$4\mathrm{.}64.$ The gas$-$phase reaction between methanol and acetic acid to form methyl acetate and water

$C{H}_{3}OH+C{H}_3\frac{C}{O}OHC{H}_3\frac{C}{O}OC{H}_3+H_{2}O$

$($A$)$

$($B$)$

$($C$)$

$($D$)$

takes place in a batch reactor. When the reaction mixture comes to equilibrium, the mole fractions of the four reactive species are related by the reaction equilibrium constant

$K_y=\frac{y_C{y}_D}{y_A{y}_B}=4\mathrm{.}87$

$($a$)$ Suppose the feed to the reactor consists of $n_{A0},n_{B0},n_{C0},n_{D0},$ and $n_{10}$ gram$-$moles of $A,B,C,D,$ and an inert gas, I, respectively. Let $$ be the extent of reaction. Write expressions for the gram$-$moles of each reactive species in the final product, $n_A(),n_B(),n_C(),$ and $n_D().$ Then use these expressions and the given equilibrium constant to derive an equation for ${}_e,$ the equilibrium extent of reaction, in terms of $n_{A0},$dots,$n_{IO}.($see Example $4\mathrm{.}6-2.)$

$($b$)$ If the feed to the reactor contains equimolar quantities of methanol and acetic acid and no other species, calculate the equilibrium fractional conversion.

$($c$)$ It is desired to produce $70$mol of methyl acetate starting with $75$mol of methanol. If the reaction proceeds to equilibrium, how much acetic acid must be fed? What is the composition of the final product?

$($d$)$ Suppose it is important to reduce the concentration of methanol by making its conversion at equilibrium as high as possible, say $99\%.$ Again assuming the feed to the reactor contains only methanol and acetic acid and that it is desired to produce $70$mol of methyl acetate, determine the extent of reaction and quantities of methanol and acetic acid that must be fed to the reactor.

$($e$)$ If you wanted to carry out the process of Part $($b$)$ or $($c$)$ commercially, what would you need to know besides the equilibrium composition to determine whether the process would be profitable? $($List several things.$)$