Thermodynamics

Maxwell Relations

From dU = TdS − pdV: (∂T/∂V)_S = −(∂p/∂S)_V — and three more from H, F, G

Maxwell's relations are four identities relating partial derivatives of thermodynamic state functions, derived from the equality of mixed second partial derivatives. From the fundamental equation dU = T dS − p dV: (∂T/∂V)_S = −(∂p/∂S)_V. Similar relations come from the differentials of enthalpy H, Helmholtz free energy F, and Gibbs free energy G: (∂T/∂p)_S = (∂V/∂S)_p, (∂S/∂V)_T = (∂p/∂T)_V, and (∂S/∂p)_T = −(∂V/∂T)_p. Originally derived by James Clerk Maxwell (1871, Theory of Heat). Applications: connect measurable quantities (e.g., (∂S/∂T)_V = C_V/T) to the equation of state, derive Clausius-Clapeyron equation for phase coexistence, calculate thermodynamic data from each other, and check consistency of measured properties.

  • CountFour relations from U, H, F, G
  • AuthorMaxwell 1871
  • Source∂²X/∂y∂z = ∂²X/∂z∂y
  • Example(∂S/∂V)_T = (∂p/∂T)_V
  • DerivesClausius-Clapeyron equation
  • UseLink heat capacity to EoS

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Why Maxwell relations matter

  • Clausius-Clapeyron. Phase boundary slope dp/dT = L/(TΔV). Predicts how boiling point shifts with altitude (water boils at 91°C at 3000 m), how ice's melting point depends on pressure, how high-pressure phases of materials emerge.
  • Heat capacity from EoS. (∂C_V/∂V)_T = T(∂²p/∂T²)_V — a direct Maxwell consequence. If you know p(T,V), you can compute the V-dependence of heat capacity without measuring it. Used to extrapolate calorimetry data to high pressures.
  • Refrigeration cycles. Joule-Thomson coefficient μ_JT = (∂T/∂p)_H = (V/C_p)·(αT − 1), where α is thermal expansion. Maxwell relations turn EoS data into cooling rates for gas liquefaction (helium, nitrogen, oxygen plants).
  • Magnetocaloric effect. Adiabatic demagnetization: (∂T/∂H)_S = −(T/C_H)(∂M/∂T)_H. Directly Maxwell-derived. Cools materials below 1 mK in laboratory cryogenics.
  • Equation of state fits. Cubic EoS (van der Waals, Peng-Robinson, Soave-Redlich-Kwong) used in process engineering rely on Maxwell relations to compute fugacity, enthalpy departures, and heat capacities self-consistently.
  • Geophysics. Pressure-temperature mineral phase diagrams in the mantle use Clausius-Clapeyron applied to olivine→spinel and other transitions; explains depth of seismic discontinuities at 410 km and 660 km.
  • Chemical equilibria. Activity coefficients, chemical potentials, and reaction equilibria derive from G(T, p, n_i); Maxwell relations among ∂μ_i/∂p, ∂V/∂n_i etc. are everywhere in chemical thermodynamics.

The four relations

  • From U(S,V): dU = T dS − p dV gives (∂T/∂V)_S = −(∂p/∂S)_V.
  • From H(S,p): dH = T dS + V dp gives (∂T/∂p)_S = (∂V/∂S)_p.
  • From F(T,V): dF = −S dT − p dV gives (∂S/∂V)_T = (∂p/∂T)_V.
  • From G(T,p): dG = −S dT + V dp gives (∂S/∂p)_T = −(∂V/∂T)_p.
  • Memory device. Born square: arrange S, V, T, p in a square. Each potential lives on a side; partials run perpendicular. Sign rule: minus when the derivatives include conjugate variables (T,S) or (p,V) "crossed".
  • Generalizations. For magnetic systems, replace (p,V) with (H,M); for chemical systems, add (μ,N) terms. Each new conjugate pair adds a potential and a Maxwell relation.

Common misconceptions

  • "Obscure mathematics." Ubiquitous in real thermodynamics. Every chemistry textbook, every refrigeration design handbook, every geophysics computation invokes them — usually without naming them.
  • "Only ideal gas." Universal — they hold for any system whose state is described by a few extensive variables and a smooth potential. Real gases, liquids, solids, magnetic systems, plasmas, even black holes all obey them.
  • "Approximate." Exact thermodynamic identities. They're not perturbative or model-dependent; they follow from the existence of state functions and Schwarz's theorem on smooth potentials.
  • "Only useful with full EoS." Even local measurements help. Knowing (∂V/∂T)_p (thermal expansion) at one state point gives you (∂S/∂p)_T at that point — which lets you compute heat absorbed/released during pressure changes without calorimetry.
  • "Sign of (∂S/∂V)_T is always positive." True for normal substances (entropy rises with volume at fixed T) but reverses for anomalous water near 4°C and for some magnetic systems. The Maxwell relation (∂S/∂V)_T = (∂p/∂T)_V tells you that anomalous (∂p/∂T)_V matches anomalous thermal behavior.
  • "You need to memorize all four." One pneumonic (Born square) generates them all. Most physicists rederive on the fly from dU = TdS − pdV plus Legendre transforms — takes 30 seconds.

History

  • 1865 Clausius. Coins "entropy"; gives ∮ dQ_rev/T = 0 — the precondition for state functions like U, H, F, G to exist.
  • 1871 Maxwell. Publishes Theory of Heat; derives the four relations as consequences of mixed-partial equality applied to thermodynamic potentials.
  • 1875 Gibbs. On the Equilibrium of Heterogeneous Substances — generalizes to multi-component systems, develops chemical potential framework, sets up the Legendre-transform structure modern thermo inherits.
  • 1882 Helmholtz. Defines free energy F = U − TS; reveals it as the natural potential at constant T,V.
  • 1909 Born. Born square (square pneumonic) for visualizing all four Maxwell relations on one diagram. Standard pedagogical device since.
  • Mid-20th c. industrial use. Cryogenic gas liquefaction (Linde, Claude processes) and refrigeration design lean heavily on Maxwell-derived Joule-Thomson coefficients and isentropic expansions.
  • Modern. Magnetocaloric refrigeration, adiabatic demagnetization to nK temperatures, mineral physics, and statistical-mechanical computations all rely on Maxwell relations to bridge experimentally accessible and theoretically natural variables.

Applications

  • Joule-Thomson cooling. Throttling a gas through a porous plug at constant H. (∂T/∂p)_H = (V/C_p)(Tα − 1) — Maxwell-relation rearrangement. Sign flips at the inversion temperature; below it, expansion cools; above, it heats. Critical for liquefying air.
  • Adiabatic compression heating. (∂T/∂p)_S = (TV α)/C_p; predicts compression-cycle exhaust temperatures in compressors. Sets duty cycle and intercooler sizing.
  • Phase diagrams. Clausius-Clapeyron solid-liquid-gas boundaries; mineral transitions in earth and planetary mantles; superconducting transition curves H_c(T).
  • Calorimetry consistency. NIST WebBook tables of C_p(T,p) checked against (∂C_p/∂p)_T = −T(∂²V/∂T²)_p — an exact Maxwell relation. Internal consistency of databases.
  • Magnetocaloric refrigeration. Gd alloys near room T; demagnetization of paramagnetic salts in cryostats. Cools to 1 mK in laboratory and to mK temperatures in dilution refrigerators.
  • Critical phenomena. Diverging response functions at second-order transitions (compressibility, susceptibility, heat capacity) constrained by Maxwell-relation amplitude ratios — universal critical exponents follow.
  • Solar interior models. Equation of state for hydrogen-helium plasma at million-K temperatures; Maxwell relations check the consistency of opacity and pressure tabulations.

Worked example: ideal gas check

  • Ideal gas EoS. pV = nRT. Compute (∂p/∂T)_V = nR/V.
  • Maxwell prediction. (∂S/∂V)_T = (∂p/∂T)_V = nR/V.
  • Direct calculation. Sackur-Tetrode entropy S = nR[(5/2) + ln(VT^(3/2)/n) + const]; (∂S/∂V)_T = nR/V. Match.
  • (∂U/∂V)_T for ideal gas. Use first law: dU = TdS − pdV; at constant T, (∂U/∂V)_T = T(∂S/∂V)_T − p = T·(nR/V) − nRT/V = 0. Confirms Joule's law: ideal gas internal energy depends only on T, not V. Maxwell did this proof in 1871.
  • Real gas correction. For van der Waals gas (p + a(n/V)²)(V − nb) = nRT, (∂p/∂T)_V = nR/(V − nb). Then (∂S/∂V)_T = nR/(V − nb), and (∂U/∂V)_T = a(n/V)² ≠ 0. The internal energy now depends on V — predicting Joule-Thomson cooling, gas-imperfection corrections, and condensation.

Frequently asked questions

What are the four Maxwell relations?

From U(S,V): (∂T/∂V)_S = −(∂p/∂S)_V. From H(S,p): (∂T/∂p)_S = (∂V/∂S)_p. From F(T,V): (∂S/∂V)_T = (∂p/∂T)_V. From G(T,p): (∂S/∂p)_T = −(∂V/∂T)_p. Each pairs an entropy derivative (hard to measure) with a mechanical derivative (easy to measure) — letting you compute one from the other.

Why does mixed-partial equality give them?

For a smooth function F(x,y), Schwarz's theorem says ∂²F/∂x∂y = ∂²F/∂y∂x. Apply it to a thermodynamic potential. From dF = −S dT − p dV, we read off (∂F/∂T)_V = −S and (∂F/∂V)_T = −p. The mixed partial ∂²F/∂T∂V = ∂²F/∂V∂T then gives −(∂S/∂V)_T = −(∂p/∂T)_V, i.e. (∂S/∂V)_T = (∂p/∂T)_V. The other three potentials yield the other three relations.

How do you derive the Clausius-Clapeyron equation?

At a phase boundary, dG = 0 along the coexistence curve since G is continuous across the transition. From dG = −S dT + V dp: −S_α dT + V_α dp = −S_β dT + V_β dp, so dp/dT = (S_β − S_α)/(V_β − V_α) = ΔS/ΔV. Using L = TΔS at the transition gives dp/dT = L/(TΔV) — the Clausius-Clapeyron equation. Tells you how fast the boiling point shifts with pressure (cooking at altitude) or melting point with pressure (ice skates).

What is a thermodynamic potential (U, H, F, G)?

Each potential is a state function whose natural variables make differential expressions exact. Internal energy U(S,V): dU = TdS − pdV. Enthalpy H(S,p) = U + pV: dH = TdS + Vdp — natural for constant-pressure processes. Helmholtz free energy F(T,V) = U − TS: dF = −SdT − pdV — natural for constant T, V (canonical ensemble). Gibbs free energy G(T,p) = U − TS + pV: dG = −SdT + Vdp — natural for constant T, p (chemistry's home base). Legendre transformations connect them.

How are they used in phase transition analysis?

Phase boundaries follow Clausius-Clapeyron, which is itself a Maxwell-derived identity. Beyond that, second-order Ehrenfest transitions (no latent heat, divergent C_p) are characterized by jumps in (∂S/∂T)_p — a Maxwell relation gives the corresponding jump in (∂V/∂T)_p (thermal expansion) or (∂V/∂p)_T (compressibility). Critical points show universal scaling exponents that follow from Maxwell-relation constraints among diverging response functions.

Why are they sometimes called consistency relations?

If you measure C_V, the equation of state p(T,V), and entropy independently, the data must be self-consistent — Maxwell relations connect them. For instance, (∂C_V/∂V)_T = T(∂²p/∂T²)_V is a direct consequence: if your tabulated C_V doesn't match the second temperature derivative of your tabulated EoS, one of them is wrong. Used as a quality check on NIST and engineering thermodynamic tables.