Obtain an expression for the work done by an ideal gas during adiabatic change and explain.

Consider certain amount of gas of volume V₁ at a pressure P₁. The temperature of the gas is T₁. Let the gas expand adiabatically to the volume V₂. The pressure and temperature change to P₂ and T₂.
The workdone during a small change in volume at a pressure P, dW = Pdv
The total workdone during the change in volume from V₁ to V₂ is given as
$\mathrm{W}=\int \mathrm{dW}={\int }_{{\mathrm{V}}_1}^{{\mathrm{V}}_2} \mathrm{Pdv}$
but ${\mathrm{pv}}^{\gamma }=\mathrm{const}\left(\mathrm{k}\right);\mathrm{p}=\frac{\mathrm{k}}{{\mathrm{v}}^{\gamma }}$

$\begin{array}{l}\therefore \mathrm{w}={\int }_{{\mathrm{V}}_1}^{{\mathrm{V}}_2} \frac{\mathrm{k}}{{\mathrm{v}}^{\mathrm{\gamma }}}\mathrm{dv}={\int }_{{\mathrm{V}}_1}^{{\mathrm{V}}_2} {\mathrm{kv}}^{-\mathrm{\gamma }}\mathrm{dv}=\mathrm{k}{\left[\frac{{\mathrm{v}}^{-\mathrm{\gamma }+1}}{-\mathrm{\gamma }+1}\right]}_{{\mathrm{V}}_1}^{{\mathrm{V}}_2}\\ =\frac{\mathrm{k}}{1-\mathrm{\gamma }}\left[{\mathrm{V}}_2^{-\mathrm{\gamma }+1}-{\mathrm{V}}_1^{-\mathrm{\gamma }+1}\right]\\ =\frac{1}{1-\mathrm{\gamma }}\left[{\mathrm{kV}}_2^{-\mathrm{\gamma }+1}-{\mathrm{kV}}_1^{-\mathrm{\gamma }+1}\right]\\ =\frac{1}{1-\mathrm{\gamma }}\left[{\mathrm{p}}_2{\mathrm{V}}_2^{\mathrm{\gamma }}{\mathrm{V}}_2^{-\mathrm{\gamma }+1}-{\mathrm{p}}_1{\mathrm{V}}_1^{\mathrm{\gamma }}{\mathrm{V}}_1^{-\mathrm{\gamma }+1}\right]\\ =\frac{1}{1-\mathrm{\gamma }}\left[{\mathrm{p}}_2{\mathrm{V}}_2-{\mathrm{p}}_1{\mathrm{V}}_1\right]\end{array}$
but ${\mathrm{p}}_1{\mathrm{V}}_1={\mathrm{RT}}_1\text{ and }{\mathrm{p}}_2{\mathrm{V}}_2={\mathrm{RT}}_2$
$\therefore \mathrm{w}=\frac{1}{1-\mathrm{\gamma }}\left[{\mathrm{RT}}_2-{\mathrm{RT}}_1\right];\mathrm{w}=\frac{\mathrm{R}}{1-\mathrm{\gamma }}\left[{\mathrm{T}}_2-{\mathrm{T}}_1\right]$