Air at $1$ bar and $298\mathrm{.}15K(25C)$ is compressed to $5$ bar and $298\mathrm{.}15K$ by two different mechanically reversible processes:

$($a$)$ Cooling at constant pressure followed by heating at constant volume.

$($b$)$ Heating at constant volume followed by cooling at constant pressure.

Calculate the heat and work requirements and $U$ and $H$ of the air for each path. The following heat capacities for air may be assumed independent of temperature: $C_v=20\mathrm{.}78$ and $C_p=29\mathrm{.}10Jmo{l}^{-1}{K}^{-1}$

Assume also for air that PV$/$T is a constant, regardless of the changes it undergoes. At $298\mathrm{.}15K$ and $1$ bar, the molar volume of air is $0\mathrm{.}02479m^{3}mo{l}^{-1}.$