Tank $A,$ having a volume $V,$ is filled with an ideal gas having a mass $m,$ pressure of $5$ bar as shown in the left$-$hand panel. The tank is in equilibrium with the environment at temperature $T.$ A frictionless piston initially rests at the bottom of cylinder $B.$ When the valve is opened, the gas flows slowly into cylinder $B.$ The mass of the piston is such that a pressure of $3$ bar is required to raise the piston. When the valve is opened, gas flows slowly into cylinder B$.$ Eventually, heat transfer with the surroundings brings the gas into equilibrium with the environment as shown in the right$-$hand panel. For this problem, you may assume that the specific heat of the gas at constant volume and pressure are $c_v=\frac{3}{2}R$ and $c_p=\frac{5}{2}R$ respectively.

$($a$)(5$ points$)$ Determine the final mass in volumes A and $B$ in terms of $m.$

$($b$)(5$ points$)$ Determine the volume of cylinder $B$ occupied by the gas in terms of $V.$