Mean free path

$v_\text{rms}$ tells us how fast the molecules are, but not how far they get between one collision and the next. A gas molecule does not travel in a straight line: it keeps bouncing off other molecules. The average distance travelled between two consecutive collisions is called the mean free path ($\lambda$; not to be confused with wavelength).

Principle — Mean free path

For a gas with $n = N/V$ molecules per unit volume, modelling the molecules as rigid spheres of diameter $d$, the mean free path is $\ev{\lambda = \frac{1}{\sqrt{2}\,n\,\pi\,d^2}}$

Idea behind the formula. In a time $\Delta t$ a molecule sweeps out a cylinder of cross-section $\pi d^2$ (the effective collision area) and length $v\,\Delta t$. The target molecules inside that cylinder are on average $n\,\pi d^2\,v\,\Delta t$. The factor $\sqrt 2$ appears when accounting for the fact that the targets are also moving, increasing the “encounters”. The average number of collisions per unit time is therefore $\sqrt 2\,n\,\pi d^2\,v$, and the distance between two collisions is $\lambda = v/(\sqrt 2\,n\pi d^2)$, independent of the speed $v$ (Halliday 2014).

Example — Air at room temperature

For nitrogen at $T = 300$ K and $P = 1$ atm, with $d \approx 3{,}7\cdot 10^{-10}$ m, we have $n = P/(k_B T) \approx 2{,}4\cdot 10^{25}\;\text{m}^{-3}$ and therefore $\lambda = \frac{1}{\sqrt 2\cdot 2{,}4\cdot 10^{25}\cdot \pi\cdot (3{,}7\cdot 10^{-10})^2} \approx 6{,}8\cdot 10^{-8}\;\text{m} \approx 70\;\text{nm}$ Hundreds of times the molecular diameter, but millions of times shorter than the distance a molecule would travel in one second at $v_\text{rms}\approx 500$ m/s. A N$_2$ molecule undergoes about $7\cdot 10^9$ collisions per second.

At low pressure $\lambda$ grows as $1/P$: at $10^{-6}$ atm molecules travel whole metres before colliding. This is the basis of the vacuum used in cathode-ray tubes and in surface-physics laboratories.

In summary

$\lambda$ is small for a dense gas, large for a rarefied gas. It determines the gas’s viscosity, thermal conductivity and diffusion.

Collegamenti

Argomenti: Teoria cinetica dei gas Concetti: Velocità quadratica media Competenze: Interpretazione micro-macro

Esercizi collegati: Problem — Escape temperature of hydrogen · Problem — Ranking of root-mean-square speeds · Problem — Why the atmosphere has no helium