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 $!,$#$$\%=1\mathrm{.}2208,$ determine the K$-$values of hexane and heptane in this binary system.