Illustrated Physics
physics, drawn
Concept

Third law — where it appears

The statement that the entropy change of any isothermal process goes to zero as the temperature does, so that a substance's entropy curves converge at absolute zero. Its content is about cooling: each stage of adiabatic demagnetisation removes a fixed fraction of what is left rather than a fixed amount, so no finite number of stages arrives.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

Two costs, and the width that balances them. The energy of a particle in a harmonic well against how tightly its wavefunction is squeezed, in units of ħω and of the width that minimises the total. Two terms compete. Squeezing the particle into a smaller region raises its kinetic energy, because the uncertainty relation makes a narrow position spread a wide momentum spread and momentum is squared in the energy; that term rises as the inverse square of the width and goes to infinity as the particle is localised. Letting it spread out raises its potential energy, since the well gets steeper away from the bottom; that term rises as the square of the width. The sum has a minimum at a width of 1.0000 in these units, where the total is 0.5000 ħω and the two terms are equal at a quarter each. That number is exactly the true ground-state energy of a quantum harmonic oscillator, obtained here with nothing but the uncertainty relation and a minimisation. What the figure shows and the formula does not is why there is a floor at all: it is not that the particle happens to keep moving, but that every way of stopping it costs more than it saves.

The motion that cannot be stopped

A particle in a well cannot sit at the bottom of it. Squeezing it into a smaller region costs kinetic energy faster than it saves potential energy, so there is a width that minimises the total — and the minimum is not zero. Helium never freezes because of it.

quantum · Uncertainty Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

The staircase that never reaches the floor

Absolute zero is unreachable, and the reason is not that the apparatus is not good enough. Every stage of cooling removes a fixed fraction of what is left rather than a fixed amount, so the steps shrink in proportion to the distance remaining — and the fixed fraction cannot be made one, because the entropy curves at two field strengths are required to meet where the axis is.

thermodynamics · Third law 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 melting curve that turns back, then lies flat. The pressure at which liquid and solid helium-3 coexist, against temperature from 0.9 millikelvin to 1 kelvin on a logarithmic axis, from the international provisional low-temperature scale PLTS-2000, which is defined by this curve. Above 315.2 mK the curve behaves like any other: a higher temperature needs a higher pressure to freeze the liquid. Below it the curve turns back — at its minimum, 2.93113 MPa, it takes a higher pressure to freeze colder helium — and it rises to 3.4391 MPa at 1 mK, where it lies almost flat, as the third law requires of every coexistence line. The kinks that mark the liquid becoming superfluid and the solid's spins ordering sit on that flat end, at 2.444, 1.896 and 0.902 mK.

The melting curve that has to arrive flat

The slope of every line on which two phases coexist is their difference in entropy divided by their difference in volume. The third law says that entropy differences vanish at absolute zero, so every such line — melting, boiling, a superconductor's critical field — must reach the bottom of the temperature scale lying flat. Helium-3 reaches it in the strangest way available: below a third of a kelvin its solid is more disordered than its liquid, the melting curve runs backwards, and squeezing the liquid into solid absorbs heat. That backwards curve was the refrigerator in which superfluid helium-3 was found, and it is now the definition of temperature below one kelvin.

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 Why dissolving beats evaporating near absolute zero. Cooling power against temperature on logarithmic axes, for pumping on liquid helium-4, pumping on liquid helium-3 — both with a pump of 100 litres a second at room temperature — and a dilution refrigerator circulating 100 micromoles of helium-3 a second (the T² law, dashed above 0.1 K where it no longer holds). Evaporative cooling is proportional to the vapour pressure, which collapses as e^(−L/T): helium-4 gives 7·10⁻¹¹ W at 0.3 K, helium-3 10⁻⁴ W at 0.3 K and 2·10⁻²⁴ W at 50 mK. The dilution refrigerator's "vapour" is the dilute phase, whose density of helium-3 stays at 6.6% however cold it gets, so its cooling falls only as T²: 2·10⁻⁵ W at 50 mK. Below about 0.3 K nothing that evaporates can compete, and most experiments that must stay continuously below a tenth of a kelvin — superconducting quantum computers, bolometers for the cosmic microwave background, searches for dark matter — run on dilution.

The mixture that stays mixed at absolute zero

The third law says a mixture's entropy must vanish at absolute zero, so every mixture must either separate completely or order into a compound as it is cooled. Liquid helium-3 dissolved in helium-4 does neither. Below 0.87 kelvin it splits into two liquids, but the one that is mostly helium-4 keeps 6.6 per cent helium-3 however cold it gets — and the third law permits it, because the dissolved atoms are a degenerate Fermi gas with a single ground state and no entropy of mixing to lose. Pushing helium-3 across that boundary absorbs heat the way evaporation does, from a 'vapour' that never thins out. That is the dilution refrigerator, the machine that holds every superconducting quantum computer at ten millikelvin.

thermodynamics · Third law

Named alongside it

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

EntropyResidual entropyEquilibriumHeat capacityQuantum statisticsDegeneracyErgodicityFermi liquidGround stateHelium-3MagnetisationMicrostates

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