Cross-Term Diffusion Coefficients by Capillary Cell Method
2281
On the Determination of Cross-Term Diffusion Coefficients in Ternary Systems by the Capillary Cell Method azuo Toukubo and Koichiro Nakanishi* Department of Industrial Chemistry, Kyoto University. Kyoto 606,Japan (Received April 22, 1974)
The extended Fick equation for the diffusion in ternary systems has been solved with the initial and boundary conditions pertinent to the open-ended capillary cell method. An analytical solution has been obtained which makes it possible to convert the experimentally measurable quantities to the four (main and cross) diffusion coefficients without introducing either mathematical or physical approximations. Application OF this technique to the determination of the diffusion coefficients in dilute solutions is discussed.
Introduction Coupled diffusion with interacting flow in ternary systems may be expressed by the extended Fick equation1
where J , is the flux relative to the volume averaged velocity and c, is the concentration, both of species i. The Dv's represent the ternary diffusion coefficients. The four DV's appearing in eq 1 have been determined by the Gouy dif. ~ an earlier fusiometer1S2or the diaphragm cell m e t h ~ d In stage in the studies on ternary diffusion, effort had been devoted to the verification of the validity of the Onsager reciprocal relation ( Rli) which insists the equality of the cross-term phenomenological coefficients, L1g E Lgl. The principle contrihutlon comes from the studies of Gosting and coworkers who established elaborate experimental technique for Gouy's diffusiometer and the method of calculation of DY7sfrom the experimentally measurable quantities.l Recently much attention has been paid to the systems where there is a strong interacting flow and accordingly the absolute values of the cross diffusion coefficients, D1g and/ or Dgl, are fairly large.* It seems therefore desirable to establish simple ),et reasonably accurate method to obtain the cross coefficnents. In this paper, we present an analytical solution of eq 1 with a boundary condition pertinent to the open-ended Capillary cell e ~ p e r i m e n t . This ~ , ~ solution has been obtained witlnout any mathematical and physical approximations and allows the calculation of four diffusion coefficients from two independent measurements with different times of diffusion.
Soluticm of the Diffusion Equation Equation 1 may be ;solved with the boundary conditions o f the capillary cell method as follows. Let the components 1 and 2 be the diffusing species, the component 3 be a solvent, and the concentration of each component outside the capillary is assumed constant throughout the measurement. Then the initial and boundary conditions become: initial conditions-, t = 0
boundary conditions, t
x =
>0
0; acl/ax = acz/ax =
o
x > = a; cl r= el'? c2 = c2' where a is the length of capillary. In order to satisfy the boundary conditions, c1 and may be expressed by the following equations
(3)
cg
where A,, B,, a,, and b, are the coefficients to be determined from the initial conditions, and
R =
[@ii
+ %) +
J D I I- Dz,)' +
4D12&,]/2
(6) =
+
022)
-
J@II
D z , ) ~+ 4L),2D21]/2
(7) Substituting the initial conditions, c1 = e10 and c2 = czo at t = 0 into eq 4 and 5 and using the Fourier analysis, the following expressions are obtained
An +
an
4(-1)" = (2n 1)n (C*O -- c1')
+
(8 )
On the other hand, substitution of eq 4 and 5 into eq 1 must lead to equations valid for any values of x and t. From this condition (10) A& = DiiAn + DIZBn
+ D12bn
(11) If we set the ratio A,/& and an/b, as P and Q, then the following equations are obtained by eliminating R and S from eq 6 and 7 .
a,S = Dllan
The Journal of Physical Chemistry, Vol. 78. No. 22, 7974
2282
Kazuo Toukubo arid Koichiro Nakanishi
TABLE 11: Cross Coefficients and Corresponding Concentration Decrease in the Capillary Cell Experiment Input data Diza
4i@~ - ~4 2 ) ' + ~ D ~ , D Z ~ I /(13) ~ D Z ~a Equations 8,9, 12, and 13 can easily be solved to give A,, B,, a,, and b,. These are then substituted into eq 4 and 5 to obtain the following solution which satisfies the original equation (eq 1).
= 7.00cm 0.01 t = 1.728 X 105sec 0.05 cI" = 0.25 0.10 c20 = 0.20 0.20 c1' = cz' = 0 0.50 Dii = 0 2 2 = 1.00" 0.50
106 cm2 sec -1.
Output results*
Diza
c1 " Y
ce ""
0.10 0.10 0.10
0.1928 0.1919 0.1908 0.1886 0.1819 0.1831
0.1516 0.1516 0.1516 0.1517
0.10
0.10 0.50
0.1518
0.1412
Calculatedfrom eq 19 and 20.
Equations 19 and 20 are valid irrespective of whether c? > cE'or e? < et'. For the measurements in dilute solutions, either e? or c,' can be set to zero in all the equations given in this section.
where
I , = 4(-1)"/(2n +I*
= (2n
+ 1)71
+ l)a/2a
(16)
(17)
In the capillary cell experiments, it is a common practice to determine the average concentration of solutes c,, in the capillary after the time of duration t. Since
(18) eq 14 and 15 are integrated to give the final equation
where kin = 8/[(22n
+ 1)'a2]
(21)
and P, Q, R, S, and k, are given by eq 12, 13, 6, 7, and 17, respectively. The four diffusion coefficients to be calculated can be expressed by the combination of P, Q, R, and S as in the fallowing equations. The Journal cf Physical Chemistry, Vol. 78, No. 22, 1974
Calculation of Diffusion Coefficients In the capillary cell experiments, the initial and average concentrations of components 1 and 2, c1O and ezo (or el' and cz') at t = 0, and ~ 1 , and ~ " ~ 2 , at ~ "t = t , are experimentally determined. Since one diffusion run yields two equations (eq 19 and 20), two runs with the same initial eoncentration and different t suffice to determine four unknown variables, i.e., P, Q, R, and S. Equations 19 and 20 indicate that the e,, can be expressed as the sum of an infinite series. Preliminary calculation shows that this series converges rather rapidly under usual experimental conditions and only the first four terms are necessary for calculating the D,'s with an accuracy of 0.1%. In order to obtain the four diffusion coefficients from the experimentally measurable quantities, the four-variable Newton-Raphson method may be used. A computer program in fortran language has been prepared as shown in Table L6 This program was proved to work well with reverse calculated data. Discussion The capillary cell method has been widely used for the determination of the intradiffusion coefficients with radioactive t r a ~ e r .It~ is , ~also applied to the determination of the mutual diffusion coefficients of diluted solute in binary or ternary solution^.^ The measurements with this method may be classified into two types according to the relative magnitude of co to c'. If co > c', then it is conventional "diffuse out" type measurement. On the other hand, co < c' may be termed as "diffuse in" type, which has been adopted by Bockris and Hooper.8 Relative merit of one of these two techniques over another would be judged by the ease with which the method chosen can be fitted more rigorously to the boundary conditions imposed. In ternary nonideal solutions, it is ultimately desirable to obtain the values of the four diffusion coefficients over the whole composition range. However, most of the available data, especially those for aqueous solutions of electrolytes
2283
Diffusion Coefficientsof Liquid Binary Nonelectrolytes
or nonelectrolytes, have been obtained in a corner area of triangle coordinate for ternary systems where two solutes are diluted in a solvent. Since the mutual diffusion coefficient is essentially equal to the intradiffusion coefficient of solute in this corner area, the present method may best be applied to the measurements in such dilute solutions. I t is interesting to examine how far the small difference in concentration should be detected for the cases that the cross coefficients, D 1 2 and 0 2 1 , are relatively smaller than the main coefficients, D l l and 0 2 2 . Table I1 shows several examples of test calculations, in which Cav)s are directly calculated by ey 19 arid 20 with appropriate input data. It is seen that the errors in concentration determination should not exceed 0.196, when D 1 2 and Dzl are less than one tenth of D l l and 0 2 2 This is a very difficult assignment and special experimentation must be necessary. However, if the main and cross coefficients are of comparable magnitude, such demands for higher accuracy will be greatly reduced. In conclusion, if the absolute values of the cross coefficients are fairly large, the capillary cell method should furnish a simple device to estimate the cross diffusion coefficients in ternary systems. Furthermore, a test of the ORR for ternary molten salt mixtures with a common ion9 may be possible, provided that very high accuracy at higher temperature could be achieved. To this end, preliminary measurements are currently in progress.
Acknowledgment. The authors thank Professor S. Ya-
bushita for his kind guidance in the mathematical problem and Professor N. Watanabe for his encouragement. Supplementary Materials Available. Table I will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JPC-74-2281.
References and Notes (1) D. D. Fitts, “Nonequilibrium Thermodynamics,” McGraw-Hill, New York, N. Y., 1962, Chapter 8. (2) For review of the method, see ref 1 and A. L. Geddes and R. B. Pontius, in “Physical Methods of Chemistry,” A. Weissberger and B. W. Rossiter, Ed., revised edltlon, Vol. 1, part IV, Wiiey-Interscience. New York, N. Y., 1971. (3) For review of the method, see R. Mills and L. A. Wooif, “The Diaphragm Cell,” Australian National University Press, Camberra, A.C.T., Australia, 1968. (4) See, for example, (a) G. R. ReinfeMs and L. F. Gosting, J. Phys. Chem., 68, 2464 (1964); (b) A. Revrln, ibid., 76,3419 (1972). (5)H. J. V. Tyrreli, “Diffusion and Heat Flow in Liquids,” Butterworths. London, 1961, Chapter 5. (6) See paragraph at end of text regarding supplementary material. (7) See, for examples, (a) E. W. Haycock, B. J. Alder and J. H. Hildebrand, J. Chem. Phys., 23, 659 (1953); (b) K. Nakanishi and T. Ozasa, J. Phys. Chem., 74, 2956 (1970). (8) J. O’M. Bockris and G. W. Hooper, Discuss. faraday SOC., 32, 218 (1961). (9) This is one of four kinds of ternary systems, for which the experimental verification of the ORR has not yet been available (see ref 1).
Diffusivities and Viscosities of Some Binary Liquid Nonelectrolytesat 25’ 13. K. Ghai“‘ and F. A. L. Dulllen Llepartment of Chemical Engineering, University of ‘Waterloo, Waterloo, Ontario, Canada N2L 36 1 (Received March 18, 7974) Publicatkm costs assisted by the National Research Council of Canada
+
+
The intradiffusivities for three binary liquid systems n- hexane toluene, carbon tetrachloride n- hexane, and toluene carbon tetrachloride and mutual diffusivities for two systems n- hexane + carbon tetrachloride and toluene carbon tetrachloride have been measured a t 25’ and ambient pressure, covering the complete composition range. The diaphragm-cell technique was employed for these measurements. The density and viscosity values were also determined for each of these systems. The data have been used to test the validity of various diffusivity equations proposed in the literature. I t has been concluded that these probably apply only in ideal solutions. The comparison of the limiting mutual and intradiffusion coefficients did not indicate an isotope effect on the rate of diffusion.
+
+
Introduction As is well known, two types of diffusion coefficients are commonly used in diffusional theories of binary mixtures: one mutual diffusion coefficient, D A B , and two intradiffusion coefficients, DA* and DB*. In a mutual diffusion process, components A and B interdiffuse into each other in such a way as to smooth out the inequalities in the concen-
trations. Intradiffusion is unlike mutual diffusion in that it takes place in a chemically homogeneous mixture; the transport due to random motion of molecules is determined with the aid of trace amounts of tagged molecules A or B. Various relations have been proposed between mutual and intradiffusion coefficients.2 Those originating from Darken3 and Hartley and Crank4 have been used most exThe Journal of Physical Chemistry, Vol. 78, No. 22, 1974
