Illustrated Physics
physics, drawn
Thermodynamics

Pressure is a rate of arrival, and the gas law falls out of counting

Nothing in a gas is pushing on the walls. Molecules arrive, bounce, and leave, and pressure is the momentum they deliver per second — from which the ideal gas law follows with no thermodynamics in it at all.
16 min read 6 figures Order out of the randomWhat stays the same

A gas in a cylinder holds the piston up, and nothing in the gas is pushing. There is no spring, no repulsion between the molecules worth mentioning, and no contact between the gas and the piston except for brief collisions separated by long emptiness.

What holds the piston up is a rate. Molecules arrive, reverse, and depart, and each one delivers a small amount of momentum in doing so. Enough of them arrive, often enough, that the delivery averages into something a gauge reads as a steady number — and this rung is about turning that sentence into the ideal gas law without borrowing anything from thermodynamics.

Pressure, counted as momentum arriving at a wallMolecules with speeds drawn from the Maxwell–Boltzmann distribution and directions drawn uniformly in the plane of the figure. Those moving toward the wall will bounce off it, reversing the component perpendicular to it and delivering twice that momentum each — and pressure is nothing but the rate at which that momentum arrives. Because the directions here lie in a plane rather than in space, the perpendicular component carries half the energy rather than the third it carries in a real gas.wallamber: heading for the wallgrey: not, this instant11 of 26approaching⟨v²⟩ = 3.03⟨vₓ²⟩ = 1.50ratio 0.49expected 0.50Σ2mvₓ = 21.1
Fig. 1 Molecules with speeds drawn from the Maxwell–Boltzmann distribution and directions drawn uniformly, some heading for the wall and some not. The figure measures, on its own sample, the quantity the derivation depends on: the share of the total kinetic energy carried by the component perpendicular to the wall.

What one collision delivers

Take a molecule of mass mm approaching a wall with velocity component vxv_x perpendicular to it. It bounces elastically, so it leaves with −vx-v_x and its other components unchanged.

Its momentum has changed by 2mvx2mv_x. Momentum is conserved, so the wall received exactly that much, and the direction is into the wall.

Two things about that expression matter more than the arithmetic. The factor of two is there because the molecule does not merely stop — it reverses, which is twice as much momentum transfer as stopping would be. And only the perpendicular component appears; a molecule skimming along the wall at enormous speed with a small vxv_x delivers almost nothing.

Counting the arrivals

Pressure is force per area, and force is momentum delivered per second, so the question is how many molecules arrive per second and with what distribution of vxv_x.

In a time Δt\Delta t, the molecules that reach a patch of wall of area AA are those within a distance vxΔtv_x\Delta t of it and moving toward it — a slab of volume AvxΔtAv_x\Delta t, of which half are moving the right way. With nn molecules per unit volume, that is 12nAvxΔt\tfrac{1}{2}nAv_x\Delta t arrivals, each delivering 2mvx2mv_x.

Multiply and divide out:

P=momentum per secondA=nm⟨vx2⟩.P = \frac{\text{momentum per second}}{A} = n m \langle v_x^2 \rangle.

The average appears because molecules have a range of speeds, and it is the mean of the square that enters — one factor of vxv_x from how hard each collision is, one from how often collisions happen. That is not a subtlety to be waved past: fast molecules count twice, and it is why pressure depends on the mean square speed rather than on the mean speed.

Where the third comes from

The final step is the one the figure is built to check.

Molecules move in three dimensions and no direction is preferred, so the mean square velocity divides equally between the three components:

⟨vx2⟩=13⟨v2⟩,\langle v_x^2 \rangle = \tfrac{1}{3}\langle v^2 \rangle,

giving

P=13nm⟨v2⟩,PV=13Nm⟨v2⟩.P = \tfrac{1}{3} n m \langle v^2 \rangle, \qquad PV = \tfrac{1}{3} N m \langle v^2 \rangle.

Pressure, counted as momentum arriving at a wallMolecules with speeds drawn from the Maxwell–Boltzmann distribution and directions drawn uniformly in the plane of the figure. Those moving toward the wall will bounce off it, reversing the component perpendicular to it and delivering twice that momentum each — and pressure is nothing but the rate at which that momentum arrives. Because the directions here lie in a plane rather than in space, the perpendicular component carries half the energy rather than the third it carries in a real gas.wallamber: heading for the wallgrey: not, this instant20 of 40approaching⟨v²⟩ = 3.18⟨vₓ²⟩ = 1.52ratio 0.48expected 0.50Σ2mvₓ = 39.9
Fig. 2 A second sample, drawn with a different seed. The measured share of energy in the perpendicular component moves around with the sample — it is an average over a few dozen molecules — and stays near the value isotropy demands, which is the check the figure is making.

The figures print ⟨v2⟩\langle v^2 \rangle and ⟨vx2⟩\langle v_x^2 \rangle for their own sample and the ratio between them. Because the drawing lays the directions out in a plane rather than in space, the expected ratio there is a half rather than a third — the energy divides between two components rather than three — and the printed number hovers near 0.5 with the scatter that a sample of a few dozen produces. Isotropy is a statement about an average, and a small sample makes that visible in a way a large one hides.

The factor of a third is therefore a statement about dimensionality, and it is the only place in the derivation where the number three appears. A gas confined to a plane would have a factor of a half; the physics is unchanged and only the counting differs, which is exactly the situation with the exponent in an inverse-square law.

What has been derived, and what has not

Comparing PV=13Nm⟨v2⟩PV = \tfrac{1}{3}Nm\langle v^2\rangle with the experimental gas law PV=NkTPV = NkT gives

12m⟨v2⟩=32kT,\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}kT,

and it is worth being careful about what that is. It is not a derivation of temperature. It is a definition of temperature in mechanical terms, made legitimate by the fact that the quantity it defines behaves the way the thermodynamic temperature behaves. The mechanics supplied PV∝Nm⟨v2⟩PV \propto N m \langle v^2\rangle; the identification of ⟨v2⟩\langle v^2\rangle with temperature is the bridge between two subjects, and it is the whole content of statistical mechanics in miniature.

What has genuinely been derived is more impressive than the formula. The mechanics alone, with no thermodynamics, predicts:

Pressure is proportional to density at fixed molecular speed — Boyle’s law, from counting arrivals.

Pressure is proportional to ⟨v2⟩\langle v^2 \rangle at fixed density — which, once the identification above is made, is the temperature dependence.

And pressure does not depend on the mass of the molecules at fixed temperature. Heavy molecules move more slowly by exactly the factor that keeps m⟨v2⟩m\langle v^2\rangle the same, so a mole of hydrogen and a mole of xenon at the same temperature exert the same pressure in the same volume. That is Avogadro’s hypothesis, which was an unexplained empirical rule for half a century, and here it is a consequence of two lines of counting.

Molecular speeds at 2 temperaturesThe distribution of molecular speeds in a gas, with each curve enclosing the same area. Raising the temperature moves the peak right and lowers it: the same molecules, spread over a wider range of speeds.012345600.20.40.6speedT = 1T = 3no molecule has the average speed; most are near it
Fig. 3 The distribution of molecular speeds at two temperatures. The pressure depends on the mean of the square of these speeds, so the broad right-hand tail contributes disproportionately — a molecule at twice the typical speed counts four times.

The scale of what is being averaged

The numbers are worth having, because they explain why a gauge reads a steady value rather than a rattle.

Air at room temperature and atmospheric pressure has about 2.5×10252.5\times10^{25} molecules per cubic metre, moving at a mean speed of about 470 m/s. The rate at which they strike a wall is 14nvˉ\tfrac{1}{4}n\bar{v}, which comes to roughly 3×10273\times10^{27} collisions per square metre per second.

On one square millimetre — the sort of area a small pressure sensor uses — that is 3×10213\times10^{21} arrivals per second. Even in a microsecond, 3×10153\times10^{15} molecules have hit it, and the relative fluctuation of a count is one over its square root, so the reading is steady to about two parts in 10810^8 over a microsecond.

That is why pressure is a smooth quantity. Not because anything about it is smooth, but because the averaging is over such an absurd number of discrete events that the graininess is pushed below every instrument. Reduce the density enough and the graininess returns: in a good vacuum a sensitive detector registers individual impacts, and the concept of pressure gives way to a counting rate.

The same picture, doing thermodynamic work

Having pressure as a mechanical quantity turns the whole of the gas’s thermodynamic behaviour into statements about molecules, and three of them are worth following because they answer questions the thermodynamic account can only restate.

A Carnot cycle on pressure–volume axesTwo isothermal steps joined by two adiabatic ones, forming a closed loop. The area enclosed is the net work done by the gas over one cycle.123400.511.5volumepressure1234net workhot isothermcold isothermadiabatic steps
Fig. 4 A Carnot cycle on pressure–volume axes, with the enclosed area as the net work. Every point on this loop is a value of 13nm⟨v2⟩\tfrac{1}{3}nm\langle v^2\rangle, and every step of it is a story about what happened to the molecules’ speeds.

Why a gas cools when it expands against a piston. A molecule bouncing off a receding wall comes back slower, because the wall was moving away during the collision — the same reason a ball thrown at a retreating bat returns more slowly. Every collision with a moving piston shaves a little from the molecule’s speed, so the mean square speed falls, so the temperature falls. Adiabatic cooling is a mechanical fact about bouncing off something that is going away.

Why compression heats. The identical argument with the sign reversed: an approaching wall returns molecules faster. That is why a bicycle pump warms, and it is the mechanism behind the compression stroke of every diesel engine.

Ways to arrange 10 coinsThe number of distinct arrangements giving each number of heads, for 10 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.1010145212032104252521061207458109110number of heads1,024 arrangements in total, all equally likelythe middle has 252 of them
Fig. 5 Arrangements of ten coins by number of heads, with the count of ways peaking in the middle. Sharing energy between two bodies is the same count with quanta in place of coins, and the peak is why the sharing ends up even.

And why heat flows from hot to cold at a contact. Fast molecules on one side strike the boundary and, on average, give energy to the slower ones on the other. No collision has a preferred direction; the net flow exists because there are more ways to share the energy evenly than unevenly.

Each of those is a thermodynamic law with a mechanical picture attached, and the picture makes each of them obvious in a way the phenomenology does not. That is the payoff of the whole kinetic programme, and it is why the ignored century of Bernoulli’s paper is worth dwelling on.

Two things the derivation did not need

The result is more robust than it has any right to be, and two omissions are worth pointing at.

Collisions between molecules never appeared. The derivation counted arrivals at the wall and said nothing about what happened in between. A gas whose molecules passed straight through one another would exert exactly the same pressure. What collisions do is maintain the distribution — they are why the speeds settle into the Maxwell–Boltzmann form and stay there — but the pressure, given a distribution, does not depend on them at all. This is why the ideal gas law holds so well in regimes where the mean free path is long, and it is the reason the law is nearly exact for a rarefied gas rather than being an approximation that degrades.

The wall’s material never appeared either. Steel, glass, ice or a soap film: the momentum delivered is 2mvx2mv_x regardless. That is a statement with real content, because it says pressure is a property of the gas alone, and it is what makes a pressure gauge a general instrument rather than one calibrated per surface.

Both omissions are the same kind of fact as the indifference of the Carnot ceiling to the working substance: an argument that never mentions a mechanism cannot be sensitive to it, and that insensitivity is where its generality comes from.

An elastic collisionTwo bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.before23.001-1.00momentum 5.00energy 9.50after20.3314.33momentum 5.00energy 9.50energy survives too — but only because e = 1
Fig. 6 An elastic collision worked out from the two conservation laws. Every molecule–wall encounter above is one of these with an infinitely massive partner: the light body reverses, the heavy one takes the momentum, and none of the kinetic energy is lost.

What the derivation assumes

Four assumptions, each of which fails somewhere that matters.

The collisions with the wall are elastic and specular. Real molecules stick briefly, exchange energy with the wall’s own vibrations, and leave in a direction largely unrelated to the one they arrived in. The pressure is unchanged if the wall is at the same temperature as the gas, because whatever leaves carries the same average momentum as whatever arrives. It is not unchanged if the wall is hotter or colder — which is how a radiometer’s vanes turn, and why the effect is a low-pressure phenomenon rather than a demonstration of light pressure.

The molecules have no volume. At high density, the space available to each molecule is less than the container’s volume, because the others are in the way, and the pressure is correspondingly higher than the ideal law predicts. That is the bb term in van der Waals’ equation.

They do not attract one another. Real molecules do, weakly, and a molecule near the wall is pulled back by the ones behind it, so it arrives with slightly less momentum than the ideal count assumes. That reduces the pressure, and it is the aa term — and the competition between the two corrections is what produces a critical point and a liquid phase.

And the gas is in equilibrium. The counting assumed a settled distribution with no net flow. A gas being pumped, heated at one end, or moving as a wind has a distribution shifted away from the equilibrium one, and pressure then depends on which direction the surface faces.

What the mechanical picture costs

The kinetic derivation is more satisfying than the thermodynamic one and it is not free, and the charges are worth listing next to each other.

It commits to a mechanism. Thermodynamics gets the gas law from measurements and stays silent about what a gas is made of, which is why Carnot’s engine argument survived being built on a wrong theory of heat. The kinetic derivation is specific: hard elastic particles, no internal structure, no forces between them. Every one of those commitments is a place it can be wrong, and three of them are wrong for every real gas.

It gives an average and hides the fluctuations. Pressure emerges as a mean, and the derivation says nothing about the spread. That spread is real, it falls as one over the square root of the count, and it becomes the dominant behaviour once the count is small — in a vacuum system, in a nanoscale cavity, or on a micron-scale membrane, where the pressure is not a number but a noisy signal with a measurable variance.

And it needs a distribution it does not derive. The counting used ⟨v2⟩\langle v^2 \rangle without saying why the speeds have the distribution they do. That distribution is the output of collisions, which the derivation was careful to point out it did not need — so the argument depends on a fact it deliberately excluded from its own machinery. Closing that loop takes the whole apparatus of statistical mechanics, and until it was closed the kinetic picture was assuming what it wanted to explain.

The honest summary is that the mechanical derivation buys explanation and pays in assumptions, and that the thermodynamic one buys robustness and pays in silence. Both are on this site for that reason.

Where the argument came from

Daniel Bernoulli published this derivation in 1738, in Hydrodynamica, and it was ignored for over a century.

It contains everything above: the molecules as tiny particles in ceaseless motion, the pressure as the accumulated impact, the correct dependence on speed, and the identification of heat with molecular motion. It was written before the conservation of energy was understood, before the atomic hypothesis had any chemical evidence behind it, and while heat was generally believed to be a fluid.

The reason it went nowhere is not that anybody refuted it. It is that the caloric theory was working well, that Carnot’s engine argument was about to succeed spectacularly without needing any molecules, and that an unobservable mechanism offering no new predictions is not a compelling purchase. Bernoulli’s account gave the same gas law that was already known empirically.

It became indispensable only when Clausius, Maxwell and Boltzmann used the same picture to predict things the thermodynamic account could not reach — viscosity, thermal conductivity, diffusion, the distribution of speeds itself — and when those predictions were confirmed. A mechanism earns its place by predicting something outside the range of the phenomenology it explains, and until it does, being right is not enough.

The ladder from here

Later rungs: the mean free path, and why a molecule at 470 m/s takes minutes to cross a room. Effusion through a small hole, and the separation of isotopes it permits. Viscosity, thermal conductivity and diffusion derived from the same transport picture, all three with the same mean-free-path factor. The van der Waals equation and the critical point. The equipartition theorem in general, and the heat capacities it predicts — including the ones it gets wrong, which is where quantum mechanics entered the subject. Real-gas equations of state. And the pressure of a photon gas, where the same counting with light in place of molecules gives a factor of a third replaced by a third of the energy density, and radiation pressure holds up the interiors of stars.

Where this goes next

0123456700.20.40.60.8speedT = 0.5T = 1T = 2T = 4no molecule has the average speed; most are near it The speeds in a still room The air in a quiet room is not still. Every molecule in it is moving at hundreds of metres per second, and temperature is a single number summarising an entire distribution. before23.001-1.00momentum 5.00energy 9.50after20.3314.33momentum 5.00energy 9.50energy survives too — but only because e = 1 Collisions are easier than forces, and momentum is the reason Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved. 1010145212032104252521061207458109110number of heads1,024 arrangements in total, all equally likelythe middle has 252 of them Entropy is a count, and the arrow of time is arithmetic Nothing in mechanics prefers a direction. Entropy is not a force pushing things toward disorder — it is the observation that some outcomes have vastly more ways of happening than others.