Illustrated Physics
physics, drawn
Concept

Knudsen number — where it appears

The ratio of a gas's mean free path to the size of whatever it is flowing through, which decides whether the gas behaves as a fluid or as independent molecules. Above about one the continuum description fails and effects with no fluid counterpart appear, including a pressure difference held by a temperature difference.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

thermodynamics · Kinetic theory The gas that leaves is not the gas inside. The distribution of molecular speeds inside a container at 300 K and in the beam that escapes through a small hole in it, each normalised to its own peak. They are not the same distribution. A molecule's chance of reaching the hole in a given time is proportional to how fast it is going, so the beam carries one more factor of speed than the gas does — v³ rather than v² times the Boltzmann factor — and the beam is therefore faster and hotter than what it came from. The mean speed inside is 476 m/s and in the beam 561 m/s, a ratio of 1.1781 against the exact 3π/8; and the mean kinetic energy is 1.500 kT inside against 2.000 kT in the beam, which are exactly 3/2 and 2. That difference is not a subtlety. A molecular beam made by effusion has a temperature, in the sense of a mean energy, a third higher than its source; a gas slowly leaking from a container leaves the remainder cooler than it would be if a fair sample had gone; and every calculation of a rate through an aperture that uses the bulk distribution is wrong by this factor.

The gas that leaves is not the gas inside

Put a small hole in a container of gas and what comes out is faster and hotter than what stays behind — its mean kinetic energy is 2kT against the 3/2 kT of the gas it came from. Nothing has heated it. A fast molecule simply reaches the hole more often than a slow one, so the sample that escapes is biased by exactly one factor of speed, and every consequence of effusion is that factor.

thermodynamics · Kinetic theory The highest note the air can carry, with height. The frequency at which a sound wave in the air loses its amplitude by a factor e in a single wavelength, in hertz on a logarithmic scale, against height, with the band of human hearing shaded. Because viscosity does not depend on pressure, the collision time grows as the pressure falls, and the ceiling falls with it. It is about 212 MHz at sea level, 0.17 MHz at 50 km and 3 kHz at 80 km. It passes the top of human hearing, 20 kHz, at about 68 km, and the bottom, 20 Hz, at about 110 km; at 100 km it is 92 Hz. Above about a hundred kilometres the air cannot carry anything a person could hear, at any loudness: the molecules are too far apart to pass a compression on before it has dispersed.

The note too high for thin air

Sound in a gas is a compression handed from molecule to molecule by collisions, and it can only be handed on if the molecules collide many times in each period of the wave. At sea level they collide seven thousand million times a second, and the limit is far above anything audible. But a gas's viscosity does not depend on its pressure, so as the air thins with height the time between collisions grows in proportion, and the highest note the air can carry falls with it: two hundred megahertz at the ground, three kilohertz at eighty kilometres, below twenty hertz above a hundred and ten. The upper atmosphere is not silent for want of loudness. It is silent because its molecules are too far apart to pass a sound along.

thermodynamics · Kinetic theory The distance at which the air takes over. Evaporation of water at 20 °C into dry air, in kilograms per square metre per second, against the thickness of the layer the vapour must cross before it is carried away, both on logarithmic scales, with the kinetic and diffusive resistances added in series: 1/u + δ/D, where u = √(RT/2πM) is the speed at which the kinetic flux carries vapour off and D the diffusion coefficient. Dashed, the kinetic ceiling. At one atmosphere the two resistances are equal at δ = D/u = 0.17 μm, about two and a half mean free paths of the air; a layer much thinner than that leaves the surface at its ceiling, and a layer much thicker throttles it in proportion to its thickness. At a tenth of an atmosphere diffusion is ten times faster and the crossover moves out to 1.7 μm. Only in vacuum, or across gaps shorter than a mean free path, does the liquid itself set the rate.

The puddle that could dry in a second

A liquid in equilibrium with its vapour loses molecules exactly as fast as the vapour delivers them, and kinetic theory says how fast that is. For water at room temperature it is two and a half kilograms per square metre every second — a puddle two and a half millimetres deep gone in a second. Puddles take hours because almost every molecule that leaves is knocked back by the air within a fraction of a micrometre. The ceiling binds only where the vapour can escape in a single free flight: in vacuum, where water freezes itself, in the smallest droplets, and on the filament of a lamp.

thermodynamics · Kinetic theory

Named alongside it

The objects these essays reach for when they reach for this one.

Mean free pathEffusionViscosityAcoustic absorptionCollision timeCollisionsContinuumContinuum limitCross-sectionDetailed balanceDiffusionEvaporation

All concepts