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

Abstract

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 to n-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: C pm = C hm X + ( 1 − 1 X ) C 1 and the isotopic signature is: ε pm C ¯ pm = ε hm C ¯ hm X + ( 1 − 1 X ) ε 1 C ¯ 1 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, ɛ is isotopic ratio, and C ¯ 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− C 1 concentration or ε pm C ¯ pm − ε 1 C ¯ 1 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: Y = ( C hm − C 2 ) − X ( C pm − C 2 ) C 1 − C 2 and: Y = ( C ¯ hm ε hm − C ¯ 2 ε 2 ) − X ( C ¯ pm ε pm − C ¯ 2 ε 2 ) C ¯ 1 ε 1 − C ¯ 2 ε 2 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.