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The coefficient of diffusion can be calculated from a modified version of Fick's second law:
where D = diffusion coefficient (m2/s); c = concentration of ions (moles/liter) as a function of distance, x, and time, t; z = charge of 1 mole of ions; E = electric field (volts/m); F = Faraday's constant (23060 cal/(mol−eV)); R = gas constant (1.987 cal/(mol K)); and T = temperature (K). For the given experimental setup, the following conditions apply:
boundary condition: c = co, x = 0, t > 0
initial condition: c = 0, x > 0, t = 0
infinite-point condition: c = 0, x = infinite, t = large number
The exact analytical solution for Eq. (1) is:
For the chloride diffusivity test the following constants apply: z = 1; T = 298 K; and E = (12/x) V/m, where x is the thickness of the sample in meters.
The coefficient of diffusion was determined based on measurements of the time delay, tD, until the first significant increase in chloride concentration was detected in the KOH solution. The ratio c/co is a small number where c is the first reliably detected (error < 2%) Cl− concentration in the downstream compartment. A value of c/co = 0.005 was used in all cases. The expression in square brackets in Eq. (2) does not depend on the concentration but only on the time tD. It supports the measurements of the Cl− downstream concentration where the sudden increase in Cl− concentration was observed to be at nearly the same time regardless of whether the upstream Cl− concentration was constant. Only after this period was the difference between these two cases manifested in the different slopes of the downstream Cl− concentration curves. Measurements of the upstream Cl− concentration indicate that the greatest ingress of Cl− ions into the sample takes place in the first 24 to 48 hours of the test and then slowly decreases. The change in Cl− concentration did not affect the time delay tD, which was used to determine the diffusion coefficient, but it does affect the rate of the Cl− concentration increase at the downstream face after the time delay. To be able to determine the coefficient of diffusion using the slope of the concentration curve after the time delay it is necessary to keep the upstream Cl− concentration constant. Not all samples were measured with constant KCl concentration; thus, only one method was used to determine the diffusivity of the samples.