Problem $4-18($Thomson text$)($only part$($a$)$ and part$($b$))$

As shown in the above sketch, gaseous species, A reacts with spherical solid, B$,$ after diffusing

through the porous product layer, C$,$ according to

$A_{(g)}+B_{(s)}{C}_{(s)}$

The reaction rate at the surface of $\text{'}B\text{'}$ is instantaneous and it can be assumed that guasi$-$steady$-$state

conditions prevail; i$.$e$.,$ accumulation terms in the conservation equations can be neglected. If the

molar flux of $\text{'}A\text{'}$ with respect to fixed coordinates is essentially equal to the molecular diffusion

flux with an effective diffusivity, $D_{\text{e}\text{f}\text{f}}.$

$($a$)$ Solve for the concentration profile of $A$ in the product layer if its concentration at $R_o=C_{A_{}}.$

$($b$)$ In reality, $R_i$ varies with time as $B$ is consumed and the molar flow rate of $A$ at $R_i(N_{A_{R_i}}^{})$ is

equal to the molar rate of disappearance of $B.$ If the molar density of $B,C_B,$ is constant, take

an unsteady$-$state molar balance on B and show that

$C_B\frac{d{R}_i}{dt}=N_{A_{i{R}_i}}$