Helmholtz free energy
F = U − TS, the work obtainable from a system held at constant temperature and volume.
Definition
The Helmholtz free energy F = U − TS is the portion of a system's internal energy available to do work at constant temperature, once the entropy term TS — the energy the environment supplies or demands for free — is subtracted. For a process at fixed temperature and volume, the maximum extractable work equals the decrease in F, and equilibrium is the state of minimum F.
Statistical mechanics gives F its sharpest meaning through the partition function: F = −k_BT ln Z. Every other thermodynamic quantity follows by differentiation — the entropy is S = −∂F/∂T, the pressure P = −∂F/∂V — so F is the bridge between the microscopic sum over states and macroscopic thermodynamics.
F is the Legendre transform of the internal energy that exchanges entropy for temperature: dF = −S dT − P dV, with natural variables T and V. It is the appropriate potential for a system in a rigid container held at constant temperature.
History
Introduced by Hermann von Helmholtz in 1882 in his work on the thermodynamics of chemical processes, distinguishing the 'free' energy available for work from the 'bound' energy TS tied up in entropy.