In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G (Gibbs free energy) or H (enthalpy). The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.

${\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V\,}$

Here, U is internal energy, T is absolute temperature, S is entropy, P is pressure, and V is volume.

This is only one expression of the fundamental thermodynamic relation. It may be expressed in other ways, using different variables (e.g. using thermodynamic potentials). For example, the fundamental relation may be expressed in terms of the enthalpy H as

${\displaystyle \mathrm {d} H=T\,\mathrm {d} S+V\,\mathrm {d} P\,}$

in terms of the Helmholtz free energy F as

${\displaystyle \mathrm {d} F=-S\,\mathrm {d} T-P\,\mathrm {d} V\,}$

and in terms of the Gibbs free energy G as

${\displaystyle \mathrm {d} G=-S\,\mathrm {d} T+V\,\mathrm {d} P\,}$.