The process during which there is no exchange of heat takes place between a system and its environment is known as an adiabatic process.

From first law of thermodynamics,

$\Delta \mathrm{Q}=\Delta \mathrm{U}+\Delta \mathrm{W}, \Delta \mathrm{Q}=0$ for adiabatic process

$\therefore \Delta \mathrm{U}=-\Delta \mathrm{W}$ or $-\Delta \mathrm{U}=\Delta \mathrm{W}$

Mean work done by gas decreases the internal energy and work done on the gas increases the internal energy.

For an ideal gas in adiabatic process,

$\mathrm{PV}^{\gamma}=$ constant

Where $\gamma$ is the ratio of specific heats at constant pressure and at constant volume.

$\therefore \gamma=\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}$

The work done in an adiabatic change of an ideal gas from the state $\left[\mathrm{P}_{1}, \mathrm{~V}_{1}, \mathrm{~T}_{1}\right]$ to the state $\left[\mathrm{P}_{2}, \mathrm{~V}_{2}, \mathrm{~T}_{2}\right]$

$\Delta \mathrm{W}=\mathrm{P} \Delta \mathrm{V}$

The total work done $\mathrm{W}$ by total change,

$\mathrm{W}=\int_{\mathrm{V}_{1}}^{\mathrm{V}_{2}} \mathrm{P} d \mathrm{~V}$

but $PVY$ = constant

$\therefore \mathrm{P}=\frac{\text { constant }}{\mathrm{V}^{\gamma}} \ldots(2)$

$\therefore$ Putting the value of equ. $(2)$ in equ. $(1)$

$\mathrm{W}=$ constant $\times \int_{\mathrm{V}_{1}}^{\mathrm{v}_{2}} \frac{d \mathrm{~V}}{\mathrm{~V}^{\gamma}}$

$=$ constant $\times\left[\frac{\mathrm{V}^{\gamma+1}}{\mathrm{~V}_{2}^{\nearrow}-\gamma+1}\right]_{\mathrm{V}_{1}}^{\mathrm{V}_{2}}$

$=\frac{\text { constant }}{1-\gamma} \times\left[\mathrm{V}_{2}^{-\gamma+1}-\mathrm{V}_{1}^{\gamma+1}\right]$

$=\frac{1}{1-\gamma}\left[\frac{\text { constant }}{\mathrm{V}_{2}^{\gamma-1}}-\frac{\text { constant }}{\mathrm{V}_{1}^{\gamma-1}}\right]$