STABILITY OF O.D.E. SYSTEMS ASSOCIATED WITH FIRST ORDER CHEMICAL KINETICS MECHANISMS WITH AND WITHOUT FINAL PRODUCTS

Year 2016, Volume: 4 Issue: 1, 80 - 87, 01.04.2016

Victor Martınez-luaces

Abstract

First order chemical reaction mechanisms are modelled through Ordinary Di erential Equations (O.D.E.) systems of the form: :X = AX , be- ing X the chemical species concentrations vector, :X its time derivative and A the associated system matrix. In previous papers, First Order Chemical Ki- netics Mechanisms (F.O.C.K.M.) involving two or three chemical species were considered and in all these cases, solutions show a weak stability (i.e., they are stable but not asymptotically). This fact implies that small errors due to mea- surements in the initial concentrations will remain bounded, but they do not tend to vanish as the reaction proceeds. In order to know if these results can be extended or not to other chemical mechanisms, a general result is obtained through an inverse modelling approach. For this purpose, theoretical mecha- nisms with and without nal products are proposed, and the corresponding F.O.C.K.M. matrices are studied. As a consequence of the particular structure of the F.O.C.K.M. matrices, the Gershgorin Circles Theorem can be applied to show that all the eigenvalues have real parts negative or zero. Moreover, it is proved as the main result of the paper, that for the null eigenvalue, alge- braic and geometric multiplicities (A.M. and G.M.) give the same number. As an application of these results, several conclusions about the stability of the O.D.E. solutions are obtained for this kind of chemical reactions, and its con- sequences on the propagation of concentrations and/or surface concentrations measurement errors are analyzed.

Keywords

Chemical mechanisms, O.D.E. linear systems, Solutions stability

References

• [1] Martinez-Luaces, V., Engaging secondary school and university teachers in modelling: some experiences in South American countries, International Journal of Mathematical Education in Science and Technology, Vol:36, No.2-3 (2005), 193-205.
• [2] Guerasimov, Y.A. et al, Physical Chemistry, 2nd Ed., Houghton-Miin, Boston,1995.
• [3] Martinez-Luaces, V., Chemical Kinetics and Inverse Modelling Problems, in Chemical Ki- netics, In Tech Open Science Eds., Rijeka, Croatia, 2012.
• [4] Martinez-Luaces, V. & Guineo Cobs, G., Un problema de Electroquimica y su Modelacion Matematica Anuario Latinoamericano de Educacion Quimica , (2002), 272-276.
• [5] Zinola, F., Mendez, E. & Martinez Luaces, V., , Modicacion de estados adsorbidos de An- hidrido Carbonico reducido por labilizacion electroquimica en supercies facetadas de platino, Proceedings of X Congreso Argentino de Fisicoqumica, Tucuman, Argentina , 1997.
• [6] Martinez-Luaces, V., Stability of O.D.E. systems associated to rst order chemical kinetics mechanisms without nal products, 4th International Eurasian Conference on Mathematical Sciences and Applications (IV IECMSA ), Athens, Greece, 2015.
• [7] Martinez-Luaces, V., Matrices with null columns in rst order chemical kinetics mecha- nisms, 4th International Eurasian Conference on Mathematical Sciences and Applications (IV IECMSA ), Athens, Greece, 2015.
• [8] Martinez-Luaces, V., First Order Chemical Kinetics Matrices and Stability of O.D.E. Sys- tems in Advances in Linear Algebra Research, Chapter 10, pp. 325-343. Nova Publishers, New York, U.S.A., 2015.
• [9] Varga, R.S., Gershgorin and His Circles, Springer-Verlag, Berlin, 2004.
• [10] Martinez-Luaces, V., Modelling and Inverse Modelling: Experiences with O.D.E. linear sys- tems in engineering courses,International Journal of Mathematical Education in Science and Technology, Vol:40, No.2 (2009), 259-268.
• [11] Martinez-Luaces, V., Modelling, applications and Inverse Modelling: Innovations in Dier- ential Equations courses,Proceedings of Southern Right Gordons Bay Delta 09, Gordons Bay, South Africa, 2009.
• [12] Martinez-Luaces, V., Problemas inversos y de modelado inverso en Matematica Educativa, Editorial Academica Espanola, Saarbrcken, Germany, 2012.
• [13] Martinez-Luaces, V., Inverse modelling problems in linear algebra undergraduate courses,International Journal of Mathematical Education in Science and Technology, Vol:44, No.7 (2013), 1056-1064.
• [14] Martinez-Luaces, V., Stability of O.D.E. solutions corresponding to chemical mechanisms based-on unimolecular rst order reactions, Mathematical Sciences and Applications E- notes,Vol:3, No.2 (2015), 58-63.