Illustrated Physics
physics, drawn
Series

Third law — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · thermodynamics
  2. 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.

    part 2 · thermodynamics
  3. The temperature that runs off the top of the scale. On the left, the entropy of a collection of two-level systems against how many of them are in the upper level. It rises, reaches 0.69314 per spin — which is ln 2, and is where half of them are up — and then falls, because a system with every spin up is as orderly as one with every spin down. On the right, the slope of that curve, which is one over the temperature. Below the maximum it is positive and ordinary: adding energy adds entropy, and the temperature is what everybody expects. At the maximum it is zero, which means the temperature is infinite. Past it the slope is negative — adding energy now removes entropy — and the temperature is negative. Such a system is not cold. It is hotter than any positive temperature whatever: put it in contact with anything at all and energy flows out of it, because that raises the total entropy. The quantity that orders systems by which way heat flows is not the temperature but its reciprocal, which runs smoothly from large and positive through zero to negative, and it is the temperature that has the discontinuity. None of this is possible unless the energy has a ceiling, which is why it happens in a spin system and not in anything that can move: a gas has no upper bound on its kinetic energy, so its entropy never turns over and its temperature is never negative.

    Hotter than any temperature there is

    A system whose energy has a ceiling can be pushed past the point where adding energy adds entropy. Its temperature is then negative — and negative temperatures are not cold. They sit above every positive temperature on the only scale that decides which way heat flows, and a working laser is at one.

    part 3 · thermodynamics
  4. 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.

    part 4 · thermodynamics
  5. 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.

    part 5 · thermodynamics
  6. Two entropies that agree until half filling. The entropy per unit of 100 two-level units against the fraction excited, by Boltzmann's definition, the logarithm of the number of arrangements at that energy, and by Gibbs's, the logarithm of the number at or below it. Below half filling they nearly coincide: at a quarter excited they are 0.538 and 0.542 per unit, and both approach the dashed large-system curve. Boltzmann's entropy then turns over and falls back to zero when every unit is excited; at three-quarters it is 0.538. Gibbs's cannot fall, because a running total cannot, and it levels off at ln 2 = 0.693, reaching 0.693 at three-quarters. The slope of each is one over its temperature.

    The count that decides which entropy is right

    There are two ways to count the states of an isolated system: the states at its energy, which is Boltzmann's entropy, and the states at or below it, which is Gibbs's. For large systems in ordinary conditions they agree to the last measurable digit. For a system whose energy has a ceiling, past the halfway point, one gives negative temperatures and the other forbids them. Definitions cannot settle which is right, but a temperature is for something — saying which way heat will flow — and putting two such systems in contact lets the count of states answer.

    part 6 · thermodynamics

All series