The Wilson equation provides the following expression for the excess Gibbs free energy for a binary system:

$(\backslash$Delta G$^$EX$)/$RT$=-$x$\_1$ ln$($x$\_1+\backslash$Lambda $\_1*$ x$\_2)-$ x$\_2$ ln$($x$\_2+\backslash$Lambda $\_21*$x$\_1)$

Here x$1$ and x$2$ are mole fractions, while $\backslash$Lambda $12$ and $\backslash$Lambda $21$ are binary interaction parameters that characterize the specific binary system $($their values are independent of x$\_$i$).$

Can the Wilson equation describe phase separation? If so$,$ under what conditions? Show your work by taking the appropriate derivative of the expression for $\backslash$Delta G$\_$Mixture. $($HINT: For phase operation, $^2$G$/$x$<0\{$instability$\}$; $\backslash$Delta G$\_$Mixture $=\backslash$Delta G$^$EX $+$ x$\_1*$lnx$\_1+$ x$\_2*$lnx$\_2)$