What is Helmholtz free energy?

Helmholtz free energy (A, sometimes F) is a thermodynamic potential defined as A = U − TS, where U is the internal energy, T is the absolute temperature, and S is the entropy. It represents the maximum useful work extractable from a closed system at constant temperature and volume.

How does it differ from Gibbs free energy?

Gibbs free energy (G = H − TS) is used for processes at constant temperature and pressure, while Helmholtz free energy (A = U − TS) applies to processes at constant temperature and volume. For gases and reactions open to the atmosphere, Gibbs is more common; for solids and sealed systems, Helmholtz is more relevant.

What does a negative Helmholtz free energy change mean?

A decrease in Helmholtz free energy (ΔA < 0) indicates a spontaneous process at constant temperature and volume. The system can do work on its surroundings. A positive ΔA means the process requires external work to proceed.

What units should I use?

Use joules (J) for internal energy, kelvin (K) for temperature, and joules per kelvin (J/K) for entropy. The result will be in joules. You can also work in kJ and kJ/K consistently — just keep units uniform.

Can temperature be zero or negative?

Temperature must be in kelvin and must be greater than zero (absolute zero is 0 K). Negative kelvin values are unphysical. The calculator will flag invalid inputs.

Helmholtz Free Energy Calculator

Enter internal energy, temperature, and entropy to compute Helmholtz free energy (A = U − TS) and interpret it relative to your chosen reference state.

Inputs

A = U − T · S

Internal Energy (U)

Temperature (T)

Entropy (S)

Helmholtz Free Energy (A)

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Summary

Enter internal energy, temperature, and entropy to compute Helmholtz free energy (A = U − TS) and interpret it relative to your chosen reference state.

How it works

Enter the internal energy (U) in joules.
Enter the absolute temperature (T) in kelvin.
Enter the entropy (S) in joules per kelvin.
The calculator applies A = U − T·S to compute Helmholtz free energy.
The sign of absolute A depends on the reference zero; spontaneity at constant T and V is determined by a decrease, ΔA < 0, between two states.

Use cases

Calculate the Helmholtz thermodynamic potential of a state from U, T, and S.
Study statistical mechanics and partition functions in physics courses.
Compare Helmholtz and Gibbs free energies for systems at fixed volume.
Verify thermodynamic cycle calculations in engineering problems.
Analyze adsorption, surface tension, and solid-state phase transitions.
Cross-check simulation outputs from molecular dynamics studies.

Frequently Asked Questions

Last updated: 2026-09-22 · Reviewed by Nham Vu