Maxwell Relations

Core Idea

Maxwell relations are a set of equations derived from the equality of mixed partial derivatives of thermodynamic potentials; for example, (∂T/∂V)_S = -(∂P/∂S)_V. They provide powerful constraints linking different measurable properties (like pressure, volume, temperature, entropy) and allow the calculation of hard-to-measure quantities from easily measured ones. Maxwell relations emerge naturally from the exactness of thermodynamic differentials and are a cornerstone of experimental thermodynamics.

How It's Best Learned

Derive Maxwell relations from the four main potentials (U, H, F, G). Practice using them to express hard-to-measure derivatives in terms of easy ones.

Common Misconceptions

• Forgetting the sign changes when reordering variables.
• Applying them to non-equilibrium states.
• Confusing the Maxwell relation from one potential with those from others.

Explainer

From your study of Legendre transformations and thermodynamic potentials, you know that the internal energy U is not the only thermodynamic potential — by performing Legendre transforms, you can construct the enthalpy H, the Helmholtz free energy F, and the Gibbs free energy G, each with its own set of natural variables. Each potential is a state function, meaning its differential is exact and path-independent. Maxwell relations are the powerful set of equations that fall out of this exactness via a single mathematical theorem: Schwarz's theorem on the equality of mixed partial derivatives.

The derivation is straightforward once you see the pattern. Take the Helmholtz free energy: dF = -S dT - P dV. This tells you that (dF/dT)_V = -S and (dF/dV)_T = -P. Because F is a state function, its differential is exact, so the mixed second partial derivatives must be equal: d(-S)/dV at constant T equals d(-P)/dT at constant V. This gives (dS/dV)_T = (dP/dT)_V — one of the four standard Maxwell relations. The same procedure applied to the other three potentials yields three more relations: from U (natural variables S, V), (dT/dV)_S = -(dP/dS)_V; from H (natural variables S, P), (dT/dP)_S = (dV/dS)_P; and from G (natural variables T, P), (dS/dP)_T = -(dV/dT)_P. Each relation equates a different pair of partial derivatives, and each belongs to a specific potential with specific natural variables.

The practical utility is immediate and profound. Entropy cannot be read directly from any instrument — there is no "entropy meter." But (dS/dV)_T = (dP/dT)_V converts an unmeasurable entropy derivative into a pressure-temperature measurement at constant volume, which is routine experimental work. Similarly, (dS/dP)_T = -(dV/dT)_P expresses an entropy derivative in terms of the thermal expansion coefficient, another standard measurement. Maxwell relations systematically bridge the gap between the quantities that thermodynamic theory requires (entropy derivatives) and the quantities that laboratory equipment can deliver (pressure, volume, temperature, and their rates of change). This is why they are indispensable in experimental thermodynamics: they make the full apparatus of thermodynamic potentials experimentally accessible.

Two technical pitfalls deserve emphasis. First, sign errors are the most common mistake. The minus signs in the differentials of each potential (dF = -S dT - P dV, dG = -S dT + V dP) propagate into the Maxwell relations, and forgetting or misplacing a sign gives the wrong relation. The mnemonic "Good Physicists Have Studied Under Very Fine Teachers" encodes the arrangement of variables around a thermodynamic square that tracks these signs. Second, Maxwell relations only hold at thermodynamic equilibrium — they are derived from state functions, which are defined only for equilibrium states. Applying them to non-equilibrium processes, where thermodynamic potentials are not well-defined, produces meaningless results. Each relation also belongs to a specific potential, and confusing which relation comes from which potential — for instance, using the Helmholtz relation when the natural variables of the problem are T and P (which call for the Gibbs relation) — is another frequent source of error.