Illustrated Physics
physics, drawn
Concept

Microstates — where it appears

The individual arrangements a system's parts can be in while its measured properties stay the same, whose count is its entropy. The count grows so fast with the number of particles that the most numerous arrangement is overwhelmingly the one observed, which is what makes thermodynamics deterministic.

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

Ways to arrange 10 coins. The number of distinct arrangements giving each number of heads, for 10 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

Entropy is a count, and the arrow of time is arithmetic

Nothing in mechanics prefers a direction. Entropy is not a force pushing things toward disorder — it is the observation that some outcomes have vastly more ways of happening than others.

thermodynamics · Entropy Heating 1 kg of water from -20°C to 130°C. Temperature against heat added for 1 kilogram of water taken from -20 to 130 degrees Celsius. The two flat stretches are the melting and the boiling, where 334 and 2260 kilojoules go in and the temperature does not move. Melting costs as much as warming the water by 80 degrees; boiling costs as much as warming it by 541, which is 73 per cent of the whole journey.

The heat that changes no temperature, and where it actually goes

A kettle reaches a hundred degrees in a minute and takes five more to boil dry. The heat going in during those five minutes changes nothing a thermometer can see, and it is most of the energy in the whole process.

thermodynamics · Phase change One mass, two entropies. The entropy of a solar mass as ordinary gas, generously counted at ten Boltzmann constants per proton, against the entropy of a solar-mass horizon. The first is 1.19·10⁵⁸ k and the second 1.05·10⁷⁷ k — a factor of 8.83·10¹⁸. This is why a horizon had to be given an entropy: without one, dropping anything at all through it destroys entropy and the second law fails.

The entropy that lives on a surface

Throw a cup of tea through a horizon and the entropy of the outside world falls. Either the second law is wrong or the horizon has an entropy of its own — and the only quantity available for it turns out to be its area, in units of a length made from gravity, quantum mechanics and the speed of light together.

thermodynamics · Entropy Ways to arrange 12 coins. The number of distinct arrangements giving each number of heads, for 12 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

What a system actually minimises

A ball falls to the bottom of a bowl and a gas fills a room, and neither of those is the rule. A system in contact with a large reservoir minimises U − TS, and the minus sign is the reservoir's own entropy written in the system's variables — which is why a rubber band pulls harder when it is heated.

thermodynamics · Entropy The entropy of mixing, and the entropy of not mixing. The entropy gained when two ideal gases at the same temperature and pressure are allowed to mix, per particle and in units of Boltzmann's constant, against the proportion of the mixture that is the first gas. The curve has no property of either gas in it — not their masses, not their sizes, not how strongly they interact, since ideal gases do not — and it is largest at 0.500, where it reaches ln 2 = 0.6931. Below it is the same quantity for two samples of the SAME gas, which is zero at every proportion: removing the partition between two halves of a box of nitrogen changes nothing that can be measured, and putting it back recovers the original state. The two results are correct and they do not join up. Make the two gases more and more alike — two isotopes, then two nuclear spin states, then nothing at all — and the upper curve does not descend to meet the lower one; it stays exactly where it is until the two species become identical, and then jumps. What the figure is really about is that the jump is in the counting and not in the gas.

Mixing what is already mixed

Let two different gases into each other's halves of a box and the entropy rises by a fixed amount that contains nothing about either gas. Do it with the same gas on both sides and it rises by nothing. Make the gases more and more alike and the answer does not converge — it jumps.

thermodynamics · Entropy The work one molecule and one bit are worth. The pressure of a gas of one molecule against its volume, at 300 K, in units of the volume it starts in. The shaded area is the work the molecule does pushing a partition out isothermally, and it is measured here by integrating the drawn curve rather than written down: expanding by 1.5× yields 0.4055 kT against ln 1.5 = 0.4055; expanding by 2× yields 0.6931 kT against ln 2 = 0.6931; expanding by 4× yields 1.3863 kT against ln 4 = 1.3863; expanding by 8× yields 2.0794 kT against ln 8 = 2.0794, agreeing to 2.0e-10. The doubling is the one that matters, because a partition inserted in the middle leaves the molecule on one side or the other, and knowing which is what lets the load be attached to the right face. That single expansion delivers kT·ln2 = 2.87 zeptojoules at 300 K. It looks like work extracted from one temperature, and it is — until the engine is asked to run again, which requires forgetting which side the molecule was on.

The bit that has to be paid for

One molecule in a box, a partition, and the knowledge of which side it went — enough, between them, to extract work from a single reservoir, which the second law forbids. The engine is real and the arithmetic is right. What closes the loophole is that the cycle does not finish until the knowledge has been thrown away, and throwing away one bit costs exactly what the expansion delivered.

thermodynamics · Entropy 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 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

Named alongside it

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

EntropyThe second lawIrreversibilityTemperatureEquilibriumFree energyLatent heatPhase transitionQuantum statisticsResidual entropyThird lawAbsolute entropy

All concepts