What a system actually minimises

A ball rolls to the bottom of a bowl and stays there. A drop of ink in a glass of water spreads until it is everywhere and stays that way. The first looks like a system finding its lowest energy, the second like one finding its highest entropy, and neither description is the rule.

Everything below is the arithmetic that reconciles the two rules, the four things it explains that neither explains alone, and where it stops being true.

Two rules that cannot both be right #

Both rules are taught, usually within a term of each other, and both as though they were general. The first is that systems fall to the lowest available energy. It works for the ball in the bowl, for a released spring, for a mass on a hill that gives its energy back, and for every minimum that behaves like a parabola near the bottom. The second is that systems move to the state of highest entropy — the gas that fills the room, the ink in the water, the spike that spreads and never re-gathers.

Two ordinary observations show that neither can be the criterion.

Ice at 1 °C melts: its energy goes up by 334 kJ for every kilogram, and its entropy goes up too. Water at −1 °C freezes: its energy goes down and its entropy goes down with it. Same substance, same pressure, surroundings differing by two degrees, and whatever decides must be a combination that changes sign near 0 °C.

Ammonium nitrate stirred into water dissolves and the beaker gets cold, absorbing 25.7 kJ for every mole: an instant cold pack is a spontaneous process running energetically uphill, while a supercooled melt left alone crystallises and runs entropically downhill.

A rule about energy alone gets one class of cases wrong; a rule about entropy alone gets the other.

The reservoir’s entropy, and where the minus sign comes from #

There is only one law, and it is about the total.

Take a system — a beaker, a crystal, a rubber band, a single molecule — in thermal contact with something enormously larger at a fixed temperature $T$: a bench, a water bath, the atmosphere, the ocean. Energy passes between the two and the total is conserved. The second law says nothing at all about the system on its own; it says that $S_{\text{tot}} = S_{\text{sys}} + S_{\text{res}}$ does not decrease. The system’s entropy is free to fall provided the reservoir’s rises by more.

The reservoir has one relevant property: it is large enough that taking energy out of it does not change its temperature. For such a body $\partial S/\partial U = 1/T$ with $T$ constant, so if the system absorbs $\Delta U$ the reservoir loses exactly that, and

$$\Delta S_{\text{res}} = -\frac{\Delta U}{T}.$$

That is the whole of the derivation. Substituting gives $\Delta S_{\text{sys}} - \Delta U/T \ge 0$, and multiplying through by $-T$, which is negative and therefore reverses the inequality,

$$\Delta U - T\,\Delta S_{\text{sys}} \le 0 .$$

The quantity on the left is the change in $F = U - TS$, and the statement is that it never increases. Maximising the entropy of system plus reservoir and minimising $U - TS$ for the system alone are the same sentence multiplied by a constant. The minus sign in front of $TS$, the part that always looks arbitrary, is the reservoir’s entropy in disguise — negative because the reservoir loses what the system gains, divided by $T$ because that is how fast its arrangement count responds to energy.

Temperature is the exchange rate between the two currencies. Energy is in joules and entropy in joules per kelvin, and $T$ converts one into the other: at 300 K a joule handed to the reservoir buys 3.33 mJ/K of entropy there, and one unit of entropy per particle is worth $RT = 2.49$ kJ/mol. Every “competition between energy and entropy” ever invoked in chemistry or biology is that conversion, applied once.

The weight itself is the exponential $e^{-E/kT}$, and everything in the derivation above is a consequence of its shape. A state twenty-five times $kT$ up is visited $e^{-25}$ as often as the ground state — one part in $10^{11}$ — which is why “the system minimises $F$” is a statement about where essentially all of the probability sits rather than about where any particular configuration is. The minimum is not preferred; it is overwhelming.

Melting, and two minima of the same depth #

The clearest place to watch $F$ do the work is a substance offered two structures.

A solid has low energy and low entropy: its atoms sit in a lattice with few arrangements available. A liquid has higher energy, because bonds have been broken, and much higher entropy, because the atoms can be almost anywhere. Their free energies differ by $\Delta F = \Delta U - T\Delta S$, with both $\Delta U$ and $\Delta S$ positive on melting. At low $T$ the energy term dominates and the solid wins; at high $T$ the entropy term dominates and the liquid wins. In between there is exactly one temperature at which the two are equal, and it is

$$T_m = \frac{\Delta U}{\Delta S}.$$

A melting point is a ratio of an energy to an entropy. That is why it is a property of the substance and cannot be altered by how vigorously the substance is heated.

Take a kilogram of water from −20 °C to 130 °C and 334 kJ go in at 0 °C and 2,260 kJ at 100 °C with no change of temperature at all, out of 3,114 kJ for the whole journey. The flat stretches are where $F$ has two minima of equal depth, so any mixture of the two phases has the same free energy and nothing decides between them. The latent heat is what it costs to move the system from one of those minima to the other.

The numbers make the ratio concrete. For water, $L_f = 334$ kJ/kg, so at 273.15 K the entropy of fusion is 334/273.15 = 1.22 kJ/kg/K, or 22.0 J/K per mole. For lead, the latent heat of fusion is only 23.0 kJ/kg — 4.77 kJ/mol, less than water’s 6.02 kJ/mol — and yet lead melts at 600.6 K, more than twice as high, because its entropy of fusion is 7.94 J/K/mol, a third of water’s.

That inverts the usual intuition. A large latent heat does not mean a high melting point: lead melts hot because melting gains it so little disorder, and ice melts cold because melting gains it so much, the hydrogen-bonded network of ice being far more constrained than a metallic lattice.

The rubber band, whose tension is made of counting #

The best experiment in domestic physics costs nothing and contradicts everything a spring teaches. A steel spring resists stretching because stretching moves atoms up the sides of an interatomic potential well: the restoring force is the slope of an energy, the stiffness is the curvature at the bottom, and the account is entirely mechanical.

A rubber band is a tangle of long polymer chains, cross-linked here and there. Stretching it does very little to the energy of any bond; what it does is reduce the number of shapes a chain can take. An unstretched chain wanders in every direction and has an enormous number of configurations; a chain pulled nearly straight has very few. Stretching reduces entropy, and the tension is therefore $f = -T(\partial S/\partial L)_T$, with no energy term worth writing. The band pulls back because there are more ways to be short than to be long, and for no other reason.

A rubber band pulls harder when it is heated. The temperature stands outside the derivative, so at fixed length the tension is proportional to the absolute temperature. Warming a stretched band from 300 K to 330 K raises its tension by about 9 per cent — 10 per cent for an ideal rubber, reduced slightly by the small energetic part. A spring does the reverse. This is the one everyday object whose restoring force is made of counting rather than of stored energy, and its consequences are all backwards.

Three of them are checkable in a kitchen.

A stretched band is warm. Pulled hard and quickly and held against the lip, it is noticeably warmer: the work done did not go into stored energy, so it had nowhere to go but heat. The rise is the work per unit volume divided by $\rho c$, and for natural rubber at large extension a strain-energy density of about 9 MJ/m³ against $\rho c = 1.8$ MJ/m³/K gives 5 K, with elastocaloric measurements reaching about 10 K. Released, the band cools by the same amount, which is the basis of a class of solid-state refrigerator.

A loaded band contracts when heated. Hang a weight from a band, warm it with a hairdryer, and the weight rises. Every other household material lengthens.

A wheel built with rubber-band spokes turns when heated on one side. The bands facing the lamp contract, pulling the hub off the axis toward the heat, so the centre of mass no longer sits on the axle; gravity then has a torque about it and the wheel rotates continuously, because the offset stays where the lamp is while the material rotates through it. Built with metal spokes the same wheel turns the other way, since a heated metal wire lengthens and pushes the hub away. A rubber-band heat engine runs the wrong way round compared with a metal one, and the sign of its rotation reads out directly whether the restoring force is entropic or energetic.

The $T$ standing in front of the derivative in that tension is the scale of the kicking. The chain segments are agitated at a scale set by $kT$ — a distribution of speeds whose whole shape scales with temperature — and the tension is that agitation rectified by a fixed end-to-end length. Which is why a rubber band at zero temperature has no tension at all: there is nothing to rectify.

How much of the tension is entropic is measurable, from the slope of tension against temperature at fixed length. Beyond about 20 per cent extension the entropic part of natural rubber’s tension is upward of 90 per cent of the total, against 100 per cent for the ideal-rubber model.

Mixing, with no energy change at all #

The sharpest case is one where the energy term is not small but exactly zero.

Two ideal gases at the same temperature and pressure, in equal volumes, separated by a partition. Ideal means the molecules do not interact, so removing the partition changes the energy of the combined system by nothing whatsoever: $\Delta U = 0$ identically, not approximately. Yet they mix, always, and never unmix.

Each gas now has twice the volume available to each of its molecules, so each gains $R \ln 2$ of entropy per mole. Per mole of the resulting mixture the gain is the same:

$$\Delta S_{\text{mix}} = R \ln 2 = 5.76 \ \text{J/K/mol},$$

and at 300 K the free-energy gain is $T\Delta S =$ 1.73 kJ/mol. That is a tiny quantity by chemical standards — half a per cent of a 350 kJ/mol covalent bond — and it is nonetheless the reason nothing in the world stays unmixed. With no energy term to oppose it, any positive entropy of mixing wins outright.

That curve carries a trap that took forty years to state properly, and it is the sharpest test of what entropy is counting.

Mix a gas with itself and the labelled count says the entropy rises, which would make removing a partition between two halves of one room a source of free energy. It is not, and the resolution is that identical molecules have no labels to permute: the count that gives the right answer is the one that does not distinguish them. That is the Gibbs paradox, and its resolution is the first place classical thermodynamics needed a fact about quantum mechanics to be self-consistent.

In a real mixture the energy term returns as the cost of putting unlike molecules next to each other, and solubility becomes the question of whether $T\Delta S_{\text{mix}}$ exceeds that cost — which is what the cold pack settles. Applied to the solvent rather than the solute, the same term is an osmotic pressure: a membrane that passes water but not solute must be held against $\Pi = cRT$, which for 1 mol/L at 300 K is 24.9 bar and for seawater, at roughly 1.1 mol/L of dissolved ions, is 27 bar. An entropy of 5.76 J/K/mol becomes twenty-seven atmospheres pushing on a membrane.

What it costs #

Desalination has a thermodynamic floor. A reverse-osmosis plant must exceed 27 bar at the membrane to move the first drop, and runs at 55 to 70 bar to move water at a useful rate. The minimum work is $\Pi V$ = 2.74 MJ per cubic metre, or 0.76 kWh/m³, rising to about 1.1 kWh/m³ at the 50 per cent recovery a plant operates at. The best plants achieve 3 kWh/m³, and the floor is not technological.

A phase-change store is sized by $T\Delta S$. Melting a kilogram of ice absorbs 334 kJ, which is what cooling a kilogram of water through 80 K would cost — so a cool box carries ice rather than chilled water, and the heat that changes no temperature is the useful capacity of every such store.

An engine’s output is a free-energy drop, which is what “free” was meant to mean. For an isothermal expansion of an ideal gas the energy does not change at all, so every joule of work comes from $T\Delta S$ — the gas being paid, in pressure against a piston, for the entropy it gains by having more room.

That is the whole of the free-energy idea reduced to one molecule. Nothing changes in the gas’s internal energy across the expansion, so every joule of work comes from $T\Delta S$, and the entropy in question is a count of where the molecule might be. The engine is being paid for information about its own contents, and the payment is exactly one $kT\ln 2$ per bit — which is where the free energy of every isothermal expansion comes from, with $10^{23}$ molecules instead of one.

Order arises spontaneously, constantly, and the books balance exactly. The claim that it cannot is the commonest piece of bad reasoning built on the second law, and one line of arithmetic retires it. A kilogram of water freezing at −10 °C loses about 1.22 kJ/K of entropy and hands 334 kJ to the surroundings, whose entropy rises by 334/263 = 1.27 kJ/K. The net is +47 J/K and the free energy falls by 12.4 kJ/kg. The pond gets more ordered because the air gets more disordered by more, and the same accounting covers a crystallising solution, a snowflake and a folding protein, which trades several hundred kJ/mol of conformational entropy for a net gain of perhaps 40.

Every interface carries a free energy per unit area, not an energy. Surface tension is $\partial F/\partial A$, which is why it falls with temperature and vanishes at the critical point, and why the skin that is not a skin behaves as it does; the angle a liquid makes with a solid is a balance of three such quantities.

One construction, three names #

$F$ is the right function when the system exchanges energy with the reservoir and nothing else. Change what is exchanged and the same construction produces a different name.

At constant pressure the reservoir supplies volume as well as energy: expanding by $\Delta V$ pushes the atmosphere back, so the reservoir’s entropy falls by $(\Delta U + P\Delta V)/T$ and repeating the derivation gives $G = U + PV - TS$, the Gibbs free energy, minimised at constant $T$ and $P$. When particles are exchanged too the reservoir also loses $\mu$ per particle, and the grand potential $\Omega = U - TS - \mu N$ is what falls.

Subtract from the energy every conserved quantity the reservoir will trade, each multiplied by its exchange rate. Entropy trades at $T$, volume at $-P$, particle number at $\mu$. Three names, one construction, and which applies is decided by what is held fixed rather than by anything about the system.

Gibbs published all three, chemical potential included, between 1876 and 1878, in a journal so obscure that Maxwell was among the few who read it. The word “free” is Helmholtz’s, from 1882, and it has caused a century of confusion: he divided the energy into freie Energie, the part available to do work, and gebundene Energie — bound energy, the $TS$ part the reservoir will not release. “Free” means available, not costless. Read as an energy the system possesses, $F$ makes no sense; read as the part a reservoir at temperature $T$ can be persuaded to let go, it is what the derivation says.

The rubber band was documented long before any of this. John Gough reported in 1805 that a stretched band warms and a loaded one contracts on heating; Joule measured it in 1859; and the explanation waited on Gibbs and then on the idea of a polymer as a chain with a configuration count, which arrived in the 1930s.

Where the model stops #

The reservoir has to be large. The derivation expanded the reservoir’s entropy to first order, which requires its heat capacity to dwarf the system’s. The neglected term is of order $C_{\text{sys}}/C_{\text{res}}$, unmeasurable for a nanoparticle in a beaker and the whole answer for two bodies of comparable size. A system as large as its surroundings has no free energy, because it has no fixed temperature to define one.

$F$ is the potential for constant temperature and volume only. Using it at constant pressure gets the answer wrong by the work done against the atmosphere. For ice melting, water contracts by $9.1 \times 10^{-5}$ m³/kg and $P\Delta V$ is 9.2 J/kg against a latent heat of 334,000 — an error of 0.003 per cent. For water boiling at 1 atm the volume change is 1.69 m³/kg and $P\Delta V$ is 171 kJ/kg against 2,260 — 7.6 per cent, which is not negligible at all.

The construction is about equilibrium and supplies no rate. Graphite is lower in free energy than diamond by 2.90 kJ/mol at 298 K, and a diamond is not observed to convert, because the barrier is several electronvolts and the traversal is governed by an exponential that has nothing to do with $F$.

The same exponential decides rates as well as populations, which is worth separating out because nothing in this essay is about rates. A barrier of 0.2, 0.35 or 0.6 eV gives wildly different temperature dependences of how fast a system crosses it, and free energy says nothing about any of them: it says which side the system ends on, not how long it takes to get there. A diamond is thermodynamically wrong at room temperature and has all the time it needs.

For small systems, “minimises” becomes “spends most of its time near”. Fluctuations in a system’s free energy are of order $kT$, which is $4.1\times10^{-21}$ J at 300 K. For a beaker that is nothing; for a single molecular motor doing perhaps twenty $kT$ of work per step it is a fifth of the budget, and such motors are regularly observed running backwards. The modern correction replaces the inequality with an equality — Jarzynski’s 1997 relation, $\langle e^{-W/kT}\rangle = e^{-\Delta F/kT}$ — checked by unfolding single RNA hairpins, whose individual pulls scatter by several $kT$ while their exponential average lands on the reversible value.

What the picture cannot show #

No figure on this page plots $F$, and that is a limitation rather than an oversight. Drawing a free energy requires an axis along which states are ordered — a reaction coordinate, a degree of crystallinity, an end-to-end length — and every such axis is chosen after the fact, not a coordinate anything moves along.

The two-minima figure comes closest and is the most misleading, because it plots a potential energy against a position. A free-energy landscape looks like that and means something else: its vertical axis already contains the entropy of everything not shown on the horizontal one, and its barriers are bottlenecks in a space of many dimensions rather than hills to be climbed. Reading it mechanically is the standard mistake in the protein-folding literature.

The coin figures can show the $S$ and never the $U$: coins have no energy, so the competition this essay is about cannot appear in them at all.

And the largest object in the argument appears nowhere. The reservoir is not drawn, because the only property used is that its temperature does not change — which is why the result applies unaltered to a bench, a bath, an atmosphere and an ocean.

The ladder from here #

Later rungs on this anchor: the partition function, and the collapse of the whole construction into $F = -kT\ln Z$; the Maxwell relations, which convert a derivative nobody can measure into one anybody can; the chemical potential, and the law of mass action; nucleation, and the barrier a new phase pays before it can grow; and the fluctuation theorems as the small-system replacement for an inequality that stops being true.

The neighbouring ladders are entropy as a count, which supplies the $S$; the exponential, the same argument applied to a single state; the latent heat, the numerator of the melting-point ratio; the ceiling on every engine, the same bookkeeping with two reservoirs; the entropy that lives on a surface, where the count stops scaling with volume; and equipartition, the source of the $kT$ in every estimate above.