COMMUNICATIONS
An Extension of the Wilson Equation to Partially Miscible Systems
A modified three-parameter form of Wilson's equation is presented by a combination of the Flory-Huggins equation and the local volume concept of Wilson. The new equation shows a better fit of vapor-liquid and liquid-liquid equilibrium data than the NRTL equation.
I t is well known that the Wilson equation (1964)is inapplicable to partially miscible systems in spite of its excellent performance in correlation of completely miscible systems. T o overcome this difficulty, Orye (1965) derived a modified version of the original Wilson equation by substituting Wilson's idea of local composition into the FloryHuggins equation for polymer solutions without an additional third parameter. The Orye equation is usually inferior t o the Wilson equation in data reduction for strongly nonideal systems such as alcohol-hydrocarbon mixtures, although the Orye equation is applicable to heterogeneous systems. Proposed Equation. We assume that the excess Gibbs free energy for a multicomponent system is given by
where /3jk = P k j , /3jj = fraction defined by
Pkk
= 0,and
E;
is the local volume
j=l
Equation 1 is similar to the Orye equation. Orye assumed t h a t the term p i j may be defined in terms of the Xij - Xj; difference. However, we set the P i j as an additional parameter to be obtained from experimental data.
Table I. Comparison of Performance of the Modified Wilson, Wilson, and NRTL Equations in Binary Data Reduction Modified Wilson parameters
Root-mean-square deviations
(caV No.
System, 1-2
1 Carbon tetr achlor ide-acetonitr iled 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Carbon tetrachloride-benzene" Carbon tetrachloride-benzene" Nitromethane-benzenef Nitromethane-carbon tetrachloridef Methanol+enzeneg Methanol-carbon tetrachlorideh Ethanol-benzene Ethanol-isooctane j Ethanol-toluenek Ethanol-toluene' Acetone-acetonitrilez Acetone-benzene Acetonitrile-benzene" n-Hexane-ethanol n-Hexane-ethanol' Acetone-n-hexane" Acetone-ethanolp Ethanol-benzene' n-Hexane-benzene' Ethyl acetate-ethanolr Ethanol-waterr
Temp No. of mol) or data ( A l 2 - (A2, p r e s s . points All) Azz)
45°C 40°C 70°C 45°C 45°C 55°C 55°C 45°C 50°C 35°C 55°C 45°C 45°C 45°C 760 m m 55°C 55°C 55°C 760 mm 760 m m 40°C 40°C
13 8 8 12 12 9 6 12 10 10 10 10 11 12 9 9 9 8 5 12 14 13
170 -121 44 650 1301 1721 2120 1360 2140 1345 1301 16 506 740 231 220 831 141 1339 168 -9 0
/j
Vapor mole fraction
(x1000)
(x1000)
MW" Wb NC MW
W
6 1 2 4 4 9 14 6 12 7
5 4 3 4 7 8 11 7 14 5 6 5 3 5 14 19 13 9 8 5 3 10
3 3 3 3 4 5 0 5 5 13 10 0 5 5 17 16 8 6 0 6 5 0 4 4 -6 0 3 2 -2 02 0 3 4 -1 00 0 6 7 1904 6 7 1740 50 163 E O 6 6 175 5 0 7 7 73 2 0 6 5 168 2 0 3 5 595 8 0 7 5 800 80 38 36 1400 194 29 55 257 76 40 60 140 -1 0 -6
2 2 2 8 80 2 20 5 20 80 8 2 5 5 2
Relative pressure
0 0 0 0
5 1 2 4
6 3 3 5 12 6 19 6 26 17 8 4 3 4 7 10 4 7 5 2 4 34
7 5
3 6 16 20 12 8 7 6 4 9
Pressure m m Hg
N MW W
8 4 3 5 1 0 8 19 6 17 1 1 7 5 3 6 16 21 11 9 8 6 4 8
1 0 1 1 1 2 8 1 4 1 2 1 1 1 5 4 5 4 5 2 1 3
1 1 2 1 1 3 6 1 4 1 1 1 1 1 5 5 5 4 4 3 1 3
N
2 1 2 1 3 4 12 2 7 2 2 1
1 1
5 6 4 4 4 1 1 3
MW = Modified Wilson equation. * W = Wilson equation. c N = NRTL equation. Data of Brown and Smith (1954). e Data of Scatchard et al. (1940). f Data of Brown and Smith (1955b). gData of Scatchard et al. (1946). "Data of Scatchard and Ticknor (1952). Data of Morachevskii and Zharov (1963). j Data of Kretschmer et al. (1948). Data of Kretschmer and Wiebe (1949). Data of Brown and Smith (1960). m Data of Brown and Smith (1957). n Data of Brown and Smith (1955a). Data of Kudryvtseva and Susarev (1963).p Data of Vinichenko and Susarev (1966).q Data of Prabhu and Van Winkle (1963). Data of Mertl (1972). 500
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Table 11. Predicted Results for Ternary Vapor-Liquid Equilibrium Data Based on Binary Data Absolute arithmetic deviation
Temp or press.
System Methanol Carbon tetrachloride (2) Benzene (3) Acetone (1) Ethanol (2) n-Hexane (3) Ethanol(1) Benzene (2) iz-Hexane(3) Ethyl acetate(1) Ethanol(2) w a t e r (3) Acetonitrile (1) Benzene@) )?-Heptane(3) a
Vapor mole fraction (x1000)
No. of data points MW"
5 5°C
8
55°C
21
760 mm
41
40°C
9
4 5°C
51
5 8 6 8 6 7 7 7 4 1O d 9 14 1le 5 9
Pressure, mm Hg
Wb
NC
6 6 7 8 6 9 6 6 4
8 9 8 12 8 8 7 7 5 9 9 14 11 4 9
MW
W
N
Data source
2
7
6
Scatchard and Ticknor (1952)
9
13
22
Vinichenko and Susarev (1966)
5
5
5
Waldo and Weber (1965)
7
7
Mertl (1972)
6
6
Palmer and Smith (1972)
MW = Modified Wilson equation. W = Wilson equation. S = XRTL equation. BIZ = 8 2 3 = cal/mol.
a13
-- 80 cal/mol
d12
= 3 2 3 = 80, P 1 3
= 300
2
We obtain for the activity coefficient of component i l n y , = In'
j=i
RT
5. xi
+
N
1-
Aj,tj
+
j=l
xi
xi
xj
j#k#i
where
A j i = ( v i / v J )e x p [ - ( X j i - X j j ) / R T ] For a binary mixture In y1 = -In (xi
+
A 1 2 x 2 )+ x 2
(4 )
-
2
Figure 1. Calculated and experimental liquid-liquid equilibria for the acetonitrile(l)-benzene(2)-n-heptane(3)system at 45OC. Concentrations are in mole fractions: 0 , experimental, Palmer and Smith (1972); --, calculated, modified Wilson equation, 412 = p 2 3 = 80, 013 = 300 cal/mol; - - - - -, calculated,NRTL equation.
Figure 2. Calculated and experimental liquid-liquid equilibria for the ethyl acetate(l)-ethanol(2)-water(3) system a t 4OOC. Concentrations are in mole fractions: 0 , experimental,Mertl (1972);--, calculated,modified Wilson equation, 912 = p 2 3 = p13 = 80 cal/mol; - - _ _-, calculated,NRTL equation. T h e analogous expression for In y2 is obtained by rotation of indices (1-2-1). Data Reduction. The capability of the present threeparameter equation is compared with t h a t of the Wilson and NRTL equation with the value of nonrandomness cons t a n t recommended by Renon and Prausnitz (1968) in vapor-liquid equilibrium data reduction and the proposed equation also is compared with the NRTL equation in the correlation of liquid-liquid equilibrium data. Vapor phase nonideality corrections were taken into consideration in accordance with the method of O'Connell and Prausnitz (1967). For binary vapor-liquid equilibrium data Table I shows t h a t the goodness of data fitting is nearly same for the modified Wilson, Wilson, and NRTL equations. T h e ternary predicted results for five systems including two partially miscible systems are presented in Table 11. The modified Wilson and NRTL equation give essentially the same results for the three sykems. However, the modified Wilson equation yields the better predicted results for the methanol-carbon tetrachloride-benzene and acetone-ethanol-n -hexane systems than the NRTL equation. In the calInd. Eng. Chem., Process Des. Dev., Vol. 14, No. 4, 1975
501
culations of ternary liquid-liquid equilibria for two systems the modified Wilson equation performs better than the NRTL equation as shown in Figures 1 and 2, although improvement is not significant for the acetonitrile-benzenen-heptane system.
Acknowledgment The authors thank the Data Computation Center, Osaka University, and the Data Processing Center, Kyoto University, for the use of their facilities.
Nomenclature g E = excess Gibbs free energy, cal/mol R = gasconstant T = absolute temperature, K u = liquid molar volume, ml/mol x = liquid phase mole fraction
Greek L e t t e r s 3( = binary interaction parameters, cal/mol y = liquid phase activity coefficient Ai = Wilson parameter defined by eq 4 X = pair interaction energy in Wilson equation, cal/mol E = local volume fraction defined by eq 2
Subscripts i,j,k = components
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Literature Cited Brown, I., Smith, F., Aust. J. Chem., 7, 264 (1954). Brown, I., Smith, F.. Aust. J. Chem., 8, 62 (1955a). Brown, I., Smith, F., Aust. J. Chem., 8, 501 (1955b) Brown. I.. Smith, F., Aust. J. Chem., 10, 423 (1957). Brown, I.,Smith, F., Aust. J. Chem., 13, 30 (1960). Kretschmer, C. B., Nowakowska, J., Wiebe, R., J. Am. Chem. Soc., 70, 1785 11948). Kretschmer. C.B.,Wiebe, R.. J. Am. Chem. SOC.,71, 1793 (1949). Kudryavtseva, L. S.,Susarev, M. P., Zh. Prikl. Kbim., 36, 1471 (1963). Mertl. I., Collect. Czech. Chem. Commun., 37, 366 (1972). Prikl. Kbim., 36, 2771 (1963). Morachevskii, A. G., Zharov, V. T.. a. O'Connell, J. P., Prausnitz, J. M., Ind. Eng. Chem., Process Des., Dev., 6, 245 (1967). Oiye, R. V., Ph.D. Dissertation, University of California, Berkeley, Calif., 1965. Palmer, D. A., Smith, B. D., J. Chem. Eng. Data, 17, 71 (1972). Prabhu, P. S.,Van Winkle, M., J. Chem. Eng. Data, 8, 210 (1963). Renon, H., Prausnitz. J. M., A.l.Ch.€. J., 14, 135 (1968). Scatchard, G..Wood, S. E., Mochel. J. M.. J. Am. Chem. Soc.. 62, 712 (1940). Scatchard, G., Wood, S. E., Mochei. J. M., J. Am. Chem. SOC., 68, 1960 ( 1946). Scatchard, G., Ticknor. L.. J. Am. Chem. Soc., 74, 3724 (1952). Vinichenko, I. G.. Susarev, M. P., Zh.Prikl. Khim., 30, 1583 (1966). Waldo, R. A.. Weber, J. H., J. Chem. Eng. Data, 8, 349 (1963). Wilson, G. M., J. Am. Chem. SOC.,66, 127 (1964).
Department of Chemical Engineering Kanazawa University Kanazawa, 920, Japan Japan Gasoline Co. Yokohama, 233, J a p a n
Isamu Nagata* Michio Ogura Masataka Nagashima
Received for review October 11,1974 Accepted M a y 13,1975
