Exchange Equilibrium and Inter-Crystalline Fractionation

The distribution of an element between coexisting phases depends on the P-T condition to which they have equilibrated and, in general, also on the composition of the mineral phases. For thermodynamic analysis, it is convenient to treat these distributions in terms of what is known as exchange equilibrium, the basic concept of which was first introduced by Ramberg and DeVore (1951) in the geologic literature. A general discussion of the theory was presented by Kretz (1961), who also made the first systematic study (Kretz 1959) of element distribution between coexisting minerals in natural rocks. For the purpose of developing the theory, let us consider the fractionation of Fe2+ and Mg between coexisting garnet and biotite, which can be expressed through the following exchange reaction: a$$ \mathop{{1/3M{g_3}A{l_2}}}\limits^{{Gt}} S{i_3}{O_{{12}}} + \mathop{{1/3KF{e_3}Al}}\limits^{{Bt}} S{i_3}{O_{{10}}}{(OH)_2} \rightleftarrows \mathop{{1/3F{e_3}A{l_2}}}\limits^{{Gt}} S{i_3}{O_{{12}}} + \mathop{{1/3KM{g_3}Al}}\limits^{{Bt}} S{i_3}{O_{{10}}}{(OH)_2} $$ At equilibrium at P and T 6.1$$ {K_{{(a)}}}\left( {P,T} \right) = \exp \left( {\frac{{ - \Delta G*\left( {P,T} \right)}}{{RT}}} \right) = \frac{{{{\left( {a_{{Alm}}^{{Gt}}} \right)}^{{1/3}}}{{\left( {a_{{Ph}}^{{Bt}}} \right)}^{{1/3}}}}}{{{{\left( {a_{{Py}}^{{Gt}}} \right)}^{{1/3}}}{{\left( {a_{{Ann}}^{{Bt}}} \right)}^{{1/3}}}}} $$ where Alm (almandine) ≡ Fe₃Al₂Si₃O₁₂, Py (pyrope) ≡ Mg₃Al₂Si₃O₁₂, Ph (phlogopite) ≡ KMg₃AlSi₃O₁₀(OH)₂, and Ann (annite) ≡ KF₃AlSi₃O₁₀(OH)₂.