Ind. Eng. Chem. Res. 1993,32, 2905-2914
2905
Space Predictor for Infinite Dilution Activity Coefficients Mitchell J. Hait, Charles L. Liotta, and Charles A. Eckert' Schools of Chemical Engineering and Chemistry and Specialty Separations Center, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
Diane L. Bergmann and Anna M. Karachewski Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801
Andrew J. Dallas, David I. Eikens, Jianjun J. Li, and Peter W. Carr Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455
Russell B. Poe and Sarah C. Rutan Department of Chemistry, Virginia Commonwealth University, Richmond, Virginia 23284-2006
A new formulation (SPACE) is presented for the prediction of infinite dilution activity coefficients in nonionic solutions. Statistical methods have been used to develop a critically evaluated ym database a t 25 "C and to determine optimum mathematical expressions. The predictive method stems from earlier work using additive contributions to the cohesive energy density, but all compoundspecific adjustable parameters have been avoided by using known molecular properties, especially the solvatochromic parameters for dipolarity/polarizabilityand for hydrogen bonding. This method has the advantages of including explicitly "chemical" interactions and of being uncoupled from any specific gE(x) expression. For the present, it is limited to monofunctional molecules a t 25 "C,and it has not yet been applied to water. Although it appears somewhat more accurate for this database than the UNIFAC method, it should be viewed as a complementary technique-an additional tool for solvent selection by the design engineer.
Introduction
Fortunately,new experimentaltechniques for measuring
The use of the infinite dilution activity coefficient ym has both fundamental and practical advantages. At the molecular level, it yields information related to the Gibbs energy for unlike-pair interactions in solution. From a practical point of view, it can greatly aid process design and screening. From an analytical and theoretical point of view, ym data provide an important insight into unlike-pair intermolecular interactions in liquids. In the past, most data have been classical (finite concentration) vapor-liquid equilibria (VLE) data, which are useful in the design of distillationcolumns,but have limited information content. For example, the current gE(x)models handle VLE well, but break down when applied to any derivativeproperties, such as liquid-liquid equilibria (LLE) or enthalpy (Nicohides and Eckert, 1978; Lafyatis et al., 1989; Trampe and Eckert, 1991). The problem is that the gE(x)expressions (and equations of state as well) are really attacking two problems simultaneously: (1)What is the force between two molecules? (2) How do we add up the forces for an assembly of molecules? These are questions that statistical mechanics must answer, but the two problems need not be approached at the same time. The key to attacking this problem is ita division into two independent effects. For this purpose, we prefer to estimate y " by a method which does not imply any specific gE(r) expression. By bifurcating the problem, we are able to study each part separately, unconfounded by uncertainties from the other portion.
* Author to whom correspondenceshouldbeaddressed. E-mail address:
[email protected] y" now offer an independent route to evaluating inter-
molecular interactions without any dependence on the statistical mechanical models, composition functionality, or mixing rules. The direct measurement of ymoffers several advantages over the traditional VLE experiments; they are in general more accurate, easier to obtain, and more meaningful in their application. They can be integrated into many types of existing models without limiting assumptions. Specifically, 1. The use of the symmetric conventionfor the standard state effectively suppresses the effect of solvent-solvent energies, and of course, at infinite dilution, the solutesolute interactions are negligible. Thus, y" offers a superior route to gaining information about unlike-pair intermolecular interactions without resort to any specific (empirical) solution of the Gibbs-Duhem equation. Furthermore, working at infinite dilution avoids any evaluation of the communal entropy, for which no exact solution is known. 2. One can construct an entire binary VLE curve accurately using only the two y" values (Schreiber and Eckert, 1971). Moreover, the multicomponent VLE data can usually be calculated as well as they can be measured. 3. A number of useful techniques now exist for accurate estimation of ym,such as ASOG (Pierotti et al., 1959; Derr and Deal, 1969), UNIFAC (Fredenslund et al., 1975, 1977a,b; Gmehling et al., 1982,1987,1990;Gmehling and Weidlich, 19871,and MOSCED (Thomasand Eckert, 1984; Howell et al., 19891, as well as this work. 4. A variety of new experimental techniques offer rapid and accurate data acquisition for -f",such as gas chromatography (Janes and Martin (1952) and many others),
0888-5885/93/2632-2905$04.00/0 1993 American Chemical Society
2906 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
gas stripping (Leroi et al., 1977), headspace chromatography (Hussam and Carr, 1985), and differential ebulliometry (Thomas et al., 1982a,b;Scott, 1986;Trampe and Eckert, 1990), and even for partial molal enthalpies at infinite dilution (the temperature derivative of y") by isothermal asymmetric flow calorimetry (Trampe and Eckert, 1991). From a practical viewpoint, ymmay be used to estimate parameters for gE(x) expressions. Often in a complex design, one can use y" to estimate binary interaction parameters in a Van Laar, Wilson, NRTL, or UNIQUAC expression, and then extend to the multicomponent case for VLE quite reliably. One may even use these results with a sensitivity analysis in a design program to determine which binary data have the greatest effect on results. This method will minimize the amount of experimental data required by pinpointing key data. The same technique can be used for LLE, but is far less reliable. In addition, one can also use ymestimates with or without a specific gE(x)expression for the selection of solvents or the tailoring of solvent mixtures. This would be useful not only in liquid extractions, but also in other processes, including extractive distillation, liquid chromatography, and optimization of either yield or selectivity in chemical reactions. Many methods have been proposed for the estimation of ym,but all have limitations. We present here a new method, which we call the SPACE (Solvatochromic Parameters for Activity CoefficientEstimation) equation, and which we feel is more firmly based on the actual intermolecular interactions occurring in dilute solutions. We have had several advantages over previous investigators: First, there are now a number of improved techniques for the measurement of y", and far more good data are available; it is no longer necessary to use unreliable extrapolated data. Second, we have a far better quantitative understanding of intermolecular interactions, and we draw strongly on the solvatochromic work of Kamlet, Taft, and others over the past decade. Finally, we have been able to bring to bear some statistical techniques from chemometrics to help us both in the evaluation of the experimental data and in the formulation of a model. The resulting SPACE model is not a theory; it is a method. It is probably not useful to attempt to determine a "best" method for ymestimation, as different methods, especiallythose derived in different ways, should be viewed as complementary. In fact, a conscientious design engineer should use multiple methods when applicable to guide the selection of solvents for separations and reactions. History of the Model Regular solution theory (RST) relates the configurational energy to the cohesive intermolecular forces approximated by the energy of vaporization; it neglects excess volume and excess entropy and assumes a geometric mean mixing rule for cohesive energy density. These assumptions work surprisingly well for a wide variety of organic systems, but may be limited for highly polar or hydrogenbonding systems. Many authors have proposed variations on RST to extend it to polar and/or hydrogen-bonding systems (Arkel and Vles, 1936;Arkel, 1946; Weimer and Prausnitz, 1965; Null and Palmer, 1969;Null, 1970;Karger et al., 1978;and many others). We previously proposed a somewhat improved version called the MOSCED (Modified Separation of CohesiveEnergy Density) equation (Thomas and Eckert 1984; Howell et al., 1989),based on expanding the cohesive energy density into a linear sum of the dispersion,
dipolar, and hydrogen-bonding interactions, with a separate term to account for size differences:
where X and 7 were the nonpolar and polar contributions to the cohesive energy density, respectively; ci and j3 were the hydrogen-bond acidity and basicity, respectively; and q was a measure of dipole-induced dipole forces. While A, and to some extent q, were determined independently, 7 , a,and j3 were essentially adjustable parameters for each compound. The parameters $ and E were asymmetry parameters, given as complex transcendental functions of 7 , q, a,and j 3 of the solvent. The term d12 was a modified Flory-Huggins term for the entropic effect of size difference: d,, = In(
;)aa
+ 1- (;)""
where aa was a function of solvent properties. Therefore, in summary, the original MOSCED was a method for calculation of ym,which was independent of choice of model for the composition dependence of gE(x), but required many physical and adjustable parameters. Also, in this first version of MOSCED, the database was rather limited. First, the ymdata available were sparse and of modest accuracy; many were only extrapolations from finite concentration VLE data. There were no really good scales available for molecular properties, especially the hydrogen bond acidity and basicity, thus requiring multiple adjustable parameters for each molecule. Finally, the functionality was determined from qualitative ideas about molecular behavior without the use of statistical methods, thus yielding an awkward form including parameters such as chemical asymmetry. Some of these drawbacks were ameliorated by the revised version of Howell et al. (1989), which improved the database by excluding extrapolated data, including new ymdata taken by improved techniques, and reducing the number of adjustable parameters by using the solvent solvatochromic a and j3 scales. A t that time, no solute solvatochromic ci and j3 scales were available for use. The number of compound-specific adjustable parameters was reduced from three to one and the database was expanded, but the predictions were slightly less accurate than with the original version. In the present work, we have further expanded the database and have evaluated the entries statistically. We have been able to use the solvatochromic scales of a,8, and ?r* for all compound-specific parameters and have included separate scales for solvents and solutes. Also, we have reformulated the equation based on our functional analysis to a simpler and more accurate form, thus eliminating the awkward induction and asymmetry parameters. The SPACE equation also assumes additivity and independence of the various contributions to the cohesive energy density: (1) dispersion, (2) dipolar interactions, (3)hydrogen-bonding interactions, and (4) size differences:
where
up
is the solute molar volume; X is the dispersion
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2907 force parameter, related to the molecule’s refractive index; 7 is the dipolar parameter; a! and 9 , are measures of the molecule’s acidity and basicity, respectively; and d12 is a modified Flory-Huggins term, eq 2. The form of eq 3 was selected from a number of physically reasonable models by factor analysis. The SPACE model uses effective values for solute parameters (mff, amy,&eff), values that are scaled between the solvent and solute values based on the interactions in solution. For example, propanol as a solvent is highly self-associated and the acidity of the oligomer is enhanced relative to that of the monomer due to electronic effects (Abboud et al., 1985). The hydrogen bond weakens the covalent bond between the oxygen and the hydrogen, thus enhancing the hydrogen-bonding acidity relative to that of the monomer. When propanol (solute) is placed in cyclohexane (solvent), the propanol exists as a monomer and was shown to be a weaker hydrogen bond acid than when it is placed in a hydrogen-bonding solvent. Propanol placed in ethanol (solvent) would yield a stronger hydrogen-bonding acid due to the inductive effect of the ethanol on the propanol since the hydrogen bonding between the ethanol and the propanol is similar to that of propanol within itself. And, of course, whatever properties are used, propanol infinitely dilute in propanol must yield y m = 1.0. The SPACE model takes into account the influence of the solvent on the solute properties and is discussed below under the subheading Closure. Group Contribution Methods. Group contribution methods treat a solution as a mixture of various structural groups, as opposed to a mixture of two or more distinct compounds. This treatment has the inherent advantage that there are significantly fewer structural groups than there are compounds, hence fewer parameters are needed. This discussion focuses on UNIFAC, a more recent model with more interaction parameters available than ASOG. UNIFAC (UNIQUAC Functional Group Activity Coefficient) is based on summing the interactions of the structural groups of each molecule. For example, the components of n-propanol are one primary alcohol group (OH), two CH2 groups, and one CH3 group. The activity coefficient is comprised of two terms, a combinatorial (entropic) and a residual (enthalpic) term. The combinatorial term is a size difference term based on calculated volumes. The residual term represents chemical interactions occurring in solution, calculated from tabulated parameters for the interactions between the different chemical groups. This model uses the UNIQUAC relationship for gE(x). There are presently 44 chemical building blocks defiied in the UNIFAC equation. Each block needs 43 interaction parameters (self-interactions are considered ideal), thus leading to a total number of 946 possible parameters. These have been regressed from experimental y data, and presently only 45% of these interaction parameters are available (Gmehling et al., 1990). Linear Solvation Energy Relationships. It is evident that if gas-liquid partition coefficients ( K ) can be predicted, then one can compute the corresponding solute activity coefficients once the solvent vapor pressure is known. Kamlet, Abraham, Taft, and their co-workershave developed a very simple predictive method based on linear solvation energy relationships. In this approach, the Gibbs energy of solution (gas to liquid) of a solute is correlated with a series of empirical parameters (eq 5) that incorporates terms that describe the cavity formation and the solute-solvent interaction processes that take place upon dissolving a gaseous solute in a liquid. In
general, the equation takes the form
+
log K = SP, + (cavity formation + dispersion) (solutesolvent interactions) (4) An equation of this very general form has been used to correlate a wide variety of phenomena, including equilibrium constants of chemical reactions, rates of reactions, and spectroscopic shifts (Kamlet and Taft, 1976a,b, 1979a,b;Kamlet et al., 1977,1979a-c, 1981;Abboud et al., 1977; Taft et al., 1982). More recently, Li et al. (1991, 1992a), and Abraham and Whitting (1992) have used an equation of the above form to correlate retention in gas chromatography. For the specificcase of dissolving a wide variety of solutes in a given liquid, the best regression equation is log K = SP,
+ I log
+ sr2* + d8, + ua2+ bP2
(5)
where SPO,1, s, d , a, and b are fitting coefficients derived at a specific temperature for a specificsolvent. In contrast LI6, u2*, 8 2 , a2, and 62 were initially derived by doing spectroscopicmeasurements in judiciously chosen solvents, and were hence termed “solvatochromic parameters”. Currently, they are more often measured from direct thermodynamic studies of hydrogen-bonding equilibra or from chromatographic data. The parameters u2*,a2, and 82 were developed as measures of solute dipolarityl polarizability, hydrogen bond donor acidity, and hydrogen bond acceptor basicity, respectively. The log L16parameter is critically important in any gasto-liquid transfer process. Fortunately, this parameter is very simple. It is the partition coefficient (Ostwald solubility) of the solute in n-hexadecane a t 25 OC. It has been found to be the best single empirical parameter that simultaneously measures the energy needed to make a hole (cavity in a solvent) upon dissolution of a gaseous solvent, plus the dispersive (London)interactions between the solute and the solvent. Equation 5 has been found to fit data for hundreds of very chemically different solutes in nearly 200 different gas chromatographic stationary phases and other liquids, with correlation coefficients of over 0.99 and average residuals of fit of 0.15 (in log K units). In eq 5, the solute parameters were designated with the subscript “2” to indicate that they pertain to the molecule acting as an infinitely dilute species. However, when one studies a given solute in a series of solvents, the general equation is best written as: log K = SP,
+ msH2+ SU* + d6KT +
+ bB
(6) Note that the subscript has been dropped and the explanatory parameters now pertain to the solvent. Specifically, u*,u , and b represent the solvent’s ability to interact with a solute via dipolar processes (Keesom and Debye), its ability to donate a proton to form a solvent to solute hydrogen bond, and its ability to accept a proton in a solute to solvent hydrogen bond, respectively. The term KT represents a “polarizability” correction factor which arises due to the fact that aliphatic, aromatic, and polyhalogenated species have quite different blends of dipolarity and polarizability. Finally, the term 8~~ is the Hildebrand solubility pararmeter of the solvent. Thus, for highly cohesive solvents, such as alcohols, it is much harder for a solute of a given size to make a cavity than it is a less cohesive solvent, such as an alkane. The question arises as to why different parameters are needed to describe the same property of a molecule, such as its ability to donate to a hydrogen bond, when it is a
2908 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
solute, and when it is a solvent. It is well-knownto physical organic chemists that dimers are simultaneously better hydrogen bond donors and hydrogen bond acceptors than are monomers (Frange et al., 1982). This occurs because of the mutual polarization that takes place upon the formation of a hydrogen bond. In the case of methanol, the removal of electron density from an oxygen atom in one molecule when it enters into a hydrogen bond with a second methanol molecule makes the attached hydrogen bond atom of the first molecule a stronger hydrogen bond acid. Methanol as a solvent is a highly hydrogen-bonded network, whereas at infinite dilution in hexane, the methanol is not hydrogen bonded. One can perceive the effects of this hydrogen bonding by simply comparing ethanol (C2HsO) with its isomer, dimethyl ether. The alcohol has a heat of vaporization of 10 kcal and a boiling point of 78 OC, compared to 4 kcal and -25 "C for the ether, primarily due to the hydrogen bonding (comparison suggested by Prausnitz (1969)).This difference in behavior has now been recognized in the Kamlet-Taft formulation, and different scales of parameters CY,8, and ?r* are being developed for many molecules as solutes (Sadek et al., 1985; Leahy et al., 1986; Carr et al., 1986; Kamlet et al., 1988; Park et al., 1988; Park and Carr, 1989; Li et al., 1991, 1992a,b). The precision of this method is good because so many thousands of data have gone into evaluating parameters; none are ever determined by one or two measurements, but generally result from many data in a variety of systems chosen for their chemical versatility. A test of the solvatochromatic scales is in the ability of the parameters to model diverse chemical and biological properties. The Kamlet-Taft (KT) scales have been shown to model successfully many properties, including, but not limited to,dipole moments, fluorescence lifetimes, reaction rates, NMR shifts, gas-liquid partition coefficients, solubilities in water and blood, and biological toxicities (Taft et al., 1985). Database This database contains 1879directly measured y" data a t 25 OC, with no data extrapolated from traditional finite concentration VLE data. These data are collected from 39 literature sources using a wide variety of techniques in the time span from 1968 to the present, with a strong emphasis on the more recent and more accurate data. All data are evaluated critically before inclusion into the database through a variety of methods including comparison with other workers, principal component factor analysis (discussed below), and checking with predictive models. The complete database, including references, is available in the supplementary material (see paragraph at end of paper regarding supplementary material).
through the use of a method called "target testing", which allows a single parameter scale to be evaluated for consistency with the experimental In ymdata. The specific approach for the factor analysis method has been described in detail (Poe et al., 1993); the key components of this method are summarized here. A data matrix comprised of In y" values for 29 solvents and 31 solutes was assembled from the complete database. Experimental values were available for 75 % of the matrix elements; the remaining values are estimated using UNIFAC (20%) or MOSCED (5%). Since direct use of t h i s data matrix would bias the results toward the UNIFAC or MOSCED models, amissing data factor analysisroutine (Dempster et al., 1977) was used to refine iteratively the values estimated by MOSCED or UNIFAC until these refined In y" values were consistent with experimental data in the data matrix. During this procedure, any experimental values that were predicted with large errors were flagged as suspect values. Several data points identified with this procedure were subsequently confirmed to be in error and corrected before continuing with the data analysis. Our analysis revealed that at least four eigenvectors are required to describe the experimental data. The eigenvectors obtained from factor analysis of the matrix described above can be fit, using linear regression, to each parameter scale of interest. This is the procedure known as "target testing". A good fit obtained from this procedure indicates that the variation described by the parameter scale is consistent with the variation in the experimental In ymvalues. Through this analysis, appropriate scales for terms involving dispersion, dipolarity, hydrogenbonding ability, and volume were identified for inclusion in the SPACE model. In particular, the chromatographically-based dipolarity and hydrogen-bonding scales developed by Li et al. (Li et al., l991,1992a,b; Li, 1992)were found to give a better description of the variations in the experimental data as compared to the values obtained from the parameters of Abraham et al. (1986, 1987, 1988a,b, 1989a-e). Details of Reformulation Dispersion Term. The dispersion term accounts for London forces and is determined from the refractive index by nonlinear least squares regression analysis for the coefficients. (7)
Application of Factor Analysis Factor analysis is a method by which the trends in a large data set may be examined by evaluating the eigenvectors and eigenvalues of a matrix. We used this method to screen the infinite dilution activity coefficient database for erroneous experimental results and typographical errors, and to evaluate potential parameters for inclusion in the SPACE equation. In particular, we wished to compare two sets of solute-based solvatochromic parameters, one based on gas chromatographic retention times (Li et al., l991,1992a,b; Li, 1992) and one based on hydrogen-bonding free energies (Abraham et al., 1986, l987,1988a,b, 1989a-e). This comparison is accomplished
Acs, = 21.85f(nD)
(11)
Different relationships are used for aliphatic and aromatic systems due to differences in the polarizability of the compounds. There is a separate relationship for the halogenated compounds because of the influence of the highly electronegative halogens on the electronic properties of the molecules. A separate value is used for CSZ than was used in Thomas' original formulation (Thomas and Eckert, 1984). It should be noted that the values for the slopes differ by at most 6 5% from a central
Ind. Eng. Chem. Res., Vol. 32, No. ll, 1993 2909 value; these values represent a fine-tuning of the relations and not radical differences in formulation. Dipolarity/Polarizability,Acidity, and Basicity. As described above, we have chosen to use the Kamlet-Taft solvatochromicparameter scale for solvents and the CarrLi chromatographic (c) scale for solutes. It is important to recognize that these are relative measures and must be scaled to the appropriate units ( ~ a l / c m ~ )for l / ~use in the SPACE equation. Both the K T and c scales are relative scales and are dimensionless. The KT and c u* scales are both normalized such that u* = 0 for cyclohexane and a* = 1for dimethylsulfoxide. The (YKT scale is normalized so that (YKT = 1 for methanol and (YKT = 0 for alkanes. The ( ~ scale is normalized so that 1 x 2 ~= 0.77 for trifluoroethanol and azc= 0 for alkanes. The BKT scale is normalized so that BKT = 1for hexamethylphosphoramide and BKT = 0 for alkanes. The Be is defined as BC = 0 for alkanes and is not specifically normalized. A listing of the solvatochromic parameters for the compounds studied are available in the supplementary material. Although the KT and c u* scales both use the same normalization, the values are not interchangeable due to differences in the solute and solvent interactions. Thus, a value of T*KT = 0.35 does not have the same physical meaning as a value of u*zC= 0.35. The dipolarity of the solvent 71, and the dipolarity of the solute 7 2 , are related to the solvatochromic parameter scales T*KTand u*zC:
(13) where A and B are group-specific parameters which are regressed using the experimental y’ data. Separate values for slopes, AI and A2, are used for the solvent and solute due to differences in the relative values of the scales. The same intercept, B, is used for both relationships. The values for the dipolarity reducing parameters are listed in Table I, and the values for 7 are listed in supplementary material. For the acidity and basicity scales,
(15)
Table I. Dipolarity Reducing Parametere? Eqm 12 and 13 family alkanes alkenes amides amines aromatics without functional groups aromatics with nitrogen groups aromatics with oxygen groups carbon disulfide cyclic aliphatics cyclic ethers ethers and esters halogenatad aliphatics 2 ~ halogenated aromatics heterocycles ketones nitriles nitroalkanes primary alcohols secondary alcohols sulfoxides Units are (cal)1/2.
rameter is set to zero if the correspondingsolvatochromic parameter equals zero in order to eliminate anomalous values for acidity and basicity. For example,the SPACE value of acidity for carbon tetrachloride is set to zero since the solvatochromatic a is zero for this compound. The specific values for these reducing parameters are included in Table 11,and the resulting SPACE parameters are listed in the supplementary material. It was necessary to separately adjust the values for carbon tetrachloride and methanol, due either to uncertainty in the solvatochromic parameter scale value or to anomalous behavior associated with the compound being the first member of a homologous series. Only five of the twelve parameters associated with these compounds were adjusted, and the adjustments are detailed in Table 111. Closure. Solute parameters must scale continuously to reflect the environmentof the solute in a specificsolvent. This means that the parameters for a solute vary continuously from the solvent values for the solute in itself to the solute parameters when the solvent is in an entirely different medium. For example, for propanol in propanol, propanol (solvent) values must be used for the solute to yield 7’ = 1; when the propanol is a solute in hexane, the propanol (solute) values are used. But when the propanol is a solute infinitely dilute in ethanol, the propanol solvatochromic parameters should be very near those for propanol (solvent). For simplicity, we determine the effective value by linearly scaling the parameter values, based on the differences in reference states as expressed by the KT solvent parameters for both molecules. A scale factor is determined on the basis of these differences, and ranges linearly from 0 to 1. A scale factor near 0 implies the reference states are similar, e.g., propanol in ethanol. A scale factor near 1implies a large difference in reference states, e.g., propanol in hexane.
(17) (Ysdefactor
The slopes and intercepts are determined by least squares regression analysis of activity coefficient data. In these relationships, only the intercepts D and Fare familydependent parameters; the slopes C and E are the same for all compounds, except a single separate value is used for C2 for both primary and secondary alcohols. It was necessary to use separate intercepts for the solvent and solute cases due to differences in the definition of the solvatochromic parameter scales. Also, the SPACE pa-
Ai Az B -8.40 -19.93 0.70 -23.77 57.61 8.45 -8.98 13.70 41.33 30.67 471.21 1.94 42.49 72.10 -6.52 13.13 -4.98 17.73 13.27 18.65 15.19 0.00 0.00 0.00 49.07 9.67 1.42 18.65 -87.67 12.68 47.88 105.32 -0.82 15.32 44.42 9.44 7.00 7.61 14.48 -14.70 0.00 17.32 15.59 47.96 14.39 28.06 71.59 8.76 36.31 73.15 1.99 -27.51 115.23 11.53 24.82 -165.57 -15.38 9.92 78.53 28.36
@scalefactor
-I
CYKT(solvent) - (YKT(solute)
1.20
(18)
-I
BKT(so1vent) - BKT(8olute) 0.95
(19)
-
**KT(solvent) - T*KT(so~u~S) (20) 1.33 Different denominators are used in these relationships since each denominator represents the maximum differTdefactor
-I -
2910 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Table 11. Hydrogen Bonding Reducing
Eqs
14-17
D1
family
0.00 alkanes 0.00 alkenes 0.00 amides 0.00 amines 0.00 aromatics without functional group aromatics with nitrogen groups 43.58 17.33 aromatics with oxygen group 0.00 carbon disulfide 0.00 cyclic aliphatics 0.00 cyclic ethers 0.00 ethers and esters 14.62 halogenated aliphatics 0.00 halogenated aromatics 0.00 heterocycles 3.11 ketones 1.56 nitriles -15.54 nitroalkanes 25.46 primary alcohols secondary alcohols 6.28 sulfoxides 0.00
Dz 0.00 0.00 0.00 0.00 0.00
FI
FZ
0.00 -2.10 21.05 16.18 3.72
0.00 0.44 0.00 -4.82 -3.68
4.14 0.00 0.00 8.28 -16.60 0.00 0.00 0.00 0.00 -49.20 0.00 -0.80 57.28 0.00 13.83 9.33 11.06 0.00 1.49 -3.97 66.67 4.06 0.00 0.00 15.73 -450.00 0.00 13.16 -31.79 28.24 105.60 10.08 -43.49 -23.39 129.28 -8.09 -10.90 275.39 -17.31 258.55 20.57 28.44 0.00 21.39 0.00
Units are (cal)lI2. C1 = 25.19; Cp = 139.96 (for primary and secondary alcohols only CZ = -449.10); E = 6.00; E2 = -21.26. Table 111. Compound-Specific Parameters special family comaound parameter value value
-
methanol methanol carbon tetrachloride carbon tetrachloride carbon tetrachloride
82 02 71 72
82
4.22 14.41 1.29 1.05 0.77
3.25” 18.57b 1.4OC 1.68d 0.4ga
%changee 30 8 16 43 71
L
I
I
J
I
t
0
1
2 3 4 5 EFF SOLUTE DlPOLARil3”(caVccyO.5
6
7
Figure 1. Influence of environment on dipolarity of nitromethane. The effective solute dipolarity parameter for nitromethane, 72@fi,is given for five solvents of varying dipolarity. This demonstrates the mechanics of the closure relationship used in the SPACE equation. Table IV. Errors of Fit of MOSCED and SPACE Models
overall error error of fit (>30%) no. of adjustable const per compd no. of data no. of families
original MOSCED”
MOSCED IIb
9% 2% 3
11% 7% 1
od
800 13
1200 10
1879 20
SPACEc 9% 3%
a Thomas and Eckert, 1984. Howell et al., 1989. This work. Except for Table 111.
Calculated using eq 17. Calculated using eq 15. Calculated using eq 12. Calculated using eq 13. e Percent difference between family value and special value.
ence in solvent solvatochromic parameters for the compounds studied for the development of this model. Next, we calculate effective solute values by a linear interpolation of the SPACE solvent (1) and solute (2) parameters for the solute: azeff
= a 1 + (“2 - a1)ascalefact.m
(21)
Pzeff
= 6 1 + (Pz - P1)Pscdefactor
(22)
= 71 + ( 7 2 - 71)Tscalefactor (23) The SPACE values for the solute parameters are representative of a hypothetical reference state, Le., at the extreme range of differences in molecular properties, whereas the effective solute parameters should indicate the chemical behavior in the solvent of interest. Figure 1shows the influence of the choice of solvent on the SPACE effective solute dipolarity parameter for nitromethane. Size Difference Term. As in MOSCED, a modified Flory-Huggins size difference term (eq 2) is used because the traditional Flory-Huggins term, developed for polymer solutions, tends to overcorrect for systems with modest volume ratios. We include the exponent aa, set to a value of 0.936, as determined by regression analysis on a data set of alkanes in alkanes.
0.75 ’ 1.1 ’ 1.5 ’ 2.5 ’ 5 15 ACTIVITY COEFFICIENT
265
Figure 2. Influence of nonideality on error of SPACE model. This histogram demonstrates the level error of fit of the SPACE model across a large range of solution nonideality.
72eff
Results and Discussion The SPACE equation is a simpler method than MOSCED, which also eliminates compound-specific adjustable constants, but it must also fit the data within their experimental accuracy, about f10 % . The results of this work are compared with the two formulations of MOSCED in Table IV. SPACE represents the data within
experimental uncertainty and drastically reduces the number of gross outliers. The performance of the method is consistent across the range of nonideality studied in this work (Figure 2). A slight reduction of error of fit is seen for nearly ideal systems, and there is no corresponding trend toward large error of fit for highly nonideal systems. The results of the behavior of the SPACE model for each of our 20 classes of compounds are presented in Table V. To avoid the influence of different levels of nonideality, two separate comparisons were made using the same reference states. The first comparison is for nonpolar, aprotic hexadecane as a solvent with 101 data records of high precision available. This analysis gives a measure of relative errors in solute dispersion and dipolarity parameters for all systems, as well as solute hydrogen-bonding parameters for amphoteric systems. The size difference term also enters into this analysis due to the large molar volume of hexadecane relative to the solutes. The errors of fit of the 101 solutes are summarized in Table VI. The mean error of fit is 7.3 % ,which is less than the 9 % overall error of fit for the model, and is probably
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2911 Table V. Space Model Result8
total no. familv
of data0
alkanes alkenes amides amines aromatics without functional group aromaticswith nitrogen group aromaticswith oxygen groups carbon disulfide napthenes cyclic ethers ethers and esters halogenated aliphatics halogenated aromatics heterocycles
1070 139
ketones nitriles nitroalLanes primary alcohols secondary almhola sulfoxides
77 45 258 111 153 49 236
99 100 353 72 27 189 140 86 310 48 36
mean overall error. ?&
no. of solvent datab
mean solvent
7.8 10.1 7.0 11.7 7.8 9.9 10.9 9.9 7.8 9.7 9.1 10.8 11.6 12.1 10.9 9.0 12.8 11.1 9.4 1.6
327 36
7.2
865
7.6
7.0 7.0
108
11.0
7.4 6.6 9.9 11.0 10.3 13.8 8.7 8.8 10.7 12.2 12.1 9.8 8.3 13.4 9.5 7.0 7.6
27 121 0 4 24 215 53 33 204 5
no. of solute datac
error. %
71 18 136 111 149 25 23 47
67 162 67 27 112 124 72 225 38 36
mean solute error. %
0 14.5 8.8 5.3 9.4 7.1 10.4 9.8 10.8 2.9
0
80
12.3 14.4 10.1 15.0 18.6
17 14 93 10 0
0.0
0 Total number of times the compound appears either 80 a solute or a solvent. Total number of times the compound a p p 80 a solvenL *Total number of times the compound appears 80 a solute.
Table VI. SPACE Error of Fit in Solvent Heudecane family alkanes alkenes amines aromatics without functional group aromatics with oxygen group cyclic diphatics cyclic ethers ethers and esters halogenated aliphatics halogenated aromatics ketones nitriles nitroalhes primary alcohols secondary alcohols overall
mean error, % no. of data 4.42 8.16 7.26 6.95 5.23 4.33 5.53 9.20 6.08
30
6 3 9 4
...............................................................................................................................
Y
0 ......................................
6
14.24 16.13
2 5 11 1 5 3 3 10 3
7.30
101
2.07 14.69 8.73 3.65
'1 E 15.m/
due to the lack of hydrogen bonding. The largest errors of fit are for primary and secondary dCOh0h and for ketonesandaldehydes. Thesefitsaregoodto 1596,which may be the order of the uncertainty in the alcohol and ketone data. These results are expected as these families had comparable errors of fit as solutes as presented in Table V. Other than for these three classes, the fitof the model is constant across compound classes. The second method involves comparing the ability of the model to predict 7- for a specific series of sohtes in several different homologous series of solvents. This type of analysisallows forthe comparison of model performance as a basis of solvent functionality. The choice of solvents and solutes is taken from the work of Park et al. (1991). who used this same series of solutes to compare the relative errors of prediction of two versions of the UNIFAC model. The sohtes used are n-octane, 2-butanone, ethanol, toluene, and 1.4-dioxane with three homologous series of solvents-alkanes, alkanenitriles, and primary alcohols. Dispersion interactions dominate for the alkanes; the alcohols are moderately dipolar, strong hydrogen bond donors, and moderate acceptors; and the alkanenitriles are highly dipolar and weak to moderate hydrogen bond acceptors. The results are summarized in Figure 3. Since all three classes of solvents fit well (all mean errors of fit are less than 13%),this analysis displays the consistency of the SPACE equation for a variety of systems.
....
.......
.....
..........
7
-
I ElHANOL -
OCTANF
@DIOXANE
~~
~
TOLUENE
2.BUTANOL
SOLUTE W E S
ALKYNITFILES
ALC0mx.S
Fignre3. SPACEmodelerroroffitfor homologouaseriesofsolv~ta The mean error of fit for five solutes in three homologous series of solvents. Table VII. Comparison of SPACE and UNIFAC. Models: Mean Error for All Solvent8 (%) solute model SPACE UNIFAC
oetane 2-hutanone ethanol toluene 6.2 10.8
6.2 9.8
11.8 30.5
9.5 8.5
1.4-diorane 9.9 43.0
Gmehling et al., 1982.
The results are compared with the UNIFAC model (Gmehling et al., 1982) in Table VII. A set of 7- data not wed in the development of either model, for a diverse groupof58organicsystemsat 25 T,wereusedtoevaluate the predictiveerror s o f themodels (Hradetzkyetal., 1990. Landau et al., 1991). Since no parameters are directly available for dioxane, we used two CHzO groups and two CHz groups with no cyclic correction. The SPACE model resultedinameanerrorofpredictionof 12% ascontrasted to a 19% error of prediction for UNIFAC. However since the largest deviations for UNIFAC were with dioxane, where the parameters have the highest uncertainty, the performance of the two methods is probably Close to comparable. Each of these two complementary models has its own set of applications, and they can be used to crone-check for anomalous results. The advantages of UNIFAC are that it can calculate activity coefficients for moleden for
2912 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
which no pure-component data are available, it yields a value for multifunctional molecules, and it is well suited for computer implementation. The first limitation of UNIFAC is that the UNIQUAC model for the composition dependence of gE(x) is built into the method, and while UNIQUAC is useful, it is not exact. Also, UNIFAC contains no separate term for specific interactions nor any understanding of the relative contributions of different intermolecular interactions. The SPACE model does differentiate the interactions and is perhaps more realistic, especially for strongly interacting systems. It does require some pure-component physical properties for use. Moreover, SPACE appears somewhat more accurate than UNIFAC for cases where both methods are applicable. For y" for multifunctional molecules, it is necessary to treat a variety of cross-interactions, both intramolecular and intermolecular. SPACE is not yet set up to do this. UNIFAC defines mathematics which will yield a result, but represents what we know of the chemistry in a less than optimum fashion. An example of this problem has been shown by Li et al. (1992b). Temperature Dependence Since many of the applications of prediction of limiting activity coefficientsinvolve chemical processing at a variety of temperatures, it is desirable to have a method to extend the SPACE predictions at 25 "C to other temperatures. The optimum method is to use experimental partial molar enthalpy data (hE"): (24)
Unfortunately, relatively few good hE"data exist. We have, in a separate work, correlated the data available (Hait et al., 1992) using linear solvation energy relationships (LSERs). Because of the nature of the thermodynamic cycle used for the correlation, there is some augmentation of uncertainty due to taking a difference of large numbers; as a result, the predicted values of hE' are only good to about *25%. Nonetheless, we find that, while experimental hE" data are preferable for extrapolating y " ( T ) (*5%),LSERpredictedvaluesworkalmost aswell (*7%). This is a t least in part due to the limited number and small temperature range of y -( 2')data available for testing. We also investigated two empirical methods for extrapolating ~"(2"):the method proposed in the original MOSCED equation, where parameters vary with temperature, and the regular solution theory assumption that In ymis linear in 1/T.
Both methods were tested using a data set of 591 y" data over a temperature range from 30 to 90 OC. The overall error of fit for both methods was the same, f 9 % . There are simply very few reliable y" data at temperatures far different than 25 "C. The available database is statistically incapable of differentiating between alternative temperature dependencies. Until more information is available, we recommend that one use hE"data when available, or if that is not possible, estimate hE" by the LSER method (Hait et al., 1992). Otherwise, one must resort to eq 25. Great care should be taken in using this relationship, as in a few cases even the sign of the slope can be wrong.
Application to Aqueous Systems The formulation as derived did not use data for y" of water or in water, nor were parameters derived for water. Because of ita structure, water as either a solvent or a solute is so different from other liquids that it must be considered separately, and ita inclusion here could bias the method. In fact, values of y" for aqueous systems vary by many orders of magnitude, from less than unity to greater than lo6or more for alkane solutes. A separate critical database has been developed for ymdata in water as a solvent, and effective SPACE parameters for water have been determined, but the fit of the ymdata at 25 OC is good to only about *70% (Trampe, 1993). Physical Property Requirements/Estimation Techniques In order to generate predictions for ym,the SPACE model (eqs 3 and 4) requires the input of the refractive index, molar volume, SPACE dipolarity 7 , and hydrogenbonding acidity (Y and basicity P parameters. These last three parameters are based on the solvatochromic parameters by eqs 12-17, and for the 133 compounds used in this study, these are presented in the supplementary material. The refractive index and molar volume are both readily available properties and are quite easy to measure experimentally. Solvent solvatochromic parameters a,8, and ?r* are available, with data reported for over 650 compounds (Carr, 1993). The solute-based solvatochromic parameter scales are a more recent addition and are available for 203 compounds (Li et al., 1992a,b;Li, 1992). Estimation techniques are being developed for both scales (Carr, 1993) and, when available, will enhance the predictive capability of the SPACE model for new compounds. Summary A database has been constructed using a combination of new data and literature data for directly measured infinite dilution activity coefficients at 25 "C in nonionic solutions. The database was evaluated by principal component factor analysis to determine optimal systems to measure, to find spurious data points, and to determine the functionality of ym. On the basis of this database and the factor analysis results, a new expression was developed for the correlation and prediction of y". The SPACE equation is a linear descendent of MOSCED, but simplifies the functionality and eliminates all compound-specific adjustable parameters. A key feature of SPACE is the use of solvatochromic parameters for molecular properties which have been determined in the literature from a wide variety of solution phenomena-for example, chromatographic retention times, spectral features, and reaction kinetics, as well as activity coefficients. SPACE has terms specificallyfor nonpolar interactions, for polar interactions, for hydrogen bonding, and for accounting for differences in size. As such, it well represents the intermolecular interactions and is capable of predicting y" to a precision as good as the experimental uncertainty. Furthermore, it has the advantage of being completely uncoupled from any specific functionality for g E ( x ) ,so that it focuses solely on the crucial unlike-pair intermolecular interactions in solution. SPACE should be a useful complement to other methods for ymprediction. Acknowledgment We gratefully acknowledge the financial support of Du Pont Chemical Co., Amoco Chemical Co., and the National ScienceFoundation. We are also grateful for the technical
Ind. Eng. Chem. Res., Vol. 32,No. 11, 1993 2913 advice received from Leon Scott of Du Pont, Gene Thomas of Exxon, and Bill Parrish of Phillips Petroleum. Furthermore, we appreciate the assistance of Ms. Cynthia Harrell and Mr. Eric Svensson in developing the 7database.
Nomenclature A = dipolarity parameter reducing slope (eqs 12 and 13) B = dipolarity parameter reducing intercept (eqs 12 and 13) C = acidity parameter reducing slope (eqs 14 and 15) D = acidity parameter reducing intercept (eqs 14 and 15) dlz = corrected Flory-Huggins term E = basicity parameter reducing slope (eqs 16 and 17) F = basicity parameter reducing intercept (eqs 16 and 17) gE(x) = excess Gibbs energy expression as a function of composition hE- = partial molar enthalpy nD = refractive index T = absolute temperature (K) u = molar volume Greek Symbols a = solvatochromic acidity parameter j3 = solvatochromic basicity parameter y” = limiting activity coefficient 6 = solvatochromicpolarity/polarizability correction parameter 6~ = Hildebrand solubility parameter X = SPACE dispersion force parameter ?r* = solvatochromic polarity/polarizability parameter T = SPACE dipolar parameter
Subscripts 1 = solvent 2 = solute c = solute solvatochromic parameters (Carr-Li scale) KT = Kamlet-Taft solvent solvatochromic parameters eff = effective solute value (closure) scalefactor = scaling factor used for closure
Supplementary Material Available: Activity coefficient database and references, and listing of the SPACE parameters for all compounds studied (132pages). Ordering information is given on any current masthead page. Literature Cited Abboud, J.-L. M.; Kamlet, M. J.; Taft, R. W. Applicationof Deuterium Magnetic Resonance to Biosynthetic Studies. 2. Rosenonolactone Biosynthesis and Stereochemistry of a Biological sN2 Reaction. J. Am. Chem. SOC. 1977,99,8327. Abboud, J.-L. M.; Sraidi, S.; Guiheneuf, G.; Negro, A.; Kamlet, M. J.; Taft, R. W. Studies on Amphiprotic Compounds. 2. Experimental Determination of the Hydrogen Bond Acceptor Basicities of ‘Monomeric’Alcohols. J. Org. Chem. 1985, 50, 2870. Abraham, M. H.; Whitting, G. S. Hydrogen Bonding XXI. Solvation Parameters for Alkylaromatic Hydrocarbons from Gas-Liquid Chromatographic Data. J. Chromatogr. 1992,594, 229. Abraham, M. H.; Duce, P. P.; Schulz, R. A.; Morris, J. J.; Taylor, P. J.; Barratt, D. G. Hydrogen Bonding. Part 1. Equilibrium Constants and Enthalpies of Complexation for Monomeric Carboxylic Acids with N-Methylpprolidinone in l,l,l-Trichloroethane. J. Chem. SOC.,Faraday Trans 1 1986,82, 3501. Abraham, M. H.; Duce, P. P.; Morris, J. J.; Taylor, P. J. Hydrogen Bonding. Part 2. Equilibrium Constants and Enthalpies of Complexation for 72 Monomeric Hydrogen-bond Acids with N-Methylpyrrolidinone in l,l,l-Trichloroethane. J. Chem. SOC., Faraday Trans. 1 1987,83 (9), 2867. Abraham, M. H.; Duce, P. P.; Prior, D. V.; Schulz, R. A,; Morris, J. J.; Taylor, P. J. Hydrogen Bonding. Part 3. Enthalpies of Transfer from l,l,l-Trichloroethane to Tetrachloromethane of Phenols,
N-Methylpyrrolidinone (NMP) and Phenol-NMP Complexes. J. Chem. SOC., Faraday Trans. 1 1988a, 84(3), 865. Abraham, M. H.; Duce, P. P.; Grellier, P. L.; Prior, D. V.; Morris, J. J.; Taylor, P. J. Hydrogen Bonding. Part 5. A Thermodynamically-basedScaleof Solute Hydrogen-bondAcidity. Tetrahedron Lett. 1988b, 29 (13), 1587. Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Morris, J. J.; Taylor, P. J.; Maria, P.-C.; Gal, J.-F. Hydrogen Bonding. Part 4. An Analysis of Solute Hydrogen-Bond Basicity, in Terms of Complexation Constants (LogK),Using Fl and FZFactors, the Principal Components of Different Kinds of Basicity. J.Phys. Org. Chem. 1989a, 2, 243. Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Morris, J. J.; Taylor, P. J.; Laurence, C.; Berthelot, M. Hydrogen Bonding. Part 6. A Thermodynamically-Based Scale of Solute Hydrogen-Bond Basicity. Tetrahedron Lett. 1989b, 30 (19), 2571. Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Duce, P. P.; Morris, J. J., Taylor, P. J. Hydrogen Bonding. Part 7. A Scale of Solute Hydrogen-bond Acidity based on log K Values for Complexation Perkin Trans. 2 1989c, in Tetrachloromethane. J. Chem. SOC., 699. Abraham, M. H.; Buist, G. J.; Grellier, P. L.; McGill, R. A.; Prior, D. V.; Oliver, S.; Turner, E.; et al. Hydrogen-Bonding 8. Possible Equivalence of Solute and Solvent Scales of Hydrogen-Bond Basicity of Non-Associated Compounds J. Phys. Org. Chem. 1989d, 2,540. Abraham, M. H.; Duce, P. P.; Prior, D. V.; Barratt, D. G.; Morris, J. J.; Taylor, P. J. Hydrogen Bonding. Part 9. Solute Proton Donor and Proton Acceptor Scales for Use in Drug Design. J. Chem. SOC., Perkin Trans. 2 19890, 1355. Arkel, A. E. van Mutual Solubility of Liquids. Trans Faraday SOC. 1946,42B, 81. Arkel, A. E. van; Vles, S. E. Solubility of Organic Compounds in Water. Red. Trav. Chem. 1936,55, 407. Carr, P. W. Solvatochromism,Linear Solvation Energy Relationships and Chromatography: A Review. J. Microchem. 1993, in press. Carr, P. W.; Doherty, R. M.; Kamlet, M. J.; Taft, R. W.; Melander, W.; Horvath, C. Study of Temperature and Mobile-Phase Effects in Reversed-Phase High-Performance Liquid Chromatography by the Use of the SolvatochromicComparison Method. Anal. Chem. 1986,57,2674. Dempster, A. P.; Larid, N. M.; Rubin, D. B. Maximum Likelihood from Incomplete Data Via the EM Algorithm. J. R. Stat. SOC. 1977, (39) 1. Derr, E. L.; Deal, C. H. Analytical Solution of Groups. Correlation of Activity Coefficients Through Structural Group Parameters. Znst. Chem. Eng. Symp. Ser., London 1969,3 (32), 40. Frange, B.; Abboud, J. L. M.; Benamou, C.; Bellon, L. AQuantitative Study of Structural Effects on the Self-Association of Alcohols. J. Org. Chem. 1982, 47, 4553. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AZChE J. 1975,21, 1086. Fredenslund, A.; Gmehling, J.; Michelsen, M. L.; Rasmussen, P.; Prausnitz, J. M. Computerized Design of Multicomponent Distillation Columna Using the UNIFAC Group Contribution Method for Calculation of Activity Coefficients. Znd. Eng. Chem. Process Des. Dev. 1977a, 16, 450. Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNZFAC; Elsevier: New York, 1977b. Gmehling,J.;Weidlich,U. A Modified UNIFAC Model. 1. Prediction of VLE, hE, and -y-. Znd. Eng. Chem. Res. 1987,26, 1372. Gmehling, J.; Fredenslund, A.; Rasmussen, P. Vapor-Liquid Equilibria by UNIFAC Group Contribution. Revision and Extension. 2. Znd. Eng. Chem. Process Des. Dev. 1982,21, 118. Gmehling, J.; Fredenslund, A.; Rasmussen, P. A Modified UNIFAC Group-Contribution Model for Prediction of Phase Equilibriaand Heats of Mixing. Znd. Eng. Chem. Res. 1987,26, 2274. Gmehling, J.; Tiegs, D.; Knipp, U. A. A Comparison of the Predictive Capability of Different Group Contribution Methods. Fluid Phase Equilib. 1990, 54, 147; 1990, 59, 337 (correction). Hait, M. J.; Eckert, C. A. Correlation of Partial Molal Enthalpies at Infinite Dilution by a Linear Solvation Energy Relationship. Submitted for publication in J. Phys. Chem. 1992. Howell, W. J.; Karachewski, A. M.; Stephenson, K. M.; Eckert, C. A.; Park, J. H.; Carr, P. W.; Rutan, S. C. An Improved MOSCED Equation for the Prediction and Application of Infinite Dilution Activity Coefficients. Fluid Phase Equilib. 1989, 52, 151.
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Received for review May 25, 1993 Accepted June 7, 1993. Abstract published in Advance ACS Abstracts, August 15, 1993.
