Question

If \( v_p, v_{\text{rms}}, v_p \) represent the mean speed, root mean square speed, and most probable speed of the molecules in an ideal monoatomic gas at temperature \( T \) and \( m \) is the mass of the molecule, then

• A. \( v_p<v_{\text{rms}}<v_p \)
• B. No molecule can have a speed greater than \( \sqrt{2} v_{\text{rms}} \)
• C. No molecule can have a speed less than \( v_p/\sqrt{2} \)
• D. None of the above

Hint:

In the Maxwell-Boltzmann distribution of molecular speeds, the most probable speed is less than the RMS speed.

Answer 1

The Correct Option is A

Step 1: Explanation of speeds.
The most probable speed, \( v_p \), is always less than the root mean square speed, \( v_{\text{rms}} \), which is in turn less than the average speed. This is a fundamental result in the Maxwell-Boltzmann distribution.

Step 2: Conclusion.
The correct inequality is \( v_p<v_{\text{rms}}<v_p \).