A chromatographic column is packed with spherical of radius R that is coated with an immobilized enzyme, lactate dehydrogenase. The enzyme converts pyruvate (A) into lactate (B) at the particle surface in a reaction that has pseudo first-order kinetics (r = kC_A moles area ^-1 t^-1). Mass transfer takes place within a stagnant film of thinness δ. Assume that δ is much smaller than the particle diameter so that curvature can be neglected. The diffusivity of pyruvate in the liquid film can be represented as D_AL and is not affected by the lactate.

a. Derive the concentration profile of pyruvate in the film.

b. Derive an expression for the flux of pyruvate and demonstrate that the flux is proportational to 1(1 + Da^-1) where Da, the Damkohler number, is given by δ/D_AL.

c. Explain qualitatively what happens to the flux of pyruvate as the reaction rate becomes very fast relative to the diffusion rate (as Da becomes large). Do increases in k have a larger effect on the flux at low or at high Da values? Give a physical explanation for your observation.