Consider the following system of gas$-$phase reactions:

A X $,r_x=k_1,k_1=0\mathrm{.}002mo\frac{l}{d}{m}^3*min$

A B

$r_B=k_2{C}_A$

$k_2=0\mathrm{.}06mi{n}^{-1}$

A $Y,r_Y=k_3{C}_A^2{k}_3=0\mathrm{.}3d\frac{m^3}{m}ol*min$

$B$ is the desired product, and $x$ and $Y$ are foul pollutants that are expensive to get rid of$.$ The specific reaction rates are at $27C.$ The reaction system is to be operated at $27C$ and $4$atm. Pure A enters the system at a volumetric flow rate of $10d\frac{m^3}{m}in.$

$($a$)$ Sketch the instantaneous selectivities $,$ and $($:$S_{\frac{B}{x}Y}=\frac{r_B}{r_x+r_Y}\}$ as a function of the concentration of $C_A.$

$($b$)$ Consider a series of reactors. What should be the volume of the first reactor?

$($c$)$ What are the effluent concentrations of A$,$ B$,$ X$,$ and Y from the first reactor.

$($d$)$ What is the conversion of $A$ in the first reactor?

$($e$)$ If $90\%$ conversion of $A$ is desired, what reaction scheme and reactor sizes should you use?

$($f$)$ Suppose that $E_1=10,000ca\frac{l}{m}ol,E_2=20,000ca\frac{l}{m}ol,$ and $E_3=30,000ca\frac{l}{m}ol.$ What temperature would you recommend for a single CSTR with a space$-$time of $10$min and an entering concentration of $A$ of $0\mathrm{.}1mo\frac{l}{d}{m}^3?$