Advertisement

Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2418–2453 | Cite as

Binary non-Markovian theory of bulk associative–dissociative reaction \( A + A \leftrightarrow C \). I: many-particle aspects of the theory

  • Alexey A. Kipriyanov
  • Alexander A. Kipriyanov
  • Alexander B. Doktorov
Original Paper
  • 30 Downloads

Abstract

It is shown that reversible reaction may be considered as irreversible one on restricted time interval. However, the time range of the applicability of the law of mass action for the case of many-particle derivation of binary kinetic equations which is valid at small density parameters also has time restrictions. If the range of binary description is narrower than that of irreversible approximation of the kinetics, then this approximation is valid over the entire time range of the applicability of the law of mass action. The necessary conditions are found for which hierarchies for reduced distribution functions (RDFs) for the reaction at hand are constructed that satisfy the required hierarchy condition, and have all the properties inherent in BBGKY hierarchies of non-equilibrium statistical mechanics. This allows one to use the advanced methods of non-equilibrium statistical mechanics in many-particle derivation of kinetic equations of chemical reactions in solutions. It is shown that such initial conditions are possible at which for complete evolution no closed kinetic equation exists. Development of RDFs for reaction systems opens the way to the employment of advanced methods of non-equilibrium statistical mechanics in many-particle derivation of kinetic equations of chemical reactions in solutions.

Keywords

Derivation of kinetic equations Binary theory Reduced distribution functions Liouville equation Fock space BBGKY Microscopic point density Diffusion Chemical kinetics Hierarchy Thermodynamic limit 

Notes

Acknowledgements

The authors are grateful to the Federal Agency of Scientific Organizations (FASO) for financial support (Projects No. 44.1.5)

References

  1. 1.
    A.B. Doktorov, in Recent Research Development in Chemical Physics, Chap. 6, vol. 6, ed. by S.G. Pandalai (Transworld Research Network, Kerala, 2012), pp. 135–192Google Scholar
  2. 2.
    T.R. Waite, Phys. Rev. 107, 463 (1957)CrossRefGoogle Scholar
  3. 3.
    T.R. Waite, J. Chem. Phys. 28, 103 (1958)CrossRefGoogle Scholar
  4. 4.
    R. Kapral, Adv. Chem. Phys. 48, 71 (1981)Google Scholar
  5. 5.
    S.A. Rice, Diffusion-Limited Reactions, Comprehensive Chemical Kinetics, vol. 25 (Elsevier, New York, 1985)Google Scholar
  6. 6.
    M. Yang, S. Lee, K.J. Shin, J. Chem. Phys. 108, 9069 (1998)CrossRefGoogle Scholar
  7. 7.
    M. Litniewski, J. Chem. Phys. 123, 124506 (2005)CrossRefGoogle Scholar
  8. 8.
    M. Litniewski, J. Chem. Phys. 124, 114501 (2006)CrossRefGoogle Scholar
  9. 9.
    J. Kim, S. Lee, J. Chem. Phys. 131, 014503 (2009)CrossRefGoogle Scholar
  10. 10.
    M. von Smoluchowski, J. Phys. Chem. 92, 129 (1916)Google Scholar
  11. 11.
    F.C. Collins, G.E. Kimbal, J. Colloid Interface Sci. 4, 425 (1949)CrossRefGoogle Scholar
  12. 12.
    H. Eyring, S.H. Lin, S.M. Lin, Basic Chemical Kinetics (Wiley, New York, 1980)Google Scholar
  13. 13.
    A.B. Doktorov, Phys. A 90, 109 (1978)CrossRefGoogle Scholar
  14. 14.
    U. Fano, Phys. Rev. 131, 259 (1963)CrossRefGoogle Scholar
  15. 15.
    T. Watanabe, Phys. Rev. A 138, 1573 (1965)CrossRefGoogle Scholar
  16. 16.
    I.Z. Steinberg, E. Katchalski, J. Chem. Phys. 48, 2404 (1968)CrossRefGoogle Scholar
  17. 17.
    M. Tachiya, Radiat. Phys. Chem. 21, 167 (1983)Google Scholar
  18. 18.
    A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Chem. Phys. 187, 251 (1994)CrossRefGoogle Scholar
  19. 19.
    A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Phys. A 205, 585 (1994)CrossRefGoogle Scholar
  20. 20.
    I.V. Gopich, A.A. Kipriyanov, A.B. Doktorov, Khim.Fizika 14, 120 (1995) (in Russian); Chem. Phys. Reports, 14, 1443 (1995)Google Scholar
  21. 21.
    V.P. Sakun, Phys. A 80, 128 (1975)CrossRefGoogle Scholar
  22. 22.
    R. Zwanzig, J. Chem. Phys. 33, 1338 (1960)CrossRefGoogle Scholar
  23. 23.
    H. Mory, Prog. Theor. Phys. 33, 423 (1965)CrossRefGoogle Scholar
  24. 24.
    H. Mory, Prog. Theor. Phys. 34, 399 (1965)CrossRefGoogle Scholar
  25. 25.
    A.A. Kipriyanov, A.B. Doktorov, A.I. Burshtein, Chem. Phys. 76, 149 (1983)CrossRefGoogle Scholar
  26. 26.
    A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Chem. Phys. 187, 241 (1994)CrossRefGoogle Scholar
  27. 27.
    A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Phys. A 255, 347 (1998)CrossRefGoogle Scholar
  28. 28.
    R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics, vol. 1 (Wiley, New York, 1978)Google Scholar
  29. 29.
    R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics, vol. 2 (Wiley, New York, 1978)Google Scholar
  30. 30.
    S. Lee, M. Karplus, J. Chem. Phys. 86, 1883 (1987)CrossRefGoogle Scholar
  31. 31.
    S. Lee, J.J. Lee, K.J. Shin, Bull. Korean Chem. Soc. 15, 311 (1994)Google Scholar
  32. 32.
    V.N. Kuzovkov, E.A. Kotomin, J. Phys. C 13, 499 (1980)CrossRefGoogle Scholar
  33. 33.
    V.N. Kuzovkov, E.A. Kotomin, Phys. State Sol. b 108, 37 (1981)CrossRefGoogle Scholar
  34. 34.
    M. Doi, J. Phys. A 9, 1465 (1976)CrossRefGoogle Scholar
  35. 35.
    A.A. Kipriyanov, O.A. Igoshin, A.B. Doktorov, Phys. A 268, 567 (1999)CrossRefGoogle Scholar
  36. 36.
    A. Szabo, J. Chem. Phys. 95, 2481 (1991)CrossRefGoogle Scholar
  37. 37.
    W. Naumann, A. Szabo, J. Chem. Phys. 107, 402 (1997)CrossRefGoogle Scholar
  38. 38.
    W. Naumann, J. Chem. Phys. 98, 2353 (1993)CrossRefGoogle Scholar
  39. 39.
    A. Molski, J. Keizer, J. Chem. Phys. 96, 1391 (1992)CrossRefGoogle Scholar
  40. 40.
    A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Chem. Phys. 191, 101 (1995)CrossRefGoogle Scholar
  41. 41.
    I.V. Gopich, A.B. Doktorov, J. Chem. Phys. 105, 2320 (1996)CrossRefGoogle Scholar
  42. 42.
    A.B. Doktorov, A.A. Kipriyanov, J. Phys. Condens. Matter. 19, 065136 (2007)CrossRefGoogle Scholar
  43. 43.
    A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (Chapman and Hall, Boca Raton, 2003)Google Scholar
  44. 44.
    V.S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1981). in Russian Google Scholar
  45. 45.
    A.A. Kipriyanov, O.A. Igoshin, A.B. Doktorov, Phys. A 275, 99 (2000)CrossRefGoogle Scholar
  46. 46.
    G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill Company, New York, 1968)Google Scholar
  47. 47.
    A.A. Kipriyanov, A.B. Doktorov, Phys. A 326, 105 (2003)CrossRefGoogle Scholar
  48. 48.
    N.N. Bogolubov, N.N. Bogolubov Jr., Introduction to Quantum Statistic Mechanics (World Scientic, Singapore, 1982)CrossRefGoogle Scholar
  49. 49.
    L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1 (Course of Theoretical Physics), vol. 5 (Pergamon Press Ltd., Oxford, 1980)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alexey A. Kipriyanov
    • 1
    • 2
  • Alexander A. Kipriyanov
    • 1
  • Alexander B. Doktorov
    • 1
    • 2
  1. 1.Voevodsky Institute of Chemical Kinetics and CombustionSB RASNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations