Urey–Bigeleisen–Mayer equation

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 ${\displaystyle A+B^{*}=A^{*}+B}$ , such that ${\displaystyle ^{*}}$  designates a molecule containing an isotope of interest, the equation can be expressed by relating the equilibrium constant, ${\displaystyle K_{eq}}$ , 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: ${\displaystyle K_{eq}={\frac {[A^{*}][B]}{[A][B^{*}]}}}$  where ${\displaystyle [A]=\prod ^{n}Q_{n,A}}$  and ${\displaystyle Q_{n}}$  is each respective partition function of molecule or atom ${\displaystyle A}$ .^[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 ${\displaystyle {\frac {s}{s'}}f}$  or ${\displaystyle ({\frac {Q^{*}}{Q}})_{r}}$ .^[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 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

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

• Timeline of the Manhattan Project
• Isotope-ratio mass spectrometry
• Hydrogen isotope biogeochemistry

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. 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. 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. 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.
7. Urey, H.C. (1947). "The Thermodynamic Properties of Isotopic Substances". Journal of the Chemical Society: 562–581. doi:10.1039/JR9470000562. PMID 20249764.
8. 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.
9. Iron, M.A.; Gropp, J. (2019). "Cost-effective density functional theory (DFT) calculations of equilibrium isotopic fractionation in large organic molecules". Physical Chemistry Chemical Physics. 21 (32): 17555–17570. Bibcode:2019PCCP...2117555I. doi:10.1039/C9CP02975C. PMID 31342034. S2CID 198491262.
10. Urey, H.C.; Greiff, L.J. (1935). "Isotopic Exchange Equilibria". J. Am. Chem. Soc. 57 (2): 321–327. doi:10.1021/ja01305a026.
11. Bigeleisen, J. (1975). "Quantum Mechanical Foundations of Isotope Chemistry". In Rock, P.A. (ed.). Isotopes and Chemical Principles. ACS Symposium Series. Vol. 11. pp. 1–28. doi:10.1021/bk-1975-0011.ch001. ISBN 9780841202252.
12. He, Y. (2018). "Equilibrium intramolecular isotope distribution in large organic molecules". High-dimensional isotope relationships (PhD thesis). Louisiana State University. pp. 48–66.
13. Li, X.; Liu, Y. (2011). "Equilibrium Se isotope fractionation parameters: A first-principles study". Earth and Planetary Science Letters. 304 (1): 113–120. Bibcode:2011E&PSL.304..113L. doi:10.1016/j.epsl.2011.01.022.
14. 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. 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.
16. Ishida, T.; Spindel, W.; Bigeleisen, J. (1969). "Theoretical Analysis of Chemical Isotope Fractionation by Orthogonal Polynomial Methods". Isotope Effects in Chemical Processes. Advances in Chemistry. Vol. 89. pp. 192–247. doi:10.1021/ba-1969-0089.ch011. ISBN 9780841200906.
17. Rosenbaum, J.M. (1997). "Gaseous, liquid, and supercritical fluid H₂O and CO₂: Oxygen isotope fractionation behavior". Geochimica et Cosmochimica Acta. 61 (23): 4993–5003. Bibcode:1997GeCoA..61.4993R. doi:10.1016/S0016-7037(97)00362-1.
18. O'Neil, J.R. (1986). "Theoretical and experimental aspects of isotopic fractionation". Stable Isotopes in High Temperature Geological Processes. Reviews in Mineralogy & Geochemistry. Vol. 16. De Gruyter. doi:10.1515/9781501508936-006.
19. Yang, J. (2018). "Mass-Dependent Fractionation from Urey to Bigeleisen" (PDF). Department of Earth, Atmospheric and Planetary Sciences. Massachusetts Institute of Technology. (PDF) from the original on 26 December 2022.
20. Bigeleisen, J.; Lee, M.W.; Mandel, F. (1973). "Equilibrium Isotope Effects". Annual Review of Physical Chemistry. 24: 407–440. Bibcode:1973ARPC...24..407B. doi:10.1146/annurev.pc.24.100173.002203.
21. Polyakov, V.B.; Kharlashina, N.N. (1994). "Effect of pressure on equilibrium isotopic fractionation". Geochimica et Cosmochimica Acta. 58 (21): 4739–4750. Bibcode:1994GeCoA..58.4739P. doi:10.1016/0016-7037(94)90204-6.
22. Bigeleisen, J. (1996). "Nuclear Size and Shape Effects in Chemical Reactions. Isotope Chemistry of the Heavy Elements". J. Am. Chem. Soc. 118 (15): 3676–3680. doi:10.1021/ja954076k.
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.
24. Prokhorov, I.; Kluge, T.; Janssen, C. (2019). "Optical clumped isotope thermometry of carbon dioxide". Scientific Reports. 9 (4765): 4765. Bibcode:2019NatSR...9.4765P. doi:10.1038/s41598-019-40750-z. PMC 6423234. PMID 30886173.
25. Webb, M.A.; Wang, W.; Braams, B.J.; et al. (2017). "Equilibrium clumped-isotope effects in doubly substituted isotopologues of ethane" (PDF). Geochimica et Cosmochimica Acta. 197: 14–26. Bibcode:2017GeCoA.197...14W. doi:10.1016/j.gca.2016.10.001.
26. "Availability of Radioactive Isotopes". Science. Manhattan Project. 103 (2685): 697–705. 14 June 1946. Bibcode:1946Sci...103..697.. doi:10.1126/science.103.2685.697. PMID 17808051.
27. Urey, H.C.; Brickwedde, F.G.; Murphy, G.M. (1932). "A Hydrogen Isotope of Mass 2". Phys. Rev. 39 (1): 164–165. Bibcode:1932PhRv...39..164U. doi:10.1103/PhysRev.39.164.
28. "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. Waldmann, L. (1943). "Zur Theorie der Isotopentrennung durch Austauschreaktionen" [On the theory of isotope separation by exchange reactions]. Naturwissenschaften (in German). 31 (16–18): 205–206. Bibcode:1943NW.....31..205W. doi:10.1007/BF01481918. S2CID 20090039.
33. Thode, H.G.; Urey, H.C. (1939). "The Further Concentration of N¹⁵". J. Chem. Phys. 7 (1): 34–39. Bibcode:1939JChPh...7...34T. doi:10.1063/1.1750320.
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.
35. Ishida, T. (2002). "Isotope Effect and Isotope Separation: A Chemist's View". Journal of Nuclear Science and Technology. 39 (4): 407–412. Bibcode:2002JNST...39..407I. doi:10.1080/18811248.2002.9715214. S2CID 95785450.
36. Saunders, M.; Cline, G.W.; Wolfsberg, M. (1989). "Calculation of Equilibrium Isotope Effects in a Conformationally Mobile Carbocation". Zeitschrift für Naturforschung A. 44 (5): 480–484. Bibcode:1989ZNatA..44..480S. doi:10.1515/zna-1989-0518. S2CID 95319151.
37. Moiseyev, N.; Rucker, J.; Glickman, M.H. (1997). "Reduction of Ferric Iron Could Drive Hydrogen Tunneling in Lipoxygenase Catalysis: Implications for Enzymatic and Chemical Mechanisms". J. Am. Chem. Soc. 119 (17): 3853–3860. doi:10.1021/ja9632825.
38. Gropp, J.; Iron, M.A.; Halevy, I. (2021). "Theoretical estimates of equilibrium carbon and hydrogen isotope effects in microbial methane production and anaerobic oxidation of methane" (PDF). Geochimica et Cosmochimica Acta. 295: 237–264. Bibcode:2021GeCoA.295..237G. doi:10.1016/j.gca.2020.10.018.
39. Wong, K.Y.; Xu, Y.; Xu, L. (2015). "Review of computer simulations of isotope effects on biochemical reactions: From the Bigeleisen equation to Feynman's path integral". Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics. 1854 (11): 1782–1794. doi:10.1016/j.bbapap.2015.04.021. PMID 25936775.
40. Giese, T.J.; Zeng, J.; Ekesan, S.; York, D.M. (2022). "Combined QM/MM, Machine Learning Path Integral Approach to Compute Free Energy Profiles and Kinetic Isotope Effects in RNA Cleavage Reactions" (PDF). J. Chem. Theory Comput. 18 (7): 4304–4317. doi:10.1021/acs.jctc.2c00151. PMC 9283286. PMID 35709391.
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. Walters, W.W.; Simonini, D.S.; Michalski, G. (2016). "Nitrogen isotope exchange between NO and NO₂ and its implications for δ¹⁵N variations in tropospheric NO_x 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.

External links

• Criss, R.E. (1991). "Temperature dependence of isotopic fractionation factors" (PDF). In Taylor, H.P.; O'Neil, J.R.; Kaplan, I.R. (eds.). Stable Isotope Geochemistry: A Tribute to Samuel Epstein. The Geochemical Society. ISBN 0-941809-02-1.