Gas obey $P^2V =$ constant. The initial temperature and volume are $T_0$ and $V_0$. If gas expands to volume $2V_0$, the final temp is
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$\mathrm{P}^{2} \mathrm{V}=\mathrm{constant}$
$\left(\frac{\mathrm{nRT}}{\mathrm{V}}\right)^{2} \mathrm{V}=\mathrm{constant}$
$\frac{\mathrm{T}^{2}}{\mathrm{V}}=\mathrm{constant}$
$\frac{\mathrm{T}_{1}^{2}}{\mathrm{V}_{1}}=\frac{\mathrm{T}_{2}^{2}}{\mathrm{V}_{2}} \quad \Rightarrow \quad \mathrm{T}_{2}^{2}=\frac{\mathrm{V}_{2}}{\mathrm{V}_{1}} \mathrm{T}_{1}^{2}=2 \mathrm{T}_{0}^{2}$
$\Rightarrow \mathrm{T}_{2}=\sqrt{2} \mathrm{T}_{0}$
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$V\propto {T^{\frac{2}{3}}}$ $[R = 1.99\ cal/mol-K]$