536
Ind. Eng. Chem. Process Des. Dev. 1902, 21 536-537 I
CORRESPONDENCE External-Degrees-of-Freedom Parameter In the UNIFAC-FV Model Sir: Recently, Oishi and Prausnitz (1978)have proposed a modification to the UNIFAC group-contribution model to allow estimation of solvent activities in polymer solutions. Their new model (UNIFAC-FV) requires the specification of two parameters for the computation of the additional free-volume term. The first parameter, b, which arises from the evaluation of the reduced volume is set equal to 1.28. The second parameter, 3cl, describes the total number of external degrees of freedom per solvent molecule. As pointed out by Beret and Prausnitz (19751, c1 = 1.0 for argon-like molecules; for all other molecules c1 > 1. Values for the number of external degrees of freedom for several fluids were obtained by Beret and Prausnitz from PVT and vapor pressure data. These values vary between 1.0 for small molecules (hydrogen, methane, etc.) to 4.52 for 9-(2-~henylethyl)-heptadecane. In their work on polymer solutions, Oishi and Prausnitz have found that good agreement is obtained with c1 = 1.1 for all the solvents considered there. For solvents with large molecules they suggest a large value for cl. Some of the solvents used in Oishi and Prausnitz's work are tabulated in Table I along with c1 values as published by Beret and Prausnitz. Arai and Iwai (1980)reported the application of the UNIFAC-FV model for the estimation of the weightfraction Henry's constant. Good agreement was obtained for a series of aromatic and paraffinic hydrocarbons in polystyrene with c1 = 2.0. The UNIFAC-FV model was also employed by Gottlieb and Herskowitz (1981)to obtain an estimate of the reduced chemical potential parameter x, for poly(dimethylsiloxane) (PDMS) solutions. The results for heptane-PDMS and for the aromatic hydrocarbons-PDMS solutions are depicted in Figure l. For the paraffinic solvent, the best fit of the data is obtained when no free-volume correction is applied (i.e., c1 = 0.0). For benzene, c1 = 5.0 is required for good data fit and a value of c1 = 7.0 is used for toluene. The value of x for the p-xylene-PDMS solution is not sensitive to the choice of cl, for 0.0 < c1 < 10.0. A summary of these results is presented in Table I. The main advantage of the quasiempirical UNIFAC method over the variety of more rigorous models available in the literature is in its ability to estimate quite accurately solution and phase equilibrium properties without the need for solution data and without resorting to the use of any adjustable parameters. Inspection of Table I reveals that this advantage does not hold for the free-volume term as propmed by Oishi and Prausnitz. The c1 values obtained from best fit of the data differ for the same solvent with the estimated property (e.g., benzene). For the same estimated property, values do not correlate with molecular size or complexity (e.g., c1 for dodecane is the same as for benzene in Arai and Iwai's work but differ considerably according to Beret and Prausnitz). Furthermore, some of the values are entirely unacceptable if the original physical 0196-4305/82/1121-0536$01.25/0
Table I. Value of c, Obtained from Pur Solvent and Polymer Solution Data
solvent
Oishi Beret and and PrausF'rausnitz nitz
argon cyclohexane benzene toluene xylene
1.00 1.66
1.1
1.54
1.1 1.1
propane pentane hexane heptane octane dodecane
1.44
Arai and Iwai
Gottlieb and Herskowitz
2.0 2.0 2.0
5.0 7.0 0.0-
10.0 1.91 2.14
0.6
1.1 1.1
0.0 0.0
2.0 2.0
2.58
x
00
05
10
POLYMER SEGMENT FRACTION
Figure 1. Comparison between experimental data and prediction of the UNIFAC-FV model with different c1 values: (a) heptanePDMS (20 "C); (b) benzenePDMS (25 "C); (c) toluene-PDMS (20 "C); (d) p-xylene-PDMS (25 "C).
interpretation of the c1 parameter is to be retained (e.g., c1 = 0.0 for heptane). Hence, one has to draw the conclusion that c1 should be 0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev.
treated as an adjustable parameter which depends on both and the particular desired property the (solvent activity, Henry’s constant, x , etc.). It is also important to realize that c1 values cannot be estimated from values of other solvents with similar chemical structure or size. Literature Cited Arai, Y.; Iwal, Y. Ind. Eng. Chem. Process D e s . Dev. 1980, 19, 508.
1982,21, 537
537
Beret, S.; Prausnitz, J. M. AIChE J. 1975, 21, 1123. Gottlieb, M.; Herskowitz, M. Macromolecules 1981, 14, 1468. Olshi, T.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 333.
Chemical Engineering Department Moshe Gottlieb* Ben Gurion University Mordechai Herskowitz Beer Sheva 84120, Israel
Response to “External-Degrees-of-Freedom Parameter In the UNIFAC-FV Model” Sir: Gottlieb and Herskowitz have pointed out what is all-too-often forgotten: the UNIFAC model provides no more than a rough approximation which should only be used to estimate thermodynamic properties when, as unfortunately happens so often, no experimental data at all are available for a particular liquid mixture. For liquid mixtures, Raoult’s law provides a zeroth approximation. UNIFAC provides a first approximation. But even that may have appreciable error. What is remarkable about UNIFAC is not that it sometimes gives poor results. What is remarkable is that so often it does surprisingly well. Reliable vapor-liquid equilibrium data are rare for polymer solutions. Oishi observed that when UNIFAC was applied to such data, using parameters obtained from ordinary liquid-mixture data, calculated solvent activities were consistently low. Since UNIFAC is based on a lattice-type theory of liquid solutions (UNIQUAC),and since Flory and Patterson have shown that for polymer solutions a lattice-type theory must be corrected for appreciable free-volume effects, it appeared reasonable to add to the UNIFAC equation a free-volume contribution based on Flory’s equation of state. Upon doing so, Oishi found that he could significantly reduce the consistent deviation between observed solvent activities and those calculated by (uncorrected) UNIFAC. Upon applying Oishi’s method to experimental data for solutions of poly(dimethylsiloxane), Gottlieb and Herskowitz obtain some appreciable deviations; they propose to remove these deviations by assigning some unrealistic values to the external-degrees-of-freedom parameter cl. This is an unsatisfactory procedure. A possible explanation for the observed deviations may be related to the well-known fact that poly(dimethy1siloxanes) are much more flexible than the polymers considered by Oishi; as shown by Lichtenthaler and Liu (1974), the configurational contribution to the Gibbs energy of poly(dimethylsi1oxane) solutions appears to be somewhat different from that calculated by Staverman’s formula, which is used in UNIFAC. The inadequacy of Staverman’s formula may therefore be responsible, in part, for the results reported by Gottlieb and Herskowitz.
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However, a more likely explanation follows from uncertainties in the residual UNIFAC group-energy parameters. Calculations are often sensitive to UNIFAC energy parameters. These parameters have been revised several times as new experimental data have become available for ordinary liquid mixtures. Since reliable data are rare for mixtures containing hydrocarbons and low-molecular dimethylsiloxanes, residual UNIFAC group-energy parameters are not known well for the group-group interactions required for UNIFAC calculations as performed by Gottlieb and Herskowitz. It is relevant to note that such parameters have not been reported by Fredenslund and co-workers (Lyngby) nor by Gmehling and co-workers (Dortmund), the two research teams that have concerned themselves most extensively with data reduction to obtain UNIFAC parameters. Indeed, it is not clear what UNIFAC parameters were used by Gottlieb and Herskowitz, nor how they were obtained. It is likely that if the data for poly(dimethylsi1oxane) solutions are examined in toto, and if relatively small adjustments are made in the (in any case uncertain) UNIFAC energy parameters, the calculated and experimental results could be brought into much better agreement. Since publication of the original UNIFAC work (Fredenslund, Jones, and Prausnitz, 1975), numerous articles have suggested modifications, and new tables of parameters have been proposed. These articles underline the approximate nature of UNIFAC which is based on essentially arbitrary choices, concerning what shall be defined as a group and concerning what shall be used as the size of a unit cell. The arbitrariness of these choices, as well as uncertainties in the parameters, can be reduced only through global comparison with extensive and reliable experimental data. Literature Cited Fredenslund, Aa.; Jones, R. L.; Prausnitz, J. M. AIChE J. 1975, 21, 1086. Lichtenthaler, R. N.; Llu, D. D.; Prausnitz, J. M. Ber. Bunsenges. W y s . Chem. 1974, 78, 470.
Department of Chemical Engineering University of California, Berkeley Berkeley, California 94720
0 1982 American
Chemical Society
J. M. Prausnitz
