Part C The yield of component B $($derived in problem $5)$ is given by

$\frac{C_B}{C_{A0}}=\frac{k_1}{k_2-k_1}(e^{-k_{1}t}-e^{-k_{2}t})$

$($a$)$ Calculate the optimum time $(t_{\text{o}\text{p}\text{t}})$ that maximizes the yield of B$.$

$t_{\text{o}\text{p}\text{t}}=\frac{1}{k_2-k_1}ln\frac{k_2}{k_1}$

$($b$)$ Find the maximum yield of B and optimal time when $($i$)k_1=2k_2($ii$)k_1=k_2.($Hint: When taking the limit $k_1{k}_2$ use Hospital's rule$)$