2.5 Collisions and the mean free path

So far the molecules have been treated as a non-interacting swarm, colliding only with the walls. They also collide with each other, and the average distance a molecule travels between collisions — the mean free path — sets the scale on which a gas behaves as a continuum and controls how it transports momentum, heat, and matter (the subject of 2.6).

The collision length

▶ ℓ = 1/(√2 n π d²) Derivation

Model each molecule as a hard sphere of diameter $d$ . Two molecules collide when their centres pass within $d$ , so a moving molecule sweeps out a collision cross-section $\sigma = \pi d^2$ : any target whose centre lies in the cylinder of cross-section $\sigma$ swept along the path is struck.

In travelling a distance $L$ the molecule sweeps a volume $\sigma L$ and so meets $n\sigma L$ targets, where $n$ is the number density. The mean distance between collisions is the path length per collision,

\ell \;=\; \frac{L}{n\sigma L} \;=\; \frac{1}{n\,\pi d^2}.

This treats the targets as stationary. Accounting for their motion replaces the molecule’s speed by the relative speed of the colliding pair, which is larger by a factor $\sqrt2$ for a Maxwell–Boltzmann gas, shortening the path:

\boxed{\;\ell \;=\; \frac{1}{\sqrt2\, n\, \pi d^2}.\;}

✓

The collision rate is then $\langle v\rangle/\ell$ . For air at standard conditions ( $n \approx 2.5\times10^{25}\,\text{m}^{-3}$ , $d \approx 0.37\,\text{nm}$ ) the mean free path is about $70\,\text{nm}$ and the collision rate about $7\times10^{9}$ per second — a molecule collides billions of times a second, which is why a gas reaches local equilibrium almost instantly and behaves as a smooth fluid on any human scale.

n (m⁻³)1.00e+25

d (m)3.16e-10

λ = 1/(√2 n π d²)2.25e-7 m

L (system)1.00e-2 m

Kn = λ/L2.25e-5

log₁₀ n = 25.0 (n = 1.00e+25 m⁻³) log₁₀ d = -9.5 (d = 0.32 nm) log₁₀ L = -2.0 (L = 1.00e-2 m)

Air at STP has n ≈ 2.5×10²⁵ m⁻³ and d ≈ 0.3 nm, giving λ ≈ 70 nm — much smaller than any acoustic system, so the continuum picture holds. Lower n (e.g. upper-atmosphere) or shrink L (e.g. MEMS) and Kn rises; once Kn ≳ 1 the gas is rarefied and the Navier-Stokes equation fails.

Slide the density and the molecular diameter and read off the mean free path together with the Knudsen number $\mathrm{Kn} = \ell/L$ for a chosen system size $L$ .

When the continuum picture fails

The Knudsen number decides whether a gas is a continuum. When $\mathrm{Kn} \ll 1$ the mean free path is tiny compared with the system, collisions keep the gas locally equilibrated, and continuum equations (the Navier–Stokes equations of fluid mechanics, Fourier’s law of heat conduction) apply. When $\mathrm{Kn}$ approaches or exceeds $1$ — rarefied upper-atmosphere flow, vacuum systems, gas in micron-scale channels — molecules cross the system between collisions, the continuum description breaks down, and the full kinetic theory of the Boltzmann transport equation is needed.