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. It is named after Josiah Willard Gibbs and Hermann von Helmholtz.
The equation is:
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/T2.

Chemical reactions

The typical applications are to chemical reactions. The equation reads:
with ΔG as the change in Gibbs energy and ΔH as the enthalpy change. The o denotes standard pressure.
Integrating with respect to T it becomes:
This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T2 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, that is
which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.

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 :
where S is the entropy, V is volume, 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.
for the differential of G, and remembering
one 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, the Gibbs-Helmholtz equation: