$1.$ Pure butanol $($C$4$H$9$OH$)$ is fed into a semibatch reactor containing pure ethyl acetate $($CH$3$COOC$2$H$5)$

to produce butyl acetate $($CH$3$COOC$4$H$9)$ and ethanol $($C$2$H$5$OH$)$ according to the following elementary

and reversible reaction:

CH$3$COOC$2$H$5+$C$4$H$9$OH$$CH$3$COOC$4$H$9+$C$2$H$5$OH

The reaction is carried out isothermally at $300$K giving an equilibrium constant of $1\mathrm{.}08$ and a forward

reaction rate constant of $9$x$10-5$ dm$3/$mol$.$s$.$ Initially there is $200$ dm$3$ of ethyl acetate in the reactor and

butanol is fed in at a rate of $0\mathrm{.}05$ dm$3/$s$.$ The feed and initial concentrations of butanol and ethyl acetate

are $10\mathrm{.}93$ mol$/$dm$3$ and $7\mathrm{.}72$ mol$/$dm$3,$ respectively.

$($a$)$ Plot the concentrations of butanol and butyl acetate as a function of time.

$($b$)$ Suggest an optimum reaction time and total reactor volume to maximise the concentration of butyl

acetate and avoid overflowing the vessel.