(a) Deduce Gibbs-Duhem relation.

(b) A binary liquid mixture of (i) benzene and (ii) toluene is at 760 torr. Vapour pressures of benzene and toluene are represented by the Antoine equation.
\[ \begin{aligned} & \log _{10} P_{1}^{\text {sat }}(\text { torr })=6.879-\frac{1196.76}{t\left({ }^{\circ} \mathrm{C}\right)+219.61} \\ & \log _{10} P_{2}^{\text {sat }}(\text { torr })=6.950-\frac{1342.310}{t\left({ }^{\circ} \mathrm{C}\right)+219.187} \end{aligned} \]
Assume bubble and dew temperature at \(95^{\circ} \mathrm{C}\). Calculate bubble and dew composition at that temperature. Total pressure is taken as 760 torr. Mole fractions of benzene in liquid and vapour are 0.4047 and 0.6263 , respectively.