Theory of Steady Reaction

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

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. 

A directly observed chemical reaction as expressed by a single stoichiometric equation was identified at that time with a single elementary reaction subject to the mass action law this constituted one of the limitations. Since the beginning of this century, however, it has been realized that a chemical reaction directly observed, if represented by a single stoichiometric equation, is in general a composite of elementary reactions, consisting of a single reaction only in special cases. The removal of this limitation has led to the theory of steady reaction, which provided a number of successful explanations of observed rate laws. 

The present article is concerned with a general review of the formulation of the rate of elementary reaction without this limitation, and of the theory of steady reaction consisting of elementary reactions with particular reference to heterogeneous ones, hence with the deduction, on this improved basis, of the rate law of steady reaction and the temperature dependence of the rate. On this basis, experimental results are discussed and accounted for, on the one hand, and the two characteristic constants of the classical kinetics subject to the above-mentioned limitations, i.e., the rate constant and activation energy, are discussed on the other hand. 

The theory of steady reaction developed below leads to a classification of steady reactions, the simplest class of them being systematically investigated in this article with regard to the relation between the rate of steady reaction and the rates of the constituent steps, on the one hand, and that between the temperature dependence of the former and the activation heats of the constituent steps, on the other hand, with special reference to heterogeneous steady reactions. 

The present section is concerned with the application of the theory of steady reaction developed in Sections III and IV to the analysis of the temperature dependence of its rate with special reference to heterogeneous reactions. 

A proper resolution of Che status of Che stoichiometric relations in the theory of steady states of catalyst pellets would be very desirable. Stewart s argument and the other fragmentary results presently available suggest they may always be satisfied for a single reaction when the boundary conditions correspond Co a uniform environment with no mass transfer resistance at the surface, regardless of the number of substances in Che mixture, the shape of the pellet, or the particular flux model used. However, this is no more than informed and perhaps wishful speculation. 

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. 

Since mass action law for elementary reactions in ideal adsorbed layers (including also adsorption and desorption processes) coincides in its form with mass action law for elementary reactions in volume ideal systems, general results of the theory of steady-state reactions are equally applicable to volume and to surface reactions. They are very useful when the reaction mechanism is complicated. 

The concepts of the theory of steady-state reactions are better explained with examples, but here they will be formulated mainly in a general form the examples will be found below in the discussion of concrete reactions. 

The critical nucleus of a new phase (Gibbs) is an activated complex (a transitory state) of a system. The motion of the system across the transitory state is the result of fluctuations and has the character of Brownian motion, in accordance with Kramers theory, and in contrast to the inertial motion in Eyring s theory of chemical reactions. The relationship between the rate (probability) of the direct and reverse processes—the growth and the decrease of the nucleus—is determined from the condition of steadiness of the equilibrium distribution, which leads to an equation of the Fourier-Fick type (heat conduction or diffusion) in a rod of variable cross-section or in a stream of variable velocity. The magnitude of the diffusion coefficient is established by comparison with the macroscopic kinetics of the change of nuclei, which does not consider fluctuations (cf. Einstein s application of Stokes law to diffusion). The steady rate of nucleus formation is calculated (the number of nuclei per cubic centimeter per second for a given supersaturation). For condensation of a vapor, the results do not differ from those of Volmer. 

Chapter 2 describes the evolution in fundamental concepts of chemical kinetics (in particular, that of heterogeneous catalysis) and the "prehis-tory of the problem, i.e. the period before the construction of the formal kinetics apparatus. Data are presented concerning the ideal adsorbed layer model and the Horiuti-Temkin theory of steady-state reactions. In what follows (Chapter 3), an apparatus for the modern formal kinetics is represented. This is based on the qualitative theory of differential equations, linear algebra and graphs theory. Closed and open systems are discussed separately (as a rule, only for isothermal cases). We will draw the reader s attention to the two results of considerable importance. 

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). 

Apart from enzyme kinetics, this new trend had also appeared in the kinetics of heterogeneous catalysis. In the 1950s, Horiuti formulated a theory of steady-state reactions [11, 12], many of the concepts of which correspond to the graph theory. Independent intermediates, a reaction route, an independent reaction route, all these concepts were introduced by Horiuti. 

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... 

A non-linear theory of steady-state kinetics of complex catalytic reactions is developed. A system of steady-state (or pseudo-steady-state) equations can always be reduced to a so called kinetic polynomial. This polynomial is a function of the steady-state reaction rate and the process parameters (concentrations of the reactants, temperature). 

The important stimuli for the development of the non-linear kinetic theory of steady-state catalytic reactions are 1) necessity to explain the critical phenomena that are experimentally observed in the steady-state kinetic experiments and 2) needs of chemical technology to understand and to apply the advantages of non-linear regimes. 

Empfrical Methods In Non-stationaiy Systems.— The extension of ignition theory of theraul reactions to non-steady states began in earnest in the 19S0 s with the studies of chemical reactor stability and latterly to simple closed reactions in the early 1960 s. We shall reserve discussion of the former until the next section. 

The material is organized as follows. In Chapters 1 and 2 the basic chemistry and physics of reactive flow are described, first simply and then in the sophisticated form required for dealing with steady flames. The theory of elementary reaction rate constants is presented in Chapters 3 and 4,... 

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

Previous theoretical kinetic treatments of the formation of secondary, tertiary and higher order ions in the ionization chamber of a conventional mass spectrometer operating at high pressure, have used either a steady state treatment (2, 24) or an ion-beam approach (43). These theories are essentially phenomenological, and they make no clear assumptions about the nature of the reactive collision. The model outlined below is a microscopic one, making definite assumptions about the kinematics of the reactive collision. If the rate constants of the reactions are fixed, the nature of these assumptions definitely affects the amount of reaction occurring. 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Vol. I The Theory of the Steady State, Clarendon, Oxford, 1975. 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Cook has propounded a rather different theory of the nature of the reaction zone. He emphasises that the demonstrable electrical conductivity of the detonation wave is evidence of a high thermal conductivity. Both these effects are ascribed to ionisation of the explosion products. In terms of the reaction zone, this implies a steady pressure with no peaks. 

Problem P7.06.02 reproduces one result from the literature. There it is apparent that in some ranges of the parameters, effectiveness can be much greater than unity, and also that at low values of Thiele modulus several steady states are possible. When both external and internal adiabatic diffusion occur, moreover, other studies find that half a dozen or more steady states can exist. Those kinds of findings involve much computer work. A book by Aris (Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,... 

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