# Gibbs–Helmholtz equation

The **Gibbs–Helmholtz equation** is a thermodynamic equation used for calculating changes in the Gibbs energy of a system as a function of temperature.

**Equation**

The equation is:[1]

where *H* is the enthalpy, *T* the absolute temperature and *G* the Gibbs free energy of the system, all at constant pressure *p*. The equation states that the change in the *G/T* ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor *H/T*^{2}.

## Chemical reactions

The typical applications are to chemical reactions. The equation reads:[2]

with Δ*G* as the change in Gibbs energy and Δ*H* as the enthalpy change (considered independent of temperature). The ~~o~~ denotes standard pressure (1 bar).

Integrating with respect to *T* (again *p* is constant) it becomes:

This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature *T*_{2} with knowledge of just the Standard Gibbs free energy change of formation and the Standard enthalpy change of formation for the individual components.

Also, using the reaction isotherm equation,[3] that is

which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.[4]

## Derivation

### Background

The definition of the Gibbs function is

where *H* is the enthalpy defined by:

Taking differentials of each definition to find *dH* and *dG*, then using the fundamental thermodynamic relation (always true for reversible or irreversible processes):

where *S* is the entropy, *V* is volume, (minus sign due to reversibility, in which *dU* = 0: work other than pressure-volume may be done and is equal to −*pV*) leads to the "reversed" form of the initial fundamental relation into a new master equation:

This is the Gibbs free energy for a closed system. The Gibbs–Helmholtz equation can be derived by this second master equation, and the chain rule for partial derivatives.[5]

Derivation Starting from the equation for the differential of

*G*, and rememberingone computes the differential of the ratio

*G*/*T*by applying the product rule of differentiation in the version for differentials:Therefore,

A comparison with the general expression for a total differential

gives the change of

*G/T*with respect to*T*at constant pressure (i.e. when*dp*= 0), the Gibbs-Helmholtz equation:

## Sources

- Physical chemistry, P. W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7
- Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
- Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0-19-855148-7
- Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3
- Physical chemistry, P. W. Atkins, Oxford University press, 1978, ISBN 0-19-855148-7