A feed of component A and B containing $55$ mole$\%$ of A is fed to a distillation tower at a rate of $40$ kmol$/$h$.$ and operating pressure of $1$ atm. The feed is at bubble point. The top product A fraction is $0\mathrm{.}975.$ Also not more than $2\%$ of A in the feed should be allowed to leave at bottom.

Determine:

a$)$ The minimum reflux ratio.

b$)$ The number of theoretical plates if actual reflux ratio is $1\mathrm{.}4$ times the minimum.

c$)$ The slopes of the two operating lines.

Equilibrium data for component A

X $0\mathrm{.}020\mathrm{.}040\mathrm{.}050\mathrm{.}070\mathrm{.}090\mathrm{.}140\mathrm{.}220\mathrm{.}430\mathrm{.}600\mathrm{.}780\mathrm{.}850\mathrm{.}930\mathrm{.}99$

y $0\mathrm{.}040\mathrm{.}090\mathrm{.}120\mathrm{.}150\mathrm{.}230\mathrm{.}320\mathrm{.}450\mathrm{.}680\mathrm{.}800\mathrm{.}910\mathrm{.}950\mathrm{.}970\mathrm{.}99$