Thermodynamic Potentials · Concept

Helmholtz Free Energy

Reading time: 9 minConfidence: HighLast verified: October 2026

The Helmholtz free energy is the thermodynamic potential F = U − TS. In a process at constant temperature, the maximum work a system can do equals the decrease in F; at fixed temperature, volume and particle number, a system evolves towards, and in equilibrium sits at, a minimum of F. In statistical mechanics, F = −k_BT ln Z, where Z is the canonical partition function.

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One sentence that captures itHelmholtz free energy is the part of a system's energy that is actually “free” to be turned into work when its temperature is held fixed.

Not all of a system's internal energy can be extracted as work: some is locked up as disorder, represented by temperature times entropy. Subtract that and what remains, F = U − TS, is the available energy at constant temperature. Hermann von Helmholtz introduced it in 1882.

Free energy explains a competition at the heart of nature: systems tend to lower their energy (U) but also to raise their entropy (S). At low temperatures energy wins and matter orders into crystals; at high temperatures entropy wins and it melts and evaporates.

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Worked example — maximum isothermal workOne mole of ideal gas at 300 K doubling its volume isothermally: U is unchanged (see Internal Energy) while S rises by R ln 2, so ΔF = −TΔS = −RT ln 2 = −1,729 J. The maximum work extractable is therefore 1,729 J, matching the reversible isothermal work (see Isothermal Process). If the gas instead expands freely into a vacuum, F falls by the same amount but no work is done: the potential work is lost as entropy production. (Computed in Python.)

Potential	Definition	Natural variables	Minimised at equilibrium when fixed
Internal energy U	U	S, V, N	S, V
Enthalpy H	U + pV	S, p, N	S, p
Helmholtz F	U − TS	T, V, N	T, V
Gibbs G	U + pV − TS	T, p, N	T, p

Common mistake — Commonly Confused“Free” does not mean the energy costs nothing; it means available. Helmholtz and Gibbs free energies differ: F applies at fixed volume and G at fixed pressure, so chemistry in open containers uses G (see Gibbs Free Energy). In older literature F is written A (from German Arbeit, work).

Deep Dive

Primary sourceHelmholtz, H. von (1882). “Die Thermodynamik chemischer Vorgänge.” Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 22–39.

Differential relations. dF = −S dT − p dV + μ dN, so S = −(∂F/∂T)_V and p = −(∂F/∂V)_T; equality of mixed second derivatives yields a Maxwell relation. F is the Legendre transform of U with respect to entropy.

Statistical mechanics. For a system in contact with a heat bath, F = −k_BT ln Z with Z = Σ e^−E_i/k_BT, which connects microscopic energy levels to all thermodynamic properties. Computing Z (or F) is the central task of statistical mechanics, and free-energy calculations by simulation are standard tools in chemistry and drug design. The minimisation principle underlies Landau theory of phase transitions (see Lev Landau).

Entropic forces. A force can arise purely from entropy: f = −(∂F/∂x)_T = T(∂S/∂x) when U does not depend on x. Rubber elasticity and the osmotic pressure of dilute solutions are classic examples, as is the depletion attraction between colloids. A stretched rubber band warms when stretched quickly and pulls harder when heated, both consequences of its entropic elasticity.

Concept Relationship Map

Requires: internal energy, entropy
Enables: Maxwell relations, partition function, Landau theory
Commonly confused with: Gibbs free energy

Sources

Helmholtz, H. von (1882). Sitzungsberichte der Akademie der Wissenschaften zu Berlin, 22–39.
Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). Wiley.
Schroeder, D.V. (2000). An Introduction to Thermal Physics. Addison-Wesley.

Last verified: October 2026 · Source: Helmholtz (1882); Callen (1985); Schroeder (2000) · Confidence: high

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