The Antoine Equation is an empirical relationship commonly used to compute the vapor pressure, $P^{\text{v}\text{a}\text{p}}(T)$ of different components.

$ln{P}^{\text{v}\text{a}\text{p}}=A-\frac{B}{T+C}$

where $P^{\text{v}\text{a}\text{p}}$ is given in kPa and $T$ is given in $K.$ The parameters for benzene are $A=13\mathrm{.}7819,B=2726\mathrm{.}81,C=-55\mathrm{.}578.$

The vapor pressure can also be computed using the generalized Peng$-$Robinson equation of state. Properties of benzene can be obtained from Table $6\mathrm{.}6-1.$

$($a$)$ Using the generalized Peng$-$Robinson equation of state and the critical parameters for benzene, generate plots of $P$ vs $V_{?}$ for $400K,450K,500K,550K$ and $600K.$ Using these graphs, what is the approximate location of the critical temperature? Explain your choice. Be sure to label your axis $($with appropriate units$).$

Hints:

The internet is your friend. If you're not familiar with these python commands look them up$.$

You will want to plot your $x-$axis $V_{?}$ on a $log$ scale. The numpy $.$ logspace $()$ function is useful for evenly spacing points in log space. You can change your $x-$axis to be on a log axis using matplotlib.pyplot.xscale$(\text{'}$log$\text{'}).$

The pressure predicted by the PR EOS explodes $($approaches infinity$)$ as $V_{}b.$ Furthermore, the PR EOS is invalid for $V_{?}{V}_{?}V=n(-4\mathrm{.}05,-1,500)V_{.}Be$ careful with the range $of{V}_{!}$ A reasonable starting guess $isto$ generate your values $of{V}_{u}singV=n.logspace(-4\mathrm{.}05,-1,500),$ though you might need $to$ tweak these ranges throughout this question.

Python functions and for loops are your friend. Use them!