As shown in the above sketch, gaseous species, A reacts with spherical solid, B$,$ after diffusing through the porous product layer, $C,$ according to

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

The reaction rate at the surface of $\text{'}B\text{'}$ is instantaneous and it can be assumed that quasi$-$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 $R_o=C_{A_v}.$

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

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