Thermodynamics · undergrad

Thermodynamic Potentials

Legendre transforms that match the variables controlled by an experiment.

Published Apr 26, 2026 · 1 min

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Natural Variables

The internal energy has natural variables $S,V,N$ :

dU=T\,dS-p\,dV+\mu\,dN.

Legendre transforms exchange inconvenient variables for controlled variables. The Helmholtz free energy

F=U-TS

has natural variables $T,V,N$ , while the Gibbs free energy

G=U-TS+pV

has natural variables $T,p,N$ .

At fixed $T$ and $V$ , equilibrium minimizes $F$ . At fixed $T$ and $p$ , equilibrium minimizes $G$ .

The change in thermodynamic potential when particle number changes.

The thermodynamic potential suited to constant temperature and pressure.

Identities between thermodynamic derivatives following from state functions.

Energy conservation written for heat, work, and internal energy.