Ind. Eng. Chem. Res. 1989,28, 324-328
324
Person, W. B. A Criteria for Reliability of Formation Constants of 1965, 87, 167. Weak Complexes. J. Am. Chem. SOC. Renon, H.; Prausnitz, J. M. On the Thermodynamics of AlcoholHydrocarbon Solutions. Chem. Eng. Sci. 1967, 22, 299. Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE. J. 1968,14, 135. Saunders, M.; Hyne, J. B. Trimer Association of &Butanol by NMR. J. Chem. Phys. 1958a, 29, 253. Saunders, M.; Hyne, J. B. Study of Hydrogen Bonding in Systems of Hydroxylic Compounds in Carbon Tetrachloride through the Use of NMR. J . Chem. Phys. 1958b, 29, 1319. Smirnova, N. A.; Kurtynina, L. M. Thermodynamic Functions of Mixing for a Number of Binary Alcohol-Hydrocarbon Solutions. Zh. Fiz. Khim. 1969, 43, 1883. Thomas, E. R.; Eckert, C. A. Prediction of Limiting Activity Coefficients by a Modified Separation of Cohesive Energy Density Model and UNIFAC. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 194. Trampe, David M. Measurement and Applications of Limiting Activity Coefficients and Liquid-Liquid Equilibria. M.S. Thesis, University of Illinois, Urbana, 1987a. Trampe, David M. University of Illinois, personal communication, 1987b. Tucker, E. E.; Becker, E. D. Alcohol Association Studies. 11. Vapor Pressure, 220-MHz Proton Magnetic Resonance, and Infrared Investigations of tert-Butyl Alcohol Association in Hexadecane. J . Phys. Chem. 1973, 77, 1783.
Van Geet, A. L. Calibration of the Methanol and Glycol Nuclear Magnetic Resonance Thermometers with a Static Thermistor Probe. Anal. Chem. 1968,40,40. Van Geet, A. L. Calibration of Methanol Nuclear Magnetic Resonance Thermometer at Low Temperature. Anal. Chem. 1970,42, 679. Varian Associates XL-Series NMR Superconducting Spectrometer Systems Basic Operation Manual, 1984. Vesely, Frantisek; Uchytil, P.; Zabransky, M.; Pick, J. Heats of Mixing of Cyclohexane with 1-Propanol and 2-Propanol. Collect. Czech. Chem. Commun. 1979,44, 2869-2881. Vonka, P.; Svoboda, V.; Strubl, K.; Holub, R. Liquid-Vapor Equilibrium. System Cyclohexane-1-Butanol a t 50 "C and 70 "C. Collect. Czech. Chem. Commun. 1971, 36, 18. Weltner, William; Pitzer, Kenneth Methyl Alcohol: The Entropy, Heat Capacity and Polymerization Equilibria in the Vapor, and 1951, Potential Barrier to Internal Rotation. J. Am. Chem. SOC. 73, 2606. Wilson, G. M. Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. SOC.1964, 86, 127. Zong, Z.; Yang, X.; Zheng, X. Correlation of Vapor-Liquid Equilibria of Associated Solutions. J . Chem. Eng. Jpn. 1984, 17, 71.
Received for review May 24, 1988 Accepted September 26, 1988
A Simple Method for Evaluating the Wilson Constants Alexander Apelblat and Jaime Wisniak* Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel
A simple method is proposed to determine the Wilson constants from binary vapor-liquid equilibrium data. The method is based on a polynomial description of the variation of G E/RTwith composition and finding the maximum value of the curve. The constants are determined by using a hand scientific calculator or a personal computer. Prediction of vapor compositions is as good or better than that obtained by complex optimization techniques. Many equations have been proposed to describe the vapor-liquid equilibrium relationships. For homogeneous systems, the two-parameter Wilson equation (Wilson, 1964) has been shown to be very suitable. In a multicomponent system, it reads
derived from the pertinent binary systems. For a binary system, the activity coefficients are In y1 =
In
where
y1 =
UjL
exp[-(Xij - Xii)/RT]
Ai, =
(2)
ui
To a first approximation the energy terms (Xi,- Xii) are assumed to be independent of temperature, although it is claimed (Nagata and Yamada, 1973) that a better fit is obtained if a polynomial dependency is assumed: Xij
-
Xii
=u
+ bT + cT' + ...
(3)
Obviously application of eq 3 carries the penalty of a larger number of constants. The Wilson equations are attractive because they have a built-in effect of temperature, lacking in previous models, and permit the calculation of multicomponent systems from a combination of parameters A,, and AIi which are
* To whom correspondence
should be addressed.
088S-5885/89/2628-0324$01.50/0
Extensive tables are available (Hudson and Van Winkle, 1970; Hirata et al., 1976; Gmehling and Onken, 1977), reporting the values of either the energy parameters (A,, - A,,) or the constants A, and A,[. The most difficult problem in using Wilson's equations is how to determine the two parameters A12and Azl from a set of data ( y , , ~ , ) .Equations 4 and 5 are a pair of transcendental equations that can only be solved numerically. Czelej (1987) has developed a mathematical procedure to transform the Wilson equations into a polynomial form that allows an easier determination of the constants. Due to some serious mathematical errors present in Czelej's equations, their use is not recommended. Several methods have been suggested for determining the optimum constants AIz and Azl. According to Hirata et al. (1976), the results of the methods depend upon the 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 325 22
=
1 - A21 1 2 + A2lXl
(10)
allows us to transform eq 8 into The corresponding expressions for the derivatives of g E with respect to concentration, in terms of z1and z z and at constant pressure and temperature, are
d2gE/dx12 =
d3gE/dX13
21(x121-
2)
+ zz(x222 - 2)
212(3- 2X1Z1) - 22(3 - 2x222)
(13)
Equations 11-13 are the starting point in the process of evaluating the Wilson parameters A12 and Azl. Combination of 11 with eq 12 and taking into account that dgE/dxl = In (y1/y2)allows the possibility of isolating z1 and z z x1 In [l + x2zl] + x 2 In [l 0
+ xlz2] -gE(x1) = 0
(14)
where
XI
Figure 1. Variation of G E/RTwith composition: (A) the common case with one extreme value, (B) one extreme value and an inflection point, and (C)the very unusual case of two extreme values.
objective functions (O.F.) employed. Gmehling and Onken (1979) use either
For example,
0.F. = C(1 - Y ~ ~ ~ 5/ cyY ~ ~ ~ (6) J ~ or 0.F. = C(PCa1, - PexpJ2 5 cp
(7)
Hirata et al. (1976) compared four compoutational techniques for minimizing the O.F., nonlinear least-squares, steepest ascent, pattern search, and complex search techniques, and concluded that the steepest ascent algorithm is generally the best. Several authors (Silverman and Tassios, 1977; Verhoeye, 1970) have shown that values of the pair of constants A12 and Azl are not unique, that they may differ substantially one from the other, and that they still have the same confidence limits. The actual pair found will depend on the algorithm used and the starting values of the parameters. Whichever of the above methods is selected, it requires availability of a moderate-to-large computer. We will now show that simpler mathematical techniques can be used to give results as accurate as the more sophisticated ones.
Theory We present here an alternative method for evaluation of the Wilson parameter, based on the data from the central concentration region, where the highest accuracy of measurements can be achieved. The method is based on the experimental fact that the value of G E / R T is zero for the pure components and that the function G E / R T ( x l ) will have one or two extreme values (Figure 1). For binary mixtures, the Wilson model expresses the excess Gibbs energy of mixing, GE, as follows: G E / R T = -3c1 In (xl + A12x2)- x 2 In (Azlxl + x 2 ) (8) Denoting g E = G E / R Tand defining the parameters z1and 22
21
=
1 - A12 x1
+ '412%
(9)
If at any value of the composition, xl, the experimental value of gE(xl)or of the activity coefficients are known, then 14 becomes a transcendental equation, with z2 as the single variable, that can be easily solved by ordinary numerical methods. Substitution of z2 into eq 15 will yield the corresponding value of zl. Knowledge of z1and z2 will allow immediate evaluation of the Wilson parameters from eq 9 and 10 as follows: 1- XlZ,
As it can be seen, the proposed method is normally based on single-point data (preferably the extreme value of g E ) but clearly the calculations can be extended to a number of points to produce a set of A12and Azl pairs, which properly treated will give the best values of the parameters with regard to the chosen objective function. The experimental g E(xl) values in the central concentration region are normally of the highest accuracy and should be preferably used in the calculations. Location of the extreme in the experimental gE(xl)curve is helped by expanding the function as a polynomial of x1 as follows: n
gE(x1) = P(xJ = Cakxlk k=O
(17)
where ao, nl, u2,...,a,, are the numerical coefficients of the polynomial P ( x l ) . Subroutines for this purpose are available in hand scientific calculators and should not be a problem. Differentiating eq 17 with respect to x1 yields n-1
dP(xl)/dxl = C k w l k - ' = In (y1/y2) k=l
(18)
Thus, the values of gE(xl)and In (y1/y2)in eq 14 and 15 can be replaced by the polynomial P ( x l ) and its first derivative, dP(xl)/dxl, respectively. Such representation
326 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 Table I. Comparison of Proposed Method with Steepest Ascent Results O.P."
A12
A21
AYIb
AYzb
O.P."
Ethyl Acetate-Benzene (760 mmHg) 2 3 4 5
Hiratac
0.95938 1.16591 1.08445 1.29352 0.43886
0.95945 0.77127 0.83942 0.66945 1.73284
0.0078 0.0083 0.0083 0.0090 0.0062
0.0016 0.0015 0.0015 0.0016 0.0043
2 3 4 5
0.0028 0.0028 0.0030 0.0037 0.0029
2 3 4 5
0.0098 0.0037 0.0035 0.0041 0.0053
2 3 4
0.033 0.028 0.025 0.025 0.031
2 3 4
0.0031 0.0027 0.0027 0.0027 0.0026
2 3 4 5
0.0048 0.0032 0.0043 0.0052 0.0033
2 3 4 5
0.022 0.028 0.026 0.023 0.027
2 3 4 5
0.021 0.022 0.023 0.022 0.017
2 3 4
0.023 0.020 0.020 0.030 0.024
2 3 4 5
0.099 0.16 0.093 0.093 0.10
2 3 4 5
0.018 0.018 0.017 0.019 0.014
2 3 4 5
Hirata
Ethyl Acetate-Toluene (760 mmHg) 2 3 4 5
0.94213 0.94334 1.23697 1.51336 0.85981
Hirata
0.84137 0.84028 0.61601 0.43880 0.96565
0.011 0.011 0.010 0.012 0.0063
0.55328 0.34718 0.33962 0.36150 0.40080
Hirata
0.64365 0.90022 0.91730 0.88681 0.83437
0.015 0.011 0.011 0.010 0.010
Hirata
5 Hirata
0.35513 0.29742 0.31766 0.37047 0.27013
0.35784 0.42566 0.44094 0.38692 0.42855
0.016 0.016 0.015 0.014 0.018
5 Hirata
4 J
Hirata
0.77454 0.60516 0.62334 0.59198 0.39981
0.77492 0.96566 0.95905 0.9998 1.30443
0.014 0.015 0.012 0.013 0.020
5 Hirata
Hirata
0.62916 0.92868 1.03198 1.06858 0.97505
0.44161 0.19301 0.15147 0.12459 0.19305
0.025 0.012 0.0085 0.013 0.0041
Hirata
5 Hirata
0.20420 0.10458 0.10221 0.13725 0.12633
0.29941 0.41537 0.44514 0.39981 0.37663
0.011 0.022 0.024 0.014 0.016
Hirata
Hirata
0.94757 1.16440 1.34244 1.18277 0.24607
0.74320 0.56697 0.45804 0.56458 1.60470
0.018 0.016 0.014 0.015 0.032
Hirata
Hirata
0.29854 0.21333 0.24326 0.46022 0.15167
0.29635 0.39796 0.41888 0.20398 0.42420
0.035 0.050 0.043 0.016 0.068
5 Hirata
3
4 5
Hirata
0.19900 0.24797 0.21003 0.30055 0.24469
0.17916 0.10702 0.18044 0.05830 0.15930
0.011 0.0088 0.011 0.011 0.0093
Hirata
Hirata
0.55577 0.56133 0.58271 0.51313 0.73176
0.51108 0.50419 0.49283 0.56107 0.51768
0.021 0.022 0.021 0.018 0.017
0.40483 0.64666 0.74845 0.66909 0.59478
0.032 0.032 0.050 0.022 0.016
0.014 0.0028 0.0073 0.0057 0.0047
0.49887 0.49627 0.51017 0.56383 0.48084
0.47784 0.48004 0.48153 0.43048 0.48129
0.0088 0.0088 0.0087 0.0099 0.0096
0.0045 0.0044 0.0045 0.0061 0.0047
0.26993 0.08460 0.07346 0.08708 0.11014
0.24845 0.78953 0.83947 0.81800 0.86087
0.041 0.022 0.023 0.019 0.011
0.041 0.0078 0.0065 0.0062 0.0071
0.22968 0.12556 0.12556 0.26993 0.12012
0.22451 0.33922 0.33922 0.24845 0.36003
0.13 0.19 0.19 0.088 0.19
0.0099 0.0089 0.0089 0.0091 0.0086
0.12781 0.06858 0.06334 0.05302 0.08938
0.12301 0.19342 0.27025 0.28407 0.20175
0.038 0.034 0.031 0.032 0.031
0.049 0.027 0.0070 0.0069 0.022
0.80939 0.67684 0.68462 0.99284 0.96830
0.84744 0.99795 0.99530 0.68518 0.67459
0.017 0.017 0.016 0.016 0.021
0.0088 0.0079 0.0082 0.010 0.0095
2.61668 2.67133 2.53832 2.79618 2.59312
0.22597 0.20743 0.25029 0.16436 0.20302
0.0021 0.0025 0.0022 0.0041 0.0078
0.0059 0.0061 0.0055 0.0066 0.0048
0.68141 0.67986 0.68086 0.58400 0.74636
0.67308 0.67336 0.67802 0.78301 0.59282
0.0064 0.0064 0.0065 0.0096 0.0062
0.0023 0.0023 0.0024 0.0032 0.0030
Cyclohexane-2-Propanol (500 mmHg)
Hirata
Methanol-Ethyl Acetate (730 mmHg) 2 3 4 5
0.26711 0.05744 0.03225 0.09170 0.10855
Methanol-Methyl Ethyl Ketone (760 mmHg)
Ethylcyclohexane-2-Propanol (400 mmHg) 2
0.030 0.023 0.028 0.025 0.015
Chloroform-Benzene (760 mmHg)
Pentane-Acetone (760 mmHg) 2 3 4 5
0.025 0.023 0.019 0.022 0.021
Methyl Acetate-Benzene (760 mmHg)
Acetic Acid-Water (760 mmHg) 2 3 4 5
0.41113 0.46404 0.50521 0.39944 0.35937
Ethanol-Hexane (760 mmHg)
Methanol-Benzene (760 mmHg) 2 3 4
0.41114 0.35808 0.34763 0.45188 0.54736
Octane-p-Cresol (760 mmHg)
Benzene-Butanol (760 mmHg) 2 3 4 5
Ayzb
Allyl Alcohol-Water (760 mmHg)
Ethyl Acetate-p-Xylene (760 mmHg) 2 3
AYlb
Methyl Ethyl Ketone-Heptane (760 mmHg)
2-Propanol-Carbon Tetrachloride (760 mmHg) 2 3 4
A21
2-Propanol-Water (95 mmHg)
Acetone-Carbon Tetrachloride (760 mmHg) 2 3 4 5
A12
Methanol-Diisopropyl Ether (730 mmHg)
0.28469 0.39837 0.43370 0.38811 0.32305
0.22970 0.11871 0.10554 0.14392 0.14811
0.032 0.023 0.019 0.016 0.032
0.014 0.022 0.024 0.0051 0.018
2-Propanol-Methylcyclohexane (760 mmHg)
Hirata
0.26353 0.15918 0.14758 0.14363 0.09067
0.27747 0.39504 0.44341 0.44776 0.43978
0.024 0.014 0.016 0.020 0.020
0.016 0.0066 0.0050 0.019 0.013
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 327 Table I (Continued) O.P."
'412
2 3 4 5 Hirata
'421
Pyridine-Tetrachloroethylene 0.44051 0.92151 0.54962 0.72208 0.56379 0.72047 0.59094 0.69128 0.66066 0.56980
AYlb (60 "C) 0.011 0.0075 0.0064 0.0061 0.012
AYZb
O.P."
'412
AY?
'421
AYZb
Hexane-1.1.1-Trichloroethane (60 "C) 0.0039 0.0054 0.0050 0.0056 0.0096
2 3 4 5 Hirata
0.84035 0.71295 0.71438 0.71463 0.61607
0.015 0.0013 0.0013 0.0013 0.0024
0.034 0.0030 0.0031 0.0031 0.0023
'
1.0000 Loo00 1.0000 1.0000 1.0950
2 3 4 5 Hirata
Benzene-Ethanol (25 "C) 0.31130 0.23214 0.026 0.44180 0.11134 0.0066 0.48281 0.09634 0.0047 0.45715 0.11735 0.0040 0.46589 0.11025 0.0034
0.013 0.0024 0.0023 0.0020 0.0018
2 3 4 5 Hirata
Hexane-2-Butanone (60 "C) 0.50355 0.47397 0.0063 0.53748 0.43928 0.0042 0.55323 0.43491 0.0042 0.53500 0.45175 0.0045 0.64871 0.36797 0.0057
0.0073 0.0062 0.0055 0.0059 0.0041
2 3 4 5 Hirata
Octane-Dioxane (80 "C) 0.50199 0.53880 0.011 0.30906 0.7 6373 0.0064 0.28419 0.81372 0.0073 0.33628 0.74199 0.0070 0.29778 0.72565 0.0071
0.015 0.0043 0.0027 0.0041 0.011
2 3 4 5 Hirata
Hexane-2-Butanol (60 "C) 0.41780 0.30432 0.030 0.53298 0.19927 0.011 0.67772 0.14278 0.018 0.67697 0.14301 0.018 0.47709 0.20058 0.025
0.0075 0.0051 0.0076 0.0076 0.0052
2 3 4 5 Hirata
Nonane-Dioxane (80 "C) 0.48403 0.54656 0.0074 0.21271 0.90980 0.0079 0.21118 0.91371 0.0079 0.38011 0.64364 0.0072 0.29778 0.72565 0.0067
0.040 0.015 0.015 0.034 0.032
2 3 4 5 Hirata
2-Propanol-Heptane (60 "C) 0.23802 0.24705 0.022 0.14373 0.35807 0.011 0.14600 0.40647 0.012 0.18953 0.35357 0.014 0.16554 0.26335 0.020
0.014 0.0069 0.0077 0.0075 0.015
2 3 4 5 Hirata
Diiaopropylamine-Water (10 "C) 0.23132 0.27149 0.051 0.09226 0.44798 0.033 0.05969 0.56110 0.032 0.11822 0.47629 0.030 0.10306 0.42809 0.034
0.059 0.054 0.053 0.054 0.054
"O.P. = order of polynomial. bAyi = Elyi - yealcl/n.cHirata et al. (1976).
of experimental data [eq 17 and 181 has obvious advantage in computer calculations. As shown in Figure 1, gE(xl)has always a t least one extreme value (maximum or minimum), its location x1 = xl* and the corresponding extreme value gE(xl*)= P(xl*) can be calculated from eq 18 by taking into account that at this point the derivative @(xl)/dxl is nil. In this case, eq 14 and 15 become xl* In [ l + x2*zl] + x2* In [ l + x1*z2] - P(xl*) = 0 (19) and
a volume fraction term to the excess Gibbs energy of mixing: G E / R T = GE/RT(Wilson) + x1 In ( x , rx,) x 2 In ( x l / r x,) (22)
+
+
+
where r denotes the ratio of the molar volumes, uZL/ulL, of the pure components and the G E/RT(Wilson) term is defined in eq 8. Equation 22 is often called the T-KWilson equation. The suggested algorithm can be used to determine the corresponding new pair of A12 and A,, parameters, if gE(xl)= GE/RT is replaced by the function Q(xl) = G E / R T - x1 In ( x , + rx,) - x 2 In ( x l / r x,) (23) where Q(xl) = x1 In (1 + xzzl) + x 2 In (1 + qz,) (24) Expressing Q(xl) in the polynomial form (17) permits calculation of the T-K-Wilson parameters from eq 19,20, and 16. Nagata et al. (1975) modified the Wilson equation by adding the Scatchard-Hildebrand term
+
- In (1 + x1*z2)]+ x2*z2 The use of eq 19 and 20 for the evaluation of A,, and A,, will be illustrated later for a number of binary systems. When the experimental data are good enough to permit representation of the second and third derivatives of gE(x,) in the polynomial form (17), evaluation of the Wilson parameters will reduce to solving the following algebraic equations: ~
1
+
GE/RT = GE/RT(Wilson)
- 221 ~ 1 ~ 2~ ~ 2 '22, - d2P(XJ/dX,2 = 0
+
3z12- 2xlzl3 - 3zZ2 2x,zZ3- d3P(xl)/dxl3= 0
(21)
Once again z1 and z2 can be isolated because the first equation in (21) is quadratic in z1 or z,. Evidently, this equation, if accurate, can replace the logarithmic relationship between z1 and z2 in eq 15 and 20. The Wilson equation in its original form has the significant disadvantage that it cannot represent liquid-liquid systems. A number of modifications have been proposed to overcome this disadvantage. The simplest one was introduced by Tsuboka and Katayama (1975), who added
+
(6, - 6,)2u1Lu,Lx1xz (X,VlL
+ X,V,L)
(25)
where a1 and 6, are the solubility parameters of pure components. Once again, the calculation procedure for eq 25 is the same as the one described for the T-K-Wilson equation, except that the auxiliary function is defined differently:
A more versatile treatment of partially miscible systems can be expected if the solubility parameters in eq 25 are
328 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
replaced by an adjustable constant b or by the strictly regular solution term, as suggested by Novak et al. (1974): G E / R T = GE/RT(Wilson) + b x 1 x 2
(27)
If one-point data are considered, then the three parameters A12, Azl, and b present in eq 25 and 27 can be evaluated from a knowledge of G E and its first and second derivatives. For the case where GE/RT is presented in the polynomial form (17), the corresponding equations for the case given in (27) become g E
agE/dxl
+ bx,x2
= P(xJ
+ b(xl - x,)
= w(xl)/dxl
d2gE/dx12- 2b = d2P(xl)/dx12
(28)
where g E and its derivatives are given by eq 11-13. The third parameter b can be easily removed from eq 28, but there is no possibility to isolate z1 and z2. They should be evaluated from the pair of transcendental equations (introduction of Q(xl) = gE/x1x2 + b where G E / R T = x1x2Q(x1)leads to more complex transcendental equations)
(29) by ordinary numerical methods. Quite clearly, the simplest form of these equations will occur when the composition is equimolar (xl = x2 = 0.5) and for the extreme value of P(xl*) where dP(xl)/dxl = 0. The case represented by eq 25 can be treated in a similar way. Results and Discussion A number of systems were selected from the Hirata (Hirata et al., 1976) collection, representing solutions with strong positive or negative deviations from ideal behavior, with or without azeotropes, a t isobaric or isothermal conditions. The values of G E/RT were calculated and then fitted with polynomials of different degrees (from 2 to 5). Equations 19 and 20 were solved and the Wilson constants were determined from relations 16. The computational procedure was done fast and efficiently in a Mac 512 computer using the Excel spreadsheet, the Statworks statistical package for the polynomial algorithm, and the Eureka program for the trial and error part. Overall computational for one complete cycle took a few minutes. A scientific hand calculator (HP 15C) was also used for comparison purposes. Results for the selected systems are presented in Table I and compared with the results of Hirata et al. Inspection of Table I leads to the following conclusions. 1. The accuracy of the suggested method (measured by the ability to predict the vapor composition) is normally as good or better than the complex search techniques. 2. A third or fourth degree polynomial will usually suffice.
3. The mathematical routine described here is very fast and avoids the need of a medium- or large-size computer. The required algorithms are normally available with personal and hand scientific calculators. 4. The procedure calls for data in the central part of the composition range, where the experimental analytical precision is the best. 5. For a given system, significantly different pairs of Wilson constants will have the same capability of predicting the composition of the gas phase. An additional important observation of the data reported in Table I is that the predictive ability for the vapor composition of component 2 is normally much higher than that for component 1. We attribute this to the fact that component 1 is usually the most volatile one and the pertinent Antoine equation is being extrapolated beyond its real range. Nomenclature Ai. = Wilson constant G k = excess Gibbs function R = gas universal constant r = ratio ulL/uZL T = absolute temperature, K u f = liquid molar volume of component i x i = mole fraction of component i in the liquid phase y i = mole fraction of component i in the vapor phase Greek S y m b o l s = solubility parameter of component i yi = activity coefficient of component i
Literature Cited Czelej, M. The Polynomial Form of the Wilson Equation for Binary and Ternary Systems. Int. Chem. Eng. 1987,27, 535-538. Hirata, M.; Ohe, S.; Nagahama, K. Computer Aided Data Book of Vapor-Liquid Equilibrium. Kodansha-Elsevier: Tokyo, 1976. Hudson, J. W.; Van Winkle, M. Multicomponent Vapor-Liquid Equilibriums in Miscible Systems from Binary Parameters. Znd. Eng. Chem. Process Des. Dev. 1970, 9,466-472. Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: New York, 1977; Vol 1. Nagata, I.; Yamada, T. Parameter-Seeking Methods of Local Compositions Equations. J . Chem. Eng. Jpn. 1973, 6, 215-219. Nagata, I.; Nagashima, M.; Ogura, M. A Comment on an Extended Form of the Wilson Equation to Correlation of Partially Miscible Systems. J . Chem. Eng. Jpn. 1975, 8, 406-408. Novak, J. P.; Vonka, P.; Suska, J.; Matous, J.; Pick, J. Applicability of the Three-Constant Wilson Equation to Correlation of Strongly Nonideal Systems. 11. Collect. Czech. Chem. Commun. 1974,39, 3593-3598. Silverman, N.; Tassios, D. The Number of Roots in the Wilson Equation in the Correlation of Vapor-Liquid Equilibrium. Znd. Eng. Chem. Process Des. Dev. 1977,16, 13-20. Tsuboka, T.; Katayama, T. Modified Wilson Equation for VaporLiquid and Liquid-Liquid Equilibria. J . Chem. Eng. Jpn. 1975, 8, 181-187. Verhoeye, L. A. J. Remarks on the Determination of the Wilson Constants in the Correlation of Vapor-Liquid Equilibrium Data. Chem. Eng. Sci. 1970,25, 1903-1908. Wilson, G. H. Vapor-Liquid Equilibrium. IX: A New Expression For the Excess Free Energy of Mixing. J . Am. Chem. SOC.1964, 86, 127-130. Received f o r review June 10, 1988 Revised manuscript received November 1, 1988 Accepted November 7, 1988
