RMS Molecular Speed Calculator
Enter absolute temperature and molar mass to compute the Maxwell-Boltzmann speeds: v_rms=√(3RT/M), v_mean=√(8RT/πM), v_p=√(2RT/M). N₂ at 273 K → vrms≈493 m/s, He ≈1300 m/s.
Input Data
Results
At a glance:In an ideal gas every molecule has a different speed, yet by the Maxwell–Boltzmann distribution you can describe it with three statistics: 'most probable speed v_p', 'mean speed v_mean' and 'root-mean-square speed v_rms=√(⟨v²⟩)'. All three are proportional to the square root of temperature and inversely proportional to the square root of molar mass — the core formulas of the kinetic theory. Term by term: v_p=√(2RT/M) is the peak of the distribution curve; v_mean=√(8RT/πM)=⟨|v|⟩ is the arithmetic mean of all speeds; v_rms=√(3RT/M) is the square root of the mean of squared speeds, equal to the speed for which KE=½mv_rms² matches the average kinetic energy 3kT/2. R=8.314 J/(mol·K) is the gas constant. Ordering: v_p<v_mean<v_rms with ratios 1:1.128:1.225. Example: nitrogen N₂ M=0.02801 kg/mol, T=273.15 K → v_rms=√(3×8.314×273.15/0.02801)≈493 m/s; helium He M=0.004 kg/mol, same T → v_rms≈1300 m/s. History: Maxwell derived the distribution in 1860, Boltzmann gave the statistical foundation in the 1870s. Applications: (1) speed of sound (∝v_rms); (2) gas diffusion and effusion rates (Graham's law uses v_rms); (3) mean free path and collision frequency; (4) deriving gas pressure pV=⅓nmv_rms²; (5) estimating reaction/pumping speed.
Formula
Most probable: v_p = √(2RT / M) (distribution peak)
Mean speed: v_mean = √(8RT / (πM)) = ⟨|v|⟩
RMS speed: v_rms = √(3RT / M)
R=8.314 J/(mol·K); T in K; M in kg/mol
Order: v_p < v_mean < v_rms (1 : 1.128 : 1.225)
$$v_p = \sqrt{\frac{2RT}{M}}, \quad \langle v \rangle = \sqrt{\frac{8RT}{\pi M}}, \quad v_{rms} = \sqrt{\frac{3RT}{M}}$$How to Use
- Step 1: enter absolute temperature T (K); if Celsius, add 273.15 first.
- Step 2: enter molar mass M (kg/mol); e.g. N₂=0.02801.
- Step 3: read the three speeds (m/s) — use them for sound speed, diffusion and mean free path.
Case Studies
Nitrogen at STP
M=0.0280134 kg/mol, T=273.15 K.
v_p=√(2×8.314×273.15/0.0280134)≈402 m/s; v_mean≈454 m/s; v_rms=√(3×8.314×273.15/0.0280134)≈493 m/s.
At room T=300 K they rise ~8% (∝√T). Oxygen is slightly slower than nitrogen (M larger); helium far faster.
Graham's effusion and isotope separation
Effusion rate ∝ 1/√M (Graham). UF₆-235 vs UF₆-238 differ by √238/√235≈1.0064 — tiny, hence thousands of cascades for uranium enrichment.
Hydrogen escapes a balloon ~4× faster than oxygen (√(32/2)=4).
Diffusion in air broadens a gas plume at a rate ∝ v_rms.
Content review: Calculatorism Editorial Team. Results are for reference only; please refer to the relevant authorities for the official figures.