Ind. Eng. Chem. Res. 2001, 40, 5029-5035
5029
Predicting Cosolvency. 3. Evaluation of the Extended Log-Linear Model An Li† Environmental and Occupational Health Sciences, School of Public Health, University of Illinois at Chicago, Chicago, Illinois 60612
In measurement of the solubility of organic compounds in mixed solvents containing water and miscible cosolvent(s), significant deviations from the widely used log-linear model have been observed. The deviations have been considered to be due mostly to the nonideality of the solvent mixture. In this paper, an approach to extend the log-linear model was evaluated. Activity coefficients of the solvent components were estimated using the UNIFAC group contribution method, and the sum of their logarithms weighted by either mole fractions or volume fractions was added to the log-linear model. Four forms of the extended model were evaluated using a large database with 13 different cosolvents and over 4000 experimental solubility data. The best form of the extended log-linear model generated improvement of 0.06-0.22 log units for 7 of the 13 water-cosolvent systems over the log-linear model in estimating the solute solubility in mixed solvents. Introduction In the previous papers of this series,1,2 we presented an approach in predicting the solubility of organic chemicals in mixed solvents containing water and cosolvent using the log-linear model, with the emphasis on predicting the model parameters from the chemical structures and easily obtainable properties of the solute and cosolvent. Prediction of aqueous-phase cosolvency using the approaches described in those papers is restricted to the situations where the ideal log-linear solubilization prevails. In reality, however, such ideal situations are rare. Most solubilization curves, which are plots of the logarithm solubility (log Sm) versus the solute-free volume fraction of a cosolvent (f), exhibit significant curvatures, which are not accounted for by the log-linear model. As presented in numerous publications,3-9 the deviation from the log-linear ideality can be either slight or severe, depending on both the solvent composition and the identity of the solute. Measured solubilization curves appear to be concave, sigmoidal, or convex. In many cases, especially with amphiprotic cosolvents, a negative deviation from the end-to-end loglinear line is often observed at low cosolvent concentrations, followed by a more significant positive deviation as the cosolvent fraction increases. Prediction of the solute behavior in complex multiplecomponent liquid systems is difficult. Several models have been derived from well-established thermodynamic concepts.10-16 Although some models give a more improved fit of the experimental data than the log-linear model, their application in predicting the solubility is often limited because of the difficulties in obtaining model parameters. Efforts to accurately predict the solubility of organic compounds in mixed solvents, especially those involving water, have achieved only limited success. The purpose of this paper is to evaluate one of the approaches to extend the log-linear model. The approach † Phone: 312-996-9597. Fax: 312-413-9898. E-mail: anli@ uic.edu.
was originally proposed by Pinal et al.,17 who added volume fraction weighted log activity coefficients to the log-linear model to account for the nonideality contributed from interactions between water and cosolvent. In this work, this approach is evaluated using a large database with over 4000 experimental solubility data obtained for various organic compounds in binary, ternary, and quinary solvent systems. Background Yalkowsky and Roseman3 introduced the log-linear model to describe the phenomenon of the exponential increase in the aqueous solubility for nonpolar organic compounds as the cosolvent concentration is increased. They showed that, for solute in binary solvent systems containing water and a cosolvent,
log Sm ) f log Sc + (1 - f) log Sw ) log Sw + σf
(1)
where S is the molar solubility of the solute in pure water (subscript w), in pure cosolvent (c), or in the mixture of water and cosolvent (m). The constant σ ) log(Sc/Sw) is the difference between logarithm solubilities of the solute in pure cosolvent and in pure water. The property and prediction of σ were discussed previously.1 If the number of cosolvents is more than 1, the solute solubility in the mixed solvent can be similarly expressed as a linear combination of its solubility in each of the pure solvent components (i), including water, weighted by the corresponding volume fractions:
log Sm ) log Sw +
∑σi fi
(2)
The ideality of the solutions was set by Raoult’s law, which states that the partial pressure of a component of a liquid mixture is the product of its mole fraction and its vapor pressure.18 Therefore, the total pressure above an ideal solution must be the sum of the saturated vapor pressures of the components weighted by their mole fractions in the solution. Similarly, the log-linear model assumes that the capability of a solution to accommodate a solute is the sum of the solubilities of
10.1021/ie010475e CCC: $20.00 © 2001 American Chemical Society Published on Web 09/29/2001
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Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
the solute in individual solvent components weighted by their volume fractions in the solvent mixture. Both Raoult’s law and the log-linear model apply exactly only when all of the components of the mixture behave identically. The nonideality of a mixture is quantitatively measured by the excess free energy of mixing
∆gE ) RT
∑(xi ln γi)
(3)
where R is the gas constant, T is the absolute temperature, and xi is the mole fraction of component i in the solution. The activity coefficient of component i, γi, is a measure of the extent of deviation from the ideal behavior. Defined by Raoult’s law, γ ) 1 for pure liquids. The nonideality of a solution containing water, solute(s), and cosolvent(s) results from the molecular interactions between its components and has to be quantified in order to accurately predict the effect of a cosolvent on the solute behavior. A generally accepted viewpoint is that the deviation from the log-linear solubilization is mainly caused by the nonideality of the solvent mixture. From this standpoint, Pinal et al.17 proposed that a term 2.303∑(fi log γi), which is the analogue to ∑(xi ln γi), be added to eq 1 to account for the effect of the solvent nonideality. A few examples were given in Pinal et al.17 to demonstrate the improvement of using the extended over the original log-linear model. This approach was further tested by Li20 using naphthalene, benzocaine, and benzoic acid in selected binary solvent mixtures. It was found that the extended log-linear model outperformed the log-linear model in more than half of the cases tested for the three solutes. The improvement was found to be highly inconsistent among different solution systems, with more occurring in regions with relatively high f values, where significant positive deviations from the log-linear pattern occur. Another approach to quantify the deviations of solubilization from the log-linear model makes use of an empirical parameter β. The modified log-linear equation then takes the form
log Sm ) log Sw + βσf
(4)
As suggested by Rao et al.,19 β reflects the extent of deviation caused by the nonideality of the solvent mixture. However, because β itself is a function of f, eq 4 does not provide additional aid for predicting the cosolvency unless the value of β can be estimated by an independent method. UNIFAC (UNIQUAC functional-group activity coefficients) is by far the most powerful and convenient tool for the prediction of activity coefficients in nonelectrolyte, nonpolymeric liquid mixtures.21 The basic assumption of UNIFAC is that a physical property of a fluid is due to the sum of contributions made by the molecule’s functional groups. UNIFAC divides the activity coefficient, γ, into a combinatorial part, γc, which reflects the size and shape of the molecules, and a residual portion γr, which depends on the functional group interactions.
ln γ ) ln γc + ln γr
(5)
The parameters of each functional group, including volume and area parameters (calculated via the normalization of van der Waals volume and surface area), and the parameters of interaction with other functional
groups (obtained by reduction of phase equilibrium experimental data) are put into a series of equations, from which γc and γr are calculated. Approach The database used for this work contains experimental data for more than 1000 ternary systems composed of water, a cosolvent, and a solute. It also contains data files of physicochemical properties of the cosolvents and solutes, as well as a few subroutine programs for different calculation purposes. In this study, data for 13 cosolvents are extracted from the database. Only binary solvent mixtures of water and a cosolvent were included. Solvent compositions originally reported in the literature have been converted to the volume fraction of the cosolvent and the units of solute solubility to moles per liter. Several forms of the extended log-linear model were tested and compared:
log Sm ) log Sw + σf + log γw + f log(γc/γw)
(6)
log Sm ) log Sw + σf + log γw + x log(γc/γw) (7) log Sm ) log Sw + σf + ln γw + f ln(γc/γw)
(8)
log Sm ) log Sw + σf + ln γw + x ln(γc/γw)
(9)
In these equations, f and x are the volume fraction and mole fraction, respectively, of the cosolvent in a solute-free solvent mixture. The logarithms of the last two terms on the right-hand sides are 10-based in eqs 6 and 7 and e-based in eqs 8 and 9. Both Sm and Sw are experimental values reported in the original literature. The cosolvency power, σ, was estimated for each dataset as the difference between the experimental logarithm solubilities of the solute in pure cosolvent (f ) 1) and those in water (f ) 0). Among these equations, eq 8 was the one originally proposed by Pinal et al.17 For multiplecomponent solvent systems, summation over all cosolvents involved is needed for the second and fourth terms on the right-hand sides of these equations. The activity coefficients of water (w) and cosolvent (c) in a solute-free mixed solvent were calculated using a UNIFAC program written in BASIC. The UNIFAC group interaction parameters were derived from vaporliquid equilibrium data.22 To use the UNIFAC program, the cosolvent volume fraction was first converted to mole fraction and the solution temperature was set to 25 °C. The calculated values of γ are summarized in Table 1. In application of these values to eqs 6-9 when the experimental cosolvent volume fraction f is not exactly those in Table 1, the values of γc and γw for the closest volume fraction were used. For instance, if the experimental volume fraction f is between 0.35 and 0.4 or between 0.4 and 0.45, the γ values at f ) 0.4 were used for that solvent composition. Results and Discussion Tables 2 and 3 summarize the results of statistical analysis of the errors produced by eqs 1 and 6-9 for binary and multiple solvent systems, respectively. Solubility data obtained at fi ) 0 or 1 were excluded in the data analysis, because in pure solvents the difference in log Sm between predicted and experimental values
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 5031 Table 1. UNIFAC-Derived Activity Coefficients for Selected Binary Water-Cosolvent Systems f 0a
0.1
0.2
0.7
0.8
0.9
1b
x γc γw
0.000 2.244 1.000
0.047 1.972 1.003
0.100 1.748 1.013
Methanol (MW ) 32.04, Density ) 0.7914) 0.160 0.229 0.308 0.400 1.564 1.413 1.289 1.189 1.029 1.055 1.091 1.140
0.509 1.135 1.197
0.640 1.052 1.298
0.800 1.014 1.424
1.000 1.000 1.604
x γc γw
0.000 7.620 1.000
0.033 5.550 1.005
0.072 4.119 1.022
Ethanol (MW ) 46.07, Density ) 0.7893) 0.117 0.171 0.236 0.316 3.120 2.416 1.917 1.564 1.051 1.097 1.163 1.256
0.418 1.318 1.386
0.552 1.152 1.570
0.735 1.050 1.854
1.000 1.000 2.661
x γc γw
0.000 20.050 1.000
0.026 12.770 1.006
0.057 8.323 1.024
1-Propanol (MW ) 60.1, Density ) 0.8053) 0.094 0.139 0.194 0.266 5.565 3.827 2.717 2.001 1.058 1.111 1.189 1.301
0.360 1.540 1.464
0.491 1.248 1.706
0.685 1.077 2.093
1.000 1.000 3.249
x γc γw
0.000 20.100 1.000
0.025 12.930 1.006
0.056 8.488 1.023
2-Propanol (MW ) 60.1, Density ) 0.7848) 0.092 0.135 0.190 0.261 5.697 3.921 2.779 2.040 1.056 1.107 1.183 1.294
0.354 1.561 1.455
0.485 1.258 1.695
0.679 1.080 2.084
1.000 1.000 3.254
x γc γw
0.000 11.470 1.000
0.026 8.786 1.004
0.058 6.724 1.015
Acetone (MW ) 58.08, Density ) 0.7899) 0.095 0.140 0.197 0.269 5.149 3.952 3.048 2.370 1.038 1.075 1.132 1.222
0.364 1.863 1.364
0.495 1.484 1.616
0.688 1.196 2.211
1.000 1.000 7.363
x γc γw
0.000 13.160 1.000
0.037 10.260 1.005
0.079 7.906 1.021
Acetonitrile (MW ) 41.05, Density ) 0.7857) 0.129 0.187 0.256 0.341 6.028 4.550 3.406 2.536 1.053 1.110 1.205 1.366
0.446 1.890 1.652
0.579 1.501 2.076
0.756 1.126 3.571
1.000 1.000 8.406
x γc γw
0.000 26.300 1.000
0.023 14.710 1.006
0.050 8.743 1.026
Dioxane (MW ) 88.11, Density ) 1.0329) 0.083 0.123 0.174 0.240 5.482 3.616 2.508 1.833 1.060 1.112 1.184 1.284
0.330 1.419 1.420
0.458 1.171 1.604
0.655 1.124 1.666
1.000 1.000 2.156
x γc γw
0.000 0.108 1.000
0.021 0.121 0.999
0.046 0.141 0.994
Dimethylacetamide (MW ) 87.12, Density ) 0.9429) 0.077 0.115 0.163 0.226 0.313 0.167 0.204 0.255 0.330 0.440 0.982 0.962 0.928 0.872 0.785
0.438 0.602 0.651
0.637 0.826 0.453
1.000 1.000 0.197
x γc γw
0.000 0.786 1.000
0.025 0.833 0.999
0.055 0.873 0.997
Dimethylformamide (MW ) 73.1, Density ) 0.9445) 0.091 0.134 0.189 0.259 0.352 0.905 0.930 0.949 0.962 0.973 0.995 0.991 0.988 0.984 0.979
0.482 0.983 0.972
0.677 0.985 0.969
1.000 1.000 0.925
x γc γw
0.000 0.057 1.000
0.027 0.080 0.996
0.060 0.110 0.981
Dimethyl Sulfoxide (MW ) 78.13, Density ) 1.1) 0.098 0.145 0.202 0.275 0.372 0.153 0.211 0.291 0.399 0.540 0.954 0.913 0.854 0.774 0.670
0.503 0.715 0.540
0.695 0.899 0.386
1.000 1.000 0.222
x γc γw
0.000 1.583 1.000
0.027 1.257 1.003
0.058 1.066 1.010
0.365 0.931 1.009
0.496 0.969 0.979
0.689 0.996 0.942
1.000 1.000 0.943
x γc γw
0.000 2.551 1.000
0.034 2.208 1.002
0.074 1.923 1.010
Ethylene Glycol (MW ) 62.07, Density ) 1.1088) 0.121 0.177 0.243 0.325 0.429 1.687 1.494 1.337 1.214 1.119 1.025 1.047 1.078 1.120 1.175
0.563 1.053 1.247
0.743 1.013 1.338
1.000 1.000 1.449
x γc γw
0.000 4.844 1.000
0.027 3.392 1.005
0.058 2.498 1.018
Propylene Glycol (MW )76.09, Density ) 1.0361) 0.095 0.140 0.197 0.269 0.364 1.932 1.567 1.331 1.177 1.092 1.039 1.069 1.104 1.145 1.185
0.495 1.044 1.224
0.688 1.019 1.267
1.000 1.000 1.437
0.3
0.4
0.5
0.6
Glycerol (MW ) 92.1, Density ) 1.2611) 0.096 0.141 0.198 0.270 0.957 0.903 0.887 0.899 1.019 1.027 1.030 1.025
a Values of γ at f ) 0 were obtained by setting the cosolvent mole fraction x ) 10-8. b Values of γ at f ) 1 were obtained by setting c w the water mole fraction 1 - x ) 10-8.
is zero by definition. Figures 1-3 illustrate comparison among eqs 2, 8, and 9, for solutes naphthalene, benzocaine, and benzoic acid, respectively, in selected binary water-cosolvent systems. By comparison of the averages summarized in Table 2, all forms of the extended log-linear model (eqs 6-9) outperform the log-linear model (eq 1) for most binary solvent systems. Equations 8 and 9 also improve the solubility prediction for a ternary system with cosolvents methanol and acetone, although there is no improvement for ternary systems containing methanol and acetonitrile (Table 3). Mixed results are obtained for quinary solvent systems containing equal amounts of
four cosolvents and varied amounts of water (Table 3). Among the four forms of the extended model, eq 9 is, in general, better than the others. It improves the prediction of the solute solubility over the original log-linear model by an overall average of 0.06-0.22 log units for 6 of the 13 cosolvents investigated including ethanol, 1-propanol, 2-propanol, acetone, acetonitrile, and dioxane. Equation 9 is based on the natural log of activity coefficients weighted by mole fractions, which is consistent with eq 2. For binary systems containing methanol or glycerol, none of the extended equations (eqs 6-9) generate more improved solubility estimates than the original log-
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Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Table 2. Error of Predictions (log Sm,estd - log Sm,expl) Using the Log-Linear and Extended Log-Linear Models for Binary Solvent Systemsa eq 1
eq 6
eq 7
cosolvent
N
aveb
SD
max
min
aveb
SD
max
min
aveb
SD
max
min
methanol ethanol 1-propanol 2-propanol glycerol ethylene glycol propylene glycol dioxane acetone acetonitrile dimethylacetamide dimethylformanide dimethyl sulfoxide overall
688 1631 116 188 124 111 503 349 220 103 97 133 131 4394
0.274 0.328 0.570 0.413 0.096 0.186 0.193 0.608 0.306 0.416 0.263 0.216 0.186 0.318
0.375 0.379 0.397 0.413 0.119 0.148 0.295 0.334 0.303 0.284 0.312 0.260 0.253 0.392
1.133 1.192 0.060 0.620 0.203 0.580 1.924 0.155 0.550 0.099 0.640 0.673 0.679 1.924
-1.524 -2.066 -1.402 -1.412 -0.284 -0.170 -1.846 -1.404 -1.034 -1.091 -0.780 -0.797 -0.634 -2.066
0.276 0.289 0.420 0.342 0.094 0.232 0.215 0.465 0.229 0.254 0.372 0.220 0.235 0.286
0.374 0.378 0.370 0.405 0.116 0.158 0.302 0.327 0.298 0.252 0.331 0.261 0.248 0.375
1.200 1.355 0.173 0.733 0.192 0.659 1.990 0.392 0.721 0.259 0.354 0.658 0.395 1.990
-1.456 -1.977 -1.221 -1.267 -0.279 -0.100 -1.813 -1.167 -0.863 -0.788 -1.093 -0.811 -0.918 -1.977
0.274 0.288 0.473 0.349 0.095 0.218 0.196 0.507 0.241 0.290 0.310 0.218 0.222 0.287
0.372 0.367 0.368 0.389 0.118 0.154 0.293 0.315 0.299 0.260 0.323 0.261 0.271 0.372
1.175 1.269 0.207 0.651 0.207 0.635 1.986 0.237 0.604 0.244 0.546 0.667 0.489 1.986
-1.460 -1.944 -1.229 -1.239 -0.278 -0.109 -1.792 -1.273 -0.980 -0.858 -0.904 -0.802 -0.825 -1.944
eq 8
eq 9
cosolvent
N
aveb
SD
max
min
aveb
SD
max
min
methanol ethanol 1-propanol 2-propanol glycerol ethylene glycol propylene glycol dioxane acetone acetonitrile dimethylacetamide dimethylformanide dimethyl sulfoxide overall
688 1631 116 188 124 111 503 349 220 103 97 133 131 4394
0.288 0.297 0.290 0.320 0.093 0.300 0.263 0.343 0.301 0.207 0.607 0.225 0.477 0.298
0.376 0.387 0.356 0.413 0.114 0.175 0.315 0.343 0.310 0.245 0.399 0.262 0.273 0.394
1.287 1.567 0.373 0.916 0.178 0.763 2.076 0.704 0.963 0.596 0.214 0.639 0.051 2.076
-1.367 -1.862 -1.020 -1.097 -0.277 -0.019 -1.771 -0.983 -0.731 -0.578 -1.501 -0.830 -1.288 -1.862
0.280 0.261 0.356 0.291 0.095 0.265 0.212 0.386 0.246 0.208 0.412 0.221 0.316 0.269
0.371 0.357 0.336 0.366 0.117 0.164 0.292 0.295 0.310 0.256 0.357 0.262 0.311 0.362
1.229 1.368 0.407 0.691 0.211 0.708 2.066 0.347 0.681 0.562 0.423 0.659 0.256 2.066
-1.375 -1.786 -1.004 -1.014 -0.273 -0.030 -1.723 -1.104 -0.910 -0.613 -1.065 -0.810 -1.073 -1.786
a
Experimental solubility data obtained at f ) 0 and 1 were excluded. b Average of the absolute errors.
Table 3. Error of Predictions (log Sm,estd - log Sm,expl) Using the Log-Linear and Extended Log-Linear Models for Ternary and Quinary Solvent Systemsa eq 2 solute
N
aveb
SD
max
eq 8
eq 9 min
aveb
0.006 0.006
0.330 0.102
0.365 0.071
1.227 0.289
0.019 0.012
Cosolvents ) Methanol + Acetonitrile 0.000 0.221 0.106 0.518
0.000
min
aveb
SD
max
SD
max
min
benzene anthracene
13 31
0.392 0.176
0.218 0.126
Cosolvents ) Methanol + Acetone 0.917 0.116 0.354 0.419 1.311 0.494 0.013 0.143 0.087 0.321
naphthalene
64
0.173
0.140
0.560
0.176
0.100
0.460
0.000
0.361 0.132 0.496
Cosolvents ) Methanol + Ethanol + Isopropyl Alcohol + Acetonitrile 0.345 0.190 -0.852 0.401 0.163 0.543 -0.358 0.315 0.084 -0.193 -0.193 0.439 0.102 0.556 0.327 0.192 0.189 -0.258 -0.744 0.127 0.081 0.204 -0.224 0.298
0.213 0.104 0.103
0.316 0.359 -0.132
-0.580 -0.083 -0.405
benzene naphthalene anthracene
4 5 5
a All experimental data are from ref 23. Data obtained at f ) 0 and 1 are excluded. The range of f is from 0 to 1 for ternary systems. i i The quinary solvent systems are composed of equal volumes of the four cosolvents (fi ) 0-0.25) with varying amounts of water. b Average of the absolute errors.
linear model. For ethylene glycol and propylene glycol, the extended model produces higher averaged errors than the log-linear model. These cosolvents are highly polar, and the log-linear solubilization mostly prevails with the mixture of water and these polar organic solvents. In all cases except those with dimethylacetamide, dimethylformanide, dimethyl sulfoxide, and glycerol, activity coefficients produced by UNIFAC are greater than unity (see Table 1). Therefore, the extended models improve the predictions when the experimental solubility is higher than that predicted on a log-linear basis. This is the situation mostly with relatively high f values. In the low-f regions, negative deviations of solubilities from the log-linear pattern are often observed but are not accounted for by the extended log-linear model. In
some cases, such as naphthalene in methanol and propylene glycol and benzoic acid in ethylene glycol, the log-linear model already overpredicts the solubility over the entire f range of 0-1. In these cases, the extended log-linear models do not offer better estimates than the original log-linear model. The bias of the original and extended log-linear models at different f values is presented in Table 4 for three binary solvent systems, for which the extended models (eqs 8 and 9) show significant (1-propanol), moderate (acetone), and no (propylene glycol) improvement over the log-linear model. While 100% of the measured solubility data deviate positively from the log-linear pattern in the 1-propanol-water system in the f range of 0.2-0.8, the corrections by the extended models are not sufficient although the average errors are largely reduced. By
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 5033 Table 4. Bias of the Log-Linear and Extended Log-Linear Models for Selected Binary Solvent Systems 1- propanol cosolvent
eq 1
eq 8
acetone eq 9
eq 1 f e 0.2 0.286 63 37
propylene glycol
eq 8
eq 9
eq 1
eq 8
eq 9
0.315 18 82 (N ) 49)
0.266 53 47
0.149 18 82
0.287 3 98 (N ) 120)
0.189 13 87
ave errora % -b % +b
0.164 91 9
0.14 0 100 (N ) 22)
0.092 73 27
ave error %%+
0.611 100 0
0.131 55 45 (N ) 22)
0.374 100 0
f ) 0.2-0.4 0.334 0.407 66 22 34 78 (N ) 50)
0.305 46 54
0.199 28 72
0.360 6 94 (N ) 120)
0.266 11 89
ave error %%+
0.808 100 0
0.272 86 14 (N ) 43)
0.463 100 0
f ) 0.4-0.6 0.37 0.315 92 18 8 82 (N ) 49)
0.218 47 53
0.218 60 40
0.248 24 76 (N ) 116)
0.232 29 71
ave error %%+
0.857 100 0
0.535 89 11 (N ) 28)
0.511 89 11
f ) 0.6-0.8 0.298 0.241 90 18 10 82 (N ) 51)
0.210 25 75
0.22 70 30
0.248 62 38 (N ) 113)
0.232 50 50
ave error %%+
0.351 87 13
0.307 57 43 (N ) 23)
0.304 48 52
0.216 5 95
0.161 74 26
0.142 68 32 (N ) 34)
0.132 50 50
f g 0.8 0.166 86 14
0.132 27 73 (N ) 22)
a Average absolute errors, with experimental solubility data obtained at f ) 0 and 1 being excluded. b % - ) percent data underestimated by the model (log Sm,estd - log Sm,expl < 0); % + ) percent data overestimated by the model (log Sm,estd - log Sm,expl > 0).
Figure 1. Deviations from the log-linear model (eq 1, circle) and the modified log-linear models (eq 8, square; eq 9, diamond) for solute naphthalene in various water-cosolvent systems. Experimental data are from ref 24.
Figure 2. Deviations from the log-linear model (eq 1, circle) and the modified log-linear models (eq 8, square; eq 9, diamond) for solute benzocaine in various water-cosolvent systems. Experimental data are from refs 25 and 26.
contrast, mostly underestimated solute solubilities in the acetone system with f > 0.6 are overcorrected by the extended models. Apparently, this approach depends on the accuracy of the activity coefficient. UNIFAC is the most widely used program for estimating activity coefficients for liquid mixtures. However, although the UNIFAC calculation algorithm was theoretically derived, its group interaction parameters have to be obtained entirely empirically from experimental phase-equilibria data. There is a possibility that the UNIFAC group interaction parameters involved are incorrect or inaccurate,
especially those between water and functional groups of dimethylacetamide, dimethyl sulfoxide, or dimethylformamide. It should be noted that there is more than one way to extend the log-linear model. The approach demonstrated in this paper resembles the excess free-energy model12 in that the solute solubility in a mixed solvent is presented as the sum of solubilities in pure solvent components weighted by their volume fractions with additional terms of extension. The differences are in the expression of the extension terms. In the excess freeenergy model, the additional terms are the infinite
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Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
log-linear pattern. More research is warranted to further understand the role of solutes. Literature Cited
Figure 3. Deviations from the log-linear model (eq 1, circle) and the modified log-linear models (eq 8, square; eq 9, diamond) for solute benzoic acid in various water-cosolvent systems. Experimental data are from refs 3 and 27.
dilution activity coefficient of a solute in a solvent mixture expressed as a power series in the volume fractions of solvent components. The extended log-linear model evaluated in this work was based on the contemplation that the “excess” activity coefficient of the solute is largely determined by the solvent nonideality, which can be expressed as weighted activity coefficients of the solvent components. An apparent limitation of the approach presented in this paper is the exclusion of any active role the solute may play on the observed deviation. The extended loglinear model claims the same magnitude of corrections for all of the solutes in the same solvent systems. This is not justified even for solutes without specific interactions with the solvent components. In this work, almost all large errors were associated with solutes with multiple polar functional groups. In ethanol-water systems, for example, errors greater than 1 log unit were found for solutes diazepam, mannite, nitrophenols, analine, triglycine, glycylglycine, aspartic acid, and aminocaproic acid. Little understanding of the influence of the solute structure and properties on deviations from the log-linear equation has been obtained. Although the patterns of deviations tend to be similar among many solutes, as mentioned in the Introduction, the extent of deviation is solute-dependent. For instance, Rubino and Obeng8 reported that C1-C4 alkyl esters of p-hydroxybenzoates and p-aminobenzoates demonstrated similar characteristics of solubilization by propylene glycol, but the magnitude of the negative deviation in the low-f region was found to be related to the length of the solute alkyl chain in each group, while that of the positive deviation in the high-f region was found to be related to the type of polar groups attached. For solutes with no polar functional groups, such as benzene, naphthalene, and phenanthrene, the extent of negative deviation diminishes as the molecular size increases.4 For nonionizable solutes, both the hydrophobicity and hydrogenbonding property of the solutes seem to be important in influencing the extent of the deviation from the ideal
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Received for review May 29, 2001 Revised manuscript received August 6, 2001 Accepted August 8, 2001 IE010475E
