A first-order heterogeneous irreversible reaction is taking place within a spherical catalyst pellet \(200 \mu \mathrm{m}\) in diameter.

\[a \xrightarrow{k^{\prime \prime}} b\]

The reactant concentration halfway between the surface and the pellet's center is \(1 / 5\) th the concentration at the surface. There is no external mass transfer resistance, so the surface concentration of \(a\) is the same as the bulk, \(c_{a s}=1 \mathrm{~mol} / \mathrm{m}^{3}\). The effective diffusivity of \(a\) within the pellet is \(0.1 \mathrm{~cm}^{2} / \mathrm{s}\). To solve the differential equation for the pellet, use the transformation \(\chi=r c_{a}\).

a. What is the concentration of the reactant at a distance of \(20 \mu \mathrm{m}\) from the pellet's surface?

b. If we need an effectiveness factor of 0.75 , what diameter should the pellet be?