How the mean free path is calculated
Treating molecules as hard spheres of diameter \(d\) in an ideal gas at temperature \(T\), the mean free path is
The \(\sqrt2\) accounts for the motion of the collision partners (Maxwell–Boltzmann velocities). At a fixed temperature the product \(\lambda p\) is constant. For N₂ at 20 °C it is 6.48 mm·Pa, or about 6.5 × 10⁻³ cm·mbar. So \(\lambda \approx 6.5\) mm at 1 Pa, and it scales as \(1/p\): about 6.5 cm at 10⁻³ mbar and 65 m at 10⁻⁶ mbar.
The other kinetic quantities follow from the Maxwell–Boltzmann distribution, with molecular mass \(m = M/N_A\):
Here \(\Phi\) is the number of molecules striking a unit area of wall per second, \(n_s \approx 10^{15}\) cm⁻² is the number of adsorption sites in a monolayer, and \(s\) is the sticking coefficient.
Knudsen number and flow regime
The Knudsen number \(\mathrm{Kn} = \lambda/D\) compares the mean free path with the size of the system. It decides whether gas behaves as a fluid or as independent molecules bouncing between walls:
| Kn | Regime | What it means |
|---|---|---|
| < 0.01 | Viscous (continuum) | Collisions between molecules dominate. Conductance depends on pressure and viscosity. |
| 0.01 – 0.5 | Transitional (Knudsen) | Gas–gas and gas–wall collisions both matter. |
| > 0.5 | Molecular | Molecules mostly hit walls. Conductance is independent of pressure. |
These boundaries are common in vacuum-industry literature. Some texts put the start of molecular flow at Kn = 1. The change between regimes is gradual, not sharp.
Gas data used
| Gas | d (Å) | M (g/mol) | λ·p at 20 °C (mm·Pa) |
|---|
The diameters are hard-sphere collision diameters derived from gas viscosity near room temperature. They are consistent with the Chapman–Enskog hard-sphere relation between viscosity and diameter to within a few percent, and published tables differ from each other by a similar amount. The value for water vapor is the least certain, because H₂O is strongly polar.
Worked example
N₂ at 10⁻³ mbar (0.1 Pa) and 20 °C, in a 100 mm diameter tube:
- \(\lambda = 6.48\ \text{mm·Pa} / 0.1\ \text{Pa} = 64.8\ \text{mm}\), so Kn = 0.65 and the flow is molecular.
- \(n = 0.1/(1.381\times10^{-23}\times293.15) = 2.47\times10^{13}\ \text{cm}^{-3}\). Even at "high vacuum" there are tens of trillions of molecules in every cubic centimetre.
- \(\bar v = 471\) m/s, and the wall is hit at \(\Phi = 2.9\times10^{17}\) cm⁻² s⁻¹. With \(s = 1\), a clean surface is covered in about 3.4 ms. At 10⁻⁶ mbar the same estimate gives 3.4 s, which is why surface-science work needs ultra-high vacuum.
Assumptions and limits
- Ideal gas with hard-sphere collisions and a single species. Mixtures need a collision diameter and reduced mass for each pair of species.
- The hard-sphere diameter depends weakly on temperature. Real molecules attract and repel, so the effective cross-section falls as T rises (a Sutherland-type correction).
- The monolayer time is an upper bound on how fast contamination builds up. Real sticking coefficients for N₂ and Ar on room-temperature surfaces are far below 1. For water and hydrocarbons they can approach 1.
References
- J. F. O'Hanlon, A User's Guide to Vacuum Technology, 3rd ed., Wiley (2003), ch. 2 (gas properties).
- K. Jousten (ed.), Handbook of Vacuum Technology, 2nd ed., Wiley-VCH (2016).
- S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed., Cambridge University Press (1970).