Why the air thins with height, and why that is the same law as the speeds

Air is not held up by anything. Each layer of the atmosphere is supported by the pressure of the layer beneath it, which is supported by the layer beneath that, and the whole column rests on the ground — where the pressure comes to 101 kilopascals, which is the weight of ten tonnes of air standing on every square metre.

From that single statement, plus the assumption that the air is all at one temperature, the density at every altitude follows. Real air is not at one temperature, and what that changes is the subject of the next rung.

The derivation, which is three lines #

Take a slab of air of thickness $\mathrm{d}h$ and unit area at height $h$. It is pushed up by the pressure below and down by the pressure above and by its own weight, and in equilibrium those balance:

$$\mathrm{d}p = -\rho g\,\mathrm{d}h.$$

Now use the ideal gas law to write the density in terms of the pressure: $\rho = pm/kT$, with $m$ the mass of one molecule. Substituting,

$$\frac{\mathrm{d}p}{p} = -\frac{mg}{kT}\,\mathrm{d}h,$$

whose solution is an exponential:

$$p(h) = p_0\,e^{-mgh/kT} = p_0\,e^{-h/H}, \qquad H = \frac{kT}{mg}.$$

The exponential appears because the equation says the fractional change in pressure per metre is constant — the same everywhere, at every altitude — and a quantity whose fractional rate of change is constant is an exponential. Nothing about the atmosphere had to be assumed beyond the gas law and hydrostatic balance.

$H$ is the scale height: the altitude gain that reduces the pressure by a factor of $e$. It is an energy over a temperature wearing the units of a length. For dry air at 288 K it is 8.4 kilometres, which is why the summit of Everest at 8.85 km sits at roughly a third of sea-level pressure, and why an airliner at 11 km cruises in air at a quarter of it.

The exponent is an energy over a temperature #

The form of the answer is the interesting part, and it is easier to see when the exponent is regrouped.

$$\frac{mgh}{kT} = \frac{\text{potential energy of one molecule at height } h}{kT}.$$

So the number of molecules per unit volume at a given height is proportional to $e^{-E/kT}$, where $E$ is the energy a molecule must have to be there. Nothing in that statement mentions gravity, height, or the atmosphere. It says: the population of a state falls exponentially with the state’s energy, measured in units of $kT$.

That is the Boltzmann factor, and it is one of the two or three most reused results in physics. The distribution of molecular speeds is the same exponential with $E = \tfrac12 mv^2$ instead of $mgh$. The vapour pressure of a liquid is the same exponential with $E$ the energy needed to escape the surface. The rate of a chemical reaction is the same exponential with $E$ the activation barrier — which is the Arrhenius law, arrived at empirically and explained by this.

The speed distribution’s high-speed tail is $e^{-mv^2/2kT}$ — the same function as the atmosphere’s altitude profile, with a different physical quantity in the numerator of the exponent. Kinetic energy in one case and potential energy in the other, both divided by $kT$, both exponentiated. That is not a resemblance between two formulae; it is one formula, applied to whichever energy the coordinate happens to carry.

The unification is worth pausing on. A column of air and a box of gas at rest look like different problems: one is about gravity and geometry, the other about collisions and speeds. They are the same problem, and the reason is that both are asking how many molecules have enough energy to be somewhere expensive. In the atmosphere the expense is height; in the distribution it is speed.

The atmosphere as an energy landscape #

The reading extends. A molecule at altitude has climbed a hill of potential energy, exactly as the hill in the energy landscape does, and the population at each height is set by how expensive that height is compared with the thermal energy available.

A potential landscape with an energy level marked is the mechanical picture, and the statistical one differs from it in a way worth stating. In a gas at temperature $T$ the molecules are not confined to the region below a single level: every height is occupied, with a population falling exponentially as the potential rises. The turning point of one molecule has been replaced by a distribution over all of them, and nothing in the atmosphere has a turning point at all.

This also answers a question the derivation does not obviously address: why the atmosphere has no top. Nothing about the exponential ever reaches zero, so there is no altitude above which no molecules are found; there is only an altitude above which they are rare. The “edge of space” at 100 km is a convention, and the exosphere extends thousands of kilometres with a population that keeps falling by a factor of $e$ every scale height.

The same reasoning says which gases a planet keeps. A molecule in the tail of the distribution with more than escape energy and no collisions ahead of it leaves. Because the exponent contains $m$, light molecules have both a larger scale height and a faster-falling energy cost of escaping, so hydrogen and helium leak away from Earth over geological time while nitrogen does not — and the Moon, whose escape energy is a twentieth of Earth’s, has retained nothing at all. One exponential, and a planet’s atmosphere is decided by it.

How wrong the isothermal assumption is #

The model’s single assumption is that the whole column is at one temperature, and it is not.

The troposphere cools with altitude at about 6.5 K per kilometre, reaching around 217 K at the tropopause — which is why the 288 K model overestimates the pressure at 20 km by 71 per cent while the 220 K one underestimates it near the ground. The real profile is an exponential whose scale height shrinks as it rises.

Two further approximations are worth naming, because both are usually left silent.

Hydrostatic equilibrium is a fiction in detail. Air moves. Weather is exactly the failure of the assumption, and the pressure at a fixed altitude varies by a few per cent with the passage of a weather system — which is what a barometer is for and is the entire reason it predicts anything.

Gravity is treated as constant. $g$ falls by 0.6 per cent over 20 km, which is negligible here and is not for an atmosphere many scale heights deep, such as a gas giant’s.

And there is a structural limitation beyond all of them: hydrostatic balance says nothing about why the temperature profile is what it is. That comes from convection and radiation, and the observed 6.5 K per kilometre is close to the adiabatic lapse rate a rising parcel of moist air follows on its own. The isothermal model on this page is not merely inaccurate — it describes an atmosphere that would not be stable against convection.

The mountain that settled it #

The claim that air has weight and that there is less of it higher up was not obvious, and the experiment that established it is a model of how to test a prediction that a theory makes and its rival does not.

The question in the 1640s was what holds up the mercury in a Torricelli tube. Two accounts were live. One was that the atmosphere presses on the open reservoir and pushes the column up, so the height measures the weight of the air above. The other was that nature abhors a vacuum and the mercury is held up by the space above it refusing to exist — a property of the tube rather than of the air.

The two agree on everything observable at one place, which is what makes the case interesting. Pascal’s insight was that they disagree about altitude: if the column is held up by the weight of the air above, carrying the apparatus up a mountain leaves less air above it and the column must fall. If it is held up by an abhorrence of vacuum, nothing about a mountain should matter, since the abhorrence is presumably the same everywhere.

His brother-in-law Périer carried a barometer up the Puy-de-Dôme in September 1648, with a second barometer left at the bottom being watched all day as a control against the weather. The column fell by about 8 centimetres of mercury over 1,000 metres of ascent and returned to its original height on the way down, while the control did not move.

That is a decline of about 11 per cent, and the exponential on this page predicts $1 - e^{-1000/8400} = 11.2$ per cent. The seventeenth century’s decisive experiment agrees with the twentieth century’s formula to within the accuracy of a mercury column carried up a volcano — and the reason to state it that way is that the agreement is not a coincidence of two models. It is the same physics, and the derivation at the top of this page is what Périer was measuring.

Made into an instrument #

The exponential is used backwards more often than forwards.

Altimeters. Every aircraft altimeter is a barometer with the formula on this page engraved into its scale. Because the real pressure at sea level varies with weather, the scale is offset by the pilot against a reported local setting; without that, two aircraft flying the same indicated altitude in different air masses would not be at the same height. Above a transition altitude all aircraft set the same reference precisely so that their errors agree — the separation that matters is between aircraft, not against the ground.

Centrifuges. Replace $mgh$ with the potential of a rotating frame and the same exponential describes how a suspension separates by mass in an ultracentrifuge. Perrin used exactly this in 1908 on microscopic particles in water, counting how their number fell with height under a microscope, and extracted Avogadro’s number from the rate of the fall — a measurement of the size of a molecule made by counting specks at different depths, and one of the decisive pieces of evidence that atoms exist.

Semiconductors and stars. The population of electrons above a band gap, the ionisation state of a stellar atmosphere, and the fraction of molecules able to react are all the same factor with a different energy in the numerator.

Two derivations that must agree #

There is a second route to the same exponential, and the fact that it does not mention hydrostatics is the strongest evidence that the result belongs to statistical mechanics rather than to meteorology.

The first route is the one at the top of this page: a slab of air, a weight, a balance of pressures. It is a mechanical argument about a continuous fluid, and it never mentions an individual molecule.

The second route counts molecules. In thermal equilibrium the probability of finding a molecule in a state of energy $E$ is proportional to $e^{-E/kT}$, from the counting of microstates and nothing else. A molecule at height $h$ has potential energy $mgh$, so the density at height $h$ is proportional to $e^{-mgh/kT}$. That is the same answer, obtained without any mention of pressure, weight, or mechanical balance.

Two arguments with no shared machinery reaching an identical result is the pattern this site keeps returning to — the three falloff exponents arrived at both by flux counting and by summing Coulomb’s law, the pendulum’s period by an elliptic integral and by an iteration. Here it also settles a question of ownership: the exponential is not a fact about air. It is a fact about thermal equilibrium in any potential, and the atmosphere is one instance.

The disagreement between the two routes is instructive as well. The statistical argument requires equilibrium and gives no scale height that varies with altitude; the hydrostatic one accommodates a temperature profile without complaint. Where the real atmosphere departs from the model, it is the equilibrium assumption that has failed, not the mechanics — the air is being stirred faster than it can settle.

The number that barely varies across the solar system #

The scale height is $kT/mg$, and it is worth noticing what is not in it: the amount of gas. A planet’s surface pressure does not appear anywhere, so two atmospheres of wildly different mass can thin out at the same rate.

They do. Venus has ninety-two bars of carbon dioxide at its surface and Mars has six millibars of the same gas, a ratio of fifteen thousand — and their scale heights are 16 and 11 kilometres. Earth’s is 8.4, Titan’s about 21, Jupiter’s about 28. Across bodies whose atmospheres differ by four orders of magnitude in mass and by hundreds of kelvin in temperature, every scale height in the solar system falls between about eight and thirty kilometres.

The reason is that the three quantities that do appear tend to compensate. Venus is hot, which raises $H$, and has a heavy molecule and a substantial gravity, which lower it. Jupiter’s gravity is enormous and its molecule is the lightest there is. Titan is very cold and very weak. What the formula measures is the ratio of a thermal energy to a gravitational one, and that ratio is a much tamer quantity than either of its parts.

It is also the number that decides whether a planet around another star can be examined at all. When such a planet crosses its star, the fraction of light it blocks depends slightly on wavelength, because the atmosphere is opaque at some colours and clear at others — and the size of that variation is set by the scale height, since that is the thickness of the annulus doing the absorbing. For a hot, low-gravity planet made mostly of hydrogen the scale height is hundreds of kilometres and the effect is a hundred parts per million, which is measurable. For an Earth-sized planet with an Earth-like atmosphere it is a fraction of a part per million, which is not. Every atmosphere characterised so far belongs to the first category, and the reason is one line of this page’s algebra.

Counting specks to weigh an atom #

Perrin’s use of the formula deserves more than the sentence it got, because it is one of the few measurements that settled an argument about whether atoms exist.

The exponential holds for any particle in thermal equilibrium in a potential, and a microscopic grain suspended in water is such a particle. Its effective weight is reduced by buoyancy — what matters is the excess of its density over the fluid’s — so a grain of resin a fifth of a micrometre across, in water, has an effective mass some ten million times smaller than the same volume of solid would suggest.

That makes its scale height enormous by molecular standards and tiny by human ones: tens of micrometres, a distance a microscope resolves by refocusing. So Perrin filled a shallow cell with a suspension, focused at a series of depths, counted the particles in view at each, and plotted the numbers.

They fell exponentially, and the rate of the fall gave $kT$ directly, because everything else in the exponent — the particle’s size, its density, the water’s — he had measured. Dividing the gas constant by that $k$ gives Avogadro’s number, which came out near $7\times10^{23}$: the right value, from counting specks through a microscope.

What made the result decisive was not its precision but the company it kept. Perrin obtained the same number from several unrelated routes — this one, Brownian displacement, the blue of the sky, radioactive decay counting — and the agreement of methods with nothing in common was what convinced the remaining sceptics that molecules were objects rather than a way of speaking.

What the figure cannot show #

The curves are drawn against pressure as a fraction of sea level, which is a ratio, and that hides something: the number of molecules involved falls by a factor of ten every 19 kilometres, so a linear axis showing three scale heights shows almost nothing of what is above it.

The figure also cannot show the mean free path, which is the quantity that decides whether “pressure” means anything at all. At sea level a molecule travels 68 nanometres between collisions; at 100 km it travels about 100 metres, and at 400 km — the altitude of the space station — several kilometres. Somewhere in there the gas stops being a fluid and becomes a collection of independent particles on ballistic trajectories, hydrostatic balance stops being the right description, and the exponential that was derived from it survives only because a different argument gives the same answer.

That crossover is invisible on a pressure plot, and it is where the model on this page genuinely ends rather than merely becoming inaccurate.

The mixture that does not separate #

An exponential whose scale height contains the molecular mass raises an obvious question: why has the atmosphere not sorted itself, with the heavy gases at the bottom and the light ones on top?

Each component would have its own scale height — 8.7 km for nitrogen, 7.6 for oxygen, 5.5 for carbon dioxide, 60 for helium — so the composition of the air at 20 km ought to differ noticeably from the composition at sea level. It does not. Below about 100 km the proportions of nitrogen, oxygen and argon are the same everywhere to a fraction of a per cent.

Why the stirring wins is a race between two timescales. Separation by molecular mass proceeds at the pace of diffusion, whose spread grows only as the square root of time; convection turns the lower atmosphere over in days. So the equilibrium calculation is correct about the destination and hopeless about the schedule — and above the turbopause, at about 100 km, the stirring stops and the composition does separate exactly as the calculation says.

The reason is that the sorting is a slow equilibrium and the atmosphere is stirred faster. Convection and wind mix the lower atmosphere on a timescale of days; diffusive separation would take far longer than that, so the mixture never gets the chance. The region where mixing wins is the homosphere, and it ends at the turbopause around 100 km — above which the stirring dies away, each gas does settle into its own exponential, and the composition changes with height as the naive reading predicts.

That is a good illustration of the difference between a thermodynamic prediction and a real one. The equilibrium calculation is correct about where the atmosphere is heading, and useless about where it is, because the arrival time is longer than the time available.

Where the ladder goes next #

The rungs from here: the adiabatic lapse rate, and why a real atmosphere’s temperature profile is set by convection; atmospheric stability and inversions; the Boltzmann distribution derived from counting microstates rather than asserted; the partition function, which turns the factor into a machine for computing every thermodynamic quantity; escape velocity and atmospheric loss over geological time; and the exponential atmosphere of a star, where the same balance is between gravity and a radiation pressure that is not an ideal gas at all.

The claim to carry forward is the one in the middle. The atmosphere thinning with height and the tail of the speed distribution are the same exponential — and once that is seen, the correct question about any population in thermal equilibrium is not what it is made of but what the energy costs.