2.8 Maxwell relations

Fundamentals of Maxwell Relations

Maxwell relations are equalities between certain partial derivatives of thermodynamic state variables. They arise because thermodynamic potentials are well-behaved functions whose mixed second derivatives are interchangeable. In practice, they let you calculate quantities that are hard to measure directly (like how entropy changes with volume) from quantities that are straightforward to measure (like how pressure changes with temperature).

Thermodynamic Potentials

Four primary thermodynamic potentials appear throughout statistical mechanics:

• Internal energy $U$, with natural variables $S, V, N$
• Enthalpy $H = U + PV$, with natural variables $S, P, N$
• Helmholtz free energy $F = U - TS$, with natural variables $T, V, N$
• Gibbs free energy $G = U - TS + PV$, with natural variables $T, P, N$

Each potential is connected to the others through Legendre transforms, which swap a variable for its conjugate (e.g., replacing $S$ with $T$). The key idea: at equilibrium, a system minimizes the appropriate potential for its constraints. For instance, a system at fixed $T$ and $V$ minimizes $F$, which maps directly to the canonical ensemble.

Partial Derivative Relationships

Because each thermodynamic potential is a state function, its differential is exact. That means the potential depends only on the current state, not on the path taken to get there. The partial derivatives of these potentials correspond to measurable quantities: heat capacities, compressibilities, thermal expansion coefficients, and so on.

Maxwell relations exploit this exactness. If a quantity is difficult to measure directly, you can often express it as a different partial derivative that's easier to access experimentally.

Symmetry in Mixed Derivatives

The mathematical backbone of Maxwell relations is Clairaut's theorem: for a sufficiently smooth function $f(x, y)$,

$\frac{\partial^2 f}{\partial x \, \partial y} = \frac{\partial^2 f}{\partial y \, \partial x}$

Thermodynamic potentials satisfy the smoothness requirements (they're continuous and twice-differentiable in their natural variables away from phase boundaries). So the order of differentiation doesn't matter, and equating the two mixed partials gives you a Maxwell relation.

Derivation of Maxwell Relations

From the Fundamental Equation

The most direct route uses the fundamental thermodynamic relation. Here's the procedure for the internal energy:

1. Write the differential: $dU = TdS - PdV$ (holding $N$ fixed for simplicity).

2. Identify the coefficients: $T = \left(\frac{\partial U}{\partial S}\right)_V$ and $-P = \left(\frac{\partial U}{\partial V}\right)_S$.

3. Take the mixed second derivatives and apply Clairaut's theorem:

$\frac{\partial}{\partial V}\left(\frac{\partial U}{\partial S}\right)_V = \frac{\partial}{\partial S}\left(\frac{\partial U}{\partial V}\right)_S$

1. Substitute the identified coefficients:

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$

That's one Maxwell relation. Repeat this process for each thermodynamic potential to get the other three.

Using Legendre Transforms

To derive the relations for $H$, $F$, and $G$:

1. Apply the appropriate Legendre transform to $U$. For example, $F = U - TS$ gives $dF = -SdT - PdV$.

2. Identify the coefficients of the differential (here, $-S$ and $-P$).

3. Equate the mixed second partial derivatives, just as above.

This generates one Maxwell relation per potential.

Jacobian Method

The Jacobian approach provides a systematic, almost mechanical way to generate Maxwell relations and is especially useful when you need to convert between different sets of independent variables.

• Express thermodynamic derivatives as ratios of Jacobians: $\left(\frac{\partial A}{\partial B}\right)_C = \frac{\partial(A, C)}{\partial(B, C)}$
• Use properties of Jacobians (antisymmetry, chain rule) to rearrange and simplify.
• This method is particularly powerful for deriving identities beyond the four standard Maxwell relations.

Four Primary Maxwell Relations

Each relation corresponds to one thermodynamic potential and its natural variables. The pattern is always the same: write the exact differential, identify the two coefficient functions, and equate their cross-derivatives.

Internal Energy Relation

From $dU = TdS - PdV$:

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$

This tells you how temperature changes with volume in an isentropic (adiabatic, reversible) process, relating it to how pressure changes with entropy at constant volume.

Helmholtz Free Energy Relation

From $dF = -SdT - PdV$:

$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$

This is one of the most frequently used Maxwell relations. The left side (how entropy changes with volume at constant $T$) is hard to measure. The right side (how pressure changes with temperature at constant $V$) is straightforward from an equation of state. This relation connects naturally to the canonical ensemble, where $T$ and $V$ are the controlled variables.

Enthalpy Relation

From $dH = TdS + VdP$:

$\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$

This is useful for analyzing isentropic compression or expansion, common in engineering applications like turbines and compressors.

Gibbs Free Energy Relation

From $dG = -SdT + VdP$:

$-\left(\frac{\partial S}{\partial P}\right)_T = \left(\frac{\partial V}{\partial T}\right)_P$

The right side is directly related to the thermal expansion coefficient $\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$. This relation is central to understanding phase transitions and chemical equilibria, since $G$ is the natural potential at constant $T$ and $P$.

Applications of Maxwell Relations

Thermodynamic Property Calculations

Maxwell relations let you trade hard-to-measure derivatives for easy ones. Some important examples:

• Entropy from an equation of state: Using $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$, you can compute entropy changes during isothermal expansion if you know $P(T,V)$.
• Internal energy dependence on volume: Combining a Maxwell relation with the first law gives $\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P$. For an ideal gas, this equals zero, confirming that $U$ depends only on $T$.
• Joule-Thomson coefficient: Maxwell relations help express $\mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_H$ in terms of $C_P$, $V$, and $\alpha$, all of which are measurable.

Equation of State Derivations

Maxwell relations impose consistency constraints on equations of state. If someone proposes a new equation of state, you can check whether it satisfies the Maxwell relations. They also help connect different formulations: for example, relating the pressure-based and volume-based descriptions of a van der Waals gas.

Phase Transition Analysis

The Clausius-Clapeyron equation follows directly from the Gibbs free energy Maxwell relation. Along a phase boundary, $G$ is equal for both phases, and the relation

$\frac{dP}{dT} = \frac{\Delta S}{\Delta V} = \frac{L}{T \Delta V}$

(where $L$ is the latent heat) describes the slope of the coexistence curve. This is how you predict, for instance, how the boiling point of water shifts with pressure.

Maxwell Relations in Different Ensembles

Microcanonical Ensemble

In the microcanonical ensemble (fixed $E, V, N$), the fundamental quantity is the entropy $S(E, V, N)$. Maxwell relations here connect derivatives of $S$: for example, the rate of change of $1/T = \left(\frac{\partial S}{\partial E}\right)_V$ with respect to $V$ equals the rate of change of $P/T = \left(\frac{\partial S}{\partial V}\right)_E$ with respect to $E$. These relations connect to foundational statistical concepts like the equipartition theorem.

Canonical Ensemble

The canonical ensemble (fixed $T, V, N$) is governed by the Helmholtz free energy $F$, computed from the partition function via $F = -k_BT \ln Z$. The Helmholtz Maxwell relation $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$ is the most natural one here. Since $Z$ encodes all thermodynamic information, you can derive both sides of this relation directly from the partition function.

Grand Canonical Ensemble

For open systems (fixed $T, V, \mu$), the relevant potential is the grand potential $\Omega = -k_BT \ln \mathcal{Z}$, where $\mathcal{Z}$ is the grand partition function. Its differential is $d\Omega = -SdT - PdV - Nd\mu$, which generates its own set of Maxwell relations, such as:

$\left(\frac{\partial S}{\partial V}\right)_{T,\mu} = \left(\frac{\partial P}{\partial T}\right)_{V,\mu}$

These are especially useful for systems with variable particle number: adsorption, chemical reactions, and multi-component phase equilibria.

Experimental Verification

Measurement Techniques

• Calorimetry measures heat capacities and entropy changes, providing one side of a Maxwell relation.
• PVT measurements (pressure-volume-temperature data) supply equations of state, giving the other side.
• Dilatometry determines thermal expansion coefficients, directly related to $\left(\frac{\partial V}{\partial T}\right)_P$.
• Spectroscopic methods can probe molecular-level properties that connect to bulk thermodynamic derivatives through statistical mechanics.

Verifying a Maxwell relation means measuring both sides independently and checking that they agree within experimental uncertainty.

Accuracy and Limitations

Maxwell relations assume that the thermodynamic potentials are smooth, well-defined functions of their variables. This breaks down in several situations:

• Near critical points, where fluctuations diverge and derivatives can become singular.
• In strongly interacting systems, where mean-field descriptions (which assume smooth potentials) may fail.
• At very low temperatures, where quantum effects modify the classical thermodynamic picture.
• Practically, maintaining truly isothermal or adiabatic conditions is always approximate, introducing systematic errors.

Advanced Topics

Higher-Order Maxwell Relations

Taking third or higher derivatives of thermodynamic potentials generates additional identities. These higher-order relations describe how response functions (like heat capacity or compressibility) themselves change with thermodynamic variables. They become particularly important near critical points, where response functions diverge, and connect to fluctuation theorems in statistical mechanics through relations like $C_V = \frac{k_BT^2}{\langle(\Delta E)^2\rangle}$.

Non-Equilibrium Extensions

Maxwell relations are strictly equilibrium results. However, for systems slightly perturbed from equilibrium, linear response theory (Onsager reciprocal relations) provides analogous symmetry relations between transport coefficients. Far from equilibrium, these symmetries generally do not hold, and the framework of Maxwell relations must be replaced by more general non-equilibrium statistical mechanics.

Quantum Statistical Mechanics Connections

In quantum systems, the partition function involves a trace over quantum states rather than a classical phase-space integral, but the thermodynamic structure (and therefore the Maxwell relations) carries over formally. The differences appear in what the potentials look like as functions of their variables. Quantum phase transitions (occurring at $T = 0$ as a parameter is varied) and quantum fluctuations modify the behavior of thermodynamic derivatives, but the Maxwell relations themselves remain valid as mathematical identities whenever the potentials are smooth.

Computational Methods

Numerical Implementation

For systems where analytical solutions aren't available, computational methods step in:

• Monte Carlo simulations sample configurations according to the Boltzmann distribution and compute thermodynamic averages (energy, pressure, etc.) from which derivatives can be extracted numerically.
• Molecular dynamics integrates equations of motion to generate time-averaged properties, which (assuming ergodicity) equal ensemble averages.
• Finite difference methods approximate partial derivatives from simulation data at nearby state points.

Maxwell relations serve as internal consistency checks: if your simulation gives $\left(\frac{\partial S}{\partial V}\right)_T$ and $\left(\frac{\partial P}{\partial T}\right)_V$ that don't agree, something is wrong with the simulation or the sampling.

Software Tools and Packages

• NIST Chemistry WebBook provides validated thermodynamic property data for comparison.
• LAMMPS and GROMACS handle molecular dynamics simulations for computing thermodynamic properties of condensed-phase systems.
• Gaussian and VASP perform ab initio calculations of energetics and thermodynamic quantities from first principles.
• Custom scripts in Python (using NumPy/SciPy) or MATLAB are commonly used for numerical differentiation, data fitting, and consistency checks against Maxwell relations.

Historical Context

Development of Thermodynamics

Thermodynamics grew from practical concerns about steam engine efficiency in the early 19th century into a rigorous theoretical framework. Carnot established the limits of heat engine efficiency, Clausius formalized entropy, and Gibbs unified the field by introducing thermodynamic potentials and their geometric interpretation. The subsequent development of statistical mechanics by Boltzmann and others provided the microscopic foundation for these macroscopic relations.

Contributions of James Clerk Maxwell

Maxwell's contributions to thermodynamics were part of his broader program connecting macroscopic phenomena to microscopic dynamics. His work on the kinetic theory of gases (the Maxwell-Boltzmann distribution) provided a statistical basis for temperature and pressure. The thermodynamic relations that bear his name formalized connections between state variables that had been used implicitly, making them systematic tools. Maxwell's emphasis on mathematical rigor and his geometric thinking (he used diagrams extensively) shaped how thermodynamics is taught and practiced to this day.