Illustrated Physics
physics, drawn
Concept

Critical exponent — where it appears

The power with which a quantity vanishes or diverges as a critical point is approached. Surface tension, for example, falls as the 1.26th power of the distance from the critical temperature, and the value is shared by whole classes of unrelated systems, which is what universality means.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Two straight lines that are not the same line. Half the difference between the liquid and vapour densities, in units of the critical density, against the distance from the critical temperature — both logarithmic, over 5 decades. The van der Waals curve is solved for the coexisting pair at each temperature and its slope in the last decade is 0.500, which is the mean-field ½. Real fluids give 0.326. The two differ by 11 per cent at t = 0.1 and by a factor of 7.9 at the bottom of the axis, which is why an equation with the wrong exponent in it looked right for eighty years.

The point at which the two become one

Heat a sealed tube of carbon dioxide and the meniscus inside it does not boil away — it fades, the two densities converging until there is nothing to separate. Twenty millikelvin before that happens the fluid turns milky, and the exponent describing the last approach is a number van der Waals got wrong and could not have got right.

thermodynamics · Phase change 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.

Correlation lengthCritical pointSurface tensionUniversalityCompressibilityFluctuationsInterfaceMean-field theoryOrder parameterPhase changePhase transitionScattering

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