Mean free path and vacuum regimes

The mean free path is the average distance a molecule travels between collisions, and since it is inversely proportional to pressure it stretches from about 70 nanometres in room air to kilometres in a good ultra-high vacuum. Comparing it with the size of the vessel is what defines the regimes: while it is much smaller than the chamber the gas behaves as a fluid and is pumped by pushing it along, and once it is much larger the molecules fly wall to wall without meeting each other, so the gas conducts heat less and less well as pressure falls and a thin-film source can reach the substrate without being scattered. That crossover is why a vacuum system needs two kinds of pump in series and why a thermos flask works at all.

Practise this concept

A molecule in a gas travels in a straight line until it hits another one. How far that is, on average, depends on how much room there is: the mean free path grows as the gas thins out, in inverse proportion to pressure. In room air it is about 70 nanometres — a molecule collides billions of times a second — while in a good ultra-high vacuum it can exceed the size of the building. The molecules themselves are no slower; at a fixed temperature they still move at hundreds of metres per second. There are simply far fewer of them to run into.

What makes the number useful is comparing it with the apparatus rather than reading it on its own. While the mean free path is much smaller than the chamber, molecules mostly talk to each other rather than to the walls, the gas behaves as a continuous fluid, and a pump works by pushing that fluid along — this is the viscous regime, where a rotary or scroll pump belongs. Once the mean free path is much larger, molecules fly from wall to wall and essentially never meet, so pushing them is meaningless and the pump must instead catch each molecule that happens to arrive: that is the free-molecular regime and the province of turbo and cryo pumps. A vacuum system therefore needs both in series, which is why there is always a backing pump behind the fast one.

Two practical consequences fall out. First, thermal conduction through a gas is roughly independent of pressure while the mean free path is short, because removing half the molecules leaves each of the rest carrying heat twice as far — the losses only collapse once the path exceeds the gap, which is exactly the condition a vacuum flask is built to reach. Second, in thin-film deposition the mean free path must be longer than the distance from source to substrate, or the evaporated atoms scatter on the way and arrive from every direction instead of in a straight line. Both are the same comparison, made against a different length each time.

Key ideas

Requires: Kinetic theory of gases, Ideal gas law

Unlocks: Thin-film deposition and growth

Formulas

\lambda = \frac{k_\mathrm{B} T}{\sqrt{2}\, \pi d^{2} p}

Symbol	Name	Unit
$L$	distance	m (metre)
$T$	temperature	K (kelvin)
$d$	diameter	m (metre)
$p$	pressure	Pa (pascal)

Common misconceptions

Thinking a lower pressure means slower molecules. Pumping removes molecules; it does not cool the ones left behind. At a fixed temperature the speed distribution is unchanged and the average speed is the same at a millionth of an atmosphere as at one atmosphere — what falls is how many molecules there are, and therefore how often they collide.
Treating the mean free path as a property of the gas alone, so that a number quoted for it settles which regime you are in. The regime is a comparison: the same 1 cm mean free path is free-molecular inside a 1 mm capillary and viscous inside a 10 m chamber. Halving the vessel size changes the flow regime just as surely as halving the pressure does.
Expecting the thermal conductivity of a gas to fall as soon as you start pumping. While the mean free path is still short compared with the gap, removing molecules leaves each one carrying heat further, and the two effects cancel almost exactly — a gas conducts about as well at a tenth of an atmosphere as at one. Conduction only collapses once the mean free path exceeds the gap, which is why a vacuum flask needs a hard vacuum rather than a partial one.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Vacuum and cryogenic technologyLOFY.01.026
Thin-film technology ILOFY.02.013