Prediction of limiting activity coefficients by a modified separation of

Ind. Eng. Chem. Process Des. Dev. 1984, 23,

194

electricity. The penalty of using nuclear energy for purposes other than generating electricity is typical of all such uses. At present, electricity is sold for the equivalent of $12 per MMBtu electricity or $6 per MMBtu of heat generated from an HTGR. If it is converted to H2, the product sells at present at a much cheaper price per MMBtu than electricity. If that H2is used as fuel the price still drops, as the cost of high quality fuels is still only $6-7 per MMBtu. Many energy sources which are attractive for electricity generation lose their attractiveness if the electricity is converted to a fuel or H2. Some may dispute here the cost of nuclear fuel as it does not include any costs related to the fuel cycle, such as waste disposal, etc. The real cost of nuclear fuel is not of interest in this analysis as it affects equally all nuclear applications. One might also ask, how will this comparison be affected if one reduces the required rate of return on investment (see Naphthali and Shinnar, 1981)? Reducing the rate of return will reduce both the cost of hydrogen and the cost of electricity by the same ratio. Our general conclusion, therefore, is not affected. 9. S u m m a r y and Discussion

The results can be summarized as follows. (1)Use of high temperature heat in production of H2 from coal is less attractive than the use of the same heat to generate electricity and split water into H2 and 02. It is even under the most optimistic assumptions more expensive per unit of coal saved and has less potential for saving coal. It also suffers from inherent thermodynamic disadvantages. (2) Neither hydrogen from electrolysis nor from direct use of nuclear heat is attractive in the near future, but it can become attractive as coal becomes expensive and scarce. (3) If an HTGR will become attractive in cost, supplying nuclear heat, it will become more attractive as a source of electricity. (4) Use of nuclear energy to replace coal by generating electricity is inherently more attractive than ita use in generating hydrogen or synthetic fuels.

194-209

These conclusions are similar to those reached by Shinnar et al. (1981) for generation of Hzby thermochemical or hybrid cycles, using high temperature heat from an HTGR. From a policy point of view these conclusions have some interesting consequences. Development of the HTGR is justified only if it leads to either cheaper electricity or better and safer electricity generation. Conclusion 1 implies that if one wants to be prepared for the long-range future, where hydrogen from nonfossil sources becomes important for synthetic fuel production, one should concentrate on developing better plants for electrolysis of water. This can be done totally uncoupled from any development of nuclear reactors. If nuclear H2 becomes attractive in the future, it will be easy to phase it into synthetic fuel production. Acknowledgment

The work was performed under DOE Contract DEAC01-79ET14811. The authors want to acknowledge with thanks the support of DOE and the help extended by the division of coal gasification. However, all opinions expressed in the paper are solely those of the authors and in no way reflect the position of DOE. Registry No. Hydrogen, 1333-74-0; water, 7732-18-5. L i t e r a t u r e Cited Corneil, H. G. “Production Economics for Hydrogen Ammonia and Methanol During the 1980-2000 Period”, Exxon Research and Engineering Co., April 1977, BNL-50663. Kugeler, K. Chem. Eng. Sci. 1880, 358 2005-2028. Naphthali, L. M.; Shinnar, R. Chem. Eng. Prog. Feb 1881, 65. Nuttle, L. J. Int. J . Hydrogen Energy 1977, 2 , 395-403. Shlnnar, R. CHEMECH 1978, 8 , 686-693. Shinnar. R.; Fortuna, G.: ShaDira, D. Ind. Eng. Chem. Process Des. Dev. i-13. Shinnar, R.; Shapira, D.: Zakai, S. Ind. Eng. Chem. Process Des. Dev. iaai.20.581-593. , ~ . ~. , . Stoneand Webster “Application Study of a Nuclear Coal Solution Gasification Process for Oklahoma Coal”. May 1972, GA-A12068. ~

Received for review May 3, 1982 Accepted July 18, 1983

PredOction of Limiting Activity Coefficients by a Modified Separation of Cohesive Energy Density Model and UNIFAC Eugene R. Thomas and Charles A. Eckert” Department of Chemical Engineerlng, University of Illinois, Urbana, Illinois 6 180 1

A modified separation of cohesive energy density (MOSCED) model was developed to predict limiting activity coefficients (7”’s) using only pure component parameters. For 3357 y”s, an average error of 9.1 % was achieved with very few errors greater than 30%. The data included activity coefficients of both protic and aprotic (but nonaqueous) systems over wide temperature ranges. The results compared very well with the predictions of the leading predictive technique in the literature, the UNIFAC solution of groups approach (average error 20.5 %).

I. I n t r o d u c t i o n Many industrial processes involving separation and transfer of chemical species require knowledge of multicomponent equilibria. As the direct experimental deter0196-4305/84/1123-0194$01.50/0

mination of the necessary equilibrium data is both costly and time-consuming, some method to predict these data would be of great industrial utility. Present thermodynamic theory allows for the accurate prediction of multi0 1984 American Chemical Society

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984 195

component vapor-liquid equilibrium (VLE)data for completely miscible systems from binary data only (e.g., Eckert et al., 1965; Abrams and Prausnitz, 1975; Eckert et al., 1980). Recent investigators (Schreiber and Eckert, 1971; Tassios, 1971; Eckert et al., 1981) have demonstrated that a more efficient method than classical VLE studies for generating these binary data is the evaluation of activity coefficients at infinite dilution. Both binary and multicomponent VLE data predicted with parameters reduced from moderately accurate infinite dilution activity coefficients (7”s) have been found comparable to predictions with parameters reduced from binary data at all compositions. There are many other advantages in determining 7”s. From the 7”s of a binary system one can predict the occurrence of a binary azeotype. If yl-

PZS

< -< P18

1 Y2

of mixing are negligible, the excess Gibbs energy of a solution is proportional to the difference between the cohesive energy density, c, of the mixture and that of its pure components

In this expression, 4 is the component volume fraction, x is the mole fraction, and u is the molar volume. The configurational energy density is reasonably approximated (well below the critical point) by c=

AH”-RT U

where AH” is the molar enthalpy of vaporization. The theory assumes that for nonpolar mixtures the primary forces are dispersion forces and hence, following London’s formulation, c12 may be taken as the geometric mean of cll and c22. The excess Gibbs energy becomes

or

gE = h#d3C1Ul + XZU2)(C11 w,-

1

> -> p28

then an azeotype exists at some finite concentration (neglecting vapor phase nonidealities). Furthermore, for immiscible binary systems where 7”s are greater than 50, the y”s can yield reasonable estimates of the mutual solubilities. The evaluation of 7-also allows for the accurate calculation of kinetic solvent effects with the Bronsted-Bjerrum relationship (Wong and Eckert, 1971; Eckert et al., 1974; Newman, 1977; Thomas, 1980). Finally, it provides more incisive information for the statistical thermodynamicist due to the absence of solute-solute interactions in the limiting dilute region, resulting in the disappearance of the order-disorder problem. The advantages of working with activity coefficients at infinite dilution were first recognized by Gatreaux and Coates in 1955. Some reasonably useful empirical methods have been developed for predicting 7-values, but none has (e.g., Pierotti et al., 1959; Null and Palmer, 1969) successfully treated the highly polar solutions frequently encountered in industry. Problems in the accurate experimental evaluation of y”s for nonideal systems were at least partially responsible. Through improvements in the gas chromatographic and ebulliometric techniques for measuring y”s, researchers (Eckert et al., 1981; Thomas et al., 1981a,b) have demonstrated that activity coefficients at infinite dilution may be readily obtained experimentally for many systems. However, there are a large number of mixtures for which existing techniques are inapplicable. Thus an expression which could predict the limiting activity coefficients from pure component properties would be of great use to the chemical engineer. In this study a modified form of the separation of cohesive energy density model (MOSCED) was used to correlate 3357 7”s with an average error of 9.1% using between zero and three adjustable parameters per molecule. It is believed that the above correlation yields the best predictions of 7”’s for nonionic solutions solely from pure component information to date. Also presented is a comparison of the 7-data with the predictions of the UNIFAC solution-of-groupsmethod (Fredenslund et al., 1977a,b). 11. Background The MOSCED model is an extension of regular solution theory (Scatchard, 1931; Hildebrand and Wood, 1933) to polar and associating systems. Regular solution theory predicts that when the excess entropy and excess volume

(2)

+ c22 - 2[C,,C221’/2)

(3)

which yields for the infinite dilution activity coefficient of component 2 in a binary mixture

Finally, a solubility parameter 6 is defined as the square root of the cohesive energy density which gives

While the regular solution equation gives a good semiquantitative representation of the excess Gibbs energy of nonpolar or moderately polar solutions, it is not applicable to mixtures containing highly polar or protic components. To extend the usefulness of this expression, later researchers (Arkel and Vlex, 1936; Arkel, 1946; Blanks and Prausnitz, 1964; Weimer and Prausnitz, 1965; Gordon, 1966; Hansen, 1967; Helpinstill and van Winkle, 1968; Nelson et al., 1970; Keller et al., 1971; Hsieh, 1973; Barton, 1975; Koenhen and Smolders, 1975; Karger et al., 1976; Tijssen et al., 1976; Karger and Snyder, 1978) assumed that the forces contributing to c act independently and are additive. Most such extensions yield forms for the cohesive energy density similar to

c12

=

c11

=

c22

=

XI2

6272

1

XlXZ

+ 71’ + Q171 + a161 + 7z2 + + f f 2 &

+ 7172 + -2 (u172 + Q27l + a162 + aZ61)

(6) (7) (8)

where the &Aj, 7 i 7 j , ui7j, and aiPj terms represent the dispersion, orientation, induction, and hydrogen bonding forces, respectively. X is a measure of a molecule’s polarizability, 7 represents its polarity, Q reflects the ability of the nonpolar part of a molecule to interact with a dipole, and a and 6 are acidity and basicity parameters, respectively. Using the above expression for the cohesive energy densities, the activity coefficient for component 2 infinitely dilute in solvent 1 becomes

196

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

Table I . Previous Formulations of Multicomponent Solubility Parameter Equationsn In

7, =

VI

+ (7)

-

-

r2)2

L’I

+ In(:)

- 2v,l

+ 1 - -; u l~ = c ( r ,

- 7?)2

U?

In

7,- = % E A l

Inr,*=”[(h, RT

+ -

- 7?)2

+ (7,

-T2)2

A (0,

-

(ct,P,

cJ2)(Tl

]

- 7 ? ) +In(:)

+

+in(:)

+ I - -v ,

-o,p2)’]

I-; “ 1

Newman (1977); Hsieh (1973) Hansen (196 7)

u2

-A,)~

RT

(TI

Helpinstill et al. (1968); Weimer and Prausnitz (1965)

+

- h , ) ~A

(TI

-

T

~

+) Z(a, ~ - aJP,

(TI - 7 > ) 2 a

2(ct,

-&:)(TI

Tijssen e t al. (1976)

-

- 7:)

-

z(ct,

- Oi,)(p,

- o2;1

Karger et al. (1976)

These equations are written with the notation of eq 11. The third, fourth, and fifth equations were not explicity written for activity coefficients, but they are obtainable from their solubility parameter representations,

To account for differences in molecular size, a FloryHuggins (Flory, 1941, 1942; Huggins, 1941) term d12was added where d12 = In -

v2

v2 + 1- -

u1

u1

The final expression then becomes

Table I is a summary of some previous attempts at a model of this type. Several techniques have been used to evaluate the parameters for this model. Bondi and Simkin (1956) proposed the homomorph technique for generating the A’s. They suggested that the dispersion component of a polar molecule be taken as the energy of vaporization of the equistructural hydrocarbon at the same reduced temperature, the polar contribution obtainable from subtracting the dispersion contribution from the total cohesive energy density as calculated by eq 2. Anderson (1961) added the further constraint that the hydrocarbon be chosen with the same molar volume as well as shape. This approach was used with moderate success by Weimer and Prausnitz (1965) and later by Helpinstill and van Winkle (1968) in correlating 7”s for hydrocarbons in polar solvents using a slightly different form of the model. However, this approach was not applied to mixtures containing more than one polar species nor was its applicability tested in pre~ both ends of a binary mixture. The cordicting y ” at relation also suffered from a limited data base utilizing primarily 7”s extrapolated from VLE data, values which were probably accurate only to 20-50%. Meyer and co-workers (Meyer and Wagner, 1966 Meyer and Ross, 1971; Meyer et al., 1971) gave an independent method for quantitative estimation of the separate forces in polar organic liquids. Vaporization energy vs. chain length was plotted for a homologous series of compounds, e.g., n-alkyl nitriles. From the slope of the line and the difference between that slope and one determined from saturated hydrocarbons, contributions of polar, induced, and dispersion interactions were calculated. From those contributions the nonhydrogen bonding terms may be determined. This method showed that the induction forces in liquids were much larger than was previously recognized (Moore, 1972). Values predicted for the parameters using his results were found insufficiently quantitative. A discussion of this may be found elsewhere (Thomas, 1980).

A few techniques to calculate the values of the parameters from physical properties have been proposed (e.g., Tijssen et al., 1976; Koenhen and Smolders, 1975; Karger et al., 1976). Typically, all parameters except one are determined from physical properties. The remaining parameter, usually 7,is determined from the other parameters and an experimentally determined cohesive energy density from an equation such as (6). Unfortunately, this method, while intuitively appealing, could rarely be made quantitative. Hsieh (1973) noted that due to the existence of more than one type of interaction there are cross terms which limit the additivity of interaction energy assumption. Thus he concluded that the parameters should be considered only as pure component constanta obtainable from accurate experimental data. Newman (1977), using this conclusion, employed regression analysis on 170 7”s for 28 nonhydrogen bonding molecules to generate the A’s, u’s, and 7’s for those molecules. His correlation benefitted from the more reliable 7-data that he determined using the gas chromatographic and ebulliometric techniques, which are probably accurate to 10%. His average error of 14.2% is reasonable since Schreiber and Eckert (1971) and Tassios (1971) have demonstrated that the use of y”s to generate parameters for entropic equations to fit binary or multicomponent data accurately at all concentrations is fairly insensitive to modest perturbations in each ym. However, Newman’s correlation has the limitation of using 47 adjustable parameters to fit 170 data points at a single temperature, 20 “C.That the data were fit as well as they were is thus not too surprising and the real utility and validity of those parameters is questionable. Newman’s correlation also fails to predict the 7“’s for the polarnonpolar systems where data were available at both ends. For example, examine the acetonitrilecarbon tetrachloride and acetonitrile-benzene binary mixtures. The measured and calculated values for these systems are reproduced below. 7” >

acetonitrile in CCl, CC1, in acetonitrile acetonitrile in benzene benzene in acetonitrile

7

measd

calcd

(1977)

(1977)

13.4 6.67 3.47 3.19

3.45 10.7 2.32 4.39

This problem does not lie with Newman’s parameter estimation, but is rather a result of the confines of the separation of cohesive energy density model, which is essentially symmetric in volume fraction. Since acetonitrile has about 55% the molar volume of CC14,the model predicts the ymof acetonitrile in C C 4to be much lower than the

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984 197 Table 11. Listing of ~ ~at ’Both s Ends of a Binary Mixture and How They Compare with the Symmetry in Volume Fraction Assumption c

CC1, (1)-cyclohexane ( 2 ) chloroform (1)-hexane ( 2 ) benzene (1)-pentane (2) acetone (l)-CCl, (2) acetonitrile (l)-CC14 (2) acetonitrile (1)-hexane (2) ethanol (1)-heptane ( 2 )

350 1.07 1.10 1.08 348 1.85 1.78 2.22 1.97 320 1.52 1.79 4.21 304 3.03 2.16 6.7 106 293 13.4 298 26.0 30.7 3500 298 49.0 16.0 17000

a Data and references given in Table VI. Using symmetry of volume fraction assumption with Y fixed.

converse. The measured values show the reverse trend. Similar, though not as striking behavior is observed in the benzene-acetonitrile system. Thus, we conclude that although there remain some problems in most efficiently determining the parameters for a model of this type, a more critical problem lies in modifying the model to account for the observed asymmetry in volume fraction. The development of such a model is now addressed.

111. Modified Separation of Cohesive Energy Density Model To simplify eq 11, it is rewritten in a form analogous to Weimer and Prausnitz (1965) h’l 7 2 - =

u2

--[(xi

RT

- hz)‘

+

q12q22(71

-

72)’ (P1

+

(CY1

-

CY2)

x

- P2)l + d l 2 (12)

Weimer assumed that q is a function only of the class of compound of the solute, i.e., paraffin, olefin, or aromatic, where q is a measure of the dipole-induced dipole energy. A similar approach is used here and is described later. The above form is suggested by the observation that the polar (7-7) and induction (7-6)terms may be linearly related with little, if any, loss in the equation’s ability to correlate and predict data. This modification of eq 11 is minor, but some more serious changes in eq 11and 12 are necessary due to their approximate symmetry in volume fraction. Neglecting the combinatorial term, which is usually minor anyway, one has (13) which is symmetric in volume fraction. For nonpolar molecules this is satisfactory; however, as Wilson (1974) noted, for solutions containing polar molecules this is a poor approximation. Wilson suggested a group contribution approach to avoid this difficulty. Let us now examine the symmetry in volume fraction assumption in more detail. Table I1 lists several ym’s where data were obtained at both ends and a comparison with the symmetry in volume fraction assumption is given. One observes from the table that the limiting activity coefficient of the nonpolar solute in the polar or protic solvent is smaller than one would expect from eq 13. This is due to the polar or protic molecule being further from its reference state when dilute in the nonpolar solvent than vice versa. For example, ethanol is highly hydrogen bonded in its pure state, but dilute in cyclohexane it is incapable of hydrogen bonding; thus, relative to its pure state, ethanol is in a greatly different environment when dilute in cyclohexane. This accounts for the large change in y with respect to com-

$

3L

A

2

Mole Froci ion Cyclohexane

Figure 1. VLE data for the system cyclohexane (l)-ethanol(2)at 30 “C (data of Naga and Ishii, 1935).

position for ethanol dilute in nonpolar solvents since adding a small amount of ethanol to the solution enables more hydrogen bonds to be formed and the more “confortable”the ethanol becomes. Cyclohexane does not interact strongly with itself, and hence, when dilute in ethanol, it is not as far from its reference state. Thus, as one would expect, adding a little cyclohexane to a solution dilute in ethanol does not change the activity coefficient as much. This is shown graphically for the system ethanol-cyclohexane in Figure 1. Similar arguments may be made for polar-nonpolar, yet aprotic, systems. Note that the greater the differences in polarity or degree of hydrogen bonding, the more the symmetry in volume fraction assumption appears to fail. For this reason eq 12 is modified to include a term $ and a term ,$,to account for this asymmetry effect. The new equation becomes In

y2-

= RT u2 [(A,

- A2)2

+

912422(71

-

72Y

+

$1

(a1

- .z)(P1 - 02) 51

I

+4

2

(14)

As one would expect, the solvent asymmetry parameters increase with increasing polarity and degree of association. These parameters are not adjustable but are functions of the other parameters. The advantages of these parameters and how they are calculated are examined in later sections. A slightly different form for the combinatorial term (d1J was also used and is discussed in the section on parameter estimation. Methods of Correlation Several approaches were used in correlating the ym’s using the above equation. The need to obtain an accurate correlation was counterbalanced by the desire to keep the method as simple as possible. Each approach was evaluated by its intuitive appeal, the number of parameters and pure component properties needed, and its ability to predict the experimental 7”’s not only in terms of the average error but also in the number and magnitude of “bad” points (considered from both physical and numerical viewpoints). It should be noted that the inclusion of the asymmetry parameters reduced the average percent error by 16% absolute and that the number of especially troublesome points was reduced considerably. The most intuitively appealing approach would be to correlate all the parameters in eq 14 with physical properties. Unfortunately, there does not exist any universal acidity and basicity scale or even any widely used index which can be used to classify these properties. The next logical approach would be to use physical properties to estimate the nonhydrogen bonding parameters and then

198

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

to use those values and the cohesive energy density to obtain the a x p product. The relative acidity and basicity may then be estimated based on some sort of experimental evidence, such as spectroscopic data, and then the final values of all parameters obtained. This is similar to the approach of Tijssen et al. (1976)and others. This approach proved unsuccessful also. Even for aprotic molecules, trying to correlate the nonhydrogen bonding parameters with physical properties, or using the values of all but one of the parameters and the cohesive energy to generate the fiial one, was unsatisfactory. The problems in using solely physical properties are discussed in the section on parameter estimation. Using the cohesive energy density to reduce the number of adjustable parameters by one entailed both numerical and theoretical problems. The major numerical difficulty is that one is generally taking the difference of two large numbers to generate the final parameter. Ignoring the hydrogen bonding contribution, to obtain T~ one needs the quantity cll - X12. Both terms are accurate to about 10%. Thus for ell = 100 and XI2 = 80, the above difference could conceivably be anywhere from 2 to 38 cal/cm3. As that difference is essentially a measure of the molecule’s polarity, its estimation is critical. This difficulty could not be overcome even when q was adjustable (Thomas, 1980). A theoretical problem lies in the misestimation of the cohesive energy density due to hindered rotation in the liquid phase. As this problem is discussed at length in other sources (Bagley et al., 1971a,b; Thomas, 1980), it will not be addressed here. It is simply stated that for highly polar or nonspherical molecules, one should not expect the cohesive energy density to give a quantitative representation of the intermolecular forces in the liquid state. Without the cohesive energy density five parameters remained to be estimated. It was found that using eq 14 with q and X calculated a priori and T , a, and P treated as adjustable parameters was most convenient. Exactly why the correlation was done in this way requires knowledge of how the parameters were estimated, so attention shall now be turned to the hard details of the correlations.

Parameter Estimation i. The Liquid Molar Volume, v. The liquid molar volume was taken as that at 20 “C and was assumed constant at all temperatures. The exceptions to this rule were aniline, CS2, and iodides. For aniline and CS2 marked improvements in the overall correlation seemed to justify changing the volumes. These improvements were probably due to problems in determining the other parameters for these molecules. For iodides, the problem may be more fundamental. It was found that the predicted activity coefficients of iodide in large molecules, either polar or nonpolar, were consistently around 30% low due to a large negative combinatorial contribution. The best solution for accurate estimation proved to be raising the molar volume of iodides and this gave markedly superior results. Whether this is due to the inapplicability of a FloryHuggins type expression for molecules containing large atoms or just an artifice is not known. Further discussion of this problem is given in the section on the estimation of d12. The values of u for all molecules studied are included in Table 111. ii. The Dispersion Parameter, A. The A’s were initially generated in three different ways to see what the most efficient approach would be. The first method was to apply the homomorph technique of Bondi and Simkin (1956), which was later used by Weimer and Prausnitz (1965) and Helpinstill and van Winkle (1968). The homomorph technique treats the dispersion term as equal

to that of the equistructural hydrocarbon at the same molar volume and reduced temperature. Molecules were split into three categories: aliphatic, cyclic, and aromatic. The A’s were then taken from the corresponding homomorph plot given by Weimer and Prausnitz. The values for nonpolar molecules were compared to those obtained from X = cl/*, where the c’s were calculated from vapor pressure data and virial coefficients (Thomas, 1980). It was found that the technique worked moderately well for molecules containing only carbon and hydrogen but failed for CC14and CS2. Thus the homomorph technique was deemed inadequate. The other two methods calculated X from physical properties. Other researchers have used this idea (e.g., Karger et al., 1976; Tijssen et al., 1976; Koenhen and Smolders, 19751, calculating X from various functions of the refractive index. Values of X using these functions were not sufficiently quantitative for our purposes. To improve these functions two approaches were tried. One was to fiid one function which could correlate the X’s for all molecules. The other was to use slightly different functions for the aromatics and nonaromatics. The best single correlation for all molecules proved to be

+

= 1.58 1 . 8 0 n ~ ~ 1 0 ’ ~ ~ (15) where nD is the refractive index at 20 O C and Z is the ionization potential in electron volts. The inclusion of Z dependence was found necessary for calculating X satisfactorily for aromatics. The major drawback with using I is that it is not readily available for all molecules and even for those compounds for which data are plentiful, different sources frequently give values which differ by as much as 10%. It was found that the inclusion of an ionization potential dependence could give superior results for some molecules, such as tertiary amines and nitriles, but that poorer results were obtained for other molecules, especially polar aromatics and halogenated compounds. Since the overall results were not sufficiently accurate and since the ionization potentials are not always easy to find or estimate, the simplicity of a single overall correlation was sacrificed. The method that was used in the overall correlations split molecules into two categories: nonaromatic and aromatic. The A’s were determined as follows

= 20.3f(nD)

+ 3.02

+

(17) (18)

= 19.5f(nD) 2.79 No dependence on ionization potential was found necessary, although it was again noted that for tertiary amines and nitriles better results could be obtained if some Z dependence were included. Since it turned out that accurate estimation of the A’s for these molecules was important, their values were modified somewhat to account for their unusual rs. In addition, the X for CS2was altered to improve the correlation. The values of the refractive index and X for the molecules of this study are included in Table 111. More detailed discussions may be found in the work of Thomas (1980). A comparison of the X values from the literature with those determined in this study is also presented there. However, two pertinent observations should be noted. First, unlike the correlations of many researchers, the X’s for nonpolar molecules were not taken to be the square roots of the cohesive energy densities; these were used only as starting AAR

Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 2, 1984

points in the overall correlations. This was done for three reasons: (1) the cohesive energy densities are accurate only to 510% and the correlated values of X were nearly always within that range; (2) it is easier to calculate a X using an equation like (17) or (18) than a cohesive energy density, as no vapor pressure data and virial coefficients are needed, and (3) equation of state effects in the liquid state are too significant to ignore. Secondly, it should be noted that while others have argued that proper estimation of X is not critical when polar forces and/or hydrogen bonding is important, it has been observed that whenever the activity coefficients are large, when the differences between the x's is large, or when the molar volume of the solute is over about 120 cm3 that small changes in X could lead to markedly different predicted activity coefficients. This also is a problem when trying to use one researcher's polar parameters with a different one's nonpolar parameters. iii. The Induction Parameter, q . The parameter q (like a) reflects the ability of the nonpolar part of a molecule to interact with a polar part. Differences between various compounds are due primarily to the relative degrees of unsaturation. To a smaller extent differences in size and polarity play a role as reflected in the expressions of Meyer and Ross (1971),but for simplicity q was treated solely as a function of unsaturation. For saturated molecules q was given a value of 1.0. A value of 0.9 was given for aromatics and for unsaturated aliphatics q was found by q = l . O - ( no. C=C no. Catoms

)

As no molecules involving carbon-carbon triple bonds were included in this study they were not taken into account, although based on the above equation, estimates could easily be made. The value of q for all molecules studied may be found in Table 111. iv. The Polar Parameter, 7. Ideally, 7 would be calculated from suitable sets of physical properties as X was. Unfortunately, no function of properties (including dipole moment, dielectric constant, refractive index, and liquid molar volume) could be found which could quantitatively predict the value of 7 for all molecules. Earlier researchers (e.g., Tijssen et al., 1976) experienced similar difficulties. Thus it was concluded that the 7's had to be treated as adjustable parameters. To help ensure that no artifices were generated, a groups approach was used to calculate 7 when possible. For most monofunctional, primary alkanes the form used was 4.5 3.5 + no. C

(no. C- 1)

where C , is a constant for a particular group and no. C is the number of skeletal carbon atoms. This equation worked very well for bromides, iodides, chlorides, nitroalkanes, nitriles, alcohols, esters, and ketones. It was not generally used for the first member of a homologous series or for ethanol. The constant C, was 2.47 for bromides, 2.02 for iodides, 2.69 for chlorides, 5.84 for nitriles, 5.87 for nitroalkanes, and 1.65 for alcohols. The values for esters and ketones were 4.03 and 3.93, respectively, where methyl formate and acetone were considered the first members of the homologous series. For these two groups, positioning of the carbons around the functional groups was largely ignored; e.g., propyl formate and ethyl acetate were considered equivalent. In general, the effects of multiple functionalities, secondary or tertiary, positioning of the functional group, chain branching, and cyclic or aromatic

199

backbones are qualitatively understood but will require additional data to predict them quantitatively by a groups approach. Some rules of thumb which might provide reasonable estimates are as follows: secondary positioning, reduce 1-2% ; tertiary positioning, reduce 1-4%; chain branching, reduce e l % ; cyclic backgone, increase 10-25%. Without the groups approach, the equation 7

= 33w/u3/4

(21)

will give reasonable approximations for nonalcohols and nonaromatics. However, this should only be used as rough guide as deviations as large as 2.0 units from the "true" values may be obtained. Nonzero 7 values were given to all aromatics in the manner of Alessi et al. (1975) as that was found to give markedly superior results. Also, carbon tetrachloride with its high quadrupole moment was given a nonzero 7 value. The values of 7 at 293'K used in this study may be found in Table 111. A comparison of these values with those in the literature may be found elsewhere (Thomas, 1980). While u , A, and q were treated as temperature independent, it was necessary to use a temperature dependence for 7i. 777

= ~293(293/T)O'~

(22)

v. The Acidity and Basicity Parameters, CY and 8. No satisfactory method of accurately correlating a and /3 could be found. It was hoped that some spectroscopic data could lead to accurate predictions, but unfortunately different studies nearly always yielded quantitatively inconsistent though qualitatively correct results. Part of the problem was definition of a neutral solvent and the total disregard of other types of forces in these studies. Correlations of basicity scales, such as the gas phase proton affinity (Long and Munseon, 1973) did not give sufficiently accurate results. Thus it was decided that a and /3 would also have to be treated as adjustable parameters. As in the work of Karger et al. (1976), the acidity and basicity for saturated hydrocarbons were assumed to be 0 and the a and 0 for primary alcohols were taken to be equal. The choice of scale does not affect the predictions. Initial estimates were taken from the literature, either from other lists (Tijssen et al., 1976; Karger et al., 1976) or from estimates obtained using the aforementioned methods. Proper care was taken to ensure that physically unreasonable values for these parameters were not obtained within the optimization. For most molecules CY was kept at zero if physically reasonable. For ketones and esters, for which nonzero CY values might be expected due to the slight acidity of the CY hydrogen, it turned out that an CY value of 0 was sufficient. For mild Lewis acids such as halogenated compounds, aromatics, and carbon disulfide, a small nonzero a value was found necessary although the overall contribution to the activity coefficient was usually small except in the presence of a strong base. For acetonitrile and nitromethane it was found that nontrivial a values gave significantly better results in the overall correlations. It was found that small a values were adequate for the higher nitriles and nitroalkanes. It seems that the a hydrogens on the smaller molecules are much more acidic because the electron withdrawal on that carbon by the nitro or nitrile group cannot be dispersed by the presence of a carbons as in the larger molecules. A groups approach with the same form as that used for 7 was used for a and /3 for the same functional groups. The values for the constant C , defined the same as C , in eq 19, are given in Table IV. The limitations to the groups approach discussed for 7,also apply here.

.

200

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

Table 111. Parameters for MOSCED Model at 20 "C 1.628 1.460 1.506 1.446 1.496 1.424 1.542 1.328 1.381 1.531 1.344 1.438 1.471 1.445 1.416 1.392 1.513 1.424 1.361 1.366 1.359 1.360 1.361 1.402 1.430 1.394 1.503 1.388 1.434 1.290 1.386 1.379 1.378 1.407 1.377 1.372 1.422 1.397 1.402 1.386 1.440 1.333 1.399 1.510 1.430 1.422 1.407 1.387 1.383 1.371 1.338 1.364 1.388 1.453 1.392 1.390 1.384 1.388 1.384 1.445 1.358 1.354 1.410 1.552 1.525 1.501 1.551 1.586 1.447 1.451 1.392 1.426 1.388 1.383 1.410 1.400 1.396

96.5 98.5 80.7 82.7 64.1 69.6 40.5 53.7 79.1 52.2 99.6 92.7 79.1 84.2 72.0 93.6 76.5 58.4 70.4 74.4 80.4 79.3 89.0 77.0 90.1 111.6 88.1 90.9 88.1 14.8 89.6 96.3 81.1 97.3 97.8 84.2 106.0 104.5 109.9 107.4 100.4 91.5 80.5 100.8 100.0 94.1 105.9 107.0 109.6 107.8 111.8 107.3 90.9 105.8 106.5 115.0 113.7 114.5 124.0 115.3 116.4 108.1 102.3 101.8 89.1 89.0 91.1 101.4 103.6 123.8 1Q8.l 125.0 126.7 112.4 122.6 125.1

8.58 9.05 8.43 8.95 8.20 9.41 7.14 7.73 9.30 7.43 8.35 8.70 8.42 8.12 7.85 9.12 8.20 7.51 7.57 7.49 7.50 1.52 7.96 8.26 7.88 8.83 7.81 8.31 6.70 7.79 7.71 7.69 8.02 7.69 7.64 8.08 7.91 7.96 7.79 8.37 7.20 7.93 8.57 8.27 8.18 8.01 7.80 7.76 7.62 7.25 7.55 7.81 8.51 7.86 7.83 7.77 7.81 7.77 8.42 7.48 7.43 8.05 8.95 8.71 8.49 8.94 9.25 8.44 8.48 7.85 8.22 7.81 7.75 8.05 7.95 7.89

0.87 1.00 1.95 2.00 2.79 2.90 2.55 6.24 1.95 5.99 1.40 2.40 3.13 ,2.20 4.85 1.66 2.04 1.36 4.82 4.10 3.32 3.32 4.15 4.62 4.13 1.40 1.90 1.74 0.00 1.16 3.25 2.84 2.30 2.84 2.84 3.32 1.62 1.66 1.60 1.52 0.00 1.02 3.16 0.58 0.52 0.00 0.26 0.25 0.25 0.26 0.25 2.50 1.10 2.77 2.77 2.48 2.48 2.48 1.36 0.00 0.00 0.90 3.78 1.84 1.95 2.16 4.22 0.28 3.05 0.23 0.00 0.23 0.23 0.00 0.23 2.42

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 0.80 0.80 1.00 0.90 0.90 0.90 1.00 0.90 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 0.90 0.90 0.90 0.90 0.92 1.00 0.92 1.00 0.92 0.92 1.00 0.92 1.00

__01---

P

0.29 0.58 0.70 3.05 2.90 2.49 2.00 7.45 1.30 0.35 0.86 0.50 1.50 0.79 1.00 0.29 0.31 0.32 6.19 0.38 0.00 0.00 0.00 0.25 0.65 0.23 0.14 0.30 0.27 0.00 5.28 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.26 0.25 0.24 0.00 4.62 0.68 0.00 0.00 0.00

0.16 0.15 0.10 0.06 0.10 0.38 0.10 7.45 2.40 0.29 3.98 0.20 0.20 0.51 0.30 2.20 0.27 0.23 6.19 3.31 4.87 3.83 3.83 1.88 10.30 1.85 0.15 0.18 0.20 0.00 5.28 4.05 3.28 4.58 3.28 3.28 4.14 0.15 0.16 0.15 0.17 0.00 4.62 6.70 0.37 0.34 0.00 0.21 0.21 0.20 0.21 0.21 3.20 4.50 3.43 3.43 2.86 2.86 2.86 0.15 0.00 0.00 4.12 1.36 0.45 0.56 1.64 2.13 0.25 4.82 0.18 0.00 0.18 0.18 0.00 0.18 3.00

0.00 0.00 0.00

0.00 0.00 0.00 4.50 0.00 0.00 0.00 0.00 0.00 0.21 0.00 0.00 4.12 0.63 0.75 0.22 16.20 3.81 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00

+ 1.00 1.02 1.02 1.16 1.17 1.42 1.45 1.94 2.18 1.16 2.17 1.06 1.28 1.53 1.22 2.04 1.10 1.18 1.48 2.04 1.86 1.60 1.60 1.88 2.06 1.87 1.06 1.15 1.12 1.00 1.34 1.57 1.42 1.25 1.42 1.42 1.60 1.09 1.10 1.09 1.08 1.00 1.26 1.40 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.31 1.25 1.40 1.40 1.30 1.30 1.30 1.06 1.00 1.00 1.20 1.51 1.09 1.11 1.43 1.68 1.00 1.50 1.00 1.00 1.00 1.00 1.00

1.00 1.28

r;

aa

ID

1.00 1.01 1.02 1.11 1.12 1.33 1.31 3.62 2.06 1.11 2.10 1.04 1.20 1.37 1.16 1.73 1.07 1.12 3.43 1.78 1.58 1.41 1.41 1.61 2.46 1.61 1.04 1.10 1.08 1.00 3.34 1.39 1.29 1.17 1.29 1.29 1.41 1.06 1.07 1.06 1.05 1.00 3.17 1.72 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.21 3.13 1.27 1.27 1.21 1.21 1.21 1.04 1.00 1.00 2.93 1.36 1.07 1.07 3.39 2.39 1.00 1.34 1.00 1.00 1.00 1.00 1.00 1.00 1.19

0.952 0.945 0.943 0.914 0.911 0.868 0.870 0.353 0.546 0.915 0.573 0.93 3 0.984 0.854 0.903 0.719 0.926 0.912 0.564 0.71 6 0.790 0.846 0.846 0.782 0.682 0.784 0.934 0.918 0.923 0.95 3 0.670 0.851 0.87 5 0.902 0.875 0.875 0.846 0.927 0.926 0.928 0.930 0.953 0.736 0.812 0.950 0.950 0.953 0.952 0.952 0.952 0.952 0.952 0.893 0.74 5 0.879 0.876 0.893 0.893 0.893 0.935 0.953 0.953 0.781 0.806 0.917 0.91 5 0.651 0.70 2 0.952 0.863 0.952 0.953 0.952 0.952 0.953 0.952 0.896

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 2 2 1 2 1 1 1

1 1

compound carbon disulfide carbon tetrachloride bromotrichloromethane chloroform bromodichloromethane dichloromethane dibromomethane methanol nitromethane methyl iodide acetonitrile 1,1,1trichloroethane 1,1,2-trichloroethane 1,2-dichloroethane 1,l-dichloroethane ni troethane ethyl iodide ethyl bromide ethanol propionitrile acetone ethyl formate methyl acetate 1-nitropropane dimethylformamide 2-nitropropane 2-propyl iodide n-propyl chloride n-propyl bromide propane propanol bu tan one methyl propionate tetrahydrofuran propyl formate ethyl acetate dioxane 2-chlorobutane n-butyl chloride tert-butyl chloride n-butyl bromide butane butanol pyridine trans-1,3-pentadiene isoprene cyclopentane isopentene 2-pentene 1-pen tene 3-methyl-1-butene 3-methyl-1-butene methyl isopropyl ketone cyclopentanol 3-pentanone 2-pentanone propyl acetate methyl butylate ethyl propionate n-pentyl bromide pentane isopentane 1-pentanol nitrobenzene chlorobenzene benzene phenol aniline cyclohexene cyclohexanone 4 -methyl-1-pentene cyclohexane 1-hexene 2-methyl-1-pentene meth ylc yclopentane 2-methyl-2-pentene 4-methyl-2-pentanone

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

201

Table I11 (Continued)

RI 17467 1.392 1.375 1.375 1.377 1.369 1.372 1.418 1.401 1.528 1.539 1.564 1.497 1.517 1.420 1.423 1.400 1.444 1.406 1.388 1.389 1.392 1.381 1.534 1.496 1.496 1.506 1.459 1.409 1.433 1.391 1.397 1.430 1.627 1.527 1.523 1.427 1.420 1.412 1.405 1.418 1.658 1.490 1.481 1.430 1.441 1.421 1.506 1.412 1.437 1.422 1.44 5 1.441 1.451 1.462 1.435 1.449 1.439 1.443 1.444 1.448 1.452 1.499 1.454 1.453 1.455 1.457

V

105.4 132.0 130.8 130.3 129.7 132.9 131.9 124.7 139.0 102.6 117.2 128.4 106.3 108.1 137.2 124.4 140.9 121.3 139.5 146.6 14 5.8 148.7 149.0 116.9 122.5 123.3 120.6 134.4 157.0 142.4 165.1 162.6 157.8 118.1 132.9 142.4 163.8 173.1 176.6 178.7 189.6 139.6 156.0 154.0 186.9 175.5 189.3 143.6 194.9 190.8 227.5 286.0 287.3 301.5 305.6 292.8 288.0 327.1 358.3 374.6 423.8 489.4 478.5 522.2 522.0 555.0 604.4

h

8.65 7.85 7.67 7.67 7.68 7.60 7.63 8.14 7.52 8.73 8.84 9.06 8.45 8.64 8.16 8.19 7.94 8.40 8.00 7.81 7.82 7.75 7.74 8.79 8.49 8.44 8.58 8.56 8.04 8.30 7.84 7.91 8.26 9.60 8.73 8.69 8.23 8.15 8.07 8.00 7.63 9.85 8.43 8.80 8.26 8.38 8.17 9.05 8.07 8.34 8.17 8.42 8.38 8.49 8.60 8.32 8.47 8.36 8.40 8.41 8.45 8.50 8.98 8.51 8.50 8.53 8.55

7

0.90 2.21 0.00 0.00 .O.OO

0.00 0.00 0.82 0.53 3.30 2.71 2.20 1.56 2.77 2.90 0.00 0.24 0.00 2.17 0.00 0.00 0.00 0.00 3.02 1.28 1.27 1.65 0.00 0.23 0.00 0.00 0.00 0.69 2.80 2.74 3.26 0.00 1.78 1.60 0.00 0.45 1.95 1.00 0.00 2.27 0.00 0.21 1.56 0.00 0.60 0.00 1.62 0.10 0.72 0.66 0.00 0.44 0.00 0.00 0.00 0.00 0.00 0.28 0.00 0.00 0.00 0.00

Q 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.90 0.90 0.90 0.90 0.90 1.00 1.00 0.93 1.00 1.00 1.00 1.00 1.00 1.00 0.90 0.90 0.90 0.90 1.00 0.94 1.00 1.00 1.00 1.00 0.90 0.90 0.90 1.00

1.00 1.00 1.00 1.00 0.90 0.90 0.90 1.00 1.00 0.95 1.00 1.00 1.00 1.00 1.00 0.97 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 0.87 1.00 1.00 1.00 1.00

a

4.00

0.00 0.00 0.00 0.00 0.00 0.00 3.73 0.00 0.64 0.74 0.80 0.15 0.28 0.22 0.00 0.00 0.00

0.00 0.00

0.00 0.00 0.00 0.86 0.05 0.07 0.04 0.00 0.00

0.00 0.00 0.00 3.13 0.34 0.69 0.23 0.00 0.00 0.00 0.00 0.00 0.57 0.05 0.00 0.17 0.00 0.00 0.00 0.00 2.72 0.00 0.12 0.00 0.11 0.10 0.00 1.99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

P 4.00 2.55 0.00 0.00 0.00 0.00 0.00 3.73 4.98 2.94 0.93 3.06 0.60 1.60 1.99 0.00 0.22 0.00 2.69 0.00 0.00

0.00 0.00 3.13 0.50 0.74 0.90 0.00 0.21 0.00 0.00 0.00 3.13 3.10 2.91 2.51 0.00 2.21 2.40 0.00 3.30 0.38 0.40 0.00 1.56 0.00 0.19 1.81 0.00 2.72 0.00 1.11 0.11 0.07 0.07 0.00 1.99 0.00 0.00 0.00 0.00 0.00 0.70 0.00 0.00 0.00 0.00

The values for a and j3 at 293 K are given in Table 111. The temperature dependence for a is

An identical expression is used for j3. vi. The Asymmetry Factors, $ and t. Both I) and [ were considered functions only of 7 , q, and the degree of

rii

1.19 1.22 1.00 1.00 1.00 1.00 1.00 1.17 1.00 1.41 1.26 1.17 1.06 1.27 1.45

1.00 1.00 1.00 1.21 1.00 1.00 1.00 1.00 1.35 1.03 1.03 1.07 1.00 1.00 1.00 1.00 1.00 1.12 1.28 1.28 1.38 1.00 1.12 1.09 1.00 1.00 1.11 1.02 1.00 1.24 1.00 1.00 1.08 1.00 1.09 1.00 1.10 1.00 1.01 1.01 1.00 1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

aa

ID

0.790 0.906 0.953 0.953 0.953 0.953 0.953 0.812 0.950 0.829 0.875 0.882 0.929 0.874 0.867 0.953 0.952 0.953 0.907 0.953 0.953 0.953 0.953 0.839 0.937 0.937 0.926 0.953 0.952 0.953 0.953 0.953 0.854 0.867 0.861 0.845 0.953 0.922 0.928 0.953 0.951 0.914 0.943 0.953 0.901 0.953 0.953 0.929 0.953 0.878 0.953 0.926 0.953 0.948 0.949 0.953 0.913 0.953 0.953 0.953 0.953 0.953 0.952 0.953 0.953 0.953 0.953

2 1 1 1 1 1

t

2.86 1.15 1.00 1.00 1.00 1.00 1.00 2.68 1.00 1.40 1.20 1.30 1.04 1.19 1.32 1.00 1.00

1.00 1.14 1.00 1.00 1.00 1.00 1.45 1.02 1.02 1.04 1.00 1.00 1.00 1.00 1.00 2.22 1.24 1.33 1.28 1.00 1.08 1.06 1.00 1.00 1.08 1.01 1.00 1.17 1.00

1.00 1.06 1.00 1.89 1.00 1.07 1.00 1.01 1.00 1.00 1.40 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00

association, CY as follows.

X

1 1 1 3 3 3 3 3 1 1 1 2 1 1 1 1 1 3 3 3 3 2 1 2 1 1 1 3 3 3 2 1 1 1 1 3 3 3 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1

compound cy clohexanol butyl acetate hexane 2,3-dimethylbutane 3-methy lpentane 2,2-dimethylbutane 2-methylpentane hexanol triethylamine benzoni trile benzyl chloride bromoanisole toluene anisole heptanenitrile methylcyclohexane 1-heptene cycloheptane ethyl butyl ketone heptane 3-methy lhexane

2,2-dimethylpentane 2,4-dime thylpentane acetophenone ethylbenzene p-xylene 0-xylene cyclooctane 1-octene ethylcyclohexane isooctane octane octanol quinoline propiop henone benzyl acetate

1,3,5-trimethylcyclohexane dibutyl ketone diisobutyl ketone nonane tripropylamine bromonaphthalene butylbenzene decalin decanenitrile butyl cyclohexane 1-decene ethyloctanoate decane decanol dodecane palmi tani trile hexadecene n-hexadecyl chloride n-hexadecyl bromide hexadecane hexadecanol octadecane eicosane heneicosane tetracosane oc tacosane squalene triacontane squalane dotriacontane pentatriacontane

8, of the solvent. They were calculated

POL = q4(1.15- 1.15 exp(-0.020rT3)) + 1

(24)

t = 293/T

(25)

fi = POL + O . O ~ ~ C Y T @ T

(26)

202

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

E = 0.68(POL - 1) + (3.4- 2.4 e ~ p ( ( a ~ P ~ ) ’ ~ ~ ( ~ . O 2 3 ) Table ) ) ~ ~IV (27) In the above expressions, a subscript 0 refers to 20 OC and a subscript T represents the temperature of the system. The values of J/ and E at 20 OC are also included in Table 111. As J/ and E are temperature sensitive, they should be evaluated at each temperature of interest. The values of all temperature-dependent parameters (7,a,P, J/,and E ) are tabulated at various temperatures in the Supplementary Material. (See paragraph at end of article regarding Supplementary Material.) Utilizing those values can reduce calculation time of a ymto a few seconds. vii. T h e Combinatorial Term, d l z . The simplest function for the combinatorial term is the Flory-Huggins equation (Flory, 1941,1942;Huggins, 1941),which at infinite dilution is VZ

dlz = In *1

V2 + 1- u1

However, it overpredicts somewhat as is discussed elsewhere (Hildebrand, 1947;Donohue and Prausnitz, 1975; Guggenheim, 1952;Staverman, 1950;Sayegh and Vera, 1980). To alleviate this, the exponent of uz/ul was lowered from 1.0 to 0.953. As noted by Hildebrand (1947)and Donohue and Prausnitz (1975),the value of this exponent must be between 0 and 1. However, this, too, was found insufficient as the combinatorial contribution for highly polar or associated molecules was still overestimated. This is not surprising as Guggenheim (1952)had observed that the lattice theories (like the Flory-Huggins equation) are inapplicable to highly associated solutions or solutions where there is a large tendency for complex formation. To account empirically for this observed effect, the exponent of u 2 / u I (aa) was calculated as aa = 0.953 - 0.00968(7z2 a2&) (2 = solute) (29)

+

then

This gave much improved, though still imperfect, results. The values of aa for the molecules are also included in Table I11 and they are tabulated as a function of temperature in the Supplementary Material as well. Note that while II. and pertain to the solvent, aa refers only to the solute. As noted earlier, the combinatorial contribution for iodides was found to be much too large unless their molar volumes were increased. This may be due to the large I atoms being too large to “fit”in a lattice site, and thus the effective volume of the iodides is larger than their observed molar volume. This could be accounted for by either changing the molar volume or by lowering the exponent in the above expression. The former approach gave slightly superior results and was somewhat easier to implement. The Staverman potential (1950)for calculating the combinatorial contribution was also tried. It is used in the UNIQUAC (Anderson and Prausnitz, 1978)and UNIFAC (Fredenslund et al., 1975)solution models. The equation is given in the Appendix. Unfortunately, its results were significantly inferior to those obtained with the above method. For about 10% of the systems studied, positive combinatorial contributions were found, a clearly impossible result. The predictions were consistently worse for iodides, aromatics, alcohols, and for small molecules in large ones. This equation will be discussed further in the UNIFAC section.

chlorides bromides iodides nitriles nitroalkanes alcohols esters ketones

0.42 0.39 0.37 0.33 0.35 7.49 0.00 0.00

0.25 0.38 0.30 4.00 2.66 7.49 4.64 4.87

Data Compilation The most difficult part of the correlation was deciding which data should be included. Ideally, only chromatographic and ebulliometric data would be involved. However, the limited data available meant that some extrapolated VLE data had to be included for completeness. Previously extrapolated values or values obtained graphically by this researcher were used. Data were rejected if they were deemed physically unreasonable, if more believable data on the same system could be found elsewhere, or if accurate extrapolation was impossible. A good deal of chromatographic and a small amount of ebulliometric data were also used from the literature. Much of the chromatographic data were reported as retention volumes (e.g., Littlewood, 1964) or partition coefficients (e.g., Rohrschneider, 1973)but could easily be transformed to activity coefficients (neglecting compressibility and vapor phase nonideality effects). Unfortunately, a significant amount of these data had to be discarded or modified. The ebulliometric data of Tochigi and Kojima (1976)did not account for the vapor and liquid holdup and hence their data for alcohols in alkanes proved to be consistently too low. The chromatographicvalues for toluene and propanol in 1,2-dichloroethane of Barker and Hilmi (1967)were raised by about 9% owing to a slightly inaccurate choice of standard. The chromatographic values for alcohols in hexadecyl derivatives by Littlewood (1964)proved consistently unreasonable and hence were ignored. Littlewood did note in his work that some possible problems with alcohol-packinginteractions may have been present. The gas chromatographic data of Monfort et al. (1970)in 1nitropropane were increased by about 20% to bring them into agreement with the data of Marsh et al. (1979).Only about 30% of the applicable data of Rohrschneider (1973) were used as many of his data points were in significant disagreement with values obtained from other sources. For example, his datum for octane infinitely dilute in carbon tetrachloride was 1.96 at 25 OC. Literature values for this point vary from 1.35 to 1.55. Rohrschneider’s technique was quoted as being accurate to about 15% but in many cases the error seemed much larger. Data were selected from his compilation if they provided information about certain molecules (such as the polarity of dioxane) which had been previously lacking. Results and Discussion The necessary pure component properties for all molecules studied are listed in Table 111. In Table V the equations used to calculate these parameters are reproduced. Table VI compares the measured activity coefficients with the predictions of the MOSCED model and UNIFAC for many systems. A complete listing of all the data studied and their predictions may be obtained in the Supplementary Material. The UNIFAC predictions are discussed later. The average error for the 3357 data points is 9.1%. This error is quite reasonable, since as others have shown (Shreiber and Eckert, 1971;Tassios, 19711,the fit of VLE data from the 7”s is insensitive to 10% perturbations in

Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 2, 1984 203 Table V. Equations to Calculate Parameters for MOSCED (Eq 14)

derpredictions of 50-75% for the 7”s in CHCl, and CH2C12. Similar but not as striking behavior was observed for TEA in other chlorinated compounds. As a result any predictions for systems involving TEA and a moderate to strong acid should be viewed with discretion. Problems such as these could be alleviated by introducing steric factors into the acidity and basicity terms, but this added level of complication is beyond the scope of the present treatment. Another limitation is the model’s inapplicability to aqueous systems. While extrapolated data involving water are abundant, no satisfactory set of parameters could be found which would work reasonably well for all systems. Entropic considerations play far too great a role for these systems to be easily handled by a primarily enthalpic model such as the MOSCED. The following systems (with values of ym2soc from Deal and Derr, 1964) demonstrate this point: n-butyl alcohol-heptane, 37.1; tert-butyl alcohol-heptane, 18.0; n-butyl alcohol-water, 48.2;tert-butyl alcohol-water, 10.1. Based on the alkanealcohol data, one would conclude that the polarity and hydrogen bonding terms for n-butyl alcohol would be larger than those for tert-butyl alcohol. This is reasonable as the steric hindrance of the methyl groups in the tertiary compound would effectively reduce the magnitude of these parameters. However, in water, tert-butyl alcohol behaves much more ideally. This is due to the fact that the number of configurations of the water molecules that give rise to favorable hydrogen bonding is much fewer around the elongated n-butyl alcohol (with its appreciable nonpolar chain), than around the more spherical tert-butyl alcohol (and ita smaller hydrocarbsn backbone). Not surprisingly, in water, n-propyl alcohol and tert-butyl alcohol behave almost identically. Problems of this nature also occur for the smaller alcohols-especially when mixed with strong acids or bases-but to a significantly lesser extent. Finally, the model will yield consistent quantitative predictions only for systems with activity coefficients below 100 or so. For a system with an activity coefficient of 500, a 10% error in In 7-(what is actually correlated) will yield an 86% error in ym. For systems of this type, a predictive scheme like that of Pierotti, Deal, and Derr (1959), involving a number of group interaction parameters, may be more effective. Despite these problems, the success of the MOSCED model is most encouraging. As noted before, there are very few unsatisfactory predictions, and these can generally be attributed to either questionable data or the simplicity of the model. The model lends itself either to hand calculation or to computation with a short and inexpensive subroutine. That only pure-component parameters were used is also a large positive aspect. The lack of requirements for any binary parameters greatly reduces the amount of data necessary to determine how a new substance will interact with other species in general. For example, if the 7”s for the unknown substance are measured in octane (nonpolar), chloroform (acidic), methanol (polar, associated),acetone (polar, basic), and nitromethane (highly polar), then the pure-component parameters which correlate these 7”s well should accurately predict most other systems involving this substance also. This approach is similar in principle to that of Rohrschneider (1973) in predicting chromatographic retention behavior. The physical significance of the parameters gives the practitioner a feel for the relative magnitudes of the different types of forces in a solution. It also lends itself to intuitively appealing temperature dependences. Hydrogen bonding has a stronger dependence than polar forces,

_-__________~__I__-_I__--

h

nonaromatics:a

h=

10.3 [ ( n D z - l ) / ( n D ’

+

2)]

t

3.02

aromatics: A = 19.5 [(nD2 - l ) / ( n D 2 + 2)] + 2.79 adjustable or groups approach (monofunctional, primary position or ketones, esters)c

7,01,P:

chlorides bromides iodides nitriles nitroalkanes alcohols esters ketones

2.69 2.47 2.02 5.84 5.87 1.65 4.03 3.93

0.42 0.39 0.37 0.33 0.35 7.49 0.00 0.00

0.25 0.38 0.30 4.00 2.66 7.49 4.64 (no. C = 1 methylformate) 4.87 (no. C = 1 acetone)

temperature dependence 7~

=

T~~~

~ , P T= O$,, q =

[ 293/TJo*1

[293/T]

1.0 saturated compounds

= 0.9 aromatics = 1.0 - 0.5 (

””

’

no. C=C bonds no. c atoms

POL = q4(1.15- 1.15 e x p ( - 0 . 0 2 @ ~ ~+) ) 1 t = 293lT $ = POL + 0.011aTPT E = 0.68(POL - 1) + (3.4 - 2.4 e x ~ ( ( a , P , ) ~ ” (-0.023))) t 2 d,, = In ( u , / u , ) ” QCJ

+

1-(U,/U,)~~

= 0.953 - 0.00968(7,z +

oi,P,)

a Except tertiary amines, nitriles, CS, . Except first member of a homologous series and ethanol. No. C is number of skeletal carbon atoms.

those 7”s. Only 2 % of the data have errors greater than 30% and the great majority of these points are from extrapolated data whose accuracy is questionable. The temperature dependences and asymmetry in volume fraction are for the m a t part characterized quite well. The chromatographic data of small molecules in large ones are predicted reasonably well. The groups approach at predicting a,p, and 7 (when applicable) proved successful. However, there were a number of unsuccessful predictions of seemingly accurate data. Before the strengths of the model are further discussed, ita limitations are examined. As noted earlier, a number of modifications of nonadjustable parameters were necessary. These included the molar volumes of aniline, CSz, and iodides as well as the A’s of nitriles, tertiary amines, and CS2. Physical justifications for most of those modifications were discussed, but such changes limit the general applicability of the model. The predictions were poorest for systems where steric considerations predominate. The two worst points were for highly basic triethylamine (TEA) in acidic chloroform (CHC1,) and methylene chloride (CH,Cl,). These are due to the basic nitrogen in TEA being somewhat inaccessible because of the three ethyl groups surrounding it. Thus while TEA can interact strongly with the “free” proton in methanol, it cannot as effectively interact with hindered protons in CHC&and CH2C12. Using a basicity parameter for TEA consistent with the 7”s in alcohols yielded un-

204

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

Table VI. Limiting Activity Coefficient Predictions by MOSCED and UNIFAC error UNIFAC error refb 1.48 -6.7 -2.2 A 1.56 -11.7 -11.17 B 11.83 -16.6 C -3.2 -6.5 2.83 -2.93 17.1 D 4.5 5.25 4.68 C -9.9 2.88 4.13 -37.2 C 3.0 1.13 3.8 1.14 P 68.9 0.9 0.35 0.21 C 8.3 3.6 1.85 1.94 C -57.1 0.12 -17.6 0.22 E -0.7 26.82 -44.1 1.5.09 N.A. 10.2 N.A. B 2.42 E -12.1 14.10 42.20 -70.6 5.17 B 8.14 22.0 -22.5 F -19.5 17.22 22.49 5.1 50.1 A 13.51 -10.5 8.06 A -1.8 3.13 3.08 -3.6 E -5.6 24.06 31.30 22.7 E 61.4 17.43 8.57 -20.7 G 13.31 -16.8 9.70 -39.4 C -3.6 2.17 0.3 2.08 C -4.2 0.50 0.56 -14.0 -10.0 H 4.45 7.45 -46.3 -9.9 N.A. 1.48 N.A. B 2.8 B 7.71 3.18 -57.6 -2.2 -6.2 4.03 B 4.21 C 40.8 3.3 1.56 2.13 E 6.9 6.63 -39.3 3.76 7.5 3.73 B 8.6 3.17 -6.6 2.10 B 1.89 -15.9 -39.8 5.87 A 3.28 -66.4 G 26.3 84.01 9.27 -86.1 -0.7 2.37 B 0.70 -70.8 1.8 26.17 B 8.24 -68.0 -2.9 0.65 0.38 B 66.8 2.0 4.55 B 2.04 -54.2 1.12 C 4.4 9.1 1.17 C 4.8 -5.4 1.60 1.45 5.2 29.03 C 28.70 4.0 5.5 4.62 C 22.5 5.36 26.15 C -20.8 17.73 -46.3 -4.0 13.25 A 25.6 17.33 -22.7 C 2.71 5.49 -61.9 -3.4 I 20.1 2.16 2.6 9 G -20.8 38.80 -46.1 26.39 G -4.7 0.45 0.18 -61.8 17.6 5.94 5.6 B 5.33 11.8 2.41 A 4.93 128.3 -0.1 49:34 8.88 A 455.0 41.42 7.6 -3.7 B 37.07 -9.5 17.10 B 13.23 -30.0 11.82 13.6 5.19 B -50.1 -0.2 1.27 N.A. N.A. B 1.11 J -2.1 0.77 -31.6 33.29 G -16.8 -57.5 17.01 3.35 1.9 76.8 K 5.82 9.89 -5.8 L 4.68 -55.4 6.13 -2.7 L 3.81 -39.6 1.7 2.85 L 2.06 -26.5 35.3 3.11 111.4 K 4.86 -2.6 0.83 M 0.6 7 -20.9 6.5 1.34 N -22.2 0.98 -4.6 0.98 N 1.57 -40.5 3.4 0.82 0.46 N -41.3 -1.9 2.15 4.32 0 -51.2 0.62 L 5.2 0.43 -26.7 -11.2 0.55 L 0.38 -38.4 References: (A) Gmehling and Onken (1977); ( B ) Thomas et al. (1981a); (C) Thomas et al. *Extrapolated data. (1981b); ( D ) Schreiber and Eckert (1971); (E) Deal and Derr (1964); ( F ) Locke (1969); ( G )Pierotti et al. (1959); (H) Monfort et al. (1970); (I) Nakanishi et al. (1967); (J)Littlewood (1964); (K) Comanita et al. (1976); (L) Kwantes and Rijnders (1958); ( M ) Snyder and Thomas (1968); (N) Martire and Pollara (1965); (0)Harris and Prausnitz (1969); (P) Bser et al. (1973). solute cyclohexane acetonitrile acetone 1-nitropropane 2-nitropropane cyclohexane tetrahydrofuran hexane triethylamine hexane chloroform hexane carbon tetrachloride pentane pentane benzene hexane hexane heptane carbon tetrachloride chloroform hexane n-propyl chloride hexane hexane acetonitrile hexane acetonitrile pentane octane hexadecane methanol hexane dichloromethane hexane carbon tetrachloride chloroform acetonitrile bu tanone butanol aniline cyclohexanone met han ol ethanol dotriacontane acetonitrile acetonitrile phenol nitrome thane nitroethane nitromethane ethyl bromide cyclohexane ethanol ethanol propionitrile acetone ethyl acetate butanol pentane 1,2-dichloroethane hexane dichlorome thane propionitrile pentane pentane

solvent carbon disulfide carbon tetrachloride carbon tetrachloride carbon tetrachloride carbon tetrachloride carbon tetrachloride chloroform chloroform chloroform methanol nitromethane nitromethane acetonitrile acetonitrile acetonitrile acetonitrile acetonitrile acetonitrile ethanol acetone acetone 1-nitropropane 2-nitropropane 2-nitropropane butanone ethyl acetate pyridine benzene benzene phenol phenol aniline aniline cyclohexanone cyclohexanone cyclohexane hexane hexane hexane hexane hexane hexane triethylamine heptane heptane p-xylene p-xylene octane isooctane isooctane octanol quinoline palmitani trile hexadecane hexadecane hexadecane hexadecane hexadecane hexadecane hexadecane hexadecanol hexadecanol squalane squalane squalane pen tatriacontane

temp 298.2 293.2 303.9 298.2 314.9 349.8 303.2 319.8 323.0 298.2 293.2 298.2 293.2 298.2 363.3 293.2 298.2 373.2 298.2 304.0 327.4 298.2 293.2 293.2 293.2 347.8 298.2 293.2 293.2 455.0 298.2 293.2 293.2 293.2 293.2 353.8 315.3 295.0 298.0 301.0 340.2 298.0 362.3 298.2 298.2 293.2 411.5 398.8 293.2 293.2 293.2 298.2 31 3.2 298.2 413.2 313.2 293.2 298.2 413.2 363.2 36 7.1 326.3 326.3 353.2 303.2 353.2

dataa 1.59 13.40 3.03 4.48* 4.58 1.10 0.21 * 1.79 0.27 27 .OO * 2.20 48.00* 6.67 21.40 9.00 3.19 25.50* 10.80* 16.00 2.16 0.58 8.28* 1.64 7.50 4.30 1.51 6.20* 3.47 2.25 9.76 66.50* 2.39 25.70 0.39 4.46 1.07 1.53 27.60 4.38 33.00 13.80 7.10 2.24 49.00 0.47 5.05 2.16 8.89 38.50 18.90 10.40 1.27 1.13 40.00* 3.29 10.50 6.30 2.80 2.30 0.85 1.26 1.65 0.79 4.40 0.59 0.62

_______-__-------_____----___-___-------_____-_____-__

which in turn have a greater one than dispersion interactions, as one would anticipate. These distinctions in forces allow for more accurate calculation of temperature effects

than models which do not do so, UNIFAC for example. The physical insight also aids in designing solvent mixtures for a given process. For example, a mixture of acidic

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984 205

chloroform and basic acetone would solvate a large molecule with acidic and basic functional groups better than either pure solvent. While the success of the model is promising, additional refinements are possible upon further expansion of the data base. Focus shall now be turned to possible methods of extending the utility of the model.

Future Considerations Some possible improvements were noted earlier. Steric factors can easily be introduced into the acidity and basicity terms. A possible form might be as follows (neglecting the asymmetry factor) @l

1+ S A l S B l

+

a2P2

-

1+ SAZSB2

a1Pz

1+ S A l S B 2

-

a281

1+ SAZSBl (31)

where S A 2 is a steric factor for the acidity of the solute. The steric factors would be 0 for small molecules such as methanol, but greater than 0 for hindered compounds such as triethylamine or chloroform. Thus, the system methanol-triethylamine would not have a steric effect while TEA-CHCl, would, as the data and intuition seem to support. The possibility of handling aqueous systems might also be increased. However, the limited available data for which these steric factors are applicable do not yet justify the further complication. The groups approach could also be further expanded to characterize cyclic and aromatic backbones, multifunctionalities, and secondary or tertiary positioning of groups. Here again, the present data base is limiting. The biggest problem will probably be with aromatics in which positioning of the groups on the ring can make significant differences. One would also have to consider the electron withdrawing or donating effect of a group on the ring in addition to just the characteristics of the group. The model can be extended to calculating directly the ymof a solute in a solvent mixture. The model in its present form cannot be used for finite concentrations since the asymmetry factors keep it from satisfying the GibbsDuhem relation. However, the prospect of calculating 7”s in solvent mixtures hinges primarily on the proper selection of composition dependence of the various parameters. The simplest approach would be to make all terms simply a function of volume fraction as is done in regular solution theory. Equation 14 could then be recast (neglecting the combinatorial term) as In y2- =

where

x = 4 1 x 1 + ... + 4,,An T=

$171

+ ... + (b,,A,

(33)

etc. However, different composition dependences would probably be necessary for the polar parameters. Unfortunately, very little data of this type exists in the literature and only one system could be found for which the necessary parameters are available. Thus no firm conclusions can as yet be drawn. It should be noted that this extension is more a convenience than a practical necessity. If the relevant pure-component parameters are available for all species involved, then all the constituent binary y”s can be generated. From these, the binary interaction param-

Table VII. Comparison of MOSCED and UNIFAC in Predicting 7”’s for 3357 Values

--

-

average error fraction of data with error >30% temperature dependence molecules of greatly different sizes nonpolar-nonpolar polar aromatics associated-associated associated-solvated polar-nonpolar polar-polar aqueous systems

MOSCED

UNIFAC

9.1% 2.0%

20.5% 23%

very good very good excellent good good good very good very good poor

fair fair excellent poor good good fair very good fair

eters for the Wilson equation can be obtained and the desired ymin the solvent mixture can then be accurately calculated. All the improvements discussed above are contingent upon a more extensive and accurate data base. As more data are collected, more molecules can be treated and more improvements may be made without worry of generated artifacts.

Comparison with UNIFAC UNIFAC (Fredenslund et al., 1977a,b) is currently a popular technique for predicting activity coefficients. Like the earlier ASOG model (Derr and Deal, 1969; Kojima and Tochigi, 1979), it considers each molecule as the sum of the functional groups (e.g., alkyl, keto, nitro) which constitute that molecule. The thermodynamic properties of a solution are then calculated based on the interactions between the functional groups which comprise the mixture. UNIFAC is compared to the MOSCED model in terms of its ability to predict the y”s. Furthermore, its ability to predict finite concentrations from parameters obtained solely from 7”s is studied. The pertinent equations for utilizing UNIFAC are given in the Appendix. The UNIFAC groups and their parameters are found in the monograph of Fredenslund et al. (1977b). Additional groups and their parameters have been found and were used (Skjold-Jorgenson et al., 1979). In Table VI UNIFAC’s predictions of the ymdata are compared to those of the MOSCED model for a few systems (complete comparison is in the Supplementary Material). For all systems studied Table VI1 compares the relative merits of the two approaches in handling various types of systems. As one can see from the table, the MOSCED model was superior to UNIFAC in predicting the 7”s. A number of problems with UNIFAC were apparent. These included an unsatisfactory combinatorial term, temperatureindependent parameters, the lack of a defined cyclic group, unsatisfactory predictions for the first member of a homologous series, inapplicability to molecules with functional groups in secondary or tertiary positions, and frequent inability to predict simultaneously both ends of the composition range for polar aromatic molecules. The original version of UNIFAC assumes temperature-independent parameters, although current work is underway to relax this restriction (Fredenslund, 1982). The temperature functionality of the parameters for alcohols in paraffins will obviously be much different than that for nonpolar aromatics in paraffins. UNIFAC forecasts the latter correctly, but is usually poor at predicting the former-the predictions are generally too low at lower temperatures and too high at higher ones. For example, the predicted value of 1-butanol in hexane at 301 K is 46% low, while the value of 1-butanol in hexadecane at 413 K is over 110% high.

206

Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 2, 1984

Table VIII. Comparison of Various Combinatorial Functions for UNIFAC for 132 7 "s in Squalane exponent in Staverman Flory-Huggins correction term included errora _ I _ _ _ _

1.0 0.66 0.66 0.75 0.75

Yes Yes no

Yes no

8.62 4.6 7 3.83 2.02 1.78

a Error calculated as 100 x (In (yexptl/ycalcd))z/number of data points.

A second problem is UNIFAC's combinatorial term which accounts for the molecules different sizes and shapes. One difficulty is that for about 10% of the systems studied, a positive combinatorial contribution was obtained, a theoretically impossible result. While this is discouraging,a more serious problem is that it is ineffective at predicting the data for systems with molecules of greatly differing sizes. For small molecules in much larger ones, UNIFAC generally underpredicted by 2 6 3 5 % . There are insufficient data for large molecules in small ones, but one obvious datum CZ in C, was groasly underpredicted (62%1. This problem is primarily due to the limitations of the group contribution approach in which all groups (e.g., -CH2-) are considered equivalent in all molecules. Larger hydrocarbons are more polarizable than smaller ones as seen by their refractive indices and solubility Parameters. Thus, as researchers in polymers have known for years, a nonzero interaction parameter is necessary even for systems where the species differ only in size (e.g., pentane and eicosane). Two attempts to circumvent these problems have been made for UNIFAC. One was through free volume considerations (Oishi and Prausnitz, 1978),but the technique is not sufficiently established for general application. A later attempt (Kikic et al., 1980) manipulated exponents in the Flory-Huggins portion of UNIFAC's combinatorial term. The exponent of (ul/uz) (or 41/x1 for finite concentrations, see Appendix) was lowered from 1.0 to 0.66 to fit data for saturated hydrocarbons. This overcorrects in some systems, at least in the dilute region. An exponent of about 0.75 seems better, but this creates an additional problem of more systems exhibiting positive combinatorial contributions. The optimum solution, at least for y"s, may be to use an exponent of around 0.75 in the Flory-Huggins part and then simply drop the Staverman correction (see Appendix). Table VI11 compares some of the proposed modifications. As one can see, modifying the exponent greatly enhances the predictive capabilities for molecules of greatly differing sizes. Dropping the Staverman correction seems to help also. As this provides more physically consistent values, it is preferable. Although the capabilities for finite concentrations remain untested, this final modification should yield approximately the same composition dependence as a (Kikic et al., 1980) combinatorial term which has been shown to work in some cases (Aleasi et al., 1981). This will necessitate recalculating the interaction parameters for groups with unusual surface area to volume characteristics (e.g., water, phenol, alcohol). Some of the group volumes and areas may also need to be modified. However, a simplier and more internally consistent model would result. Still another problem is with cyclic compounds. The data for ketones and paraffins in Table VI demonstrate how UNIFAC is incapable of handling polar molecules with cyclic backbones. While the data for acetone and butanone with p a r a f f i are generally predicted quite well,

those for cyclohexanone are grossly underpredicted. This is reasonable as the ring structure ties back the neighboring methyl groups in cyclohexanone thereby making the C=O group significantly more exposed than in the corresponding straight chain compound. This leads to both a higher effective polarity and basicity for cyclohexanone than one would calculate from a simple groups approach. Defining a cyclic group should lead to significantly improved predictions and may be worth the added complication. As the creators of UNIFAC recognized earlier (Fredenslund et al., 1977b),UNIFAC has a great deal of difficulty with the first member of a homologous series and the data bear this out. Note the systems nitromethane and nitroethane in isooctane. It is suggested that for highly polar functional groups (e.g., -OH, -CN, -NOz) that a separate group be defined for the smallest molecule with that group. This has already been done for methanol (Skjold-Jorgenson et al., 1979) and should be done for acetonitrile and nitromethane as well (Thomas, 1980). Still another problem lies in properly handling secondary or tertiary positions of functional groups. For example, the prediction of l-nitropropane in CCh is 17% high, while that for 2-nitropropane at a comparable temperature is 37% low. The data seem to indicate that the difference in polarity due to positioning of the functional group is significantly less than that predicted by UNIFAC. In this example, while the Q values (area parameters) of the two nitro compounds are about the same, the relative contributions of the polar and nonpolar parts are significantly different. This results in the secondary compound seeming much less polar than the primary compound, although the data indicate that they behave only slightly differently. Similar problems were seen when examining primary and tertiary chlorides in polar compounds (Thomas, 1980). There does not seem to be any simple and accurate way of alleviating this problem within the UNIFAC framework. Lastly, UNIFAC in its present form is not appropriate for polar aromatic molecules. For these molecules UNIFAC frequently cannot predict the limiting activity coefficients at both ends of the composition range, even if the parameters are obtained solely from the limiting activity coefficients. With the Wilson equation, one can always generate interaction parameters which can predict the limiting activity coefficients for a binary system, although there may be problems with numerical stability or multiple roots when solving for the parameters. With a groups approach, such as UNIFAC, this is not necessarily the case. For example, in the aniline-alkane system there are the following interactions: CH2- ACH, ACH - ACNH2, and ACNH, - CH2. Given UNIFAC's interaction parameters for the first two, it is not possible to obtain interaction parameters for the third which can simultaneously predict hexane in aniline and aniline in hexane. This is clearly demonstrated elsewhere (Thomas, 1980),but the obvious problems UNIFAC has with polar aromatics in Table VI should be evidence enough. One possible remedy is to define separate groups for all polar aromatics. This of course greatly limits the utility of UNIFAC as a groups approach. Another is to define a polar aromatic group which would provide the backbone for molecules such as aniline, phenol, and nitrobenzene. This could help somewhat but probably is not the final answer. In any event, UNIFAC's predictions for systems involving polar aromatics should be used with care. Despite the many problems with UNIFAC, it gave satisfadory predictions for most systems. One must consider that the parameters were obtained from almost solely finite concentration data. As noted by Zarkarian et al. (19791,

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984 207

h

VLE Dolo Puri (1974) Dolo from Originol UNIFAC Porometers Doto from New UNIFAC Porometers

Mole Froction Acetone

Figure 2. Comparison of the original and new UNIFAC parameters with experimental data for the acetone-cyclohexane system.

UNIFAC frequently encounters problems in the dilute region as it cannot duplicate the large change in slope of y over small concentration intervals seen in many highly nonideal systems. That the limiting activity coefficients were not predicted exceedingly well is thus not surprising. While the temperature dependence was not predicted accurately for most systems, activity coefficient predictions were generally best around 50-80 “C, temperatures more frequently encountered industrially than 20 OC-where many of the data were taken. For moderately polar-nonpolar systems, UNIFAC frequently gave slightly better results than the MOSCED model. For many associatedsolvated systems, UNIFAC generally had better predictions-especially for the chloroformalcohol systems. UNIFAC was also evaluated in terms of its ability to predict finite concentration data from group parameters obtained solely from 7”s. For moderately polar systems involving molecules of comparable sizes, the finite concentration data were predicted equally well using the original UNIFAC parameters and those obtained from the y”s. One such system is shown in Figure 2. Other systems are discussed elsewhere (Thomas, 1980). The highly polar mixtures are more interesting, but accurate data over the entire composition range are scarce. That both ends of the composition range were rarely predicted well for these systems preempts the attempt to compare UNIFAC‘s finite concentration predictions with the limited (and generally suspect) data available. It seems that if UNIFAC can predict the 7”’s well, at least for moderately polar or nonpolar mixtures without a large combinatorial contribution, then it can handle finite concentrations as well. Results for other systems might be worse, but at present it is difficult to estimate by how much. It has been suggested (Fredenslund, 1982) that if UNIFAC parameters were developed from ymdata only instead of including many data at finite concentration in evaluating parameters, UNIFAC would function better for predicting ymdata. We concur. We do feel that there is some advantage in uncoupling the 7-prediction from any forced composition dependence for Agethat is the composition dependence of the UNIQUAC equation may not optimum and the MOSCED method avoids any such constraint. But reevaluation of the UNIFAC parameters from ymdata would be valuable, though certainly beyond the scope of this work.

Summary and Conclusions A modified separation of cohesive energy density (MOSCED) model was used to correlate 3357 7”s of nonaqueous systems with an average error of 9.1% and very few unsatisfactory predictions. The modifications included

asymmetry parameters in the polar and hydrogen bonding terms to calculate better both ends of binary system, a slightly different function for the combinatorial contribution, and differing temperature dependences for the various types of interactions. Methods of calculating or estimating parameters from physical or group properties were also presented. The UNIFAC group contribution method of Fredenslund et al. (197713) was studied in terms of its ability to predict the available limiting activity coefficients. While reasonably quantitative predictions (20.5% average error) were generally obtained, some problems with UNIFAC became readily apparent. These problems included an unsatisfactory combinatorial term, the assumption of temperature independent parameters for all interactions, the lack of a defined cyclic group, unsatisfactory predictions for the first member of a homologous series, general inapplicability to molecules with functional groups in secondary or tertiary position, and inability to predict simultaneously both ends of the composition range for polar aromatic molecules. Possible improvements of UNIFAC were also discussed. The MOSCED model offers four advantages. The first and foremost is its quantitative ability. It seems to offer the best representation of nonionic, nonaqueous y mdata to date. Another advantage is its intuitive appeal. The parameters, if not calculated from physical properties, at least have some physical significance. This yields insight into the molecular forces within a system. Thus the parameter values themselves can offer significant screening information in selecting a solvent for an extractive distillation, azeotropic distillation, or extraction process. A third point is its simplicity. Using a hand calculator or a simple subroutine, one can calculate the 7”s for many systems. The Supplementary Material tabulates the temperature dependent variables over a 180 K temperature range which can make hand calculation even easier. Finally, the model needs relatively little data to generate parameters for a new compound. Occasionally, no data are necessary as the groups approach is applicable. If not, one can often measure the 7”s of a few representative solutes in that compound as a solvent on a chromatographic column (e.g., Thomas et al. 1980a). The representative solutes would include a nonpolar, an aromatic, a polar, a basic, an acidic, and an associated compound. The parameters capable of representing these six points should work for other data as well. When studying solution behavior in the past, scientists and engineers frequently embranced solubility parameter approaches because of their simplicity and intuitive appeal, but they often shunned them because of their qualitative predictive abilities. The more quantitative MOSCED model mitigates the primary problem in using multicomponent solubility parameters and hence should find many useful applications. Acknowledgment The authors gratefully acknowledge the financial support of the Phillips Petroleum Company. Gratitude is expressed to B. Banapour, D. Husa, K. Cox, and P. Cunningham for their help in the computations. Appendix Equations for UNIFAC (Fredenslund et al., 1975). UNIFAC considers the natural logarithm of the activity coefficient as the sum of a combinatorial part due to size and shape differences and a residual part due to energy differences. Thus for a molecule i in any solution In y i = In y? + In yiR (A)

208

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 2, 1984

where superscript C denotes combinatorial and R is residual. The combinatorial activity coefficients are calculated using Staverman's potential in the same manner as Abrams and Prausnitz (1975) did in the UNIQUAC model. The necessary equations are as follows.

The UNIFAC model also calculates the residual part from the UNIQUAC method. The rK's used in the above equation are obtained as follows m

m

n

In r K = Q K [ ~- In (Eom$mK) - C ( e m I t ~ m / C O n $ n m ) I (K) where m, n = 1, ..., N (all groups) Qmxm

om = CQnXn

z = 10 (coordination number)

where (C) is molecular surface fraction and (D)is molecular volume fraction j = 1, ..., m (number of components)

where (L) is the group surface fraction and (M) is the group fraction j = 1, ..., m Itnm

= exp(-anm/T)

anm= temperature independent group

interaction parameter (N)

where (E) is van der Waals volume and (F) is van der Waals surface area Q) = number of groups of type K in molecule i

K = 1, ..., N (number of groups in molecule i) where RK and QKare the group volumes and surface areas obtained from atomic and molecular structure data found in Bondi (1968). Kikic et al. (1980) modified this expression to correct for the overly large combinatorial contribution. For the term in square brackets in eq B, 4i was replaced by &* where r?f3xi

di* =

-

Crj2f3xj

(GI

They claimed that when obtaining UNIFAC parameters from 7"'s involving molecules of greatly differing sizes, this expression yielded superior results. This work suggests first to raise the exponent of ri from 0.66 to 0.75 and then drop the other terms in the equations, i.e., those preceded by -'12zqi. The resulting equations would be In

Y;C

4i

= In - - 1

4i +-

(H)

The residual part is taken to be equal to the sum of the individual contributions of each solute group less the s u m of the individual contributions in the pure-component environment. Thus N

In

T ~ R=

C @)[In rK- In r p ]

k=l

(J)

where N = number of different groups in the mixture, rK = residual activity coefficient of group K in a solution, r K S = residual activity coefficient of group K in a reference solution containing only molecules of type i, and u K ( ~ )= number of groups of type K in molecule i. This representation of the residual activity coefficient is common to almost all solution-of-groups approaches.

Equation I also holds for r#. The group interaction parameters, a,,,,,, represent the difference in interaction energy between a group n and a group m and two groups m. Note that anm# a,,,,, and that they are assumed temperature independent. The group interaction parameters were obtained by reduction of VLE data from over 2500 isotherms and isobars. Nomenclature a,ACID = acidity parameter in MOSCED model p, BASE = basicity parameter in MOSCED model CEDI, C, C11 = cohesive energy density, cal/cm3 C,, C,, C, = constants used in calculating 7,a,or j3 by groups approach no. C=C = number of carbon-carbon double bonds in a molecule no. C, = number of carbon atoms (outside of those in a functional group) in a molecule aa = interaction term in expression for d12 dlz = Flory-Huggins type term in account for mixing of molecules of different sizes in MOSCED 6 = square root of cohesive energy density hEv = energy of vaporization, cal/mol f(nD)= Lorentz-Lorenz function of refractive index used in calculating polarizability yw = infinite dilution activity coefficient yc = combinatorial activity coefficient for UNIFAC model yR = residual activity coefficient for UNIFAC model r = residual activity coefficient of group K in a solution g#= excess Gibbs energy (intensive) GE = excess Gibbs energy (extensive) AHv = enthalpy of vaporization ID = identification number: 1 = alkane, 2 = naphthene, 3 = aromatic h = dispersion parameter in MOSCED ( ~ a l / c m ~ ) ' / ~ LLE = liquid-liquid equilibrium MOSCED = Modified Separation of Cohesive Energy Density (model) N.A. = UNIFAC parameters not available nD = refractive index v k = number of groups of a given type in a molecule for UNIFAC model 4 = volume fraction I) = polar asymmetry parameter in MOSCED I)nm = energy parameter in UNIFAC model q = induction parameter in MOSCED model qK, QQ, Q = van der Waals molecular surface area in UNIFAC model QK = group surface area in UNIFAC R = gas constant

Ind. Eng. Chem.

REF = reference rK, R!, 3 = van der Waals molecular volume in UNIFAC model R K = group volume for UNIFAC SA,SB = possible steric factors in MOSCED u = induction parameter in one form of SCED model t = 293/T T = Temperature K or "C T (subscript) = at temperature of the system AT = temperature difference 7 = polar parameter in MOSCED model Bi = group surface area fraction for UNIFAC Bi = molecular surface area fraction for UNIFAC u, V, VI = molar volume, cm3/mol VLE = vapor-liquid equilibrium x. = liquid mole fraction 5 = hydrogen bonding asymmetry factor for MOSCED z = coordination number (= 10) for UNIFAC Literature Cited Abrams. D. S.; Prausnb, J. M. A I C M J . 1975, 27, 116. Alessl, P.; Klklc, 1.; Fredenslund, A.; Rasmussen, P. 2nd World Congress of Chemlcal Englneerlng, Montreal, Oct 4-9, 1961. Alessi, P.; Klkic. I.; Tonlano, G. J . Chromatogr. 1975, 706, 17. Anderson, R. Ph.D. Thesis, University of Callfornla, Berkeley, CA, 1961. Anderson, T. F.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1976, 77, 552. Arkel. V. A. E. Trans. Faradey SOC. 1846, 7428, 61. Arkel, V. A. E.; Vlex, S. E. Red. Trav. Chem. 1936, 55, 407. Bagley, E. B.; Nelson, T. P.; Barlow, J. W.; Chen, S.-A. Ind. Eng. Chem. Fundam. 1871, 9 . 93. Bagley, E. 0.; Nelson, T. P.; Scigllano, J. M. J . Paht Techno/. 1971, 43(555), 35. Barker, P. E.; Hllmi, A. K. J . Gas Chromatogr. 1887, 5 , 119. Barton, A. F. M. Chem. Rev. 1875, 75(6), 731. Blanks, R. F.; Prausnb, J. M. Ind. Eng. Chem. Fundam. 1964, 3 , 1. Bondl. A. "Physical Properties of Molecular Crystals. Liquids. and Glasses"; Wiley: New York. 1966. Bondl, A.; Slmkln, D. J. J . Chem. fhys. 1856, 25, 1073. Byer, S. M.; Gibbs. R. E.; van Ness, H. C. A I C M J . 1973. 79, 245. Comanlta, V. J.; Greenkorn, R. A.; Chao, K. J . Chem. Eng. Data 1976, 27(4), 491. Deal, C. H.; Derr, E. L. Ind. Eng. Chem. Frocess Des. Dev. 1964, 3 , 394. Derr, E. L.; Deai, C. H. Inst. Chem. Eng. Symp. Ser. London 1968, 3(32), 40. Donohue. M. D.; Prausnltz. J. M. Can. J . Chem. 1975, 53, 1586. Eckert, C. A.; Hsleh, C. K.; McCabe, J. R. AIChE J . 1874, 20(1), 20. Eckert, C. A.; Newman, B. A.; Nlcolaldes, 0. L.; Long, T. C. AIChE J . 1961, 727. 33. Eckert, C. A.; Prausnltz, J. M.; Otye, R. V.; O'Connell, J. P. AIChE-lnd. Chem. Eng. Symp. Ser. 1865, 7 , 75. Eckert, C. A.; Thomas, E. R.; Johnston, K. P. Proceedings of the 2nd International Conference on Phase Equlllbrla and Fluid Properties In the Process Industries, Berlin, Germany, 11, 399, 1980. Flory, P. J. J . Chem. M y s . 1841, 9 , 660. Fredenslund, A. Personal communlcatbn, 1982. Fredenslund, A.; Gmehling, J.; Mlchelsen, M. L.; Rasmussen, P.; Prausnltz, J. M. Ind. . Eng. " 2 ( frocessDes. Dev. 1877a. 76, 450. Fredenslund, A.; Gmehllng, J.; Rasmussen, P. "Vapor-LiquM Equilibria Using UNIFAC"; Elsevier: New York, 1977b. Fredenslund, A.; Jones, R. L.; Prausnltz, J. M. AIChE J . 1875, 2 7 , 1086. Gatreaux, M. F.; Coates, J. AI&€ J . 1855, 7 , 496. Gmehllng, J.; Onken, U. "Vapor-Liquld Equilibrium Data Collection"; DECHEMA: Frankfurt, West Germany, 1977. Gordon, J. L. J . faint Technol. 1966, 38, 43. Guggenheim. E. A. "Mixtures"; Clarendon Press: Oxford, 1952. Hansen, C. M. J . falnt Technol. 1867, 39, 505. Harris. H. G.; Prausnltz, J. M. J . Chromat. Sci. 1868, 7 , 685. Helpinstill, J. G.; van Winkle, M. Ind. Eng. Chem. Process Des. Dev. 1866, 7 , 213. Hildebrand, J. H. J . Chem. fhys. 1847, 75, 225.

Process Des. Dev., Vol. 23, No. 2, 1984 209

Hildebrand, J. H.; Wood, S. E. J . Chem. fhys. 1933, 7 , 817. Hsieh, C. K. R. Ph.D. Thesis, Unlverslty of Illinois, Urbana, IL, 1973. Hugglns. M. L. J . Chem. fhys. 1941, 9 , 440. Karger, E. L.; Snyder, L. R.; Eon, C. J . Chromatogr. 1976, 725, 71. Karger, B. L.; Snyder, L. R.; Eon, C. Anal. Chem. 1876, 50(14), 2126. Keller, R. A.; Karger, B. L.; Snyder, L. R. G a s . Chromatogr., Roc. Int. Symp. (Eur.) 1971, 8 , 125. Kiklc, I.; Aliesl, P.; Rasmussen, P.; Fredenslund, A. Can. J . Chem. Eng. 1980, 58, 253. Koenhen, D. M.; Smolders, C. A. J . Appl. folym. Sc;. 1975, 79, 1163. Kojima, K.; Tochigi, K. "Predictions of Vapor-Liquid Equllibrla by the ASOG Method"; Elsevier Publishlng Co.: New York, 1979. Kwantes, A.; RiJnders. G. W. A. "Gas Chromatography"; Buttersworth: London, 1958. and references cited therein. Littlewood, A. 6. Anal. Chem. 1864, 36, 1441. Locke, D. C. Adv. Chromatogr. 1868, 8 , 47. Long, J.; Munseon. B. J . Am. Chem. SOC. 1973. 95, 2427. Marsh, K. N.; French, H. T.; Rogers, H. P. J . Chem. Thermodyn. 1979, 7 7 , 897. Martire, D. E.; Pollara, L. 2. J . Chem. Eng. Data 1965, 70, 40. Meyer, E. F.; Ross, R. A. J . fhys. Chem. 1971, 75, 831. Meyer, E. F.; Wagner, R. E. J . fhys. Chem. 1966, 70, 3162. Meyer, E. F.; Renner, T. A.; Stec, K. S. J . fhys. Chem. 1971, 75, 642. Monfort, J. P.; Vidal, J.; Renon, H. J . Chlm. fhys. 1970, 67, 746. Moore, W. "Physical Chemistry"; PrenticaHall: Englewood Cliffs, NJ, 1972. Naga, J.; Ishll, N. J . Soc. Chem. Ind. Jpn. 1935, 38, 66. Nakanishi, K.; Shirai, H.; Minahiyama, T. J . Chem. Eng. Data, 1967, 72, 591. Nelson, R. C.; Hemwail, R. W.; Edwards, G. D. J . faint Techno/. 1970, 42, 636. Newman, B. A. Ph.D. Thesis, Unlverslty of Illinois, Urbana, IL, 1977. Null, H. R.; Palmer, D. A. Chem. €ng. Rogr. 1969, 65(9), 47. Olshi, T.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1976, 17, 333. Pierottl, G. J.; Deal, C. H.; Derr, E. L. Ind. Eng. Chem. Fundam. 1959, 5 7 , 95, and references cited therein. Prausnb, J. M.; Eckert, C. A.; Orye, R. V.; O'Connell. J. P. "Computer Caiculatlons for Multicomponent Vapor-Liquld Equilibria"; Prentice-Hall: Englewood Cllffs, NJ, 1965. Rohrschneider, L. Anal. Chem. 1873, 45, 1241. Sayegh, S. G.; Vera, J. H. Chem. Eng. J . 1960, 79, 1. Scatchard, G. Chem. Rev. 1831, 8 , 321. Schrelber, L. B.; Eckert, C. A. Ind. Eng. Chem. Process Des. Dev. 1871, 70, 5 12, and references clted therein. SkjoldJorgensen, S.; Koibe, B.; Gmehling, J.; Rasmussen, P.; Prausnitz, J. M. Ind. Eng. Chem. frocess Des. Dev. 1979, 18, 450. Snyder, P. S.;Thomas, J. F. J . Chem. Eng. Data 1866, 73,527. Staverman, A. J. Rec. Trav. Chlm. fays-Bas 1950, 69, 163. Tasslos, D. AIChE J . 1971, 77(6), 1367. Thomas, E. R. Ph.D. Thesis, University of Illinois, Urbana, IL, 1980. Thomas, E. R.; Newman, B. A.; Long, T. C.; Wood, D. A. J . Chem. Eng. Data 19828, 2 7 , 399. Thomas, E. R.; Newman, B. A.; Nicoialdes, G. L.; Eckert, C. A. J . Chem. Eng. Data 1882b, 27, 233. Tijssen, R.; Elllet, H. A. H.; Schoenmakers, P. J. J . Chromatogr. 1976, 722. 185. Tochlgi, K.; Kojima, K. J . Chem. Eng. Jpn. 1878, 9 , 267. Welmer, R. F.; Prausnltz, J. M. Hydrocarbon Process. Petrol. Refiner. 1965, 44, 237. Wilson, G. M. AIChE Symp. Ser. 1874, 740(70), 120. Wong. K. F.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1971, 70, 20. Zarkarlan, J.; Anderson, F. E.; Boyd, J. A. Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1979, 78, 657.

Received f o r review September 8, 1981 Accepted April 13, 1983 Supplementary Material Available: The supplementary material includes the 3357 y"s and references along with the predictions of the MOSCED model and UNIFAC. Also tabulated are the values of all temperature dependent variables ( T , a,p, +, 5, aa) from 0 to 180 OC at 20° intervals (74 pages). Ordering information is given on any current masthead page.