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}$