A New Expression Similar to the Three-Parameter Wilson Equation Mitsuyasu Hiranuma Tomakomai Technical College, Nishikioka, Tomakomai, Hokkaido, Japan
The new expression of excess Gibbs energy of mixing which is similar to the three-parameter Wilson equation but is different from it, is derived by evaluating the number of ways of arranging the molecules for sites; they are g E / R T = -cixi In ( X i Aijxj) c j x j In (Xj A j i x i ) , where the parameter A i j is analogous to Wilson's parameter. They are applicable to partially miscible systems as well as to completely miscible systems.
+
Introduction In 1964, Wilson (1964) proposed an equation for gE, the molar excess Gibbs energy of a liquid mixture. For a binary mixture
gF/RT
= - x , In ( x ,
+ A12x2)- x 2 In ( x , + AjIx1) (1)
A,, = ( u , / v , ) exp(-(A,,
- A,,) 1RT
(2)
where X L J is an energy parameter characterizing the interaction between molecules i andj. A study by Orye and Prausnitz (1965) indicated that Wilson's equation, using only two adjustable parameters, is useful for representing excess Gibbs energies of a wide variety of liquid mixtures including strongly nonideal mixtures. However, as pointed out by Wilson, eq 1 is not applicable to liquid mixtures which are only partially miscible. To overcome this limitation, Wilson suggested that his expression for gE be multiplied by an arbitrary constant, c. Wilson thereby introduced a third adjustable parameter. The significance of this parameter has been discussed by Renon and Prausnitz (1969). They found that c = 212 where z is the coordination number of the liquid. Unfortunately, there appears to be no way to extend Wilson's three-parameter model to multicomponent solutions except for the case where the third parameter is the same for all constituent binaries. It is unclear how to estimate the value of c when binary parameters are determined from binary liquid phase solubility data only. The following is an attempt to resolve the difficulty. Derivation of an Equation Similar to Wilson's An equation similar to Wilson's can be derived as follows. Let us calculate, for a linear arrangement, the number of ways of arranging the molecules, or a number proportional to the number of ways, by evaluating the number of molecules available for the first site, then the second, third, etc. In the mixture there are N1 molecules of the first component and Nz molecules of the second component. Each occupies one lattice site. The total number of sites is therefore N1 + N2. The parameter A [ j is defined as the probability of finding a molecule of type j , next to a molecule i, compared to that of finding a molecule of type i when the numbers of molecules N[ = Nj. According to the definition, the number of molecules available around the first site is proportional either to V I = NI + A l 2 N 2 or to V I = N2 + hzlN1, if the molecule on the site before the first site is supposed to be type 1 or 2, respectively. Sup1)th site pose we wish to place a molecule in the (n when n sites have been occupied already by nl molecules of 1 and nz molecules of 2. The number of molecules available for the ( n + 1)th site will be proportional either to
+
+
-
vnl
+1 = N ,
- n1 + AI,(N, - n,)
or to vn,+1=
N, -
n 2 + AJN,
- n,)
(3j
according as a type of the molecule on the nth site is 1 or 2. No matter how small the numbers nl and n~ are, when the concepts of thermodynamic quantities are applicable to them, nl/N1 = n 2 / N 2 . Therefore, eq 3 can be written as vnI+l
= ( N , + A12N2)((Nl + N L- n)/(N,
+ N J ) (la)
or where n becomes lJ"'+l
(N,
+ A2,NI)((Nl+ N , - n ) / ( N , + A;?)) q(4b) = nl + n2. Of course, for random mixtures, eq 4
= (N?
U,?+I
5
vnz+l=
+ N , ) ( ( N , + N 2 - n ) / ( N , + N 2 ) ) =N,+ N? -
(4c)
The number of ways of arranging all the molecules is represented by N,
,I*\
n,
vnIlTv% 1
=
-
-
N , !N,! ( N , N,)! N, Al,N,)'"i ( N 2 A21N,).v2 N,! "I ( N , N,)"' ( N 2 NJV'
+
{- +
+
+ +
The factor ( N l + Nz)!/N1!Nz! is the normalizing factor, which is the number of arrangements for an ideal mixture, characterized by A 1 2 = A21 = 1. Calculated in a similar way, the number of ways of arranging the molecules of the pure liquid is given by 01 = N l ! / N l ! = 1 for N2 = 0 and O2 = Nz!/Nz! = 1 for Nl = 0. Then, OM = 0,/O~!2z = 0,. I assume that the molar Gibbs energy of a liquid mixture for non-athermal mixture, gM is given by a relation (see Appendix)
g M = -kT In Q M
(6)
Substitution of eq 5 into eq 6 gives the Wilson equation.
A New Expression similar to the Three-Parameter Wilson Equation Now, to allow for a free volume (or contact number) difference on the number of configurations, MM, a mixture is assumed to contain ( r l / r o ) N l molecules of 1 and ( r 2 / Ind. Eng. Chem., Fundam., Vol. 13, No. 3, 1974
219
Table I System no.
System components
Temp, "C
E thanol-benzene Ethanol-water Acetonechloroform Acetone-water Acetone-methyl acetate Acetonitrilebenzene Benzenen-heptane Benzene-water C hloroform-water Methyl acetate-water Acetonitrile-n-heptane Ethanol (1)-benzene (2)-water (3) Acetone (1)-chloroform (2)-water (3) Acetone (1)-methyl acetate (2)-water (3) Benzene (1)-n-heptane (2)-acetonitrile (3)
1 2
3 4 5
6 7 8
9 10 11
I I1 I11 IV
ro)N2 molecules of 2 instead N1 molecules and N2 molecules. Each prime component may have a different free volume, where r, and ro are, respectively, proportional to free volumes of molecule i and a key molecule. The number of ways is
+
cl.\ 1 K r l / r @ ) N l (r21r@)N21!( r 1 / r 2 ) N+ l AJV2 K r l / r o ) N l lkr2/ro)N2tl ! ( r lI r2)lVl+ N 2 N , ( r 1 / r 2 ) i i e l N'>1'N , + (r,/r2)N,
{
{
}
30
45 45
25 23
30 45
25 25 30
45
Bussei Jyosu (1965) Kagaku Benran (1966) Kagaku Benran (1966) Kagaku Benran (1966) Bussei Jyosu (1970) Palmer and Smith (1972) Palmer and Smith (1972) Griswold and Klecka (1950) Kagaku Benran (1966) Venkataratnam, et al. (1957) Palmer and Smith (1972) Hand (1930) Hand (1930) Yenkataratnam, et al. (1957) Palmer and Smith (1972)
The activity coefficients are found by appropriate differentiation; they are In y,= - c, In ( x ,
+ Allxi) + c,x, + cJx, c,x, XI
1 .
+
50
39.76 35.17 25
Source exptl vapor-liquid and liquid-liquid equilibrium data
(7)
where the number in the exponent is represented by c, instead of r, Ira for convenience. The number of ways of arranging the molecules calculated above is of course inexact, because it does not give the correct number of configurations for an athermal mixture. The normalizing factor must be introduced so that the value Q M is the correct total number of configurations for an athermal mixture for which -112 = A 2 1 = 1 (Hildebrand and Scott, 1964). Then, the number of ways of arranging the molecules is given by
+
- c]11 x J I
XI
j
+ AjJxi
(12)
Equation 11 is similar to the three-parameter Wilson equation, but is different from it. Equation 11 is easily extended to a mixture of more than two components
Here. we assume
c, =
(u,/uO)l'm
(15)
supposing that c, may depend on a ratio of component volumes, where uo is the molar volume of a key molecule. We further assume that the smallest of the molecules in the mixture is the key molecule; i.e., for aqueous solutions water is the key molecule.
gE 4, - = x, In RT Xf
+ x J In A4 + c,x, In 4)5, + clxJ In XI
-
51
- (9)
41
where
Equation 9 yields the Flory-Huggins equation, provided that ALJ = A J t = 1 and r,/rJ = u,/u,. Equation 9 yields Wilson's equation, provided that c, = 1andr,/r, = u l / u J . For substances of not too great difference in molar volume, we may, as a good approximation, regard rL/r, as unity (Shinoda and Hildebrand, 1958, 1961). It is not so unreasonable to assume that u,/ul>rl/rl = 1. On the other hand, though the power, c,, is nearly unity, it is necessary to make c, # 1 to obtain a solution which permits the existence of two liquid phases, when eq 9 is substituted into the classical equations for phase instability. Thus, we attempt to simplify eq 9, assuming that rl/rl = 1and c, # 1
gE/RT = -c,x, In (x, 220
Ind. Eng.
+ A,,xj) - c , x J In ( x , + '1
Chern., Fundarn., Vol. 13, No. 3, 1974
x ) (11)
Test of the Proposed Equation Partially miscible mixtures provide the best test for the proposed equations, because liquid-liquid equilibrium data are much more sensitive to the values of the parameters than are vapor-liquid data. Data for four ternary partially miscible systems were chosen for the test along with all the necessary binary data as in Table I: (I) the system acetone-chloroform-water, which includes the system acetone-chloroform having negative deviations from ideality; (11) the system ethanol-benzene-water, which includes the system benzene-water with very small mutual solubilities; (111) the system acetone-methyl acetate-water, which includes the system methyl acetate-water with rather large mutual solubilities; and (IV) the system benzene-n-heptane-acetonitrile, which does not contain water. Renon and Prausnitz (1968) proposed the NRTL equations for mixtures
Table 11. Parameters and Related Data for NRTL Equation, the Proposed Equation, and Wilson Equation ~~
~~
~
NRTL equation System no.
Gz L
1 2 3 4 5 6 7 8 9 10 11
0.4154 0.5687 1.008 0.7178 0.7224 0.7910 1.154 0.3026 0.3599 0.4609 0.5958
GI2
Proposed equation OQ
CY
0.7292 1.027 1.250 0.7263 1.232 0.8870 0.6755 0.4320 0.4882 0.8212 0.5126
0.2430 0.2421 0.9262 0.4477 0.6380 0,4808 1,604 0,01792 0.02678 0.1789 0.1145
8.1 20.6 3.2
0.47 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.3 0.3
10.1
7.4 12.5 7.0
... ...
...
...
m
Xl?
A12
Wilson equation
0.8476 1.172 1,6532 0.3804 1.323 0,6490 0,3319 0.008347 0.01347 0.5359 0,1512
3 3 3 3 3 3 3 3 3 3 3
9.7 22.7 3.2 11.4 6.7 11.5 6.0
...
--
-
Exptl m=2 .-3-m=3 NRTL
Cz-
/
V
V
V
4.3 24.8 3.4 11.5 6.7 11.2 5.9
...
Exptl -3-m = 3 NRTL
V
&'
UQ
\3
Mole %. Figure 1 . Calculated and observed liquid-liquid equilibria for the system ethanol(l)-benzene(2)-water(3) at 25°C.
The NRTL equation gives a good representation of excess functions for a variety of liquid mixtures, including those with limited miscibility. Equations 11 and 14 were compared with the NRTL equation. The parameters (221 and GI2 for the NRTL equation and A12 and - 1 2 1 for the proposed equation were used as the adjustable parameters. Values of the parameters were determined from the binary liquid phase solubilities or binary gas-liquid equilibrium data. When the parameters were obtained from the gas-liquid equilibrium data, a nonlinear least-squares fitting method was used which minimizes the sum of squares of deviations in activity coefficients for all data points. In all cases, cy12 was selected according to the rules given by Renon and m was changed from 2 to 5 in the proposed equation. Table I1 shows the root-mean-square deviations obtained between the predicted compositions of the vapor and the experimental values for the constituent binary systems. For ternary mixtures, the tie lines on the triangular phase diagrams were calculated with the use of binary parameters only and without temperature corrections since temperature corrections for parameters are not always reliable and the results were not sensitive to temperature
2
V
-\
V
V
V
3
Mole %. Figure 2. Calculated and observed liquid-liquid equilibria for the system acetone(l)-chloroform(2)-water(3) a t 25°C.
corrections applied to the parameters of miscible binary systems. The resulting predictions of activity coefficients depend on the following: (1) the quality of the data; ( 2 ) most sensitively, the right selection of m or a,, for the most nonideal binary; (3) the choice of u'; and (4) the fitting method used when determining binary parameters from gas-liquid equilibrium data. Figures 1-4 illustrate typical agreement between experimental data and results calculated with the NRTL equation and with the proposed equation. Generally, m = 3 is recommended when experimental ternary data are not available. However, for systems which include a constituent binary system having very small mutual solubilities, it appears that m less than 3 is more suitable; for systems which include a constituent binary with rather large mutual solubilities, it appears that m larger than 3 is more suitable. Appendix In statistical thermodynamics of mixtures, the molar free energy of mixing gM for a binary mixture is usually assumed to be given by gM= -kT In QM; QM = Qm/Q,Q2 where Q is the configurational partition function. For athermal mixtures, the ratio of ways of arranging Ind. Eng. Chern., Fundarn., Vol. 13,
No. 3, 1974 221
--
-
Exptl -m=3 NRTL
Exptl
-
NRTL
a V
2
3
Mole %.
Mole %. F i g u r e 3. Calculated a n d observed liquid-liquid equilibria system acetone (1)-methyl acetate(2)-water(3) a t 30°C.
for t h e
the molecules between the mixture and the pure liquids R m , which is Qm/R1Q2,is the same as, or a t least proportional to, the configurational partition function or configuration integral, QM (Hildebrand and Scott, 1964). That such a conclusion is applicable even to non-athermal mixtures seems not unreasonable when we compared eq 5 with the equation derived by Guggenheim (1952) from statistical thermodynamics. By using eq Al, Guggenheim has shown that
-Rg -TM - x1 In
x1
N, - + x 2 In x2 + '{ x1 In 2 ( N l - x"*) -
-t
+ +
where K = 2/(1 (1 4xlx2(~2- 1))112) and l / q 2 corresponds to A l d z l . If z = 2, by using (A3), eq A2 is rewritten asgM = -kTln QM,where
N"N,
(N,
+ (1 - K ) N J "
F i g u r e 4. Calculated a n d observed liquid-liquid equilibria for t h e system benzene(l)-t~-heptane(2)-acetonitrile(3) a t 45°C.
Furthermore, if 1112 = A21 = 1/q, it follows that QMat X I = x2, and QM = RM = 1at XI = Oandxz = 0.
QM
=
Nomenclature k = Boltzmann's constant m = empirical constant in eq 15 Ni = total number of i molecules in mixture ni = number of i moles arranged on lattice sites Qm,Q1,Q2 = configurational partition function of the mixture (m) and the pure liquids (1and 2) ri = corresponding to a free volume of molecule i R = gasconstant T = absolute temperature ut = molarvolumeofi xi = mole fraction of component i Greek Letters ai, = empirical constant in the NRTL equation yi = activity coefficient of component i in the liquid phase Y = number of molecules available to a site Qm,Q1,R2 = number of ways of arranging the molecules of the mixture (m) and the pure liquids (1and 2)
For comparison, eq 5 can be also written as Literature Cited
N2" ( N 2 + (1 - KZ)N,IN2 where
It can be shown that
Kagaku Kogaku Kyokai, "Bussei Jyosu," 3, 205 (1965); 8, 112 (1970). Griswold, J., Klecka, M. E., ind. Eng. Chem.. 42, 1250 (1950). Guggenheim, E. A., "Mixtures," Clarendon Press, Oxford, 1952. Hand, D. B., J. Phys. Chem., 34, 1980 (1930). Hildebrand, J. H., Scott, R. L., "The Solubility of Nonelectrolytes," Dover Publications, Inc., New York. N. Y . . 1964. Nippon Kagaku Kai, "Kagaku Benran, Kisohen 1 1 , " 596 (1966). Orye, R. V., Prausnitz, J. M., ind. Eng. Chem., 5 7 , 18 (1965) Palmer, D.A., Smith, 8 . D., J. Chem. Eng. Dafa. 17, 73 (1972). Renon, H., Prausnitz, J. M.,A.I.Ch.E.J . , 1 4 , 135 (1968). Renon, H., Prausnitz, J. M., A./.Ch.E.J . . 15, 785 (1969). Shinoda, M . , Hildebrand, J. H.. J . Phys. Chem., 62, 481 (1958). Shinoda, M . . Hildebrand, J. H., J . Phys. Chem.. 65, 1885 (1961). Venkataratnam, eta/.. Chem. Eng. Sci., 7 , 104 (1957). 86, 127 (1964). Wilson, G. M . , J. Amer. Chem. SOC.,
Received for review September 5 , 1972 Accepted January 29,1974
222
Ind. Eng. Chem., Fundam., Vol. 13, No. 3, 1974
