A distillation column composed of theoretical plates is used to separate a benzene$-$toluene mixture. The

feed is $500mo\frac{l}{h}$ introduced as saturated liquid and contains $40$mol$\%$ benzene. During a run, liquid and

vapor samples of streams leaving a stage in the rectifying section are taken, and the analyses are y$=0\mathrm{.}63$ mole fraction benzene, x$=0\mathrm{.}51$ mole fraction benzene

The total condenser produces a stream of $96$mol$\%$ benzene, and the thermosyphon reboiler produces a liquid

$($bottom$)$ stream with $5$mol$\%$ benzene. The equilibrium curve for the benzene follows the equation y$=(2\mathrm{.}05-$x$)$

$($a$)$ At what reflux ratio is the column operating?

$($b$)$ Determine the distillate and bottoms flow rates.

$($c$)$ Determine the number of theoretical stages needed and feed plate location.

$($d$)$ What is the ratio of reflux ratio $($under operating conditions$)$ to that under minimum reflux condition.

In other words what is $\frac{R}{R_m}?$

$($e$)$ Estimate the condenser heat duty.