Q$2.(10$ marks$)$

Extending the fugacity for a pure species to solution, you are given the following relations:

Where is the partial molar residual Gibbs free energy defined as the difference between the partial molar Gibbs free energy for a real gas or liquid and the corresponding ideal gas or liquid state. is the fugacity coefficient of a species $i$ in solution.

Using the residual property relations for $G_R,$ justify the following equations:

$(\frac{dellnhat({)}_i}{\text{d}\text{e}\text{l}\text{P}}{)}_{T,x}=\frac{\frac{?}{\text{b}\text{a}\text{r}}(V{)}_i^R}{RT},(\frac{dellnhat({)}_i}{\text{d}\text{e}\text{l}\text{T}}{)}_{P,x}=-\frac{\frac{?}{\text{b}\text{a}\text{r}}(H{)}_i^R}{R{T}^2}$

$\frac{G^R}{RT}={\displaystyle \underset{i}{\overset{?}{}}}{x}_{i}lnhat({)}_i,{\displaystyle \underset{i}{\overset{?}{}}}{x}_i$dlnhat$({)}_i=0,($ const $T,P)$