(a) A gas obeys the equation of state \(P(V-b)=R T\). For this gas, \(b=0.0391 \mathrm{~L} / \mathrm{mol}\). Calculate the fugacity and fugacity coefficient for the gas at \(1000^{\circ} \mathrm{C}\) and \(1000 \mathrm{~atm}\).

(b) For a binary liquid mixture at constant temperature and pressure, the excess Gibbs energy is given by \[ \frac{G^{\mathrm{E}}}{R T}=\left(-2.6 x_{1}-1.8 x_{2}\right) x_{1} x_{2} \]
Show that this expression satisfies the Gibbs-Duhem equation.