$($b$)$ A mixture of hydrocarbons containing $32$ mole $\%$ n$-$hexane, $38\%$ n$-$heptane and $30\%$ n$-$octane

is to be distilled to obtain a distillate product with $0\mathrm{.}05$ mole fraction of $n-$heptane and a bottom

product with $0\mathrm{.}05$ mole fraction of $n-$hexane. The column operating pressure is $1\mathrm{.}2$atm and $T=$

$102\mathrm{.}2C\mathrm{.}40\%$ of the feed is to be vaporized. Calculate the product composition and the minimum

number of ideal plates.