When a system contains more than one chemical species, its Gibbs free energy depends on composition as well as temperature and pressure. The chemical potential $\mu_i$ of species $i$ is its partial molar Gibbs free energy: it measures how much the system's Gibbs free energy changes per mole when a very small amount of that species is added while temperature, pressure and the amounts of the other species are held fixed.

For an ideal gas or ideal dilute solution, chemical potential can be written as

$$\boxed{\mu_i=\mu_i^\circ+RT\ln a_i},$$

where $\mu_i^\circ$ is the chemical potential in the chosen standard state and $a_i$ is the dimensionless activity. In an ideal dilute solution,

$$a_i\approx\frac{c_i}{c^\circ}.$$

Thus increasing concentration generally increases the chemical potential of a dissolved species.

Matter moves down chemical-potential differences

Suppose the same species can move between two regions at the same temperature and pressure. Moving one mole from region 1 to region 2 changes Gibbs free energy by

$$\Delta G_{1\to2}=\mu_{i,2}-\mu_{i,1}.$$

Transfer is thermodynamically favored toward the region of lower chemical potential.

For an ideal dilute solute,

$$\mu_{i,2}-\mu_{i,1}=RT\ln!\left(\frac{c_2}{c_1}\right).$$

Example

If $c_1=0.10,\mathrm M$ and $c_2=0.010,\mathrm M$, then

$$\ln\left(\frac{c_2}{c_1}\right)=\ln(0.10)<0,$$

so

$$\mu_{i,2}-\mu_{i,1}<0.$$

Transfer from region 1 to region 2 lowers Gibbs free energy. This is the thermodynamic origin of the familiar tendency for diffusion from higher toward lower concentration in an ideal system.

Equilibrium means equal chemical potential

If a species can exchange freely between two regions, equilibrium requires

$$\boxed{\mu_{i,1}=\mu_{i,2}}.$$

Equal chemical potential does not always mean equal concentration. Pressure, nonideal interactions, phase identity or electrical potential can also contribute to the total free-energy balance.

Chemical potential therefore provides the common thermodynamic language behind diffusion, phase equilibrium, osmosis, chemical equilibrium and transport of ions across membranes.