A gas with $\dfrac {C_{P}}{C_{V}} = \gamma$ goes from an initial state $(p_{1}, V_{1}, T_{1})$ to a final state $(p_{2}, V_{2}, T_{2})$ through an adiabatic process. The work done by the gas is

$w=\int_{0}^{\infty} d \omega=\int_{v_{1}}^{v_{2}} P d V$

for adiabatic change.

$\begin{array}{l} P V^{V}=\text { const }=K \text {. } \\ \text { or, } P=\frac{k}{v^{r}} \\ W=\int_{V_{1}}^{V_{2}} \frac{k}{V^{r}} d V=k \int_{V_{1}}^{v_{2}} V^{-r} d V \\ \text { So } \omega=k\left|\frac{v^{1-r}}{1-r}\right| \\ =\frac{k}{1-\gamma}\left[v_{2}^{1-v}-v_{1}^{1-v}\right] \\ \end{array}$

Now $\quad P_{1} V_{1}{ }^{2}=P_{2} V_{2}{ }^{2}=k$.

$\begin{array}{l} w=\left[\frac{1}{1-\gamma}\left[P_{2} V_{2}^{V} \cdot V_{2}^{1-V}-P_{1} V_{1}^{\gamma} \cdot V_{1}^{1-V}\right]\right. \\ W=\left[\frac{1}{1-\gamma}\right]\left[P_{2} V_{2}-P_{1} V_{1}\right] \\ W=\frac{1}{1-\gamma}\left[n R T_{2}-n R T_{1}\right] \\ \omega=\frac{n R}{1-\gamma}\left[T_{2}-T_{1}\right] \end{array}$