Roll a ball into a bowl and it settles at the bottom: nature, left alone, slides downhill in energy. That single instinct — equilibrium sits at a minimum — is the backbone of thermodynamics too. But it hides a trap. A chemist mixing reagents in an open beaker on the bench, a gas sealed in a rigid flask, a star's core held at fixed entropy — each reaches equilibrium, yet they are not all minimising the same energy. Which "energy" a system minimises depends entirely on what you are holding fixed while it settles.

A gas free to soak up heat from the room and to push its piston outward does not minimise its internal energy — it happily borrows energy from the surroundings if that lets it spread out. What it actually minimises is a different bookkeeping quantity, one that already has the heat traffic and the pushing-back of the atmosphere folded in. These tailored quantities are the thermodynamic potentials, and this page is about the family of four — internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy — how they are built from one another, and the beautiful web of identities (the Maxwell relations) that falls out for free once you have them.

The one idea to carry the whole way: for every set of experimental constraints there is a potential that is minimised at equilibrium, and the four potentials are just the same physics re-expressed in whichever variables you actually control in the lab.

Which knobs are you holding? — the beaker on the bench

Picture a reaction fizzing in an open beaker. The lab's thermostat pins the temperature T; the atmosphere pins the pressure p. Those are the two knobs held fixed — not the entropy, not the volume. Heat flows freely across the glass and the liquid level rises or falls against the air. Under those constraints the quantity that runs downhill to a minimum is the Gibbs free energy G. That is why chemists live in "G-land": constant T and p is the natural habitat of bench chemistry, and a reaction is spontaneous exactly when it lowers G.

Seal that same reaction in a rigid, insulated bomb instead — fix the volume V and let no heat escape (fix entropy S) — and now the potential that is minimised is the plain internal energy U. Put it in a rigid flask in a heat bath (fix T and V) and it is the Helmholtz free energy F that bottoms out. Same molecules, same chemistry — three different "energies" minimised, because you clamped three different pairs of knobs.

The four potentials and their fundamental differentials

Each potential is a function of its own natural variables — the very knobs held fixed when it is the one being minimised. Written in those variables, the total differential of each potential is exact and clean. Start from the first law wrapped up in the combined statement for a simple substance:

so the natural variables of U are (S, V, N). Reading the coefficients straight off gives the intensive quantities: T = \left(\partial U/\partial S\right)_{V,N}, -p = \left(\partial U/\partial V\right)_{S,N}, and the chemical potential \mu = \left(\partial U/\partial N\right)_{S,V}. The other three potentials are built by adding or subtracting a product of a conjugate pair:

Notice the pattern in the differentials. Every time you swap a variable — say entropy S for temperature T — the term +T\,dS turns into -S\,dT: the coefficient and the differential trade places and pick up a minus sign. Swap volume V for pressure p and -p\,dV becomes +V\,dp. That is no accident — it is a Legendre transform, and it is the engine that generates the whole family.

Legendre transforms: trading a variable for its slope

The internal energy U(S,V,N) knows everything, but it is written in S — a quantity you cannot dial on a bench. What you can control is T, the slope of U with respect to S. A Legendre transform rewrites a function in terms of its slope instead of its original variable, without losing any information.

The recipe is mechanical. To swap a natural variable for its conjugate (S\leftrightarrow T are a conjugate pair, as are V\leftrightarrow p), subtract or add the product of that conjugate pair:

• Trade S for T: form F = U - TS. Then dF = dU - T\,dS - S\,dT = -S\,dT - p\,dV + \mu\,dN — the T\,dS cancels and S is gone, replaced by T.
• Trade V for p: form H = U + pV, giving dH = T\,dS + V\,dp + \mu\,dN.
• Do both trades: G = U - TS + pV = H - TS = F + pV, whose natural variables are the two bench knobs (T, p, N).

So the family is a little lattice of transforms — two independent trades, giving 2\times 2 = 4 potentials:

Both routes from U arrive at the same G — the transforms commute. This is exactly the same operation that turns a Lagrangian into a Hamiltonian in mechanics: give up a variable, gain its slope.

A picture that remembers all of it: the thermodynamic square

There is a lovely mnemonic — the thermodynamic square (Guggenheim's square) — that packs all four potentials, their natural variables, their differentials, and the Maxwell relations into one little diagram. Each edge carries a potential; the two corners flanking that edge are its natural variables. The corners on each diagonal are conjugate pairs: S opposite T, and V opposite p. Reveal it step by step.

To read off a differential — say for F on the right edge — its flanking corners are V and T, so dF is built from dT and dV. Each differential is multiplied by the variable diagonally opposite its corner (T opposite S, V opposite p), with signs read from the arrows, recovering dF = -S\,dT - p\,dV. The mnemonic sentence for the border letters, starting at bottom-middle and going round, is "Good Physicists Have Studied Under Very Fine Teachers": G, p, H, S, U, V, F, T.

Maxwell relations: entropy on the cheap

Here is where the potentials earn their keep. Each differential d\Phi is an exact differential (a genuine change in a state function), and for an exact differential the mixed second partial derivatives are equal — the order of differentiation does not matter. Apply that to dF = -S\,dT - p\,dV: since -S = (\partial F/\partial T)_V and -p = (\partial F/\partial V)_T, equating \partial^2 F/\partial V\,\partial T = \partial^2 F/\partial T\,\partial V gives

Do the same for each potential and you get the four Maxwell relations:

Why is this a big deal? The left-hand sides all involve entropy — a quantity you cannot stick a meter on. The right-hand sides involve only pressure, volume and temperature — the easy, mechanical, everyday measurables. Maxwell relations let you trade a derivative of entropy you can never measure directly for a derivative of the equation of state you can. For an ideal gas pV = nRT, for instance, (\partial p/\partial T)_V = nR/V, so immediately

telling you exactly how the entropy of a gas grows as it expands isothermally — deduced without ever measuring entropy at all.

Worked examples

Example 1 — squeezing the equation of state out of F. For an ideal gas one can show (via the partition function — see the partition function) that the Helmholtz free energy has the form F(T,V,N) with \left(\partial F/\partial V\right)_T = -nRT/V. Because dF = -S\,dT - p\,dV, the pressure is just the (negative) volume slope:

which is the ideal-gas law, recovered by differentiating a single potential. Likewise the entropy is minus the temperature slope, S = -\left(\partial F/\partial T\right)_V. One potential, and every thermodynamic property tumbles out.

Example 2 — a Maxwell relation in anger. Find how the entropy of one mole of ideal gas changes with volume at fixed temperature. Direct measurement is hopeless, so use the F-Maxwell relation:

From p = nRT/V with n = 1, the right side is R/V. Integrating at fixed T from V_1 to V_2 gives \Delta S = R\ln(V_2/V_1) — the familiar entropy of isothermal expansion, pulled straight from the equation of state.

Example 3 — a Legendre mini-exercise. You are handed U(S,V,N) and asked for the potential natural to constant pressure. You want to trade V for p, so add the conjugate product pV:

The -p\,dV and +p\,dV cancel, leaving dH written in dS and dp. So the enthalpy's natural variables are (S, p, N), and at constant pressure (dp = 0) with no particle change, dH = T\,dS = \delta Q — enthalpy change is the heat absorbed at constant pressure, which is why reaction enthalpies are what a bench calorimeter reads.

Two phases, one winner: minimising G foreshadows melting

Because a system at fixed T and p minimises G, whichever phase of a substance has the lower Gibbs free energy is the stable one. Since G = H - TS and (\partial G/\partial T)_p = -S, a plot of G against temperature is a line whose slope is -S: the higher-entropy phase (the liquid) falls off more steeply. The two lines cross at the melting temperature T_m, and below it the solid wins, above it the liquid wins.

This is the seed of the whole theory of phase transitions: the substance always occupies the lower-G branch, jumping from one to the other where the lines cross. The kink in the stable-phase curve at T_m — where the slope (-S) jumps — is precisely the entropy of melting, the signature of a first-order transition.