$18-2$ The hydrolysis of methyl acetate $($A$)$ in dilute aqueous solution to form methanol $($B$)$ and acetic acid $(C)$ is to take place in a batch reactor operating isothermally. The reaction is reversible, pseudo$-$first$-$order with respect to acetate in the forward direction $($:$1\mathrm{.}82{10}^{-4}{s}^{-1}\},$ and first$-$order with respect to each product species in the reverse direction $)=(4\mathrm{.}49{10}^{-4}(L)mo{l}^{-1}{s}^{-1}.$ The feed contains only $A$ in water, at a concentration of $0\mathrm{.}050mol{L}^{-1}.$ Determine the size of the reactor required, if the rate of product formation is to be $100mol{h}^{-1}$ on a continuing basis, the down$-$time per batch is $30$min, and the optimal fractional conversion $($i$.$e$.,$ that which maximizes production$)$ is obtained in each cycle.

$18-3$ The reaction in problem $18-2$ is to be conducted in an isothermal $60-$L PFR$,$ with a feed rate $(F_{Ao})$ of $0\mathrm{.}75molmi{n}^{-1}.$ All other data $(c_{Ao},k_r,k_f)$ remain the same.

$($a$)$ Calculate the fractional conversion $f_A$ at the outlet of the reactor.

$($b$)$ What is the maximum possible value of $f_A,$ regardless of reactor type or size?