RMS / Avg / Most-Probable Speeds

RMS (Root Mean Square) speed: v_rms = √⟨v²⟩.

v_rms = √(3k_BT/m) = √(3RT/M). Lighter molecule or higher T ⇒ faster v_rms.

Three commonly cited speeds: v_rms = √(3RT/M); v_mean = √(8RT/πM); v_mp (most probable) = √(2RT/M).

Ratio: v_mp : v_mean : v_rms = √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22.

Sound speed in gas ≈ v_rms/√3 — same order of magnitude as molecular speeds.

v_rms in air at 300 K: oxygen ~480 m/s, nitrogen ~510 m/s, hydrogen ~1930 m/s, helium ~1370 m/s.

Maxwell-Boltzmann distribution: f(v) ∝ v²·exp(−mv²/2kT) — gives all three speeds from one curve.