We can rig up a mass transfer fin so that there is a reaction occurring within the volume of the fin.

a. Derive the differential equation for the concentration in the fin assuming a zero-order chemical reaction in the volume with a reaction rate given by: \(r_{a}=-k_{0}^{\prime \prime} \frac{\text { mole }}{\mathrm{m}^{3} \mathrm{~s}}\)

b. Derive the differential equation for the concentration in the fin assuming a first-order chemical reaction in the volume with a reaction rate given by: \(r_{a}=-k^{\prime \prime} c_{a} \frac{\text { mole }}{\mathrm{m}^{3} \mathrm{~s}}\)

c. If the fins have a constant base concentration held at 0 and an insulated tip, determine the concentration profiles within each fin and the flux of \(a\) at the base in each case.