Kinetic polynomial equation

We have termed the resultant of the overall reaction rate as the kinetic polynomial. Equation (3) is just the particular form of kinetic polynomial for the linear mechanism. 

In this chapter, we will try to answer the next obvious question can we find an explicit reaction rate equation for the general non-linear reaction mechanism, at least for its thermodynamic branch, which goes through the equilibrium. Applying the kinetic polynomial concept, we introduce the new explicit form of reaction rate equation in terms of hypergeometric series. 

Equation (3) is linear with respect to the reaction rate variable, R. In the further analysis of more complex, non-linear, mechanisms and corresponding kinetic models, we will present the polynomial as an equation, which generalizes Equation (3), and term it as the kinetic polynomial. We will demonstrate that the overall reaction rate, in the general non-linear case, cannot generally be presented as a difference between two terms representing the forward and reverse reaction rates. This presentation is valid only at the special conditions that will be described. 

The right-hand side of Equation (25) is the kinetic polynomial. Assuming V] 7 0, we can define the resultant with respect to R as... 

Kinetic polynomial as the generalized overall reaction rate equation 2.2.4.7 The root count. The reader interested first in deriving explicit reaction rate equation may omit this section and start to read Section 2.2.4.T)... 

This linear affinity approximation does not always correspond to the linear approximation of kinetic polynomial - (Bo)/(Bj). This happens only when degree p of cyclic characteristic in Proposition 1 (see Equation (34)) is one. If p>l, linear approximation of the kinetic polynomial does not correspond to... 

Note that both Equations (56) and (60) result in the same series for the root of kinetic polynomial corresponding the "thermodynamic branch" (see Appendix 4 for the proof). 

Eley-Rideal mechanism. Kinetic polynomial here is quadratic in R (see Equation (48)). There is only one feasible solution (49) here. The feasible branch should vanish at the thermodynamic equilibrium. Thus, the only candidate for the feasible branch expansion is R = — [Bq/Bi] because the second branch expansion is R — —B2/Bi+[Bq/Bi] and it does not vanish at equilibrium. First terms of series for reaction rate generated by formula (55) at = 1 are... 

The four-term overall reaction rate equation. It follows from Propositions 1, 3 and the fact that the kinetic polynomial defined by formula (26) is a rational function of reaction weights fs and that we can write Equation (67) as... 

In the case of the quadratic equation, the convergence condition for the "thermodynamic branch" series is simply positive discriminant (Passare and Tsikh, 2004). For kinetic polynomial (48) this discriminant is always positive for feasible values of parameters (see Equation (49)). This explains the convergence pattern for this series, in which the addition of new terms extended the convergence domain. 

The following case study demonstrates the convergence behavior for the LH mechanism (50) with irreversible first stage (i.e. r i = 0). In this case the kinetic polynomial (51) always has (structurally unstable with respect to feasibility) zero root whereas three other roots could be found from the cubic equation... 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

In order to make the comparison between Ep and Ep/2 measurements summarized in Table 9, the two quantities were measured in separate experiments. A recent study by Eliason and Parker has shown that this is not necessary [57]. Analysis of theoretical LSV waves by second-order linear regression showed that data in the region of Ep are very nearly parabolic. The data in Fig. 9 are for the LSV wave for Nernstian charge transfer. The circles are theoretical data and the solid line is that described by a second-order polynomial equation. It was concluded that no detectable error will be invoked in the measurement of LSV Ep and Ip by the assumption that the data fit the equation for a parabola as long as the data is restricted to about 10 mV on either side of the maximum. This was verified by experimental measurements on both a Nernstian and a kinetic system. 

Finally, it appears that the kinetic models of complex reactions contain two types of components independent of and dependent on the complex mechanism structure [4—7]. Hence the thermodynamic correctness of these models is ensured. The analysis of simple classes indicates that an unusual analog arises for the equation of state relating the observed characteristics of the open chemical system, i.e. a kinetic polynomial [7]. This polynomial distinctly shows how a complex kinetic relationship is assembled from simple reaction equations. 

It is not surprising that simple kinetics do not model browning accurately, since measuring browning is determining the concentration of a product after a series of reactions. Polynomial equation have been used to characterize these absorbance-time relationships. These are developed from idea that a reactant is produced during the reaction and are based on a consecutive 3-step mechanism... 

As mentioned in Section 2.3.2, glycidol and DCP are produced only in small amounts during the HKR reaction. Upon completion of the HKR, the equilibrium equations for DCP and glycidol formation are far from equilibrium and a non-equilibrated condition is assumed. Based upon these assumptions, the macroscopic kinetic law governing each reaction is expressed as a typical polynomial equation (Eqs. 9-11). 

Hempenstall et al.333 reported a calculation method that can be performed by simple computers, whereby the degradation curve is represented by a polynomial equation in order to easily obtain a rate constant kT at a temperature T. Using Eq. (2.90), kT can be represented by Eq. (2.91) in the case of first-order degradation kinetics. Inserting the coefficients a0, ah. . . , an, [which are obtained by fitting the drug concentration versus time data to Eq. (2.90)]... 

A non-linear theory of steady-state kinetics of complex catalytic reactions is developed. A system of steady-state (or pseudo-steady-state) equations can always be reduced to a so called kinetic polynomial. This polynomial is a function of the steady-state reaction rate and the process parameters (concentrations of the reactants, temperature). 

The kinetic polynomial is a non-linear implicit equation. The physically meaningful solutions correspond to the different steady-states. Using the kinetic polynomial it is convenient to specify the region of critical kinetic behaviour, e. g. region of multiplicity of steady-states and self-oscillations. As an example of kinetic polynomial CO oxidation was analyzed. 

The analysis of non-linear mechanisms and corresponding kinetic models are much more difficult than that of linear ones. The obvious difficulty in this case is the follows an explicit solution for steady-state reaction rate R can be obtained only for special non-linear algebraic systems of steady-state (or pseudo-steady-state) equations. In general case it is impossible to solve explicitly a system of non-linear steady-state (or pseudo-steady-state) equations. However, in the case of mass-action-law-model it is always possible to apply to this system a method of elimination of variables and reduce it to a polynomial in one variable [4], i.e., a polynomial in terms of the steady-state reaction rate. We refer a polynomial in the steady-state reaction as a kinetic polynomial. The idea of this polynomial was firstly emphasized in [5]. 

The kinetic polynomial (K.P.) can be defined as a state equation of the open isothermal chemical system related to the reaction rate and the parameters of the process (composition of the mixture, temperature). 

Siojan, f.. Golicnik, M., and Fournier, D, (2004). Rational polynomial equation as an unbiased approach for the kinetic studies of Dm.sophila melanogasier acetylcholinesterase reaction mechanism. Bhxiiim. Biophy.s. Acta 1703, 53-61. 

The information in Figure 3.7 can be converted to molar concentrations, after which we can apply the usual mathematical methods used in a chemical kinetics investigation. Figure 3.8 shows sucrose concentration as a function of time for Run 19. The polynomial equation fitting sucrose concentration to time is displayed in Figure 3.8. Note that it has the general form... 

The clearing up of this question is important from the point of view of practical chemistry as well as of mathematics. In the first place, if a polynomial differential equation has been fitted to experimental data, then it is a question, whether this equation can be considered as a model of reactions. In the second place, utilising the special structure of the kinetic differential equations, surprisingly strong theorems exist for the qualitative properties of the solutions. More precisely, certain systems of differential equations can be studied more efficiently if they can be models of chemical reactions. 

Section 4.7 treats an inverse problem whose solution can be formulated. This is the problem of the characterisation of kinetic differential equations among the polynomial differential equations. In other words how can one decide if a... 

Polynomial differential equations, kinetic differential equations, kinetic initial value problems... 

It is an astonishing fact that the converse of the theorem holds as well. If the right-hand side of a differential equation is an (M, M)-polynomial without negative cross-effects then it may be considered as the induced kinetic differential equation of a reaction, or, in other words, if there is no negative cross-effect in the right-hand side then there exists a reaction with the given equation as its deterministic model. 

Polynomial differential equations without negative cross-effects will usually be called kinetic differential equations from now on. 

Is the lack of negative cross-effects not too strong a restriction in the sense that a randomly selected polynomial differential equation is usually nonkinetic. How dense is the set of kinetic differential equations within the set of polynomial ones ... 

According to several different real situations different definitions have been given of the random event that a polynomial differential equation is kinetic (Toth, 1981b, pp. 44-8). The results can be summarised as follows. If one selects a polynomial differential equation with fixed coefficients and the random selection only concerns the exponents than the probability of getting a kinetic differential equation is 1. If the exponents are fixed and the coefficients are randomly chosen then the probability of getting a kinetic differential equation is 0. Finally, as a consequence of the statements above, if both the coefficients and the exponents are randomly selected then the probability of getting a kinetic differential equation is again 0. 

The right-hand side of the induced kinetic differential equation of reaction (4.21) is the (2, 2)-polynomial 0. Such a case cannot occur with reversible, or even with weakly reversible reactions. Neither can it occur with acyclic reactions. 

The right-hand side of the induced kinetic differential equation of an acyclic reaction cannot be the polynomial 0. As the reaction is acyclic it should contain at least one point without arrows pointing towards it. If one of these points corresponds to a component then the derivative of this component surely contains a (negative) term that cannot be counterbalanced as a result of other elementary reactions as no other point points toward this point. If all these points correspond to elementary reactions, than the components formed in this elementary reactions have a constant inflow that cannot be counterbalanced. 

In Section 4.8.6 we shall turn to the problem of how is it possible or not possible to obtain a kinetic differential equation from a polynomial nonkine-tic one. 

Every change of scale (including every change of unit of measure) preserves the character of polynomial equations in the sense that kinetic differential equations remain kinetic, while nonkinetic ones remain nonkinetic. 

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