Q.A fresh air is composed of nitrogen $N_2$ (78%) and oxygen $O_2$ (21%). Find the rms speed of $N_2$ and $O_2$ at $20^\circ$C.

Definition and formula. The root mean square speed, $v_{rms}=\sqrt{\overline{v^2}}$, is the square root of the average of the squared speeds of every molecule in a gas sample. From the kinetic-theory temperature result $kT=\tfrac13m\overline{v^2}$,

where $M$ is the molar mass and $R$ the universal gas constant.

Dependence on temperature and mass. $v_{rms}\propto\sqrt T$ and $v_{rms}\propto1/\sqrt M$: heating a gas always raises its $v_{rms}$, and at any fixed temperature, lighter molecules (hydrogen, helium) always have a higher $v_{rms}$ than heavier ones (oxygen, nitrogen, carbon dioxide).

Where it matters physically. $v_{rms}$ is the speed that appears directly in the pressure formula, $P=\tfrac13nmv_{rms}^2$, since $\overline{v^2}=v_{rms}^2$ by definition -- so of the three characteristic speeds, it is the one most directly tied to a gas's pressure. It is also the speed compared against a planet or moon's escape speed to decide whether a given gas can be retained in that body's atmosphere: if $v_{rms}$ for a gas exceeds the local escape speed, that gas will steadily leak away into space over geological time (which is why the Moon, with its very low escape speed, has essentially no atmosphere at all, and why Earth has lost almost all its primordial hydrogen and helium while retaining heavier nitrogen and oxygen).

Distinguishing it from the other two speeds. $v_{rms}$ is always the largest of the three characteristic speeds ($v_{rms}>\bar v>v_{mp}$) -- it is not the same number as the plain average speed, which is consistently about $0.92$ times smaller.