Maxwell relations

The Maxwell relations or Maxwell relations of thermodynamics ( after the physicist James Clerk Maxwell ) make important connections between different state variables.

Statement

The Maxwell's relations allow to express changes of state variables (eg temperature T or entropy S) as amendments to other state variables (eg, pressure p and volume V):

Exemplary derivation

The relations can be derived by the Characteristic Functions ( total differentials ) we consider the functions of state Internal energy U, enthalpy H, free energy F and Gibbs free energy G.

For example, the total differential of the internal energy U, depending on entropy S and volume V:

Substituting a sufficiently smooth function on U advance, the set of black indicates that

This is the first Maxwell relation.

Guggenheim scheme

For practical work, you can use the so-called Guggenheim square. This yields all the above Maxwell relations.

We find the relation by two variables reads from the corners of a ( horizontal or vertical ) of the diagram, so that one side of the Maxwell equation formulated and removes the other side of the equation from the opposite side in the same manner.

For example, you and from which follows the expression takes. Opposite are now and then, leading to the expression. Differential quotient, which also included both as to get a negative sign, since both (!) Symbols are located on the edge with the minus sign ( in above-mentioned example). The held constant variable of a page is always rediscover in the denominator of the other side.

Mnemonics for the square can be found at: Guggenheim square ( aphorisms )

General Maxwell relation

If a function z ( x, y), by the theorem on implicit function at a point clearly resolvable both to x and to y, as can be shown, among other things, that

To show this, it is where the total of the differentials functions x and z.

Insertion results in

The partial differentials can be reduced if the retained variables are the same.

For more examples of this kind, the English language version of this article offers.

• Thermodynamics