HelmholtzFreie

Helmholtz free energy, denoted F, is a thermodynamic potential defined by F = U − T S, where U is internal energy, T is temperature, and S is entropy. It is a function of the extensive variables volume V, particle number N, and the temperature T: F = F(T, V, N). In statistical mechanics, for a system described by the canonical ensemble with fixed N, V, and T, F relates to the partition function Z by F = −k_B T ln Z.

Differential form: dF = −S dT − P dV + μ dN, where P is pressure and μ chemical potential. Consequently, S = − (∂F/∂T)_V,N, P = − (∂F/∂V)_T,N, and μ = (∂F/∂N)_T,V. At fixed T and V, F attains a minimum at equilibrium with respect to internal degrees of freedom; a lower F implies a more stable state at those conditions.

Applications: The Helmholtz free energy is the natural potential for systems held at constant temperature and volume, such as many laboratory or condensed-matter contexts. It is used to compute phase behavior, reaction spontaneity (via μ and the derivatives), and to derive equations of state. It contrasts with the Gibbs free energy G, which is the natural potential at fixed temperature and pressure, G = F + P V.

History and naming: The concept is named after Hermann von Helmholtz, who contributed to thermodynamics in the 19th century; in German literature it is often called Freie Energie or Helmholtzsche Freie Energie. In many physics texts F is used, while chemistry sometimes uses A or F depending on conventions. It remains a central tool in thermodynamics, statistical mechanics, and physical chemistry.