At constant entropy and volume, which thermodynamic potential is minimized at equilibrium?

At constant entropy and volume, internal energy is minimized at equilibrium. This is the energy-minimum principle for an isolated system when entropy, volume, and particle numbers are fixed.

The result follows from the first and second laws. For a simple compressible system, dU = T dS − p dV. If both S and V are fixed, allowable spontaneous changes cannot lower entropy or alter volume; a stable equilibrium state is therefore the state with the lowest compatible U. Equivalently, an isolated system reaches equilibrium by maximizing entropy at fixed U, which is mathematically equivalent to minimizing U at fixed entropy.

The other options apply under different constraints. Helmholtz free energy is minimized at constant temperature and volume, Gibbs free energy at constant temperature and pressure, and enthalpy has a more specialized minimum principle involving constant pressure and entropy. The potential is chosen according to the variables controlled by the surroundings.