A spherical organism consisting of an active core covered with an inert membrane like layer is exposed to air. Oxygen $($A$)$ dissolves at the surface of the organism and then diffuses radially through the inert membrane like layer without any reaction. At the air$-$layer $($i$.$e$.,r=R_2)$ interface the concentration of dissolved oxygen in the layer remains constant at CAs. At the layer$-$core interface $($i$.$e$.,r=R_1)$ oxygen transfers from the layer to the core, and then diffuses radially through the core while being consumed metabolically. The oxygen metabolism rate in the core can be described by a zero$-$order

homogeneous chemical reaction with a rate $r_A=k_0((mol)\frac{O_2}{m^3}(s)),$ where $k_0$ is a constant. At the layer$-$core interface $($i$.$e$.,r=R_1),$ an equilibrium exists between the concentration of oxygen in the layer and core phase, that is$,C_{\text{A}\text{I}\text{I}}{|}_r=R_1=K{C}_A^{\text{I}}{|}_r=R_1$ here $C_A^{\text{I}}$ is the local concentration of oxygen in the layer, and $K$ is the equilibrium constant. The diffusion coefficients of oxygen in the layer and the core are $D_A^{\text{I}}$ and $D_A^{II}$ respectively.

a$)$ From steady state material balances on oxygen $($A$)$ over differential spherical elements develop differential equations that describe the concentration profiles of oxygen in the layer and the core as functions of the radial position $r.$

b$)$ Express the boundary conditions associated with the differential equations.

c$)$ Assume that the concentration of oxygen $($A$)$ in the layer at $R=R_1$ is $C_A^H{|}_r=R_1=C_{A1}.$ Solve the differential equations in terms of $C_{A1}$ to obtain concentration profiles of oxygen in the layer and the core.

d$)$ Develop an expression for $C_{A1}$

e$)$ Derive an expression for $R_4,$ the consumption rate of oxygen in the organism.