Equilibrium thermodynamics of multiply substituted isotopologues of molecular gases

Isotopologues of molecular gases containing more than one rare isotope (multiply substituted isotopologues) can be analyzed with high precision (1σ <0.1‰), despite their low natural abundances (∼ ppm to ppt in air), and can constrain geochemical budgets of natural systems. We derive a method for calculating abundances of all such species in a thermodynamically equilibrated population of isotopologues, and present results of these calculations for O₂, CO, N₂, NO, CO₂, and N₂O between 1000 and 193 to 77 K. In most cases, multiply substituted isotopologues are predicted to be enriched relative to stochastic (random) distributions by ca. 1 to 2‰ at earth-surface temperatures. This deviation, defined as Δ_i for isotopologue i, generally increases linearly with 1/T at temperatures ≤ 500 K. An exception is N₂O, which shows complex temperature dependences and 10’s of per-mill enrichments or depletions of abundances for some isotopologues. These calculations provide a basis for discriminating between fractionations controlled by equilibrium thermodynamics and other sorts of isotopic fractionations in the budgets of atmospheric gases. Moreover, because abundances of multiply substituted isotopologues in thermodynamically equilibrated populations of molecules vary systematically with temperature, they can be used as geothermometers. Such thermometers are unusual in that they involve homogeneous rather than heterogeneous equilibria (e.g., isotopic distribution in gaseous CO₂ alone, rather than difference in isotopic composition between CO₂ and coexisting water). Also, multiple independent thermometers exist for all molecules having more than one multiply substituted isotopologue (e.g., thermometers based on abundances of ¹⁸O¹³C¹⁶O and ¹⁸O¹²C¹⁸O are independent); thus, temperatures estimated by this method can be tested for internal consistency.