LEAST-SQUARES METHOD
FOR CALCULATING
436 1
DIFFUSIONCOEFFICIENTS
A Least-Squares Method for Calculating Diffusion Coefficients for Ternary Systems
by R. L. Dunn and J. D. Hatfield Division of Chemical Development, Tennessee Valley Authority, Wilson Dam, Alabama (Received August 2, 1966)
A least-squares iterative method for determining,'l for n measurements of diffusion in a ternary system with the Gouy interferometer is described. The method minimizes the sums of squares of the differences between experimental and calculated relative fringe deviations. When n > 2, u+, u-, 61, and 6 2 are also determined by least squares which minimize the variances of calculated and experimental values of l / a and ~ the quantity ( 1 / 4 - (1 Pn)[(l/u-) -- (l/U+)l.
-
Several procedures' -s for calculating the four relevant diffusion coefficients that apply to isothermal diffusion in a ternary system were reviewed recently16 and a general analytical method for their computation was described. Maqy of the earlier reported results have been recalculated7 by the analytical method.6 The earlier methods included simplifications that made the calculations practicable without high-speed computers. I n one method13 the calculation was simplified by restricting experiments to concentrations in which diffusion is ganssian-a serious limitation for ternary systems. Wendts has programmed the an& lytical methods for the Bendix G-15 computer. The analytical method of Fujita and GostingB requires a plot of the relative fringe deviations, Set, against the reduced fringe number, f(p), from which the area, &, is obtained. I n this paper, a method is described that evaluates the Ci,-f(r) curve from the most precise relationship available by the method of least squares. This obviates the somewhat subjective construction of the curve and permits a complete analytical treatment for the calculation of the diffusion coefficientsby an IBM 704 computer. Throughout this paper, the symbols have the meanings used in related publications.1-6 Published equations are referred to by their numbers in the original reference; for example, eq. 23-1 refers to eq. 23 in ref. 1.
Method The theory relating Gouy diffusiometer data to the
diffusion coefficients of ternary systems has been published by Gosting and his collaborators.1-6 Here a method is described which yields the values of the quantities F-, u+, and u-, which minimizes 2( L ~ O ~ and ) ~ , which, when more than two experiments are made a t the same average concentration, minimizes 2[A(l/&)J2 and ~ ( A U ) where ~, A denotes the difference between calculated and experimental values. I n the method, the best fit of the experimental and calculated relative fringe deviations and reduced-height/ area ratios are obtained simultaneously with determination of the best values for I'-, u+, and u-. Values of the functions 61 and 02 and of the four diffusion coefficients are calculated by substituting values of the experimental quantities ai (refractive fraction) and R, (refractive index derivative) along with the bestfit values of I?-, u+, and u- into eq. 25-2 and 26-2, and 30a-5,31a-5,51-5, and 64-5. The least-squares treatments for calculating the dif-
(1) D. F. Akeley and L. J. Gosting, J . Am. Chem. Soc., 75, 5085 (1963). (2) R. L. Baldwin, P. J. Dunlop, and L. J. Gosting, ibid., 77, 5235 (1955). (3) P. J. Dunlop, J . Phys. Chem., 61, 994 (1957). (4) P. J. Dunlop and L. J. Gosting, J . Am. Chem. SOC.,77, 5238 (1955). (6) H.Fujita and L. J. Gosting, ibid., 78,1099 (1956). (6) H. Fujita and L. J. Gosting, J. Phys. Chem., 64, 1256 (1960). (7) P. J. Dunlop, ibid., 68, 3062 (1964). (8) R. D. Wendt, ibid,, 66, 1279 (1962).
Volume 60,Number 18 December 1966
4362
R. L. DUNNAND J. D. HATFIELD
fusion coefficients for ternary systems are developed below. Equations f o r Two Experiments. When the calcule tion is based on data from only two experiments a t the same average composition, the expressions used to minimize the sum of the squares of the differences between the experimental and the calculated relative fringe deviations are obtained from the following equations. Here, and in all subsequent equations, all T’s are I’-,but the subscript minus sign is dropped for convenience. 1
~- - (1 -
6
Q
=
e-r2
-
(1
r ) G ++ r.\/a_
iPIIt = bQt/brIIA,etc. The terms A r I and ArII are corrections to the original estimates, P I A and r I I A =
ru
= rIIA
f ArI
+A
- r)d
