Question

$v_{rms} , v_{av}$ and $v_{mp}$ are root mean square, average and most probable speeds of molecules of a gas obeying Maxwellian velocity distribution. Which of the following statements is correct?

• A. $v_{ rms } < v_{ av } < v_{ mp }$
• B. $v_{ rms }>v_{ av }>v_{ mp }$
• C. $v_{ mp } < v_{ rms } < v_{ av }$
• D. $v_{ mp }>v_{ rms }>v_{ av }$

Answer 1

Answer: (B)

Root mean square speed. The root mean square speed is used to measure the velocity of particles in a gas. It is given by
$v_{ rms }=\sqrt{\frac{3 R T}{M}}=1.732 \sqrt{\frac{R T}{M}}$ ... (1)
where $M$ is molar mass and $R$ is gas constant, $T$ is temperature.
Most probable speed $v_{ p }$, is the speed most likely to be possessed by any molecule in the system.
$v_{ av }=\sqrt{\frac{2 R T}{M}}=1.41 \sqrt{\frac{R T}{M}}$ ... (2)
whereas mean speed is
$v_{ mp }=\sqrt{\frac{8 R T}{\pi M}}=1.6 \sqrt{\frac{R T}{M}}$ ... (3)
From Eqs. (1), (2) and (3), we conclude that
$V_{rms} > V_{av} > V_{mp}$