P. F. MIJNLIEFFAND H. A. VREEDENBERC
2158
A Study of Concentration-Dependent Three-Component Diffusion
by P. F. Mijnlieff and H. A. Vreedenberg KoninklijkelShell-Laboratorium, Amterdam, Holland
(Received November 53,1966)
The continuity equations for isothermal and isopiestic diffusion in systems of three components, solutes I and I1 and a solvent, have been solved on an analog computer for the case of infinitely long diffusion cells (- OD < x < a). The concentration dependences of the four diffusion coefficients, DI-I, DI-11 etc., were chosen such as to agree closely with those holding for aqueous solutions of two electrolytes with one common ion. The concentrations, c ~ ( z , t )and CII(~,~), in the diffuse boundaries tend to become a function of x/ 0, and the smaller to - tref, the greater the resemblance of such an initial distribution to a supersharp one. The supersharp distribution (step function) itself conforms with the (and with any) final distribution, for a t to = tref all finite values of 5 are “compressed” a t x = 0. The final distributions, therefore, are the solutions of eq 16 for the initial and boundary conditions: (with i = I, 11)
< 0} a t t = tref >0 for x + }at t
c( = ci+ for x ct = ci,+- for x
ct +ci+
ci +ci,+- for x +
+
00 00
Itret
(18b)
and, naturally, for the particular functions DI-I(c~,c~I), etc. For the two-component case this was already stated by Boltzmann (see section I). Comparison of eq 18 and 3 shows that tref used here is identical with &harp used in section I. If the distribution a t t = to differs from the one which would result, a t t = to, from the initial distribution (18) R. P.Wendt, J. Phys. Chem., 66, 1740 (1962).
CONCENTRATION-DEPENDENT THREE-COMPONENT DIFFUSION
1)
for the total increase of the entropy of a finite system remains finite. However, as long as the “final distribution” exists, eq 24 prevails and as such provides an answer, albeit a partial one, to questionsm on the way in which nonequilibrium systems “move” toward equilibrium. We note that the inverse proportionality with the square root of time is a property of the total entropy production utot,.obtaining as soon as the concentra-
(29) For the value to be attributed to t see ref 25. (30) G . V. Cheater, Rept. PTOQT. Phys., 26, 411 (1963). (31) See ref 28, Chapter 5.
Volume 70,Number 7 J d y 1966
P. F. MIJNLIEFFAND H. A. VREEDENBERG
2168
for n
=
1, 2, . . ., 16. In this formula 0
a
NaCI
UCI
Ci,n
rt,n=
+
YCI,~
h . n
I, 11; c I , ~and ~ 1 1are , ~ the values of CI and to 5: = 5:n; A = En - f n - 1 = 0.8. The second eq 16 yields a similar result. The quantities and C i J 7 which occur in the equations for n = 1 and n = 16 can be eliminated by means of the boundary conditions where i
=
CII pertaining
ci,O = c i , 2 ; ci,17
=
(All
ci.16
where i = I, 11. For a chosen initial distribution of CI and CII all the 32 quantities cI,%/cOand PCII, J yco were simultaneously generated as functions of 9. At f n = *6.0, cI and cI1 were always found to be independent of 9. Thus it was justified to restrict the computation to the range I f ( 5 6 with the aid of the boundary conditions (Al). According to the above procedure, difference equations instead of differential equations were solved on the computer. The accuracy of these solutions was high; for concentration-independent two-component diffusion the difference equation can also be solved by hand, and the computer values of C/CO (at selected 4: values, for 9 > about 6) for such a case agreed within 0.001 with those of the solution obtained by hand. The comparison, for the same simple case, between the solution of the difference equation and the physically sound differential equation was less favorable. This is demonstrated in Table V. Table V : Solution of the Difference Equation and the Differential Equation for NaCl in Water c/co as the solution of
E
-6.0 -5.2 -4.4 -3.6 -2.8 -2.0 -1.2 -0.4
Difference equation
0.020 0.020
0.020 0.020 0.027 0.063
0.176 0.382
as the solution of Differential Difequation ference (error equation function)
C/CO
Differentia1 equrttion (error function)
6
0.020 0.020 0.020 0.022 0.033 0,076 0.188 0.388
0.4 1.2 2.0 2.8 3.6 4.4 5.2 6.0
0.635 0.841 0.955 0.992 0.999
0.632 0.832 0.944 0.987 0.998
1.000 1.000 1.000
1.000 1.000 1.000
The discrepancies shown in this table will decrease the closer the 5: values selected are and the larger their number is. If one were interested in solutions nearer The Journal of Physical Chemistry
-4t
V
-10
Figure 4. A(CI/CO) and A(CII/CO)(see text) a s a function of for initial distribution e of Table 111.
to those of the differential equations, the analog computer should be used in a different way or a digital computer could be used. There the limitation with regard to the number and distance of 5: values, inherent in the analog computation outlined above, hardly exists. Nevertheless, the influences of the concentrations and the concentration gradients of the two electrolytes on each other's diffusion manifest themselves clearly in the solutions of the difference equations. They result in differences, at corresponding E values A(C~/CO)
5 (Ci/CO)real
- (Ci/CO)indep
(i = 1, 11)
where ( C ~ / C & ~ ~ I indicates the computer solutions (derived from plots of the type of Figure 1, for 9 > about 6) for the realistic case of coupled concentrationdependent diffusion (initial distributions c, d, e, and f of Table I11 therefore), while (ci/co)indep denotes the computer solutions for the imaginary case that I and (32) D.R. Hafeman, J . Phya. Chem., 69,4226 (1965). (33) J. L. Duda and J. 9. Vrentas, ibid., 69, 3305 (1965).
HEATSOF IMMERSION IN WATEROF CHARACTERIZED SILICAS
11, though present together, diffuse independently; in the latter process the diffusion coefficients would be
D1-11 = D11-1 = 0; DI-I
=
a
-D1; Y
D11-11=
?Da
P
(A21 The solutions of the difference equations are then either equal to those for initial distributions a and b of Table 111,or constant. For one initial distribution the values of A(cJc0) have been plotted in Figure 4. The general character of the results for the initial distributions c and f is the same. At least as far as
2169
their sign is concerned, the results are as can be expected qualitatively from eq 13 and Tables I and 11. In agreement with qualitative and quantitative expectations, A(cf/co) for initial distribution d came out to be zero for all E. It might well be, therefore, that the influences of I and I1 on each other's diffusion are adequately described by the A(cf/co) as found with the computer. If so, the complete distributions, (ct/co) a t chosen 5 values and at t -t m , should be found by adding these A(cf/co) values to the exact solutions for independent diffusion (preferably to be derived from error functions, using D1-I and D11-11, as given in eq A2).
Heats of Immersion in Water of Characterized Silicas of Varying Specific Surface Area
by J. A. G. Taylor' and J. A. Hockey Chemistry Department, Faculty of Technology, University of Manchester, Manchsster, England (Received November 29, 1965)
The heat of immersion ( M I )in water of annealed, fully hydroxylated amorphous silica is 160 =k 3 ergs/cm2. This value is independent of the specific surface area in the range 8.5-147.5 m2/g. Higher values were obtained for samples containing microporous defects.
Introduction Although absolute values for the surface energy of solids cannot be obtained from heat of immersion ( A H I ) studies, this technique has been widely used in recent years to measure variations between surfaces of The results obtained have in differing some cases been interpreted in terms of the crystallinity of the solid. It has been suggested that as the particle size of the solid increases, the surface corresponds more closely to that of the pure crystalline material and so the energy change on formation of the solid-liquid interface increases. This hypothesis has been used to interpret the marked increase in AH1 on passing from high to low surface area solids.2 I n
the present work, values for AH1 in water of wellcharacterized silicas of similar surface properties but varying particle size have been determined. (1) Chemical Physics Division, Unilever Research Laboratory, Port Sunlight, Cheshire, England. (2) W. H. Wade, H. D. Cole, D. E. Meyer, and N. Hackerman, Advances in Chemistry Series, No. 33, American Chemical Society, Washington, D. C., 1961,p 35. (3) (a) A. C. Makrides and N. Hackerman, J. Phys. Chem., 63, 594 (1959); (b) W. H.Wade, R. L. Every, and N. Hackerman, ibid., 64, 355 (1960). (4) R. L. Venables, W. H. Wade, and N. Hackerman, ibid., 69, 317 (1965). (5) J. W. Whalen, ref 2, p 281. (6) M. M. Egorov and V. F. Kiselev, Zh. Fiz. Khim., 36, 158 (1962). (7) D. Kolar, Croat. C h m . Acta, 35, 123,289 (1963).
Volume YO, Number 7 July 1966
