Skip to main content
SpringerLink
Log in

Analytical solutions for the rate equations of irreversible two-step consecutive processes with second order later steps

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

Analytical solutions are presented for four consecutive two-step kinetic schemes, which involve a first reaction with zeroth, first, second or mixed second order dependence and a second step, which is second order with respect to the intermediate formed in the first step. Two of the analytical solutions found use elementary functions not very commonly encountered in chemistry, as the rate equations are shown to be related to the Legendre or modified Bessel differential equations. The solutions are analyzed not only as a function of time, but by plotting two concentrations as a function of each other as well. The dependence of the kinetic traces on the parameter values is also investigated. In all cases, two scaling parameters are identified. Three of the four cases are characterized by a single shape parameter, which is basically the ratio of the rate constant scaled with a suitable concentration unit if necessary. The mixed second order–second order scheme has an additional shape parameter, which is the ratio of the initial concentrations of the two reactants.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. J.H. Espenson, Chemical Kinetics and Reaction Mechanisms, 2nd edn. (McGraw-Hill, New York, 1995)

    Google Scholar 

  2. P. Érdi, J. Tóth, Mathematical Models of Chemical Reactions (Manchester University Press, Manchester, 1989)

    Google Scholar 

  3. P. Érdi, G. Lente, Stochastic Chemical Kinetics. Theory and (Mostly) Systems Biological Applications (Springer, Heidelberg, 2014)

    Google Scholar 

  4. C.W. Gear, Commun. ACM 14, 176–179 (1971)

    Article  Google Scholar 

  5. T.P.J. Knowles, C.A. Waudby, G.L. Devlin, S.I.A. Cohen, A. Aguzzi, M. Vendruscolo, E.M. Terentjev, M.E. Welland, C.M. Dobson, Science 326, 1533–1537 (2009)

    Article  CAS  Google Scholar 

  6. F. Garcia-Sevilla, M. Garcia-Moreno, M. Molina-Alarcon, M.J. Garcia-Meseguer, J.M. Villalba, E. Arribas, R. Varon, J. Math. Chem. 50, 1598–1624 (2012)

    Article  CAS  Google Scholar 

  7. D. Vogt, J. Math. Chem. 51, 826–842 (2013)

    Article  CAS  Google Scholar 

  8. P. Miškinis, J. Math. Chem. 51, 1822–1834 (2013)

    Article  Google Scholar 

  9. G. Milani, J. Math. Chem. 51, 2033–2061 (2013)

    Article  CAS  Google Scholar 

  10. V. Vlasov, React. Kinet. Mech. Catal. 110, 5–13 (2013)

    Article  CAS  Google Scholar 

  11. R.M. Torrez Irigoyena, S.A. Giner, J. Food Eng. 128, 31–39 (2014)

    Article  Google Scholar 

  12. D.K. Garg, C.A. Serra, Y. Hoarau, D. Parida, M. Bouquey, R. Muller, Macromolecules 47, 4567–4586 (2014)

    Article  CAS  Google Scholar 

  13. D. Belkić, J. Math. Chem. 52, 1201–1252 (2014)

    Article  Google Scholar 

  14. A.A. Khadom, A.A. Abdul-Hadi, React. Kinet. Mech. Catal. 112, 15–26 (2014)

    Article  CAS  Google Scholar 

  15. A. Izadbakhsh, A. Khatami, React. Kinet. Mech. Catal. 112, 77–100 (2014)

    Article  CAS  Google Scholar 

  16. H. Vazquez-Leal, M. Sandoval-Hernandez, R. Castaneda-Sheissa, U. Filobello-Nino, A. Sarmiento-Reyes, Int. J. Appl. Math. Res. 4, 253–258 (2015)

    Article  Google Scholar 

  17. J. Sun, D. Li, R. Yao, Z. Sun, X. Li, W. Li, React. Kinet. Mech. Catal. 114, 451–471 (2015)

    Article  CAS  Google Scholar 

  18. G. Milani, T. Hanel, R. Donetti, F. Milani, J. Math. Chem. 53, 975–997 (2015)

    Article  CAS  Google Scholar 

  19. G. Lente, J. Math. Chem. 53, 1172–1183 (2015)

    Article  CAS  Google Scholar 

  20. M.L. Strekalov, J. Math. Chem. 53, 1313–1324 (2015)

    Article  CAS  Google Scholar 

  21. G. Lente, Deterministic Kinetics in Chemistry and Systems Biology the Dynamics of Complex Reaction Networks (Springer, Heidelberg, 2015)

    Google Scholar 

  22. http://mathworld.wolfram.com/Sign.html

  23. P. Muller, Pure Appl. Chem. 66, 1077–1184 (1994)

    Article  Google Scholar 

  24. K.J. Laidler, Pure Appl. Chem. 68, 149–192 (1996)

    Article  CAS  Google Scholar 

  25. http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html

  26. http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

  27. http://mathworld.wolfram.com/LegendreFunctionoftheFirstKind.html

  28. http://functions.wolfram.com/HypergeometricFunctions/LegendreQ3General/

  29. http://mathworld.wolfram.com/HypergeometricFunction.html

  30. http://mathworld.wolfram.com/GammaFunction.html

  31. J. Kalmár, É. Dóka, G. Lente, I. Fábián, Dalton Trans. 43, 4862–4870 (2014)

    Article  Google Scholar 

  32. P.P. Levin, A.F. Efremkin, I.V. Khudyakov, Photochem. Photobiol. Sci. 6, 891–896 (2015)

    Article  Google Scholar 

  33. N.D. Gomez, V. D’Accurso, V.M. Freytes, F.A. Manzano, J. Codnia, M.L. Azcárate, Int. J. Chem. Kinet. 45, 306–313 (2013)

    Article  CAS  Google Scholar 

  34. G. Milani, F. Milani, J. Math. Chem. 51, 1116–1133 (2013)

    Article  CAS  Google Scholar 

  35. G. Milani, A. Galanti, C. Cardelli, F. Milani, J. Appl. Polym. Sci. 131, 40075 (2014)

    Article  Google Scholar 

  36. C. Brandt, I. Fábián, R. van Eldik, Inorg. Chem. 33, 687–701 (1994)

    Article  CAS  Google Scholar 

  37. É. Dóka, G. Lente, I. Fábián, Dalton Trans. 43, 9596–9603 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

The research was supported by the EU and co-financed by the European Social Fund under the Project ENVIKUT (TÁMOP-4.2.2.A-11/1/KONV-2012-0043). The author also thanks the Hungarian Science Foundation for financial support under grant No. NK 105156.

Author information

Authors and Affiliations

  1. Department of Inorganic and Analytical Chemistry, University of Debrecen, P.O.B. 21, Debrecen, 4010, Hungary

    Gábor Lente

Authors
  1. Gábor Lente
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gábor Lente.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (docx 88 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lente, G. Analytical solutions for the rate equations of irreversible two-step consecutive processes with second order later steps. J Math Chem 53, 1759–1771 (2015). https://doi.org/10.1007/s10910-015-0517-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-015-0517-3

Keywords

Search

Navigation