determination of New ASOG Parameters. J. Chem. Eng. Japan 1990, 23, 453-463. (8) Hansen, C. M. Three-dimensional Solubility Parameter-Key to Paint-Component Affnities. 1. Solvents, Plasticizers, Polymers, and Resins. J. Paint. Technol. 1967, 39, 104-117. (9) Hansen, C. M. Solubility Parameters: A Users Handbook CRC Press: Boca Raton, 2000. (10) Thomas, E. R.; Eckert, C. A. Prediction of Limiting Activity Coefficients by a Modified Separation of Cohesive Energy Density Model and UNIFAC. Ind. Eng. Chem. Proc. Des. Dev. 1984, 23, 194-209. (11) Haile, J. M. Molecular Dynamics Simulation: Elementary Methods; Wiley: New York, 1992. (12) Sadus, R. J. Molecular Simulation of Henry’s Constant at Vapor-Liquid and Liquid-Liquid-Phase Boundaries. J. Phys. Chem. B 1997, 101, 3834-3838.

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Received for review September 10, 2004 Revised manuscript received March 11, 2005 Accepted March 23, 2005 IE049122G