The speeds in a still room

The air in a closed, quiet room is doing nothing. Nothing is blowing, nothing is being stirred, and the whole volume of it is at rest.

At the scale of a molecule the same room is violent beyond any everyday comparison. A typical nitrogen molecule in it is travelling at about five hundred metres per second — faster than sound, comparable to a rifle bullet — and it collides with another molecule roughly seven billion times a second. The reason the room seems still is that all of this motion is in every direction at once, and the average of a vast number of vectors pointing everywhere is very close to zero.

Temperature is not a speed #

The first thing the figure says is that a gas at one temperature does not have one speed. It has a distribution — some molecules crawling, some travelling several times the typical speed, and a broad hump in between.

That matters because temperature is routinely described as “the average kinetic energy of the molecules”, which is true as a definition and misleading as a picture. It gives the impression of a population all doing much the same thing, and the truth is a population doing a wide range of things whose average happens to be a useful number.

Three different averages can be extracted, and they are genuinely different: the most probable speed at the peak, the arithmetic mean somewhat above it, and the root-mean-square speed higher still. The asymmetry causes it. Speeds cannot be negative, so the distribution is bounded below at zero and unbounded above, and a long right tail pulls the mean past the peak.

The one that enters thermodynamics is the root-mean-square, because energy goes as the square of speed and it is energy that is being averaged:

$$\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}kT.$$

Read that as a definition of temperature rather than a discovery about it. What makes it substantial is the constant $\tfrac{3}{2}$: one half of $kT$ for each independent direction of motion, which is the equipartition theorem, and one of the few results in physics that hands out energy by counting rather than by calculating.

Two consequences follow from the same equation. Since the energy per molecule depends only on $T$, a heavy molecule at a given temperature moves slower than a light one — as the square root of the mass ratio. Hydrogen at room temperature averages about 1,900 metres per second, carbon dioxide about 380. That is the mechanism behind gaseous diffusion, which is how uranium isotopes were first separated: two molecules differing by one percent in mass differ by half a percent in speed, and half a percent, repeated through thousands of stages, is enough.

Where the shape comes from #

The curve is not an empirical fit. It follows from two assumptions and no others.

The first is that the probability of a molecule having energy $E$ falls off as $e^{-E/kT}$ — the Boltzmann factor, which is the single most reused expression in statistical physics and which comes from counting the ways energy can be shared out. The second is purely geometric: speeds near $v$ occupy a shell in velocity space whose area grows as $v^2$, so there are more ways to have a large speed than a small one.

Multiply the two:

$$f(v) \propto v^2 e^{-mv^2/2kT}.$$

The $v^2$ pushes the curve up from zero, the exponential pulls it down at large speeds, and the peak is where the two effects balance. Every feature of the shape is one factor or the other winning.

That structure — a growing count multiplied by a falling probability — recurs constantly. It is why the line-counting argument and the entropy argument have the same feel, and it is the same competition that produces the peak in the binomial distribution of arrangements: many ways, each unlikely.

The tail does the work #

The most consequential part of the distribution is the part with almost nothing in it.

A great many processes have a threshold: a molecule needs a certain minimum energy to escape a liquid surface, to break a bond, to ionise, to get over a reaction barrier. Whether the process happens at all depends on how many molecules are past that threshold, which is the area under the far right of the curve.

That area is exponentially sensitive to temperature. Because the tail falls as $e^{-E/kT}$, a modest rise in $T$ multiplies the population above a high threshold by a large factor. This is the Arrhenius law in chemistry, and it is the reason reaction rates roughly double for every ten-degree rise near room temperature — not because molecules are moving twice as fast, which they are not, but because the fraction of them past the barrier has doubled.

Evaporation is the everyday case. A puddle at fifteen degrees dries even though the energy needed to leave the liquid corresponds to a much higher temperature, because the tail of the distribution always contains some molecules with enough. And the molecules that leave are the fastest ones, so what is left behind has a lower average energy: evaporation cools, and the cooling is a direct consequence of the shape of a curve. Sweating works because of the right-hand tail.

The same tail explains why the Earth has kept its nitrogen and lost its hydrogen. Escape from the atmosphere requires about 11 km/s, which is far out in the tail for any gas at atmospheric temperatures. For hydrogen the tail reaches; over geological time the reaching is enough. For nitrogen it does not, by a margin that grows exponentially with the mass. The composition of a planet’s atmosphere is set by which curves have appreciable area past the escape speed, which is why the Moon has none and Titan, colder and no more massive, has a thick one.

What the curve cannot show #

Three things are missing from every version of this figure.

It shows speeds, not velocities. The direction information has been integrated away, and it is the directions that make the gas isotropic and the room still. A gas in a wind has the same speed distribution about a shifted mean, and the plot cannot tell the difference.

It shows a snapshot, not a history. No individual molecule sits anywhere on the curve for long. Each collision reassigns its speed, and a molecule that is momentarily in the fast tail is typically back near the peak within nanoseconds. The distribution is stationary; its members are not. This is the crucial difference between a distribution being in equilibrium and anything in it being at rest.

It shows the gas, not the collisions that maintain it. The shape is not imposed from outside — it is what collisions produce and re-produce. Start a gas with every molecule at exactly the same speed and it will relax to this curve within a few collision times, which is a few nanoseconds. That relaxation is irreversible in the same sense an inelastic collision is, and it is the microscopic content of the second law.

Where the shape comes from, again #

The competition producing the peak is not particular to velocities. It is the same one that produces every distribution in statistical physics.

Counting arrangements is what the $v^2$ factor is doing: it is a count of directions, and there are more ways to move fast than slow simply because a larger sphere in velocity space has more surface. The exponential is the price of energy. Peak where the two balance, tails on both sides, and a sharpness that grows with the number of particles.

The correspondence extends to a detail that is easy to miss. The coin figure and the speed figure are both distributions over macrostates, and in both cases every underlying microstate is equally likely. Nothing prefers a molecule to move at the most probable speed. There are simply more ways to do it.

The gas as an engine’s working substance #

Pressure is not a property molecules have. It is what a wall experiences when $10^{23}$ of them strike it per second, each delivering a small momentum change, and the total is steady only because the number is large. Counting those impacts gives $pV = NkT$ directly from mechanics, with no thermodynamics assumed.

That derivation is worth knowing because it makes the gas laws stop being empirical. Doubling the temperature doubles the average kinetic energy, which raises both the rate of impacts and the momentum delivered per impact — each by a factor of the square root — so the pressure doubles. And once the gas laws are mechanical, the ceiling on every engine becomes a statement about what can be done with a population of moving particles rather than about steam.

Where the model stops #

The kinetic picture assumes an ideal gas: point particles, no forces between them except during instantaneous collisions, and no internal structure.

Molecular volume matters at high density. Real molecules occupy space, so the volume available for movement is less than the container’s, and the pressure at a given density is higher than ideal. That is the $b$ term in the van der Waals equation.

Attraction between molecules matters at low temperature. Molecules pull on each other slightly, so one approaching the wall is retarded by those behind it, and the pressure is lower than ideal — the $a$ term. Between them the two corrections predict a liquid phase, which the ideal gas cannot do at all: no attraction, no condensation.

Internal structure matters for the energy budget. A diatomic molecule can rotate and vibrate as well as translate — the vibration being the same harmonic oscillator that governs a pendulum — and equipartition gives each of those modes its share. So the heat capacity of nitrogen is higher than that of argon, and — the historically important part — it changes with temperature in steps, as modes switch on. Classical physics has no mechanism for a mode to be switched off, and the failure to explain those steps was one of the earliest indications that something was badly wrong with the classical account of matter.

Distinguishable particles is assumed throughout — the same assumption that makes the coin-counting picture of entropy subtly wrong for a real gas. At low temperature and high density the quantum statistics of identical particles take over and the Maxwell–Boltzmann distribution is replaced. Electrons in a metal are the standard example: they obey a completely different distribution at room temperature, which is why a metal’s electrons contribute far less to its heat capacity than counting them would suggest.

The ladder from here #

Later rungs: the pressure of a gas derived by counting momentum delivered to a wall, which produces the ideal gas law from mechanics alone. The mean free path, and why a molecule moving at 500 m/s takes minutes to cross a room. Effusion and Graham’s law. Viscosity, conduction and diffusion as three faces of the same molecular transport. The van der Waals equation and the critical point. Brownian motion, which made molecules visible and settled the argument about whether they existed. The equipartition theorem and its failure. And the Boltzmann factor derived from counting arrangements rather than assumed.

Maxwell derived the distribution in 1860 from a symmetry argument of startling economy, at a time when the existence of molecules was still seriously disputed. He was describing the statistics of objects nobody could demonstrate were there.