Urey–Bigeleisen–Mayer equation

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In

stable isotope geochemistry, the Urey–Bigeleisen–Mayer equation, also known as the Bigeleisen–Mayer equation or the Urey model,[1] is a model describing the approximate equilibrium isotope fractionation in an isotope exchange reaction.[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.[2][7][8]

Description

Since its original descriptions, the Urey–Bigeleisen–Mayer equation has taken many forms. Given an isotopic exchange reaction , such that designates a molecule containing an isotope of interest, the equation can be expressed by relating the equilibrium constant, , to the product of partition function ratios, namely the translational, rotational, vibrational, and sometimes electronic partition functions.[10][11][12] Thus the equation can be written as: where and is each respective partition function of molecule or atom .[12][13] It is typical to approximate the rotational partition function ratio as quantized rotational energies in a rigid rotor system.[11][14] The Urey model also treats molecular vibrations as simplified harmonic oscillators and follows the Born–Oppenheimer approximation.[11][14][15]

Isotope partitioning behavior is often reported as a reduced partition function ratio, a simplified form of the Bigeleisen–Mayer equation notated mathematically as or .[16][17] The reduced partition function ratio can be derived from power series expansion of the function and allows the partition functions to be expressed in terms of frequency.[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.[15] Some common, significant modifications to the equation include accounting for pressure effects,[21] nuclear geometry,[22] and corrections for anharmonicity and quantum mechanical effects.[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]

History of discovery

One aim of the

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][7][11] Bigeleisen and Mayer had been working on similar work since at least 1944 and, in 1947, published their model independently from Urey.[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]

Applications

Initially used to approximate chemical

computational systems biology often uses the Bigeleisen–Mayer model as a baseline in the development of more complex models of biological systems.[39][40] Isotope fractionation modeling is a critical component of isotope geochemistry and can be used to reconstruct past Earth environments as well as examine surface processes.[41][42][43][44]

See also

Notes

  1. ^ Bigeleisen & Mayer (1947) contains the addendum:

    After this paper had been completed, Professor W.F. Libby kindly called a paper by L. Waldmann[32] to our attention. 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. These are the exchange between NH3 and NH4+ and HCN and CN- studies by Urey[33][34] and co-workers.

References

  1. ^
    doi:10.1016/j.gca.2010.09.014
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  2. ^
    doi:10.1146/annurev.ea.05.050177.000433
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  3. doi:10.1016/j.chemgeo.2014.12.013
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  4. doi:10.1146/annurev-earth-060115-012157
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  5. S2CID 100190768
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  6. ^
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  7. ^
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  8. ^
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  9. S2CID 198491262
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  10. doi:10.1021/ja01305a026
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  11. ^
    ISBN 9780841202252
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  12. ^ a b He, Y. (2018). "Equilibrium intramolecular isotope distribution in large organic molecules". High-dimensional isotope relationships (PhD thesis). Louisiana State University. pp. 48–66.
  13. doi:10.1016/j.epsl.2011.01.022
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  14. ^
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  15. ^
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  16. ^
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  17. doi:10.1016/S0016-7037(97)00362-1
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  18. doi:10.1515/9781501508936-006
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  19. ^ Yang, J. (2018). "Mass-Dependent Fractionation from Urey to Bigeleisen" (PDF). Department of Earth, Atmospheric and Planetary Sciences. Massachusetts Institute of Technology. Archived (PDF) from the original on 26 December 2022.
  20. doi:10.1146/annurev.pc.24.100173.002203
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  21. doi:10.1016/0016-7037(94)90204-6
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  22. ^
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  23. PMID 9560183
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  24. PMID 30886173
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  25. doi:10.1016/j.gca.2016.10.001
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  26. PMID 17808051. {{cite journal}}: Unknown parameter |agency= ignored (help
    )
  27. doi:10.1103/PhysRev.39.164
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  28. ^ a b "Guide to the Harold C. Urey Papers 1932-1953". University of Chicago Library. 2007. Retrieved 25 December 2022.
  29. ^ Hewlett, R.G.; Anderson, O.E. (1962). "In the beginning". The New World, 1939/1946 (PDF). A History of the United States Atomic Energy Commission. Vol. I. The Pennsylvania State University Press. pp. 9–52.
  30. ^ "Jacob Bigeleisen: 1919–2010" (PDF). National Academy of Sciences. Biographical Memoirs. 2014.
  31. ^ "Maria Goeppert Mayer - Biographical". The Nobel Prize.
  32. S2CID 20090039
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  33. doi:10.1063/1.1750320
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  34. doi:10.1063/1.1750707
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  35. ^
    S2CID 95785450
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  36. S2CID 95319151
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  37. doi:10.1021/ja9632825
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  38. doi:10.1016/j.gca.2020.10.018
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  39. PMID 25936775
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  40. PMID 35709391
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  41. ^ Kendall, C.; Caldwell, E.A. (1998). "Chapter 2: Fundamentals of Isotope Geochemistry". In Kendall, C.; McDonnell, J.J. (eds.). Isotope Tracers in Catchment Hydrology. Elsevier Science B.V.
  42. ^ Otake, Tsubasa (2008). Understanding Redox Processes in Surface Environments from Iron Oxide Transformations and Multiple Sulfur Isotope Fractionations (PhD thesis). The Pennsylvania State University.
  43. S2CID 55819382
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  44. doi:10.1016/j.gca.2017.10.017
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External links
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