- Email: [email protected]
Thermodynamic modeling of activity coefficient and prediction of solubility: Part 1 predictive models
Thermodynamic Modeling of Activity Coefficient and Prediction of Solubility: Part 1. Predictive Models MAHMOUD MIRMEHRABI, SOHRAB ROHANI, LUISA PERRY Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, Ontario, N6A 5B9, Canada
Received 4 May 2005; revised 17 November 2005; accepted 18 November 2005 Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jps.20560
ABSTRACT: A new activity coefficient model was developed from excess Gibbs free energy in the form Gex ¼ cAa xb1 . . . xbn. The constants of the proposed model were considered to be function of solute and solvent dielectric constants, Hildebrand solubility parameters and specific volumes of solute and solvent molecules. The proposed model obeys the Gibbs–Duhem condition for activity coefficient models. To generalize the model and make it as a purely predictive model without any adjustable parameters, its constants were found using the experimental activity coefficient and physical properties of 20 vapor-liquid systems. The predictive capability of the proposed model was tested by calculating the activity coefficients of 41 binary vapor-liquid equilibrium systems and showed good agreement with the experimental data in comparison with two other predictive models, the UNIFAC and Hildebrand models. The only data used for the prediction of activity coefficients, were dielectric constants, Hildebrand solubility parameters, and specific volumes of the solute and solvent molecules. Furthermore, the proposed model was used to predict the activity coefficient of an organic compound, stearic acid, whose physical properties were available in methanol and 2-butanone. The predicted activity coefficient along with the thermal properties of the stearic acid were used to calculate the solubility of stearic acid in these two solvents and resulted in a better agreement with the experimental data compared to the UNIFAC and Hildebrand predictive models. ß 2006 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 95:790–797, 2006
Keywords:
activity coefficient; predictive models; solubility; bulk properties
INTRODUCTION Crystallization process is a key unit operation in the production of pharmaceutical solids, which has a major effect on drug quality in terms of crystal size distribution, solid bulk and tap density, polymorphism, filterability, and flowability. Supersaturation, which is a function of solubility, is a kinetic and thermodynamic parameter that influences the crystal size distribution, Mahmoud Mirmehrabi’s present address is Wyeth Pharmaceuticals, Montreal, Canada, H4R 1J6. Correspondence to: Sohrab Rohani (Telephone: 519-6614116; Fax: 519-661-3498; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 95, 790–797 (2006) ß 2006 Wiley-Liss, Inc. and the American Pharmacists Association
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morphology, and polymorphic formation. Thus, knowing the solubility is essential in the study of a crystallization process. Solubility is needed to determine the crystallization throughput or yield. It is evident that the solubility prediction is necessary in solvent selection and control of crystallization processes especially for the drugs and fine chemicals where the experimental data are scarce. Using thermal properties of a solute, for example, heat of fusion and melting temperature, the ideal solubility of the solute can be calculated.1 To determine the real solubility, the activity coefficient of the solute in the solvent, an indication of the degree of nonideality, is needed. There are many factors, which affect the activity coefficient
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including the molecular size, polarity, and interaction forces. In predicting the activity coefficient using the bulk properties of solute and solvent, the Hildebrand solubility Eq. 1 can be used only for the regular or near-ideal solutions2
ln 2 ¼
v2 f21 ð1 2 Þ2 RT
ð1Þ
where d is the Hildebrand solubility parameter (J/cm3)0.5, g2 is the activity coefficient of the solute, v is the specific volume cm3/mol, Tis the absolute temperature, and f is the volume fraction of the solvent. Subscripts 1 and 2 refer to the solvent and solute, respectively. This equation is invalid for a wide range of conditions including the cases in which there are interactions or even nonsimilarities between solvent and solute molecules. Most pharmaceuticals are complicated compounds, which have various functional groups that interact with the solvent and with other functional groups of the solute molecule. Therefore, the regular solution theory does not work properly. The advantage of the Hildebrand equation is a predictive model that uses the bulk properties, which are available in handbooks or easily measurable. The UNIFAC model, a group contribution model,3 is another predictive model. In many cases, however, the configuration of the neighboring groups restricts the predictive capability of this model.4 In Part 1 of this two-part contribution, a new model is proposed for the prediction of the activity coefficient of a solute in a solution using the bulk properties of the solute and the solvent. The proposed model, once its constants are determined using the experimental activity coefficients of 20 pairs of vapor-liquid systems, becomes a purely predictive model similar to Hildebrand equation and the UNIFAC model. The activity coefficient in a solid-liquid or in a vapor-liquid system, depends on the interaction of the dissolved solute or vapor molecules with the solvent molecules, in the liquid phase. The solid phase or the vapor phase, therefore, does not play an important role in the nonideality of the solution. Therefore, the experimental vapor-liquid systems can be used in the development or validation of an activity coefficient model. In the present work, the predictive capability of the proposed model is examined by comparing the predicted and the experimental activity coefficients of 41 vapor-liquid binary systems. The results of the UNIFAC and Hildebrand DOI 10.1002/jps
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predictive models are also presented for comparison. In part 2, the proposed model will be revised by retaining two adjustable parameters, as a semipredictive model, to enhance and generalize its predictive capability. In Part 2, two other semipredictive models, the UNIQUAC and NRTL will also be used for modeling and prediction of solubility of a few pharmaceutical compounds.
RESULTS AND DISCUSSION Proposed Activity Coefficient Model There are various intermolecular forces between molecules of a pure compound, such as Van der Waals forces, dipole–dipole forces, and hydrogen bonding. The bulk physical properties of a chemical are functions of these intermolecular forces. Hildebrand solubility parameter is a direct function of the heat of vaporization. The other bulk properties that distinguish a compound are molar volume and dielectric constant. Usually the smaller molecules possess higher polarity. Dielectric constant is an indication of the degree of polarity and polarizability of a molecule. Many attempts have been made to correlate the solubility of a solid compound with its dielectric constant and specific volume.5–9 The proposed predictive activity coefficient model is based on the excess Gibbs free energy defined as: Gex ¼ cAa xb1 . . . xbm
ð2Þ
ex
where G is the excess Gibbs free energy, A is a constant which is a function of the bulk properties of the molecules in the solution, xi is the mole fraction of the mth component, a, b, and c are constants. This model is similar to Margules model, Gex ¼ Ax1 x2, in which A can be found from the experimental activity coefficient values. The activity coefficient can be calculated from the following equation ex
ln i ¼
Gi RT
ð3Þ
where ex Gi
@ðNGex Þ ¼ @Ni T; P; Nj
ð4Þ
ex
And Gi is the partial excess Gibbs free energy, Ni is the number of moles of each component and N is the total number of moles in the solution. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 95, NO. 4, APRIL 2006
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Table 1. The Binary VLE Systems that were used to Determine the Constants of the New Model Vapor-Liquid Equilibrium Component 1 Ethanol Ethanol Ethanol Ethanol Benzene Ethanol Ethanol Methanol Ethanol Dichloromethane
Vapor-Liquid Equilibrium
Component 2
Component 1
Component 2
1,2-Propanediol Acetonitrile 2-Propanol 2-Methyl-1-propanol Hexane Cyclohexanol 2-Butanone Acetone 1,2-dichloroethane Ethanol
Ethanol Methyl acetate Propyl acetate Ethyl acetate Ethanol Chloroform Ethanol MEK Ethanol Ethanol
Aniline Ethanol Ethanol Ethanol Chlorobenzene Ethanol Butyl acetate Heptane Ethyl benzene 1,4-dioxane
Equilibrium vapor mole fractions at different temperatures were available for each pair (9).
Eqs. 2–4 result in h i cA b b ðb1Þ x1 :::xm j6¼i bxi ln i ¼ þ ð1 mbÞxbi ð5Þ RT a
where m is the number of components in the system. A is also a function of the bulk properties. In fact, this term represents the difference between the properties of the solute and solvent A ¼ ðsolute Þ2 ðvsolute vÞ ð"solute "Þd ð6Þ where d is a constant, dsolute and are the solute and average Hildebrand solubility parameters, vsolute and v are the solute and average molar volumes, esolute and " are the solute and average dielectric constants. All of these parameters are at 258C. The parameter A is not a function of composition as it is showing how the solute and solvent molecules are different in nature. However, introducing the composition into Eq. 6 may enhance the generalizability of the equation for systems with more than two components. If the physical properties of the solvent and solute; the solubility parameter, specific volume, and the dielectric constant are similar, then the term A approaches zero and the activity coefficient becomes unity. This model obeys the GibbsDuhem equation for activity coefficients. In general any activity coefficient model that is derived from excess Gibbs free energy (Eq. 2) and satisfies the condition Gex ! 0 as xi ! 0, would obey the Gibbs–Duhem law. Appendix 1 presents the proof that the new model obeys the Gibbs–Duhem condition. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 95, NO. 4, APRIL 2006
To find parameters a, b, c, and d of the proposed model, the vapor-liquid equilibrium data10 of 20 systems (Tab. 1) along with the modified Raoult’s law were used. The 20 pairs of compounds, cover a wide range of properties from nonpolar to polar and hydrogen bond forming chemicals. At low pressures, the fugacity coefficient in the gas phase approaches unity and the modified Raoult’s equation can be used as follows: xi i Pi ¼ yi Pt
ð7Þ
where Pi* is the vapor pressure of component i at the studied temperature that can be calculated from the Antoine equation,10 yi is the mole fraction in the vapor phase, and Pt is the total pressure of the system. The parameters a, b, c, and d were varied to minimize the difference between the calculated (using the new model) and the experimental mole fractions (of the two components) in the vapor phase. The regression was carried out using fmincon function of MATLABTM. The algorithm of fmincon is based on the interior-reflective Newton method, which is described by Coleman and Li11,12. The estimated parameters are listed in Table 2. One of the major molecular interactions is dipole–dipole interaction especially for the nonregular solu-
Table 2. Constants of the Proposed Predictive New Activity Coefficient Model a 0.1329
b
c
d
0.9567
67.9
10
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tions. In our proposed equation, the dielectric constant is an empirical parameter that addresses this interaction. The estimated parameters show that the dielectric constant, which is a direct indication of the degree of polarity, has the largest effect on deviation from the ideal state. This may be a reason that the Hildebrand equation, which does not consider the degree of polarity, results in poor prediction for polar systems. Other methods have been developed to extend the Hildebrand equation and also Hansen equation13–16, which showed significant improvement from the original equations. However, these extended methods did not consider the dielectric constant as an effective parameter. Using the parameters of Table 2, the new model becomes a predictive model. In the next section the predictive capability of the new model will be examined to estimate the activity coefficients of 41 vapor-liquid systems.
order to compare the models, the average absolute error between the experimental and the calculated mole fractions ( yi, exp and yi, calc) defined by n P
yi;exp yi;calc
Average absolute error ¼ i¼1
ð8Þ
n
was used. The average absolute errors for all models for the systems listed in Table 3 are relatively small. Table 4 lists the average absolute error for highly nonideal vapor-liquid systems. The average absolute error and the standard deviation of the new model are smaller than the other predictive models. The Hildebrand regular solution model is the worst.
Solubility Prediction of Organic Solids For nonideal solutions we have ln
Prediction of Activity Coefficient using Vapor-Liquid Equilibrium Data To test the ability of the predictive new model, the activity coefficients of 41 pairs of compounds with a wide range of properties were estimated. The solutes’ activity coefficients and the vapor phase mole fractions were calculated and compared with the experimental mole fractions listed in Tables 3 and 4. Raoult’s law, Hildebrand, and the UNIFAC models were also used to predict the vapor phase mole fractions of the 41 vapor-liquid systems. In
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f2L DHfus Ttp Dcp Ttp 1 1 ¼ RTtp T R T f2S Dcp Ttp þ ln R T
ð9Þ
where f2L and f2S are the fugacities of pure subcooled liquid and pure solid forms of the solute, respectively, and Ttp is the triple point temperature which can be considered as the melting point. DHfus is heat of fusion and Dcp is the difference in the heat capacities of liquid and solid at absolute temperature, T. The first term in
Table 3. The Average Absolute Error of Vapor Mole Fraction from the Experimental Data using Raoults’s Law, the New Model, UNIFAC and Hildebrand Equation
Average Methanol–Ethanol Acetone–Pyridine MEK–Toluene Chlorobenzene–Decane Hexane–Benzene Ethanol–MEK Methanol–1,2-Propanediol 2-Butanone–1-Butanol Acetone–2-Propanol Acetone–Benzene Absolute average Standard Deviation
Number of Data Points 12 12 23 11 10 10 10 13 13 13
Raoult
New Model
UNIFAC
Hildebrand
0.012 (0.025) (0.032) (0.035) (0.041) (0.054) 0.056 (0.059) (0.061) (0.069) 0.044 0.018
0.040 0.018 (0.008) 0.022 0.040 0.037 0.056 0.058 0.060 (0.019) 0.036 0.019
0.050 0.030 (0.021) 0.050 0.010 0.011 (0.047) 0.036 0.025 (0.024) 0.030 0.014
0.032 (0.014) (0.031) 0.064 0.017 0.110 0.056 0.057 (0.016) (0.065) 0.046 0.030
The predicted mole fractions in parentheses indicate that the prediction is less than the experimental value. DOI 10.1002/jps
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Table 4. The Average Absolute Error of the Calculated Vapor Mole Fractions From the Experimental Data Using Raoults’s Law, the New Model, UNIFAC, and Hildebrand Equation
Average Ethyl acetate–Ethanol Methyl acetate–Ethanol Methanol–Acetonitrile Chloroform–1-Butanol Ethanol–Nitromethane Methanol–Carbon tetrachloride Pyridine–Nonane Ethanol–Benzene Ethyl acetate–1 Butanol Chloroform–Methanol Chloroform–Ethanol Methanol–2 Pentanone Methanol–Hexane Acetone–1 Butanol 1,2 Dichloroethane–1 Propanol MEK–Heptane Toluene–1 Butanol Benzene–1 Propanol 1,2 Dichloroethane–1 Butanol Methanol–Toluene Dichloromethane–Ethanol Acetone–Heptane Ethanol–Heptane Ethanol–Chlorobenzene Ethanol–Aniline Cyclohexane–Furfural Heptane–Furfural Methanol–p-Xylene Ethanol–Pentane Methanol–Heptane 2 Methylbutane–Ethanol Absolute average Standard deviation
Number of Data Points
Raoult
New Model
UNIFAC
Hildebrand
13 9 20 24 18 36
(0.092) (0.105) (0.105) (0.117) (0.119) (0.125)
(0.072) (0.042) 0.092 0.036 0.095 (0.050)
0.063 0.039 0.104 0.171 (0.050) (0.015)
0.152 0.122 (0.052) (0.025) (0.119) 0.179
13 21 18 12 14 9 21 15 20 8 15 12 25 10 12 9 17 11 8 10 9 11 11 10 11
(0.128) (0.134) (0.135) (0.138) (0.139) (0.140) (0.143) (0.150) (0.161) (0.163) (0.167) (0.196) (0.202) (0.203) (0.206) (0.237) (0.245) (0.259) (0.300) (0.344) (0.364) (0.368) (0.383) (0.411) (0.457) 0.208 0.103
0.085 0.039 0.069 (0.176) (0.032) 0.047 (0.049) 0.137 0.137 0.072 0.103 0.122 0.177 (0.076) 0.127 0.072 0.110 0.111 0.107 (0.149) (0.242) (0.134) 0.108 (0.086) 0.116 0.099 0.048
0.019 0.011 0.011 0.278 0.083 0.177 0.160 (0.066) (0.029) (0.038) (0.006) (0.042) (0.032) 0.207 (0.052) (0.077) (0.039) (0.056) (0.296) (0.040) 0.036 0.438 (0.103) 0.542 (0.031) 0.107 0.128
0.069 (0.041) 0.027 0.328 0.086 0.459 0.318 (0.092) (0.116) (0.101) (0.081) (0.106) (0.165) 0.212 (0.100) (0.130) 0.354 (0.099) (0.268) 0.278 0.153 0.377 1.153 1.211 5.918 0.416 1.058
The predicted mole fractions in parentheses indicate that the prediction is less than the experimental value.
Eq. 9 has the largest effect and the next two terms with the opposite signs tend to compensate each other.1 Melting of many pharmaceutical compounds is accompanied with exothermic degradation so that measuring the liquid heat capacity is a formidable task. In these cases, just the first term will be used fS Fugacities are related by : x2 2 ¼ 2L f2
ð10Þ
where g2 is the activity coefficient of solute in the solution. The two last terms in Eq. 9 are not considered here. For ideal systems the activity JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 95, NO. 4, APRIL 2006
coefficient of the solute is equal to unity so that the equation will be truncated to 1 DHfus Ttp 1 ð11Þ ln ideal ¼ RTtp T x2 Ideal and experimental mole fractions are related by x2 ¼
xideal 2 2
ð12Þ
Equation 11 calculates the ideal saturation mole fraction (solubility in terms of mole fraction) using the thermal properties of the solid. To obtain the real saturation mole fraction, the activity DOI 10.1002/jps
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the specific volume of both solute and solvent. 3. Calculate the new mole fraction from Eq. 12 using the predicted activity coefficient obtained and ideal mole fraction. 4. Repeat steps 2 and 3 until the new calculated mole fraction does not differ from the previous iteration by more than a set tolerance limit. 5. Calculate the solubility from Eq. 13 using the obtained mole fraction.
coefficient of the solute in the solution should be replaced in Eq. 12. The following equations yield the solubility in terms of grams solute per 100 g solvent using the saturation mole fraction, x2 x¼
x2 1 x2
and then
S¼
x MWsolute 100 MWsolvent ð13Þ
Where S is the solubility in terms of grams solute per 100 g solvent and MW is the molecular weight, g/mole. Hildebrand, the UNIFAC and the new models were used to predict the activity coefficient and the solubility of stearic acid polymorph C in methanol and 2-butanone. To predict the mole fraction of dissolved solute in the solvent, an iterative procedure was performed: 1. Calculate the ideal mole fraction of the solute from Eq. 11. 2. Calculate the corresponding activity coefficient from the activity coefficient models using the calculated ideal mole fraction. The group interaction parameters are required for the UNIFAC model. Hildebrand equation uses the solubility parameters of both solute and solvent. The new model requires the Hildebrand solubility parameters, dielectric constants, and also
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Figures 1 and 2 illustrate the experimental solubility data along with the predicted results. The Hildebrand equation resulted in a very large number for the activity coefficient of stearic acid in methanol (approximately 107). Therefore the solubility approached zero. The estimated solubility in both solvents using the new activity coefficient model offered a better prediction compared with other predictive models, UNIFAC and Hildebrand. However, similar to the Hildebrand equation, the new model does not recognize the difference between positive and negative deviations from the Raoult’s law (see also Tabs. 3 and 4). If the coefficients of Eq. 5 (a, b, c, and A) are considered as adjustable parameters then the model will also work for negative deviations as the overall coefficient can be negative which lead to an activity coefficient less than 1.
Figure 1. DOI 10.1002/jps
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Gibbs–Duhem condition requires that the following equation be satisfied x1
d ln 1 d ln 2 þ x2 ¼0 d x1 d x1
ðA-3Þ
Rearranging Eqs. A-1 and A-2 and differentiating results in d ln 1 A ½að1 x1 Þa1 faxa1 ¼ þ ð1 2aÞxa1 g 1 RT d x1 ðA-4Þ þ ð1 x1 Þa faða 1Þxa2 1 þ ð1 2aÞaxa1 1 g
Figure 2.
.
and d ln 2 A a1 faxa1 ¼ 1 ½að1 x1 Þ RT d x1 þ ð1 2aÞð1 x1 Þa
CONCLUSIONS Activity coefficient of a solute in a solution is required for determining the solubility of the solute in the solvent. In Part 1 of this two-part contribution, we developed a new predictive activity coefficient model that only uses the bulk properties of the solute and solvent. The proposed model is much simpler than the existing predictive activity coefficient models in the literature with comparable or better predictive capability. The predictive capability of the model was tested in 41 vaporliquid equilibrium systems and compared with the results of the UNIFAC and Hildebrand predictive models. The solubility of stearic acid polymorph C in methanol and 2-butanone was also predicted using the new activity coefficient model. The advantage of the proposed activity coefficient model, in addition to its superior predictive capability, is its simplicity.
ð1 2aÞað1 x1 Þa1 g Replacing Eqs. A-4 and A-5 in Eq. A-3 results in d ln 2 d ln 2 þ x2 d x1 d x1 A a1 faxa1 ¼ x1 2 ½ax1 RT þ ð1 2aÞxa1 þ xa2 ½aða 1Þxa2 1 A a1 faxa1 þ ð1 2aÞaxa1 1 g þ x2 1 ½ax2 RT þ ð1 2aÞxa2 þ xa1 ½aða 1Þxa2 2
x1
ð1 2aÞaxa1 2 g ¼
ð1 2aÞaxa1 þ x2 xa1 ½aða 1Þxa2 2 2 g
The authors would like to thank Apotex Pharmachem, Inc. (Brantford, Canada) and the NSERC (Canada) for partial financial support.
A aþ1 a1 fa2 xa1 xa1 axa1 þ 2a2 xaþ1 2 1 x2 1 x2 RT a a a 2 a1 a þ aða 1Þxa1 1 x2 þ að1 2aÞx1 x2 þ a x1 x2 að1 2aÞxa1 xa2
For a binary mixture
a1 a1 ¼ a2 xa1 xa1 axaþ1 þ 2a2 xaþ1 2 1 x2 1 x2
ðA-1Þ
a a1 a 2 a1 a a1 aþ1 þ a2 xa1 1 x2 ax1 x2 þ a x1 x2 þ ax1 x2 aþ1 2a2 xa1 a2 xa1 xa1 þ axa1 xa1 2a2 xa1 xa1 1 x2 2 2 2
and A a a1 x ½ax ln 2 ¼ þ ð1 2aÞxa2 RT 1 2
¼
aþ1 aþ1 þ axa1 2aa xa1 aða 1Þxa1 xa1 1 x2 1 x2 2
Appendix 1-Gibbs–Duhem Test fo the New Model
A a a1 x ½ax þ ð1 2aÞxa1 RT 2 1
A a1 fax1 xa1 þ ð1 2aÞxa1 2 ½ax1 RT þ ð1 2aÞaxa1 þ x1 xa2 ½aða 1Þxa2 1 1 a1 þ ð1 2aÞxa2 þ ax2 xa1 1 ½ax2
ACKNOWLEDGMENTS
ln 1 ¼
ðA-5Þ
þ xa1 ½aða 1Þð1 x1 Þa2
a1 a1 a axaþ1 þ 2a2 xaþ1 þ 2a2 xa1 1 x2 1 x2 1 x2
ðA-2Þ
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a a1 aþ1 aþ1 axa1 2a2 xa1 þ axa1 xa1 1 x2 þ ax1 x2 1 x2 2
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a1 a a1 aþ1 ¼ ð2a 4a2 Þ½xa1 xa1 xaþ1 xa1 2 1 x2 þ x1 x2 1 x2 a1 a ¼ ð2a 4a2 Þ½xa1 xa1 2 ð1 x1 Þ x1 x2 ð1 x2 Þ
¼ ð2a 4a2 Þ xa1 xa2 xa1 xa2 ¼ 0
ðA-6Þ
The right hand side (RHS) of Eq. A-6 is zero so it obeys the Gibbs–Duhem condition.
REFERENCES 1. Prausnitz JM, Lichtenthaler RN, Azevedo EG. 1999. Molecular thermodynamics of fluid-phase equilibria. Upper Saddle River, New Jersey: Prentice-Hall, pp 59–77. 2. Hildebrand JH, Prausnitz JM, Scott RL. 1970. Regular and related solutions. New York: Van Nostrand Reinhold. 22p. 3. Fredenslund A, Jones RL, Prausnitz JM. 1975. Group-contribution estimation of activity coefficient in nonideal liquid mixtures. AIChE J 21: 1086–1099. 4. Gracin S, Brinck T, Rasmuson AC. 2002. Prediction of solubility of solid organic compounds in solvents by UNIFAC. Ind Eng Chem Res 41:5114– 5124. 5. Paruta AN, Irani SA. 1965. Dielectric solubility profiles in dioxane-water mixtures for several antipyretic drugs. J Pharm Sci 54:1334– 1338. 6. Paruta AN, Sciarrone BJ, Lordi NG. 1964. Solubility of salicylic acid as a function of dielectric constant. J Pharm Sci 53:1349–1353.
DOI 10.1002/jps
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7. Sorby DL, Bitter RG, Webb JG. 1963. Dielectric constants of complex pharmaceutical solvent systems I. J Pharm Sci 52:1149–1153. 8. Oaruta AN, Sciarrone BJ, Lordi NG. 1965. Dielectric solubility profiles of acetanilide and several derivatives in dioxane-water mixtures. J Pharm Sci 54:1325–1333. 9. Fung HL, Higuchi T. 1971. Molecular interactions and solubility of polar nonelectrolytes in nonpolar solvents. J Pharm Sci 60:1782–1788. 10. Gmehling J, Onken U, Arlt W. 1979. Vapourliquid equilibrium data collection. Frankfurt: DECHEMA. 11. Coleman TF, Li Y. 1994. On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds. Math Program 67:189–224. 12. Coleman TF, Li Y. 2004. An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J Optim 6:418–445. 13. Hansen CM, Skaarup K. 1967. The three dimensional solubility parameter—key to paint component affinities. III. Independent calculation of the parameter components. J Paint Technol 39:511– 520. 14. Martin A, Newburger J, Adjei A. 1980. Extended Hildebrand solubility approach: solubility of theophylline in polar binary solvents. J Pharm Sci 69: 487–491. 15. Adjei A, Newburger J, Martin A. 1980. Extended Hildebrand approach: solubility of caffeine in dioxane-water mixtures. J Pharm Sci 69:659–661. 16. Subrahmanyam CVS, Prakash KR, Rao PG. 1996. Estimation of the solubility parameter of trimethoprim by current methods. Pharm Acta Helv 71: 175–183.
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Received 4 May 2005; revised 17 November 2005; accepted 18 November 2005 Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jps.20560
ABSTRACT: A new activity coefficient model was developed from excess Gibbs free energy in the form Gex ¼ cAa xb1 . . . xbn. The constants of the proposed model were considered to be function of solute and solvent dielectric constants, Hildebrand solubility parameters and specific volumes of solute and solvent molecules. The proposed model obeys the Gibbs–Duhem condition for activity coefficient models. To generalize the model and make it as a purely predictive model without any adjustable parameters, its constants were found using the experimental activity coefficient and physical properties of 20 vapor-liquid systems. The predictive capability of the proposed model was tested by calculating the activity coefficients of 41 binary vapor-liquid equilibrium systems and showed good agreement with the experimental data in comparison with two other predictive models, the UNIFAC and Hildebrand models. The only data used for the prediction of activity coefficients, were dielectric constants, Hildebrand solubility parameters, and specific volumes of the solute and solvent molecules. Furthermore, the proposed model was used to predict the activity coefficient of an organic compound, stearic acid, whose physical properties were available in methanol and 2-butanone. The predicted activity coefficient along with the thermal properties of the stearic acid were used to calculate the solubility of stearic acid in these two solvents and resulted in a better agreement with the experimental data compared to the UNIFAC and Hildebrand predictive models. ß 2006 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 95:790–797, 2006
Keywords:
activity coefficient; predictive models; solubility; bulk properties
INTRODUCTION Crystallization process is a key unit operation in the production of pharmaceutical solids, which has a major effect on drug quality in terms of crystal size distribution, solid bulk and tap density, polymorphism, filterability, and flowability. Supersaturation, which is a function of solubility, is a kinetic and thermodynamic parameter that influences the crystal size distribution, Mahmoud Mirmehrabi’s present address is Wyeth Pharmaceuticals, Montreal, Canada, H4R 1J6. Correspondence to: Sohrab Rohani (Telephone: 519-6614116; Fax: 519-661-3498; E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 95, 790–797 (2006) ß 2006 Wiley-Liss, Inc. and the American Pharmacists Association
790
morphology, and polymorphic formation. Thus, knowing the solubility is essential in the study of a crystallization process. Solubility is needed to determine the crystallization throughput or yield. It is evident that the solubility prediction is necessary in solvent selection and control of crystallization processes especially for the drugs and fine chemicals where the experimental data are scarce. Using thermal properties of a solute, for example, heat of fusion and melting temperature, the ideal solubility of the solute can be calculated.1 To determine the real solubility, the activity coefficient of the solute in the solvent, an indication of the degree of nonideality, is needed. There are many factors, which affect the activity coefficient
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including the molecular size, polarity, and interaction forces. In predicting the activity coefficient using the bulk properties of solute and solvent, the Hildebrand solubility Eq. 1 can be used only for the regular or near-ideal solutions2
ln 2 ¼
v2 f21 ð1 2 Þ2 RT
ð1Þ
where d is the Hildebrand solubility parameter (J/cm3)0.5, g2 is the activity coefficient of the solute, v is the specific volume cm3/mol, Tis the absolute temperature, and f is the volume fraction of the solvent. Subscripts 1 and 2 refer to the solvent and solute, respectively. This equation is invalid for a wide range of conditions including the cases in which there are interactions or even nonsimilarities between solvent and solute molecules. Most pharmaceuticals are complicated compounds, which have various functional groups that interact with the solvent and with other functional groups of the solute molecule. Therefore, the regular solution theory does not work properly. The advantage of the Hildebrand equation is a predictive model that uses the bulk properties, which are available in handbooks or easily measurable. The UNIFAC model, a group contribution model,3 is another predictive model. In many cases, however, the configuration of the neighboring groups restricts the predictive capability of this model.4 In Part 1 of this two-part contribution, a new model is proposed for the prediction of the activity coefficient of a solute in a solution using the bulk properties of the solute and the solvent. The proposed model, once its constants are determined using the experimental activity coefficients of 20 pairs of vapor-liquid systems, becomes a purely predictive model similar to Hildebrand equation and the UNIFAC model. The activity coefficient in a solid-liquid or in a vapor-liquid system, depends on the interaction of the dissolved solute or vapor molecules with the solvent molecules, in the liquid phase. The solid phase or the vapor phase, therefore, does not play an important role in the nonideality of the solution. Therefore, the experimental vapor-liquid systems can be used in the development or validation of an activity coefficient model. In the present work, the predictive capability of the proposed model is examined by comparing the predicted and the experimental activity coefficients of 41 vapor-liquid binary systems. The results of the UNIFAC and Hildebrand DOI 10.1002/jps
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predictive models are also presented for comparison. In part 2, the proposed model will be revised by retaining two adjustable parameters, as a semipredictive model, to enhance and generalize its predictive capability. In Part 2, two other semipredictive models, the UNIQUAC and NRTL will also be used for modeling and prediction of solubility of a few pharmaceutical compounds.
RESULTS AND DISCUSSION Proposed Activity Coefficient Model There are various intermolecular forces between molecules of a pure compound, such as Van der Waals forces, dipole–dipole forces, and hydrogen bonding. The bulk physical properties of a chemical are functions of these intermolecular forces. Hildebrand solubility parameter is a direct function of the heat of vaporization. The other bulk properties that distinguish a compound are molar volume and dielectric constant. Usually the smaller molecules possess higher polarity. Dielectric constant is an indication of the degree of polarity and polarizability of a molecule. Many attempts have been made to correlate the solubility of a solid compound with its dielectric constant and specific volume.5–9 The proposed predictive activity coefficient model is based on the excess Gibbs free energy defined as: Gex ¼ cAa xb1 . . . xbm
ð2Þ
ex
where G is the excess Gibbs free energy, A is a constant which is a function of the bulk properties of the molecules in the solution, xi is the mole fraction of the mth component, a, b, and c are constants. This model is similar to Margules model, Gex ¼ Ax1 x2, in which A can be found from the experimental activity coefficient values. The activity coefficient can be calculated from the following equation ex
ln i ¼
Gi RT
ð3Þ
where ex Gi
@ðNGex Þ ¼ @Ni T; P; Nj
ð4Þ
ex
And Gi is the partial excess Gibbs free energy, Ni is the number of moles of each component and N is the total number of moles in the solution. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 95, NO. 4, APRIL 2006
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Table 1. The Binary VLE Systems that were used to Determine the Constants of the New Model Vapor-Liquid Equilibrium Component 1 Ethanol Ethanol Ethanol Ethanol Benzene Ethanol Ethanol Methanol Ethanol Dichloromethane
Vapor-Liquid Equilibrium
Component 2
Component 1
Component 2
1,2-Propanediol Acetonitrile 2-Propanol 2-Methyl-1-propanol Hexane Cyclohexanol 2-Butanone Acetone 1,2-dichloroethane Ethanol
Ethanol Methyl acetate Propyl acetate Ethyl acetate Ethanol Chloroform Ethanol MEK Ethanol Ethanol
Aniline Ethanol Ethanol Ethanol Chlorobenzene Ethanol Butyl acetate Heptane Ethyl benzene 1,4-dioxane
Equilibrium vapor mole fractions at different temperatures were available for each pair (9).
Eqs. 2–4 result in h i cA b b ðb1Þ x1 :::xm j6¼i bxi ln i ¼ þ ð1 mbÞxbi ð5Þ RT a
where m is the number of components in the system. A is also a function of the bulk properties. In fact, this term represents the difference between the properties of the solute and solvent A ¼ ðsolute Þ2 ðvsolute vÞ ð"solute "Þd ð6Þ where d is a constant, dsolute and are the solute and average Hildebrand solubility parameters, vsolute and v are the solute and average molar volumes, esolute and " are the solute and average dielectric constants. All of these parameters are at 258C. The parameter A is not a function of composition as it is showing how the solute and solvent molecules are different in nature. However, introducing the composition into Eq. 6 may enhance the generalizability of the equation for systems with more than two components. If the physical properties of the solvent and solute; the solubility parameter, specific volume, and the dielectric constant are similar, then the term A approaches zero and the activity coefficient becomes unity. This model obeys the GibbsDuhem equation for activity coefficients. In general any activity coefficient model that is derived from excess Gibbs free energy (Eq. 2) and satisfies the condition Gex ! 0 as xi ! 0, would obey the Gibbs–Duhem law. Appendix 1 presents the proof that the new model obeys the Gibbs–Duhem condition. JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 95, NO. 4, APRIL 2006
To find parameters a, b, c, and d of the proposed model, the vapor-liquid equilibrium data10 of 20 systems (Tab. 1) along with the modified Raoult’s law were used. The 20 pairs of compounds, cover a wide range of properties from nonpolar to polar and hydrogen bond forming chemicals. At low pressures, the fugacity coefficient in the gas phase approaches unity and the modified Raoult’s equation can be used as follows: xi i Pi ¼ yi Pt
ð7Þ
where Pi* is the vapor pressure of component i at the studied temperature that can be calculated from the Antoine equation,10 yi is the mole fraction in the vapor phase, and Pt is the total pressure of the system. The parameters a, b, c, and d were varied to minimize the difference between the calculated (using the new model) and the experimental mole fractions (of the two components) in the vapor phase. The regression was carried out using fmincon function of MATLABTM. The algorithm of fmincon is based on the interior-reflective Newton method, which is described by Coleman and Li11,12. The estimated parameters are listed in Table 2. One of the major molecular interactions is dipole–dipole interaction especially for the nonregular solu-
Table 2. Constants of the Proposed Predictive New Activity Coefficient Model a 0.1329
b
c
d
0.9567
67.9
10
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tions. In our proposed equation, the dielectric constant is an empirical parameter that addresses this interaction. The estimated parameters show that the dielectric constant, which is a direct indication of the degree of polarity, has the largest effect on deviation from the ideal state. This may be a reason that the Hildebrand equation, which does not consider the degree of polarity, results in poor prediction for polar systems. Other methods have been developed to extend the Hildebrand equation and also Hansen equation13–16, which showed significant improvement from the original equations. However, these extended methods did not consider the dielectric constant as an effective parameter. Using the parameters of Table 2, the new model becomes a predictive model. In the next section the predictive capability of the new model will be examined to estimate the activity coefficients of 41 vapor-liquid systems.
order to compare the models, the average absolute error between the experimental and the calculated mole fractions ( yi, exp and yi, calc) defined by n P
yi;exp yi;calc
Average absolute error ¼ i¼1
ð8Þ
n
was used. The average absolute errors for all models for the systems listed in Table 3 are relatively small. Table 4 lists the average absolute error for highly nonideal vapor-liquid systems. The average absolute error and the standard deviation of the new model are smaller than the other predictive models. The Hildebrand regular solution model is the worst.
Solubility Prediction of Organic Solids For nonideal solutions we have ln
Prediction of Activity Coefficient using Vapor-Liquid Equilibrium Data To test the ability of the predictive new model, the activity coefficients of 41 pairs of compounds with a wide range of properties were estimated. The solutes’ activity coefficients and the vapor phase mole fractions were calculated and compared with the experimental mole fractions listed in Tables 3 and 4. Raoult’s law, Hildebrand, and the UNIFAC models were also used to predict the vapor phase mole fractions of the 41 vapor-liquid systems. In
793
f2L DHfus Ttp Dcp Ttp 1 1 ¼ RTtp T R T f2S Dcp Ttp þ ln R T
ð9Þ
where f2L and f2S are the fugacities of pure subcooled liquid and pure solid forms of the solute, respectively, and Ttp is the triple point temperature which can be considered as the melting point. DHfus is heat of fusion and Dcp is the difference in the heat capacities of liquid and solid at absolute temperature, T. The first term in
Table 3. The Average Absolute Error of Vapor Mole Fraction from the Experimental Data using Raoults’s Law, the New Model, UNIFAC and Hildebrand Equation
Average Methanol–Ethanol Acetone–Pyridine MEK–Toluene Chlorobenzene–Decane Hexane–Benzene Ethanol–MEK Methanol–1,2-Propanediol 2-Butanone–1-Butanol Acetone–2-Propanol Acetone–Benzene Absolute average Standard Deviation
Number of Data Points 12 12 23 11 10 10 10 13 13 13
Raoult
New Model
UNIFAC
Hildebrand
0.012 (0.025) (0.032) (0.035) (0.041) (0.054) 0.056 (0.059) (0.061) (0.069) 0.044 0.018
0.040 0.018 (0.008) 0.022 0.040 0.037 0.056 0.058 0.060 (0.019) 0.036 0.019
0.050 0.030 (0.021) 0.050 0.010 0.011 (0.047) 0.036 0.025 (0.024) 0.030 0.014
0.032 (0.014) (0.031) 0.064 0.017 0.110 0.056 0.057 (0.016) (0.065) 0.046 0.030
The predicted mole fractions in parentheses indicate that the prediction is less than the experimental value. DOI 10.1002/jps
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Table 4. The Average Absolute Error of the Calculated Vapor Mole Fractions From the Experimental Data Using Raoults’s Law, the New Model, UNIFAC, and Hildebrand Equation
Average Ethyl acetate–Ethanol Methyl acetate–Ethanol Methanol–Acetonitrile Chloroform–1-Butanol Ethanol–Nitromethane Methanol–Carbon tetrachloride Pyridine–Nonane Ethanol–Benzene Ethyl acetate–1 Butanol Chloroform–Methanol Chloroform–Ethanol Methanol–2 Pentanone Methanol–Hexane Acetone–1 Butanol 1,2 Dichloroethane–1 Propanol MEK–Heptane Toluene–1 Butanol Benzene–1 Propanol 1,2 Dichloroethane–1 Butanol Methanol–Toluene Dichloromethane–Ethanol Acetone–Heptane Ethanol–Heptane Ethanol–Chlorobenzene Ethanol–Aniline Cyclohexane–Furfural Heptane–Furfural Methanol–p-Xylene Ethanol–Pentane Methanol–Heptane 2 Methylbutane–Ethanol Absolute average Standard deviation
Number of Data Points
Raoult
New Model
UNIFAC
Hildebrand
13 9 20 24 18 36
(0.092) (0.105) (0.105) (0.117) (0.119) (0.125)
(0.072) (0.042) 0.092 0.036 0.095 (0.050)
0.063 0.039 0.104 0.171 (0.050) (0.015)
0.152 0.122 (0.052) (0.025) (0.119) 0.179
13 21 18 12 14 9 21 15 20 8 15 12 25 10 12 9 17 11 8 10 9 11 11 10 11
(0.128) (0.134) (0.135) (0.138) (0.139) (0.140) (0.143) (0.150) (0.161) (0.163) (0.167) (0.196) (0.202) (0.203) (0.206) (0.237) (0.245) (0.259) (0.300) (0.344) (0.364) (0.368) (0.383) (0.411) (0.457) 0.208 0.103
0.085 0.039 0.069 (0.176) (0.032) 0.047 (0.049) 0.137 0.137 0.072 0.103 0.122 0.177 (0.076) 0.127 0.072 0.110 0.111 0.107 (0.149) (0.242) (0.134) 0.108 (0.086) 0.116 0.099 0.048
0.019 0.011 0.011 0.278 0.083 0.177 0.160 (0.066) (0.029) (0.038) (0.006) (0.042) (0.032) 0.207 (0.052) (0.077) (0.039) (0.056) (0.296) (0.040) 0.036 0.438 (0.103) 0.542 (0.031) 0.107 0.128
0.069 (0.041) 0.027 0.328 0.086 0.459 0.318 (0.092) (0.116) (0.101) (0.081) (0.106) (0.165) 0.212 (0.100) (0.130) 0.354 (0.099) (0.268) 0.278 0.153 0.377 1.153 1.211 5.918 0.416 1.058
The predicted mole fractions in parentheses indicate that the prediction is less than the experimental value.
Eq. 9 has the largest effect and the next two terms with the opposite signs tend to compensate each other.1 Melting of many pharmaceutical compounds is accompanied with exothermic degradation so that measuring the liquid heat capacity is a formidable task. In these cases, just the first term will be used fS Fugacities are related by : x2 2 ¼ 2L f2
ð10Þ
where g2 is the activity coefficient of solute in the solution. The two last terms in Eq. 9 are not considered here. For ideal systems the activity JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 95, NO. 4, APRIL 2006
coefficient of the solute is equal to unity so that the equation will be truncated to 1 DHfus Ttp 1 ð11Þ ln ideal ¼ RTtp T x2 Ideal and experimental mole fractions are related by x2 ¼
xideal 2 2
ð12Þ
Equation 11 calculates the ideal saturation mole fraction (solubility in terms of mole fraction) using the thermal properties of the solid. To obtain the real saturation mole fraction, the activity DOI 10.1002/jps
SOLUBILITY OF PREDICTIVE MODELS
the specific volume of both solute and solvent. 3. Calculate the new mole fraction from Eq. 12 using the predicted activity coefficient obtained and ideal mole fraction. 4. Repeat steps 2 and 3 until the new calculated mole fraction does not differ from the previous iteration by more than a set tolerance limit. 5. Calculate the solubility from Eq. 13 using the obtained mole fraction.
coefficient of the solute in the solution should be replaced in Eq. 12. The following equations yield the solubility in terms of grams solute per 100 g solvent using the saturation mole fraction, x2 x¼
x2 1 x2
and then
S¼
x MWsolute 100 MWsolvent ð13Þ
Where S is the solubility in terms of grams solute per 100 g solvent and MW is the molecular weight, g/mole. Hildebrand, the UNIFAC and the new models were used to predict the activity coefficient and the solubility of stearic acid polymorph C in methanol and 2-butanone. To predict the mole fraction of dissolved solute in the solvent, an iterative procedure was performed: 1. Calculate the ideal mole fraction of the solute from Eq. 11. 2. Calculate the corresponding activity coefficient from the activity coefficient models using the calculated ideal mole fraction. The group interaction parameters are required for the UNIFAC model. Hildebrand equation uses the solubility parameters of both solute and solvent. The new model requires the Hildebrand solubility parameters, dielectric constants, and also
795
Figures 1 and 2 illustrate the experimental solubility data along with the predicted results. The Hildebrand equation resulted in a very large number for the activity coefficient of stearic acid in methanol (approximately 107). Therefore the solubility approached zero. The estimated solubility in both solvents using the new activity coefficient model offered a better prediction compared with other predictive models, UNIFAC and Hildebrand. However, similar to the Hildebrand equation, the new model does not recognize the difference between positive and negative deviations from the Raoult’s law (see also Tabs. 3 and 4). If the coefficients of Eq. 5 (a, b, c, and A) are considered as adjustable parameters then the model will also work for negative deviations as the overall coefficient can be negative which lead to an activity coefficient less than 1.
Figure 1. DOI 10.1002/jps
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Gibbs–Duhem condition requires that the following equation be satisfied x1
d ln 1 d ln 2 þ x2 ¼0 d x1 d x1
ðA-3Þ
Rearranging Eqs. A-1 and A-2 and differentiating results in d ln 1 A ½að1 x1 Þa1 faxa1 ¼ þ ð1 2aÞxa1 g 1 RT d x1 ðA-4Þ þ ð1 x1 Þa faða 1Þxa2 1 þ ð1 2aÞaxa1 1 g
Figure 2.
.
and d ln 2 A a1 faxa1 ¼ 1 ½að1 x1 Þ RT d x1 þ ð1 2aÞð1 x1 Þa
CONCLUSIONS Activity coefficient of a solute in a solution is required for determining the solubility of the solute in the solvent. In Part 1 of this two-part contribution, we developed a new predictive activity coefficient model that only uses the bulk properties of the solute and solvent. The proposed model is much simpler than the existing predictive activity coefficient models in the literature with comparable or better predictive capability. The predictive capability of the model was tested in 41 vaporliquid equilibrium systems and compared with the results of the UNIFAC and Hildebrand predictive models. The solubility of stearic acid polymorph C in methanol and 2-butanone was also predicted using the new activity coefficient model. The advantage of the proposed activity coefficient model, in addition to its superior predictive capability, is its simplicity.
ð1 2aÞað1 x1 Þa1 g Replacing Eqs. A-4 and A-5 in Eq. A-3 results in d ln 2 d ln 2 þ x2 d x1 d x1 A a1 faxa1 ¼ x1 2 ½ax1 RT þ ð1 2aÞxa1 þ xa2 ½aða 1Þxa2 1 A a1 faxa1 þ ð1 2aÞaxa1 1 g þ x2 1 ½ax2 RT þ ð1 2aÞxa2 þ xa1 ½aða 1Þxa2 2
x1
ð1 2aÞaxa1 2 g ¼
ð1 2aÞaxa1 þ x2 xa1 ½aða 1Þxa2 2 2 g
The authors would like to thank Apotex Pharmachem, Inc. (Brantford, Canada) and the NSERC (Canada) for partial financial support.
A aþ1 a1 fa2 xa1 xa1 axa1 þ 2a2 xaþ1 2 1 x2 1 x2 RT a a a 2 a1 a þ aða 1Þxa1 1 x2 þ að1 2aÞx1 x2 þ a x1 x2 að1 2aÞxa1 xa2
For a binary mixture
a1 a1 ¼ a2 xa1 xa1 axaþ1 þ 2a2 xaþ1 2 1 x2 1 x2
ðA-1Þ
a a1 a 2 a1 a a1 aþ1 þ a2 xa1 1 x2 ax1 x2 þ a x1 x2 þ ax1 x2 aþ1 2a2 xa1 a2 xa1 xa1 þ axa1 xa1 2a2 xa1 xa1 1 x2 2 2 2
and A a a1 x ½ax ln 2 ¼ þ ð1 2aÞxa2 RT 1 2
¼
aþ1 aþ1 þ axa1 2aa xa1 aða 1Þxa1 xa1 1 x2 1 x2 2
Appendix 1-Gibbs–Duhem Test fo the New Model
A a a1 x ½ax þ ð1 2aÞxa1 RT 2 1
A a1 fax1 xa1 þ ð1 2aÞxa1 2 ½ax1 RT þ ð1 2aÞaxa1 þ x1 xa2 ½aða 1Þxa2 1 1 a1 þ ð1 2aÞxa2 þ ax2 xa1 1 ½ax2
ACKNOWLEDGMENTS
ln 1 ¼
ðA-5Þ
þ xa1 ½aða 1Þð1 x1 Þa2
a1 a1 a axaþ1 þ 2a2 xaþ1 þ 2a2 xa1 1 x2 1 x2 1 x2
ðA-2Þ
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a a1 aþ1 aþ1 axa1 2a2 xa1 þ axa1 xa1 1 x2 þ ax1 x2 1 x2 2
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SOLUBILITY OF PREDICTIVE MODELS
a1 a a1 aþ1 ¼ ð2a 4a2 Þ½xa1 xa1 xaþ1 xa1 2 1 x2 þ x1 x2 1 x2 a1 a ¼ ð2a 4a2 Þ½xa1 xa1 2 ð1 x1 Þ x1 x2 ð1 x2 Þ
¼ ð2a 4a2 Þ xa1 xa2 xa1 xa2 ¼ 0
ðA-6Þ
The right hand side (RHS) of Eq. A-6 is zero so it obeys the Gibbs–Duhem condition.
REFERENCES 1. Prausnitz JM, Lichtenthaler RN, Azevedo EG. 1999. Molecular thermodynamics of fluid-phase equilibria. Upper Saddle River, New Jersey: Prentice-Hall, pp 59–77. 2. Hildebrand JH, Prausnitz JM, Scott RL. 1970. Regular and related solutions. New York: Van Nostrand Reinhold. 22p. 3. Fredenslund A, Jones RL, Prausnitz JM. 1975. Group-contribution estimation of activity coefficient in nonideal liquid mixtures. AIChE J 21: 1086–1099. 4. Gracin S, Brinck T, Rasmuson AC. 2002. Prediction of solubility of solid organic compounds in solvents by UNIFAC. Ind Eng Chem Res 41:5114– 5124. 5. Paruta AN, Irani SA. 1965. Dielectric solubility profiles in dioxane-water mixtures for several antipyretic drugs. J Pharm Sci 54:1334– 1338. 6. Paruta AN, Sciarrone BJ, Lordi NG. 1964. Solubility of salicylic acid as a function of dielectric constant. J Pharm Sci 53:1349–1353.
DOI 10.1002/jps
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7. Sorby DL, Bitter RG, Webb JG. 1963. Dielectric constants of complex pharmaceutical solvent systems I. J Pharm Sci 52:1149–1153. 8. Oaruta AN, Sciarrone BJ, Lordi NG. 1965. Dielectric solubility profiles of acetanilide and several derivatives in dioxane-water mixtures. J Pharm Sci 54:1325–1333. 9. Fung HL, Higuchi T. 1971. Molecular interactions and solubility of polar nonelectrolytes in nonpolar solvents. J Pharm Sci 60:1782–1788. 10. Gmehling J, Onken U, Arlt W. 1979. Vapourliquid equilibrium data collection. Frankfurt: DECHEMA. 11. Coleman TF, Li Y. 1994. On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds. Math Program 67:189–224. 12. Coleman TF, Li Y. 2004. An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J Optim 6:418–445. 13. Hansen CM, Skaarup K. 1967. The three dimensional solubility parameter—key to paint component affinities. III. Independent calculation of the parameter components. J Paint Technol 39:511– 520. 14. Martin A, Newburger J, Adjei A. 1980. Extended Hildebrand solubility approach: solubility of theophylline in polar binary solvents. J Pharm Sci 69: 487–491. 15. Adjei A, Newburger J, Martin A. 1980. Extended Hildebrand approach: solubility of caffeine in dioxane-water mixtures. J Pharm Sci 69:659–661. 16. Subrahmanyam CVS, Prakash KR, Rao PG. 1996. Estimation of the solubility parameter of trimethoprim by current methods. Pharm Acta Helv 71: 175–183.
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