A distillation column is fed with a mixture of pentane and hexane. The feed composition is $55\%$ by mole. The top product will contain $90$mol$\%$ pentane and the bottom product will contain $90$mol$\%$ hexane. Determine the $($a$)$ state of the feed, $($b$)$ the number of theoretical plates if there is a total condenser and a partial reboiler, $($c$)$ the number of actual plates required if the column efficiency is $0\mathrm{.}5,($d$)$ the tower height.

Pentane has a relative volatility of $3\mathrm{.}15$ with respect to hexane. For your convenience, the equilibrium curve obtained by plotting $y$ vs$.x$ using this value of relativity is given below. As well, the operating lines for upper and lower sections are shown in the graph.