Question $1(12)$

A $50$:$50$ mixture $($molar basis$)$ of benzene and toluene is fed into a distillation tower at a

rate of $100$ mole$/$minute$.$ The vapor stream V$,$ leaving the distillation column contains $97\%$

benzene. The vapor stream is totally condensed $($that means the liquid leaving the

condenser will also contain $97\%$ benzene$).$ This stream is split into two equal parts $(T$ and

$T.$ One stream is returned to the distillation column, the other one is the top product

stream. The top product stream contains $89\mathrm{.}2\%$ of the benzene fed to the column. A liquid

stream flows from the bottom plate in the column to the reboiler, but this is a partial

reboiler, that means not all the liquid is evaporated. Under conditions where a liquid and a

vapor co$-$exist, there is a relationship between the molar fractions in the gas phase and

liquid phase. We use $x_B$ to denote the molar fraction of benzene in the liquid phase and

$y_B$ is the molar fraction of benzene in the vapor phase. The following relation exists

between the two molar fractions:

$\frac{\frac{y_B}{1-y_B}}{\frac{x_B}{1-x_B}}=2\mathrm{.}25$

Draw a diagram of the problem and annotate it$.(2)$

Use the given information and solve for $T$ and B$.(3)$

Do a benzene balance over the total process and solve for $x_B$ in the bottoms product.

$(3)$

Find $y_B,$ the molar fraction of benzene fed to the reboiler. $(2)$