$(18\%)($a$)$ For a real gas obeying the virial equation of state, $P\frac{V}{R}T=Z=1+BP+C{P}^2+D{P}^3+$dots, prove that $ln(\frac{f}{P}){|}_P|=P={\displaystyle \underset{P=0}{\overset{P=P}{}}}\frac{Z-1}{P}dP($f: fugacity$).$

$($b$)$ The change of Gibbs free energy for $1$ mole of pure $i$

$($condensed phase$)$ to $1$ mole of $i$ in a sulution is $\frac{}{b}ar(G{)}_i^M.$ Explain why $\frac{}{b}ar(G{)}_i^M=RTln{a}_i$ and give the definition of $a_i.($c$)$ For a $A-B$ binary system, draw the $"G^{M}vs,x_B"$ plot and explain why

$\frac{}{b}ar(G{)}_B^M=G^M+x_A\frac{d{G}^M}{d{x}_B}$