HYOUNGMAN KIM
562
Procedures for Isothermal Diffusion Studies of Four-Component Systems’
by Hyoungman Kim Institute for Enzyme Research, Universaty of W$ewnsin, Madison, Whconein 63706 (Received September 13, 1966)
The basic differential equations are solved for the case of free diffusion in four-component systems. From the resulting expressions for the solute concentrations, corresponding equations are readily obtained for the solute concentrations in both restricted diffusion and steady-state diffusion. By using these expressions, procedures are developed for computing the nine diffusion coefficients from free, restricted, and steady-state diffusion experiments, respectively. It is found that in each case three different types of quantities, obtained from a t least three diffusion experiments a t given mean solute concentrations but with different initial relative solute concentration differences, are required to obtain nine relations which can be solved for the nine diffusion coefficients.
Introduction During the past decade, procedures have been developed for obtaining four diffusion coefficients from free,2--6 restricted,6 and diaphragm diffusion experiments with ternary systems. Some data were obtained for a number of systems by using the Gouy d i f f u s i ~ m e t e r ~ . l ~and - ~ ~ the diaphragm cell method.7-9v22-24These results have shown that the cross-term diffusion coefficients are significant not only in the systems containing strong electrolytes but also in systems composed entirely of nonelectrolytes. The Onsager reciprocal relationship was verified for several of these systems.13-15,26*26 From the earlier of these findings, it was emphasizedz? that generallized flow equations should be used for properly describing transport of biological rnacromolecules in buffer solutions and the transport of small molecules into and within the living cell. Theories for obtaining the molecular weights of macromolecules from sedimentation-diffusion experiments with multicomponent systems have been published recently by a number of worker^.^^-^^ The present study extends the Fujita-Gosting procedures4~6for studying free diffusion of three-component systems into a procedure applicable to diffusion experiments on four-component systems. Fujita’s procedures6 for studying restricted diffusion of ternary systems are also extended to a procedure for fourcomponent systems, and it is found that a similar procedure can also be used in diaphragm cell studies of The Journal of Physical Chemistry
four-component systems. It is hoped that these procedures will be of help in making improved diffusion studies of biological systems which can be treated as four-component systems. (1) Part of this work was presented at the 148th National Meeting of the American Chemical Society, Chicago, Ill., Sept 1964. This investigation was supported in part by Public Health Service Research Grant AM-05177 from the National Institute of Arthritis and Metabolic Diseases. (2) R. L. Baldwin, P. J. Dunlop, and L. J. Gosting, J . Am. Chem. SOC.,7 7 , 5235 (1955). (3) P. J. Dunlop and L. J. Gosting, ibid., 77, 5238 (1955). (4) H. Fujita and L. J. Gosting, ibid., 78, 1099 (1956). (5) H. Fujita and L. J. Gosting, J . Phys. Chem., 64, 1256 (1960). (6) H. Fujita, ibid., 63, 242 (1959). (7) E. R. Gilliland, R. F. Baddour, and D. J. Goldstein, Can. J . Chem. Eng., 35, 10 (1957). (8) F. J. Kelly, Ph.D. Thesis, University of New England, Armidale, New South Wales, Australia, 1961. (9) J. K. Burchard and H. L. Toor, J . Phys. Chem., 66, 2015 (1962). (10) P. J. Dunlop, ibid., 61, 994, 1619 (1957). (11) F. E. Weir and M. Dole, J . Am. Chem. Soc., 80, 302 (1958). (12) I. J. O’Donnell and L. J. Gosting in “The Structure of Electrolytic Solutions,” W. J. Hamer, Ed., John Wiley and Sons, Inc., New York, N. Y., 1959, p 160. (13) P. J. Dunlop, J . Phys. Chem., 612, 63, 2091 (1959). (14) L. A. Woolf, D. G. Miller, and L. J. Gosting, J. Am. Chem. Soc., 84,317 (1962). (15) R. P. Wendt, J. Phys. Chem., 66, 1279 (1962). (16) L. A. Woolf, ibid., 67, 273 (1963). (17) J. M. Creeth and L. A. Woolf, &id., 67, 2777 (1963). (18) G. Reinfelds and L. J. Gosting, ibid., 68, 2464 (1964). (19) P. J. Dunlop, ibid., 68, 3062 (1964). (20) P. J. Dunlop and L. 5. Gosting, ibid., 68, 3874 (1964).
ISOTHERMAL DIFFUSION STUDIES OF FOUR-COMPONENT SYSTEMS
563
Basic Differential Equations and Their Solutions I n recent years some solutions of the differential equations of diffusion for any number of components have a ~ p e a r e d . ~ Of ~ - thesk, ~ the solutions with matrix methods of Toor and of Stewart and Prober are most general and can be used for a number of boundary conditions. Although Toor’s expressions for fourcomponent systems (eq 26-28 of ref 36b) may be used in developing methods of obtaining the nine diffusion coefficients, the solution of the basic differential equations using the procedure adopted by Fujita and Gosting4 will be described briefly in order to define symbols and to relate this development to current procedures for studying ternary systems. The same final solutions can be obtained, after some manipulations, from eq 26-28 of ref 36b. The differential equations for isothermal diffusion of a four-component system in one dimension can be written2038
where t is time and x is the position coordinate measured downward from the position of the sharp initial boundary. Here it is assumed that the diffusion coefficients, Dtl, are independent of solute concentrations, CI, C2, and C3, within the diffusion cell and that no volume change occurs on m i ~ i n g . ~ + ~ ~ For free diffusion where a sharp boundary is formed a t t = 0 between the upper and lower initial solutions, the combined initial and boundary conditions are4 Ci +Ci
+ ACi/2
Ci ---t Ci - AC,/2
(y + a)
(2%)
(y + - a)
(2b)
where y = x/21/t
ci =
[(Ci)A
(3)
+ (c
