Читать книгу Geochemistry - William M. White - Страница 97

We now want to consider two important properties of the chemical potential. To illustrate these properties, consider a simple two-phase system in which an infinitesimal amount of component i is transferred from phase β to phase α, under conditions where T, P, and the amount of other components is held constant in each phase. One example of such a reaction would be the transfer of Pb from a hydrothermal solution to a sulfide mineral phase. The chemical potential expresses the change in Gibbs free energy under these conditions:

(3.15)

since we are holding everything else constant, atoms gained by α must be lost by β, so and:

(3.16)

At equilibrium, dG = 0, and therefore

(3.17)

Equation 3.17 reflects a very general and very important relationship, namely:

In a system at equilibrium, the chemical potential of every component in a phase is equal to the chemical potential of that component in every other phase in which that component is present.

Equilibrium is the state toward which systems will naturally transform. The Gibbs free energy is the chemical energy available to fuel these transformations. We can regard differences in chemical potentials as the forces driving transfer of components between phases. In this sense, the chemical potential is similar to other forms of potential energy, such as gravitational or electromagnetic. Physical systems spontaneously transform so as to minimize potential energy. Thus, for example, water on the surface of the Earth will move to a point where its gravitational potential energy is minimized – downhill. Just as gravitational potential energy drives this motion, the chemical potential drives chemical reactions, and just as water will come to rest when gravitational energy is minimized, chemical reactions will cease when chemical potential is minimized. So in our example above, the spontaneous transfer of Pb between a hydrothermal solution and a sulfide phase will occur until the chemical potentials of Pb in the solution and in the sulfide are equal. At this point, there is no further energy available to drive the transfer.

We defined the chemical potential in terms of the Gibbs free energy. However, in his original work, Gibbs based the chemical potential on the internal energy of the system. As it turns out, however, the quantities are the same:

(3.18)

It can be further shown (but we won't) that: