Consider a catalyst pellet in the shape of a Raschig ring (a hollow cylinder of inner radius, \(r_{i}\), and outer radius, \(r_{o}\) ). A first-order reaction occurs inside the catalyst.

\[a \xrightarrow{k^{\prime \prime}} b \quad k_{e f f}^{\prime \prime}=1.5 \times 10^{-5} \mathrm{~s}^{-1}\]

a. If the effective diffusivity of \(A\) inside the catalyst is \(D_{\text {eff }}=2 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\), we have dilute solutions, 1-D mass transfer in the radial direction, and there is no external mass transfer resistance, find the concentration profile and effectiveness of the catalyst.

b. If the outer radius is \(1.5 \mathrm{~cm}\) and the inner radius is \(1.0 \mathrm{~cm}\), find the effectiveness. Compare with the effectiveness for a solid cylindrical catalyst of the same outer dimensions.