Maxwell Relations

Each thermodynamic potential provides a different route for connecting state-variable changes. Internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy are written in terms of their relevant natural variables. Taking mixed second derivatives of these expressions in either order gives equal results, producing a corresponding relation between measurable and less directly accessible thermodynamic quantities.

The equality of mixed second derivatives supplies the mathematical link between two differential descriptions of the same thermodynamic potential. Because the order of differentiation does not change the result under the stated conditions, a derivative involving entropy can be related to one involving pressure, volume, or temperature. This turns formal thermodynamic relationships into experimentally useful connections.

The appropriate relation depends on which variables naturally describe the chosen potential. Internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy each organize thermodynamic information differently, so selecting one determines the variables held constant during differentiation. This choice helps match the calculation to available measurements, such as pressure, volume, or temperature.

The variables held constant define the specific partial derivatives and therefore determine the valid form of the relation being applied. Maxwell relations are derived from a potential under its relevant state-variable conditions, not from arbitrary independent changes. Keeping those conditions clear prevents a derivative from being interpreted with the wrong constraint and supports consistent comparison with measurements.

Begin by selecting the thermodynamic potential suited to the variables in the problem. Express its differential in terms of those variables, identify the first derivatives associated with measurable properties, and then compare the two possible orders of mixed differentiation. Equating those second derivatives yields the desired relation, which can then be used to replace a difficult quantity with accessible measurements.

They are useful when a thermodynamic change is difficult to measure directly but can be connected to pressure, volume, or temperature data. Applications include analyzing phase transitions, testing equations of state, characterizing material properties, and studying equilibrium processes. In each case, the relations help translate theoretical thermodynamic quantities into information obtainable through laboratory observation.