Molecular Thermodynamic Modeling of Mixed Solvent Solubility

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Ind. Eng. Chem. Res. 2010, 49, 11620–11632

Molecular Thermodynamic Modeling of Mixed Solvent Solubility Martin D. Ellegaard,† Jens Abildskov,*,† and John P. O’Connell‡ Department of Chemical and Biochemical Engineering, Technical UniVersity of Denmark, Building 229, Søltofts Plads, 2800 Kgs. Lyngby, Denmark, and Department of Chemical Engineering, UniVersity of Virginia, 102 Engineers’ Way, P.O. Box 400741, CharlottesVille, Virginia 22904-4741, United States

A method based on statistical mechanical fluctuation solution theory for composition derivatives of activity coefficients is employed for estimating dilute solubilities of 11 solid pharmaceutical solutes in nearly 70 mixed aqueous and nonaqueous solvent systems. The solvent mixtures range from nearly ideal to strongly nonideal. The database covers a temperature range from 293 to 323 K. Comparisons with available data and other existing solubility methods show that the method successfully describes a variety of observed mixed solvent solubility behaviors using solute-solvent parameters from global regression of ternary data as well as predictions based on pure solvent solubilities with an average error of about 10% on mole fractions. 1. Introduction Many separations for production of solid substances of traditional chemicals and pharmaceutical products are more effective when mixtures of solvents are employed compared to using pure solvents. This is particularly important to the design of separations, since the functionality of chemical products can depend strongly upon formulations with mixed solvents. Solubilities of solids in single solvents have been reported extensively.1-4 However, there are fewer literature values for solubilities in mixed solvents, and the large number of available solvent mixtures makes thorough experimental testing infeasible. Therefore, accurate prediction of the solubilities in mixtures using simple methods would be a powerful tool for solvent selection. However, accurate prediction is not straightforward, since the dependence of solubilities on solvent composition can be quite complex, ranging from a linear variation to multiple extrema. Previously, we developed a method for predicting the solubility of single solutes in pure and mixed solvents.5-7 That method used a “reference solvent”, for which the solubility is known, to predict the solubility of the solute in other solvents. The method estimated differences between infinite dilution activity coefficients of the solute in different solvents, such as might be calculated from group contribution methods. Although one key contribution of that method was systematic identification of a minimum number of significant parameters (and estimation of their values only), that number could still be unacceptably large in some cases. In addition, reference solvent selection, though rule-based, was essentially empirical. Several other attempts to model the solubility of solids in mixed solvents exist in the engineering literature. They tend to fall in either of two categories: (i) thermodynamic models expressing equilibrium criteria and solving these with respect to relevant state variables and (ii) models based on the “excess” solubility concept. The first approach requires a model for the activity coefficient of the solute in the mixed solvent. UNIFAC8 predicts Lewis-Randall liquid-phase activity coefficients from component molecular structures and compositions. UNIFAC is capable of describing both vapor-liquid and (low temperature) * To whom all correspondence should be addressed. E-mail: [email protected]. Tel: +45-45252905. Fax: +45-45932906. † Technical University of Denmark. ‡ University of Virginia.

solid-liquid9 equilibria using the same set of model parameters. While this is a powerful and very general way of predicting liquid-phase nonideality, it does not adequately predict the properties of polyfunctional molecular structures, as we have shown for certain cases.10-12 Also, parameters for many groups of these substances are frequently unavailable. Other methods for computing liquid-phase nonideality are based on equations of state. Amino acid solubility in aqueous alcohol solutions13,14 using the PC-SAFT15 equation of state is an example. While the results can be in good agreement with reported experimental values, this method can also require up to five parameters for each pure component. Furthermore, in addition to often being computationally expensive, the method is highly sensitive to the binary parameters used to describe interactions between unlike molecular species. Furthermore, the scheme of association for species association and cross-association must be known; choosing such a scheme is not always obvious, and may even need to be changed with composition.16 Models based on the excess solubility concept divide into two different categories. One frequently cited approach is the algebraic mixing rule (otherwise referred to as the log-linear model), where the solubility of a solid in the mixed solvent is given by a linear combination of the pure solvent solubilities, weighted with respect to solvent composition. This allows the mixed solvent solubility to be predicted from the solubilities of the solute in the pure species alone, i.e., ideal mixing. It is often associated with Yalkowsky and co-workers,17 but its essentials were employed for gas solubility (and outlined for solid solubility) studies already in the 1960s by Kehiaian18-21 and by O’Connell and Prausnitz.22 A similar form for solid solubilities was later implemented by Williams and Amidon.23-25 Most real systems do not conform to the underlying assumptions inherent in the log-linear approach, and experimental drug solubilities can differ significantly from its estimates. The other approach expresses the excess solubility as a function of solvent composition such as with polynomials ranging from one to six adjustable parameters, as reviewed by Jouyban et al.26,27 Models with less than two parameters are rarely adequate for generalizations. While for the more parametrized versions, the parameters might be interpreted as characterizing solute- and solvent-solvent interactions, their values come only from regression to experimental data on multicom-

10.1021/ie101059y  2010 American Chemical Society Published on Web 09/29/2010

Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

ponent systems. Thus, the major limitation of excess solubility models is the availability of parameters for systems with limited data. This paper deals with a simple, yet theoretically generalizable, approach to the excess solubility of a solid solute in a binary mixed solvent. The method28 uses a fundamental solution theory arising from statistical mechanics.29,30 Recently we used this approach to estimate and predict the excess solubility of a range of five solutes in mixed solvents. In the present work, we extend this method to solubilities of 11 solid pharmaceutical solutes (steroids, NSAIDs, antipyretics, etc.) in mixed solvents. We also demonstrate how three-component equilibrium can be predicted from binary data alone. We begin with some basic definitions and expressions and then present results of cases of mixed solvent solubility correlation and prediction.

ln xiγi(T, x) ) -

(

solvents

∑

ln xEi ≡ ln xi,m -

(1)

x′jln xi,j

j*i

Here x′j is the solute-free mole fraction of solvent j, xi,j is the mole fraction solubility of solute i in pure solvent j, and xi,m is the solubility of solute i in the solvent mixture. We focus on solid-liquid equilibria for which the solid phase is pure and the liquid phase is a mixture of solvents saturated with the solute. The Lewis-Randall framework for such cases involves the ratio of the solute fugacities in the pure subcooled liquid (L) and solid state (S1) at the temperature and pressure of the solution, related to the molar Gibbs energy difference between the pure solid and hypothetical subcooled liquid at the system temperature31,32 ln

f f

S1 i(pure solid)

L i(pure subcooled liquid)

) ln xiγi(T, x) ≡ ln xid i (T) )

L S1 gi(pure solid) - gi(pure subcooled liquid) RT

(2)

The expression for ln xidi (T) in terms of measurable properties is developed from a thermodynamic process on the pure solute. Appendix A derives the rigorous expression for the case of a solid with a first-order phase transition between the melting temperature, Tm,i and the system temperature, T:

(

) )

S1 L gi(pure ∆hm,i 1 solid) - gi(pure subcooled liquid) 1 )RT R T Tm,i Tm,i ∆ht,i 1 ∆cP,m,i T 1 ln -1+ R Tm,i T R Tt,i T Tt,i ∆cP,t,i ln +1R T

ln xiγi(T, x) ≈ -

[( )

]

( [( )

+

Tt,i T

]

(3)

The properties involved are the enthalpy changes for melting at Tm,i and transition at Tt,i, and the corresponding heat capacity differences between the solid and liquid phases at the melting point, ∆cP,m,i, as well as the heat capacity change from one solid phase to the other, ∆cP,t,i, both of which are assumed to be temperature independent. The full form is rarely used. For example, if there are no known phase transitions between T and Tm,i:

(4)

(

∆hm,i 1 1 R T Tm,i

)

(5)

Alternatively, if Tm,i is much greater than T, the contribution from ∆cP,m,i can be significant. Another frequently made choice is to assume that ∆cP,m,i ≈ ∆hm,i/Tm,i ) ∆sm,i, which gives ln xiγi(T, x) ≈ -

( )

∆hm,i Tm,i ln RTm,i T

(6)

Equations 5 and 6 have been evaluated extensively. Yalkowsky33 concluded that eq 5 is a good approximation for the solubilities of naphthalene, anthracene, phenanthrene, and fluorene in benzene solutions. Neau and Flynn34 concluded that eq 6 is generally better than eq 5, except for flat, rigid molecules, such as those studied by Yalkowsky. In the absence of heat capacities from either data or estimation,35,36 it is not clear how to obtain values for the right-hand side of eq 3. Fortunately, as long as the solute crystals are the same in all solvents, it is independent of the solvent structure(s), and the uncertain terms cancel out in the excess solubility, eq 1. If, in addition, the solubility in mole fraction is smaller than 0.01, the activity coefficient of eq 2 can be set to its infinite dilution value, γ∞i . Then, solvents ∞ ln xEi ≈ -ln γi,m +

∑

x'jln γ∞i,j

(7)

j*i

This connects the excess solubility to the excess Henry’s law constant, one being the negative of the other. The excess Henry’s law constant is more commonly used in gas solubility studies,22 while the excess solubility is more commonly used in solid solubility studies.23-25,28 2.2. Fluctuation Solution Theory Method. The fluctuation solution theory (FST) of Kirkwood and Buff29 relates integrals of the statistical mechanical radial distribution function to solution isothermal compressibility, component partial molar volumes, and composition derivatives of component chemical potentials. The property connections to the total correlation function integrals (TCFI) between components i and j Hij ) Hji ≡ F

+

]

If ∆cP,m,i ≈ 0, or T is close to Tm,i, a good approximation is

2. Method The concepts of “excess” and “ideal” solubility are given first, followed by the solution theory expressing the composition dependence of these quantities. 2.1. Concepts. The excess solubility22 is defined:

[( )

)

∆hm,i 1 ∆cP,m,i 1 T + ln R T Tm,i R Tm,i Tm,i 1+ T

11621

∫ ∫ [g (r, Ω) - 1]4πr

2

ij

dr dΩ

(8)

were written by O’Connell30 in a more general and convenient matrix form, including formal expressions for multicomponent systems. In eq 8, F is the bulk molecular solution density, and gij(r,Ω) is the radial distribution function between molecular centers of i and j in orientations characterized by Ω. Compact expressions for composition derivatives may be written in terms of the TCFI. For a binary system, one useful result is

( ) ∂ ln γ1 ∂x1

)-

T,P,N2

x2f12 1 + x1x2 f12

(9)

Here the matrix f is defined by fij ≡ Hii + Hjj - 2Hij

(10)

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Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

The composition dependence of Hij is generally not known and may be complex, so integrating eq 9 to obtain the activity coefficient is not normally done rigorously. Rather, the activity coefficient is expanded37 about the infinite dilution standard state to give

( )

ln γi ) ln γ∞i + xi

∂ ln γi ∂xi

∞

+

T,P,Nj*i

( )

x2i ∂2 ln γi 2 ∂x2 i

∞

+ ...

T,P,Nj*i

(11)

Depending upon the form of eq 11, which is assumed to be valid over the entire solvent composition range, the excess solubility of a solute, 1, in a mixed binary solvent (of components 2 and 3) can be expressed as

( )

x3 ∂ ln γ3 ln xE1 ) 2 ∂x3

+

T,P,N2

(

)

x2 f+ 12 - f 012 2 1 + x x f+ 2 3 23 f+ x3 13 - f 013 2 1 + x x f+

(

2 3 23

)

(12)

Here + denotes infinite dilution of the solute in the mixed 0 solvent and f 1i indicates the property when component 1 is at infinite dilution in pure solvent i. The first term on the righthand side of eq 12 is correct to all orders of eq 11. Thus, its contribution is not empirical and it may be calculated from any gE-model describing the binary solvent mixture of 2 with 3. Similarly, a rearranged form of eq 9 gives a property independent of the solute component 1,

f+ 23 ) -

( ) ( )

1 ∂ ln γ3 x2 ∂x3

+

T,P,N2

∂ ln γ3 1 + x3 ∂x3

+

(13)

T,P,N2

where f +23 can be determined from a gE-model for the 2-3 binary mixture. Our fundamental assumption for implementing eq 12 is that the parameter values for a mixture equal the corresponding parameter values for pure solvents, i.e. 0 f+ 1j: ) f 1j

(14)

The radial distribution function between components i and j is a measure of their total correlations, including indirect effects. The fact that the elements of f are differences suggests that this approximation should be reasonable except when the solution density varies strongly with solvent composition. This is rarely the case with organic solvent mixtures, though the density variations of aqueous mixtures with organics can be large. The result of eq 14 in eq 12 is that the expression of the excess solubility simplifies to ln xE1 ≈ -

( )

x3 ∂ ln γ3 2 ∂x3

+

[1 + x2 f 012 + x3 f 013]

(15)

T,P,N2

Thus, the parameters for this model (f0) are only solute-solvent parameters and their basis in a fundamental solution theory allows us to estimate their values from binary data. Note that f 01j will be the same for that solute-solvent pair, 1-j, regardless of the identity of the other solvent. While the derivative in eq 15 might suggest an asymmetry toward component 3, the Gibbs-Duhem equation applied to the 2 + 3 solvent mixture

+ is identical to x2(∂ ln shows that the product x3(∂ ln γ3/∂x3)T,P,N 2 + γ2/∂x2)T,P,N3.Otherworkershavemodeledequilibriumsituations38-41 for systems containing proteins and smaller molecules in mixed solvent systems with the theory of Kirkwood and Buff as their basis. Also, Mazo and Smith42,43 applied the Kirkwood-Buff theory to the solubility of electrolytes in mixed solvents. 2.3. Solvent Mixture Representation. Equation 15 requires a gE-model for computing the activity coefficient derivative of the solvent mixture. Such derivatives are determined using the expression

( ) ∂ ln γi ∂xj

) T,P,Nk*j

( ) ∂ ln γi ∂Nj

∂xj ∂N T,P,Nk*j j )

|

-1

T,P,Nk*j

( )

∂ ln γi N 1 - xj ∂Nj

(16)

T,P,Nk*j

We have considered three models for application in eq 15: The Wilson equation,44 the modified Margules equation,45 and UNIFAC.8,46 The relevant expressions are given in Appendix B. Wilson or Margules model parameters will not be available if no vapor-liquid data have been reported, but UNIFAC may be used in such cases. 3. Data Reduction We need to determine values of the f0 in eq 15. Below we will explore two ways of doing that. For both, we compare model performance with ternary data from various sources. While the method is written in terms of mole fractions, some experimental data are published in units other than mole fraction, e.g. molar concentrations or weight of solute per weight of solvent mixture. In the cases of volumetric units our conversion to mole fraction neglected the volumetric contribution of the solute. The total molar volume is estimated from the molar volumes of the solvent species, i.e., calculated by ignoring volumetric nonideality. Equation 7 is valid at infinite dilution (typically when the mole fraction solubility is less than 1%) but also in systems where the solute molar volume is significantly less than the solvent volumes. This is often the case with the present systems. 3.1. Binary Parameters from Ternary Data. Values of the f0 in eq 15 can be found from global regression of ternary experimental information by minimizing the sum of squared residuals, data points

min f0

∑ (ln xˆ

E 1

- ln xE1 )j2

(17)

j

where the circumflex denotes a calculated value using eq 15. The linear estimation problem in eq 17 has a computationally favorable unique solution. Variances of parameter estimates can be obtained from the underlying variance-covariance matrix.47 We have used the full variance-covariance matrix in an errorpropagation expression to calculate the standard deviations plotted as error bars in the figures below. The solute-free parameters used with the gE-model for the binary (2 + 3) mixture are assumed to have zero variance. The total number of parameters estimated is equal to the number of possible solute-solvent combinations. 3.2. Binary Parameters from Binary Data. While the regression above can provide accurate values of f0, obtaining f0 from binary data is more advantageous whenever possible. Consider eq 13 for the case of component 1 in component 2,

{

f 012 ) lim x2f1

( ) ( )

1 ∂ ln γ1 x2 ∂x1 1 + x1

T,P,N2

∂ ln γ1 ∂x1

T,P,N2

}

Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010 Table 1. Thermophysical Properties of Solutes

)-

( ) ∂ ln γ1 ∂x1

0

T,P,N2

(18)

which, with eq 15, leads to ln xE1 ) -

( ) [ ( ) ( ) ]

x3 ∂ ln γ3 2 ∂x3

+

1 - x2

T,P,N2

∂ ln γ1 ∂x1

0

-

T,P,N2

∂ ln γ1 x3 ∂x1

0

(19)

T,P,N3

This equation requires determination of the activity coefficient derivatives of solutes in the pure solvents. The solubility of the solute in a pure solvent allows a single parameter of a gE-model to be obtained. The simplest model of this form is the Porter E /RTx1x2 ) B12, equation, which for a mixture of 1 and 2 is g12 where B12 is a temperature-dependent characteristic parameter for the 1-2 pair, gE12 /RTx1x2 ) B12 ⇒ ln γ1 ) B12(1 - x1)2 ⇒ ∂ ln γ1 0 ) -2B12 ) -f 012 (20) ∂x1 T,P,N2

( )

A value of B12 can be obtained from the solubility of 1 in 2, x1,2, and the ideal solubility of a form of eq 2, giving f 01j(T) ) 2

11623

a

ln xid 1 (T) - ln x1,j(T) (1 - x1,j(T))2

(21)

Thus, eq 21 connects mixed solvent solubility data to f0, and values of these parameters may be regressed from ternary data with eq 17 or estimated from binary data with eq 21. Note that determining B1j for a solute 1 in pure solvent j requires the solidphase thermophysical constants of eq 3 in addition to the experimental solubility. Thus, the cancelation of these terms in the excess solubility equation does not occur here. Note that a more accurate gE-model for eq 20 would be expected to provide better results, particularly if the symmetric composition dependence of the Porter equation is incorrect. However, such models require more than one parameter. Unlike the parameter estimation in eq 17, the use of eq 21 with the model does not require regression of data. 4. Results Table 1 shows the thermophysical properties needed to calculate ideal solubility values. We have compiled nearly 70 data sets on 11 pharmaceutical solutes in a variety of solvent mixtures, ranging from almost ideal to strongly nonideal solutions. Table 2 summarizes the sets treated in this work with literature references for solubilities, temperatures of measurement, references for vapor-liquid equilibrium data for the binary solvent gE-model, f0 parameters with standard deviations from ternary data regression using eq 17 (columns 7-10), and f0 parameters estimated from binary data using eq 21 (columns 11-12). Again, the model requires that the f0 parameter characterizing a given molecular pair be the same for all systems where that pair occurs, as in the cases of Table 2. Representative results are shown below, while properties, parameters, and comparisons with data for all systems are can be obtained as Supporting Information.

solute, i

Tm,i (K)

∆hm,i (kJ mol-1)

cholesterol desmosterol mefenamic acid paracetamol phenacetin sulfamethazine sulfamethoxypyridazine sulfanilamide theophylline

421.4374,75,82,89 384.6474 503.55178 443.0191-93 407.4095 470.6396-98 453.5353,99,54 437.0057 546.5670-73

26.6374,82 15.9074 38.2478 27.8591,92,92 28.7565 31.1496,98 30.0753,99,54 23.6557 29.6570-73

∆cP,m,i (J mol-1 K-1) 8.890 99.894

a Values with more than one reference represent an unweighted average.

4.1. Parameter Regression from Ternary Data. The agreement of the model with data is nearly quantitative when the f0 is regressed from eq 17 using all of the available ternary solubility data for a particular solute. An illustration is the solute aminopyrine in the binary mixtures of water-ethanol and water-dioxane, as shown in Figure 1. Solubilities and excess solubilities are displayed on the vertical axis. The symbols denote experimental measurements. The solid curve gives the 0 obtained using eq 17. results with the regressed parameters f 1j The dashes give the ideal mixture solubility from equating to zero the excess solubility in eq 7. The asymmetry of the solubility is fully captured by the model using the same f0 parameter for aminopyrine with water. We were unable to find thermophysical property information 0 0 or f 13 from for aminopyrine to use in eqs 3-6 to compute f 12 binary data. Thus, only regressions were possible 4.2. Parameter Estimation from Pure Solvent Solubilities. Here we explore the capabilities of the model for describing mixed solvent solubilities using f 012 and f 013 values obtained from solubilities in pure solvents. This procedure requires x1id(T) for use in eq 21 and application of a form of eq 3. Equation 3 can rarely be maintained in its full form, due to limitations on the available thermophysical data. For each solute below, we selected a form of eq 3 partly based on the availability of thermophysical property information and partly based on past experience with the respective solutes as described in the literature. The absence of heat capacity data meant choosing 0 either eq 5 or 6. We sought consistency among values of f 12 0 and f 13 estimated by solving eq 17 and values obtained from eq 21 for parameter transferability. Therefore, we also compared which ideal solubility calculation led to the best model-data agreement, even though such an analysis would not be feasible in the general case. 4.2.1. Sulfamethazine, Sulfamethoxypyridazine, and Sulfanilamide. For sulfamethazine, comparison of f0 values obtained with eqs 5 and 6 suggests that superior results are obtained from eq 5, which also was used in previous works.50,51 For sulfamethoxypyridazine, there is significant uncertainty in the heat of melting data. Escalera et al.52 report 33 948 J/mol, while Bustamente et al.53 report 22 300 J/mol. Previous treatments of sulfamethoxypyridazine50,51,53,54 and sulfanilamide50,55 consistently employed eq 5. We find f0 values from binary data to be somewhat more consistent with the ternary fitting when eq 6 is employed, but the difference is not substantial. In the end, therefore, we have employed eq 5. All of these substances have been reported to have polymorphs,56 but insufficient information about their properties is available to deal with this situation. Figure 2 shows excess solubility estimates for sulfamethazine in water with dioxane. The solid and dashed lines are as in Figure 1, while the dot-dash lines show predicted excess 0 estimates from binary solubility solubilities obtained with f 1j 0 data, using eq 21. As Table 2 shows, the predicted value of f 12

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Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

Table 2. Solutes and Their Solvents with References to Their Origin in the Literature for Both Solubilities (SLE) and Solvent Mixture Nonideality (VLE)a reference

regressed from ternary SLE

solute (1)

solvent (2)

solvent (3)

T (K)

SLE

VLE

f 012

aminopyrine aminopyrine antipyrine antipyrine cholesterol cholesterol cholesterol cholesterol cholesterol cholesterol cholesterol cholesterol cholesterol cholesterol cholesterol desmosterol desmosterol desmosterol desmosterol desmosterol mefenamic acid mefenamic acid paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol paracetamol phenacetin phenacetin phenacetin phenacetin phenacetin phenacetin phenacetin phenacetin sulfamethazine sulfamethazine sulfamethazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfamethoxypyridazine sulfanilamide sulfanilamide sulfanilamide theophylline theophylline theophylline

water 1,4-dioxane water 1,4-dioxane hexane hexane hexane hexane hexane 1,4-dioxane benzene benzene ethanol ethanol 1,4-dioxane hexane hexane hexane hexane hexane water ethyl acetate 1,4-dioxane water water ethyl acetate water ethyl acetate methanol ethanol acetone acetone acetone water water water water water water water water water water water ethyl acetate water water ethyl acetate water ethanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol water ethyl acetate ethyl acetate water ethyl acetate water water water water

ethanol water ethanol water ethanol ethanol ethanol ethanol ethanol ethanol 1,4-dioxane hexane hexane benzene hexane ethanol ethanol ethanol ethanol ethanol ethanol ethanol water ethanol ethanol ethanol 1,4-dioxane methanol water methanol toluene toluene toluene acetone acetone acetone acetone 1,4-dioxane 1,4-dioxane 1,4-dioxane 1,4-dioxane 1,4-dioxane 1,4-dioxane ethanol ethanol 1,4-dioxane ethanol ethanol 1,4-dioxane water water water water water ethyl acetate ethyl acetate ethyl acetate ethyl acetate ethyl acetate ethanol ethanol hexane ethanol ethanol 1,4-dioxane 1,4-dioxane methanol acetonitrile

298.15 298.15 298.15 298.15 293.20 298.20 303.20 313.20 323.20 293.15 293.15 293.15 293.15 293.15 293.15 293.20 298.20 303.20 313.20 323.20 298.15 298.15 298.15 303.15 298.15 298.15 298.15 298.15 298.15 298.15 293.15 298.15 303.15 293.15 296.15 298.15 303.15 313.00 308.00 303.00 298.00 293.00 298.15 298.15 298.15 298.15 298.15 298.15 298.15 293.15 298.15 303.15 308.15 313.15 293.15 298.15 303.15 308.15 313.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15

48 49 48 49 74 74 74 74 74 76 76 76 76 76 76 74 74 74 74 74 78 78 49 105 59 59 59 60 60 60 63 63 63 63 63 63 63 55 55 55 55 55 65 65 65 51 57 57 53 53 53 53 53 53 53 53 53 53 53 52 52 52 57 57 55 70 73 73

100 101 100 101 102 102 102 102 102 46 46 103 102 46 46 102 102 102 102 102 100 104 101 100 100 104 101 104 46 46 106 106 106 107 107 107 107 101 101 101 101 101 101 100 104 101 100 104 101 100 100 100 100 100 104 104 104 104 104 100 104 46 100 104 101 101 46 108

5.64 0.01 0.44 0.01 2.47 2.47 2.47 2.47 2.47 2.58 2.11 2.11 4.22 4.22 2.58 1.34 1.34 1.34 1.34 1.34 1.44 2.34 4.05 7.19 7.19 5.22 7.19 5.22 -1.09 2.54 3.18 3.18 3.18 7.19 7.19 7.19 7.19 13.86 13.86 13.86 13.86 13.86 13.86 13.86 0.57 16.38 16.38 1.87 12.81 2.89 2.89 2.89 2.89 2.89 2.89 2.89 2.89 2.89 2.89 12.81 2.93 2.93 10.91 4.99 10.91 5.44 5.44 5.44

a

from binary SLE

SD

f 013

SD

f 012

f 013

0.21 0.27 0.11 0.14 0.18 0.18 0.18 0.18 0.18 0.48 0.37 0.37 0.21 0.21 0.48 0.15 0.15 0.15 0.15 0.15 0.57 0.97 0.50 0.22 0.22 0.82 0.22 0.82 0.46 0.61 0.39 0.39 0.39 0.22 0.22 0.22 0.22 0.45 0.45 0.45 0.45 0.45 0.45 0.45 1.87 0.31 0.31 0.63 0.53 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.53 0.92 0.92 0.80 2.11 0.80 0.43 0.43 0.43

1.99 5.64 -0.17 0.44 4.22 4.22 4.22 4.22 4.22 4.22 2.58 2.47 2.47 2.11 2.47 4.15 4.15 4.15 4.15 4.15 4.47 4.47 7.19 2.54 2.54 2.54 4.05 -1.09 7.19 -1.09 3.93 3.93 3.93 3.18 3.18 3.18 3.18 2.06 2.06 2.06 2.06 2.06 2.06 1.29 1.29 1.70 5.32 5.32 5.57 12.81 12.81 12.81 12.81 12.81 2.93 2.93 2.93 2.93 2.93 2.89 2.89 -3.52 -0.09 -0.09 3.10 5.38 7.60 1.17

0.50 0.21 0.27 0.11 0.21 0.21 0.21 0.21 0.21 0.21 0.48 0.18 0.18 0.37 0.18 0.18 0.18 0.18 0.18 0.18 0.82 0.82 0.22 0.61 0.61 0.61 0.50 0.46 0.22 0.46 0.91 0.91 0.91 0.39 0.39 0.39 0.39 0.73 0.73 0.73 0.73 0.73 0.73 1.25 1.25 0.34 0.48 0.48 0.80 0.53 0.53 0.53 0.53 0.53 0.92 0.92 0.92 0.92 0.92 0.71 0.71 2.42 1.46 1.46 0.98 1.01 2.81 0.66

n/a n/a n/a n/a 4.45 4.63 4.39 4.34 4.23 0.93 0.70 0.70 6.12 6.12 0.93 7.07 6.85 6.64 6.08 5.09 15.69 1.56 0.95 7.01 7.37 4.72 7.37 4.92 1.25 0.96 1.57 1.52 1.49 7.63 7.56 7.51 7.38 13.21 13.34 13.34 13.51 13.63 13.12 13.12 3.14 16.21 16.21 4.00 12.09 4.70 4.78 5.06 4.92 5.34 4.70 4.78 5.06 4.92 5.34 12.10 4.08 4.08 9.73 4.51 9.73 6.51 6.89 6.89

n/a n/a n/a n/a 4.96 5.05 5.03 4.89 4.88 6.12 0.93 4.46 4.46 0.70 4.46 7.71 7.51 7.31 6.77 6.35 3.06 3.06 7.49 0.27 0.72 0.72 0.97 1.25 8.64 1.25 11.39 11.57 11.93 1.57 1.55 1.52 1.49 2.65 2.66 2.78 2.92 3.01 2.93 3.26 3.26 3.82 5.20 5.20 -0.87 11.86 12.08 11.88 11.86 12.24 3.92 4.08 4.20 4.18 4.06 4.78 4.78 21.28 4.87 4.87 0.08 4.02 5.33 7.61

Parameters are listed, obtained from binary data and ternary solubility data. Standard deviations (SD) are provided for the latter.

for sulfamethazine (1) and water (2) is quite close to the value 0 for water is much greater obtained from ternary data. Also, f 12 0 than f 13 for the other solvents: Ethanol, ethyl acetate, and dioxane. The form of eq 15 suggests that the larger parameter influences the results the most. Thus, if an estimate of the

dominant parameter agrees with the value from fitting ternary data, the predictions are usually good. Here, the sulfamethazinewater parameter dominates the excess solubility calculation so the predictions are accurate. The case of the ethyl acetate-ethanol binary is different. The ethanol-sulfamethazine parameter is

Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

Figure 1. Solubilities of aminopyrine (1) in water (2)-ethanol (3)48 (left) and in dioxane (2)-water (3)49 (right), both at 298 K.

Figure 2. Solubility and excess solubility of sulfamethazine (1) in water (2)-dioxane (3) mixtures at 298 K.51

Figure 3. Sulfamethazine (1) solubilities in water (2)-ethanol (3) (left) and ethanol (2)-ethyl acetate (3) (right), both at 298 K.57

Figure 4. Sulfamethoxypyridazine (1) in water (2)-dioxane (3)51 at 298 K (left) and ethanol (2)-water (3)53 at 293 K (right).

essentially the same as the fitted value, but it is not much larger than the estimated ethyl acetate-sulfamethazine parameter, which is more than twice the ternary-based parameter. The result is a greater discrepancy of prediction and data for the ethyl acetate-ethanol system than on the water-ethanol system. This is shown in Figure 3. Figure 4 shows sulfamethoxypyridazine solubility in mixtures of water with dioxane and water with ethanol. The sulfamethoxypyridazine-water parameter predicted from eq 21 is in good agreement with that fitted to ternary data

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and in close agreement to data for both calculated lines at waterrich compositions. On the other hand, neither the ethanol nor the dioxane parameters are as well-predicted, since the estimated line deviates from the data and the fitted lines at organic-rich compositions. The dioxane parameter is greater and the dioxane disagreement is greater, giving greater discrepancy here than for the aqueous ethanol system. The system sulfamethoxypyridazine (1)-ethyl acetate (2)-hexane (3) is fitted well but predicted poorly, as indicated in Table 0 is 21.28, whereas fitting of 2. The binary-based value of f 13 ternary data gives -3.52. There can be several reasons for this discrepancy. The fit is from only two data points, which is normally insufficient, though they are near equimolar in the solvents. The excess Gibbs energy of the sulfamethoxypyridazine-hexane binary might not be symmetric, as the Porter equation requires. Finally, the cause may be from the solubility of sulfamethoxypyridazine in hexane being extremely small and the results of eq 21 being very sensitive to measurement error in such cases, as discussed later. There are three sulfanilamide cases, two aqueous and one nonaqueous. Descriptions of the aqueous systems are good because the water-sulfanilamide parameter dominates, and its values from fitting and estimation agree quite well. The excess solubilities for the ethyl acetate-ethanol case are symmetric, while the fitted and predicted results show asymmetry. However, the magnitudes are close to experiment at midrange solvent compositions, giving acceptable prediction over the whole range. 4.2.2. Paracetamol. Previously we28 have treated a limited set of paracetamol data providing values for paracetamol parameters with five different solvents. Here we have revised and extended that set, in particular by finding acetone and toluene parameters. The literature is rich in studies of paracetamol solubility, including ideal solubilities calculated from both eqs 558,59 and 6,60 as well as from eq 4 with measured ∆cP,m,i values.61 Equation 6 produces results quite close to the data, whereas eq 5 gives a significant error. With ∆cP,m,i equal to 99.8 J/(mol K) at the melting point,62 eq 4 is also accurate. Using f0 values estimated from binary data gives close agreement in solubilities with those from ternary regressions for both eq 6 and eq 4, as reported in the Supporting Information. In the pharmaceutical literature, solid paracetamol is usually found in one of two forms, form I (commercially available monoclinic) and form II (orthorhombic), although results have also appeared for alternative forms.64 Since transition enthalpies have not been reported, such effects have not been taken into account here. The solubilities have been reported for 17 paracetamol systems, nine of which are aqueous. The water cases demonstrate again the importance of good agreement between the prediction and regression results for the dominant solvent component. The paracetamol-water parameter fitted to all systems is 7.19 ( 0.22, whereas the values estimated from binary data range from 7.01 to 7.63. The water-methanol mixture is an exception where the value is 8.64, but the description of the water-rich solubility data is not very good in that case. This difference suggests experimental error as well as parameter sensitivity. Yet, all nine aqueous systems (two are shown in Figure 5) are relatively well represented by the model, irrespective of whether the f0 parameters are obtained from ternary or binary data. Figure 6 shows the solubilities of paracetamol in two nonaqueous binary systems: Methanol-ethanol and ethyl acetate-ethanol. In the first, where there is only a single data point, the excess solubility is very small, consistent with the solvent solution being nearly

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Figure 5. Paracetamol (1) in water (2)-acetone (3)63 at 293 K (left) and water (2)-ethanol (3)59 at 298 K (right).

Figure 7. Phenacetin (1) in water (2)-dioxane (3). Left55 and right65 sets both at 298 K.

Figure 6. Paracetamol (1) in ethanol (2)-methanol (3)60 (left) and ethyl acetate (2)-ethanol (3)59 (right), both at 298 K.

Figure 8. Phenacetin (1) in ethyl acetate (2)-ethanol (3) at 298 K.65

0 ideal. Thus, the results are relatively independent of the f 1j values. The other system shows greater nonideality. Interestingly, both binary parameter values are less than those ternary values from regression of the entire set of solubilities. However, since in the midrange the ternary parameters overestimate the solubility, the binary-based predictions give better overall agreement with the measured data. Among the systems investigated, the solubility of paracetemol in toluene with acetone is somewhat better predicted than expected, because the acetoneparacetamol parameter from binary data is less than that from ternary data, partly compensating for the erroneous toluene parameter. One can note from Table 2 that the parameter values of paracetamol with water, ethyl acetate, methanol, ethanol, and dioxane are in quite good agreement with our previously28 reported values, even though the database for the present values is significantly larger. 4.2.3. Phenacetin. We know of no phase transitions for phenacetin below the melting point. This is consistent with the investigations of Pena et al.65 Though Yalkowsky et al.66 used eq 5, we find eq 6 to be better. Application of the model to phenacetin solubilities in aqueous mixtures is successful, though there are some discrepancies with data from Pena et al. Figure 7 shows two different measurements for the water-dioxane system. Both binary- and ternary-based parameter values slightly underestimate the data of Reillo et al.55 (left plot) while overestimating that of Pena et al. (right plot). This is not unusual when examining data from independent investigators. Figure 8 shows results for ethanol with ethyl acetate, where parameter estimates from binary data are both greater than those from ternary data, leading to errors in the predictions. 4.2.4. Theophylline. The behavior of theophylline in the solid phase is complex. Below 340 K, monohydrous theophylline is stable, while above 340 K, crystalline theophylline is stable, as determined by Fokkens et al.67 using DSC and vapor pressure studies. This hydration behavior of theophylline is complicated by the fact that the state of the hydrate depends on the water activity of the crystallization medium.68 In contact

Figure 9. Theophylline (1) in water (2)-methanol (3) (left) and water (2)-acetonitrile (3) (right), both at 298 K.73

with methanol-water or 2-propanol-water mixtures at water activities less than 0.25, the anhydrate is the only solid phase observed, no matter which solid form was initially added. At water activities greater than 0.25 in either solvent mixture, the monohydrate is obtained as the most stable form. Finally, the monohydrous form can be observed metastably at lower temperatures, leading to the wrong solid for solubility. Most available thermophysical data on theophylline are for the anhydrous form, but the proper thermophysical data for calculating theophylline solubilities in water and in organics would not be the same if the solids differ. Most of the solubility studies of anhydrous theophylline in the literature69-72 have used eq 6, which we use, though a few66,67 have adopted eq 5. Theophylline data, in the three aqueous mixtures, are wellrepresented by both approaches, except for acetonitrile, where the solubility is very small, as seen in Figure 9, and the predictions give solubilities somewhat higher than experiment. 4.2.5. Desmosterol. The only studies of desmosterol solubility in mixed solvents are in hexane with ethanol from 293 to 323 K from Chen et al.74 They used eq 5, but there is little difference when eq 6 is used. Since the ternary regressions are only for these solvents, the fits are quantitative. However, since the parameters from pure solvent solubilities generally exceed the ternary-based fits, the desmosterol solubilities in mixtures

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Figure 10. Cholesterol (1) in ethanol (2)-benzene (3) (left) and dioxane (2)-ethanol (3) (right), both at 293 K.76

Figure 12. Distribution of average error of AARD with parameters based on regression of ternary data and those found from binary solute-solvent data.

Figure 11. Cholesterol (1) in hexane (2)-ethanol (3). Left74 and right76 both at 293.15 K.

are overpredicted. It might be that the desmosterol binaries with hexane-ethanol are not symmetric enough for the Porter model of eq 20 to apply, due to the great difference in size and shape of the solute desmosterol from the two solvents. 4.2.6. Cholesterol. Several treatments of cholesterol are reported in the literature. The heat capacity change on melting and temperatures of phase transitions below the melting point have been established by Domanska et al.,75 potentially allowing use of eq 4. Chen et al.74 also reduced data on cholesterol solutions by treating ∆cP,m,i as an adjustable parameter to be estimated from fitting solubility data. Most cholesterol systems are represented successfully. Figure 10 illustrates the results for mixtures of benzene with ethanol and dioxane with ethanol. The solubilities of cholesterol in hexane-ethanol binaries are reported by Chen et al.74 and by Weicherz and Marschik.76 The data do not agree, as can be seen from Figure 11. Regressions underestimate the results of Weicherz and Marschik while matching the data of Chen et al. On the other hand, binary predictions using eq 21 overestimate the data of Chen et al. and produce a nearly quantitative agreement with the data of Weicherz and Marschik. Such discrepancies complicate conclusions about experimental accuracy. 4.2.7. Mefenamic Acid. Mefenamic acid undergoes a phase transition from a form identified as I to form II at 140-150 °C. Form II melts at 230 °C,77 while form I is observed in solubility measurements78 at temperatures less than 140 °C. Unfortunately, the heat of transition is unknown, preventing the inclusion of its effect into predictions of the ideal solubility. The ideal solubility of mefenamic acid is very small with values from eqs 5 and 6 differing by a factor of more than four. While the solubility in water is small, solubilities in ethanol and ethyl acetate are in the same range as the ideal values. Further, there is a large difference in the binary- and ternary-based parameter values for water-mefenamic acid. The discrepancy could be due to the binary system being asymmetric or to error in the extremely low aqueous solubility, which strongly affects the solute-water parameter value. Regardless, the result is that both

regression and prediction of mixed aqueous solvent solubility are problematic. The f0 values from binary data are in quantitatively better agreement with those from ternary fitting when eq 5 is employed, but eq 6 is in better agreement with the excess solubility variation with mixed solvent composition, so we have employed eq 6. The nonaqueous system of ethyl acetate-ethanol shows quite good agreement between binaryand ternary-based estimates. 4.3. Comparison with Other Methods. The ability of the model to quantitatively describe the mixed solvent solubility data can be assessed by the average absolute relative deviation (AARD), according to AARD )

1 n

∑| nj

j

xˆ1 - x1 x1

|

(22) j

for all n data points j in a set. This statistic is identical to that employed by Jouyban et al.27 in their comparisons of a variety of models for solubility in mixed solvents. They found that for a four-parameter (two model constants plus the two pure solvent solubilities) the AARDs for different models were in the range 0.08-0.19. Using the present method, with two solute-solvent parameters regressed from mixed solvent solubility data (and two pure solvent solubilities), the values range from 0.003 to 0.58, with an average of 0.11. When estimating f0 from binary data (including only the two pure solvent solubilities), the values fall in between 0.004 and 1.12, averaging 0.23. Removing points more than two standard deviations away from the mean, these averages drop to 0.09 and 0.20, respectively. This result can be compared to that for the two-parameter log-linear model, which also employs transferrable solute-solvent parameters, for which Jouyban et al. found an AARD of 0.50. Figure 12 shows the distribution of errors, with the majority of them within 0.1-0.2. While our overall statistics are not, in principle, directly comparable to those of Jouyban et al., since they included 30 different sets and we have included 68 sets for regression of f0 and 64 sets to obtain f0 from pure solvent solubilities, the comparison suggests that the present method has advantages in accuracy for the small number of solute-solvent parameters that are transferable among systems. 5. Discussion The present method for predicting the solubility of classical pharmaceuticals in mixed solvents should be attractive to

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pharmacists and thermodynamicists, since the method may be used in a variety of forms, depending upon the available input data. Compared to previous studies using this approach,28 a key conclusion of this paper is that solubilities in mixed solvents can often be predicted from solubilities in pure solvents alone, though the quality of the estimates without multisolvent data depends upon the following factors: (i) A gE-model must be known for the binary solvent mixtures (also when regressing parameters). 0 parameters for both (ii) Estimates must be made of the f 1j solute-solvent pairs. (iii) Thermophysical properties of the pure solid solute must be obtainable. While having activity coefficient derivatives of the mixed solvent is necessary, the particular gE-model selected does not strongly affect the results. When sufficient data have been available, we have made solubility estimates with both the Wilson and modified Margules equations. Normally, the results are very similar. The model is limited in that the sign of the excess solubility must be consistent with sign of gE for the solvent pair. This means that if the solvent-solvent gE changes sign, the excess solubility in that mixture must do so also in order to qualitatively describe the data. This behavior is not observed in the pharmaceutical cases reported here. For reliability, the most important aspect seems to be accurate deter0 , especially mination of the solute-solvent parameters, f 1j parameters of larger value in magnitude. If this quantity is accurate, predictions will commonly be reliable. The bigger parameter is often associated with the more nonideal solutesolvent pair or the solvent with the lowest solute solubility. Often this is water. Thus, having accurate measurements of low solubilities, especially in water, can determine accuracy in predicting mixed solvent solubilities. It is likely that some errors arise because the assumed symmetric composition dependence of the Porter equation is incorrect. Alternatives include models such as UNIFAC, though it is frequently not directly applicable to pharmaceutical systems. Other methods to circumvent this difficulty exist.4-6 Explorations79 of the properties of UNIFAC derivatives at infinite dilution suggest that this could have several attractive features, but the methods are more complicated than the present approach. Extensions of UNIFAC parameter tables continue to appear,80,81 but several pharmaceuticals will remain untractable for quite some time. Finally, having values of the pure solute properties to estimate ideal solubility is required. First, at least Tm and ∆hm must be known. We have found that results are better when ∆cP,m is also known, especially if its value is large or if Tm - T is significant. Phase transitions below the melting point are relevant, but they seem not to make much difference according to the cases reported here, as seen in the systems containing sulfamethazine, sulfamethoxypyridazine, sulfanilamide, paracetamol, cholesterol, and mefenamic acid. All of these solutes undergo a solid-phase transition below the melting point. The apparent insensitivity is probably due to the effects being taken into account in obtaining the parameter values. Also the magnitude of the transition enthalpies are usually much less than the enthalpy of fusion. For cholesterol, Garti et al.82 reports a value of the transition enthalpy of 2.845 kJ/mol while the fusion enthalpy is 28.034 kJ/mol. However, that does not mean the effects should be ignored, particularly when there are polymorphs83 and hydrates. For example, the aqueous solubility of theophylline anhydrate is nearly double that of the hydrated solid.83

The model assumes the connection between the excess solubility and the excess Henry’s law constant; eq 7 is for infinitely dilute solutes (x1 e 0.01) in pure solvents and mixtures. Frequently, this is not the case. In fact, 48 of 68 solute-solvent mixtures have at least one experimental solubility point at higher concentrations. This is especially the case in mixtures of organic solvents, while aqueous systems typically have very dilute solubilities in the water-rich end and higher concentrations toward the organic end of the solvent composition range. Yet, the method seems to do well for those cases too. Finally, it is possible to conceive of estimating the f0 parameters in eq 10 for liquid mixtures by molecular simulation. We have explored this possibility84-88 via integration of the molecular radial distribution function. While progress is being made, achieving prediction without data is unlikely. Computer simulation of dense mixtures takes considerable computational resources and is time-consuming, and robust techniques for spatial integration are still under development. 6. Conclusions A method based on fluctuation solution theory has been developed for describing solubility of solid solutes in mixed solvents. Application to an extensive database shows good accuracy and that the one parameter for each solute-solvent pair, transferable to all systems with these substances, is best found by regression to mixed solvent solubilities. In addition, predictions for mixtures based on measured solubilities in pure solvents are often quite accurate, especially if data for solvent with the lowest solubility are highly accurate. Finally, although the expressions have been derived for infinite dilution of the solute, the method can reliably describe solubilities of higher magnitude. Acknowledgment M.D.E. is grateful to the Technical University of Denmark for financial support. Supporting Information Available: Figures for all systems in Table 2. This material is available free of charge via the Internet at http://pubs.acs.org. Appendix A: Derivation of Solid Solubility Equation The relation sought is for the solubility of a component i when solid, if pure at T and having a first-order solid-phase transition from S1 to S2 at Tt,i between T and the melting temperature, Tm,i. At phase equilibrium, the fundamental relation is the isofugacity of the solute i in the solid and liquid phases: fi(pure solid)(T) ) xiγi(T, x)fi(pure subcooled liquid)(T)

(A-1)

Here f i(pure subcooled liquid)(T) is the fugacity of the pure subcooled liquid i at T and pressure dependence has been ignored. This leads to ln

fi(pure solid) ) ln xiγi(T, x) fi(pure subcooled liquid) S1 L gi(pure solid) - gi(pure subcooled liquid) ) RT

(A-2)

The left-hand-side pure component fugacity ratio is evaluated by a thermodynamic process involving six states, as shown in Figure 13. Here, the solid is warmed from the pure solid in phase S1 through a solid phase transition at Tt,i and melted at

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11629

temperature from the solid transition temperature to the melting point can be written as hS4,i2 (Tm,i) ) hS3,i2 (Tt,i) +

∫

Tm,i S cP,i2

Tt,i

dT ≈ hS3,i2 (Tt,i) + cSP,i2 (Tm,i - Tt,i) (A-6)

By combining the terms we get

( (

Figure 13. Thermodynamic cycle for solubilization of a pure solid.

Tm,i. Finally, the liquid is cooled below the melting and transition points. In the absence of a phase transition, the dotted line from 2 to 4 indicates the path that must be followed instead of 2f3f4. The difference in Gibbs energies between states 1 and 6 is g6,i(T) g1,i(T) gi,S(1 pure solid) - gi,L(pure subcooled liquid) ) ) RT RT RT . gS1,i1 (T) gS2,i1 (Tt,i) gS3,i2 (Tt,i) - gS2,i1 (Tt,i) + + RTt,i RT RTt,i gS3,i2 (Tt,i) gS4,i2 (Tm,i) RTm,i RTt,i . gL5,i(Tm,i) gL5,i(Tm,i) - gS4,i2 (Tm,i) gL6,i(T) + + RTm,i RTt,i RTm,i . gS1,i1 (T) gS3,i2 (Tt,i) gS2,i1 (Tt,i) gS4,i2 (Tm,i) ) + + RTt,i T RTm,i RTt,i gL5,i(Tm,i) gL6,i(T) (A-3) RT RTm,i

(

(

) (

(

)

) ( ) ( (

(

)

)

) )

The Gibbs energies of transition and melting (steps 2f3 and 4f5) are zero. The difference in Gibbs energies at different temperatures is evaluated by integrating the Gibbs-Helmholtz equation at constant pressure -

∂ g h ) 2 ∂T T T

[ ]

(A-4)

P

For the terms in eq A-3 we get, by adding and subtracting S2 h4,i /R(1/T - 1/Tm,i) -

(

)

S1 L S1 h2,i gi(pure 1 solid) - gi(pure subcooled liquid) 1 ) RT R Tt,i T S1 cP,i

R

(

[( ) ln

]

S2 h3,i

(

) )

[( )

[( ) ] ( ) [( ) ] (

L S2 L h5,i cP,i Tm,i h4,i 1 1 1 T 1 ln -1+ R T Tm,i R Tm,i T R T Tm,i

]

[( )

]

If we now define the enthalpy and heat capacity differences ∆hm,i ≡ hL5,i - hS4,i2 , ∆ht,i ≡

hS2,i1

-

hS3,i2 ,

∆cP,m,i ≡ cLP,i - cSP,i2

(A-8)

∆cP,t,i ≡ cSP,i1 - cSP,i2

the final expression becomes

(

) ) ]

S1 L gi(pure ∆hm,i 1 solid) - gi(pure subcooled liquid) 1 )+ RT R T Tm,i Tm,i ∆ht,i 1 ∆cP,m,i T 1 ln -1+ + R Tm,i T R Tt,i T Tt,i Tt,i ∆cP,t,i ln +1(A-9) R T T

[( )

]

[( )

(

The assumptions concerning the derivation can be summarized as follows: (1) There are no pressure effects on the properties of condensed matter. (2) Solid-phase transitions are first-order. (3) The heat capacities are independent of temperature. Appendix B: Expressions for Activity Coefficient Derivatives The derivative of eq 9 is given by

( ) ∂ ln γi ∂xj

) T,P,Nk*j

∂ ln γi ∂Nj

|

∂xj ∂N j T,P,Nk*j )

|

-1

T,P,Nk*j

N ∂ ln γi 1 - xj ∂Nj

|

(B-1) T,P,Nk*j

All expressions are symmetric with regards to i and j. For the two-parameter Wilson equation the result is N

Tt,i Tt,i 1 1 +1+ T T R Tm,i Tt,i

S2 S2 Tm,i Tt,i h4,i cP,i 1 1 ln -1+ + + R Tt,i Tm,i R T Tm,i

) )

S1 L hL5,i - hS4,i2 1 gi(pure 1 solid) - gi(pure subcooled liquid) ) RT R T Tm,i Tm,i hS2,i1 - hS3,i2 1 cLP,i - cSP,i2 T 1 ln -1+ + R Tm,i T R Tt,i T Tt,i Tt,i cSP,i1 - cSP,i2 ln (A-7) +1R T T

-

∂ ln γi ) 1 - εij - εji + ∂Nj T,P,Nk*j

C

∑x ε ε

(B-2)

[ ]

(B-3)

k kj ki

k

Here

)

(A-5)

εij )

Λij

,

C

∑x Λ k

Λik )

Vk aik exp Vi T

ik

k)1

Here it has been assumed that the heat capacities are independent of temperature. The solid-phase heat capacities are discontinuous at the transition for a first-order transition. For the solid phase 2, the enthalpy change resulting from changing

where aik is a binary parameter characterizing the interaction between species i and k in units of kelvin, and Vi and Vk are pure solvent molar volumes.

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The five-parameter modified Margules model gives

( )

∂ ln γ3 + ) -2x2 A32 + 2(A23 - A32)x3 - F1x2x3 + ∂x3 T,P,N2 1 F (R + ηx22)x32 + x22[2(A23 - A32) + F1(x2 - x3) + 2 2 32 F2[R32x3(1 + x2) - R23x2x3 + 2η(x2 - x3)x3x2]] - F3(R32 + ηx22)(x2x3)2 (B-4)

[

]

The quantity Fk is defined as Fk ≡

2R23R32

(B-5)

(x2R23 + x3R32 + ηx2x3)k

For the UNIFAC method, the activity coefficient is given by

(

)

Ji Ji + ln + Li Li G ski ski qi(1 - ln Li) ϑk - Gki ln ηk ηk k

ln γi ) 1 - Ji + ln Ji - 5qi 1 -

∑

(

)

ri , jr

Li )

(B-6)

C

qi , qj

jr )

∑x r , m m

m

C

∑x q ,

qj )

Gki ) νkiQk

m m

m

G

ski )

(B-7)

C

∑G

ϑk )

miτmk,

m

∑x G m

km,

m

C

ηk )

∑x s

m km,

τmk ) exp -Amk/T

m

Then eq B-7 gives N

∂ ln γi ∂Nj

|

) -(1 - Ji)(1 - Jj) - 5(Ji - Li)(Jj - Lj)qj +

T,P,Nk*j G

LiLjqj +

∑ k

(

ϑk

skiskj ηk

2

- Gkj

ski skj - Gki ηk ηk

)

Nomenclature x ) mole fraction B ) constant in the Porter equation c ) molar concentration T ) temperature P ) pressure ∆g ) change in Gibbs free energy ∆h ) change in enthalpy ∆s ) change in entropy ∆cP ) change in heat capacity at constant pressure R ) gas constant f ) total correlation function integral difference H ) total correlation function integral N ) number of moles γ ) activity coefficient ε ) parameter in the Wilson equation Λ, a ) variables in the Wilson equation V ) molar volume A, R, η ) parameters in the Margules equation F ) variable in the Margules equation J, L, q, ϑ, s, η, Q, τ ) variables in UNIFAC equations

Superscripts and Subscripts E ) excess quantity ′ ) solute-free basis id ) ideal ∞ ) infinite dilution 0 ) infinite dilution in single solvent + ) infinite dilution in mixed solvent 1, 2, 3, i, j, k ) molecular species indices m ) mixed solvent m ) melting t ) phase transition f ) solid-liquid phase transition

Literature Cited

where Ji )

ν ) group stoichiometric coefficient in UNIFAC equations q, r, A ) parameters in UNIFAC equations r ) radial distance between molecular centers

(B-8)

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ReceiVed for reView May 8, 2010 ReVised manuscript receiVed September 8, 2010 Accepted September 13, 2010 IE101059Y