$($a$)$ How much work is required to compress $5\mathrm{.}05$ mol of air at $19\mathrm{.}9$C and $1\mathrm{.}00$ atm to one$-$eighth of the original volume by an isothermal process?

The work required for a tiny volume change is P$$V$.$ In this case the pressure will change as the volume decreases. How can you account for this changing pressure? kJ

$($b$)$ How much work is required to produce the same compression in an adiabatic process?

Can you determine the final temperature of the gas? How does this help determine the total work done on the gas? kJ

$($c$)$ What is the final pressure in part $($a$)?$

atm

$($d$)$ What is the final pressure in part $($b$)?$