What this calculator does
A gas molecule at room pressure travels only a tiny distance before hitting another one. That distance, the mean free path, turns out to be around 65 nanometres for air at atmospheric pressure, which is hundreds of times the molecular size but still vanishingly small.
It matters because so much gas behaviour depends on it. Viscosity, thermal conduction and diffusion all follow from how far molecules get between collisions, and vacuum technology is essentially the business of making that distance large.
The formula
Divide the Boltzmann constant times temperature by the product of the square root of two, π, the square of the molecular diameter, and the pressure. The result is in metres.
| Term | Meaning |
|---|---|
| Mean free path (λ) | The average distance a molecule covers between successive collisions. |
| Boltzmann constant | 1.380649 × 10⁻²³ J/K, exact by definition since 2019. |
| Molecular diameter | The effective collision diameter of the molecule, about 3.7 × 10⁻¹⁰ m for air. |
The inputs explained
| Field | What to enter |
|---|---|
| Temperature (K) | Absolute temperature in kelvin. |
| Pressure (Pa) | Pressure in pascals. Atmospheric pressure is 101,325 Pa; a good vacuum is many orders of magnitude lower. |
| Molecular diameter (air ≈ 3.7×10⁻¹⁰) (m) | The effective molecular diameter in metres. |
When to use it
Designing for vacuum
As pressure falls the mean free path grows, and once it exceeds the chamber size the gas stops behaving like a fluid entirely.
Understanding thin-film deposition
Coating processes need molecules to travel from source to target without colliding, which sets the pressure required.
Explaining gas transport properties
Viscosity and thermal conductivity both trace back to how far molecules travel between collisions.
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 the mean free path grow as pressure falls?
The same gas at a range of pressures from atmospheric down to vacuum.
| Pressure | Mean free path (nm) | Mean free path (m) |
|---|---|---|
| 101.3 kPa | 65.67 nm | 6.5673e-8 m |
| 10.1 kPa | 656.73 nm | 6.5673e-7 m |
| 1.0 kPa | 6,567.32 nm | 6.5673e-6 m |
| 101.3 Pa | 65,673.21 nm | 6.5673e-5 m |
Questions
Why does lower pressure mean a longer path?
Because there are fewer molecules per unit volume to collide with. Halving the number density doubles the average distance travelled before a collision occurs.
How does it compare with the molecule itself?
At atmospheric pressure the path is roughly 180 times the molecular diameter. Gas at normal pressure is mostly empty space, but not nearly as empty as the pressure alone might suggest.
What is the square root of two doing in the formula?
It accounts for the fact that the other molecules are moving too, not sitting still waiting to be hit. Treating them as stationary would overestimate the path by that factor.
When does this stop being useful?
When the mean free path exceeds the size of the container. Beyond that, molecules hit the walls more often than each other, the gas stops behaving as a continuous fluid, and a different treatment is needed.
For the speed those molecules travel at, see the RMS speed of a gas calculator. For the pressure, volume and temperature relationship, see the ideal gas law calculator.