#$19\mathrm{.}51$ :: An air pump has a cylinder $0\mathrm{.}250$ m long with a movable piston. The pump is used to compress air from the atmosphere $($at absolute pressure $1\mathrm{.}01{10}^{5}Pa)$ into a very large tank at gauge pressure $3\mathrm{.}80{10}^{5}Pa.($For air, use $C_V=20\mathrm{.}8\frac{J}{\text{m}\text{o}\text{l}\text{K}}.)$ Remember to solve as algebraically as possible before plugging in numbers.

$($a$)$ The piston begins the compression stroke at the open end of the cylinder. How far down the length of the cylinder has the piston moved when $"$ air first begins to flow from the cylinder into the tank? Assume that the compression is adiabatic.

$($b$)$ If the air is taken into the pump at $27C,$ what is the temperature of the compressed air?

$($c$)$ How much work does the pump do in putting $20\mathrm{.}0$ mol of air into the tank?

$($d$)$ Draw a $pV-$diagram of this process.

Be sure to use the $1$st Law for part c$.$