The above process belongs to the thermodynamic process known as the adiabatic process and the relation between pressure and volume in the adiabatic process is, $P{V}^{\gamma }=K$, now use ideal gas equation, $\Rightarrow P{\left[\frac{\left(nRT\right)}{\left(P\right)}\right]}^{\gamma }=K\Rightarrow \frac{T^{\gamma }}{P^{\gamma -1}}=K$

$$\begin{gathered} \Rightarrow \frac{{T_1}^{\gamma }}{{P_1}^{\gamma -1}}=\frac{{T_2}^{\gamma }}{{P_2}^{\gamma -1}}\Rightarrow \frac{T^{\gamma }}{2^{\gamma -1}}=\frac{{T_2}^{\gamma }}{1^{\gamma -1}}\Rightarrow \frac{T^{{\displaystyle \frac{{}^3}{2}}}}{2^{{\displaystyle \frac{1}{2}}}}=\frac{{T_2}^{\frac{{}^3}{2}}}{1} \\ \Rightarrow {\left(T_2\right)}^3=\frac{1}{2}\times T^3\Rightarrow \left(T_2\right)={\left(\frac{1}{2}\right)}^{1/3}T \end{gathered}$$