Problem $5(\frac{25}{100})$ : A first order reaction $AP$ is taking place isothermally within a catalytic

micro$-$channel schematically depicted below. The channel is symmetric and has two openings. The

channel walls are coated with platinum along the entire length of the channel. Neglect radial gradients.

It is known that the reactant A concentration at $z=L$ is one$-$tenth the concentration of $A$ at $z=0.$ The

concentration of $A$ at $z=0$ is $0\mathrm{.}001mo\frac{l}{L}.$ The micro$-$channel length $(2L)$ is $2{10}^{-3}cm,$ and the

diffusion coefficient is $0\mathrm{.}1c\frac{m^2}{s}.$

a$)$ Derive a reaction$-$diffusion equation describing the concentration distribution along the

channel axial dimension and solve this equation to obtain the concentration of $A$ as a function

of the channel length $($z$)$ and Thiele Modulus $(\{$:${}_1).$

b$)$ What is the concentration of the reactant $A$ at $\frac{L}{2}?$