the Maxwell relations
Some thermodynamic quantities are easy to measure — how pressure changes when you warm a fixed volume — while others are almost impossible to grab directly, like how a substance's entropy changes when you squeeze it. The Maxwell relations are a set of exact equalities that let you trade the impossible for the easy. They arise from a simple mathematical fact: for a smooth function, the order of taking two partial derivatives does not matter.
Each thermodynamic potential is an exact differential, so its mixed second partials are equal. From dU = T dS - p dV one gets (dT/dV)_S = -(dp/dS)_V; from the Helmholtz free energy dF = -S dT - p dV comes (dS/dV)_T = (dp/dT)_V; from the Gibbs free energy dG = -S dT + V dp comes (dS/dp)_T = -(dV/dT)_p; and from dH = T dS + V dp comes (dT/dp)_S = (dV/dS)_p. In each, an entropy derivative (hard) equals a mechanical derivative (easy).
These relations are the everyday machinery of thermodynamics: they let you compute an entropy change from a measured equation of state, relate C_p to C_v, and derive the Clausius-Clapeyron equation. Caveat: they hold only where the potential is smooth (twice differentiable); at a first-order phase transition, where the potential has a kink, they can fail across the boundary.
Suppose you want the isothermal entropy change of a gas as you compress it, (dS/dp)_T, which no thermometer reads directly. The Gibbs Maxwell relation says it equals -(dV/dT)_p, the thermal expansion — something a ruler and a thermometer measure easily. For an ideal gas V = N k_B T / p, so (dV/dT)_p = N k_B / p and hence (dS/dp)_T = -N k_B / p.
A Maxwell relation converts an unmeasurable entropy derivative into a routine expansion measurement.
Maxwell relations follow purely from the potentials being exact differentials of state functions; they are identities, valid for any equilibrium substance and requiring no model — but they break down at a phase boundary where the potential is not differentiable.