An elementary reaction is taking place inside a cylindrical catalyst pore. The process contains a small amount of a catalyst poison, \(P\), and you would like to know how this affects the effectiveness of your catalyst. The poison has severe diffusional limitations inside the pore and so can be modeled as deactivating the pore up to a distance, \(z_{d}\), from the pore mouth. The pore has a radius, \(r_{o}\), and a length, \(L\). The diffusivity of reactant within the pore is \(D_{a b}\) and a first-order reaction occurs on the pore walls with a rate constant \(k^{\prime \prime}\).

a. Derive expressions for the reactant concentration in the poisoned and active regions of the catalyst. Assume only diffusion takes place in the poisoned portion of the pore.

b. What is the effectiveness factor for the poisoned pore?