Gas Molecule Speed Calculator
Calculate the rms speed, average speed, and most probable speed of gas molecules at any temperature.
Based on the Maxwell-Boltzmann distribution.
The kinetic molecular theory describes gas molecules as point particles in constant random motion. At a given temperature, molecules have a distribution of speeds described by the Maxwell-Boltzmann distribution.
Three characteristic speeds:
Root mean square speed (v_rms):
v_rms = √(3RT/M) = √(3kBT/m)
Average speed (v_avg):
v_avg = √(8RT/πM) = √(8kBT/πm)
Most probable speed (v_mp):
v_mp = √(2RT/M) = √(2kBT/m)
Relationship: v_mp < v_avg < v_rms (approximate ratio: 1 : 1.128 : 1.225)
Where:
- R = 8.314 J/mol·K
- T = temperature (Kelvin)
- M = molar mass (kg/mol)
Gas molecule speeds at 25°C (298 K):
| Gas | M (g/mol) | v_rms (m/s) |
|---|---|---|
| H₂ | 2.016 | 1,920 |
| He | 4.003 | 1,363 |
| H₂O | 18.02 | 642 |
| N₂ | 28.02 | 515 |
| O₂ | 32.00 | 482 |
| CO₂ | 44.01 | 411 |
| Xe | 131.3 | 238 |
| UF₆ | 352.0 | 145 |
Why do lighter molecules move faster?
Kinetic energy is equally distributed among all molecules at the same temperature:
KE_avg = (3/2)kBT (same for all gases at same T)
Since KE = ½mv², lighter molecules must move faster to have the same energy.
Temperature effect: Speed ∝ √T. To double v_rms, temperature must quadruple. Going from 298 K to 1192 K (×4 T) only doubles the speed.
What the three speeds mean
A gas does not have one speed, it has a distribution of them, and three different averages get quoted. The most probable speed is the peak of the Maxwell-Boltzmann curve, the speed more molecules have than any other. The mean speed is the ordinary arithmetic average. The root-mean-square speed is the one that matters for energy, because kinetic energy goes as v², and it is always the largest of the three. Their ratio is fixed: v_mp : v_avg : v_rms = 1 : 1.128 : 1.225, whatever the gas and whatever the temperature.
These numbers are faster than they feel
Nitrogen at room temperature has an rms speed around 515 m/s, which is faster than a rifle bullet and slightly faster than the speed of sound. That is not a coincidence: sound travels by molecular collisions, so it cannot outrun the molecules carrying it, and the two speeds stay within about 30% of each other in any gas.
Yet a smell crosses a room in seconds, not milliseconds. The molecules are moving at hundreds of metres per second but colliding billions of times a second, so their net progress is a slow random walk rather than a straight line.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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