INDUSTRIAL AND ENGINEERING CHEMISTRY
194
Brough, H. W., Schlinger, W, G., and Sage, B. H., IND.ENG. CHBIM., 43, 2442 (1951). (5) Golding, B. H.,and Sage, B H. Ibid., 43,160 (1951) (6)Jenkins, F. A., Barton, H. A,, and Mulliken, R. S., Phys. Rev., (4)
30, 150 (1927). (7)Johnston, H.L,,and Weimer, H. R., J. Am. C h m . Soc., 56, 625 (1934). (8) Rossini, F. D., “Chemioal Thermodynamics,”p 191,New York, John Wiley & Sons, 1950.
Vol. 46*No. k
(9)Selleck, F. T.,Opfell, J. B., and Sage, B H., IND.EXG.CHEM., 45, 1350 (1953). (10)Spencer, H.M., Ibid., 40, 2152 (1948). (11) Witmer, E.E.,J . Am. Chem. Soc., 56, 2229 (1934).
RECEIVED for review December 22, 1952.
~ 4 c C E P r E pJuly 27, 1953. Copies of Figures 4 and 5, with a more detailed coordinate aystem approximately W/Z by 11 inches, may be obtained from the authom far the aaat a i duplication.
Thermodynamic Correlation of
Vapor-Liquid Equilibria PREDICTION OF MULTICOMPONENT SYSTEMS BERNARD S. EDWARDS’, FRANK HASHMALL2, ROGER GILMONT3, AND DONALD F. OTHMER The Polytechnic Institute of Brooklyn, Brooklyn, N . Y .
A
NEW method of predicting the equilibrium conditions of
multicomponent systems from binary data has been developed. Parameters determined from relative volatility data of the binaries are extended by thermodynamic principles to cover multicomponent systems. The equations for determining the parameters are generalized for use under constant-pressure as well as constanttemperature conditions. -4 correlation which would permit the prediction of vaporliquid equilibria of multicomponent systems from experimental binary data would be a valuable aid in the chemical field. A method based on thermodynamic, rather than empirical, considerations, which is applicable to both constant-pressure and constanbtemperature systems, and which would require relatively simple mathematical manipulation would be the most ideal correlation. Considerable work has been done in this field by both the thermodynamic and the empirical approach. For the most part, the work in binary (1, 8, 11, 18, 16, 19) and ternary systems (8,9,18, II,22) was primarily correlation methods which expressed data in the form of equations. The prime purposes of these equations are twofold: to determine internal inconsistency, and, where necessary, to smooth existing data. These correlations may also be considered prediction correlations only in so far as they permit interpolations betxeen experimental points, or neglect multicomponent effects. Gilmont et al. ( 4 ) have correlated binary mixtures by plotting the relative volatilities and obtaining a series of parameters which determine an analytical expression for the activity coefficient. The purpose of this paper is to develop a set of equations which may extend these parameters to predict multicomponent systems without recourse to multicomponent data.
+ or
in which log y i j represents the value of log pressed as a function of q. By defining
yi
in a binary i-j ex-
(3) and taking into consideration the Gibbs-Duhem equation %,d(g,) i
=
0
(4)
it may be shoa-n that, by operating on Equation 1
(5)
DERlVATION OF PREDICTION EQUATIONS
The total molal free energy of a multicomponent system, divided by 2.303RT, can be expressed as
or
9 = z1 log
Y1
+
22
log
YZ
+ z3 log + Y3
For an ideal binary mixture
For a nonideal binary with an ideal vapor phase
Sssuming that the excess chemical potential of a multicomponent 1 2
3
Present address, E. I. du Pont de Nemours 8: Co., Ino., Buffalo, N. Y . Present address, M. W. Kellogs, Co., New York. N. Y. Present address, The Emil Greiner Co., Xew York, N. Y .
Therefore
y1/y2
is the deviation from ideality for the relative
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1954
volatility of nonideal mixtures and will be designated as relative volatility deviation.
a!?~,
Theref ore =
gj;
log or::
-j
(log
Equation 11 is the simplest solution of the set of partial differential equations given in Equation 5 consistent with the assumption of Equation 2. The boundary conditions for each of the binary systems is satisfied by the simplest set of terms. Addi~ . which automatically tional multicomponent terms, x ~ x ~ & x .zC,, satisfy the boundary conditions have been omitted. This equation satisfies the tests for thermodynamic consistency of multicomponent systems as given by Herrington (8). It accounts for multicomponent effects as far as can be expected from binary data alone. Equation 11 when expanded for a three-component system appears as follows:
.
and from Equation 6 log yi = g
195
a!
:: )5>
(9)
+ .... ...
,
p,
Figure 1. Relation of P E @ - ~to) P R for Benzene-Chloroform-Acetone
Figure 3.
22
Lines for Benzene-Chloroform-Acetone
---
1 atmosphere Predicted Experimental
Log CY: may be evaluated by taking the difference between the partial derivatives of g with respect to x i and xi, respectively, which when expanded for three components (i,j, and k ) is: 2
0
6
4
8
1.0
22 +
Figure 2. Temperature Curves for Benzene-ChloroformAcetone
-Experimental Predicted ---
1 atmosphere
In the paper by Gilmont et al. ( 4 ) the function log y I 1is expressed in terms of a power series of x, and parameters & which were determined from relative volatility data. The function given is log y,, = g:2:+1 (10)
-
log, f f ;= gi:x: &xf 2gajzjzs - 2g:iZiZk 3g;x:xj 3g;jz;zk -
+
+
+ (8;--gzz? + +(& agiisiz; - 2gjizizj + - &)xX” + 3&x*2; il’$ y r t - $ y x f + . . . . . (12) &i’n)ZP
% ;f :
3&(i23k
,
The values of logri can now be obtained from Equation 9. GENERALIZATION OF PREDICTIOIi
In the paper by Gilmont et al. ( 4 )the parameters of the binaries were determined for constant-temperature systems. Because of this, py / p i was assumed constant and
;
i.e.,
log
ylt =
g:&
+ &x: + &’z: +
Substituting Equation 10 in Equation 2 gives a general expression for excess free energy explicitly in terms of composition:
In order to generalize the equations so that they may be applicable to constant pressure systems (within the limits imposed by the Gibbs-Duhem equation, which is strictly applicable to syatems under constant temperature and pressure), the equation:
INDUSTRIAL AND ENGINEERING CHEMISTRY
196
is used. The determination of the relative volatilit,y parameters is now made by plotting : a instead of a j i . Although for constant-temperature systems it is simpler to plot aii, the generalized method using a; will yield the same parameters. While the constant temperature systems do not require total pressure data to determine the parameters, the generalized method for constant pressure systems does require temperature data for the binaries.
Vol. 46, No. 1
TABLEI. PARAMETERS FOR BENZENE-CHLOROFORM-ACETONE SYSTEM AT 1 ATMOSPHEREPRESSURE
!Il
2:
gas
g,,
-0.1385
gil =
=a = -0,0525
8:2
-0.1400 -0.4250 = 0.0440 = 0.2440
f?,
!j2
0.0573
= -0.0573 = -0.1900
-
0.1900
= 0.1333 g I 3 = -0,1333
!$,
PREDICTION OF VAPOR COMPOSITION AND TEMPERATURE OR PRESSURE FOR MULTICOMPONENT SYSTEMS
The method of determining the total pressure and vapor composition of a constant temperature system is direct.
P = +pi
(15)
i To determine the temperature and vapor composition of a constant-pressure system requires some method of successive approxition, because pp is not constant. A number of methods are available. The method used in the test system presented here is not necessarily the most convenient, but is described as follows: Expanding Equation 15
P = YIXlpD,
+ y2xzp; + yaxrp; + ... . .
(17)
Let p f equal the vapor pressure, a t the same temperature as the system, of a reference material (usually water) which has a wellknown pressure-temperature relationship. On the basis of the Othmer plot ( 1 3 ) p t can be expressed in terms of the reference material. p: = ci(p:p Dividing Equation 17 by
1.0
0
.2
.4 fa
Figure 5.
-
.6
IO
Raoult’s Law Temperature Curves for Benzene-Chloroform-Acetone
---
1 atmosphere Raoult’s law Experimental
and rearranging, A value of pf substituted in the right side of Equation 18 leads
to a new value of pf which is closer to the correct value. By successive substitution, a value of p: is finally reached which is within the desired accuracy, The temperature is determined from this value. The number of approximations required is determined by the accuracy of the first approximation. A graphical interpolation of the binaries, as the first approximation, is usually sufficient to give a satisfactory result in three approximations. To obtain the vapor compositions, Equation 16 could be used but, after the foregoing calculations, it is simpler to use the equivalent equation
SPECIFIC EXAMPLE
Figure 4. zr Lines for Benzene-Chloroform-Acetone
---
1 atmosphere Predicted
Experimental
PREDICTION OF BENZENE-CHLOROFORM-ACETONE SYSTEMAT ATMOSPHERE PRESSURE. The binary data of the ternary system benzene-chloroform-acetoneas given by Reinders and Minjer (17)was used as a test system for the prediction of ternary data. The predicted results were compared to the ternary experimental data of the same source. The parameters for the threecomponent binaries were obtained by the method of difference plots as given by Gilmont et aZ. ( 4 ) with the substitution of log a; for log a,i, because this is a constant-pressure system. Table I lists the parameters obtained by this method. The value of log a; for this system was calculated by Equation 12, and the value of g by Equation l l a . A total of 66 points were calculated a t compositions which were multiples of 0.1 mole fraction for each
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 1954
component. These calculations were done in tabular form, as ahown in Tables I1 and 111, for ease in using a calculating machine. The tables are so arranged that the figures in the main body are functions of composition only and therefore may be used for any ternary system. Table I1 in particular is set up in terms of the general subscripts, i,j , and k. In this way the same table may be used for the three log a t ' s needed for the solution of Equation 9. In using Tables I1 and I11 all that is necessary is to place the proper parameter values in the top row and then calculate the last column by cumulative multiplication, which is easily handled by a calculating machine. Table IV (deposited with American Documentation Institute) shows the calculations by log yl,temperature, and vapor composition in accordance with Equations 9,18, and 19. To prevent the
197
necessity of taking powers of pf for the various approximations, a graph of p:(#
