Consider a distillation column operated at $1$ atm that separates a $120$ kmol$/$hr feed

containing $60$ mol$\%$ of hexane $($component $2)$ and $40$ mol$\%$ of heptane $($component $1).$ The feed is a two$-$

phase mixture with $50\%$ in vapor state and $50\%$ in liquid state. The goal is to obtain a distillate product

containing $96$ mol$\%$ of hexane and a bottoms product containing $95$ mol$\%$ of heptane. We know that $20\%$

of the overhead vapor entering the condenser is produced as distillate product. $($a$)$ Calculate the operating lines for rectifying and stripping sections, q$-$line equation, and the reflux ratio.

$($b$)$ Using the McCabe$-$Thiele method, determine the number of equilibrium stages required $($which includes the reboiler stage$).$

$($c$)$ Given that the minimum reflux ratio for this separation is $($RD$)$min $=1\mathrm{.}2208,$ determine the K$-$values of hexane and heptane in this binary system.