$5.(25$ pts$)$ This is a T$-$S diagram for a $\backslash (0\mathrm{.}012\backslash$mathrm$\{$~m$\}^\{3\}\backslash )$ vessel containing $3$ moles of helium gas that is heated quasi$-$statically from state A to state B$,$ with the temperature of the two states as labeled. There is a piston trapping the gas within this vessel, but it is not allowed to move until part d$.$

a$.$ Find the change in pressure for the gas when it goes from state $\backslash ($ A $\backslash )$ to state $\backslash ($ B $\backslash )$

b$.$ Find the area under the curve from $\backslash ($ A $\backslash )$ to $\backslash ($ B $\backslash )$

c$.$ Find the entropy difference between state A

d$.$ The system next follows a quasi$-$static process by continuing from state B down to state C $.$ Find the work done and heat exchanged during this process. Indicate whether the work is done on or by the gas and whether the heat transfers in or out of the gas.

e$.$ Find the efficiency of an engine that runs quasi$-$statically through the cycle $\backslash ($ A $\backslash$rightarrow B $\backslash$rightarrow $\backslash )\backslash ($ C $\backslash$rightarrow A $\backslash )$