Maxwell's thermodynamic relations
Maxwell's relations are four equalities between partial derivatives of pressure, volume, temperature and entropy, derived from the fact that each thermodynamic potential's mixed second partial derivatives must agree — they let a hard-to-measure quantity such as (∂S/∂V)_T be replaced by an easily measured one such as (∂p/∂T)_V.
Practise this conceptEach of the four thermodynamic potentials — internal energy U, enthalpy H, Helmholtz free energy F, and Gibbs free energy G — has a differential written entirely in terms of two natural variables (for G, that's temperature T and pressure p: dG = −S dT + V dp). Because G is a well-behaved function of state, it doesn't matter in which order you take its two partial derivatives — mixing ∂/∂T and ∂/∂p gives the same result either way. Writing that equality out is exactly a Maxwell relation.
The payoff is practical, not just formal. Entropy is awkward to measure directly, but (∂S/∂p) at constant T shows up naturally from G's differential. The matching Maxwell relation says this equals −(∂V/∂T) at constant p — the rate at which a substance expands when heated at fixed pressure, something you can read straight off a thermal-expansion measurement. Four potentials, each with two variables, give four such relations, and each swaps an entropy derivative for a directly measurable mechanical one.
None of this depends on the substance being an ideal gas, or on any particular equation of state. The relations follow from calculus applied to any system in equilibrium, which is exactly why they're called relations rather than laws: nothing new is being asserted about the physics, only that a state function's own consistency constrains how its measurable properties must relate to each other.
Key ideas
Requires: The laws of thermodynamics, Entropy and the second law
Common misconceptions
- Believing Maxwell's relations are separate physical laws that need their own experimental confirmation, rather than mathematical consequences of the thermodynamic potentials being exact differentials — once U, H, F and G are accepted as well-defined state functions, all four relations follow from calculus alone.
- Dropping or misplacing the minus sign and the held-constant subscripts, e.g. writing (∂T/∂V)_S = (∂p/∂S)_V instead of (∂T/∂V)_S = −(∂p/∂S)_V — the sign and which variable is held fixed on each side are not interchangeable, they come directly from which potential the relation was derived from.
- Believing Maxwell's relations only hold for an ideal gas, since that is the system most textbook examples use — the relations follow purely from calculus applied to any equilibrium system with a well-defined thermodynamic potential, with no equation of state assumed.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Thermodynamics and statistical physicsLOFY.04.004