Starting from the fundamental equation $dg=-sdT+vdP$ and with the aid of the figure above, show or illustrate that at phase equilibrium $($saturated liquid and saturated vapor$)$ the equation below must be satisfied:

${\displaystyle \underset{v_1}{\overset{v_v}{}}}Pdv-P^{\text{s}\text{a}\text{t}}(v_v-v_l)=0$

where $U_V$ and $v_l$ are the saturation vapor and liquid molar volumes at a given temperature, respectively.

$2.$ Calculate the vapor pressure $P^{\text{s}\text{a}\text{t}}$ and the saturated liquid and vapor densities of isobutane at $375K$ using the Redlich$-$Kwong equation of state and the equation above. Compare your calculated values with the experimental data at $375K,$ the experimental vapor pressure is $20\mathrm{.}66$ bar and the saturated vapor and liquid densities are $59\mathrm{.}7k\frac{g}{m^3}$ and $422k\frac{g}{m^3},$ respectively. Explain clearly all the steps in your calculations. You may use Excel or Matlab to solve this problem but make sure that your solution can be easily followed.

$3.$ Calculate the fugacity of isobutane in the vapor and liquid phase in problem $2.$