(a) The following equations have been proposed to represent activity coefficient data for a system at fixed \(T\) and \(P\).

\[ \begin{aligned} & \ln \gamma_{1}=x_{2}^{2}\left(0.5+2 x_{1}\right) \\ & \ln \gamma_{2}=x_{1}^{2}\left(1.5-2 x_{2}\right) \end{aligned} \]

Do these equations satisfy Gibbs-Duhem equation? Determine an expression for \(\frac{G^{\mathrm{E}}}{R T}\) for the system.

(b) Express volumes \(\left(\mathrm{cm}^{-3} \mathrm{~mol}^{-1}\right)\) for the ethanol (1)-methyl butyl ether (2) system at \(25^{\circ} \mathrm{C}\) are given by the equation \[ V^{\mathrm{E}}=x_{1} x_{2}\left[-1.026-0.220\left(x_{1}-x_{2}\right)\right] \]
If \(V_{1}=58.63\) and \(V_{2}=118.46 \mathrm{~cm}^{-3} \mathrm{~mol}^{-1}\), what volume of mixture is formed when \(1000 \mathrm{~m}^{3}\) each of pure species 1 and 2 are mixed at \(25^{\circ} \mathrm{C}\) ?