Maxwell relations

Maxwell relations are equalities connecting thermodynamic partial derivatives. They follow from the equality of mixed second derivatives of thermodynamic potentials.

Thermodynamic potentials: State functions that encode a thermodynamic system’s energy and determine its equilibrium properties. Each potential generates a distinct Maxwell relation through its natural variables.
Thermodynamic state function: A property whose value depends only on a system’s state, not the path taken to reach it. The relations apply to derivatives of state functions rather than arbitrary path-dependent quantities.
Thermal expansion coefficient: A measure of how a material’s volume changes with temperature at fixed pressure. It helps evaluate pressure derivatives that replace entropy derivatives in practical calculations.
Cyclic relation: An identity relating three cyclically arranged partial derivatives among three variables. It is another derivative constraint, but does not arise specifically from mixed derivatives of a potential.
Enthalpy: A thermodynamic state function equal to internal energy plus pressure times volume. Its natural variables yield a Maxwell relation linking temperature and pressure derivatives.
Natural variables: The independent variables in terms of which a thermodynamic potential has a simple differential. Choosing a potential’s natural variables determines the derivative pair in its relation.
Isothermal compressibility: A measure of how a material’s volume changes with pressure at fixed temperature. It provides an experimentally accessible derivative for thermodynamic property calculations.
Gibbs–Duhem equation: A thermodynamic constraint linking changes in intensive variables for a system at equilibrium. It constrains thermodynamic variations but is distinct from the derivative equalities called Maxwell relations.
Helmholtz free energy: A thermodynamic potential equal to internal energy minus temperature times entropy. Its differential produces the relation between entropy’s volume derivative and pressure’s temperature derivative.
Entropy: A thermodynamic state function that measures energy dispersal and determines reversible heat transfer. Entropy derivatives appear throughout the relations and can be replaced by measurable quantities.

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