Consider a fin incorporating facilitated transport only at its base. The reaction occurring there is given by \(-r_{a}=k^{\prime \prime} c_{a}\) (moles/s). The fin is essentially infinitely long, has a diameter, \(r_{o}\), and species \(a\) diffuses through it with a diffusivity, \(D_{a}\). It is immersed in a fluid containing species \(a\) at a concentration of \(c_{a \infty}\left(c_{a \infty}>c_{a}\right.\) throughout the fin). Assuming constant properties, dilute solutions, steady-state operation, and letting \(\chi=c_{a} / c_{a \infty}\) :

a. Determine the concentration profile within the fin.

b. What is the flux and concentration at the base of the fin?