Stoichiometric number of step

Fig. 7-8. Reaction path consisting of a series of elementary steps R s particles in the initial state of reaction P = particles produced in the final state Agx- affinity of step i, V = stoichiometric number of step i.

This matrix, if we omit the stoichiometric numbers of steps, is reduced to the so-called Christiansen matrix , named after the famous Dannish scientist Jens Anton Christiansen. 

The stoichiometric numbers of steps other than the rds are not readily accessible by experiment, as these steps are kinetically invisible. The case where there is no unique rds will be discussed later (Section 5). 

Let us consider the mechanism of the electrode process as a sequence of elementary reactions all of which, in a steady state, occur at the same velocity, equal to the overall velocity, when stoichiometry is properly taken into account. Let us designate by j the rate-determining step and let us assume that it occurs yj times during one occurrence of the overall process. The number yj is the stoichiometric number of step j. Let us consider the system as being in a state close to electrochemical equilibrium for which it can safely be assumed that the electrochemical affinities of all other... 

Table 19.8. Equations to compute stoichiometric numbers of step [19.44]...

The stoichiometric number of step 4 in Eq. (4.7) is zero, which is logical as the solvent does not take part in the chemical transformations. At the same time, the solvent concentration can influence the reaction rate by blocking some catalytic sites, which will be otherwise accessible to the reactants. 

Electrochemical processes are particularly well suited for the manufacture of fine chemicals in view of their high sjjecificity (almost comparable to that offered by enzymes), the smaller number of steps required, adoption of milder conditions, lack of scale-up problems, avoidance of effluents, etc. The ease with which oxidation and/or reduction can be carried out with the practically mass-free clean electrons makes electrochemical processes well suited for such jobs, including paired synthesis in effect, we use electricity as a reagent . Consider a standard chemical oxidant like manganic or chromic sulphate. Here, a stoichiometric amount of the reduced salt will be formed the disposal of which can be a serious problem. If we adopt an electrochemical process, then the reduced salt is converted into the desired oxidized salt. 

If it is known which of the reactions determine the rate of the overall complex electrode process, then the concept of the stoichiometric number of the electrode process v is often introduced. This number is equal to the number of identical partial reactions required to realize the overall electrode process, as written in an equation of type (5.2.2).t If the rate constant of this partial rate-determining reaction is ka, then ka = /ca/v. Thus, for example, if the first of reactions (5.1.7) is the rate-determining step in the overall electrode process (5.1.4) then the stoichiometric number has the value v = 2. 

In all but the simplest cases, the mechanism of a reaction consists of a number of steps, some of which involve reacting species that do not appear in the overall stoichiometric equation for the reaction. Some of these intermediate species are stable molecules that can be isolated in the laboratory. Others are highly reactive species that can be observed only by using sophisticated... 

Guideline 6. The great majority of known elementary steps are bimolecular, the remainder being unimolecular or termolecular. Any reaction where the stoichiometric coefficients of the reactants add up to four or more must involve a multiplicity of steps. The ammonia synthesis reaction is known to occur by a number of steps rather than as... 

Fig. 7-11. Potential energy curves for a mialtistep reaction of (a) single rate-determining step and (b) multiple rate-determining steps v = stoichiometric number of a single rate-determining step v = mean stoichiometric number of multiple rate-determining steps.

With a single rate-determining step, the affinity of elementary steps other than the rate-determining step is negligible, and the overall reaction affinity — AG approximately equals the affinity - 4gr multiplied by the stoichiometric number of the rate-determining step in Eqn. 7-44 as has been shown in Eqn. 

Ihis number v is the mean stoichiometric number of the rate-determining multiple steps. The mean stoichiometric number is thus represented by the energy average (afiiniiy average) of the stoichiometric niunbers v weighed with the step affinity 4gi in the respective rate-determining steps. 

The reaction affinity - AG and the ratio v /vj of the forward to the backward rate can be estimated, regardless of whether the reaction rate is determined by a single step or multiple steps. Thus, Eqn. 7-51 can be used to determine the mean stoichiometric number of the multiple rate-determining steps. 

Another important aspect that we have so far ignored is that reactions almost never actually proceed as represented by a stoichiometric equation (eg, Eq 1). Usually a stoichiometric equation is the algebraic sum of a number of steps called elementary reactions. Frequently one of these elementary reactions is much slower than the others, and thus con-... 

Here vT is the transposed matrix of stoichiometric numbers and Tint is the matrix of stoichiometric coefficients for intermediates. Elements of the latter are taken to be negative if substance is consumed in a given reaction step, positive if it is formed, and zero if substance is not involved in the reaction step. Multiplication of matrix vT (P-by-s) by matrix Tmt (s-by-/tot) gives the matrix vTrint whose size is (P-by-/tot) (s is the number of steps). 

Let us analyze the structure of eqn. (16). Its numerator can be written as K [A] - K [B], where K = k k + and K = A k k. In this form it accounts for the stoichiometric equation A = B obtained by adding all the steps of the detailed mechanism multiplied by unit stoichiometric numbers. It is interesting that the numerator is absolutely independent of the mechanism "details . Irrespective of the number of steps in our mechanism (a thousand, a million), the numerator of a steady-state kinetic equation always corresponds to the kinetic law of the brutto reaction as if it were simple and obeys the law of mass action. The denominator characterizes a "non-elementary character accounting for the rate of the catalytic reaction inhibition by the initial substances and products. 

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... 

The theory of steady-state reactions operates with the concepts of "a path of the step , "a path of the route , and "the reaction rate along the basic route . Let us give their determination in accordance with ref. 16. The number of step paths is interpreted as the difference of the number of elementary reaction acts in the direct and reverse directions. Then the rate for the direct step is equal to that of the paths per unit time in unit reaction space. One path along the route signifies that every step has as many paths as its stoichiometric number for a given route. In the case when the formation of a molecule in one of the steps is compensated by its consumption in the other step, the steady-state reaction process is realized. If, in the course of this step, no final product but a new intermediate is formed, then it is this... 

It can readily be shown that eqn. (27) is equivalent to the quasi-steady-state condition in its general formulation. In unit time and in unit reaction space there forms Tint w of an intermediate, where Tint is the stoichiometric intermediate matrix. Let us recall that the dimension of Fint is (/tot x S), where 7tot is the total number of independent intermediates and S is the number of steps. After substituting uj from eqn. (27), we obtain... 

A feasible reaction scheme includes all the reactants and products, and it generally includes a variety of reaction intermediates. The validity of an elementary step in a reaction sequence is often assessed by noting the number of chemical bonds broken and formed. Elementary steps that involve the transformation of more than a few chemical bonds are usually thought to be unrealistic. However, the desire to formulate reaction schemes in terms of elementary processes taking place on the catalyst surface must be balanced with the need to express the reaction scheme in terms of kinetic parameters that are accessible to experimental measurement or theoretical prediction. This compromise between molecular detail and kinetic parameter estimation plays an important role in the formulation of reaction schemes for analyses. The description of a catalytic cycle requires that the reaction scheme contain a closed sequence of elementary steps. Accordingly, the overall stoichiometric reaction from reactants to products is described by the summation of the individual stoichiometric steps multiplied by the stoichiometric number of that step, ai. 

We emphasize that this conservation of sensitivity applies in general to reaction schemes for which the stoichiometric numbers of the individual steps are not all equal to unity. 

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