$($c$)$ Derive an expression for the free energy of mixing of an ideal solution in terms of the mole fraction of the components, starting from the expression:

$G_m=G_m+RTln(\frac{p}{p})$ where $G_m$ is the molar Gibbs free energy, $G_m$ is the Gibbs free energy at standard pressure and $p$ and $p$ are the actual and standard pressures, respectively.

$($d$)$ Explain why two ideal solutions will tend to mix with one another over the whole composition range

$($i$)$ from the free energy of mixing;

$($ii$)$ in terms of the entropy and enthalpy of mixing ideal solutions.

$($e$)$ The enthalpy of mixing a regular solution is given by: $H_{\text{m}\text{i}\text{x}}={}_A{}_B$ where $$ is some interaction parameter, and $$ is the mole fraction of components A and $B.$ Derive an expression for the free energy of mixing a regular solution. Determine the value to two significant figures of $$ at which de$-$mixing will start to occur at $100C.$