Diffusion of Molecules in a Solution

Question:
Molecules are injected at the origin x=0 and at x=a, and then they diffuse in the solution in the region 0 ≤ x ≤ a with the diffusion constant D.
The molecules in the solution react and disappear in the first-order reaction with the rate constant k. The concentration of molecules at the beginning and at x=a are equal to c₀ and c₁ , correspondingly, at all times.
a) Calculate the stationary density profile of the molecules for 0 < x < a.
b) Compute the particle current at x=a and at x=0? What is the particle flux for a → ∞ in both cases?

Solution:
a) In the region 0 < x < a, with stationary density profile, concentration (or density) of molecules c, is time-independent, so the rate of reaction is equal to the rate of diffusion, i.e.,

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