a$)$ A distillation column with a total condenser is separating acetone from ethanol. A distillate concentration of $x_D=0\mathrm{.}90$ mole fraction acetone is desired. Since CMO is valid, $\frac{L}{V}=$ constant. If $\frac{L}{V}$ is equal to $0\mathrm{.}8,$ find the composition of the liquid leaving the fifth stage below the total condenser. $[5$ marks$]$

b$)$ A distillation column separating acetone and ethanol has a partial reboiler that acts as an equilibrium contact. If the bottoms composition is $x_B=0\mathrm{.}13$ mole fraction acetone and the boilup ratio $\frac{?}{\text{b}\text{a}\text{r}}\frac{V}{B}=1\mathrm{.}0,$ find the vapor composition leaving the second stage above the partial reboiler. $[5$ marks$]$

c$)$ The distillation column in parts a and $b$ is separating acetone from ethanol and has $x_D=0\mathrm{.}9,x_B=0\mathrm{.}13,\frac{L}{V}=0\mathrm{.}8$ and $\frac{?}{\text{b}\text{a}\text{r}}\frac{V}{B}=1\mathrm{.}0.$ If the feed composition is $z=0\mathrm{.}3($all concentration are mole fraction of more volatile component$),$ find the optimum feed plate location, total number of stages, and required $q$ value of the feed. marks