I
K. C. CHAO'
%
Department of Chemical Engbneering, University of Wisconsin, Madison, Wis.
Isobaric Vapor-Liquid Equilibria Modified equations correlate isobaric vapor-liquid equilibria when the system shows a wide boiling range and considerable heat of solution
IN
4
THE correlation of vapor-liquid equilibria, it is the common practice to use the same functional form of equation for either isothermal or isobaric conditions. Examples of such forms are the Margules, the van Laar, and the Redlich-Kister equations. These equations satisfy the usual Gibbs-Duhem relation.
position to liquid composition for a solution is given by [YdDyifiv~ = Y 2 Lx , f , L o ~ p , T (2) t'p,",
= total pressure p , = vapor pressure of pure liquid com?r
Pressure, Mm. H g 760 760 760 760 300 150 760 300 100
ponent i at the temperature of the system
,
Equilibrium between the vapor and liquid phases of a mixture is attained when the same pressure, temperature, and component fugacities prevail throughout the system. Under thepe conditions the relation of vapor com1 Present address, California Research Gorp., Richmond, Calif.
2 Benzene Heptane Benzene Toluene Toluene Toluene Ethylbenzene Ethylbenzene Ethylbenzene
= hgacities of pure component i in the respective vapor and
this relation may usually be simplified to give y i ~ T i X I P, (3) where
Theory
1 Pentane Benzene Hexane Hexane Hexane Hexane Heptane Heptane Heptane
f t ~ '
liquid phases. y s v ,7 %=~activity coefficients At atmospheric pressure and below,
=
System
Differentiating Equation 4 and combining with Equation 6 gives
where
[%
de23 (l) dlnxz p , ~ This is a hypothetical condition which cannot be attained experimentally; for isothermal conditions the pressure changes with composition, and for isobaric conditions the temperature changes with composition. These variations of temperature or pressure are taken care of in the experimental measurements of activity coefficients where for each composition, equilibrium conditions of temperature, pressure, and composition must be maintained. T o account also for the variation of blnylblnx under isobaric or isothermal conditions Ibl and Dodge (3) have modified the GibbsDuhem equation. The modification for isothermal conditions is usually negligible, but the modification for isobaric , conditions may be appreciable. I n this article the corresponding corrected modifications of the RedlichKister equations are presented. One application of these modified equations has been presented for publication by Chao and Hougen (7) for the ternary system, benzene-cyclohexane-ethyl acetate, and its three binaries.
respectively, each in the liquid phase
The activity coefficient of any component Z' is directly related to the molal exces8 free energy of the solution; thus, for a binary System
GE = RT
(XI
In
y1
+ x z In yz)
GE
x1x2
[B
+ C(x1 - + D(x1 + ..1 x2)
x2)2
*
(5)
The number of terms required in Equation 5 depends on the complexity of the system and the accuracy of experimental data. The correct form of the Gibbs-Duhem relation for isobaric conditions has been shown by Ibl and Dodge (3) to be dln 71 dln y~ X Z -= z x1 ___ dxl dx1 where - AH dT RT2 dxi AH = H xi Hi" ~2 Hzo (7) H, HI', Hz0 = molal enthalpies of mixture, and components
+
2
-
=
YZ
B(x2 -
+
D ( X Z- x i ) (1
C(6xlx2 - 1 )
-
+
+ ...
8~1x2)
J1 In
dxl = 0
(10)
Herrington ( 2 ) has shown that Equation 10 usually does not hold for isobaric systems. Let the values of this integral be designated as a where a is a function of pressure, then
The following modification was made of Equation 9 for isobaric conditions allowing for the effect of the Z term and the requirements of Equation 11.
2
In
= a
Y2
d(xz
+ b(x2 - +
- xi)
(1
XI) ~ ( 6 ~ 1 x2 1) + - 8 x 1 ~ 2 )+ . . . (12)
The two sets of constants appearing in Equations 5 and 12 must be established from experimental data. The activity coefficients of the individual components may be obtained by combining Equations 5 and 12, thus, In
+ C(XI - + + ...1+ + + - 1) f d(xz - x i ) (1 + . . . ] (13) x m [B + C(xi + D(x1 + . . . ] - x ~ [ a+ b(x2 + - 1) + d(xz - xi) (1 +... 1
Y I = ~ 1 x 2[B D(x1 - x 2 ) Z b(x$ - x i )
22)
x2
[a
~(6x1~2
8~1x2)
In
yz =
x2)
X P ) ~
XI)
~ ( 6 ~ 1 ~ 2
8~1x2)
Table 1. Recommended Least Squares Constants Constants in Eqs. 5 and 12 Constants in Eq.9 B' C D a b C B
... ... 0.1824
.,. ... 0.2030 ... 0 .;'586
... e..
-0.0381
... ... 0 ... ... 0
... ... 0 ... ... 0 ... ... 0
(9)
According to this equation
- (-)
-
-
(4)
The variation of excess free energy with composition under isobaric conditions may be expressed according to Redlich and Kister (5)by a series function. j f f=
[& (&)I
(8) lnz = 2 P Where Z is negligible, substitution of Equation 5 into 8 gives the Redlich and Kister equation
C 0
-0.0355 -0.0149
0.2377 0.1473
0 0.0408
0.2736 0.1454
0.0519
- 0.0284
0.1579 0.1992
0 0
0.1945 0.2204
0 0
0
0.1252 0.1370
- 0.0129
... ... 0.1241
-0.0111
0.1449
...
-0.0234
...
...
...
...
... 0 ...
VOL. 51, NO. 1
...
9 . .
JANUARY 1959
...
... 0
0
... 93
'
The relative accuracy of experimental data becomes poorer as the composition approaches either pure component. For this reason. experimental data in which the composition of either component is below 5 mole in either the liquid or vapor phase were not used in the correlation. I n the numerical calculation logarithms were based upon 10. For simplicity the same symbol for the constants based upon natural logarithms was retained. Results and Discussion
02 04 016 018 MOLE FRACTION OF PENTANE, x p
-O4;
'lb
Correlation i s improved b y use of modified equations x = 760 mm. Hg Testing Modified Equations
The experimental data of Myers ( 4 ) were used for establishing the magnitude of these corrections in calculating tem0 perature and vapor-liquid composition. Temperatures and compositions were calculated from both modified and unmodified equations and compared with experimental values by the root mean square deviations. The method of least squares was used to fit experimental data with Equations 5 and 12. These equations were used successively xrith one, two, and three terms and the constants evaluated for each combination. These expressions were then combined according to Equation 13 to give the individual activity coefficients. The conventional Redlich-Kister equation was tested on the same systems and in the same manner using Equation 9, instead of Equation 12. The constants in Equation 5 were thus automatically determined, as they \
