Joule and Thomson experimented with expansion valves to study deviations from ideal gas behavior. Consider propane at an initial temperature of $T_1$ and pressure of $P_1$ that is expanded to a pressure of $P_2.$

c$.(5$ points$)$ Determine the exit temperature of the gas if it is an ideal gas.

d$.(15$ points$)$ Write the expression$($s$)$ needed to solve for the exit temperature if the PVT behavior is described by the van der Waals equation.

e$.(5$ points$)$ Describe how you would solve the expression$($s$)$ to get the final exit temperature.