Question

At a given temperature if $V_{rms}$ is the root mean square velocity of the molecules of a gas and $V_{s}$ the velocity of sound in it, then these are related as $\left( \gamma = \frac{C_{P}}{C_{v}} \right)$

• A. $V_{rms} = V_{s}$
• B. $V_{rms} = \sqrt{\frac{3}{\gamma}} \times V_{s}$
• C. $V_{rms} = \sqrt{\frac{\gamma}{3}} \times V_{s}$
• D. $V_{rms} = \left( \frac{3}{\gamma} \right) \times V_{s}$

Answer 1

Answer: (B)

$V_{rms} = \sqrt{\frac{3}{\gamma}} \times V_{s}$

Answer 2

$V_{rms} = \sqrt{\frac{3P}{\rho}},$ and $V_{Sound} = \sqrt{\frac{\gamma P}{\rho}}$

So $\frac{V_{rms}}{V_{Sound}} = \sqrt{\frac{3}{\gamma}} \Rightarrow V_{rms} = \sqrt{\frac{3}{\gamma}} \times V_{Sound}$