Why do the Maxwell relations hold?

Because each thermodynamic potential is a true state function with an exact differential, its mixed second derivatives are equal regardless of the order of differentiation, and writing out that equality for each potential produces the relations.

Maxwell Relations

The Maxwell relations are equalities among partial derivatives of thermodynamic variables that follow from the exactness of the differentials of the thermodynamic potentials.

Definition

The Maxwell relations are a set of identities equating partial derivatives of conjugate thermodynamic quantities, obtained because the mixed second partial derivatives of any thermodynamic potential are independent of the order of differentiation.

Scope

This topic covers the derivation of the four standard Maxwell relations from the equality of mixed second partial derivatives of the internal energy, enthalpy, Helmholtz, and Gibbs potentials, the use of these relations to express inaccessible derivatives in measurable terms, and systematic methods such as the thermodynamic square for recalling them.

Core questions

How does the equality of mixed second derivatives yield the Maxwell relations?
Which pairs of derivatives does each of the four standard relations connect?
How are the relations used to replace experimentally inaccessible derivatives with measurable ones?
What mnemonic schemes organize the relations and their signs?

Key concepts

Exact differentials and equality of mixed partials
The four standard Maxwell relations
Conjugate variable pairs
Replacing inaccessible with measurable derivatives
Thermodynamic square mnemonic

Clinical relevance

Maxwell relations are essential in experimental thermodynamics for converting hard-to-measure entropy derivatives into volume and pressure measurements, and they underpin the derivation of equations of state and the analysis of cooling processes such as the Joule-Thomson effect.

History

Named for James Clerk Maxwell, who presented them in his Theory of Heat in the 1870s, the relations follow directly from Gibbs's potential formalism and have since become a standard tool for manipulating thermodynamic derivatives.

Key figures

James Clerk Maxwell
J. Willard Gibbs

Seminal works

callen1985

Frequently asked questions

Why do the Maxwell relations hold?: Because each thermodynamic potential is a true state function with an exact differential, its mixed second derivatives are equal regardless of the order of differentiation, and writing out that equality for each potential produces the relations.