Illustrated Physics
physics, drawn
Concept

Residual entropy — where it appears

Entropy remaining at absolute zero, which the third law permits when the ground state is degenerate. Ice keeps 3.41 joules per kelvin per mole because its hydrogen arrangements are a count nobody can change; a glass keeps whatever it had when it stopped relaxing, which depends on how fast it was cooled.

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

Entropy that is still there at absolute zero, counted and measured. Four substances whose entropy does not go to zero when they are cooled as far as anybody can cool them, with the entropy counted from the arrangements they froze into beside the entropy measured by integrating their heat capacities. Ice is the famous one: every oxygen has four hydrogen bonds and the rule is that two hydrogens sit near it and two far, which leaves six of the sixteen placements legal, and Pauling's count of the whole crystal collapses to R ln(3/2) = 3.371 J per mole per kelvin against a measured 3.41 — an agreement to one per cent from an argument on one line. Carbon monoxide and nitrous oxide are the easy cases, molecules that can lie either way round in the lattice and have too little to gain by choosing. The worst of the four is off by 25 per cent, which is the honest state of this subject: the counts are crude, they ignore the correlations between neighbouring choices, and they still land within sight of a calorimeter. What the figure is really about is that the third law has an escape clause and the escape clause is measurable. A perfect crystal has one arrangement and zero entropy; a crystal that ran out of time while it had many has the logarithm of however many it stopped at, permanently.

The entropy that is still there at zero

The third law says a perfect crystal has no entropy at absolute zero. Ice has 3.41 joules per kelvin per mole left over, and the number can be recovered from one line of counting — two hydrogens near each oxygen and two far, six legal arrangements out of sixteen, R ln(3/2). The law has an escape clause and the escape clause is measurable.

thermodynamics · Third law The entropy a substance keeps depends on how fast it was cooled. The entropy of a supercooled liquid in excess of its crystal's, against temperature, for a substance melting at 305 K with an entropy of fusion of 43 joules per kelvin per mole and a liquid heat capacity exceeding the crystal's by 62. The equilibrium curve — the one the liquid follows while it can still relax — falls steadily and would reach the crystal's entropy at 152.4 kelvin. It never gets there, because the liquid falls out of equilibrium first, at a temperature that depends on how long it is given: 0.01 s/K freezes at 196.1 K with 15.61 left, 1 s/K freezes at 188.1 K with 13.03 left, 100 s/K freezes at 182.1 K with 11.03 left. Ice's residual entropy is a count and does not move; a glass's is whatever it happened to have when it stopped being able to change, and that is a property of the experiment.

The entropy that depends on how fast it was cooled

Ice's residual entropy is a count, and it comes out the same whoever measures it. A glass's does not. A glass keeps whatever entropy it happened to have when its own relaxation time crossed the experiment's, so cooling ten times more slowly leaves less behind — and extrapolating the equilibrium liquid below that point takes its entropy under the crystal's at a finite temperature, which cannot happen and does not, for a reason that is still argued about.

thermodynamics · Third law The entropy each degree of freedom has not yet given up. The entropy carried by each kind of degree of freedom, drawn against the temperature at which it orders and hands that entropy over. Lattice vibrations freeze out around room temperature; electron spins in a paramagnetic salt order in the millikelvin range, which is what makes adiabatic demagnetisation work; and nuclear spins hold R ln(2I+1) — 11.5 joules per kelvin per mole for copper — down to some tens of nanokelvin, where their own dipolar interactions finally sort them out. A copper sample at a microkelvin therefore has a large entropy and violates nothing: its nuclear spin system has not reached its ground state, and the third law is a statement about ground states rather than about thermometers.

A law about spectra, not about heat

The third law is usually met as a statement about cooling. Its statistical form is a statement about a spectrum: the entropy of a system in its ground state is k ln g, and it vanishes only when the ground state is unique. Every apparent exception is a degeneracy or a system that never reached its ground state — and copper nuclei carry eleven joules per kelvin per mole down to a hundred nanokelvin without violating anything.

thermodynamics · Third law 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

Named alongside it

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

Third lawEntropyEquilibriumErgodicityHeat capacityMicrostatesAbsolute entropyCurie lawDebye modelDegeneracyEnergy levelsFermi gas

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