Please compute python code for the following problem:

$($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 $2$nd$.$ 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$=0.$Generate a new plot of P vs V for $450$K $($for the PR EOS$).$ Add to this plot a horizontal line at

P

vap$(450$K$)$ as predicted from the Antoine equation. Confirm that this value of P

vap 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

vap 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

$.$