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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

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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

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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][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

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Initially used to approximate chemical reaction rates,[7][8] models of isotope fractionation are used throughout the physical sciences. In chemistry, the Urey–Bigeleisen–Mayer equation has been used to predict equilibrium isotope effects and interpret the distributions of isotopes and isotopologues within systems, especially as deviations from their natural abundance.[35][36] The model is also used to explain isotopic shifts in spectroscopy, such as those from nuclear field effects or mass independent effects.[1][22][35] In biochemistry, it is used to model enzymatic kinetic isotope effects.[37][38] Simulation testing in 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

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Notes

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  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

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  1. ^ a b c d e Liu, Q.; Tossell, J.A.; Liu, Y. (2010). "On the proper use of the Bigeleisen–Mayer equation and corrections to it in the calculation of isotopic fractionation equilibrium constants". Geochimica et Cosmochimica Acta. 74 (24): 6965–6983. Bibcode:2010GeCoA..74.6965L. doi:10.1016/j.gca.2010.09.014.
  2. ^ a b c d e Richet, P.; Bottinga, Y.; Javoy, M. (1977). "A Review of Hydrogen, Carbon, Nitrogen, Oxygen, Sulphur, and Chlorine Stable Isotope Fractionation Among Gaseous Molecules". Annual Review of Earth and Planetary Sciences. 5: 65–110. Bibcode:1977AREPS...5...65R. doi:10.1146/annurev.ea.05.050177.000433.
  3. ^ Young, E.D.; Manning, C.E.; Schauble, E.A.; et al. (2015). "High-temperature equilibrium isotope fractionation of non-traditional stable isotopes: Experiments, theory, and applications". Chemical Geology. 395: 176–195. Bibcode:2015ChGeo.395..176Y. doi:10.1016/j.chemgeo.2014.12.013.
  4. ^ Dauphas, N.; Schauble, E.A. (2016). "Mass Fractionation Laws, Mass-Independent Effects, and Isotopic Anomalies". Annual Review of Earth and Planetary Sciences. 44: 709–783. Bibcode:2016AREPS..44..709D. doi:10.1146/annurev-earth-060115-012157.
  5. ^ Blanchard, M.; Balan, E.; Schauble, E.A. (2017). "Equilibrium Fractionation of Non-traditional Isotopes: a Molecular Modeling Perspective" (PDF). Reviews in Mineralogy and Geochemistry. 82 (1): 27–63. Bibcode:2017RvMG...82...27B. doi:10.2138/rmg.2017.82.2. S2CID 100190768.
  6. ^ a b Li, L.; He, Y.; et al. (2021). "Nitrogen isotope fractionations among gaseous and aqueous NH4+, NH3, N2, and metal-ammine complexes: Theoretical calculations and applications". Geochimica et Cosmochimica Acta. 295: 80–97. Bibcode:2021GeCoA.295...80L. doi:10.1016/j.gca.2020.12.010. S2CID 233921905.
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  8. ^ a b c d e f Bigeleisen, J.; Mayer, M.G. (1947). "Calculation of Equilibrium Constants for Isotopic Exchange Reactions". The Journal of Chemical Physics. 15 (5): 261–267. Bibcode:1947JChPh..15..261B. doi:10.1063/1.1746492. hdl:2027/mdp.39015074123996.
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  14. ^ a b Webb, M.A.; Miller, T.F. III (2013). "Position-Specific and Clumped Stable Isotope Studies: Comparison of the Urey and Path-Integral Approaches for Carbon Dioxide, Nitrous Oxide, Methane, and Propane". J. Phys. Chem. A. 118 (2): 467–474. doi:10.1021/jp411134v. PMID 24372450.
  15. ^ a b Liu, Q.; Yin, X.; Zhang, Y.; et al. (2021). "Theoretical calculation of position-specific carbon and hydrogen isotope equilibriums in butane isomers". Chemical Geology. 561: 120031. Bibcode:2021ChGeo.56120031L. doi:10.1016/j.chemgeo.2020.120031. S2CID 230547059.
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  23. ^ Bigeleisen, J. (1998). "Second-order correction to the Bigeleisen–Mayer equation due to the nuclear field shift". PNAS. 95 (9): 4808–4809. Bibcode:1998PNAS...95.4808B. doi:10.1073/pnas.95.9.4808. PMC 20168. PMID 9560183.
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  34. ^ Hutchison, C.A.; Stewart, D.W.; Urey, H.C. (1940). "The Concentration of C13". J. Chem. Phys. 8 (7): 532–537. Bibcode:1940JChPh...8..532H. doi:10.1063/1.1750707.
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  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. ^ Walters, W.W.; Simonini, D.S.; Michalski, G. (2016). "Nitrogen isotope exchange between NO and NO2 and its implications for δ15N variations in tropospheric NOx and atmospheric nitrate". Geophysical Research Letters. 43 (1): 440–448. Bibcode:2016GeoRL..43..440W. doi:10.1002/2015GL066438. S2CID 55819382.
  44. ^ Balan, E.; Noireaux, J.; Mavromatis, V.; et al. (2018). "Theoretical isotopic fractionation between structural boron in carbonates and aqueous boric acid and borate ion". Geochimica et Cosmochimica Acta. 222: 117–129. Bibcode:2018GeCoA.222..117B. doi:10.1016/j.gca.2017.10.017.
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