Mass balance calculations with end member compositional variability: applications to petrologic problems

Although end member compositional variability is quite common, most quantitative mass balance procedures cannot accommodate this variability in a systematic manner. By rearranging the traditional mass balance relations, a series of equations can be derived to account for such heterogeneity. While this approach can be applied ton-component systems, the results are difficult to represent graphically. Accordingly, the procedure is most useful for two- and three-component systems. For two components, the concentration of an element in a mixture is: and the isotopic signature is: where pm, hm and 1 refer to parental magma, hybrid magma and component 1, respectively, X is the proportion of parental magma, C is concentration, is isotopic ratio, and the concentration of the denominator isotope in the particular isotopic ratio of interest. These equations describe straight lines, termed isoproportional or IP lines, of fixed mixing proportion in the C_pm−C₁ concentration or planes. Only those IP lines intersecting the rectangle defined by observed end member compositions represent viable mixing proportions. For three-component systems, the mass balance equations are: and: where Y is now the proportion of component 1 and the proportion of component 2, Z, is simply 1 − X − Y. Using major, trace and rare earth element as well as isotopic data, a region of the X-Y plane representing possible mixing combinations can be defined. Due to the compositional variability of most magmatic end members, this new mass balance procedure should be applicable to a diverse range of petrologic problems. This procedure has been applied to three different petrologic processes: Aleutian parental magma genesis, assimilation/contamination, and crust-mantle differentiation.