Q: 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 surrounding, its temperature increases by

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

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

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

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

Ans: (d)

Sol: According to First Law of Thermodynamics ,

∆Q = ∆U + ∆W

0 = ∆U + ∆W

$\large 0 = n C_v \Delta T + (0 – \frac{1}{2}M v^2)$

$\large \Delta T = \frac{\frac{1}{2}M v^2}{n C_v} $

$\large \Delta T = \frac{\frac{1}{2}M v^2}{n \frac{R}{\gamma -1}} $

$\large \Delta T = \frac{\gamma -1 }{2 n R} M v^2 $

$\large \Delta T = \frac{\gamma -1 }{2 R} M v^2$ ; (n = 1)