Mean Free Path & Knudsen Number Calculator
How far does a gas molecule travel before it hits another one? Enter gas, pressure and temperature to get the mean free path, and add a pipe or chamber size to find the flow regime.
Formulas
k is Boltzmann’s constant, T absolute temperature, d the molecular (collision) diameter, p the pressure, D the characteristic dimension, R the gas constant and M the molar mass. Collision diameters are viscosity-derived values commonly used in vacuum technology.
Mean free path of air vs. pressure (20 °C)
| Pressure (mbar) | Pressure (Pa) | Mean free path | Molecules / cm³ |
|---|---|---|---|
| 1,013 | 101,300 | 65 nm | 2.5×10¹⁹ |
| 100 | 10,000 | 658 nm | 2.47×10¹⁸ |
| 10 | 1,000 | 6.58 µm | 2.47×10¹⁷ |
| 1 | 100 | 65.8 µm | 2.47×10¹⁶ |
| 0.1 | 10 | 658 µm | 2.47×10¹⁵ |
| 0.01 | 1 | 6.58 mm | 2.47×10¹⁴ |
| 0.001 | 0.1 | 65.8 mm | 2.47×10¹³ |
| 1×10⁻⁴ | 0.01 | 658 mm | 2.47×10¹² |
| 1×10⁻⁶ | 1×10⁻⁴ | 65.8 m | 2.47×10¹⁰ |
| 1×10⁻⁸ | 1×10⁻⁶ | 6.58 km | 2.47×10⁸ |
| 1×10⁻¹⁰ | 1×10⁻⁸ | 658 km | 2.47×10⁶ |
References & further reading
- K. Jousten (ed.). Handbook of Vacuum Technology, 2nd ed.. Wiley-VCH (2016). Standard reference on gas flow, conductance, pumps and gauges.
- J. F. O’Hanlon & T. A. Gessert. A User’s Guide to Vacuum Technology, 4th ed.. Wiley (2023). Practical design of vacuum systems, pump-down and outgassing.
- CERN Accelerator School. CAS Vacuum in Accelerators, Platja d’Aro, Spain, 2006 — proceedings (CERN-2007-003). CERN (2007), free to read. Lectures on gas dynamics, conductance, pumps and outgassing.
Frequently asked questions
What is the mean free path in a vacuum?
The mean free path λ is the average distance a gas molecule travels between collisions with other molecules: λ = k·T / (√2 · π · d² · p). For air at 20 °C it is about 65 nm at atmospheric pressure, 6.6 mm at 1 Pa (0.01 mbar) and 66 m at 10⁻⁴ Pa.
What is the Knudsen number?
Kn = λ / L, the ratio of mean free path to a characteristic size such as the pipe diameter. Kn < 0.01 means viscous (continuum) flow where molecules mainly hit each other; Kn > 0.5 means molecular flow where they mainly hit the walls.
What is the rule of thumb for mean free path of air?
λ [cm] ≈ 6.6 × 10⁻³ / p [mbar], or λ [mm] ≈ 6.6 / p [Pa], for air at 20 °C. It scales inversely with pressure and proportionally with absolute temperature.
What is monolayer formation time?
The time it takes for a clean surface to be covered by one layer of gas molecules, assuming every molecule sticks. It is only a few seconds at 10⁻⁶ mbar but around 10 hours at 10⁻¹⁰ mbar, which is why surface science requires ultra-high vacuum.
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