Root mean square speed

If temperature measures the average kinetic energy, how fast are the molecules really moving? The quantity that answers this is the root mean square speed $v_\text{rms}$: the square root of the average of the squares of the molecular speeds. It is the “typical” speed of thermal agitation.

Principle — Agitation speed

$\ev{v_{\text{rms}} = \sqrt{\frac{3\,k_B\,T}{m_{\text{part}}}} = \sqrt{\frac{3\,R\,T}{m_{\text{mol}}}}}$

Here too the two forms are equivalent: one uses the mass of the single particle $m_\text{part}$ with $k_B$, the other the molar mass $m_\text{mol}$ with $R$. Two immediate readings: at the same temperature, lighter molecules are faster ($v_\text{rms}\propto 1/\sqrt{m}$); and at the same mass, heating the gas increases the speed ($v_\text{rms}\propto\sqrt{T}$).

Note the factor $3$: it is always $3$, regardless of whether the gas is mono- or diatomic. This is because only the three translational degrees of freedom contribute to the speed of translation through space, and these are always three. Any rotations (which take $f$ from 3 to 5 in the internal energy) do not move the centre of the molecule and so do not enter $v_\text{rms}$.

Common mistake

Do not confuse the factor $\frac{f}{2}$ of the internal energy with the factor $3$ of $v_\text{rms}$. The first counts all the degrees of freedom; the second counts only the three translational ones, always three.

Links

Topics: Teoria cinetica dei gas Concepts: Velocità quadratica media · Temperatura Skills: Interpretazione micro-macro

Related exercises: Problema — Temperatura di fuga dell’idrogeno · Problema — Temperatura da v_rms · Problema — Velocità quadratica media dell’azoto