Illustrated Physics
physics, drawn
Concept

Entropy of mixing — where it appears

The entropy gained when two distinguishable substances are allowed to occupy the same volume, depending on the proportions and on nothing else about them. It vanishes when the two are the same substance, which makes it a statement about what can be told apart rather than about matter.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

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 column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

The pressure that comes from counting

Dissolve a teaspoon of anything in a litre of water, put a membrane between it and pure water, and the solution will hold up a column of water two and a half metres tall. Nothing is pulling. The pressure does not depend on what was dissolved, only on how many particles it made — which is the ideal gas law, with the solute in place of the gas.

fluids · Osmosis The least energy that fresh water can cost. The work needed to take fresh water out of a feed of 1150 mol/m³ of dissolved particles — seawater — against how much of the feed is taken, in kilowatt-hours per cubic metre of product. The lower curve is the reversible minimum, in which the pressure is raised continuously as the remaining feed gets saltier; the upper one is a single stage held throughout at the pressure the final, saltiest concentrate needs. At vanishing recovery both tend to the feed's own osmotic pressure, 28.5 bar or 0.79 kWh/m³, which is the floor for the first drop and is checked here against the limit of the formula. Taking more of the feed costs more per unit taken, because what is left behind is saltier: at 10% recovery 0.83 kWh/m³ reversibly and 0.88 in one stage, 30% recovery 0.94 kWh/m³ reversibly and 1.13 in one stage, 50% recovery 1.10 kWh/m³ reversibly and 1.58 in one stage, 60% recovery 1.21 kWh/m³ reversibly and 1.98 in one stage, 75% recovery 1.46 kWh/m³ reversibly and 3.17 in one stage. The horizontal line is what a good seawater plant actually uses, 3 kWh/m³, so the second-law efficiency of the industry is about 37 per cent. None of this is about membranes. It is the free energy of mixing salt into water, read backwards, and no technology of any kind can go below the lower curve.

What it costs to take the salt out

Salt dissolves in water because mixing is overwhelmingly the more probable arrangement, and separating the two again means paying back what the mixing gave away. The bill can be computed before any apparatus is chosen — 0.79 kilowatt-hours for the first cubic metre from seawater — and it rises with every further cubic metre taken, because what is left behind is saltier than what was started with.

fluids · Osmosis Every reaction's free energy has its lowest point inside. An ideal reaction A ⇌ B at 298 K. Across: how far it has gone, from pure A towards pure B. Up: the Gibbs energy of the mixture per mole, relative to pure A. Left, for standard reaction Gibbs energies ΔG° of −4, 0, +4 kJ/mol: the dashed straight lines are what the energy would be if A and B did not mix, and the solid curves add the entropy of mixing them. For ΔG° = −4 kJ/mol the lowest point is at 83.4 per cent B; for ΔG° = 0 kJ/mol the lowest point is at 50.0 per cent B; for ΔG° = +4 kJ/mol the lowest point is at 16.6 per cent B. Right, magnified near pure A, a reaction with ΔG° = +10 kJ/mol, whose straight line climbs from the start and which looks as if it should not proceed at all: its curve first falls, to a minimum of −43 J/mol at 1.74 per cent B, because the mixing term falls infinitely steeply away from a pure end. Each minimum was found by search and sits where the ratio of B to A equals exp(−ΔG°/RT).

The reaction that cannot go all the way

Chemistry speaks of reactions that go to completion and reactions that do not happen, and at equilibrium there are neither. The reason is a logarithm. The free energy of a half-finished reaction contains the entropy of mixing, whose slope is infinite at both pure ends, so every reaction's lowest point lies strictly inside — and the slope of that free energy, the chemical potential, is to particles what temperature is to heat.

thermodynamics · Chemical potential 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.

Free energySemipermeable membraneChemical potentialOsmotic pressureBinomial distributionChemical equilibriumColligativeCooling powerDiffusionDilution refrigeratorEfficiencyEntropy

All concepts