Chemical Potential and Phase Equilibria

The chemical potential measures the energy change of adding a particle, and equality of chemical potentials across phases sets the conditions for phase and chemical equilibrium.

Definition

The chemical potential is the change in a thermodynamic potential when one particle of a given species is added at fixed conjugate variables, and phase equilibrium is the state in which temperature, pressure, and the chemical potential of each species are equal across all coexisting phases.

Scope

This topic covers the definition of the chemical potential as the derivative of the appropriate potential with respect to particle number, the conditions of mechanical, thermal, and chemical equilibrium between coexisting phases, the Gibbs phase rule, the Clausius-Clapeyron equation for coexistence curves, and the equilibrium of chemical reactions. The order-parameter description of transitions is treated under phase transitions.

Core questions

• How is the chemical potential defined and related to the Gibbs free energy per particle?
• Why must the chemical potential be equal across phases at equilibrium?
• How does the Gibbs phase rule count the degrees of freedom of a multiphase system?
• How does the Clausius-Clapeyron equation determine the slope of a coexistence curve?

Key concepts

• Chemical potential and particle number
• Conditions for phase coexistence
• Gibbs phase rule
• Clausius-Clapeyron equation
• Chemical reaction equilibrium

Key theories

Gibbs conditions for phase equilibrium

Two phases coexist in equilibrium when their temperatures, pressures, and the chemical potential of each component are equal, minimizing the total Gibbs free energy of the heterogeneous system.

Clinical relevance

The chemical potential governs diffusion, osmosis, solubility, and reaction equilibria across chemistry, biology, and materials science, while phase-equilibrium conditions underpin the construction of phase diagrams used in metallurgy, geology, and chemical engineering.

History

Gibbs introduced the chemical potential and the phase rule in his 1870s memoir on heterogeneous equilibrium, unifying earlier results such as the Clausius-Clapeyron relation into a general theory of coexisting phases and reactions.

Key figures

• J. Willard Gibbs
• Rudolf Clausius
• Benoit Clapeyron

Seminal works

• gibbs1876
• callen1985

Frequently asked questions

What does it mean for chemical potentials to be equal across phases?

It means there is no net driving force to move particles from one phase to another; if the potentials differed, particles would flow toward the phase of lower chemical potential until equality, and hence equilibrium, was restored.