$(25$ pts$)$ I promised you a Dave$-$ane problem $($a made up hydrocarbon molecule$).$ As mentioned before, the molecular weight, critical temperature, critical pressure, accentricity, and heat capacity parameters of Dave$-$ane are molar mass $=38\mathrm{.}0\frac{g}{m}ol,Tc=335K,Pc=4\mathrm{.}6$MPa, omega $=0.$ For heat capacity please use the following parameters: $20,15,0,0.$ This corresponds to a molar heat capacity of Dave$-$ane $80\frac{J}{m}$olK at $400K$ and $95\frac{J}{m}$olK at $500K$

You have a continuous stream of Dave$-$ane at a pressure of $10$MPa and a temperature of $450K.$ The Dave$-$ane goes through a valve at a rate of $2$mole$\frac{s}{sec},$ and at the downstream side of the valve the pressure of the Dave$-$ane is $5$MPa. The valve acts adiabatically.

a$)(5$ pts$)$ Which thermodynamic variable is the same for the Dave$-$ane upstream and downstream of the valve?

b$)(20$ pts$)$ Using the info from part a$)$ and the Peng Robinson EOS $($the PYTHON code$),$ find the temperature of the exiting Dave$-$ane. For partial credit, please show the upstream value of the important thermodynamic quantities that you obtaine from the PREOS.