In this question we start with a gas cylinder with a volume of $10$ dm$3$ containing N$2$ at a pressure of $10$ bar at $300$ K$.$ We call this condition T$1.$ The heat capacity of the gas is Cv$\backslash$deg $=10\mathrm{.}4($J$/($mol$$K$)$; $\backslash$mu $\_($J$-$T$)($K$/$MPa$)=2\mathrm{.}15$; Pext $=1\mathrm{.}0$ bar.

a$)$ Calculate the amount of moles in the cylinder; you can assume that the gas behaves ideally.

b$)$ Initially the volume of the gas is increased to $50$ dm$3$ while the pressure remains the same. This is state T$2.$ Assume that the gas behaves ideally. Calculate w$,$ q$,$ DU and DH for this trajectory $($T$1$ T$2).$ Watch the sign!

c$)$ We then reduce the pressure of the ideal gas, keeping the volume the same, until we reach the original temperature of $300$K$.$ This is state T$3.$ Calculate w$,$ q$,$ DU and DH for this trajectory $($T$2$ T$3).$

d$)$ In the next step we apply thermal insulation. From T$3$ we allow the ideal N$2$ gas to expand reversibly adiabatically and compress it to the original volume of T$1.$ Calculate the final temperature.

In another experiment, all the N$2$ gas from state T$1$ is completely transferred via a porous connection to an expandable $2$nd cylinder under isoenthalpic conditions. The expandable cylinder ultimately has a volume of $50$ dm$3.$

e$)$ Calculate the final temperature assuming that the gas behaves ideally.

f$)$ Calculate the final temperature assuming that the gas does not behave ideally.