Root-mean-square speed
v_rms = √(3k_BT/m) — the speed whose square equals the mean square molecular speed, and the one tied directly to temperature.
Definition
The root-mean-square speed of gas molecules is v_rms = √(⟨v²⟩) = √(3k_BT/m), where m is the molecular mass. It is the square root of the average of the squared speeds, and it is the speed that appears naturally in kinetic theory because mean kinetic energy depends on ⟨v²⟩, not on ⟨v⟩. It is slightly larger than the mean speed and the most-probable speed.
Because v_rms scales as 1/√m, light molecules move faster at the same temperature. Nitrogen at room temperature has v_rms ≈ 510 m/s, while hydrogen reaches ≈ 1900 m/s. This mass dependence explains why Earth's gravity retains nitrogen and oxygen but slowly leaks hydrogen and helium to space, and why the Moon, with weaker gravity, retains almost no atmosphere at all.
The rms speed sits inside the broader Maxwell–Boltzmann distribution of speeds; it is one of three characteristic speeds (most-probable < mean < rms) that mark the distribution. Its direct proportionality to √T makes it a convenient quantitative handle on the meaning of temperature.