Question

A thermally insulated vessel containing a gas whose molar mass is $M$ and ratio of specific heat $\gamma$ moves with velocity $v$. What is the change in temperature of the gas if the vessel is suddenly stopped?

• A. $\frac{\left(\gamma-1\right)}{2\left(\gamma+1\right)R}Mv^{2}$
• B. $\frac{\left(\gamma-1\right)}{2\gamma R}Mv^{2}$
• C. $\frac{\gamma Mv^{2}}{2R}$
• D. $\frac{\left(\gamma-1\right)Mv^{2}}{2R}$

Answer 1

Answer: (D)

$\frac{\left(\gamma-1\right)Mv^{2}}{2R}$

Answer 2

When the vessel is suddenly stopped, its kinetic energy is used to increase the temperature of the gas. $\therefore\frac{1}{2}mv^{2} = nC_{V}\Delta T$ $\frac{1}{2}mv^{2} = \left(\frac{m}{M}\right)C_{V}\Delta T\quad\left(\because n=\frac{m}{M}\right)$ $\Delta T =\frac{Mv^{2}}{2C_{V}}$ $= \frac{Mv^{2}\left(\gamma-1\right)}{2R}\quad\left(\because C_{V} = \frac{R}{\gamma-1}\right)$