$1.(15$ pts$)$ Containers $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ in the diagram each contain different ideal gases at equilibrium. Container $\backslash ($ A $\backslash )$ holds its volume fixed, while container $\backslash ($ B $\backslash )$ has a movable piston with a fixed weight. There is no heat transfer$/$loss with the external environment. Between the chambers, there is a barrier that is initially insulating, but later that doesn't allow the gases to mix, but allows very slow heat transfer between the containers $($so that the gases in each chamber change states quasi$-$statically while the heat transfers between them$).$ Container A holds $3$ moles of helium, while container $\backslash ($ B $\backslash )$ holds $4$ moles of molecular $($diatomic$)$ nitrogen, and with the insulating barrier in place, both systems are at equilibrium. When the barrier is removed, the piston slowly begins to sink$,$ and comes to rest when the two containers enclose the same volume.

a$.$ After the piston comes to rest, find the ratio of the pressures of the two containers, $\backslash (\backslash$frac$\{$P$\_\{$A$\}\}\{$P$\_\{$B$\}\}\backslash )$

b$.$ Find the ratio of the temperature changes of the two cases, $\backslash (\backslash$frac$\{\backslash$Delta T$\_\{$A$\}\}\{\backslash$Delta T$\_\{$B$\}\}\backslash ),$ including the sign

c$.$ Find the fraction of work done on the system that is exchanged between the two containers as heat