In thermodynamics, the thermodynamic free energy is one of the state functions of a thermodynamic system (the others being internal energy, enthalpy, entropy, etc.). The change in the free energy is the maximum amount of work that the system can perform in a process at constant temperature, and its sign indicates whether the process is thermodynamically favorable or forbidden. Since free energy usually contains potential energy, it is not absolute but depends on the choice of a zero point. Therefore, only relative free energy values, or changes in free energy, are physically meaningful.
The free energy is the portion of any first-law energy that is available to perform thermodynamic work at constant temperature, i.e., work mediated by thermal energy. Free energy is subject to irreversible loss in the course of such work. Since first-law energy is always conserved, it is evident that free energy is an expendable, second-law kind of energy. Several free energy functions may be formulated based on system criteria. Free energy functions are Legendre transforms of the internal energy.
The Gibbs free energy is given by G = H − TS, where H is the enthalpy, T is the absolute temperature, and S is the entropy. H = U + pV, where U is the internal energy, p is the pressure, and V is the volume. G is the most useful for processes involving a system at constant pressure p and temperature T, because, in addition to subsuming any entropy change due merely to heat, a change in G also excludes the p dV work needed to "make space for additional molecules" produced by various processes. Gibbs free energy change therefore equals work not associated with system expansion or compression, at constant temperature and pressure, hence its utility to solution-phase chemists, including biochemists.
The historically earlier Helmholtz free energy is defined in contrast as A = U − TS. Its change is equal to the amount of reversible work done on, or obtainable from, a system at constant T. Thus its appellation "work content", and the designation A (from German Arbeit 'work'). Since it makes no reference to any quantities involved in work (such as p and V), the Helmholtz function is completely general: its decrease is the maximum amount of work which can be done by a system at constant temperature, and it can increase at most by the amount of work done on a system isothermally. The Helmholtz free energy has a special theoretical importance since it is proportional to the logarithm of the partition function for the canonical ensemble in statistical mechanics. (Hence its utility to physicists; and to gas-phase chemists and engineers, who do not want to ignore p dV work.)
Historically, the term 'free energy' has been used for either quantity. In physics, free energy most often refers to the Helmholtz free energy, denoted by A (or F), while in chemistry, free energy most often refers to the Gibbs free energy. The values of the two free energies are usually quite similar and the intended free energy function is often implicit in manuscripts and presentations.

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