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A thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by :
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The correct answer is:
$\frac{(\gamma-1)}{2 \mathrm{R}} \mathrm{Mv}^2 \mathrm{~K}$
$\frac{(\gamma-1)}{2 \mathrm{R}} \mathrm{Mv}^2 \mathrm{~K}$
$\mathrm{W}=\Delta \mathrm{U}$
$\frac{1}{2} \mathrm{mv}^2=\mathrm{nC}_{\mathrm{v}} \mathrm{dT}$
$=\frac{\mathrm{m}}{\mathrm{M}} \frac{\mathrm{R}}{\gamma-1} \mathrm{dT}$
$\mathrm{dT}=\frac{\mathrm{M}(\gamma-1) \mathrm{v}^2}{2 \mathrm{R}} \mathrm{K}$
$\frac{1}{2} \mathrm{mv}^2=\mathrm{nC}_{\mathrm{v}} \mathrm{dT}$
$=\frac{\mathrm{m}}{\mathrm{M}} \frac{\mathrm{R}}{\gamma-1} \mathrm{dT}$
$\mathrm{dT}=\frac{\mathrm{M}(\gamma-1) \mathrm{v}^2}{2 \mathrm{R}} \mathrm{K}$
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