Stoichiometric vector

To write another form of Equation (27) we use stoichiometric vectors for a reaction A, Ay the stoichiometric vector yy, is a vector in concentration space with zth coordinate —1, yth coordinate 1 and other coordinates 0. The reaction rate... 

Acyclic auxiliary dynamical system with one attractor have a characteristic property among all auxiliary dynamical systems the stoichiometric vectors of reactions Aj A q form a basis in the subspace of concentration space with c, = 0. Indeed, for such a system there exist n—1 reactions, and their... 

Let us assume that the auxiliary dynamical system is acyclic and has only one attractor, a fixed point. This means that stoichiometric vectors form a basis in a subspace of concentration space with — 0. For every reaction A,- A the following linear operators Qu can be defined ... 

Each (sth) step corresponds to its stoichiometric vector ys whose components are... 

Here the step rate wsu acts as a coefficient for the stoichiometric vector yM, and it is this coefficient that is found when processing data of kinetic experiments. 

The other way to describe this property is as follows. A convex envelope for the multitude of stoichiometric vectors for the edges in the graph of predominant directions written as if they were direct reaction steps does not contain zero, i.e. there are no such non-negative Alt. . Aq as... 

The number of complexes minus that of connected components of the graph for their conversions equals the number of linearly independent reactions (stoichiometric vectors). A second Horn and Jackson condition for quasi-thermodynamic behaviour is the weak reversibility of the graph for complex conversions. This graph is called weakly reversible if any of its connected components contain a route to get from any node to any other moving in the direction of its arrows. For example, the scheme... 

Let us check these conditions for the oxidation of CO on Pt. The Eley-Rideal (impact) mechanism has four complexes, M = 4, two connected components [2Z] [2ZO], [ZO] — [Z], and two stoichiometric vectors... 

In this case, M = 3, l = 1, and S = 3 but among the stoichiometric vectors only two are linearly independent. Thus 3 — 1 = 2 and the first Horn and Jackson condition is fulfilled. Also fulfilled is the second condition, i.e. weak reversibility of the graph for complex conversions. Since both conditions are fulfilled, a steady state is unique and stable despite the mechanism having two interaction steps for various intermediates. 

The stoichiometric vector is o=[2 2 1]. The rates of the elementary steps are given by vector R and the generation rates of gas-phase (r) and surface components (r ) are calculated from... 

Aa° is the column vector containing the changes of concentrations of all the reactants as elements. The stoichiometric vector v corresponds to its transpose v° defined above. Eq. (2.2) is explicitly written as follows ... 

We now check the consistency between the overall reaction (eq. (10.15)) and the elementary steps (eq. (10.14)), we need to find out how many of each of the elementary steps (eq. (10.14)) are necessary for the generation of the overall reaction. We designate the stoichiometric vector attached to the overall reaction (eq. (10.15)) as... 

An important insight from Figure 8.16 is that the direction of the reaction (the stoichiometric vector, v) in essence determines what the direction of progression of Xaz/Xa is to be (evident from Equation 8.34). Thus, when v=[l, 1, -2] (the reverse reaction of Figure 8.16), the X z/Xas progression will proceed in the opposite direction of that shown in Figure 8.16. Notice too that, as with all other complex columns with multiple CSs, difference points are linearly linked to one another. 

If for a given reaction mechanism, the vectors are linearly independent, the rest point of Eq. (7.43) is a point of the detailed balance. Adding a new step with a stoichiometric vector that is a linear combination of the stoichiometric vectors of the original mechanism does not change the point of the detailed balance. This is a very important property of a closed chemical system. 

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