Mean Free Path

The molecules of an ideal gas are in continuous, random motion of the same qualitative character as the Brownian motion studied earlier -- ceaseless, erratic, undirected motion with no molecule favouring any particular direction over another.

Because molecules move in straight lines between collisions and change direction abruptly at each collision (Section 3.3), it is useful to define the mean free path, denoted $\lambda$, as the average distance traversed by a molecule, moving with constant velocity, between two successive collisions.

The mean free path is not a fixed, universal number -- it depends on how crowded the gas is and how large its molecules are:

• $\lambda$ is expected to vary inversely with the number density of the gas, $\dfrac{N}{V}$ (where $N$ is the number of molecules in volume $V$): the higher the density, the more frequent the collisions, and so the smaller the mean free path.
• $\lambda$ is also inversely proportional to the square of the molecular diameter $d$ -- not simply to $d$ itself -- because what actually matters for the chance of a collision is the molecule's effective cross-sectional area, which scales as $d^2$, not its linear size.

Putting these dependences together (with a numerical constant obtained from a fuller derivation) gives

A worked numeric application of this formula for nitrogen gas at ordinary temperature and pressure is given in Example 3.1, where the computed mean free path turns out to be roughly 247 times the molecular diameter itself -- a useful reminder of just how sparsely the molecules of an everyday gas are actually packed relative to their own size, even though the gas appears macroscopically continuous. …