Equation brutto

Let us analyze the structure of eqn. (70). Its numerator can be written as K+ [A] - K [B], where K+ = Aq 2 3 and K = k 1k 2k 3. In this form, it corresponds to the brutto-equation of the reaction A = B obtained by adding all the steps of the detailed mechanism with unit stoichiometric numbers. The numerator is a kinetic equation for the brutto-reaction A = B considered to be elementary and fitting the mass action law. The denominator accounts for the "non-elementary character due to the inhibition of the complex catalytic reaction rate by the initial substances and products. 

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). 

The stoichiometric (brutto) equations for the conversion of gas-phase substances are considered. According to them, linear kinetic equations can be obtained, apparently, if only the time scale is changed. These reactions are pseudo-monomolecular and comprehensively treated by Wei and Prater [11]. An example is the familiar reaction of butene isomerization. 

Due to the fulfilment of this law of conservation, the number of linearly independent intermediates is not three but one fewer, i.e. it amounts to two. To the right of mechanism (1) we gave a column of numerals. Steps of the detailed mechanism must be multiplied by these numerals so that, after the subsequent addition of the equations, a stoichiometric equation for a complex reaction (a brutto equation) is obtained that contains no intermediates. The Japanese physical chemist Horiuti suggested that these numerals should be called "stoichiometric numerals. We believe this term is not too suitable, since it is often confused with stoichiometric coefficients, indicating the number of reactant molecules taking part in the reaction. In our opinion it would be more correct to call them Horiuti numerals. For our simplest mechanism, eqn. (1), these numerals amount to unity. 

This can also be said about the Horiuti number (or, as Horiuti called it himself, the stoichiometric number) discussed previously. The Horiuti numbers are the numbers chosen such that, after multiplying the chemical equation for every step by the appropriate Horiuti number and subsequent adding, all intermediates are cancelled. The equation thus obtained is the stoichiometric (brutto) equation. Each set of stoichiometric numbers leading to the elimination of intermediates is called a reaction route. In the general case, the Horiuti numbers form a matrix and its vector columns are the routes. 

The sense of the cyclic characteristic is simple. It is a kinetic equation of our brutto-reaction as if it were a step and consists of elementary reactions obeying the law of mass action. For the cycle with the brutto-equation 0 = 0, the cyclic characteristic is C = 0. If all cycles have the same "natural brutto-equations, their cyclic characteristics are represented as... 

Thus cyclic characteristics of various cycles will differ only in values of the factors (n ). Cyclic characteristics for two different cycles with the same"natural brutto-equations are proportional to each other... 

We have already said that, in principle, the "natural brutto-equation can... 

Mechanism I accounts for the "natural brutto-equation A = B obtained by adding steps of the detailed mechanism, whereas mechanisms II and III correspond to the equation 2 A = 2 B. Cyclic characteristics will, apparently, differ. In the former case C = K+ CA - K CB, in the latter C = K+Cl -K-Cl... 

The mechanism for the synthesis of vinyl chloride, eqns. (24), whose graph is given in Fig. 3(d), also has two routes with one "natural brutto-equation. Without taking into account the reversibility of steps (1) and (3), the rate of product formation will be... 

Let us note that in eqn. (59) the expressions f+ (c) and f (c) are the kinetic dependences that are written according to the law of mass action for the "natural brutto-reaction, i.e. for the reaction obtained by a simple addition of all cycle steps, and K fT) is the equilibrium constant for this reaction. However, as we mentioned above for the reaction of catalytic isomerization, the "natural brutto-equation should not necessarily have integer-valued coefficients. For the mechanism... 

Let us consider the case in which the rate of the step (or steps) of interest is expressed as eqn. (59) or (62). This step participates in simple cycles at a non-zero rate (non-zero cycles) and these cycles correspond to the same "natural brutto-equation. 

The cycle will be characterized by a zero rate in the two cases (a) the cycle corresponds to the "natural brutto-equation 0 = 0 then C = 0 and (b) the cycle is in equilibrium then C = 0. Cycles with zero rates (zero cycles) do not provide any additional summands in the numerator, whereas the denominator will now have summands accounting for the reaction retardation by the intermediates of these cycles. 

Let two cycles have similar "natural brutto-equations. Then their cyclic characteristics will be expressed as... 

Thus eqn. (65) takes the form of eqn. (59). It is evident that the same statement can also be made in the case when some step takes part in many cycles with the same "natural brutto-equation. The representation of type of eqn. (65) will also be valid for the steady-state rate of concentration variation for substance A [see eqn. (51)] if this substance participates in non-zero cycles with the same "natural brutto-equation. 

It is possible that the reaction with the only brutto-equation will follow several routes. For example, the reaction of vinyl chloride synthesis... 

But, in principle, it is possible that, for the reaction with the only brutto-equation, its different routes correspond to the "natural brutto-equations having different multiplicities [see eqn. (62)]. Then eqn. (59) would not be valid. The literature lacks studies in which this problem has been examined on the basis of experimental data. 

In this case not all the parameters can be determined. An estimate for the number of these indeterminable parameters is obtained as follows. The number is equal to the number of graph outpoints. (A proof of this results from the Giles theorem [55].) Thus, for the case illustrated in Fig. 5(b), the factors in the denominator of eqn. (46) being known, one cannot determine one constant, whereas in the case shown in Fig. 5(d) two constants cannot be found. This estimate will decrease if the parameters are determined on the basis of the coefficients not only from the denominator but also from the numerator. It can be done since we can also apply some expressions for the rates of variation of substances (in this case reaction cycles differ in their brutto-equations). 

As an example, let us describe a two-route mechanism with different brutto-equations. 

Every graph s cycle corresponds to its "natural brutto-equation. We will assume that the stoichiometric coefficients in this equation are minimum integer-valued, i.e. for simplicity the multiplicity is taken to be equal to unity (see Sects. 2.3 and 2.4). We suggest that, as in all the above examples, only one molecule of each observed substance (either initial or product) is either consumed or formed. 

Then, for a mechanism of class 1, one can give a simple estimate of the number of steps corresponding to a given brutto-equation... 

Let us apply estimate (94) to the various mechanisms of class 1 given above. For the Michaelis-Menten mechanism, the brutto-reaction is of the form S = P, nln = 1, nprod = 1, and s = 1 + 1 = 2. For CO conversion, the brutto-equation takes the form... 

The brutto-equation depends on the structure of the kinetic equation and its parameters. In Sect. 2.3 we have already spoken about cyclic characteristics in the numerator of the steady-state kinetic eqn. (46). It is the kinetic equation of the brutto-reaction as if it were a simple step. The form of the cyclic characteristics is independent of the detailed mechanism. But under... 

The complex K3 is non-Arrhenius and is the sum of three products. The reason for this is that the brutto-equation involves three molecules of H2, and the three steps of the detailed mechanism must be subject to the same type of kinetic law. It is due to this fact that such spanning trees appear. 

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