Horiuti stoichiometric number

Here v is the matrix of the Horiuti (stoichiometric) numbers and v and w the vector-columns of the rates along basic routes and of the step rates, respectively. Thus the rate of every step is represented as a linear combination of the rates along the basic routes. Here it is recommended that a simple hydrodynamic analogy be used. The total liquid flow along the tube (step) is the reaction rate. This flow consists of individual streams which are the rates along the routes. 

One must not underestimate, however, the importance of the general results obtained in terms of the steady-state reaction theory. Its informative concepts are used in theoretical kinetics, in particular the concept of Horiuti (stoichiometric) numbers and a new formulation for the steady-state... 

J. Horiuti, and T. Nakamura, Stoichiometric number and the theory of steady reaction, Z. Phys. Chem. Neue Folge 11, 358-365 (1957). 

A reaction consisting of a single elementary step alone is uncommon, and most reactions involve a number of elementary steps with reaction-intermediates (miiltistep reactions). For a reaction consisting of a series connection of several elementary steps, the munber of repetitions that an elementary step proceeds in a unit extent (advancement) of the overall reaction is defined as the stoichiometric number y v, of the step [Horiuti-Dcushima, 1939]. For example, if the cathodic reaction of hydrogen electrode consists of the following two steps,... 

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. ... 

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. 

Introduction of stoichiometric number concept and linear transformation of the "conventional" QSSA equations (16) to the equivalent system (20) was essentially the major (and, possibly, only) result of theory of steady reactions developed independently by J. Horiuti in 1950s and M. I. Temkin in 1960s. 

This displays the convention, tacitly assumed later, that the positive direction of a step corresponds to the advancement from left to right of the stated chemical equation. The matrix of stoichiometric coefficients for these reactions is shown in Table II. The diagonalization of the matrix in Table II gives the matrix in Table III, from which the steady-state mechanism is S + 2s2 + 2s3 + 2s4. In Horiuti s terminology the stoichiometric numbers are 1 for Sj and 2 for s2, s3, and s4. 

Hence, in this example, two occurrences of the rds are required to produce one occurrence of the overall reaction. One says that the stoichiometric numberp of such a reaction scheme is 2. Thus, a stoichiometric number of v, as introduced by Horiuti in 1939, indicates that there is a mo-one correspondence between the occurrences of the rds and the overall reaction. 

The concept of stoichiometric number was first introduced in electrochemistry by Horiuti and Ikusima [48, 49] for the hydrogen evolution reaction and discussed by Bockris and Potter [50, 52] and Parsons [47]. 

Overall reaction equations are linear combinations of chemical equations of stages, i.e., they are obtained by the addition of chemical equations of stages multiplied by certain numbers (positive, negative, or zero). The numbers must be chosen in such a way that the overall equations contain no intermediates. According to Horiuti, such numbers are called stoichiometric. To illustrate this notion by a simple example, let us assume that the mechanism of oxidation of S02 on the surface of a solid catalyst, e.g., platinum, is described by the following scheme ... 

A set of stoichiometric numbers of the stages producing an overall reaction equation is called, after Horiuti, a reaction route. Thus, in scheme (35) there are two reaction routes, N(l) and iV(2) route Na> cannot be obtained from N(i) through multiplication by a number. Therefore, routes N(1) and N(2) are essentially different, although their respective overall equations are identical. 

To obtain stoichiometric equations for the reaction steps (brutto-equa-tions) that do not involve intermediates, one must add the steps of the detailed mechanism, first multiplying them by the numbers specified. For the simple mechanism of eqns. (44) these numbers equal unity and are placed on the right-hand side of the equations of the reaction steps. Horiuti specified these numbers as stoichiometric numbers (not to confuse them with stoichiometric coefficients, which indicate the number of molecules of a reacting substance). Stoichiometric numbers must fit the equation... 

It means that we consider only mono-, bi- and (rarely) termolecular reactions. The coefficients stoichiometric coefficients and stoichiometric numbers observed in the Horiuti-Temkin theory of steady-state reactions. The latter indicate the number by which the elementary step must be multiplied so that the addition of steps involved in one mechanism will provide a stoichiometric (brutto) equation containing no intermediates (they have been discussed in Chap. 2). 

Horiuti stoichiometric rule. This rule is applied to find the number of linearly independent routes. Stoichiometric numbers must satisfy the equation... 

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... 

This relationship for the determination of the number for the linearly independent routes is called the Horiuti stoichiometric rule. Let us apply it. 

Stoichiometric number— The concept of stoichiometric number, introduced by Horiuti and Ikusima in 1939, was initially applied to reactions associated with the synthesis of NH3. In electrochemistry, stoichiometric number was used in the kinetics of hydrogen and oxygen evolution. Generally, stoichiometric number of a substance B is the z/p coefficient in the stoichiometric equation of the multistep -> chemical reaction... 

Like the kinetic concepts of Christiansen and Horiuti, those of Temkin were far ahead of their common acceptance by the catalytic community. Even today, more than 50 years after the Temkin-Pyzhev paper, the idea of virtual fiigacity is not well understood by the majority of workers in catalytic kinetics. It is safe to predict that many of the other ideas of Temkin, like that of average stoichiometric number or reaction routes, will influence younger catalytic ki-neticists who now have access to powerful computers. 

The stoichiometric number uj, introduced by Horiuti (1962) for reactions with a single rate-controlling step, is the number of times that step occurs in an overall reaction j as written. This number is computable with reasonable precision from simultaneous observations of the forward and reverse reactions, and has been investigated for many reaction.systems. Note, however, that (jj is necessarily unity if one constructs the overall reaction j as a composite event in the manner just described. Thus, a reported stoichiometric number is simply the ratio of the investigator s chosen stoichiometric coefficients to those of the actual composite event, and is not needed further once the composite event has been constructed. 

Horiuti. J., Significance and experimental determination of stoichiometric number, J. Catai, 1, 199-207 (1962). 

The idea of n was first put forward by Horiuti (30) and designated as the stoichiometric number ( Kagaku-Ryosu ). 

The determination of the stoichiometric number was made by Horiuti and Ikushiraa (102) for the hydrogen electrode process on platinum and more recently by Horiuti and Enomoto (103) for the ammonia synthesis... 

Horiuti, J., Theory of reaction rates as based on the stoichiometric number concept. Ann. N.Y. Acad. Sci. 213, 5-30 (1973). 

The coefficient n is defined by Horiuti as the stoichiometric number of the particular elementary process which controls the rate. This is the number of times this step must occur to accomplish the overall reaction. Suppose that the mechanism of the reaction we have chosen consists of the elementary steps... 

Finally, here it should be noted formally that a is, of course, identical with in cases where the initial step in a reaction sequence is a one-electron charge transfer process and is itself the rate-controlling process. Also b can be related to the total number of electrons passed in the overall reaction or in the rate-controlling step through a and the stoichiometric number v. This matter has, however, been treated in various earlier works by, e.g., Horiuti and Ikusima, Bockris, and Gileadi, and so need not be examined again here since no involvement of the temperature variable arises apart from that in b itself. 

The stoichiometric number, v, of a given reaction step is defined as the number of times that step must occur for one turnover of the whole reaction. The overall stoichiometric number of a reaction is specifically the number of times the rds has to occur. This is an important quantity with respect to mechanism elucidation and was originally defined by Horiuti for the hydrogen reaction.45,46 According to the stoichiometric number concept of Horiuti, V for each reaction step will multiply the Gibbs energy change (or as it is also called, its electrochemical affinity) for that step, Agi, and the affinity, AG, for the overall reaction is... 

However, Manes and co-workers noted (19, 20) that the potential term could assume a more complicated form in some cases. Horiuti (10, II, 12, 13, 14) introduced the concept of the stoichiometric number of the rate-controlling step and showed how it could be used to develop a more rigorous form for the potential term. The stoichiometric number of each of the elementary steps which make up an over-all reaction is defined as the number of times it occurs for each occurrence of the over-all reaction. 

Horiuti and Enomoto have shown (3) in extension of the theory of the stoichiometric number (4, 5) that V,/—V, = Ka /i.e., that the exponent to the equation obtained by eliminating k/—k from (2) must be the reciprocal of the stoichiometric number v r) of r, i.e., the number of r occurring for every act of the over-all reaction in the steady state. This conclusion would lead along with the above sufficient assumption (2b) to the relation,... 

Horiuti and Horiuti and Nakamura developed the concept of stoichiometric number nearly 50 years ago. In this concept, each reaction is assumed to be the sum of elementary steps. One or more of the steps will be rate-controlling, and all remaining... 

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