Читать книгу Geochemistry - William M. White - Страница 121

Many crystalline solids can be successfully treated as ideal solutions. Where this is possible, the thermodynamic treatment and assessment of equilibrium are greatly simplified. A simple and often successful model that assumes ideality but takes account of the ordered nature of the crystalline state is the mixing-on-site model, which considers the substitution of species in sites individually. In this model, the activity of an individual species is calculated as:

(3.78)

where X is the mole fraction of the i^th atom and ν is the number of sites per formula unit on which mixing takes place. For example, ν = 2 in the Fe–Mg exchange in olivine, (Mg,Fe)₂SiO₄. One trick to simplifying this equation is to pick the formula unit such that ν = 1. For example, we would pick (Mg,Fe)Si_½O₂ as the formula unit for olivine. We must then consistently choose all other thermodynamic parameters to be half those of (Mg,Fe)₂SiO₄.

The entropy of mixing is given by:

(3.79)

where the subscript j refers to sites and the subscript i refers to components, and n is the number of sites per formula unit. The entropy of mixing is the same as the configurational entropy, residual entropy, or “third law entropy” (i.e., entropy when T = 0 K). For example, in clinopyroxene, there are two exchangeable sites, a sixfold-coordinated M1 site, (Mg, Fe²⁺, Fe³⁺, Al³⁺), and an eightfold-coordinated M2 site (Ca²⁺, Na⁺). Here j ranges from 1 to 2 (e.g., 1 = M1, 2 = M2), but n = 1 in both cases (because both sites accept only one atom). i must range over all present ions in each site, so in this example, i ranges from 1 to 4 (1 = Mg, 2 = Fe²⁺, etc.) when j = 1 and from 1 to 2 when j = 2. Since we have assumed an ideal solution, ΔH = 0 and ΔG_ideal = −TΔS. In other words, all we need is temperature and eqn. 3.79 to calculate the free energy of solution.

In the mixing-on-site model, the activity of a phase component in a solution, for example, pyrope in garnet, is the product of the activity of the individual species in each site in the phase:

(3.80)

where a_φ is the activity of phase component φ, i are the ion components of pure φ, and ν_i is the stoichiometric proportion of i in pure φ. For example, to calculate the activity of aegirine (NaFe³⁺Si₂O₆) in aegirine-augite ([Na,Ca][Fe³⁺,Fe²⁺,Mg]Si₂O₆), we would calculate the product: X_Na. Note that it would not be necessary to include the mole fractions of Si and O, since these are 1 (see Example 3.4).

A slight complication arises when more than one ion occupies a structural site in the pure phase. For example, suppose we wish to calculate the activity of phlogopite (KMg₃Si₃AlO₁₀(OH)₂) in a biotite of composition K_0.8Ca_0.2(Mg_0.17Fe_0.83)₃Si_2.8Al_1.2O₁₀(OH)₂. The tetrahedral site is occupied by Si and Al in the ratio of 3:1 in the pure phase end members. If we were to calculate the activity of phlogopite in pure phlogopite using eqn. 3.80, the activities in the tetrahedral site would contribute only in the pure phase. So we would obtain an activity of 0.1055 instead of 1 for phlogopite in pure phlogopite. Since the activity of a phase component must be one when it is pure, we need to normalize the result. Thus, we apply a correction by multiplying by the raw activity we obtain from 3.92 by 1/(0.1055) = 9.481, and thus obtain an activity of phlogopite of 1.