hydrogen, hydroxyl, and carbonate ions, so they need not be among our minimum set of components.
In igneous and metamorphic petrology, components are often the major oxides (though we may often choose to consider only a subset of these). On the other hand, if we were concerned with the isotopic equilibration of minerals with a hydrothermal fluid, 18O might be considered as a different component than 16O.
Perhaps the most straightforward way of determining the number of components is a graphical approach. If all phases can be represented on a one-dimensional diagram (that is, a straightline representing composition), we are dealing with a two-component system. For example, consider the hydration of Al2O3 (corundum) to form boehmite (AlO(OH)) or gibbsite Al(OH)3. Such a system would contain four phases (corundum, boehmite, gibbsite, water), but is nevertheless a two-component system because all phases may be represented in one dimension of composition space, as shown in Figure 3.1. Because there are two polymorphs of gibbsite, one of boehmite, and two other possible phases of water, there are nine possible phases in this two-component system. Clearly, a system may have many more phases than components.
Similarly, if a system may be represented in two dimensions, it is a three-component system. Figure 3.2 is a ternary diagram illustrating the system Al2O3–H2O–SiO2. The graphical representation approach reaches its practical limit in a four-component system because of the difficulty of representing more than three dimensions on paper. A four-component system is a quaternary one, and can be represented with a three-dimensional quaternary diagram.
Figure 3.1 Graphical representation of the system Al2O3−H2O.
Figure 3.2 Phase diagram for the system Al2O3–H2O–SiO2. The lines are called joins because they join phases. In addition to the end-members, or components, phases represented are g: gibbsite, by: bayerite, n: norstrandite (all polymorphs of Al(OH)3), d: diaspore, bo: boehmite (polymorphs of AlO(OH)), a: andalusite, k: kyanite, s: sillimanite (all polymorphs of Al2SiO5), ka: kaolinite, ha: halloysite, di: dickite, na: nacrite (all polymorphs of Al2Si2O5(OH)4), and p: pyrophyllite (Al2Si4O10(OH)2). There are also six polymorphs of quartz, q (coesite, stishovite, tridymite, cristobalite, α-quartz, and β-quartz).
It is important to understand that a component may or may not have chemical reality. For example in the exchange reaction:
we could alternatively define the exchange operator KNa−1 (where Na−1 is −1 mol of Na ion) and write the equation as:
In addition, we can also write the reaction:
Here we have four species and two reactions and thus a minimum of only two components. You can see that a component is merely an algebraic term.
There is generally some freedom in choosing components. For example, in the ternary (i.e., three-component) system SiO2−Mg2SiO4−MgCaSi2O6, we could choose our components to be quartz, diopside, and forsterite, or we could choose them to be SiO2, MgO, and CaO. Either way, we are dealing with a ternary system (which contains MgSiO3 as well as the three other phases).
3.2.1.4 Degrees of freedom
The number of degrees of freedom in a system is equal to the sum of the number of independent intensive variables (generally temperature and pressure) and independent concentrations (or activities or chemical potentials) of components in phases that must be fixed to define uniquely the state of the system. A system that has no degrees of freedom (i.e., is uniquely fixed) is said to be invariant, one that has one degree of freedom is univariant, and so on. Thus, in a univariant system, for example, we need specify the value of only one variable, for example, temperature or the concentration of one component in one phase, and the value of pressure and all other concentrations are then fixed and can be calculated (assuming the system is at equilibrium).
3.2.2 The Gibbs phase rule
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
