A plant distills $100$mole$\frac{s}{h}$ of a mixture containing $50$mol$\%$ methanol and $50$mol$\%$ water. The overhead product is to contain $98$mol$\%$ methanol and the bottom product $0\mathrm{.}02$mol$\%$ methanol. Calculate the following:

$($a$)$ Slope of q$-$line if the feed is cold and for each mole of feed, $0\mathrm{.}4$ moles of vapor is condensed on the feed plate

$($b$)$ Minimum reflux ratio, if the feed is an equimolar mixture of liquid and vapor.

$($c$)$ No$.$ of theoretical stages required, if the reflux ratio is $2\mathrm{.}5$ times the minimum in case $($b$).$

$($d$)$ Minimum number of stages required for case $($b$).$

Equilibrium data:

$\backslash$table$[[x,0,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9,1\mathrm{.}0],[y,0,0\mathrm{.}417,0\mathrm{.}579,0\mathrm{.}669,0\mathrm{.}729,0\mathrm{.}78,0\mathrm{.}825,0\mathrm{.}871,0\mathrm{.}915,0\mathrm{.}959,1\mathrm{.}0]]$

where $x$ and $y$ are the mole percentages of $A$ in the liquid and vapor respectively.