Читать книгу Geochemistry - William M. White - Страница 90

A common problem in geochemistry is to know how a phase boundary varies in P–T space, for example, how a melting temperature will vary with pressure. At a phase boundary, two phases must be in equilibrium, so ΔG must be 0 for the reaction Phase 1 ⇌ Phase 2. The phase boundary therefore describes the condition:

Thus, the slope of a phase boundary on a temperature-pressure diagram is:

(3.3)

where ΔV_r and ΔS_r are the volume and entropy changes associated with the reaction. Equation 3.3 is known as the Clausius–Clapeyron equation, or simply the Clapeyron equation. Because ΔV_r and ΔS_r are functions of temperature and pressure, this is, of course, only an instantaneous slope. For many reactions, however, particularly those involving only solids, the temperature and pressure dependencies of ΔV_r and ΔS_r will be small and the Clapeyron slope will be relatively constant over a large T and P range (see Example 3.1).

Because ΔS = ΔH/T, the Clapeyron equation may be equivalently written as:

(3.4)

Slopes of phase boundaries in P–T space are generally positive, implying that the phases with the largest volumes also generally have the largest entropies (for reasons that become clear from a statistical mechanical treatment). This is particularly true of solid–liquid phase boundaries, although there is one very important exception: water. How do we determine the pressure and temperature dependence of ΔV_r and why is ΔV_r relatively T- and P-independent in solids?

We should emphasize that application of the Clapeyron equation is not limited to reactions between two phases in a one-component system but may be applied to any univariant reaction.