Liquid species A and C fed to a CSTR are expected to undergo the following reactions:

$(1)A+CD($Desired$),r_D=(800e^{-\frac{2000}{T}})C_A^{0\mathrm{.}5}{C}_C,T$ in $K$

$(2)A+C{U}_1($Undesired$)r_{U1}=(10e^{-\frac{300}{T}})C_A{C}_C,T$ in $K$

$($a$)$ What is the corresponding expression of the instantaneous selectivity and what are the temperature

and concentration strategies to maximize the selectivity?

$($b$)$ In reality, there is a subsequent undesired reaction taking place in the reactor:

$(3)C+D{U}_2($Undesired$)r_{U2}=({10}^6{e}^{-\frac{8000}{T}})C_C{C}_D,T$ in $K$

Calculate and plot the actual instantaneous selectivity $S_{\text{D}\text{U}\text{U}},U=U_1+U_2$ and the overall yield tilde$(Y{)}_D$

for a CSTR with equimolar feed and space time $=2$min at $0C,100C,200C,300C,400$

$C,$ and $500C.$ Determine the temperatures that maximize $S_{DU}$ and tilde$(Y{)}_D.$