Thermodynamics

The Thermodynamic Potentials

Legendre transforms of the energy. Each picks out the variables you can actually hold fixed in a lab.

Updated July 2026

The internal energy is a natural quantity to start from, and an awkward one to use. Its differential

d U = T d S - P d V

tells you that $U$ is really a function of entropy and volume. Those are its natural variables, the ones it wants to be written in. The trouble is that entropy is not something you can set on a dial. Volume you can fix with a piston, temperature with a bath, pressure with the weight of the atmosphere, but not entropy.

So the energy, in the variables it prefers, is written in terms of a quantity no experiment controls directly. The thermodynamic potentials are the fix. Each one trades an inconvenient variable for a convenient one, and the trade has a name.

Trading a variable is a Legendre transform

The move is the same each time. To swap $S$ for its conjugate $T = (\partial U / \partial S)_{V}$ , subtract the product of the two:

F = U - T S, d F = - S d T - P d V .

The $T d S$ term has been cancelled and replaced by a $- S d T$ , so $F$ is now a function of $T$ and $V$ . This is the Helmholtz free energy. Do the same to volume, or to both:

H = U + P V, G = U - T S + P V .

$H$ is the enthalpy, a function of $S$ and $P$ ; $G$ is the Gibbs free energy, a function of $T$ and $P$ , the two variables you can hold fixed most easily in a lab.

Each potential is minimised under its own constraints

The second law says the entropy of the universe increases. Rewriting that in terms of a system and a reservoir turns each potential into the thing that is extremised at equilibrium under a particular set of held-fixed variables.

Consequence

Which potential to watch

At fixed $T$ and $V$ , a system settles to the minimum of $F$ . At fixed $T$ and $P$ , it settles to the minimum of $G$ . The internal energy is minimised at fixed $S$ and $V$ . The rule is always the same: hold a potential's natural variables fixed, and that potential is what the system minimises.

This is why $G$ does so much work in chemistry. Reactions on a bench happen at the temperature of the room and the pressure of the air, so it is the Gibbs energy that decides which way they run.

Maxwell relations

Because each $d U$ , $d F$ , $d H$ , $d G$ is an exact differential, its second mixed partials are equal regardless of the order of differentiation. Applied to $d F = - S d T - P d V$ ,

(\frac{\partial S}{\partial V})_{T} = (\frac{\partial P}{\partial T})_{V} .

The left side asks how entropy changes when you expand at fixed temperature, which is hard to measure. The right side is just how pressure changes with temperature at fixed volume, which is easy. The four potentials give four such identities, and their whole value is this: they turn a quantity you cannot get at into one you can.

Definition

A worked relation

From $d G = - S d T + V d P$ the same equality of mixed partials gives $(\partial S / \partial P)_{T} = - (\partial V / \partial T)_{P}$ . The right side is the thermal expansion coefficient, tabulated for every material. The left side, the entropy released on compression, would otherwise need a calorimeter.

None of this adds physics to the first and second laws. It is bookkeeping, and the bookkeeping is the point: the potentials are four views of the same surface, chosen so that whatever you happen to be able to hold fixed, one of them is written in exactly those variables.