Question:
The ratio of velocity of sound to the rms velocity of gas molecules in a diatomic gas is:
The ratio of velocity of sound to the rms velocity of gas molecules in a diatomic gas is:
Show Hint
The ratio \( \frac{v_s}{v_{\text{rms}}} = \sqrt{\frac{\gamma}{3}} \) is a universal relationship for any ideal gas. This means the speed of sound will always be slightly less than the average thermal speed of the molecules carrying it.
Updated On: Jun 8, 2026
- \( \sqrt{\frac{9}{5}} \)
- \( 5/9 \)
- \( \frac{7}{15} \)
- \( \frac{15}{7} \)
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The Correct Option is C
Solution and Explanation
Concept:
The velocity of a sound wave passing through a gas medium and the root-mean-square speed of its individual molecules are given by the standard kinetic equations:
\[
v_s = \sqrt{\frac{\gamma RT}{M}}, \quad v_{\text{rms}} = \sqrt{\frac{3RT}{M}}
\]
Step 1: Determining the specific heat ratio \( \gamma \) for a diatomic gas.
A diatomic gas molecule has 5 degrees of freedom at standard temperatures (3 translational and 2 rotational). Its specific heat ratio is: \[ \gamma = 1 + \frac{2}{f} = 1 + \frac{2}{5} = \frac{7}{5} = 1.4 \]
Step 2: Calculating the structural ratio.
Dividing the sound velocity equation by the rms speed equation causes the common gas constants to cancel out: \[ \frac{v_s}{v_{\text{rms}}} = \frac{\sqrt{\frac{\gamma RT}{M}}}{\sqrt{\frac{3RT}{M}}} = \sqrt{\frac{\gamma}{3}} \] Substitute \( \gamma = \frac{7}{5} \): \[ \frac{v_s}{v_{\text{rms}}} = \sqrt{\frac{7/5}{3}} = \sqrt{\frac{7}{15}} \] This corresponds to option choice (C).
Step 1: Determining the specific heat ratio \( \gamma \) for a diatomic gas.
A diatomic gas molecule has 5 degrees of freedom at standard temperatures (3 translational and 2 rotational). Its specific heat ratio is: \[ \gamma = 1 + \frac{2}{f} = 1 + \frac{2}{5} = \frac{7}{5} = 1.4 \]
Step 2: Calculating the structural ratio.
Dividing the sound velocity equation by the rms speed equation causes the common gas constants to cancel out: \[ \frac{v_s}{v_{\text{rms}}} = \frac{\sqrt{\frac{\gamma RT}{M}}}{\sqrt{\frac{3RT}{M}}} = \sqrt{\frac{\gamma}{3}} \] Substitute \( \gamma = \frac{7}{5} \): \[ \frac{v_s}{v_{\text{rms}}} = \sqrt{\frac{7/5}{3}} = \sqrt{\frac{7}{15}} \] This corresponds to option choice (C).
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