E. L. CUSSLER, JR.,AND PETER J. DUNLOP
1880
An Experimental Comparison of the Gouy and the Diaphragm Cell
Methods for Studying Isothermal Ternary Diffusion
by E. L. Cussler, Jr., and Peter J. Dunlop Department of Physical and Inorganic Chemistry, The University of Adelaide, Adelaide, South Australia (Receiaed December 9, 1966)
Ternary diffusion coefficients a t 25' for one composition of the system water-sucrosepotassium chloride have been measured both with the Gouy diffusiometer and with the diaphragm cell. The same solutions were used for both types of experiments to ensure measurement of coefficients averaged over the same range of concentrations; at the completion of each diaphragm cell experiment the differences in both the refractive index and the density of the solutions in each cell were measured. By combining these data with the initial values of the refractive index and density differences, the four ternary diffusion coefficients were obtained by a new analytical technique which is free from the experimentally inconvenient restrictions of earlier theories. The diffusion coefficients measured by each method agree within the estimated error although the estimated error for the diaphragm cell experiments is roughly four times the error for the Gouy experiments. The coefficients based on measurements of both refractive index and density differences showed significantly smaller errors than those based on either refractive index or density measurements.
Two experimental geometries which are sufficiently developed to study ternary diffusion accurately are free diffusion measured with the Gouy diffusiometer and pseudo-steady-state diffusion measured with the diaphragm cell. However, no satisfactory comparison of these two techniques exists in the literature; the partial comparison which has been made is incomplete.' This study was undertaken to compare these two methods. The same solutions were used for both types of experiments so that the diffusion coefficients obtained would be averaged over the same concentration intervals. The ternary system chosen was water-sucrosepotassium chloride since previous diffusion studies with this system using either the Gouy diffusiometer or the diaphragm cell have shown that the cross-term diffusion coefficients are large and can be measured acc ~ r a t e l y . In ~ ~addition, ~ since both solutes are easily purified and nonhygroscopic, the solution concentrations can be determined very accurately. In the early stages of this work it was realized that previous ternary diffusion theories for the diaphragm T h e Journal of Physical Chemistry
cell were limited by inconvenient experimental restrictions. A theory unrestricted by such limitations is developed in the following section.
Theory Diaphragm Cell. Ternary diffusion in one dimension may be described by the flux equation^^-^
( 1 ) L. A. Woolf, J . Phys. Chem., 67, 273 (1963). (2) P. N. Henrion, Trans. Faraday SOC.,60, 75 (1964). (3) (4) (5) (6)
G. Reinfelds and L. J. Gosting, J. Phys. Chem., 68,2464 (1964).
L. Onsager, Ann. N . Y . Acad. Sei., 46, 241 (1945). 0. La", Arkiv K e m i Mineral. Geol., 18B, No. 2 (1944). L. S. Darken and R. W. Gurry, "Physical Chemistry of Metals," McGraw-Hill Book Co., Inc., New York, N. Y., 1953, p 458. (7) R. L. Baldwin, P. J. Dunlop, and L. J. Gosting, J . Am. Chem. SOC.,77, 5235 (1955). (8) 0. Lamm, J . Phys. Chem., 61,948 (1957).
R~ETHODS FOR STUDYING ISOTHERMAL TERNARY DIFFUSION
where j P is the mass flux of component “i” relative to the local volume average v e l ~ c i t y ,the ~ ~ ~p i ~ are the mass concentrations of the components per unit volume, and the D2; are the ternary diffusion coefficients“ at constant temperature, T , and pressure, P . The limits of these coefficients as the pi tend to zero have been given in the literature.12-15 If, in a diaphragm cell experiment the initial concentration differences, Apzo,for a given composition of a ternary system are sufficiently small, the diffusion coefficients measured correspond to the differential values at the mean solute concentrations, pi, and the volumeaverage velocity is zero. For the boundary conditions used in the diaphragm cell these flux equations have been solved by both a l g e b r a i ~ ’ and ~ , ~ ~matrix techniques’*to yieldlg
(Dii -
ez)Apio
u1 - uz
[(Dii
I+ +
4- DizApzo
- gi)Apio 02
-
DizApzo
e1
DziAPio 4- (Dzz - g2)Apzo e1
-
u2
DziApio
e-Polt
I +
+ (D22 62
-
e-Bolt
1881
0, they obtained excellent results. However, ternary diffusionieter has shown that it is advantageous at a given solution concentration to perform some experiments where neither Aplo nor Ap20 equals zero. To avoid the restriction imposed on the experiments of Kelly and Stokes, Kimz7 has developed a theory which allows all values of Aplo to be used but which integrates Ap, over Pt. This integration is sometimes difficult to realize experimentally and leads to errors if the ut are dissimilar. In order to solve eq 2 and 3 for the D,,it is convenient to define the reduced variables X and I’ T V O ~ with ~ ~ ~the J Gouy ~
(7) where the W , are weighting factors which depend on the experimental method used to determine the concentration differences. Three methods commonly used for binary diaphragm cell experiments, which are suitable for ternary systems, measure refractive index differences.28density difference^,*^-^^
ei)Ap2o
e1
(9) S. R. de Groot and P. XIazur, “Non-Equilibrium Thermodynamics,” North-Holland Publishing Co., Amsterdam, 1962. (10) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, J . Chem. Phys., 33, 1505 (1960). (11) By convention, 0 2 2 is chosen as greater than D ~ I . (12) I. J. O’Donnell and L. J. Gosting, “The Structure of ElectroSolutions,” W. J. Hamer, Ed., John Wiley and Sons, Inc., New e1 = ‘/z[Dii Dzz d ( D i i - Dzz)’ 4DizDzil (4) lyte York, N. Y., 1959, p 160. (13) F. 0. Shuck and H . L. Toor, J . Phys. Chem., 67, 540 (1963). ez = ‘/z [Dii Dzz - d ( D i i - Dzz)’ 4DizDzil ( 5 ) (14) E. L. Cussler, Jr., and E. N. Lightfoot, ibid.,69, 1135 (1965). (15) P. J. Dunlop. ibid., 69, 1693 (1965). Equations 2 and 3 relate the four diffusion coefficients (16) E. R. Gilliland, R. F. Baddour, and D. J. Goldst,ein, Can. J . Dll, D12, DZ1, and Dzz to the five experimentally measChem. Eng., 35, 10 (1957). urable quantities Aplo, Apl, Apz0, Apz, and Pt. To ob(17) F. J. Kelly, Ph.D. Thesis, University of New England, Armitain four independent equations relating the Dij, one dale, N.S.R., Australia, 1961. (18) E. hi. Lightfoot and E. L. Cussler, Jr., Chem. Engr. Progr. must measure a minimum of foul. values of Apa with Symp. Sei-., 61, 66 (1965). different values of Apto and j3t. I n the following, a (19) The superscript v on the Dilv has been omitted for convenience general method is given for solving eq 2 and 3 for the in the remainder of this paper. D,, for four or more values of Ap
