Polynomial kinetics

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

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. 

Non-linear mechanisms the kinetic polynomial 2.2.1 The resultant in reaction rate... 

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. 

Authors founded the kinetic polynomial theory in early 1980s (Lazman and Yablonskii, 1991 Lazman et ah, 1985a, 1985b, 1987a, 1987b Yablonskii et ah, 1982, 1983). It was further developed in collaboration with mathematicians Bykov and Kytmanov (Bykov et ah, 1987,1989). Later, applying computer algebra methods. 

Remarkably, the development of kinetic polynomial stimulated obtaining pure mathematical results that became the "standard references" in mathematical texts (see, for instance, WWW sources as E. W. Weisstein. "Resultant". From MathWorld — A Wolfram Web Resource, http //mathworld.wolfram.com/Resultant.html, Multidimensional logarithmic residues. Encyclopaedia of Mathematics — ISBN 1402006098 Edited by Michiel Hazewinkel CWI, Amsterdam, 2002, Springer, Berlin). 

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... 

This means that we could have a situation when more than one root of kinetic polynomial vanishes at the thermodynamic equilibrium. However, only one of these roots would be feasible. 

We have found recently the topological interpretation of property (34). The stoichiometric constraints (24) can be interpreted in terms of the topological object, the circuit. Existence of the circuit "explains" the appearance of the cyclic characteristic in the constant term of kinetic polynomial. Thus, we can say that in some sense the correspondence between the detailed mechanism and thermodynamics is governed by pure topology. 

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)... 

Eley-Rideal mechanism. Kinetic polynomial corresponding to the mechanism... 

Figure 1 Dependence of overall reaction rate on the parameter 2 (LH mechanism). Branches Rl, R2, R3 and R4 represent the roots of kinetic polynomial. Solid line indicates feasible steady states. Branches Re(Rl), Re(R2) and Re(R3) correspond to the real parts of conjugated complex roots of kinetic polynomial. Parameter values fi = 1.4, — 0.1, t2 = 0.1, fj = 15 and rj = 2.

The cyclic characteristic C is small in the vicinity of thermodynamic equilibrium. We can find the overall reaction rate approximation in the vicinity of equilibrium either directly from kinetic polynomial or by expanding the reaction rate in power series by the small parameter C. The explicit expression for the first term is presented by Lazman and Yablonskii (1988, 1991). It is written as follows ... 

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... 

In this case, we can derive (53) from linear approximation of kinetic polynomial (Lazman and Yablonskii, 1991). See also Appendix 3, (A3.11) for details. 

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... 

Figures 4 and 5 compare the exact solutions of the kinetic polynomial (51) (i.e. the quasi-steady-state values of reaction rate) to their approximations. Even the first term of series (65) gives the satisfactory approximation of all four branches of the solution (see Figure 4). Figure 5, in which "brackets" from Appendix 2 are compared to the exact solution, illustrates the existence of the region (in this case, the interval of parameter /2), where hypergeometric series converge for each root.

Figure 4 All roots of kinetic polynomial (dots) and their first-term (lines) approximations for LH mechanism. Parameters f = 1.4, r — 0.1, r2 — 0.1, — 0.1 and — 0.01.

Figure 5 All roots of the kinetic polynomial from Figure 4 (dots) and their higher-order (m = 3) approximations (lines).

Proposition 3. If p> 1 and property (35) is valid, then the cyclic characteristic C is contained in the coefficient Bi of kinetic polynomial with an exponent equal... 

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... 

Figure 8 A convergence domain (rhomboid) and coefficients of the kinetic polynomial (ovals). The ovals represent the coefficients b2 and b3 as parametric functions of parameter fj at different values of parameter Parameters f = 1.4, = 0.9 and = 0.4.

Applying "kinetic polynomial" approach we found the analytical representation for the "thermodynamic branch" of the overall reaction rate of the complex reaction with no traditional assumptions on the rate limiting and "fast" equilibrium of steps. 

Lazman, M. Z., and Yablonskii, G. S., Kinetic polynomial A new concept of chemical kinetics. Patterns and Dynamics in Reactive Media, The IMA Volumes in Mathematics and its Applications, pp. 117-150, Berlin, Springer (1991). 

Por Keq > 0, condition (A4.6) could be satisfied only, if j = 0. There is a bijection between solution (A4.1) and condition (A4.6), and the case 7 = 0 corresponds to the only feasible solution (A4.2) (see Proposition A4.1). However, when p = 1, there is only one branch of solutions of kinetic polynomial vanishing at the equilibrium. As the thermodynamic branch satisfies the equilibrium condition (A4.0) and there are no other branches vanishing at the equilibrium (we proved in Appendix 3 that Bi O at the equilibrium (see also Lazman and Yablonskii, 1991), this branch should be feasible. By continuity, this property should be valid in some vicinity of equilibrium. 

Let now p> 1. In this case, we have p branches of kinetic polynomial zeros vanishing at the equilibrium. Which one corresponds to the thermodynamic branch ... 

By definition of the resultant, every root of kinetic polynomial solves the system and vice versa. Formula (A3.4) is presented schematically here. 

---