The entropy that is still there at zero

The third law of thermodynamics is usually stated as a fact about entropy and is usually taught as a fact about temperature. Its working content is a fact about heat capacity, and the cleanest way to see all three at once is to look at a substance that breaks the first version and obeys the other two.

What the law asserts #

Nernst’s original statement is about changes: the entropy change of any isothermal process goes to zero as the temperature does. Planck sharpened it to the version usually quoted — the entropy of a perfect crystalline substance is zero at absolute zero — and the sharpening is what makes absolute entropies tabulatable, because it supplies the constant of integration that classical thermodynamics leaves free.

Entropy as a count of arrangements makes the reason obvious: a perfect crystal at zero temperature is in one microstate, the logarithm of one is zero, and there is nothing left to say. The interesting cases are the ones where the crystal is in more than one.

The half of it that is about heat capacity #

Before the exceptions, the part of the law that does the daily work.

An absolute entropy is obtained by integrating $C/T$ from zero, and that integral only exists if $C$ falls to zero fast enough. It does, and the reason is that a solid at low temperature has only long-wavelength sound waves available to excite.

The count is a counting argument of the same family as the rest of this ladder. The number of vibrational modes with frequency below $\omega$ goes as $\omega^3$ in three dimensions, only modes with $\hbar\omega < kT$ are excited, so the excited count goes as $T^3$ and so does the energy per mode’s worth of it — giving $C \propto T^3$ and, integrating, $S \propto T^3$ as well.

That the same solid reaches $3R$ at high temperature and $\propto T^3$ at low, with one parameter between them, is the whole of Debye’s model, and the two limits are computed from the integral rather than quoted: $3.000R$ at the top and $12\pi^4/5 = 233.78$ times $(T/\theta)^3$ at the bottom. The four solids drawn reach half of Dulong and Petit at 26, 85, 160 and 555 kelvin, a spread of a factor of twenty from one number each.

Metals add a term linear in $T$ from the conduction electrons, which the exclusion principle makes very small but which dominates below a few kelvin. It integrates too. Nothing in the low-temperature behaviour of any ordinary material threatens the convergence, and that is the third law doing its job.

The ice count #

Ice is the famous exception and its explanation is one of the most economical arguments in physical chemistry.

Each oxygen in ice has four neighbours. Between each pair of oxygens sits one hydrogen, nearer to one of them than the other, and the rule — the ice rule — is that every oxygen has exactly two hydrogens near it and two far. That makes each oxygen a water molecule rather than an ion, which is what the chemistry requires.

Pauling’s count goes like this. There are $2N$ hydrogen bonds and each has two positions, giving $2^{2N}$ arrangements before any rule is applied. Each oxygen has $2^4 = 16$ ways for its four bonds to be near or far, of which $\binom{4}{2} = 6$ satisfy the rule, so each oxygen imposes a factor of $6/16$. The total is

$$2^{2N}\left(\tfrac{6}{16}\right)^N = \left(\tfrac{3}{2}\right)^N$$

and the entropy is $R\ln(3/2) = 3.371$ joules per kelvin per mole against a measured 3.41.

An agreement to one per cent from an argument that treats the oxygens as independent — which they are not, since a hydrogen shared between two oxygens is counted in both their factors — is better than the argument deserves, and the corrections have been computed since and are indeed about a per cent. The count is a mean-field estimate that happens to be very good.

The other three cases are easier. Carbon monoxide and nitrous oxide are nearly symmetric molecules that can sit either way round in the lattice, giving $R\ln 2 = 5.76$; the measured values are 4.6 and 4.8, so about a fifth of the molecules do manage to order. That gap is honest: the count assumes complete disorder and the reality is partial.

What has to fail #

Nothing here is a violation of anything. Every one of these substances has a unique ground state, in which the hydrogens or the molecules are ordered, and its entropy there is zero. What fails is the ability to reach it.

Reaching the ordered state requires rearranging molecules, and the barrier to rearranging them is many times $kT$ at the temperature where the ordering would become favourable. So the substance falls out of equilibrium, freezes into whichever of the enormous number of arrangements it happened to be in, and stays there.

That is a failure of ergodicity, not of thermodynamics. The system stops exploring its own state space, and the entropy that a calorimeter measures — which is a sum over the states actually accessible during the measurement — includes the frozen-in multiplicity for as long as it stays frozen. The number is real, reproducible and predictable, and it is a property of the substance plus the timescale.

Hydrogen, which fails in a different way #

The largest residual entropy in the table belongs to hydrogen, and its origin is not a lattice arrangement at all but a pair of nuclear spins that cannot turn over.

The two protons in a hydrogen molecule are spin-halves and combine into a triplet and a singlet exactly as two electrons do. The molecule’s whole state must change sign when the two protons are swapped, rotating it by half a turn is that swap, and rotation multiplies the state by $(-1)^J$ — so the nuclear triplet, of weight three, goes with odd rotational levels and the singlet with even.

Nothing in an ordinary collision turns a nuclear spin over. So a gas prepared at room temperature, where the ratio is three to one, keeps that ratio when cooled, and the low-temperature heat capacity is the frozen mixture’s rather than the equilibrium one’s. The measured curve is the flat one, Dennison identified this in 1927, and the discrepancy it settled had stood since 1912.

The residual entropy that goes with it is $R\ln 9$ per mole of ortho-hydrogen — three nuclear spin states for each of two nuclei, less the ones the symmetry removes — which is 18.27 against a measured 18.3. It disappears entirely if a catalyst is provided, which is why liquid hydrogen is stored over one: the conversion releases more heat than boiling the tank dry.

How the measurement is actually made #

The measured column in the first figure is not measured directly. Entropy has no meter, and every number in it is an integral.

The procedure is: cool the substance as far as the apparatus reaches, then warm it in small steps, measuring the heat needed for each and the temperature it produced. That gives $C(T)$. Divide by $T$ and integrate upward, adding the latent heat divided by the transition temperature at every phase change on the way. The result is the entropy at the top, relative to the entropy at the bottom.

The measurement is made by integrating, and that is where the difficulty enters. Heat going in against temperature coming out gives a heat capacity, and the entropy is the integral of $C/T$ from zero upwards — so the measurement needs data all the way down, and $1/T$ diverges. Every plateau on such a curve is a phase change absorbing energy at no temperature change at all, contributing a finite jump, and the whole enterprise depends on the heat capacity going to zero fast enough at the bottom for the integral to converge. That it does is the third law’s other half.

Then compare with the entropy the gas phase is known to have, from statistical mechanics: the Sackur–Tetrode expression, which is computed from the molecule’s mass and the temperature and contains no adjustable anything. If the two agree, the substance had zero entropy at the bottom of the integration. If the calorimetric value falls short, the difference is what was left behind — and that difference is the residual entropy.

So every number in the table is a discrepancy between two routes to the same quantity, one experimental and one theoretical, and the ability to state it at all rests on the gas-phase calculation being trustworthy. That it is trustworthy is one of the quieter triumphs of the subject: a count of arrangements computed from first principles, agreeing with a calorimeter, to a fraction of a joule per kelvin.

The unattainability half #

The other face of the law is the statement that absolute zero cannot be reached in finitely many steps, and the previous rung on this ladder is what draws it: a cooling process whose stages each remove a fixed fraction of the remaining temperature never arrives.

A cooling staircase makes the unattainability concrete: each pair of steps takes the temperature down by a fixed ratio rather than a fixed amount, so the floor is approached and never reached. The reason is the third law itself — the two entropy curves the process shuttles between converge as the temperature falls, so each cycle buys less than the last, and an infinite number of steps is required to arrive.

The two statements are connected and the connection is exact. If the entropy at zero temperature were different for two settings of a control parameter — a magnetic field, say — then a single isothermal step at zero temperature between them would carry heat, and a finite process would reach the floor. The requirement that the two curves meet is what closes off the last step, and it is Nernst’s original formulation rather than Planck’s.

Which is why a substance with residual entropy does not break the unattainability. Its entropy at zero is not zero, but it is the same number whatever the field or pressure — the count of ice-rule arrangements does not depend on either — so the two curves still meet and the staircase is still infinite.

The transitions that do the ordering, when they happen #

A substance that reaches zero entropy does so by having, on the way down, a transition at which the multiplicity is removed. Watching where those transitions sit is the best way to see what the third law is really demanding.

Cooling is a walk down a curve of arrangements, and a substance reaches zero entropy only if the walk arrives at a single one. What decides whether it does is a transition — an ordering that removes the last degeneracy — and whether that transition has time to happen before the material stops being able to rearrange. Glasses fail on the second count, hydrogen on the first, and a perfect crystal succeeds on both. The third law is a statement about the destination, and the residual entropies are all failures to arrive.

Ordinary crystals do it in one step at the freezing point. Magnetic materials do it again lower down, when the spins order. Nuclear spins order lower still — in copper, at about a nanokelvin — and the entropy $R\ln 4$ they carry is present in every gram of the metal at every temperature anybody has worked at.

So the honest picture is a ladder of ordering transitions, each removing one multiplicity, with the lowest of them below anything reached. The third law asserts that the ladder continues to the bottom; it does not assert that anybody has been down it. Every substance has a residual entropy relative to a temperature not yet attained, and the ones in this essay are simply the cases where the missing transition happens to be blocked rather than merely cold.

That distinction — blocked against cold — is the one that matters, and it is a distinction about a rate rather than about an energy. Ice’s hydrogens are not prevented from ordering by a lack of driving force; they are prevented by the time it takes, which is the same reason a liquid can be cooled below its own freezing point and stay liquid.

The constant of integration is Planck’s constant #

The comparison that makes every number in this essay possible deserves a closer look, because the formula being compared against contains something no thermodynamic argument could have supplied.

The absolute entropy of a monatomic gas is Sackur and Tetrode’s, and written out it contains the mass of the atom, the temperature, the volume — and Planck’s constant, cubed. It has to. Entropy counts states, counting states means dividing phase space into cells, and nothing classical says how big a cell is. Make the cells smaller and the count rises without limit; the entropy of a classical gas is defined only up to an additive constant, and that constant is exactly what the third law is supposed to fix.

What fixes it is that a cell is $h$ per degree of freedom. So an absolute entropy is not a classical quantity at all, and the third law’s clean statement — zero at zero — is a quantum statement wearing thermodynamic clothes.

The historical order makes the point sharply. Sackur and Tetrode wrote the expression in 1912, before quantum mechanics existed, by inserting an unknown cell size and then determining it from measurement: they took the calorimetric entropy of mercury vapour, compared it with their formula, and solved for the constant. What came out agreed with the value Planck had obtained from the blackbody spectrum.

Two entirely unconnected measurements — the colour of a hot cavity and the heat capacity of a metal warmed in a calorimeter — returning the same fundamental constant. That is the reason the gas-phase calculation can be trusted well enough to call the shortfall in ice’s integral a residual entropy rather than an experimental discrepancy.

The same count, in a magnet #

Pauling’s ice argument uses nothing about hydrogen. It uses a lattice of four-coordinated sites and a rule that two of the four bonds at each site point one way and two the other, and any system with that structure must have the same entropy.

There is one, and it is magnetic. In certain pyrochlore oxides the magnetic moments sit on the corners of corner-sharing tetrahedra and are forced by the crystal to point either directly into or directly out of each tetrahedron. The lowest-energy arrangement has two in and two out of every tetrahedron — the identical combinatorial rule, on the identical lattice, with a spin direction in place of a proton position.

So the prediction is $R\ln(3/2)$, the same 3.37 joules per kelvin per mole, for a substance with no hydrogen in it. It was measured in 1999, by integrating the magnetic heat capacity in exactly the way this essay describes, and it came out at the Pauling value.

These materials are called spin ices for that reason, and their other consequence — that their excitations behave like magnetic charges — comes from the same rule being violated locally. One combinatorial statement produces a residual entropy and an emergent particle.

They also supply the sharpest available test of the distinction this essay draws between blocked and cold. A residual entropy is a statement about a timescale, and one long-running experiment on a spin ice reported that the entropy does slowly drain away when the sample is held near its freezing point for weeks rather than hours — a claim that is disputed and not settled. If it is right, the substance is not degenerate at zero temperature after all; it is merely very slow, which is what the third law says every such substance must be.

What it costs #

Absolute entropies are only as good as the extrapolation below the lowest measurement. A calorimeter reaches a few kelvin at best, and everything below that is Debye’s $T^3$ extrapolated. That is a good extrapolation for a simple solid and a poor one for a glass, a magnetic material with a low-temperature transition, or anything with a lot of low-lying levels, and the tabulated entropies of such substances carry a real uncertainty in their first digit.

“Perfect crystal” is doing more work than it looks. Isotopic disorder is a multiplicity too, and a natural element with several isotopes in its lattice has a mixing entropy that never goes away. It is conventionally excluded from tabulated entropies because it cancels out of every chemical reaction, which is a convention rather than a physical statement.

And a glass has no defensible residual entropy at all. A supercooled liquid falls out of equilibrium at a temperature that depends on how fast it was cooled, so the frozen-in multiplicity — and therefore the entropy — depends on the experimenter’s schedule. That is not a failure of measurement; the number genuinely is not a property of the material alone, and the extrapolation of a glass’s entropy to zero temperature is one of the longest-running open questions in the subject.

What the picture cannot show #

The counted entropies are for perfectly disordered arrangements and the substances are not perfectly disordered. The gap between $R\ln 2$ and the measured 4.6 for carbon monoxide is real partial ordering, and there is no clean way to compute how much of it there should be.

The measured values are differences dressed as absolutes. A calorimeter measures $C$ and the integral gives the entropy rise from the lowest temperature reached. Calling the residue an absolute entropy is an inference that depends on the extrapolation below that point and on the third law being right.

Debye’s curve is a model with one parameter and real solids need more. The $T^3$ law is exact in the limit and the approach to it is not universal; a layered solid crosses over to $T^2$ over a range of temperature, and a chain-like one to $T$, because the mode counting that produced the cube depends on the dimensionality of the vibrations rather than of the crystal.

And nothing here reaches absolute zero. Every number is an extrapolation to a temperature nothing has been at. The law’s own second face guarantees that this will remain true, which is a pleasing sort of consistency and also a permanent limitation on the evidence.

The ladder from here #

Later rungs on this anchor: the glass transition and why its residual entropy depends on the cooling rate; the Kauzmann paradox, where extrapolating a supercooled liquid’s entropy below the glass transition takes it under the crystal’s; the electronic and nuclear contributions that survive to microkelvin temperatures and are ordered by their own interactions; negative absolute temperatures, which are reached by systems with a bounded energy spectrum and which the third law has to be restated for; and the statistical-mechanical proof of the law, which needs the ground state to be non-degenerate and is therefore a statement about spectra rather than about heat.

The neighbouring ladders are the cooling staircase, which is the unattainability half of the same law, and entropy as a count, without which the residual numbers here would be facts with no explanation. A latent heat is the ordinary way a substance loses entropy on cooling, and the ones in this essay are the substances that ran out of time to do it.