Some Applications of Thermodynamics to Coexisting Minerals of Variable Composition. Examples: Orthopyroxene-Clinopyroxene and Orthopyroxene-Garnet

Consideration is given to the relationship between the chemical potential of a species and its concentration in solids of the type (A, B, C)M and (A, B)(C, D)qM, where A, B, C, D, and M are chemical species. On this basis the distribution coefficient is derived for different combinations of chemical mixtures. For coexisting phases α = (A, B)M, and β = (A, B)N, the distribution coefficient with reference to species A is {aastex} {amsbsy} {amsfonts} {amssymb} {bm} {mathrsfs} {pifont} {stmaryrd} {textcomp} {portland,xspace} {amsmath,amsxtra} {wasysym} {empty} {10}{9}{7}{6} {document}$$KD = {XαA(1-XβA)}{(1 - XαA)XβA} = {fαB fβA}{fαA fβB} {λBMB λANA}{λAMA λBNB},$${document} where X = A /(A + B), the terms in f denote activity coefficients, and the terms in λ denote absolute activities. For coexisting phases α = (A, B, C)M and β = (A, B,Ç)N the distribution coefficient with reference to A has a similar form and is now dependent on the B: C ratio in either phase. The pressure dependence and temperature dependence of the distribution coefficient can be expressed by the equations: {aastex} {amsbsy} {amsfonts} {amssymb} {bm} {mathrsfs} {pifont} {stmaryrd} {textcomp} {portland,xspace} {amsmath,amsxtra} {wasysym} {empty} {10}{9}{7}{6} {document}KD\,2= KD\,1\ exp[( - VαA + VαB + VβA - VβB)(P₂-P₁)RT],{document} and {aastex} {amsbsy} {amsfonts} {amssymb} {bm} {mathrsfs} {pifont} {stmaryrd} {textcomp} {portland,xspace} {amsmath,amsxtra} {wasysym} {empty} {10}{9}{7}{6} {document}KD\,2= KD\,1\ exp[(HαA - HαB - HβA + HβB) (1T₁ - 1T₂)R],{document} where P = pressure, T = absolute temperature, R is the gas constant, and the terms in V denote partial molar volumes and those in H, partial molar enthalpies. The coexistence of natural orthopyroxene and clinopyroxene is considered as an example of two phases of the type (A, B)M and (A, B)N. KD is evidently independent of X, indicating that both phases are ideal mixtures. In rocks of presumed metamorphic crystallization, {aastex} {amsbsy} {amsfonts} {amssymb} {bm} {mathrsfs} {pifont} {stmaryrd} {textcomp} {portland,xspace} {amsmath,amsxtra} {wasysym} {empty} {10}{9}{7}{6} {document}KD 0.54{document} and in rocks of presumed igneous crystallization, {aastex} {amsbsy} {amsfonts} {amssymb} {bm} {mathrsfs} {pifont} {stmaryrd} {textcomp} {portland,xspace} {amsmath,amsxtra} {wasysym} {empty} {10}{9}{7}{6} {document}KD 0.73{document}. The difference in KD is attributed to a difference in crystallization temperature. The coexistence of natural orthopyroxene and garnet is considered as an example of two phases of the type (A, B, C, D)M and (A, B, C, D)N. With reference to Mg, KD is correlative with the Mn-concentration in garnet. This relationship provides empirical confirmation of the chemical theory.