Reversible and irreversible compression of a thermally isolated system.

a$)$ Irreversible Process: an ideal gas given$)$ is compressed by a jump of external pressure from $P_0-P_1$ with $P_1>P_0.$ Calculate the final volume, $V_1$ and temperature, $T.$

b$)$ Reversible Process: repeat the same calculations as above, except now assume $P_{\text{e}\text{x}\text{t}}$ is gradually increased from $P_0{P}_1.$ Find $V_1$ and $T,$ and compare with the above results.