Ind. Eng. Chem. Res. 1993,32, 240
240
Simple Graphical Method for Calculating the Constants of the Redlich-Kister Equation An alternative method is presented for calculating the constants of the three-suffix Redlich-Kieter model, based on the plot of Ge/RTxlx2as a function of composition and a single reading a t x1 = 0.5.
Some years ago Guggenheim (1952) suggested that the excess Gibbs function Ge for a binary system may be represented by the following infinite series: m
G e / R T = ~ 1 x 2 C , ( X ~ -xp)" n=o
(1)
In eq 1the term xlx2 assures that the value of Gewill be nil at x1 = 0 and x1 = 1.0. Equation 1truncated after three terms is also called the Redlich-Kister equation and is widely used to fit vapor-liquid equilibrium data. The Redlich-Kister equation is normally written G e / R T = x l x z [ B + C ( X-~~ p +) D(x1 - ~ 2 ) ' + ...I (2) Using the relation d ( G e ) / h l = R T In y1/y2
(3)
we obtain R T 71/72 = B ( x -~ X I ) + C ( 6 ~ 1 ~- 21) + D ( x -~x J ( l - 8x1~2)(4) A simple method of determining the constants B, C, and
D is to plot log y1/y2as a function of x1 and read off log y1/y2from the curve at preselected compositions. For example, when xl = 0.5 we have log y1/y2= CJ2.303,when x1 = 0.1464 we get log 71/y2= 0.7072B - CJ4, and so on. The above method is quick but gives equal statistical weight to all points, particularly those in the low concentration range where the analytical accuracy is low. In addition it requires reading the ordinate at awkward abscissa values as well as taking the difference between the read values to calculate the separate values of the constants. We propose here a more accurate and fast procedure based on the plot of g = Ge/RTxlxzas a function of the composition, as indicated in Figure 1. From eq 2 we get that the values of g at infinite dilution are X l = 0: g" = B - C + D = ab (5) x1
= 1:
g" = B
+ C + D = cd
b 0L
h
0.5
L
1.0 d
X1
Figure 1. Graphical construction for determining the RedlichKister constants.
and hence ef = e m - f m = (C-D ) - C = -D (11) In other words, a single reading at xl = 0.5, where the analytical accuracy is best, allows the immediate calculation of all three constants B, C, and D. As indicated by Van Ness (1964),the plot of Ge/RTxlx, has the advantage of being very sensitive to the precision of the data. If the data are of poor quality, this is usually obvious from the graph; if the data are reasonably good, this plot provides an excellent method of smoothing them.
Literature Cited
(6)
Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952. Van Ness, H. C. Partial Molal Properties. Classical Thermodynamics of Non-Electrolyte Solutions Pergamon Press: Oxford, 1964; Chapter 5, p 102.
(7)
Jaime Wisniak
(8)
Department of Chemical Engineering Ben-Gurion University of the Negeu Beer-Sheva, Israel 84105
and at x1 = 0.5 g=B=eh so that the traces ck = cd - ab = (B + C + D ) - (B - C + D ) = 2C
fm = C em = eh - m h = B - ( B - C
(9)
+ D) = C - D
(10)
0888-5885/93/2632-0240$04.00/0
Received for review August 6, 1992 Accepted November 3, 1992
0 1993 American Chemical Society
