Adiabatic reversible expansion of a gas is represented by

$$\begin{gathered} {\mathrm{T}}_1{\mathrm{V}}_1^{\left(\mathrm{\gamma }-1\right)}={\mathrm{T}}_2{\mathrm{V}}_2^{\left(\mathrm{\gamma }-1\right)} \\ \mathrm{or} \frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}={\left(\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1}\right)}^{\left(\mathrm{\gamma }-1\right)} ....\left(1\right) \end{gathered}$$

According to ideal gas equation for 1 mole

PV = RT

$\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1}=\left(\frac{{\mathrm{T}}_2}{{\mathrm{T}}_1}\right) \left(\frac{{\mathrm{P}}_1}{{\mathrm{P}}_2}\right) ....\left(2\right)$

From equations (1) and (2), we get

$$\begin{gathered} \frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}={\left(\frac{{\mathrm{T}}_2}{{\mathrm{T}}_1}\right)}^{\mathrm{\gamma }-1} {\left(\frac{{\mathrm{P}}_1}{{\mathrm{P}}_2}\right)}^{\mathrm{\gamma }-1} \\ {\left(\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}\right)}^{\mathrm{\gamma }} = {\left(\frac{{\mathrm{P}}_2}{{\mathrm{P}}_1}\right)}^{\left(1-\mathrm{\gamma }\right)} \end{gathered}$$