Gibbs–Helmholtz equation

The Gibbs - Helmholtz equation (also Gibbs Helmholtz equation) is an equation of thermodynamics. It is named after the American physicist Josiah Willard Gibbs and the German physiologist and physicist Hermann von Helmholtz.

The equation states among other things:

T: absolute temperature

G: Gibbs free

H: enthalpy

P: pressure

: Number of moles of component j

The curly brackets around the material components are intended to indicate that this enumeration of all the different components j is meant in the system. This only applies to an open system in which an exchange of material is possible. In closed systems, this dependence is omitted.


The general relationship between the Gibbs free enthalpy and internal energy of a system can be produced by a corresponding Legendre transformation:

Or expressed in differential form:

Is the chemical potential of each component j.

The Legendre transformation between the enthalpy H and the Gibbs free energy G is:

Now, if the expression for the entropy S of the differential form here substituted the following:

After application of the quotient rule of differential calculus:

This corresponds to the aforementioned equation.

Other spellings

After another summaries can be yet another form of the equation specify:


Using the chain rule of differential calculus can now also show that:

The relationship, really just a Legendre transformation, which describes the relationship between the enthalpy H and Gibbs energy G is in some references called the Gibbs - Helmholtz equation (see Gibbs energy ). This is probably due to the fact that one has the basis to perform the Legendre transformation exclusively mathematical operations of the calculus, to get to the equation, which is commonly referred to as the Gibbs - Helmholtz equation.

In this simplification has been introduced indicating the change of the entropy of the system under consideration. The entropy is a measure of the number of micro-states, which can adopt a system. Thus, the Gibbs - Helmholtz equation also includes the statement of the second law of thermodynamics, according to which nature strives for possible low-energy states ( entropiereichere states, since the same amount of energy over more particles distributed more microstates allows ).

Processes are used as positive endergonic, those in which the change in the Gibbs free energy is negative, called exergonic. Exergonic processes can run voluntarily, while endergonic processes take place only with the supply of free energy.

A special position taking: The system is in equilibrium.