Reaction$-$Diffusion in a spherical catalyst pellet

Consider a spherical catalyst pellet. Assume that transport of product is by diffusion, with

diffusivity $D_{\text{e}\text{f}\text{f}}.$ We assume that the transport of reactant within the pellet is decoupled from

the transport of product. Mass transfer in the gas is sufficiently rapid so that the reactant

concentration at the catalyst surface is maintained at $c_{}.$ Suppose that $D_{\text{e}\text{f}\text{f}}=D_0($independent

of position$)$ and the reaction is the first order reaction, i$.$e$.,$

$R=-k_1^{\text{e}\text{f}\text{f}}c.$

$($a$)$ Define Damkholer number for this reaction rate.

$($b$)$ To determine the concentration distribution of the reactant inside the pellet assuming that

the Damk$$hler number is small,

Obtain a first approximation $($i$.$e$.O(l))$ for $C(r)$ in this limit using an appropriate scaling.

$($DO solve this$)$

$($c$)$ Now suppose that $Da>1.$ After scaling for proper nondimensionalization, write two

governing equations to solve and appropriate boundary conditions for $C(r)$ up to $O(\frac{1}{D}a)$

$($Do NOT solve this$)$

$($d$)$ How long does it take to get to steady state in the above two limiting cases?