Question

During an experiment, an ideal gas is found to obey an additional law $V{P}^2=constant$. The gas is initially at a temperature $T$ and volume $V$. When it expands to a volume $2V$, the temperature becomes ______ ${}^{\circ }C$.

Answer 1

$PV=RT$ (ideal gas equation)
$\Rightarrow P=\frac{RT}{V}....\left(i\right)$
Given that
$V{P}^2=constant.....\left(ii\right)$
From Eqs. (i) and (ii),
$V\times \frac{R^2{T}^2}{V^2}=constant$
$\therefore \frac{T^2}{V}=constant$
$\therefore \frac{T_1^2}{V_1}=\frac{T_2^2}{V_2}$
$\Rightarrow T_2=T_1\sqrt{\frac{V_2}{V_1}}=T\sqrt{\frac{2V}{V}}=\sqrt{2}T$.