A $50$:$50$ mixture $($molar basis$)$ of benzene and toluene is fed into a distillation tower at a rate of $100$mol$\frac{e}{m}i$

$ute.$ 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. $(2)$