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Since μ is the partial molar Gibbs free passim energy, the Gibbs free energy of a system is the sum of the chemical potentials of each component:

(3.19)

The differential form of this equation (which we get simply by applying the chain rule) is:

(3.20)

Equating this with eqn. 3.14, we obtain:

(3.21)

Rearranging, we obtain the Gibbs–Duhem relation:

(3.22)

The Gibbs–Duhem equation describes the relationship between simultaneous changes in pressure, temperature, and composition in a single-phase system. In a closed system at equilibrium, net changes in chemical potential will occur only as a result of changes in temperature or pressure. At constant temperature and pressure, there can be no net change in chemical potential at equilibrium:

(3.23)

This equation further tells us that the chemical potentials do not vary independently but change in a related way. In a closed system, only one chemical potential can vary independently. For example, consider a two-component system. Then we have n₁dμ₁ + n₂dμ₂ = 0 and dμ₂ = −(n₁/n₂)dμ₁. If a given variation in composition produces a change in μ₁ then there is a concomitant change in μ₂.

For multiphase systems, we can write a version of the Gibbs–Duhem relation for each phase in the system. For such systems, the Gibbs–Duhem relation allows us to reduce the number of independently variable components in each phase by one. We will return to this point later in the chapter.

We can now state an additional property of chemical potential:

In spontaneous processes, components or species are distributed between phases so as to minimize the chemical potential of all components.

This allows us to make one more characterization of equilibrium: equilibrium is the point where the chemical potential of all components is minimized.