Single-route steady-state reaction

Let a single-route steady-state reaction occur in the spherical porous catalyst grain. It is convenient to transpose all terms of (1) to the right side, i.e., to represent the reaction equation in the following form ... 

We shall denote the rate of a single-route reaction as r and the stoichiometric numbers of its stages as vs. The equation of single-route steady-state reactions follows directly from (51) ... 

Another case occurs if a mechanism of a many-route reaction includes a block that has only one basic route then, as was explained in Section VII, the equation of single-route steady-state reactions and, consequently, the concept of forward and reverse rates can be applied. 

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

A single-route complex catalytic reaction, steady state or quasi (pseudo) steady state, is a favorite topic in kinetics of complex chemical reactions. The practical problem is to find and analyze a steady-state or quasi (pseudo)-steady-state kinetic dependence based on the detailed mechanism or/and experimental data. In both mentioned cases, the problem is to determine the concentrations of intermediates and overall reaction rate (i.e. rate of change of reactants and products) as dependences on concentrations of reactants and products as well as temperature. At the same time, the problem posed and analyzed in this chapter is directly related to one of main problems of theoretical chemical kinetics, i.e. search for general law of complex chemical reactions at least for some classes of detailed mechanisms. 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

For linear mechanisms we have obtained structurized forms of steady-state kinetic equations (Chap. 4). These forms make possible a rapid derivation of steady-state kinetic equations on the basis of a reaction scheme without laborious intermediate calculations. The advantage of these forms is, however, not so much in the simplicity of derivation as in the fact that, on their basis, various physico-chemical conclusions can be drawn, in particular those concerning the relation between the characteristics of detailed mechanisms and the observable kinetic parameters. An interesting and important property of the structurized forms is that they vividly show in what way a complex chemical reaction is assembled from simple ones. Thus, for a single-route linear mechanism, the numerator of a steady-state kinetic equation always corresponds to the kinetic law of the overall reaction as if it were simple and obeyed the law of mass action. This type of numerator is absolutely independent of the number of steps (a thousand, a million) involved in a single-route mechanism. The denominator, however, characterizes the "non-elementary character accounting for the retardation of the complex catalytic reaction by the initial substances and products. 

The outcome ri = f2 = = r is true for any reaction that proceeds via a single-route mechanism. At steady state, the rates of all elementary steps are equal and also equal to the rate of the overall reaction. There is a simple analogy from hydrodynamics illustrating this if the flow through a closed pipeline system consisting of a set of different tubes is at steady state, the flow rate in any section of the pipeline is the same. 

A simple graph recipe for deriving the steady-state rate equation of a single-route catalytic reaction is the following ... 

Since for a single-route mechanism the numerator of the rate does not depend on the details of the mechanism, it can be written directly based on the overall reaction according to the mass-action law. In such a fast derivation of the steady-state rate equation, steps (7) and (8) are omitted. 

The example of temporal behavior of a single-route catalytic reaction shows promising results. Apparently, many results obtained for single-route two-step catalytic reactions can be transferred into results for single-route reactions with a linear mechanism under the assumption of a quasi-steady-state kinetic regime regarding the surface intermediates. 

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