An insulated box containing a diatomic gas of molar mass $M$ is moving with speed $v$. If the box is suddenly stopped, the temperature of the gas is increased by $\raisebox{1ex}{$M{v}^2$}\!\left/ \!\raisebox{-1ex}{$xR$}\right.$. Value of $x$ is _____

When the box suddenly stops, the KE of the gas due to motion is absorbed by the gas itself increasing its temperature by Δ𝑇 i.e., its internal energy increases by ${\mathrm{\mu C}}_{\mathrm{V}}\mathrm{\Delta T}=\frac{\mathrm{\mu R\Delta T}}{\mathrm{\gamma }-1}$,

 where 𝜇 is the number of moles of the gas.
$\Rightarrow \frac{\mathrm{\mu R\Delta T}}{\mathrm{\gamma }-1}=\frac{1}{2}{\mathrm{mu}}^2$

$\Rightarrow \frac{\mathrm{mR\Delta T}}{\mathrm{M}(\mathrm{\gamma }-1)}=\frac{1}{2}{\mathrm{mu}}^2$

$\Rightarrow \mathrm{\Delta T}=\frac{{\mathrm{Mu}}^2(\mathrm{\gamma }-1)}{2\mathrm{R}}=\frac{{\mathrm{Mu}}^2\left(\frac{7}{5}-1\right)}{2\mathrm{R}}=\frac{{\mathrm{Mu}}^2}{5\mathrm{R}}$