Читать книгу Geochemistry - William M. White - Страница 89

The Gibbs‡ phase rule is a rule for determining the degrees of freedom, or variance, of a system at equilibrium. The rule is:

(3.2)

where ƒ is the degrees of freedom, c is the number of components, and φ is the number of phases. The mathematical analogy is that the degrees of freedom are equal to the number of variables minus the number of equations relating those variables. For example, in a system consisting of just H₂O, if two phases coexist, for example, water and steam, then the system is univariant. Three phases coexist at the triple point of water, so the system is said to be invariant, and T and P are uniquely fixed: there is only one temperature and one pressure at which the three phases of water can coexist (273.15 K and 0.006 MPa). If only one phase is present, for example just liquid water, then we need to specify two variables to describe completely the system. It does not matter which two we pick. We could specify molar volume and temperature and from that we could deduce pressure. Alternatively, we could specify pressure and temperature. There is only one possible value for the molar volume if temperature and pressure are fixed. It is important to remember this applies to intensive parameters. To know volume, an extensive parameter, we would have to fix one additional extensive variable (such as mass or number of moles). And again, we emphasize that all this applies only to systems at equilibrium.

Now consider the hydration of corundum to form gibbsite. There are three phases, but there need be only two components. If these three phases (water, corundum, gibbsite) are at equilibrium, we have only one degree of freedom (i.e., if we know the temperature at which these three phases are in equilibrium, the pressure is also fixed).

Rearranging eqn. 3.2, we also can determine the maximum number of phases that can coexist at equilibrium in any system. The degrees of freedom cannot be less than zero, so for an invariant, one-component system, a maximum of three phases can coexist at equilibrium. In a univariant one-component system, only two phases can coexist. Thus, sillimanite and kyanite can coexist over a range of temperatures, as can kyanite and andalusite, but the three phases of Al₂SiO₅ coexist only at one unique temperature and pressure.

Let's consider the example of the three-component system Al₂O₃–H₂O–SiO₂ in Figure 3.2. Although many phases are possible in this system, for any given composition of the system only three phases can coexist at equilibrium over a range of temperature and pressure. Four phases (e.g, a, k, s, and p) can coexist only along a one-dimensional line or curve in P–T space. Such points are called univariant lines (or curves). Five phases can coexist at invariant points at which both temperature and pressure are uniquely fixed. Turning this around, if we found a metamorphic rock whose composition fell within the Al₂O₃–H₂O–SiO₂ system, and if the rock contained five phases, it would be possible to determine uniquely the temperature and pressure at which the rock equilibrated.