Ideal multisite mineral activity ($a=X^q$)

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.