$($b$)$ Generate a new plot of $P$ vs $V_{?}$ for $450K($for the PR EOS$).$ Add to this plot a horizontal line at $P^{\text{v}\text{a}\text{p}}(450K)$ as predicted from the Antoine equation. Confirm that this value of $P^{\text{v}\text{a}\text{p}}$ crosses the $P$ vs $V_{?}$ plot at $3$ points. You might need to tweak the range of your $x-$axis to see this. Then, using the equal$-$area construction show that the $P^{\text{v}\text{a}\text{p}}$ obtained from the Antoine equation satisfies the thermodynamic criteria for phase equilibrium $T^L=T^V,P^L=P^V,$ and $G_{{?}^{?}}(L)=G_{{?}^{?}}(V).$

Hints:

Be careful with your units. You might also need to tweak the range of $V_{?}$ to see the three intersections.

As discussed in class, you will want to divide your $P$ vs $V_{?}$ curve into two regions, one part of the "loop" that is less than $P^{\text{v}\text{a}\text{p}}$ and another part of the "loop" that is greater than $P^{\text{v}\text{a}\text{p}}.$

To calculate the area of each region numerically use the numpy $.$ trapz$()$ function. Note that this function takes the $y-$values as the $1$st argument, and the $x-$values as then $2nd.$ If you're having trouble with numpy. trapz try using it to calculate the area beneath a simple curve $($i$.$e$.y=x$ or $y=sin(x)$ and then move onto calculating the area within the "loop".

Don't forget to add$/$subtract the value of $P^{\text{v}\text{a}\text{p}}$ prior to performing the integration. The "height" of your trapezoids in the integration step should be relative to $P^{\text{v}\text{a}\text{p}},$ notP$=0.$

$($c$)$ Determine the vapor pressure of pure benzene at $300K,350K,400K,450K$ and $500K$ using only the Peng$-$Robinson equation of state. For iterations, use a convergence criterion of ${10}^{-5}$ for the iteration error. Next, use the Antoine equation to predict $P^{\text{v}\text{a}\text{p}}$ at these temperatures, and generate a plot comparing the predictions from the PR EOS and the Antoine equation. How do the predictions of $P^{\text{v}\text{a}\text{p}}$ obtained from each method compare for each of these temperatures?

Hint: Use the Python from the Vapor$-$Liquid Phase Equilibrium class example as a starting point for these calculations.

Note: This problem should be solved entirely with Python and$/$or a Jupyter notebook.