Since we now know how to relate temperature and kinetic energy, we can relate temperature to the velocity of gas molecules. Note since these are distributions the values (\(E_{\rm k}\) or velocity) that we are talking about are always averages.
\[E_{\rm k}={3\over 2}\,R\,T\]
\[E_{\rm k}={1\over 2}\,m\,v^2\]
setting these two equal and solving for the average square velocity we get
\[v^2 = {3\,R\,T \over m}\]
The root mean square velocity or \(v_{\rm rms}\) is the square root of the average square velocity and is
\[v_{\rm rms} = \sqrt{3\,R\,T\over M}\]
Where \(M\) is equal to the molar mass of the molecule in kg/mol. The root mean square velocity is the square root of the average of the square of the velocity. As such, it has units of velocity. The reason we use the rms velocity instead of the average is that for a typical gas sample the net velocity is zero since the particles are moving in all directions. This is a key formula as the velocity of the particles is what determines both the diffusion and effusion rates.
Below is a problem examining the ratio of the velocities for two different molecules at the same temperature.