Catalytic heterogeneous reactions reaction rate constant

After in the foregoing chapter thermodynamic properties at high pressure were considered, in this chapter other fundamental problems, namely the influence of pressure on the kinetic of chemical reactions and on transport properties, is discussed. For this purpose first the molecular theory of the reaction rate constant is considered. The key parameter is the activation volume Av which describes the influence of the pressure on the rate constant. The evaluation of Av from measurement of reaction rates is therefor outlined in detail together with theoretical prediction. Typical value of the activation volume of different single reactions, like unimolecular dissociation, Diels-Alder-, rearrangement-, polymerization- and Menshutkin-reactions but also on complex homogeneous and heterogeneous catalytic reactions are presented and discussed. 

The example of consecutive, irreversible heterogeneous catalytic reaction of the type A —> B — C has been solved in a more general way by Thomas et al. (16). The authors considered scheme (III) with the listed values of the rate constants of surface reactions along with the constants of adsorption and desorption of the reactant A and of the product C. 

The quantitative solution of the problem, i.e. simultaneous determination of both the sequence of surface chemical steps and the ratios of the rate constants of adsorption-desorption processes to the rate constants of surface reactions from experimental kinetic data, is extraordinarily difficult. The attempt made by Smith and Prater 82) in a study of cyclohexane-cyclohexene-benzene interconversion, using elegant mathematic procedures based on the previous theoretical treatment 28), has met with only partial success. Nevertheless, their work is an example of how a sophisticated approach to the quantitative solution of a coupled heterogeneous catalytic system should be employed if the system is studied as a whole. 

An analogous situation occurs in the catalytic cracking of mixed feed gas oils, where certain components of the feed are more difficult to crack (less reactive or more refractory) than the others. The heterogeneity in reactivities (in the form of Equations 3 and 5) makes kinetic modelling difficult. However, Kemp and Wojclechowskl (11) describe a technique which lumps the rate constants and concentrations into overall quantities and then, because of the effects of heterogeneity, account for the changes of these quantities with time, or extent of reaction. First a fractional activity is defined as... 

The kinetic factor is proportional to the energetic state of the system and (for heterogeneous catalytic systems) the number of active sites per unit volume (mass) of catalyst. The driving-force group includes the influence of concentration and distance from chemical equilibrium on the reaction rate, and the hindering group describes the hindering effect of components of the reaction mixture on the reaction rate. The kinetic factor is expressed as the rate constant, possibly multiplied by an equilibrium constant(s) as will be shown later. 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

Electrochemistry is in many aspects directly comparable to the concepts known in heterogeneous catalysis. In electrochemistry, the main driving force for the electrochemical reaction is the difference between the electrode potential and the standard potential (E — E°), also called the overpotential. Large overpotentials, however, reduce the efficiency of the electrochemical process. Electrode optimization, therefore, aims to maximize the rate constant k, which is determined by the catalytic properties of the electrode surface, to maximize the surface area A, and, by minimization of transport losses, to result in maximum concentration of the reactants. 

The interpretation of slopes also requires meaningful rate data. When the reaction consists of a series of elementary steps (and this is always so with heterogeneous catalytic reactions), the rate coefficients obtained from a superficial treatment of a limited set of measurements may be composites of several rate and equilibrium constants for individual steps, in favorable cases constituting a product. As every step may be influenced by the substituents, the resulting effect can be easily attributed to a false elementary step. 

Rate equations for simple reversible reactions are often developed from mechanistic models on the assumption that the kinetics of elementary steps can be described in terms of rate constants and surface concentrations of intermediates. An application of the Langmuir adsorption theory for such development was described in the classic text by Hougen and Watson (/ ), and was used for constructing rate equations for a number of heterogeneous catalytic reactions. In their treatment it was assumed that one step would be rate-controlling for a unique mechanism with the other steps at equilibrium. 

Two important ways in which heterogeneously catalyzed reactions differ from homogeneous counterparts are the definition of the rate constant k and the form of its dependence on temperature T. The heterogeneous rate equation relates the rate of decline of the concentration (or partial pressure) c of a reactant to the fraction / of the catalytic surface area that it covers when adsorbed. Thus, for a first-order reaction,... 

The apparent rate constant in (2.10), which is obtained by multiplying a true rate constant kc and the square root of an equilibrium constant, Keq, can show a law of dependence on temperature different from the simple Arrhenius law. In some cases, even a negative temperature dependence can be observed. Moreover, if both mechanisms (2.6) and (2.7)-(2.8) are active in parallel, the observed reaction rate is the sum of the single rates, and an effective reaction order variable from 1 /2 to 1 can be observed with respect to reactant A. Variable and fractionary reaction orders can be also encountered in heterogeneous catalytic reactions as a consequence of the adsorption on a solid surface [6],... 

A knowledge of the kinetics of the reaction at the active sites is of primary importance in determining the nature of catalytic action in heterogeneous catalysis. Information about the nature of the catalyst-substrate interaction can be obtained from the way in which the rate constants in the kinetics change on variation of such parameters as temperature, catalyst treatment, and catalyst composition. In addition, these constants are the quantities which should correlate with structural information such as that obtained by the methods of solid state physics. However, the true kinetics at the active sites is not always obtained unless certain precautions are taken, as has been pointed out in a recent volume of Advances in Catalysis (1). 

In summary, it can be seen for the three-step reaction scheme of this example that the net rate of the overall reaction is controlled by three kinetic parameters, KTSi, that depend only on the properties of the transition states for the elementary steps relative to the reactants (and possibly the products) of the overall reaction. The reaction scheme is represented by six individual rate constants /c, and /c the product of which must give the equilibrium constant for the overall reaction. However, it is not necessary to determine values for five linearly independent rate constants to determine the rate of the overall reaction. We conclude that the maximum number of kinetic parameters needed to determine the net rate of overall reaction is equal to the number of transition states in the reaction scheme (equal to three in the current case) since each kinetic parameter is related to a quasi-equilibrium constant for the formation of each transition state from the reactants and/or products of the overall reaction. To calculate rates of heterogeneous catalytic reactions, an addition kinetic parameter is required for each surface species that is abundant on the catalyst surface. Specifically, the net rate of the overall reaction is determined by the intrinsic kinetic parameters Kf s as well as by the fraction of the surface sites, 0, available for formation of the transition states furthermore, the value of o. is determined by the extent of site blocking by abundant surface species. 

As a first approximation a convective term in the film region has been negleted, u is the superficial gas velocity and u f denotes the gas velocity at minimum fluidization conditions. Tne specific mass transfer area a(h) is based on unit volume of the expanded fluidized bed and e OO is the bubble gas hold-up at a height h above the bottom plate. Mathematical expressions for these two latter quantities may be found in detail in (20). The concentrations of the reactants in the bubble phase and in film and bulk of the suspension phase are denoted by c, c and c, respectively. The rate constant for the first order heterogeneous catalytic reaction of the component i to component j is denoted... 

Catalytic reactions consist of a reaction cycle formed by a series of elementary reaction steps. Hence the rate expression is in general a function of many parameters. In heterogeneously catalysed reactions reactant molecules are adsorbed on the catalyst surface (characterized by equilibrium constants Kj), undergo chemical modifications on the surface to give adsorbed products with rate constants fc, and these products finally desorb. The overall catalyst activity and selectivity is determined by the composition and structure of its surface. Hence it is important to relate constants, such as fc and K with the chemical reactivity of the catalyst surface. 

The experiments with the inert tracer may only show that the time, necessary for the fluid in the reactor to be well mixed, is much smaller than the average residence time. When a chemical reaction takes place, an additional time-scale, the time constant of the chemical reaction, appears. This time characterizes the reaction rate and can be defined as the time in which the reaction proceeds to a certain conversion, say 50%. For many practical heterogeneous catalytic reactions, the reaction time is so short that reactants entering the reactor may be converted without being mixed, for example, during the first cycle. For such fast reactions, of course, the reactor cannot be considered as gradient-free, whatever the recirculation ratio is. 

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