Horiuti numbers

Within the Horiuti s approach, the physical meaning of the molecularity is clear. Horiuti introduced the concept of stoichiometric numbers (Horiuti numbers, v) Horiuti numbers are the numbers such that, after multiplying the chemical equation for every reaction step by the appropriate Horiuti number v, and subsequent adding, all reaction intermediates are cancelled. The equation obtained is the overall reaction. In the general case, the Horiuti numbers form a matrix. Each set of Horiuti numbers (i.e. matrix column) leading to elimination of intermediates corresponds to the specific reaction route. ... 

For typical one-route linear mechanisms all the Horiuti numbers can be selected to be equal to 1. This is not necessarily true for non-linear reaction mechanism, e.g. for SO2 oxidation mechanism... 

It is more convenient, in our opinion, to use the term "Horiuti number" instead of the "stoichiometric number" as the latter could be mistakenly identified with the term "stoichiometric coefficient" which designates the number of molecules participating in the reaction. 

Obviously, Horiuti numbers are defined up to non-singular linear transformation. 

According to the Temkin s classification, this mechanism is non-linear, because in the first step oxygen reacts with two catalyst sites K. Numbers to the right of stoichiometric equations are Horiuti numbers vi = 1 and V2 — 2. An analysis of data by Boreskov showed that the power M in Equations (13) and (14) is... 

The Horiuti numbers defining the cyclic characteristic in Equation (34) are relatively prime i.e. GCD(vi,..., v ) — 1. The exponent p in Equation (34) is the natural number. If we assume additionally that... 

Here vT is the transposed matrix of the Horiuti numbers (stoichiometric numbers) and Tint the matrix of the intermediate stoichiometric coefficients. The size for the matrices vT and rint is (P x S) and (S x Jtot), respectively, where S is the number of steps, Jtot the total number of independent intermediates, and P the number of routes. Due to the existence of a conservation law (at least one), the catalyst quantity and the number of linearly independent intermediates will be... 

Horiuti numbers v (S x P) is the route of a complex reaction. The rank of the matrix rint cannot be higher than (S - P) since, according to eqn. (19) there are P linearly independent rows of Tint. As usual, we have... 

As shown above, the stoichiometric (Horiuti) numbers must satisfy the equality... 

In Chapter 2, a new original approach is presented based on a new unified C-matrix, which generates three main matrices of chemical systems representing both chemical composition and complex chemical transformation. These three matrices are the molecular matrix, the stoichiometric matrix, and the matrix of Horiuti numbers, and the original algorithm is derived in this chapter. 

For a set of reactions involving intermediates, which typically cannot be measured, determine the overall reactions, that is, the ones not involving such intermediates, and find the numbers by which these can be written as combinations of the given reactions (the so-called Horiuti numbers). This will be addressed by augmenting a specially crafted stoichiometric matrix, as explained in Section 2.4. 

The overall reaction is obtained by multiplying the reactions with certain coefficients, the so-called Horiuti numbers <7, and then adding the results. 

The rows in which all intermediates have zero entries provide a basis of the overall reactions. Note that if no such rows occur, there is no overall reaction because intermediates are always involved. There can be several different overall reactions and the same overall reaction may be found by two different sets of Horiuti numbers, but in that case their difference will produce a zero overall reaction 0 0, which will show up as such in... 

In the present case, we ignore the first row but consider the second H2O + CO H2 + CO2, which is the overall reaction, its Horiuti numbers are the coefficients affecting the original reactions Ri and R2 in that row, that is, 1 and 1. If we bundle these numbers in a so-called Horiuti matrix, o, and consider only the intermediate part Sint of the stoichiometric matrix S,... 

Determine the overall reaction(s) and Horiuti numbers. Solution ... 

The first two rows are discarded, since the intermediates occur in these rows, but the third and fourth are kept. The overall reactions are 2C02< 2C0 + 02 and the zero reaction the respective Horiuti numbers are —1, 0,-2, 0 and 0, 1,-1, 1, and the Horiuti matrix is given by... 

In practice, the Horiuti numbers of the zero overall reaction are not considered as such, but indicate the freedom that remains in choosing a set of Horiuti numbers any linear combination of zero-reaction Horiuti numbers can be added to or subtracted from those of any overall reaction. For instance, subtracting them from the overall ones in this case yields —1,-1, 1,-1,... 

We consider only the third row, and read out the overall reaction as AB< A + B with Horiuti numbers 1, — 1, — 1. In practice, these numbers would be more likely reported equivalently as 1, 1,1 for the reverse reaction, A + B AB.This is also the form that would have been obtained if the component AB had been listed before A and B instead of after them. 

Temkin-Boudart mechanism. Catalytic intermediate Z is consumed in the first reaction and produced in the second one, while intermediate AZ (or OZ in the WGS mechanism) is produced in the first reaction and consumed in the second. Multiplying these reactions by the corresponding Horiuti numbers o, which here equal one, and adding them, yields the overall reaction. 

Horiuti number (-) characteristic time (s) space time (s)... 

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