= Common-potential test for a three-mineral activity diagram
{title2=$\Phi_C=0,\quad\Phi_1=\ell_1(x)-y,\quad\Phi_2=\ell_2(x)-y$}
Suppose two dissolution reactions to a reference <mineral> $C$ have been normalized to the same amount of a conserved reference component. After division by the same positive energy factor, let their <Gibbs energies of reaction> be $y-\ell_1(x)$ and $y-\ell_2(x)$. Their relative thermodynamic potentials can then be written as displayed. The stable <mineral> has the smallest potential.
It follows that $C$ is stable below $y=\min(\ell_1,\ell_2)$, mineral 1 is stable where $\ell_1<\ell_2$ and $y>\ell_1$, and mineral 2 is stable where $\ell_2<\ell_1$ and $y>\ell_2$. The two nonreference minerals coexist stably where $\ell_1=\ell_2<y$. Their equality below that height is a <metastable> continuation. This comparison proves the stable lower envelope and selects which portions of the pairwise mass-action lines belong on the <mineral activity diagram>.
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