J. C. Dennis
Physics Department Stephen F. Austin State College Nacogdoches, Texas
II
Diffusion Coefficients of Electrolytes in G ~ I S
Diffusion coefficients of electrolytes in gels are normally determined with the aid of radioactive tracers. Radioactive materials, and in particular the appropriate ones, are not always available in the lab. This paper demonstrates how diffusion coefficients can be determined from changes in the concentration of a solution placed above a gel. A gel column of height a contains a known mass of diffusant, with the distribution initially uniform. An equal depth of water is placed above the gel a t t (time) = 0, and is stirred constantly as diiusion proceeds from the gel into the solution. The concentration of diffusant in the gel will be denoted by u = u(x,t) and the concentration in solution by v = v(t). The diiusion equation for one-dimensional diffusion is
"
[1/(2n + son with the well-known series The series
C
n
converges by compari-
0
-
1 = *' C n< 6
n=1
Since "
"
1
1
n =1
n
1
r
-
1
'
Thus v(t) becomes "
where D is the diffusion coefficient. The solution must be continuous for all values of t . The boundary conditions are ( 1 ) bulb = 0, z = 0 , t > 0 ( A t x = 0 the cylinder is sealed.)
( 2 ) u(a,t) = u(t), t > 0 (The gel-solution interface is at z = a,) (3) u(z,O) = j ( z ) , where j ( z ) is the initial distribution.
Let f (x)
=
uo
=
with R
4
= Dt/a2, and u(x,t) is
2 ~ ( ~ = , t -) r
!sP
2n+1
cos Bze-it
-C
+
4 ( 2 n f l ) W exp
1 (Unit initial concentration), and let
(-(2n4af,
1)'*' ~ t ) ]
w ( z , t ) = u ( z , t ) - u(t)
Then
The well-known solution to this problem is
+
where@= (2n l)s/2a, and X = DOZ. The mass flux through x = a is given by -D(bu/ The mass flux is the mass per unit area per ax),. unit time. Thus the concentration in the solution is given by the total flux up to time t that crosses x = a, divided by the length of the solution column, a. The expression is V(Rl versus R in the ranger R = 0.0-0.4, 0.4-0.8.
and as
the final expression becomes 432
/
Journal of Chemical Education
The figure shows how v(R) looks. Refractive index versus concentration graphs can be determined for a particular electrolyte with an immersion refractometer. The concentration in the solution as a function of time can be taken from the calibration curves, or from the table. The diffusion process is not affected because
only one or two drops of the solution are sampled a t one time. At least ten values of R should be obtained as v(R) approaches '/s For each v(R), R, and hence D, can be taken from the figure. D is averaged for the best value. Diffusion coefficients are presently being determined by the proceeding method here at the College.' The dependence of the diffusion coefficient on temperature is also being investigated. The author wishes to thank W. D. Clark, Math Department, for his assistance.
V(R) versus R R
V(R)
R
V(R)
R
V(R)
R
V(R)
'A table of results will be supplied upon request.
Volume 45, Number 6, June 1968
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433
