STABILITY OF O.D.E. SOLUTIONS CORRESPONDING TO CHEMICAL MECHANISMS BASED-ON UNIMOLECULAR FIRST ORDER REACTIONS

Year 2015, Volume: 3 Issue: 2, 58 - 63, 30.10.2015

Victor Martınez-luaces

https://doi.org/10.36753/mathenot.421332

Abstract

The most simple unimolecular first order chemical reaction mechanism
that involves two species, can be exemplified by the Mutarotation of
Glucose [1]. The corresponding mathematical model is an O.D.E. linear system
which solutions are stable, but not asymptotically [2]. When three chemical
compounds are considered, the mechanism can vary from a simple two
reactions sequence to a complicated one as the adsorption of Carbon Dioxide
(CO2) over Platinum (P t) surfaces [2]. Although in these examples the
mechanisms are very different, in both cases the O.D.E. system has two negative
eigenvalues and the other one is zero. Once again, solutions show a weak
stability which implies that small errors due to measurements in the initial
concentrations will remain bounded, but they do not tend to vanish as the
reaction proceeds. In this paper, a general result for reversible reactions is
stated through an inverse modelling approach [3] [4], proposing theoretical
mechanisms and showing algebraically that all the eigenvalues are negative,
except one, which is zero. From this fact, the conclusions about the stability
of the solutions are obtained and their consequences on the propagation of
measurements errors are analysed.

Keywords

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

References

• [1] Guerasimov,Y.A.etal, Physical Chemistry 2nd Edition, Houghton-Mifflin,Boston,1995.
• [2] Martinez-Luaces,V.,Chemical Kinetics and Inverse Modelling Problems, in Chemical Kinetics, In Tech Open Science, Rijeka, Croatia, 2012.
• [3] Martinez-Luaces,V., Modelling and Inverse Modelling: Experiences with O.D.E. linear systems in engineering courses, International Journal of Mathematical Education in Science and Technology, 40 (2009), no.2, 259-268.
• [4] Martinez-Luaces, V., Problemas inversos y de modelado inverso en Matemtica Educativa, Editorial Acadmica Espaola, Saarbrcken, Germany, 2012.
• [5] Martinez-Luaces,V.,First Order Chemical Kinetics Matrices and Stability of O.D.E. Systems, in Advances in Linear Algebra Research, Chapter 10. Nova Science Publishers, New York, U.S.A., 2015.