Illustrated Physics
physics, drawn
Concept

Van der waals — where it appears

An equation of state that adds two corrections to the ideal gas: a term for the attraction between molecules and a term for the room they take up. It gets the critical point, the coexistence curve and the sign of the Joule–Thomson effect qualitatively right with two parameters, and no quantity in it to better than tens of per cent.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The part of the curve no fluid follows. Van der Waals' isotherms in reduced units, at 5 temperatures either side of the critical one, so that nothing about any particular substance appears. Above the critical temperature the pressure falls monotonically as the volume grows, which is what a fluid does. Below it the curve develops a loop with a rising section in the middle, and that section says the pressure increases as the substance expands — a material with negative compressibility, which cannot exist, because any fluctuation would run away. What happens instead is drawn as the horizontal line: the substance separates into two phases at one pressure, and the volume moves along the line as the proportions change. Its height, 0.6470 of the critical pressure, is fixed by requiring the two areas the line cuts off to be equal, which is the condition that the two phases have the same Gibbs energy. It meets the curve at volumes 0.603 and 2.349, a ratio of 3.9, and those are the densities of the liquid and its vapour. The two turning points of the loop, at 0.72 and 1.53, bound the part that is not merely unobserved but impossible; between them and the construction the substance can be made to sit, superheated or supercooled, until something nucleates.

The part of the curve no fluid follows

One equation for a real gas produces isotherms with a rising middle section, which says a substance would expand as the pressure on it grows. Nothing does that. What replaces it is a horizontal line whose height is fixed by making two areas equal, and the condition is not a convenience.

thermodynamics · Phase change Throttling a gas, and the curve that says which way it goes. Curves of constant enthalpy for a van der Waals gas, in temperature and pressure both measured against the critical values. A gas pushed slowly through a plug or a valve keeps its enthalpy, so it moves along one of these curves — from right to left, since the pressure falls. Where a curve slopes upward to the right the gas cools as it expands; where it slopes downward it warms. An ideal gas would give horizontal lines and no change at all, because its enthalpy depends on the temperature alone; every curve here is bent, and the bending is the attraction between molecules and the room they take up, fighting. The dashed line through the tops of the curves is the inversion curve, and the maxima were found on the drawn points rather than put there — they lie on the closed form to 0.65 per cent. Which side of it a gas starts on decides the sign of the effect, and that is the whole of why air can be liquefied by throttling at room temperature and hydrogen cannot: hydrogen has to be pre-cooled below its own inversion temperature first, which is why Dewar needed liquid air before he could get liquid hydrogen, and why Onnes needed liquid hydrogen before he could get helium.

The gas that cools by being let go

Push a gas through a plug and its temperature changes, although no work is done on anything and no heat goes anywhere. Which way it changes depends on where it starts: inside a dome in the pressure–temperature plane it cools, outside it warms, and hydrogen at room temperature is outside — which is why liquid hydrogen needed liquid air first.

thermodynamics · Phase change 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 colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly.

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

fluids · Surface tension London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm.

The attraction that weakens when light is too slow

Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

electromagnetism · Retardation Water's surface tension, all the way to the critical point. The surface tension of water against its liquid vapour, from the international standard correlation, against temperature from 0 °C to the critical point at 373.946 °C. 20 °C: 72.7 mN/m; 100 °C: 58.9 mN/m; 200 °C: 37.7 mN/m; 300 °C: 14.4 mN/m; 370 °C: 0.4 mN/m. It falls almost in a straight line for most of the range and then bends down to reach zero exactly at the critical point, where liquid and vapour stop being two things. The bend is a power law, the gap to the critical temperature raised to 1.256.

The tension that vanishes where the phases meet

Surface tension is usually treated as a property of a liquid, like its density. It is a property of a boundary between two things, and it can only exist while there are two things. Heat water in a sealed tube and its surface tension falls steadily, then faster, and reaches exactly zero at 374 °C, where liquid and vapour stop being different. On the way the surface stops being a surface: the step in density spreads from a molecule's width to thousands, and the rate at which the tension dies turns out to be a number the whole family of critical points shares and the simplest theory gets wrong.

fluids · Surface tension

Named alongside it

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

Critical pointEquation of stateCompressibilityIntermolecular forcesSurface tensionThe Boltzmann factorCapillary pressureCasimir polderCorrelation lengthCritical exponentDisjoining pressureDispersion force

All concepts