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)

The rms speed of molecules of gas is
$v_{rms}=\sqrt{\frac{3kT}{m}}=1.732\sqrt{\frac{kT}{m}}...(i)$
The average speed of molecules of gas is
$v_{av}=\sqrt{\frac{8kT}{\pi m}}=1.6\sqrt{\frac{kT}{m}}...(ii)$
also the most probable speed of molecules of gas is
$v_{mp}=\sqrt{\frac{2kT}{m}}=1.414\sqrt{\frac{kT}{m}}...(iii)$
By observing Eqs. (i), (ii) and (iii), we conclude that
$v_{rms}>v_{av}>v_{mp}$