$($a$)$ Derive the equation for the effluent concentration from a CFSTR with a $1$st$-$order decay reaction. Assume steady state and that the contaminant chemical is removed in the chemical reaction. Start your answer by writing the mass balance equation. Write the final equation with $C_{in},,$ and $k_0($not with $Q$ or $V$ even though you have $Q$ and $V$ in the derivation$).$

$($b$)$ Repeat Part $($a$)$ for a PFR that has a length of $L.$ Start your answer by defining a control volume and writing the mass balance equation. Note that you are to derive the equation for effluent concentration at $x=L).$ Write the final equation as a function of $C_{in},,$ and $k_0($not with $v,A,Q,L).$

$($c$)$ Using your answers in Parts $($a$)$ and $($b$),$ plot the effluent concentration from CFSTR and PFR vs$.$ the mean hydraulic retention time $()$ up to $60$min. Assume: $C_{in}=150m\frac{g}{L}$;$k_1=0\mathrm{.}25mi{n}^{-1}.$ What is the minimum retention time for $90\%$ removal? Discuss which reactor is more efficient in removing the contaminant between CFSTR and PFR$.$