Q74 / 90 · JEE Main 2011 · Physics

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 :

  • A
    $\frac{(\gamma-1)}{2 \gamma \mathrm{R}} \mathrm{Mv}^2 \mathrm{~K}$
  • B
    $\frac{\gamma M v^2}{2 \mathrm{R}} \mathrm{K}$
  • Correct answer
    C
    $\frac{(\gamma-1)}{2 \mathrm{R}} \mathrm{Mv}^2 \mathrm{~K}$
  • D
    $\frac{(\gamma-1)}{2(\gamma+1) R} \mathrm{Mv}^2 \mathrm{~K}$

Solution

$\mathrm{W}=\Delta \mathrm{U}$ $\frac{1}{2} \mathrm{mv}^2=\mathrm{nC}_{\mathrm{v}} \mathrm{dT}$ $=\frac{\mathrm{m}}{\mathrm{M}} \frac{\mathrm{R}}{\gamma-1} \mathrm{dT}$ $\mathrm{dT}=\frac{\mathrm{M}(\gamma-1) \mathrm{v}^2}{2 \mathrm{R}} \mathrm{K}$

Hence, the correct answer is option: 'C'.

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