#$5.$ The following temperature$-$composition data is for a toluene$-$octane mixture at $1$atm pressure. $x_T$ is the mole fraction of toluene in the liquid, and $y_T$ is the mole fraction of toluene in the vapor phase.

$\backslash$table$[[{(}^{0}C),110\mathrm{.}9,112\mathrm{.}0,114\mathrm{.}0,115\mathrm{.}8,117\mathrm{.}3,119\mathrm{.}0,120\mathrm{.}0,123\mathrm{.}0],[x,0\mathrm{.}908,0\mathrm{.}795,0\mathrm{.}615,0\mathrm{.}527,0\mathrm{.}408,0\mathrm{.}300,0\mathrm{.}203,0\mathrm{.}097],[yT,0\mathrm{.}923,0\mathrm{.}836,0\mathrm{.}698,0\mathrm{.}624,0\mathrm{.}527,0\mathrm{.}410,0\mathrm{.}297,0\mathrm{.}164]]$

The boiling points are $110\mathrm{.}6C$ for toluene and $125\mathrm{.}6C$ for octane.

a$)$ Plot the liquid$-$vapor phase diagram for this mixture. Use graph paper or a computer plotting program. Label the phases present in the different regions of your graph.

b$)$ What is the composition of the vapor in equilibrium with liquid of composition $x_T=0\mathrm{.}250?$

c$)$ Suppose we start with a liquid mixture containing a mole fraction $0\mathrm{.}300$ of toluene. At what temperature will this liquid begin to boil $($at $1$atm pressure$)?$ What is the composition of the first drop of distillate? At what temperature would one achieve equal amounts of liquid and vapor? Approximately how many theoretical plates are needed in a fractional distillation process if we wish to reach $x_T=0\mathrm{.}900?($hint: the answers are not the result of a calculation; they are just estimates by looking at your graph$)$