A rectification column is fed with an equimolal mixture of pentane and hexane. The feed is $50\%$ vaporized. The top product should contain $90$mol$\%$ pentane and the bottom product should contain $90$mol$\%$ hexane. Determine the $($a$)$ minimum reflux ratio, $($b$)$ the minimum number of theoretical plates, $($c$)$ the number of theoretical plates required if $R=2R_{\text{M}\text{I}\text{N}}$ and optimal feed location for the operation. $($d$)$ the number of actual plates required if the column efficiency is $0\mathrm{.}35.$

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.