Comparative Study of Semitheoretical Models for Predicting Infinite

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Comparative Study of Semitheoretical Models for Predicting Infinite Dilution Activity Coefficients of Alkanes in Organic Solvents Cecilia B. Castells and Peter W. Carr* Department of Chemistry, Kolthoff and Smith Hall, University of Minnesota, 207 Pleasant Street, SE Minneapolis, Minnesota 55455

David I. Eikens 3M General Offices, 3M Center, St. Paul, Minnesota 55144

David Bush and Charles A. Eckert School of Chemical Engineering and Specialty Separations Center, Georgia Institute of Technology, Atlanta, Georgia 30332-0100

Five nonelectrolyte solution models are used to predict infinite dilution activity coefficients (γ∞) of five linear, four branched, and two cyclic alkanes in 67 solvents at 25 °C, and the results are compared with experimental data. The models use two distinct approaches to the prediction of γ∞. The solution of groups concept provides the basis for three versions of the UNIFAC model: original UNIFAC, γ∞-based UNIFAC, and modified UNIFAC (Dortmund). The MOSCED and the SPACE models avoid the group concept and use only pure component parameters. For a database of 737 limiting activity coefficients, the SPACE model gave an average absolute error of 8.1%, and in only 13.3% of the cases were the errors worse than 15%. The modified UNIFAC model gave an absolute average error of 9.8%, and 32% of the predicted γ∞ had errors larger than 15%. The SPACE approach also produced the most reliable estimations over a wide range of activity coefficient values. Introduction The application of γ∞ values to the synthesis and design of many bulk separation processes has been widely demonstrated.1-5 Limiting activity coefficients are also valuable for understanding the underlying physical and chemical interactions between unlike molecules in solution. Several techniques for the direct measurement of γ∞ have been developed. A comprehensive and detailed comparison of the degree of precision and accuracy achieved with these methodologies can be found in the literature.6-8 As more direct experimental data have been gradually introduced, more reliable databases have been acquired. Currently, the Dortmund Data Bank9 contains more than 30 000 γ∞ data from 450 bibliographic references.10 At the same time, the development and subsequent improvement of a variety of semiempirical models of activity coefficients have facilitated great advances in the prediction of γ∞ values. Among these models, group contributional methods, basing molecular data on the contributions of the constituent groups, are the most widely known. Using this approach, Deal and Derr proposed the ASOG method for VLE calculations.11-13 A related approach was developed by Fredenslund et al. in 1975,14 who presented the UNIFAC (UNIQUAC functional group activity coefficient) model. Since the first version of UNIFAC, several revisions and extensions have been proposed to * To whom correspondence should be addressed. Phone: 612-624-0253. Fax: 612-626-7541. E-mail: [email protected].

improve the prediction of γ∞ and excess enthalpies for highly asymmetric mixtures. In 1988, Bastos et al.15 generated a new UNIFAC parameter table based exclusively on experimental γ∞ data especially suited for the prediction of γ∞. This γ∞-based UNIFAC also includes the modification of the combinatorial term suggested by Kikic et al.16 to correct for significant differences in the molar volumes of components. Weidlich and Gmehling17 modified UNIFAC, changing the combinatorial term, defining new groups, and introducing temperature-dependent parameters. Since then, two revisions of the parameter matrix have been published.18,19 The MOSCED (modified separation of cohesive energy density) model3 is an extension of regular solution theory to mixtures that can contain polar and hydrogenbonding components. The cohesive energy density is separated into dispersion forces, dipole forces, and hydrogen bonding, with small corrections made for asymmetry. The dipolarity and hydrogen bond basicity and acidity parameters were correlated on the basis of a limited database of activity coefficients. The SPACE (solvatochromic parameters for activity coefficients estimation) method, developed by Hait et al.,20 used a much larger database and recently established scales of solvent and solute dipolarity and hydrogen bonding.21-25 In addition, the adjustable parameters for each compound were removed. In this paper, we discuss the strengths and weaknesses of these five approaches by comparing the predicted and experimental γ∞ values at 25 °C of 11 alkanes in 67 common solvents belonging to diverse families. Numerous studies have examined the parti-

10.1021/ie990096+ CCC: $18.00 © 1999 American Chemical Society Published on Web 09/11/1999

Ind. Eng. Chem. Res., Vol. 38, No. 10, 1999 4105 Table 1. Absolute Average Model Error for Several Homologous Seriesa parameters

UNIFAC

γ∞-based UNIFAC

modified UNIFAC

MOSCED

SPACE

E%a nb max. errorc fraction with error > 20%d fraction with error > 15%e

20.0 671 (91%) 96.1 40% 49%

20.7 616 (84%) 170.5 41% 52%

9.8 643 (87%) 78.2 21% 32%

8.8 432 (59%) 44.9 7.2% 18%

8.1 539 (73%) 24.6 2.6% 13%

a Average absolute errors were computed as the average of relative errors: E(%) ) 100 × |γ∞ ∞ ∞ b model - γ experimental|/γ experimental. Number of data points. c Maximum error, E(%) between the model and experimental values. d Percentage of data having errors larger than 20%. e Percentage of data having errors larger than 15%.

tioning of alkane solutes, but most have been limited to only a few solvents such as water26-30 or high molecular weight gas chromatography stationary phases.31 γ∞ data for a group of straight, branched, and cyclic alkanes in organic common solvents have been rather scarce. After the rare gases, alkanes are the simplest compounds to use in studying solute-solvent interactions. The only possible interactions are dispersive and, with polar solvents, solvent dipole-solute-induced dipole interactions. Attempts to predict these types of mixtures represent an important challenge to predictive models. Database Infinite dilution activity coefficients of 11 alkanes in 67 solvents at 25 °C were obtained using automated headspace gas chromatography, with a methodology similar to that of Hussam,32 Park,33 and Cheong.34 The main differences from previous work are the simplification of the sampling valve, the replacement of metal by fused silica capillary transfer lines, and the method of varying the concentration of solutes in the equilibrium cell.35 The solutes include linear (n-pentane to nnonane), branched (2-methylpentane, 2,4-dimethylpentane, 2,5-dimethylhexane, and 2,3,4-trimethylpentane), and cyclic (cyclohexane and ethylcyclohexane) alkanes. The solvents include several homologue series (hydrocarbons, alcohols, ketones, nitriles, acetates) and several other common solvents. The estimated accuracy of the data is 5% and the long-term reproducibility is 5%.36 The experimental database is included as Supporting Information, along with the predicted γ∞ values from the five models. The equations used to calculate the combinatorial and the residual contributions in the UNIFAC approaches are those reported previously. The UNIFAC groups and their interaction parameters presented by Rasmussen et al.37-40 were used for the UNIFAC approach. The interaction parameters determined by Bastos et al.15 were used for the γ∞-based UNIFAC method, and those obtained by Weidlich and Gmehling17,18 were employed to examine the modified UNIFAC model. The parameters for the MOSCED estimations were obtained from the original paper by Thomas.3 SPACE γ∞ predictions were also obtained following the original paper;20 solvent polarizability, acidity, and basicity parameters were taken from Kamlet21 and solute polarizabilities from Li.22,25 Results and Discussion The performance of the five models for the γ∞ prediction of 11 alkanes in 67 solvents at 25 °C is summarized in Table 1. The original UNIFAC model is a poor performer with 20.0% average absolute error; however,

Figure 1. Absolute average errors between the experimental and predicted values of γ∞ of 11 alkanes in six homologous series.

modified UNIFAC shows substantial improvement with an average error of 9.8%. Although the γ∞-based UNIFAC average errors are similar to those reported by Bastos (20.2%),15 several predictions show gross individual errors (up to 170%). Thomas et al.3 found a 9.1% average absolute error in the original MOSCED model for 3357 data tested. In the present study of 432 γ∞ data, the relative difference from the experimental values was 8.8%, and fewer than 8% of the estimates have errors greater than 20%. The SPACE model predicts γ∞ with 8.1% average deviations, and only 2.6% of the data have errors greater than 20%. The original SPACE paper reported an overall error of 9% for 1879 data points.20 It should be mentioned that the linear and cyclic alkane data were included into the fitting used to parametrize the original SPACE model. We also examined the predictions obtained with the ASOG technique using the most recent parameter tables,41 but the results are not presented here since the average absolute deviations were larger than 29% and several individual predictions were inconsistent. Figure 1 shows the absolute average error between the experimental and predicted values of γ∞ grouped by homologous series of solvents. Activity coefficients of alkanes in alkane and alkene solvents are well-predicted by all five models. Alcoholic solvents are more difficult to model due to self-association driven by strong hydrogen-bonding interactions. UNIFAC and γ∞-based UNIFAC show the greatest differences, with errors of 21.2 and 26.3%, respectively. This indicates that the errors in these systems cannot be attributed either to the exponent used in the estimation of the combinatorial term or to the extrapolated experimental data used to obtain the interaction parameters in the original UNIFAC. The predictions based on the modified UNIFAC method are notably better (6.9% average absolute error). MOSCED and SPACE predictions are quite good: the average absolute errors are below the overall average

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Figure 2. Residual plots of ln γ∞ predicted and experimental: (A) UNIFAC; (B) γ∞-based UNIFAC; (C) modified UNIFAC; (D) MOSCED; (E) SPACE; (b), linear alkanes; (O), branched alkanes; (4), cyclic alkanes.

errors. For ketone and acetate solvents, the original UNIFAC model gives somewhat better predictions. For the nitriles, the SPACE model provides remarkably good predictions of γ∞ with an average absolute error of 7.8%, while errors with MOSCED and modified UNIFAC are somewhat larger (11.6 and 12.2%, respectively). The errors observed for alkane/nitrile mixtures with UNIFAC and γ∞-based UNIFAC are about twice those obtained using modified UNIFAC. Since the database includes a wide variety of solvents, most particularly a number of homologue series of solvents, it allows for a critical evaluation of the models in terms of their predictive abilities over a wide range of limiting activity coefficients. The deviations between the experimental and calculated ln γ∞ values are presented in Figure 2. The group contribution methods show systematic deviations at higher ln γ∞ values, indicating that estimates of the activity coefficients in highly nonideal systems by these approaches would be extremely low. The SPACE and MOSCED models present a much more uniformly distributed pattern. The availability of data for the normal, branched, and cyclic hydrocarbons allowed us to study the effect of branching and ring formation on the activity coefficients. Experimental γ∞ values for the 11 hydrocarbons in methanol (see Figure 3) show a strong correlation between ln γ∞ and the molar volume of n-alkanes (r ) 0.9995). The branched and cyclic isomers deviate from the linear least-squares plot for the n-alkane data. In Figure 3B, the predicted γ∞ values for linear hydrocarbons in methanol are compared with the experimental values. The MOSCED and SPACE models reproduce the experimental slope particularly well: excess Gibbs free

Figure 3. (A) Plot of experimental ln γ∞ of 11 hydrocarbons in methanol against solute molar volume: (b) linear alkanes; (2) branched alkanes; (9) cyclic alkanes. (B) Comparison of experimental and predicted ln γ∞ of linear alkanes in methanol against solute molar volumes. (b) experimental; (O), UNIFAC; (0), γ∞based UNIFAC; (4), modified UNIFAC; (3), MOSCED; (]), SPACE.

energy per methylene group values estimated from the experimental slope of ln γ∞ against carbon number and those obtained from the MOSCED and SPACE models differ only by 1.5 and 0.8%, respectively. Figure 4A-D displays the ratios of activity coefficients between linear and branched isomeric pairs in 1-alcohols, γ∞(l)/γ∞(br). Also, the ratios γ∞(n-hexane)/γ∞(cyclohexane) in the same alcohols are included in Figure 4E. The experimental data (filled symbols) show a larger positive γ∞ for straight chain species relative to the equivalent branched and cyclic isomers. The plots also suggest that molecules with a higher degree of branching (i.e., 2,3,4-trimethylpentane) form more ideal mixtures than do the linear alkanes. The γ∞ ratios are also larger in methanol and tend to decrease as the solvent polarity decreases. The same trends are also seen in most of the other polar solvents. More profound differences in γ∞ between normal and branched alkane molecules in water have also been reported.42-44 Since the group contribution models do not differentiate between CH3, CH2, CH, and C groups, the estimates of the contribution to residual γ∞ for all alkane isomers in a given solvent are exactly the same. Thus, in the context of the UNIFAC models, the deviations of the activity coefficients for these systems from ideality are due only to the combinatorial contribution. This term is entropic and depends on the difference in molecular sizes. It reaches a maximum of unity for components of the same size. The molar volumes of the alkane isomers differ less than 3%; thus, combinatorial contributions

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tions, and size difference. The SPACE equation for the γ∞ is

ln γ∞2 ) (V2/RT)[(λ1 - λ2)2 + (τ1 - τ2eff)2 + (R1 - R2eff)(β1 - β2eff)] + d12 (1)

Figure 4. Plots of activity coefficient ratios of isomer alkanes in 1-alcohols against solvent carbon number: (A) γ∞(n-hexane)/γ∞(2-methylpentane); (B) γ∞(n-heptane)/γ∞(2,4-dimethylpentane); (C) γ∞(n-octane)/γ∞(2,5-dimethylpentane); (D) γ∞(n-octane)/γ∞(2,3,4trimethylpentane); (E) γ∞(n-hexane)/γ∞(cyclohexane); b, experimental; O, UNIFAC; 0, γ∞-based UNIFAC; ∆, modified UNIFAC; (∇), MOSCED; ], SPACE.

to γ∞ of isomeric alkane molecules in a given solvent are quite similar and the predicted γ∞ values are practically the same for branched and linear aliphatic hydrocarbons. This shortcoming of UNIFAC has been recognized by Gmehling et al.10 To overcome this drawback, different group interaction parameters for primary, secondary, and tertiary carbon atoms are needed. Gani et al. recently introduced a second-order term to both the combinatorial and the residual UNIFAC equations.45 This modification successfully provided additional structural information that improved the prediction of activity coefficients of a few branched alkanes. Figure 4A-D (open symbols) also shows the ratios of γ∞ calculated by the MOSCED and SPACE approaches. With a single exception, both models predict a minor increase in γ∞ due to branching; that is not in agreement with the experimental facts. This is not the case for the cyclic alkanes. Figure 4E shows that basically the five models predict a decrease in γ∞ for the cyclic structure over the straight chain isomer. This is the expected trend given that the cyclic compounds have significantly smaller molar volumes than the linear isomer: the hexane/cyclohexane molar volume ratio at 25 °C is 1.21; therefore, the γ∞ predictions are expected to be about 20% larger for the linear compared to the cyclic isomer. Consider the individual contributions to γ∞ in the context of regular solution theory models. Four types of independent and additive interactions are assumed: dispersive, induction/polarizability, chemical interac-

where V2 is the solute molar volume and λi, τi, Ri, and βi represent the molecule’s dispersion, induction/polarity, hydrogen-bond acidity, and hydrogen-bond basicity, respectively. The last term is a Flory-Huggins combinatorial term that accounts for differences in molecular size. A similar expression is applicable to the MOSCED model. Neglecting the combinatorial term, since it is usually of minor importance in highly nonideal systems, two factors determine the excess Gibbs free energy of an alkane in a polar solvent: the physical interactions (dispersion and polarizability of the solute molecules) and cavity formation. According to the refractive indices and the solubility parameters, the polarizabilities of hydrocarbons increase with the number of carbon atoms, and the n-alkanes are more polarizable than are the branched isomers. However, the terms (λ1 - λ2)2 and (τ1 - τ2eff)2, which account for London forces and for dipole-induced dipole interactions, respectively, are not significantly different for any isomer relative to the n-alkane in any solvent. Therefore, predictions of the excess Gibbs free energy will be roughly proportional to the solute molar volume. The solute volume is scalar of the cavity formation. Except for 2,3,4-trimethylpentane, branching increases the solute molar volume compared to the linear chain, albeit only slightly; thus predictions of γ∞ of the branched hydrocarbons are higher than those corresponding to the normal isomer. Molecular volume has been used to predict the solubility of aliphatic hydrocarbons in aqueous solvents,42,46 and empirical corrections have been proposed to explain differences between normal and branched isomers. Leinonen successfully correlated the solubility of hydrocarbons in water with the “effective molar volume,” defined as the actual molar volume reduced by an increment of 3.1424 cm3/mol for each degree of branching.46 On the other hand, the solute surface area as the parameter to model cavity formation has also been extensively used to correlate the solubility of hydrocarbons in water.26,47,48 According to Hermann,26 branched structures significantly decrease the contact surface area with the solvent molecules, while molar volumes are essentially constant. Moreover, both normal and branched alkanes were incorporated on the same linear plot of solubility vs surface area. Therefore, the solvent accessible surface area is a better parameter for correlating the solubilities of all types of alkanes in aqueous solvents than is the experimental molar volume. The success of the various methods used to predict γ∞ depends on the mixture. Thus, while the SPACE approach was consistently good for most mixtures, the UNIFAC-based models were more accurate for alkane/ alkane, alkane/alkene, and the alkane/acetate solutions. In any event, we recommend that several approaches be used to predict γ∞ for the systems of interest and that chemical judgment come into play when interpreting the results. Conclusions Alkane/solvent mixtures, where specific interactions are absent, constitute excellent test systems for the

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critical comparison of solution models. On the whole, UNIFAC and γ∞-based UNIFAC methods perform comparably. Although it was created specifically for the task of estimating γ∞, the γ∞-based UNIFAC method does not give as good predictions as might be expected. Even though the SPACE model was primarily developed to describe systems in which hydrogen-bonding interactions take place, we conclude that the approach is also the more reliable method for alkane solutes in polar solvents. In contrast, the modified UNIFAC approach performs better than does the SPACE model in predicting γ∞ for alkane-alkane and alkane-acetate mixtures. Finally, the analysis of branched chain molecules demonstrates that none of the models are sensitive enough to deal with the experimental differences of γ∞ between branched and linear isomer in polar solvents. Acknowledgment We thank the National Science Foundation for financial support. C.B.C. acknowledges her fellowship from CONICET (Argentina). Supporting Information Available: Additional information includes the experimental and predicted γ∞ data. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Schreiber, L. B.; Eckert, C. A. Use of Infinite Dilution Activity Coefficients with Wilson’s Equation. Ind. Eng. Chem. Process Des. Dev. 1971, 10, 572. (2) Eckert, C. A.; Newman, B. A.; Nicolaides, G. L.; Long, T. C. Measurement and Application of Limiting Activity Coefficients. AIChE J. 1981, 27, 33-40. (3) Thomas, E. R.; Eckert, C. A. Prediction of Limiting Activity Coefficients by a Modified Separation of Cohesive Energy Density Model and UNIFAC. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 194-209. (4) Karger, B. L.; Snyder, L. R.; Horvath, C. An Introduction to Separation Science; John Wiley & Sons: New York, 1973. (5) Sandler, S. I. Infinite Dilution Activity Coefficients in Chemical, Environmental and Biochemical Engineering. Fluid Phase Equilib. 1996, 116, 343-353. (6) Tiegs, D.; Gmehling, J.; Medina, A.; Soares, M.; Bastos, J.; Alessi, P.; Kikic, I. Activity Coefficients at Infinite Dilution; DECHEMA Chemistry Data Series; DECHEMA: Franckfurt/ Main, 1986; Part 1. (7) Park, J. H. Headspace Gas Chromatographic Measurement and Applications of Limiting Activity Coefficients. Ph.D. Thesis, University of Minnesota, Minneapolis, 1988. (8) Eckert, C. A.; Sherman, S. R. Measurement and Prediction of Limiting Activity Coefficients. Fluid Phase Equilib. 1996, 116, 333-342. (9) Gmehling, J.; Menke, J.; Schiller, M.; Tiegs, D.; Medina, A.; Soares, M.; Bastos, J.; Alessi, P.; Kikic, I. Activity Coefficients at Infinite Dilution; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, starting 1986, 4 Parts. (10) Gmehling, J. Present Status of Group-Contribution Methods for the Synthesis and Design of Chemical Processes. Fluid Phase Equilib. 1998, 144, 37-47. (11) Pierotti, G. J.; Deal, C. H.; Derr, E. L. Activity Coefficients and Molecular Structure. Ind. Eng. Chem. 1959, 51, 95. (12) Wilson, G. M.; Deal, C. H. Activity Coefficients and Molecular Structure. Ind. Eng. Chem. Fundam. 1962, 1, 20-23. (13) Derr, E. L.; Deal, C. H. Analytical Solutions of Groups. Correlation of Activity Coefficients Through Structural Group Parameters. I. Inst. Chem. Eng. Symp. Ser. (London) 1969, 3, 40. (14) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086-1099.

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Received for review February 8, 1999 Revised manuscript received July 13, 1999 Accepted July 22, 1999 IE990096+