Maxwell Relations
The Maxwell relations are four identities between partial derivatives of the thermodynamic variables S, T, p and V, obtained from the equality of mixed second derivatives of the potentials U, H, F and G. For example, from dF = −S dT − p dV follows (∂S/∂V)T = (∂p/∂T)V. They allow entropy and energy changes, which cannot be measured directly, to be computed from measurable equation-of-state data.
Curious
Entropy cannot be read off a meter. But the Maxwell relations, discovered by James Clerk Maxwell in his 1871 textbook Theory of Heat, show that certain changes in entropy are exactly equal to changes in pressure, volume or temperature, which can be measured (see James Clerk Maxwell).
They follow from a simple mathematical fact: for any smooth function of two variables, the order of differentiation does not matter. Applied to the four free energies and related potentials, this gives four useful identities.
Exploring
| Potential | Differential | Maxwell relation |
|---|---|---|
| U(S,V) | dU = T dS − p dV | (∂T/∂V)S = −(∂p/∂S)V |
| H(S,p) | dH = T dS + V dp | (∂T/∂p)S = (∂V/∂S)p |
| F(T,V) | dF = −S dT − p dV | (∂S/∂V)T = (∂p/∂T)V |
| G(T,p) | dG = −S dT + V dp | (∂S/∂p)T = −(∂V/∂T)p |
Deep Dive
Uses. From (∂S/∂V)T = (∂p/∂T)V follows the “energy equation” (∂U/∂V)T = T(∂p/∂T)V − p, which is zero for an ideal gas (proving its internal energy depends on T only; see Internal Energy) and equals a/Vm² for a van der Waals gas (see Van der Waals Equation). From the G relation, (∂S/∂p)T = −Vα, so compressing a substance that expands on heating lowers its entropy. The Joule–Thomson coefficient and the Clausius–Clapeyron equation are derived in the same way.
Mnemonics and structure. The thermodynamic square (Born square) is a common device for recalling the relations. Geometrically, they reflect the fact that the potentials are Legendre transforms of one another and that the equilibrium states form a surface on which the potentials are smooth functions. Extensions include magnetic systems, with (∂S/∂B)T = (∂M/∂T)B, which underlies magnetic refrigeration (adiabatic demagnetisation) used to reach millikelvin temperatures.
Limits. The relations hold for equilibrium states and require the potentials to be differentiable; at first-order phase transitions some derivatives are discontinuous, and the relations must be applied within each phase.
Concept Relationship Map
Requires: Helmholtz and Gibbs free energies, entropy
Enables: energy equation, Joule–Thomson coefficient, magnetic cooling
Commonly confused with: Maxwell's equations
Sources
- Maxwell, J.C. (1871). Theory of Heat. London: Longmans, Green.
- Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). Wiley.
- Wagner, W., Pruß, A. (2002). “The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance.” Journal of Physical and Chemical Reference Data, 31, 387–535.