Mean Free Path

Between one collision and the next, a gas molecule travels in a straight line at essentially constant velocity (Section 9.2's assumption 3) -- but because the gas is full of other molecules to collide with, this straight-line stretch is typically very short before the next collision cuts it off and sends the molecule off in a new direction. The MEAN FREE PATH, denoted $\lambda$, is defined as the average distance a molecule travels between two successive collisions.

To derive an expression for $\lambda$, picture a single molecule of diameter $d$ moving through a gas of number density $n$ (molecules per unit volume). A collision occurs whenever the CENTRE of another molecule comes within a distance $d$ of the moving molecule's own centre (since each molecule has radius $d/2$, and they touch when their centres are $d$ apart) -- so the moving molecule effectively sweeps out a cylindrical tube of cross-sectional area $\pi d^2$ as it travels (see the figure for this section). In a time $t$, moving at average speed $\bar{v}$, this molecule sweeps out a cylindrical volume $\pi d^2 \bar{v} t$, and the number of collisions it suffers in this time equals the number of OTHER molecules whose centres lie inside this swept volume, namely $n\pi d^2\bar{v}t$.

A first (simplified) estimate of the mean free path, ignoring the motion of the other molecules, would divide the total distance travelled, $\bar{v}t$, by this number of collisions, giving $\lambda \approx 1/(n\pi d^2)$. However, this ignores the fact that the OTHER molecules are also moving, not sitting still -- so the moving molecule's speed RELATIVE to the ones it is about to collide with is, on average, larger than its own speed relative to the container. A careful statistical treatment (accounting for the full distribution of relative velocities between molecules moving in random directions) shows that the correct average relative speed is a factor of $\sqrt{2}$ larger than the individual molecular speed, which reduces the correct mean free path by that same factor of $\sqrt{2}$:

$\boxed{\lambda = \frac{1}{\sqrt{2}\,\pi d^2 n}}$ …