Maxwell Relations

Introduction

Maxwell relations are a set of equations in thermodynamics that establish connections between various partial derivatives of thermodynamic properties. They arise from the symmetry of second derivatives of thermodynamic potentials, facilitating calculations of difficult-to-measure quantities using more accessible parameters. Core to classical thermodynamics, Maxwell relations simplify analysis of state functions and support prediction of system behavior.

"Thermodynamics is a branch of physics that deals with heat and temperature and their relation to energy and work." -- Rudolf Clausius

Thermodynamic Potentials

Internal Energy (U)

State function; total energy contained within a system. Natural variables: entropy (S), volume (V).

Helmholtz Free Energy (F)

Defined as F = U - TS; useful for systems at constant temperature (T) and volume (V).

Enthalpy (H)

Defined as H = U + PV; natural variables: entropy (S), pressure (P). Applicable to constant pressure processes.

Gibbs Free Energy (G)

Defined as G = H - TS = U + PV - TS; natural variables: temperature (T), pressure (P). Used extensively in chemical thermodynamics.

Mathematical Foundations

State Functions and Exact Differentials

Thermodynamic potentials are state functions; their differentials are exact, implying path-independence.

Symmetry of Second Derivatives

For well-behaved functions, mixed partial derivatives commute: ∂²f/∂x∂y = ∂²f/∂y∂x.

Maxwell’s Relations Basis

Derived from the equality of mixed second derivatives of thermodynamic potentials with respect to their natural variables.

Derivation of Maxwell Relations

Starting from Differential Forms

Use total differentials of potentials; e.g., dU = TdS - PdV.

Apply Equality of Mixed Derivatives

Set ∂/∂V (∂U/∂S) = ∂/∂S (∂U/∂V) to generate relations.

Repeat for Each Potential

Generate Maxwell relations by applying same procedure to F, H, and G.

dU = TdS - PdV∂T/∂V|_S = -∂P/∂S|_V

Maxwell Relations from Internal Energy

Starting Expression

dU = TdS - PdV

Natural Variables

S and V

Maxwell Relation

∂T/∂V|_S = -∂P/∂S|_V

Interpretation: relates how temperature changes with volume at constant entropy to how pressure changes with entropy at constant volume.

Maxwell Relations from Helmholtz Free Energy

Starting Expression

dF = -SdT - PdV

Natural Variables

T and V

Maxwell Relation

∂S/∂V|_T = ∂P/∂T|_V

Meaning: entropy-volume variation at constant temperature equals pressure-temperature variation at constant volume.

Maxwell Relations from Enthalpy

Starting Expression

dH = TdS + VdP

Natural Variables

S and P

Maxwell Relation

∂T/∂P|_S = ∂V/∂S|_P

Interpretation: temperature-pressure derivative at constant entropy equals volume-entropy derivative at constant pressure.

Maxwell Relations from Gibbs Free Energy

Starting Expression

dG = -SdT + VdP

Natural Variables

T and P

Maxwell Relation

∂S/∂P|_T = -∂V/∂T|_P

Physical meaning: entropy-pressure variation at constant temperature equals negative volume-temperature variation at constant pressure.

Applications

Thermodynamic Property Estimation

Calculate difficult derivatives using accessible measurements (e.g., ∂S/∂V via ∂P/∂T).

Phase Transition Analysis

Relate changes in entropy and volume during phase changes without direct measurement.

Material Science

Evaluate material response functions like compressibility and thermal expansion coefficients.

Limitations and Assumptions

Reversibility

Maxwell relations assume reversible processes and equilibrium states.

Continuity and Differentiability

Potentials must be continuous and twice differentiable functions.

Closed Systems

Derived relations generally apply to closed systems without particle exchange.

Examples

Example 1: Calculating ∂S/∂V|_T

Using Helmholtz free energy relation: ∂S/∂V|_T = ∂P/∂T|_V. Measure P vs T at fixed V.

Example 2: Entropy Change with Pressure

Using Gibbs relation: ∂S/∂P|_T = -∂V/∂T|_P. Measure volume change with temperature at constant pressure.

Example 3: Temperature Change with Volume

Internal energy relation: ∂T/∂V|_S = -∂P/∂S|_V. Useful in adiabatic expansion calculations.

Tables Summary

References

• Callen, H.B., Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, pp. 35-50.
• Reif, F., Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1965, pp. 120-130.
• Moran, M.J., Shapiro, H.N., Fundamentals of Engineering Thermodynamics, 7th ed., Wiley, 2010, pp. 150-160.
• McQuarrie, D.A., Statistical Mechanics, University Science Books, 2000, pp. 200-210.
• Atkins, P.W., de Paula, J., Physical Chemistry, 10th ed., Oxford University Press, 2014, pp. 260-270.