Problem — Collision frequency

Problem

At atmospheric pressure and $T = 300$ K, the mean free path of air molecules is $\ell \approx 70$ nm. Estimate the collision frequency of a single molecule, knowing $v_\text{rms} \approx 500$ m/s.

Solution

Between one collision and the next the molecule travels on average the mean free path $\ell$ at speed $v_\text{rms}$. The collision frequency is the number of collisions per second: $\nu = \frac{v_\text{rms}}{\ell}$ I convert the mean free path: $\ell = 70\ \text{nm} = 70\cdot10^{-9}$ m. $\nu = \frac{500}{70\cdot10^{-9}}$ $\ev{\nu \approx 7{,}1\cdot10^{9}\ \text{Hz}}$ Each molecule undergoes billions of collisions per second: this is why the gas reaches thermal equilibrium so quickly.

Links

Topics: Kinetic theory of gases Concepts: Root-mean-square speed Skills: Dimensional analysis