RMS / Avg / Most-Probable Speeds
Compare H₂, He, N₂, O₂, CO₂.
Key Notes
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.
Formulas
RMS speed
M = molar mass.
Mean speed
Average of |v|.
Most probable speed
Peak of Maxwell-Boltzmann distribution.
Ratio
Three speeds always in this ratio for ideal gas.
Important Points
v_rms is what enters kinetic-theory pressure formula.
Higher T ⇒ faster v_rms. Doubling T ⇒ × √2.
Lighter mass ⇒ faster speed. H₂ moves ~4× faster than O₂ at same T.
Sound speed ≈ v_rms/√3 — comparable to molecular speeds.
Highest mean KE = highest v_rms (heavier vs lighter at same T).
Maxwell distribution gives full range of speeds — most molecules near v_mp, some much faster.
RMS / Avg / Most-Probable Speeds notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.