$(1)$ Nitrogen in a piston is expanded at a constant pressure of $100$ bar from a molar volume of $0\mathrm{.}0001$ m$3/$mol to a molar volume of $0\mathrm{.}0003$ m$3/$mol$.$ Treat the nitrogen as a van der Waals gas with a $=0\mathrm{.}1368$ and b $=3\mathrm{.}864$ x $10-5$

$($a$)$ What is the temperature of the gas at the beginning and end of the process?

$($b$)$ Derive the enthalpy departure function for the van der Waals equation of state. Hint: you will want to use the compressibility$-$factor form of the equation of state along with equation $8\mathrm{.}24$ in the text.

$($c$)$ Compute the amount of heat that must be added in this process. Use the polynomial heat capacity for nitrogen in the book.

$($d$)$ How much work is done by this process?

$($e$)$ What is the thermal efficiency of this process $($i$.$e$.$ what fraction of heat in is converted to work?$)$