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The first Maxwell relation is derived from the internal energy differential and relates entropy and volume to temperature and pressure.

The second Maxwell relation is obtained from the enthalpy differential and connects the specific heat at constant pressure to the expansion coefficient and temperature.

The third Maxwell relation comes from the Helmholtz free energy and relates temperature and volume with entropy and pressure.

The fourth Maxwell relation stems from the Gibbs free energy and connects temperature and pressure changes with respect to entropy and volume.

This equation provides the relationship between the pressure and temperature at phase boundaries and is used to describe the slope of phase transition lines on a P-T diagram.

This formula relates heat capacities at constant pressure (Cp) and at constant volume (Cv), highlighting their difference in terms of the gas constant and the volume expansion coefficient.

This differential form highlights the changes in Gibbs energy as a result of changes in temperature and pressure, being fundamental in understanding chemical reactions and phase changes.