What this calculator does
Gas molecules move fast. Air at room temperature has molecules travelling at roughly 500 metres per second on average, which is faster than a commercial airliner, yet the air in a room goes nowhere because the motion is random in every direction.
The speed depends on temperature and on molecular mass. Lighter molecules move faster at the same temperature, which is why hydrogen and helium escape from a planet's atmosphere while heavier gases are retained.
The formula
Take the square root of three times the gas constant times absolute temperature, divided by the molar mass in kilograms per mole. The average kinetic energy per molecule follows as three halves of the Boltzmann constant times temperature.
| Term | Meaning |
|---|---|
| RMS speed | The root mean square of molecular speeds, which is the value that relates directly to kinetic energy. |
| Molar mass (M) | The mass of one mole of the gas in grams. Air averages 28.97; hydrogen is about 2. |
| Maxwell-Boltzmann distribution | The spread of molecular speeds in a gas, of which the RMS speed is one characteristic measure. |
The inputs explained
| Field | What to enter |
|---|---|
| Temperature (K) | Absolute temperature in kelvin. |
| Molar mass (air ≈ 28.97) (g/mol) | Molar mass in grams per mole. Air is about 28.97; hydrogen about 2.016; carbon dioxide about 44.01. |
When to use it
Comparing gases at the same temperature
Lighter molecules move faster, which explains diffusion rates and why helium escapes a balloon so quickly.
Understanding atmospheric escape
A planet retains a gas only if molecular speeds stay well below escape velocity, which is why Earth keeps nitrogen but not hydrogen.
Linking temperature to molecular motion
Temperature is essentially a measure of average molecular kinetic energy, and this makes that relationship concrete.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does molecular speed rise with temperature?
The same gas at a range of temperatures.
| Temperature | RMS speed | Average kinetic energy per molecule |
|---|---|---|
| 200 K | 415.0 m/s | 4.1419e-21 J |
| 293.15 K | 502.4 m/s | 6.0711e-21 J |
| 500 K | 656.1 m/s | 1.0355e-20 J |
| 1000 K | 927.9 m/s | 2.0710e-20 J |
Questions
If molecules move at 500 m/s, why does a smell take time to cross a room?
Because the motion is random and constantly interrupted by collisions. A molecule covers only about 65 nanometres between collisions, so its net progress is a slow random walk rather than a straight run.
Why do lighter gases move faster?
Because at a given temperature every gas has the same average kinetic energy per molecule. Since energy is half mass times speed squared, a smaller mass must come with a higher speed to match.
Why square root of temperature rather than proportional?
Because temperature is proportional to kinetic energy, and energy depends on speed squared. Reversing that gives speed proportional to the square root of temperature.
Is RMS speed the same as average speed?
Not quite. The RMS speed is slightly higher than the simple mean, because squaring weights faster molecules more heavily. RMS is the one used in energy calculations because it relates directly to kinetic energy.
For how far those molecules travel between collisions, see the mean free path calculator. For the gas laws themselves, see the ideal gas law calculator.