First slide

Specific heat and degrees of freedom

Question

Certain amount of an ideal gas are contained in a closed vessel. The vessel is moving with a constant velocity v. The molecular mass of gas is M. The rise in temperature of the gas when the vessel is suddenly stopped is $(γ = C_{P} / C_{V})$

Moderate

A

$\frac{{Mv}^{2}}{2 R (γ + 1)}$
B

$\frac{{Mv}^{2} (γ - 1)}{2 R}$
C

$\frac{{Mv}^{2}}{2 R (γ + 1)}$
D

$\frac{{Mv}^{2}}{R (γ + 1)}$

Solution

If m is the total mass of the gas then its kinetic energy $= \frac{1}{2} {mv}^{2}$
When the vessel is suddenly stopped then total kinetic energy will increase the temperature of the gas. Hence
$\begin{array}{l} \frac{1}{2} {mv}^{2} = {μC}_{v} ΔT = \frac{m}{M} C_{v} ΔT [As C_{v} = \frac{R}{γ - 1}] \\ \Rightarrow \frac{m}{M} \frac{R}{γ - 1} ΔT = \frac{1}{2} {mv}^{2} \Rightarrow ΔT = \frac{{Mv}^{2} (γ - 1)}{2 R} \end{array}$

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