Free Path and Mean Time Between Collisions

The ideal gas relation was derived under the assumption that each molecule travels undisturbed from wall to wall, which is certainly not true at common pressures and temperatures. To see this, we need to get some estimate of the mean free path k (the mean distance a molecule travels before it undergoes a collision), and the mean time between collisions. 

If you double the concentration N/V (= P/ kn T), on average you expect to go only half as far before you encounter another molecule. We thus predict that k is proportional to V/N = kn 7 / P. If you double the size of the target (expressed as the collisional cross-section a2, where a is the size in the hard-sphere potential), on average you will also only go half as far before you undergo a collision. Thus we also predict that k is inversely proportional to a2. 

These predictions are correct. The precise expression turns out to be  

We also sometimes evaluate the mean time between collisions r, which is the 

Intermolecular collisions do not cause large deviations from the ideal gas law at STP for molecules such as N2 or He, which are well above their boiling points, but they do dramatically decrease the average distance molecules travel to a number which is far less than would be predicted from the average molecular speed. Collisions randomize the velocity vector many times in the nominal round trip time, leading to diffusional effects as discussed in Chapter 4. If all of the molecules start at time t = 0 at the position x = 0, the concentration distribution C(x,t) at later times is a Gaussian  

---

Mean Free Path and Mean Time Between Collisions... 

---