$4-(25$ P$.)$ In a mixture of toluene and benzene, the vapor pressures of pure toluene $($substance A$)$ and pure benzene $($substance B$)$ at different temperatures are listed in the table below. The standard bolling temperature of toluene is $384K$ and that of benzene is $353K.$

$\backslash$table$[[T(K),353,358,363,368,373,378,384],[P_{\text{T}\text{o}\text{l}\text{u}\text{e}\text{n}\text{e}}^{**}(kPa),-,46,54,63,74,86,-],[P_{\text{B}\text{e}\text{m}\text{z}\text{e}\text{n}\text{t}}^{**}(kPa),-,116,133,155,179,206,-],[,,,,,,,],[Y_{\text{B}\text{e}\text{n}\text{t}\text{e}\text{n}\text{e}},,,,,,,]]$

a$)$ To create the temperature$-$composition diagram of the toluene $/$ benzene mixture, calculate the mole fractions of benzene in the liquid and vapor phases for the temperatures given in the table above and add them to the table $($The constant total pressure $=100$kPa $).$

b$)$ Draw the temperature$-$composition diagram using the data in the table and identify the phases in the diagram. Label the regions of the diagram according to which phases are present. $(x_B$ represents the mole fraction of benzene$)$

When the mixture of toluene and benzene, where the mole fraction of benzene is $x_B=0\mathrm{.}54,$ is heated to $367K,$ find the composition of the liquid and vapor phases in equilibrium.

When $x_{\text{t}\text{o}\text{l}\text{v}\text{e}\text{n}\text{e}}=0\mathrm{.}76,$ what is the composition of the vapor in equilibrium with the liquid in the composition?