Isotopologues of molecular gases containing more than one rare isotope (multiply substituted isotopologues) can be analyzed with high precision (1sigma <0.1parts per thousand), despite their low natural abundances (similar to 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 O-2, CO, N-2, 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 2parts per thousand at earth-surface temperatures. This deviation, defined as Delta(i) for isotopologue i, generally increases linearly with 1/T at temperatures less than or equal to 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 geothermorneters. Such thermometers are unusual in that they involve homogeneous rather than heterogeneous equilibria (e.g., isotopic distribution in gaseous CO, alone. rather than difference in isotopic composition between CO, 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 (OCO)-O-18-C-13-O-16 and (OCO)-O-18-C-12-O-18 are independent); thus, temperatures estimated by this method can be tested for internal consistency. Copyright (C) 2004 Elsevier Ltd.