In the first$-$order, isothermal, aqueous$-$phase reaction $AB,$ the half$-$life of $A$ is $1000s.$A conversion of $90\%$ is expected to minimize waste. It is desired to process $1000k\frac{g}{h}$ of $A.$ The molecular weight of $A$ is $100\frac{g}{m}ol$ and the molar concentration of $A$ is $1mo\frac{l}{L}.$

$($a$)$ Determine the rate constant for the reaction together with the units.

$($b$)$ If the reactor is operated as a batch reactor and it is initially charged with pure A$,$ calculate the reactor volume required, neglecting the charge and discharge times?

$($c$)$ If the reactor is to be operated as a continuous$-$stirred tank reactor $($CSTR$)$ with a pure A feed, calculate the required volume for the same conditions given?

$($d$)$ If the batch reactor has a charge time of $3h$ and a discharge time of $3h,$ which mode of operation of the reactor will have the smallest volume? Batch or CSTR or both? Hint: Total batch time $=$ reaction time $+$ downtime $.$

$($e$)$ If instead of one CSTR$,$ five equal$-$sized CSTRs$-$in$-$series were to be utilized, what would be their total volume?