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For ideal independent mixing on crystallographic sublattices, an endmember’s thermodynamic activity is the product of its required site fractions, with multiplicities. If one component occupies $q$ equivalent mixing positions per formula unit and has site fraction $X$, then $a=X^q$, not generally $X$.

For a binary example, the mixing Gibbs free energy per mole of formula units is $qRT[X\ln X+(1-X)\ln (1-X)]$. Differentiating the total free energy with respect to the amount of the all-component endmember gives a mixing chemical-potential term $qRT\ln X$. Comparison with $RT\ln a$ proves $a=X^q$. This derivation assumes ideal mixing and equal site fractions; nonideal interactions and preferential site occupancy require a different model. Mg endmembers of biotite and garnet commonly use $X_{\mathrm{Mg}}^3$ in this simplified three-site approximation.

A specified limiting composition used to describe a solid solution in a mineral. Its thermodynamic activity depends on the chosen standard state and activity model. For example, pyrope is a magnesian garnet endmember, while phlogopite is a magnesian biotite endmember.

An activity model is required in addition to the two compositions. Use ideal multisite mineral activity: independent ideal Fe–Mg mixing on the three relevant sites per formula unit givesThe garnet/biotite activity ratio is $0.008/0.512=1/64$. All other solids are pure, so the dimensionless reaction quotient isWith the same pressure-corrected numerator as in part b,Diluting the magnesian product endmember more strongly than the reactant favors the forward dehydration and lowers its equilibrium temperature. The powers of three count Fe–Mg mixing sites, not three formula units of either mineral. Nonideal site interactions or site partitioning would change these activities. If ideal mixing of whole formula units rather than sites were intended, activities would instead be $0.8$ and $0.2$; composition alone does not select that different model. The calculation explicitly adopts the conventional ideal-site interpretation.

The compositions alone do not specify a unique thermodynamic activity model. Adopt the usual ideal, independent mixing on crystallographic sites approximation: three Mg–Fe positions per formula unit, with the other sites fixed at their stated endmember compositions. Then the mineral endmember activities areThe exponent counts the three mixing cations; treating each whole mineral formula as a single mixing site would be a different model. In particular, the magnesian biotite and garnet components here are phlogopite and pyrope.

For the written reaction, the reaction quotient iswith the other solid activities equal to one. The Gibbs energy of reaction balance therefore givesThe smaller product-endmember activity lowers the reaction chemical potential and lowers the equilibrium temperature in this model. Nonideal activity coefficients or unequal occupancies of crystallographic sites would require further information rather than a revised universal numerical answer.^[ref]

The magnesian biotite mineral endmember. Iron substitution on its magnesium-bearing positions produces a solid solution requiring an activity model when used in chemical equilibrium calculations.