Illustrated Physics
physics, drawn
Concept

The ideal gas law — where it appears

The relation pV = nRT, exact for a gas whose molecules have no size and no interactions and good to a per cent for air at ordinary conditions. It fails where either assumption does — near condensation, where attraction matters, and at high density, where the molecules' own volume does.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Pressure, counted as momentum arriving at a wall. Molecules 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.

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.

thermodynamics · Kinetic theory Newton's answer, Laplace's, and the measurement. The speed of sound in 4 gases at 273.15 K. The short bar is Newton's √(RT/M), which assumes the compressions stay at one temperature; the long one is the same multiplied by √γ, which is what they come to if no heat crosses between a compression and the rarefaction beside it; the upright mark is the measured value. Newton's is 15.5% low for air, 22.5% low for helium, 22.5% low for argon, 12.0% low for carbon dioxide, and the corrected one is right to 0.05% for every gas here. Turning it round: γ read off each pair of bars is 1.400 for air, 1.665 for helium, 1.666 for argon, 1.290 for carbon dioxide, which is 1 + 2/f with f = 5.0, 3.0, 3.0, 6.9 ways of holding energy — three for the monatomic gases, five for the diatomic ones, and nearly seven for carbon dioxide, whose bending modes have begun to take a share at this temperature and its stretch has not. A speed measured with a stopwatch counts the ways a molecule can move.

The correction that took a century

Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.

waves · Wave motion The column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

The pressure that comes from counting

Dissolve a teaspoon of anything in a litre of water, put a membrane between it and pure water, and the solution will hold up a column of water two and a half metres tall. Nothing is pulling. The pressure does not depend on what was dissolved, only on how many particles it made — which is the ideal gas law, with the solute in place of the gas.

fluids · Osmosis The pair potential, and the two things it does to a gas. The Lennard-Jones potential between two molecules, in units of its own depth and range, with the Mayer function it produces at 1, 3, 8 times the well depth in temperature. The virial coefficient is minus the integral of that function over volume, so the two parts of the potential contribute with opposite signs: the steep repulsive core makes the function minus one there, giving a positive contribution — molecules take up room — and the attractive well makes it positive, giving a negative one. At low temperature the attraction dominates and a gas is easier to compress than an ideal one; at high temperature the core dominates and it is harder. Between them is one temperature at which they cancel.

The first correction to the gas law

An ideal gas has no forces between its molecules. The first correction to what it does is computable from those forces alone — one integral over the pair potential — and its sign flips at a temperature where a real gas obeys the ideal law without being ideal at all.

thermodynamics · Kinetic theory The gas that has to stop expanding. The thermal expansion coefficient at constant pressure times the temperature, αT, against temperature over the Fermi temperature, on logarithmic axes, for an ideal classical gas and an ideal gas of spin-half fermions at the same density. For the classical gas αT is exactly one at every temperature, so α = 1/T grows without limit as the gas is cooled, which the third law forbids: by a Maxwell relation α is minus the pressure derivative of the entropy per unit volume, and that must vanish at absolute zero. The Fermi gas agrees with the classical one when hot and departs from it near the Fermi temperature; at a tenth of it αT is 0.048, and at a hundredth 0.002, falling as π²/2 times the square of the temperature over the Fermi temperature. Its expansion vanishes as the law requires.

The two laws absolute zero will not allow

The ideal-gas law and Curie's law are the first two equations of state most people meet. The third law of thermodynamics says both must be wrong at low temperature, and it says so without any model of what replaces them. A Maxwell relation turns every derivative of the entropy into something measurable with a ruler or a magnetometer: the thermal expansion, the slope of the magnetisation with temperature. If the entropy stops depending on pressure and field at absolute zero, those must vanish. The classical gas expands as 1/T and Curie's magnetisation climbs as 1/T, and both slopes grow without limit instead.

thermodynamics · Third law The lift of a zero-pressure balloon as it climbs. The gross lift, in tonnes, of a zero-pressure balloon whose envelope holds at most 1.1 million cubic metres, against height, for launches with 3,500 and 5,000 cubic metres of helium; the dashed line is its 3 tonnes of envelope, payload and ballast. At launch the envelope is mostly empty. As the balloon climbs the gas expands exactly as fast as the air thins, so the mass of air it displaces stays the same and the lift stays at 3.69 or 5.28 tonnes all the way up. The envelope fills at 37.5 km with the larger fill and 39.9 km with the smaller; above that, gas vents through the open base, the volume is fixed and the lift falls with the air's density. Both reach the same float, 41.3 km, where the lift has fallen to the weight. More helium only makes the climb faster; the ceiling is set by the envelope's size and the mass it carries.

The balloon that floats at a density

A helium balloon let go rises until something stops it, and what stops it depends entirely on how it is built. A party balloon or a weather balloon stretches until it bursts. An open-bottomed science balloon climbs at constant lift — its gas expanding exactly as fast as the air thins — until its envelope is full, then floats where the air is thin enough. A sealed, unstretchable balloon floats where its whole density equals the air's and bobs back if pushed away. None of them floats at a height; each floats at a density. The difference between the three is how compressible the balloon is compared with the air around it, the same comparison that decides whether a Cartesian diver sinks.

fluids · Buoyancy What the heat shares out, and what it does not. Two averages per particle in the Maxwell–Jüttner gas, in units of kT, against kT/mc² on a logarithmic scale: the momentum times the velocity, p·v, whose average over the three directions gives the pressure, and the kinetic energy. The first is 3 at every temperature, computed here by averaging over the distribution and equal to 3 to four decimal places across the whole figure — so the pressure is exactly nkT, the ideal gas law, from a gas colder than a star to one hotter than the early universe. The kinetic energy is not shared out the same way. It is 1.506 kT when the gas is cold, Boltzmann's 3/2, and rises to 3.00 kT when it is ultra-relativistic, because for a fast particle the energy is p·c and p·v, which is what equipartition hands out, is the whole of it rather than twice it.

The gas law relativity leaves exact

Heat a gas until its particles move at nearly the speed of light and almost everything Maxwell and Boltzmann said about it changes. The speeds pile up just below c instead of spreading out. The energy per particle climbs from three halves of kT to three. The heat capacity doubles, and the ratio of heat capacities falls from five-thirds to four-thirds. One law survives untouched: the pressure is still exactly nkT, at a thousand degrees or a hundred thousand million. The reason it survives is the reason the others change. What heat hands out equally to every way of moving is not energy but momentum times velocity, and for slow particles those happen to differ only by a factor of two.

relativity · Relativistic thermodynamics The height at which hot air stops being light enough. Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load.

The balloon a cold morning lifts

A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.

fluids · Buoyancy

Named alongside it

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

EquipartitionArchimedes principleBuoyancyCompressibilityDiffusionHeat capacityPressureStandard atmosphereAdiabatic indexAdiabatic processThe Boltzmann factorBulk modulus

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