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

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

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