The elementary liquid$-$phase series reaction

$ABC$

is carried out in a $500L$ batch reactor. The initial concentration of $A$ is $1\mathrm{.}6mo\frac{l}{L}.$ The desired product is $B,$ and separation of the undesired product $C$ is very costly$.$ Because the reaction is carried out at a relatively high temperature, the reaction is easily quenched. Given $k_1=0\mathrm{.}4h^{-1}$ and $k_2=0\mathrm{.}01h^{-1}$ :

a$).($Solving this part by hand$)$ Calculate the maximum $C_B$ and the optimum time to achieve maximum $C_B.$ Start from $\frac{1}{2}\frac{?}{3}$ and the reverse product rule.

b$).($Solving this part using MATLAB$)$ Plot and analyze the concentration of $A,B,$ and $C$ as a function of time. Create your own code or modify Fogler's code from his website if you want.