Application of Quotient Rings for Stability Analysis in Chemical Systems

Langmuir-Hinshelwood mechanism [12]. In Section 2 we carry out the stability analysis; the algebraic concepts (and the notation) follow the book by Cox et al. [13]. As a real reaction system the wellknown carbon monoxide oxidation is investigated by the same procedure in Section 4. 2. Langmuir-Hinshelwood Mechanism Most surface reactions obey the well-known Langmuir-Hinshelwood (LH) mechanism: two educts have to adsorb on the surface before reacting: X1+νx1∗ X1,ad, X2+νx2∗ X2,ad, X1,ad +X2,ad → (νx1 +νx2)∗+X4, where ∗ denotes a vacant surface site, νx1 and νx2 give the number of sites which are required for adsorption of the corresponding species X1 and X2. The productX4 has to desorb sufficiently fast, otherwise self-poisoning of the catalyst occurs. In [9] an abstract model of an LH mechanism was shown which exhibits bistability, except for the degenerated case when the site requirements and adsorption kinetics of X1 and X2 are exactly the same [12]. Due to its fast desorption, X4 constitutes an inert product and needs not be taken into account. Setting νx1 = 2, νx2 = 1, x3 = ∗, and assuming constant pressures of X1 and X2, the pseudo reaction equations of this model are given by X1 k1 −→ 2X3, 2X3 k2 −→ X1, X2 k3 −→ X3, X3 k4 −→ X2, X1+X2 k5 −→ 3X3, (1) and can be represented by the network diagram in Figure 1 (left). The stoichiometric matrix N, the flux vector v(x,k) comprising the reaction rates v j, and the kinetic matrix κ of the chemical reaction system in (1) are given by N =   1 0 0 −1 0 0 −1 1 −1 2 −2 1 −1 3   , v(x,k) =   k1x1 k2x3 k3x2 k4x3 k5x1x2   , κ =   0 0 0 1 0 0 1 0 1 0 2 0 1 0   . (2) The reaction rates v j in v(x,k) are modelled with a mass action rate law v j(k j,x) = k j ∏i=1 xκi j , whereby κi j correspond to the kinetic exponents of species i in the reaction j which form the kinetic matrix κ . Then the kinetic equations are given by ẋ = Nv(x,k): ẋ1 =−k1x1+ k2x3− k5x1x2, ẋ2 =−k3x2+ k4x3− k5x1x2, ẋ3 = 2k1x1− 2k2x3+ k3x2− k4x3+ 3k5x1x2 with a conservation relation of the total number of adsorption sites of 2x1+ x2+ x3 = const. (3) In the following sections the kinetic equations are solved via SNA and toric geometry. Note that we work in the reaction rate space at the beginning. 2.1. Reduction of the Reaction Rate Space Clarke has shown that the non-negative stationary reaction rates lie in a convex polyhedral cone Kv [1]. S. Sauerbrei et al. · Application of Quotient Rings 233 This cone is obtained by the intersection of the kernel of the stoichiometric matrix N in (2) with the nonnegative orthant of the r-dimensional reaction space R r ≥0 (r is the number of reactions): Kv = {v ∈ Rr|Nv = 0,v ≥ 0} = (kerN ∩R≥0) = { ∑ i=1 jiEi, ji ≥ 0∀i }