During an experiment, an ideal gas is found to obey an additional law \(\mathrm{VP}^2=\) constant. The gas is initially at temperature ' \(T\) ' and volume ' V '. What will be the temperature of the gas when it expands to a volume 2 V ?

\(\mathrm{PV} \doteq \mathrm{nRT} \)
\( \mathrm{P}=\frac{\mathrm{RT}}{\mathrm{~V}} ...(i)\)
\( \mathrm{VP}^2=\text { constant } ...\text{(given)}\)
\( \mathrm{V}\left(\frac{\mathrm{RT}}{\mathrm{~V}}\right)^2=\text { constant } ...\text{(from(i))}\)
\( \frac{\mathrm{R}^2 \mathrm{~T}^2}{\mathrm{~V}}=\text { constant } \Rightarrow \frac{\mathrm{T}^2}{\mathrm{~V}}=\text { constant }\) \(\ldots\left(\text{given }T_1=T\right) \)
\( \left(\frac{\mathrm{T}_1}{\mathrm{~T}_2}\right)^2=\frac{\mathrm{V}_1}{\mathrm{~V}_2} \)
\( \frac{\mathrm{~T}_1}{\mathrm{~T}_2}=\sqrt{\frac{\mathrm{V}}{2 \mathrm{~V}}} \)
\( \therefore \mathrm{~T}_2=\sqrt{2} \mathrm{~T}\)