$(35$ points$)$ A catalytic converter is used to catalyze the following chemical reaction:

$CO+\frac{1}{2}{O}_{2}C{O}_2$

The concentration of $CO($species A$)$ is reduced by passing the exhaust gases over the catalyst surface, and the rate of reduction for $CO$ species consumption at the catalyst is governed by a first$-$order reaction with rate constant $k_1^{\text{'}\text{'}}=0\mathrm{.}050\frac{m}{s}.$ As a first approximation, assume that CO reaches the surface by one$-$dimensional diffusion through a thin gas film above the catalyst surface. The gas film thickness is $L=1\mathrm{.}0mm.$ The diffusion coefficient of $CO$ in the gas is $8\mathrm{.}75{10}^{-5}\frac{m^2}{s}.$ The exhaust gas is at $400C$ and $1\mathrm{.}1$ bars and the mole fraction at $x=L$ is

$x_{CO,L}=0\mathrm{.}0045.$ As a first approximation, assume the film is stagnant $($the medium is stationary$)$

and an ideal gas. Evaluate the system at steady state.

$($a$)$ What is the CO concentration in the gas film at height $L,$ meaning $C_A(x=L)$ in $\frac{\text{m}\text{o}\text{l}}{m^3}?$

$($b$)$ Start from the general mass diffusion equation and determine the simplified mass diffusion

equation for this system. Solve for the general solution of the concentration profile $C_A(x).$

$($c$)$ What are the boundary conditions to solve $C_A(x)$ in $($a$)$ written in terms of $C_A?$

$($d$)$ Solve for the $CO$ concentration profile $C_A(x)$ in the gas film above the catalyst.

$($e$)$ What is the molar concentration of $CO$ in the gas at the catalyst surface?

$($f$)$ Catalytic converters have very high catalyst surface area using honeycomb structures. What

would be the molar rate of $CO$ removal for a catalyst surface area $A=12,000m^2?$