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HIROSHI FUJITA
Vol. 63
RESTRICTED DIFFUSION 1N THREE-COMPONENT SYSTEMS WITH INTERACTING FLOWS BY HIROSHI FUJITA~ Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received August 16,I968
The exact solutions are derived for one-dimensional restricted diffusion in a three-component system with interacting flows, under assumptions similar to those employed previously for the corresponding study of free diffusion for such a system. It is also assumed that the system contains two ionized solutes and that the specific conductance of the solution can be represented as a linear function of both solute concentrations. Combining this assumption with our exact solutions of the basic diffusion equations, two procedures are described which permit evaluation of the main and cross-term diffusion coefficients of the system. For the electrolyte solutes the measurement selected is the specific conductance of the solution as a function sf time a t some fixed positions along the cell. The equations presented may be applied to ternary solutions of non-electrolyte solutes, provided$ some quantity which depends linearly on solute concentrations can be accurately measured.
I n a recent article Fujita and Gosting2 presented tions for the evaluation of studies of restricted difprocedures for the computation of the four diffu- fusion as applied to three-component systems with sion coefficients of three-component systems from interacting flows, and to present two procedures appropriate sets of experimental data obtained with which may be used to determine the four diffusion the Gouy diffusiometer. These methods were coefficients of such systems with a reasonably high based on the exact solution2 of a modified form3 of accuracy. The equations are applicable in the Onsager’s phenomenological flow equations, sub- case of any experimental method which can project to the boundary conditions of free diffu~ion.~vide the concentrations of the solution (or linearly Gosting and co-w0rkers~35-~have applied them to related quantities) at two fixed levels in the diffuseveral systems, electrolytes and non-electrolytes, sion cell as functions of time. For the present we with the ultimate aim to obtain data which may be shall restrict our discussion to systems which conused to verify Onsager’s reciprocal relationships8 in tain ionized solutes, as in the experiments of Harned and co-workers for binary solutions. the thermodynamics of irreversible processes. I n view of the increasing interest in experimental Solutions to the Diffusion Equations studies of the interaction of flows in multi-compoBasic Equations.-We shall consider one-dimennent systems, it seems worthwhile to develop additional approaches to the accurate evaluation of sional diffusion of two solutes in a liquid medium both the main and cross-term diffusion coefficients contained in a cylindrical cell of length 21, mounted in such systems. Ip this connection it is to be vertically. The general equations for describing noted that the current optical methods for the ob- this diffusion process are3 servations of free diffusion are inapplicable to solutions a t very low concentrations. Harned and cow o r k e r ~have ~~~ developed ~ a precise conductome tric method for measuring the diffusion coefficient of a solution containing an electrolyte; they achieved remarkable success in studies a t the very where C1 and Cz are solute concentrations for comlow concentrations. Their method is based on an ponents 1 and 2 (solvent is conveniently defined as approximate (series) relation for conductance cs. component 0), expressed in moles per unit volume time when the redistribution of components takes of solution; Dn and D22 are the main diffusion coplace under the conditions of restricted diffusion. efficients; and D12 and D21 are the cross-term difThe purpose of this report is to develop equa- fusion coefficients. The positive direction of coordinate x is taken downward along the cell, with (1) On leave from the Physical Chemistry Laboratory, Departits origin a t the mid-point between the top and ment of Fisheries, University of Kyoto, Maizuru, Japan. (2) H. Fujita and L. J. Gosting, J . Am. Chem. Soc., 76, 1099(195G). bottom of the cell, and t is the time variable. It is (3) R. L. Baldwin, P. J. Dunlop and L. J. Gosting, ibid., 77, 5235 noted that equations 1 and 2 have been derived on (1955); P. J. Dunlop and 1,. J. Gosting, ibid,, 7 7 , 5238 (1955). the same assumptions as those used previously for (4) It was further assumed t h a t the diffusion coefficients are all the earlier similar study on free diffusion2 (see footindependent of solute concentrations and t h a t no volume ahange occurs on mixing. These conditions are satisfied if the concentranote 4). tion differences across the diffusing boundary are taken sufficiently Because neither solute can flow through the top *mall; cf. L. J. Gosting and H. Fujita, J . Am. Chem. Soc., 79, 1359 and bottom of the cell, solutions Cl(a,t) and C2(x,t) (1957); H. Fujita, THISJOURNAL, in preparation. to equations 1 and 2 must satisfy the boundary (5) P. J. Dunlop, i b i d . , 61,994 (1957). (6) P. J. Dunlop, ibid., 61, 1619 (1957). conditions
(7) I. J. O’Donnell and L. J. Gosting, a paper in a Symposium a t the 1957 meeting of the Electrochemical Society in Washington, D. C., John Wiley and Sons, N. Y., 1959. (8) L. Onsager, P h y s . Rev., 37,405; 38,2265 (1931); S. R. de Groot, “Thermodynamics of Irreversible Processes,’’ North-Holland Publishing Co., Amsterdam, 1952. (9) E.S. Harned and D. M. French, Ann. N . Y. Acad. Sci., 46, 267 (1945). (10) H. 8. Harned and R. L. Nuttall, J . Am. Chem. Soc., 69, 736 (1947).
a s= 0 dX
(x = +l, t
> 0)
(3)
where i = 1 and 2 . The initial condition chosen here is that a sharp boundary is formed a t t = 0 between the solutions A and B which are placed above and below the position x = 0, respectively. Thus for the two solutes (i = 1,2)
Feb., 1959
243
RESTRICTED DIFFUSION IN THREE-COMPONENT SYSTEMS
+ (ACi/'2) ( 0
