Mean Free Path Calculator: Molecular Collisions & Knudsen Number

simulator beginner ~8 min
Loading simulation...
λ = 0.066 mm — viscous flow regime (Kn = 0.0013)

At 100 Pa and 300 K with N₂ molecules (d = 372 pm), the mean free path is 0.066 mm. With a 50 mm tube, Kn = 0.0013 — firmly in viscous flow where gas behaves as a continuous fluid.

Formula

λ = kT / (√2 × π × d² × P)
Kn = λ / L
n = P / (kT) — number density

Molecules in Flight

Gas molecules are in constant chaotic motion, colliding with each other billions of times per second at atmospheric pressure. The mean free path — the average distance between collisions — is the single most important parameter in vacuum physics. It determines whether gas behaves as a continuous fluid or as a collection of independent ballistic particles, fundamentally changing how vacuum systems must be designed and analyzed.

The Knudsen Number

Martin Knudsen recognized that the ratio of mean free path to system size determines gas behavior. When Kn < 0.01, molecules collide so frequently that the gas acts as a continuum fluid described by Navier-Stokes equations. When Kn > 1, molecules fly freely between wall collisions with no intermolecular scattering — this molecular flow regime has completely different conductance laws.

Pressure Regimes

Vacuum is classified by pressure ranges: rough vacuum (1000–1 Pa), medium vacuum (1–0.001 Pa), high vacuum (10⁻³–10⁻⁷ Pa), and ultra-high vacuum (below 10⁻⁷ Pa). Each regime has characteristic mean free paths ranging from fractions of a millimeter to kilometers, and correspondingly different engineering approaches for pumping, sealing, and measurement.

Simulation Details

This visualization shows molecules as particles moving through a tube. At high pressure, you see dense, collision-rich behavior. As you lower the pressure, collisions become rare and molecules bounce ballistically between walls. The Knudsen number display updates in real time, showing the transition between flow regimes as a visual and quantitative experience.

FAQ

What is mean free path?

Mean free path (λ) is the average distance a gas molecule travels between consecutive collisions with other molecules. It depends on pressure, temperature, and molecular size: λ = kT / (√2 × π × d² × P). At atmospheric pressure, λ for air is about 68 nm; at high vacuum (0.001 Pa), it exceeds 60 meters.

What is the Knudsen number?

The Knudsen number Kn = λ/L is the ratio of mean free path to the characteristic dimension of the system. It determines the flow regime: Kn < 0.01 is viscous flow, Kn > 1 is molecular flow, and 0.01–1 is the transition regime.

Why does flow regime matter for vacuum design?

Pipe conductance formulas differ completely between regimes. In viscous flow, conductance depends on pressure and pipe diameter to the 4th power (Poiseuille). In molecular flow, conductance is pressure-independent and scales with diameter cubed. Using the wrong formula leads to undersized pumping systems.

How does temperature affect mean free path?

Higher temperature increases mean free path linearly (λ ∝ T) because faster molecules spread out more. However, in most vacuum systems, temperature effects are secondary compared to pressure changes spanning many orders of magnitude.

Sources

Embed

<iframe src="https://homo-deus.com/lab/vacuum-science/mean-free-path/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub