Please answer completely and legibly.

For the kinetics scheme A $BC,$ each step being first$-$order, for reaction occurring in a constant$-$

volume batch reactor $($only A present initially$),$

$($a$)$ At what time, $t,$ in terms of $k_1$ and $k_2,$ are $C_A$ and $C_B$ equal $($other than $t),$ and what is the

condition for this to happen?

$($b$)$ What is the value oft, when $k_1=k_2?$

$($c$)$ Show that $C_B$ has an inflection point at $2t_{\text{m}\text{a}\text{x}}.$

$($d$)$ Calculate $k_1{t}_{\text{m}\text{a}\text{x}}$ and $\frac{C_{\text{B}\text{m}\text{a}\text{x}}}{C_{Ao}}$ for each of the cases $($i$)K=\frac{k_2}{k_1}=10,($ii$)K=1,$ and $($iii$)K=0\mathrm{.}1.$

$($e$)$ From the results in $($d$),$ describe how $t_{\text{m}\text{a}\text{x}}$ and $\frac{C_{\text{B}\text{m}\text{a}\text{x}}}{C_{A_0}}$ change with decreasing $K.$