[2][3][4][5][6] While the equation itself can be written in numerous forms, it is generally presented as a ratio of partition functions of the isotopic molecules involved in a given reaction.
[7][8] The Urey–Bigeleisen–Mayer equation is widely applied in the fields of quantum chemistry and geochemistry and is often modified or paired with other quantum chemical modelling methods (such as density functional theory) to improve accuracy and precision and reduce the computational cost of calculations.
[1][6][9] The equation was first introduced by Harold Urey and, independently, by Jacob Bigeleisen and Maria Goeppert Mayer in 1947.
designates a molecule containing an isotope of interest, the equation can be expressed by relating the equilibrium constant,
[11][14] The Urey model also treats molecular vibrations as simplified harmonic oscillators and follows the Born–Oppenheimer approximation.
[16][18][19] It can be used to relate molecular vibrations and intermolecular forces to equilibrium isotope effects.
[20] As the model is an approximation, many applications append corrections for improved accuracy.
[1][2][23][24] For example, hydrogen isotope exchange reactions have been shown to disagree with the requisite assumptions for the model but correction techniques using path integral methods have been suggested.
[1][8][25] One aim of the Manhattan Project was increasing the availability of concentrated radioactive and stable isotopes, in particular 14C, 35S, 32P, and deuterium for heavy water.
[26] Harold Urey, Nobel laureate physical chemist known for his discovery of deuterium,[27] became its head of isotope separation research while a professor at Columbia University.
[28][29]: 45 In 1945, he joined The Institute for Nuclear Studies at the University of Chicago, where he continued to work with chemist Jacob Bigeleisen and physicist Maria Mayer, both also veterans of isotopic research in the Manhattan Project.
[11][28][30][31] In 1946, Urey delivered the Liversidge lecture at the then-Royal Institute of Chemistry, where he outlined his proposed model of stable isotope fractionation.
[2][8][11] Their calculations were mathematically equivalent to a 1943 derivation of the reduced partition function by German physicist Ludwig Waldmann.
[8][11][a] Initially used to approximate chemical reaction rates,[7][8] models of isotope fractionation are used throughout the physical sciences.
In this paper, Waldmann discusses briefly the fact that the chemical separation of isotopes is a quantum effect.
He gives formulae which are equivalent to our (11') and (11a) and discusses qualitatively their application to two acid base exchange equilibria.