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 :

(C) $\frac{(\gamma-1)}{2 R} M v^{2} K$

(A) $\frac{(\gamma-1)}{2 \gamma R} M v^{2} K$

(B) $\frac{\gamma M v^{2}}{2 R} K$

(D) $\frac{(\gamma-1)}{2(\gamma+1) R} M v^{2} K$

JEE Main · 2011 · 10 Apr 2011 · Offline · Heat & Thermodynamics

A thermally insulated vessel contains an ideal gas of molecular mass M and ratio of specific heats

γ

. 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) $\frac{( γ - 1 )}{2 γ R} M v^{2} K$
(B) $\frac{γ M v ^{2}}{2 R} K$
(C) $\frac{( γ - 1 )}{2 R} M v^{2} K$ ✓ Correct
(D) $\frac{( γ - 1 )}{2 ( γ + 1 ) R} M v^{2} K$

Answer: (C)

Solution

When the moving vessel is suddenly stopped, the ordered kinetic energy of the gas as a whole becomes internal energy, with no heat exchanged and no external work, so

\frac{1}{2} m v^{2} = Δ U

. For

n

moles the mass is

m = n M

and the internal-energy rise is

Δ U = n C_{v} Δ T = n \frac{R}{γ - 1} Δ T

. Equating

\frac{1}{2} n M v^{2} = n \frac{R}{γ - 1} Δ T

and cancelling

n

gives

Δ T = \frac{( γ - 1 ) M v ^{2}}{2 R}

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option

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JEE Main · 2011 · 10 Apr 2011 · Offline · Heat & Thermodynamics

Solution

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