Spherical polymer pellets impregnated with hormones or other proteins are being investigated as implants to deliver therapeutic agents directly to an affected area. In general, when the agent is released from the polymer matrix, it diffuses from the matrix and is decomposed by enzymes in the body. Assuming a first-order decomposition reaction, \(k^{\prime \prime} c_{a}\), a diffusion coefficient, \(D_{a b}\), and a dilute solution of \(a\) in the body:

a. Derive the steady-state differential equation representing the concentration of \(a\) in the body in the neighborhood of the implant.

b. Using the boundary conditions, \(r=r_{o}, c_{a}=c_{a o}, r=r_{\infty}\), and \(c_{a}=0\), solve the differential equation for the concentration profile. You may find it useful to use the variable transformation, \(\chi=r c_{a}\).

c. Show that when \(r_{\infty} \rightarrow \infty\), the Sherwood number becomes:

\[S h_{d}=2+\sqrt{\frac{4 k^{\prime \prime} r_{o}^{2}}{D_{a b}}}\]