Problem $2.(50$ points$).$ The compressor of a closed Brayton cycle operates with argon as the working fluid $($see figure below$).$ The argon enters the compressor after the low$-$temperature heat exchanger at a temperature of $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and a pressure of $0\mathrm{.}2$ MPa $.$ It is compressed to $15$ MPa before it enters the combustion chamber $($the high$-$temperature heat exchanger where it is heated to the maximum temperature of this power cycle$).$ What is the temperature of the argon leaving the compressor, if you can assume that this is a reversible compression, and that argon obeys the Van der Waals equation of state:

$\backslash [$

P$=\backslash$frac$\{$R T$\}\{\backslash$underline$\{$V$\}-$b$\}-\backslash$frac$\{$a$\}\{\backslash$underline$\{$V$\}^\{2\}\}$

$\backslash ]$

To simplify this problem slightly, once you have determined an appropriate value for $\backslash (\backslash$boldsymbol$\{$C$\}\_\{\backslash$boldsymbol$\{$p$\}\}\backslash )$ or $\backslash (\backslash$boldsymbol$\{$C$\}\_\{\backslash$boldsymbol$\{$v$\}\}\backslash )$ you are allowed to assume they are constant in this problem. However, you will need to determine what an appropriate value for the heat capacity is$,$ and also comment on the accuracy of this simplification.

You can earn $5$ bonus points if you do not assume that $\backslash ($ C$\_\{$p$\}\backslash )$ or $\backslash ($ C$\_\{$v$\}\backslash )$ is constant and solve for the final temperature using the full equation for $\backslash ($ C$\_\{$p$\}\backslash )$ or $\backslash ($ C$\_\{$v$\}\backslash ).$