The gas$-$phase reaction between methanol and acetic acid to form methyl acetate and water

$C{H}_{3}OH($A$)+C{H}_3$COO $H(B)C{H}_3$COO $C{H}_3(C)+H_{2}O($D$)$

$($A$)=$ CH$3$OH

$($B$)=$ CH$3$COOH

$($C$)=$ CH$3$COOCH$3$

$($D$)=$ H$2$O

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 $n_{AO},$dots,$n_{10}.$

Please refer to Example $4\mathrm{.}6-2$ in the lecture note.

$($b$)$ If the feed to the reactor contains $n_{A0}=1$mol and $n_{B0}=1$mol and no other species, calculate

the equilibrium fractional conversion.