RMS Velocity Calculator
Root-mean-square, mean and most probable molecular speeds.
Work out RMS Velocity. Root-mean-square, mean and most probable molecular speeds. Names the sign error before you make it.
N₂ 28.014, O₂ 31.998, H₂ 2.016, He 4.003, CO₂ 44.01.
Root-mean-square speed
515.2 m/s
1,855 km/h — the average molecule, at room temperature
Three different averages of the same Maxwell-Boltzmann distribution, always in the order most probable < mean < rms. They differ because the distribution has a long tail of fast molecules that drags the higher averages up. Speed depends only on temperature and molar mass — not pressure, not volume. Kinetic energy per mole depends only on temperature, identically for every gas, which is why lighter molecules must move faster to carry the same energy. That is Graham's law of effusion in its underlying form. These speeds are large, but molecules do not travel far between collisions: at atmospheric pressure the mean free path is under a micrometre.
How the RMS Velocity Calculator works
Enter a molar mass and temperature to get all three speeds from the Maxwell-Boltzmann distribution, plus the kinetic energy per mole. They always come out in the order most probable, mean, then root-mean-square, because the distribution has a long fast tail.
Also known as: root mean square speed calculator · molecular speed calculator · average speed of gas molecules · kinetic molecular theory calculator
Frequently asked questions
How do I calculate RMS velocity?
v_rms = √(3RT/M), with R at 8.314 J/(mol·K), T in kelvin and M in kilograms per mole. Nitrogen at 25 °C comes out around 515 m/s.
Why are there three different average speeds?
Because the Maxwell-Boltzmann distribution is skewed. The most probable speed is the peak, the mean is the arithmetic average, and the root-mean-square is higher again because squaring weights the fast molecules more heavily.
Why do lighter gases move faster?
Kinetic energy per mole depends only on temperature, identically for every gas. To carry the same energy with less mass, a lighter molecule must move faster — which is Graham's law of effusion in its underlying form.
If molecules move at 500 m/s, why does a smell take time to cross a room?
Because they do not travel in straight lines. At atmospheric pressure the mean free path between collisions is well under a micrometre, so diffusion is a slow random walk however fast the individual steps are.
Put this calculator on your own site
Free to use, on any site, commercial or not. Paste this where you want it to appear. It is a plain iframe, so it works in WordPress, Squarespace, Wix, Webflow, Ghost and anything else that accepts HTML.
<iframe src="https://www.thecalclibrary.com/embed/rms-velocity-calculator" width="100%" height="640" style="border:1px solid #e2e8f0;border-radius:12px" loading="lazy" title="RMS Velocity Calculator"></iframe>
<p style="font:13px/1.5 system-ui,sans-serif;margin:6px 0 0;color:#64748b">Powered by <a href="https://www.thecalclibrary.com/rms-velocity-calculator" style="color:#64748b">RMS Velocity Calculator</a> from The Calc Library</p>The only condition is that the credit line below the frame stays in place. That one line is what pays for the tool being free — it is how anyone else finds it.
Related calculators
Graham's Law Calculator
Effusion rates, as the inverse square root of molar mass.
OpenGas Density Calculator
Density from molar mass, pressure and temperature.
OpenVan der Waals Equation Calculator
Real gas pressure, and how far it strays from ideal.
Open