Starting with $v_{1}$ moles of $B_{1}$ and $v_{2}$ moles of $B_{2}$, show that:
(a) At any value of $\epsilon$,
$$
G=\epsilon\left(v_{3} \mu_{3}+v_{4} \mu_{4}-v_{1} \mu_{1}-v_{2} \mu_{2}\right)+v_{1} \mu_{1}+v_{2} \mu_{2}
$$
(b) At equilibrium,
$$
G(\min )=v_{1} \mu_{1},+v_{2} \mu_{2}+
$$
where the subscript e denotes an equilibrium value.
(c)
$$
\frac{G-G(\min )}{R T}=\epsilon\left(\ln \frac{x_{3}^{n_{3}} x_{4}^{y_{4}}}{x_{1}^{n_{1}} x_{2}^{v_{2}}}-\ln \frac{x_{3 e}^{n} x_{4}^{v_{e}}}{x_{1 e}^{v_{1}} x_{2 e}^{v_{2}}}\right)+\ln x_{1}^{n} x_{2}^{v_{2}}-\ln x_{1}^{*_{1}} x_{2 e^{-}}^{n}
$$
(d) At $e=0$,
$$
\frac{G_{0}-G(\min )}{R T}=\ln \left(\frac{v_{1}}{v_{1}+v_{2}}\right)^{n_{1}}\left(\frac{v_{2}}{v_{1}+v_{2}}\right)^{n_{2}}-\ln x_{i e}^{v_{1}} x_{2 e^{-}}^{k_{2}}
$$
(e) At $\epsilon=1$,
$$
\frac{G_{1}-G(\min )}{R T}=\ln \left(\frac{v_{3}}{v_{3}+v_{4}}\right)^{n}\left(\frac{v_{4}}{v_{3}+v_{4}}\right)^{n_{4}}-\ln x_{3 e}^{n_{3}} x_{4 e^{+}}^{n_{4}}
$$