Geochemistry. William M. White

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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.

Example 3.4 Calculating activities using the mixing-on-site model

      Sometimes it is desirable to calculate the activities of pure end-member components in solid solutions. Garnet has the general formula X₃Y₂Si₃O₁₂. Calculate the activity of pyrope, Mg₃Al₂Si₃O₁₂, in a garnet solid solution of composition:

      Answer: The chemical potential of pyrope in garnet contains mixing contributions from both Mg in the cubic site and Al in the octahedral site:

      The activity of pyrope is thus given by:

      In the example composition above, the activity of Mg is:

      and that of Al is:

      The activity of pyrope in the garnet composition above is 0.002 × 0.956 = 0.00191. There is, of course, no mixing contribution from the tetrahedral site because it is occupied only by Si in both the solution and the pure pyrope phase.

      3.8.2 Local charge balance model

Yet another model for the calculation of activities in ideal solid solutions is the local charge balance model. A common example is the substitution of Ca for Na in the plagioclase solid solution (NaAlSi₃O₈–CaAl₂Si₂O₈). To maintain charge balance, the substitution of Ca²⁺ for Na⁺ in the octahedral site requires substitution of Al³⁺ for Si⁴⁺ in the tetrahedral site. In this model, the activity of the end-member of phase component is equal to the mole fraction of the component (see Example 3.5).

Example 3.5 Activities using the local charge balance model

      Given the adjacent analysis of a plagioclase crystal, what are the activities of albite and anorthite in the solution?

      Plagioclase Analysis

Oxide	Wt. percent
SiO2	44.35
Al2O3	34.85
CaO	18.63
Na2O	0.79
K2O	0.05

      Answer: According to the local charge balance model, the activity of albite will be equal to the mole fraction of Na in the octahedral site. To calculate this, we first must convert the weight percent oxides to formula units of cation. The first step is to calculate the moles of cation from the oxide weight percentages. First, we can convert weight percent oxide to weight percent cation using the formula:

      Next, we calculate the moles of cation:

      Combining

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