In a adiabatic process, the work involved during expansion or compression of an ideal gas is given by:

W_adiabatic  = $\Delta U=n{C}_v\Delta T$

$C_V=\frac{R}{\gamma -1}\Rightarrow \omega =\frac{nR}{\gamma -1}[T_2-T_1]$

$\omega =-P_{ext}[V_2-V_1]=-P_{ext}[\frac{nR{T}_2}{P_2}-\frac{nR{T}_1}{P_1}]=-nR{P}_{ext}[\frac{T_2}{P_2}-\frac{T_1}{P_1}]$

W_isothermal $= -2.303RT\log \frac{V_2}{V_1}.$ applicable for reversible isothermal process