Chemical Potential

Quantity governing the spontaneous flow of matter between phases or components; µ = (∂G/∂n)_{T,p}.

Conceptual context

Chemical potential $\mu$ is the thermodynamic quantity governing matter exchange between phases or components of a multicomponent system, in the same way that temperature governs heat exchange and pressure governs mechanical work exchange. The concept was introduced by Josiah Willard Gibbs in the 1870s as the missing piece to extend classical thermodynamics to systems of variable composition.

In practice, $\mu$ answers questions such as: in which direction does a solute spontaneously flow between two solutions? At what temperature do liquid and vapor coexist? Under what conditions does a crystal grow rather than dissolve? All these questions share a unified criterion: equality or inequality of $\mu$ between the phases or components involved.

Within the tree of thermodynamic potentials, $\mu$ is the partial derivative of Gibbs free energy $G$ with respect to the number of moles, evaluated at constant T and p. This positions it as the intensive potential that completes $G$ in systems of variable composition.

🟢 Essential level

The central idea of chemical potential is surprisingly simple: matter flows spontaneously from where $\mu$ is higher to where $\mu$ is lower, exactly as heat flows from higher to lower temperature. When $\mu$ is the same in all phases of a system, net matter exchange stops and the system has reached phase equilibrium.

This analogy with temperature is the most powerful intuitive key. Just as one does not need to know exactly what entropy is to use temperature as a criterion for thermal equilibrium, one does not need to explicitly compute $G$ to use $\mu$ as a criterion for matter equilibrium. What matters is the difference in $\mu$ between regions of the system.

A useful way to picture $\mu$ is as a molar energetic height: if a molecule can move from a phase with high $\mu$ to one with low $\mu$, the system lowers its total free energy. That is why $\mu$ is not interpreted by its absolute sign, but by comparison between phases, solutions, or compartments.

🔵 Formal level

Basic formulas

The precise definition of $\mu$ starts from the Gibbs free energy $G$ of the system:

Definition of µ as partial derivative of G

In the leaf validator, this derivative is identified as $\partial G/\partial n_i$ to distinguish it from the total $G$. This partial derivative evaluates how much free energy the system gains or loses when an infinitesimal mole of component i is added, holding T, p and the amounts of all other components fixed. Its unit is J/mol.

In real systems, $\mu$ is computed from its standard value and the activity $a$ of the component:

Chemical potential as a function of activity

Real student questions

Why does $\mu$ decrease when a component is diluted? Because dilution increases the entropy of mixing: there are more possible configurations for the dilute system than for the concentrated one. The entropic contribution to chemical potential is negative (R·$T$ multiplied by a negative logarithm of $a$ < 1), reducing $\mu$ below the pure standard state value.

Does phase equilibrium require equal concentration in both phases? No. It requires equal $\mu$, which depends on both $\mu^\circ$ (which may differ between phases) and on activity. At liquid-vapor equilibrium the component concentration in the vapor is very different from that in the liquid, yet the $\mu$ values are equal.

What happens if $\mu$ of a component differs greatly between two phases in contact? The system is far from equilibrium. The component flows massively and spontaneously toward the phase of lower $\mu$ until the difference is reduced. The rate depends on kinetics, but thermodynamics guarantees that the final equilibrium corresponds to equal $\mu$.

How is $\mu$ related to the vapor pressure of a liquid? At liquid-vapor equilibrium, $\mu$ of the component in the vapor equals $\mu$ in the liquid. For ideal vapor, $\mu$ is $\mu^\circ$ plus R·$T$·ln($a$) where $a$ is the partial pressure in bars. Equating with $\mu$ of the liquid yields Raoult's law for vapor pressure.

Final synthesis

$\mu$ is the universal criterion for matter equilibrium: equality of $\mu$ between phases means no net component transfer. Its logarithmic expression in terms of $a$ connects the formal thermodynamic definition with experimental data on concentration, partial pressure and activity, making $\mu$ the central tool for analyzing phase diagrams, solubilities and equilibrium reactions.