Quantification of Lambda (Λ) in multi-elemental compound-specific isotope analysis

In multi-elemental compound-specific isotope analysis the lambda (Λ) value expresses the isotope shift of one element versus the isotope shift of a second element. In dual-isotope plots, the slope of the regression lines typical reveals the footprint of the underlying isotope effects allowing to distinguish degradation pathways of an organic contaminant molecule in the environment. While different conventions and fitting procedures are used in the literature to determine Λ, it remains unclear how they affect the magnitude of Λ. Here we generate synthetic data for benzene δ²H and δ¹³C with two enrichment factors ε_H and ε_C using the Rayleigh equation to examine how different conventions and linear fitting procedures yield distinct Λ. Fitting an error-free data set in a graph plotting the δ²H versus δ¹³C overestimates Λ by 0.225%⋅ε_H/ε_C, meaning that if ε_H/ε_Cis larger than 22, Λ is overestimated by more than 5%. The correct fitting of Λ requires a natural logarithmic transformation of δ²H versus δ¹³C data. Using this transformation, the ordinary linear regression (OLR), the reduced major-axis (RMA) and the York methods find the correct Λ, even for large ε_H/ε_C. Fitting a dataset with synthetic data with typical random errors let to the same conclusion and positioned the suitability of each regression method. We conclude that fitting of non-transformed δ values should be discontinued. The validity of most previous Λ values is not compromised, although previously obtained Λ values for large ε_H/ε_C could be corrected using our error estimation to improve comparison.