Maxwell relations
/ MAKS-welz /
The Maxwell relations are four surprising equalities that connect quantities you would never guess were related — letting you trade a measurement you can't easily make for one you can. They are like exchange-rate tables for thermodynamics: instead of measuring how a system's entropy changes when you squeeze it (very hard to do directly), a Maxwell relation tells you it equals how its pressure changes when you warm it (easy to measure).
They arise from a simple mathematical fact. For a smooth quantity that depends on two variables, the order in which you take its two slopes doesn't matter — the mixed second derivatives are equal. Apply this to the internal energy and the three free energies, and out drop four tidy equalities linking the slopes of entropy, temperature, pressure, and volume against one another.
Why it matters: the Maxwell relations are the engine that lets thermodynamics convert easy benchtop measurements into hard-to-reach quantities, and they tie all of a substance's thermal and mechanical responses into one consistent web. The honest caveat: they are exact bookkeeping, not new physics — they tell you how properties must hang together, never the actual value of any one property, which still has to be measured or modeled.
How a gas's entropy falls when you compress it is fiendish to measure directly. A Maxwell relation says it exactly equals how the gas's pressure climbs when you heat it at fixed volume — which any pressure gauge and thermometer can read off.
Trade a hard-to-measure slope for an easy one — exactly.
Despite the shared name, these are not the famous Maxwell equations of electromagnetism — both honor James Clerk Maxwell, but these four belong entirely to thermodynamics. A common derivation aid is the 'thermodynamic square' mnemonic.