Consider a semi$-$batch reactor used to carry out the following liquid$-$phase reaction:

$2$A $+$ B $->$ C $+(1)/(2)$ D r $=$ kCA$^2$; k $=5$ x $10-4$ L$/$mol$$s

Initially, the reactor is filled with A at a concentration of $2\mathrm{.}1$ mol$/$L and B at a concentration of $1\mathrm{.}6$ mol$/$L$.$ The initial volume is $4\mathrm{.}1$ m$3.$ B is fed to the reactor at a concentration of $5\mathrm{.}2$ mol$/$L and with a flowrate of $4$ L$/$min$.$

a$.$ Write all of the necessary material balances.

b$.$ Graph the concentrations of A$,$ B$,$ C$,$ and D over time.

c$.$ At what time is a conversion of A of $0\mathrm{.}82$ reached? What is the reactor volume at this time?

d$.$ Using your plot and equations, explain why CB initially decreases and then increases with time.