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Abstract We provide a new algorithm for mass‐balance calculations in petrology and geochemistry based on the log‐ratio approach championed initially by John Aitchison (e.g., Aitchison, 1982,https://doi.org/10.1111/j.2517-6161.1982.tb01195.x; Aitchison, 1984,https://doi.org/10.1007/bf01029316) along with the underlying principles, mathematical frameworks, and data requirements. Log‐ratio Inversion of Mixed End‐members (LIME) is written in MATLAB and calculates phase proportions in an experiment or rock given a bulk composition, the composition of each phase, and the associated compositional uncertainties. An important advantage of LIME is that performing the mass‐balance calculation in inverse log‐ratio space constrains phase proportions to be between 0 and 100 wt.%. Further, the resulting LIME phase proportions provide realistic estimates of uncertainty regardless of data distribution. These two characteristics of LIME improve upon standard multiple linear regression techniques, which may yield negative values for phase proportions if non‐constrained or report oversimplified symmetric errors. Primary applications of LIME include estimating phase abundances, calculating melting and metamorphic reaction stoichiometries, and checking for open system behavior in phase equilibria experiments. The technique presented here covers whole‐rock analysis, mineralogy, and phase abundance, but could be extended to isotopic tracers, trace element modeling, and regolith component un‐mixing. We highlight the importance of uncertainty estimations for phase abundances to the fields of petrology and geochemistry by comparing our results from LIME to previously published mass‐balance calculations. Furthermore, we present case studies that demonstrate the role of mass‐balance calculations in determining magma crystallinity and defining melting reactions. 

Abstract Oxygen fugacity is an important but difficult parameter to constrain for primitive arc magmas. In this study, the partitioning behavior of Fe3+/Fe2+ between amphibole and glass synthesized in piston-cylinder and cold-seal apparatus experiments is developed as an oxybarometer, applicable to magmas ranging from basaltic to dacitic composition. The partitioning of Fe2+ is strongly dependent on melt polymerization; the relative compatibility of Fe2+ in amphibole decreases with increasing polymerization. The Fe2+/Mg distribution coefficient between amphibole and melt is a relatively constant value across all compositions and is, on average, 0.27. The amphibole oxybarometer is applied to amphibole in mafic enclaves, cumulates, and basaltic tephra erupted from Shiveluch volcano in Kamchatka with measured Fe3+/FeTotal. An average Fe3+/Fe2+ amphibole-glass distribution coefficient for basalt is used to convert the Fe3+/FeTotal of amphibole in samples from Shiveluch to magmatic oxygen fugacity relative to NNO. The fO2 of primitive melts at the volcano is approximately NNO+2 and is faithfully recorded in amphibole from an amphibole-rich cumulate and the basaltic tephra. Apparently, higher fO2 recorded by amphibole in mafic enclaves likely results from partial dehydrogenation of amphibole during residence in a shallow andesite storage region. We identify three pulses of mafic magma recharge within two weeks of, a month before, and two to three months before the eruption and find that, at each of these times, the host andesite was recharged by at least two magmas at varying stages of differentiation. Application of the amphibole oxybarometer not only gives insight into magmatic fO2 but also potentially details of shallow magmatic processes.