How far a molecule gets

Everything this collection has said about gases treats them as continuous — a pressure at each point, a density at each point, a temperature at each point. None of that is literally true. A gas is molecules with nothing between them, and the question worth answering is why the fiction works, and where it stops.

The estimate, and what is in it #

A particle moving through a field of targets sweeps out a volume as it goes. If each target presents a cross-sectional area $\sigma$ and there are $n$ of them per unit volume, then travelling a distance $L$ sweeps a volume $\sigma L$ containing $n\sigma L$ targets. Setting that count to one gives the distance at which a collision becomes likely:

$$\lambda = \frac{1}{n\sigma}.$$

That is the mean free path. It contains a number density and a size, and nothing else — no temperature, no mass, no speed. A faster molecule collides more often per second and travels exactly as far between collisions.

For air at room conditions: $n \approx 2.5 \times 10^{25}$ per cubic metre, and a nitrogen molecule’s collision diameter is about 0.37 nm, giving $\sigma = \pi d^2 \approx 4.3 \times 10^{-19}$ m². The result is about 68 nanometres.

Measuring it rather than asserting it #

The figure does not quote that formula. It lays out a seeded field of discs, flies a probe through it on a torus so nothing escapes, records the length of every straight segment, and reports the mean of four thousand of them.

The agreement is to a few per cent, and the residual is informative rather than embarrassing. The closed form is derived for a vanishingly dilute field, and the drawn discs cover nine per cent of the plane, because a field dilute enough for the formula to be exact would be a field with nothing visible in it. Two finite-density effects pull in opposite directions: crowding shortens the path, and discs shadowing one another lengthen it. At nine per cent coverage the first wins and the measurement sits below the formula; at four per cent the second wins and it sits above.

An earlier version of this figure asserted that the measurement lay between the dilute law and a crowding correction, and that is false for half the densities it draws. The claim now requires only that the departure stay small — and that the departures not all share a sign, which is a statement the confident version could not have made.

The √2 that is sometimes there #

The estimate above treated the targets as stationary. In a real gas they are moving as fast as the projectile, so what matters is the relative speed, and averaging the relative speed of two Maxwell-distributed molecules over all pairs gives $\sqrt{2}$ times the mean speed of one. The collision rate is higher by that factor and the free path shorter:

$$\lambda = \frac{1}{\sqrt{2}\, n \sigma}.$$

That is the number quoted for gases, and the 68 nm above includes it.

It is worth being careful about which case is at hand, because the factor is often applied reflexively. A fast particle crossing a gas of much slower ones — a beam through a chamber, a neutron through a moderator — sees effectively stationary targets and the $\sqrt{2}$ is absent. The figure on this page is that case, which is why it is checked against the formula without it.

Two hundred times its own size #

Divide the free path by the molecular diameter: $68 / 0.37 \approx 184$. A molecule travels about two hundred times its own width between collisions.

Compare that with the spacing between neighbours, which is $n^{-1/3} \approx 3.4$ nm — about nine molecular diameters. So the free path is roughly twenty times the typical spacing, which says something slightly surprising: a molecule passes close to many others before actually hitting one. Molecules are small targets and mostly miss each other.

The volume occupied is the other way of saying it. At atmospheric pressure the molecules of air fill about a thousandth of the space available. A gas is overwhelmingly empty, which is why it compresses easily, why its density can be changed by a factor of a thousand without anything unusual happening, and why a liquid, where the molecules are in contact, behaves so differently.

The same molecules arriving at a wall are where pressure comes from, and the connection is worth making explicit. Between one arrival and the next each molecule has travelled a free path — so pressure, which reads as a perfectly steady push, is the average of an enormous number of independent journeys, each of them ending at a randomly chosen moment. Steadiness at the scale of a room is a statement about how many of them there are.

Why the continuum works #

Now the payoff. Fluid mechanics assigns a density, a velocity and a pressure to each point. A point contains no molecules, so what is meant is an average over a small region — small compared with the apparatus, large enough that the fluctuations are negligible.

Both requirements can be met when the free path is far smaller than anything being measured. Take a cube one micron on a side in ordinary air: it contains about $2.5\times10^7$ molecules, so the fractional fluctuation in that count is about $1/\sqrt{N} \approx 0.02$ per cent. It is also a thousandth of a millimetre, which is a point as far as any ordinary apparatus is concerned.

That is the whole justification. The continuum picture is not an approximation to be apologised for; it is the consequence of a ratio of about $10^{-5}$ between the free path and the apparatus, and it is why pressure at a point and viscosity as a diffusivity are meaningful ideas at all.

The ratio has a name — the Knudsen number, $\mathrm{Kn} = \lambda / L$ — and the regimes it separates are sharp enough to be worth listing. Below about 0.01 the continuum equations hold with no-slip at walls. Between 0.01 and 0.1 they hold in the bulk but the fluid slips at walls. Between 0.1 and 10 neither description works and the problem must be solved molecule by molecule. Above 10 molecules cross the apparatus without meeting each other at all, and the gas is a set of independent projectiles.

How often, rather than how far #

The free path is a distance, and its companion is a time. Divide by the mean molecular speed — about 470 m/s for nitrogen at room temperature — and the interval between collisions comes out at roughly $1.4 \times 10^{-10}$ seconds. A molecule of air collides about seven billion times a second.

That number is why thermodynamic equilibrium is such a good assumption in ordinary circumstances. Any disturbance to the velocity distribution is erased within a few collision times, which is a fraction of a nanosecond, so a gas subjected to almost anything a laboratory can do to it is Maxwell-distributed at every instant. The distribution of speeds is not an idealisation the gas approaches slowly; it is re-established continuously and faster than anything else in the problem.

It also puts a floor under how fast a gas can respond. Sound in air travels at about 340 m/s and the free path is 68 nm, so a sound wave of a gigahertz would have a wavelength comparable with the free path — and above roughly that frequency sound stops propagating in air at all, because there are not enough collisions per wavelength to carry it. The gas simply cannot support the wave.

The same arithmetic gives the speed of sound its rough value. Sound is a disturbance carried by molecules bumping into one another, so it cannot travel much faster than the molecules themselves — and indeed $c = \sqrt{\gamma kT/m}$ against $\bar{v} = \sqrt{8kT/\pi m}$ differ only by a factor near 0.75. The speed of sound in a gas is essentially the molecular speed, which is not a coincidence.

Seven billion collisions a second is what makes equilibrium inevitable rather than merely likely. With that many rearrangements, a gas explores its accessible configurations so rapidly that the overwhelmingly most numerous ones are the only ones ever observed — which is the counting argument supplied with a rate. The counting says which arrangements are common; the collision frequency says how quickly the gas gets to them, and without the second the first would be a statement about eternity rather than about the next microsecond.

Where the gas stops being a fluid #

The four regimes are not academic, and each has an engineering home.

Vacuum systems. At $10^{-3}$ mbar the free path is about 5 cm; at $10^{-6}$ it is 50 metres. So in a high-vacuum chamber molecules travel from wall to wall without colliding, and the whole language of flow and viscosity is inapplicable. Pumping is then a matter of individual molecules finding their way out, and the design rules are entirely different.

Microfluidics and MEMS. Air in a one-micron channel has $\mathrm{Kn} \approx 0.07$ — the slip regime — so the no-slip condition fails and the flow exceeds what the pipe law predicts. Micromachined devices are designed with slip corrections as a matter of routine.

Re-entry. A spacecraft at 100 km altitude meets air whose free path is metres, so it is in free-molecular flow: heating and drag must be computed by counting molecular impacts. Lower down it becomes continuum flow and ordinary aerodynamics applies. A vehicle passes through every regime on this list in a few minutes.

Chemical vapour deposition and thin-film growth. Whether an arriving atom travels to the substrate in a straight line or diffuses through a gas decides how uniformly a film covers a stepped surface, and the free path against the chamber dimension is what selects between them.

Aerogels and fine insulation. Trap air in pores smaller than 68 nm and the molecules collide with the pore walls more often than with each other, which suppresses the gas’s ability to conduct heat. Insulation of that kind beats still air, which sounds impossible until the free path is taken into account.

Temperature, incidentally, is not what moves that boundary. It changes how fast a molecule crosses its free path and not how long the path is, so the collision rate rises with temperature while the mean free path does not — a distinction the estimate makes cleanly and intuition usually does not.

What else it decides #

The free path is the length scale in every transport property of a gas, and each one comes out as a density times a speed times $\lambda$.

Viscosity is $\mu \sim \tfrac13 \rho \bar{v} \lambda$, which is momentum being carried across a flow by molecules travelling one free path between deliveries. Thermal conductivity has the same form with heat capacity in place of momentum. Diffusion has it with concentration.

The consequence Maxwell noticed and disbelieved is that $\rho\lambda$ is independent of density — twice as many carriers each going half as far — so a gas’s viscosity does not depend on its pressure. Halve the air in a vessel and its viscosity is unchanged. He tested it and it held, over a range of pressures, and it is one of the earliest quantitative triumphs of the molecular picture — a prediction that follows from nothing but counting, made before there was any independent evidence that molecules existed at all. That evidence arrived later, and from a different direction.

The independence fails at both ends, and the free path says where. At very high density the molecules are no longer sparse and the derivation’s assumptions collapse; at very low density $\lambda$ exceeds the apparatus, the carriers travel wall-to-wall, and the viscosity becomes proportional to pressure after all. That low-pressure failure is what a Pirani gauge measures.

Changing it deliberately #

Because $\lambda = 1/n\sigma$, the free path can be moved by changing either factor, and both are used.

Lowering the density. This is what a vacuum system does, and the range available is enormous: from 68 nm at atmospheric pressure to kilometres in ultra-high vacuum. Every decade of pressure is a decade of free path, exactly, until the walls intervene. The whole design of vacuum equipment — what kind of pump, what kind of gauge, whether flow can be spoken of at all — is a question of which decade the free path is in.

Confining the gas. If the container is smaller than the free path would be, the walls take over as the collision partner and the effective free path becomes the container’s size. That is what makes fine-pored insulation work, and it is why the thermal conductivity of a gas in a small enough pore falls below its bulk value while the gas itself is unchanged.

Changing the cross-section. Larger molecules have shorter free paths at the same density. A xenon atom’s collision diameter is about 40 per cent larger than helium’s, so its free path is about half — which contributes to xenon’s low thermal conductivity and its use as an insulating fill gas.

There is a pleasing consistency check available in the first of these. A gas expanded to a tenth of its density has ten times the free path and a tenth of the carriers, so the transport properties are unchanged — until the free path reaches the vessel, at which point they fall. The pressure at which that happens is a measurement of the vessel’s size, and it is how a Pirani gauge is calibrated.

How the number was got before anyone could count #

The free path is quoted here from a known density and a known molecular size, which makes it look like a derived convenience. Historically the arrow ran the other way: the free path was the measured quantity, and it is how the size of a molecule was first found.

Loschmidt did it in 1865, with two facts and no new experiment. The first was the viscosity of air, which kinetic theory writes as roughly $\tfrac13\rho\bar v\lambda$ — so measuring a viscosity measures a free path, and the free path is $1/n\sigma$, giving one equation in the number density and the molecular diameter. The second was that air can be liquefied, and that in the liquid the molecules are essentially touching, so the ratio of liquid volume to gas volume is the fraction of the gas’s volume the molecules themselves occupy — which is $n$ times the volume of one molecule, a second equation in the same two unknowns.

Two equations, two unknowns, and out come both. Loschmidt obtained a molecular diameter of about a nanometre and a number density of order $10^{24}$ per cubic metre. The modern values are 0.37 nanometres and $2.7\times10^{25}$, so he was within a factor of three on the size and rather worse on the count — and he was the first person in history to state either.

What makes it a good argument rather than a lucky one is that neither input mentions molecules. A viscosity is measured with a capillary and a liquid density with a balance, and the molecular quantities appear only because the theory relates them. That is the same move Perrin later made with Brownian motion, and it is why the free path deserves to be thought of as an observable rather than as an estimate.

The height at which the air stops colliding #

The barometric figure runs the density down through several decades, and if it were continued far enough it would reach the altitude where this essay’s whole framework changes character.

The free path grows as the density falls, and the atmosphere’s density falls exponentially, so somewhere the free path becomes comparable with the scale height itself — the distance over which the density changes appreciably. Above that level a molecule launched upward does not collide again on the way; it follows a ballistic trajectory and comes back down, or does not. That level is the exobase, about five hundred kilometres up, and above it the atmosphere is not a gas in any sense this collection has used.

What happens there is decided by the tail of the speed distribution. A molecule leaving the exobase upward with more than the escape speed is gone. At the thousand or so kelvin of that region a hydrogen atom’s typical speed is about four kilometres a second against an escape speed near eleven, so a small but real fraction of the tail clears it and hydrogen leaks away continuously.

For nitrogen the same arithmetic is annihilating. Nitrogen is twenty-eight times heavier, so its typical speed is $\sqrt{28}$ times smaller, the ratio to the escape speed is fourteen rather than two and a half, and the surviving fraction of the distribution is of order $10^{-86}$. Not small: absent.

So a planet’s atmospheric composition is decided by an exponential of the square of a mass ratio, evaluated at one altitude — the altitude where the free path stops being small. Earth keeps its nitrogen and oxygen for ever and loses its hydrogen; a body with a lower escape speed loses more; and the reason there is a sharp answer at all is that there is a definite height at which molecules stop meeting each other.

Where the model stops #

Molecules are hard spheres. They are not; they attract at a distance and repel at contact, so the effective cross-section depends on the collision speed and therefore on temperature. Measured free paths shrink more slowly with temperature than the hard-sphere estimate predicts.

Collisions are binary and uncorrelated. At high density three-body collisions matter and successive collisions are correlated, so the free path becomes a poorer description of the transport than the number alone suggests.

One species. A mixture has a free path per pair of species, and light molecules in a heavy gas behave quite differently from the reverse.

The scatterers are fixed in the figure and moving in a gas. That is the √2 above, and the figure is honest about which case it draws rather than importing a factor from the other one.

Nothing is charged. In a plasma the interaction has infinite range, so the whole idea of a cross-section has to be rebuilt around small-angle deflections accumulating, and the resulting length can be very different.

The gas is dilute enough for a mean to mean something. The figure’s own residual is a small demonstration of that boundary, and it is measured rather than assumed.

The ladder from here #

Later rungs on this anchor: transport coefficients derived properly from the Boltzmann equation rather than estimated. Effusion, where molecules leave through a hole smaller than the free path and the rate depends on molecular mass — the basis of isotope separation. Thermal transpiration, where two vessels at different temperatures connected by a fine tube reach different pressures at equilibrium, which is impossible in the continuum picture. And the Knudsen regime as an engineering discipline, where a gas is simulated as individual particles because nothing else works.