Catalytic reactions kinetic equations

To conclude the discussion of some technological aspects of the theory of DS, we shall touch upon the question of its role in the catalytic reaction kinetics. Since Langmuir s time, the kinetic laws of a heterogeneous catalytic process have been described exclusively by models involving ordinary differential equation sets. Our results indicate also that under experimental conditions, the researcher is most likely to run into the stratification phenomena, the domain structure formation in a kinetic reactor (stationary. 

Related to the concentration of the active sites, the concentration of the reactant at the catalyst is part of the reaction rate equation, hi order to be able to differentiate between extensive (concentration of adsorbed reactant species) and intensive (impact of the catalytically active site on the reactant molecule) factors, the reaction kinetic equation has to be known in detail and to be carefully analyzed or valuable information would be wasted. 

The so-called two-step sequence method is that the derivation of reaction rate expression only requires to consider two key steps for a reaction involving multielementary steps. Only the rate constants or equilibrium constants of the two key steps appear in rate expression, which are of clear physical meaning. In order to determine the key steps, a concept of most abundant reaction intermediate (Mari) must be introduced. Mari is an intermediate of maximum concentration among all reactive intermediates invovled in the reaction, and the concentration of other intermediates can be ignored. Based on the concepts of both rate determining step and most abundant reaction intermediate, the mechanisms of many catalytic reactions can be simplified to two-step sequences for the derivation of kinetic equations. In order to explain the rules for the treatment of heterogeneous catalytic reaction kinetics by simplest two-step sequences method, two examples are given as follows ... 

It can be seen from the above-mentioned discussion that the Temkin s theory of catalytic reaction kinetics on heterogeneous surfaces is confirmed not only by overall reaction kinetic data of ammonia synthesis, isotherms and adsorptive rates of nitrogen on iron catalysts, more importantly, perhaps, but also some very useful generalization results derived from this theory. For instance, Temkin s equation is obtained based on two steps or simplified two steps mechanism. So, it can apply to any kind of catalytic reactions. The problem is Are some simplifications required by reasonable formation of two step mechanism, and can the assumption of rate determining step be made for non-uniform surface The answer of Temkin s theory is positive, and especially Temkin s theory of catalytic reaction kinetics on non-uniform plays an important role in solving the selection and improvement of catalysts. 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

All catalytic reactions involve chemical combination of reacting species with the catalyst to form some type of inteniiediate complex, the nature of which is the subject of abundant research in catalysis. The overall reaction rate is often determined by the rate at which these complexes are formed and decomposed. The most widely-used nonlinear kinetic equation that describes... 

The development of methods for the kinetic measurement of heterogeneous catalytic reactions has enabled workers to obtain rate data of a great number of reactions [for a review, see (1, )]. The use of a statistical treatment of kinetic data and of computers [cf. (3-7) ] renders it possible to estimate objectively the suitability of kinetic models as well as to determine relatively accurate values of the constants of rate equations. Nevertheless, even these improvements allow the interpretation of kinetic results from the point of view of reaction mechanisms only within certain limits ... 

The simultaneous determination of a great number of constants is a serious disadvantage of this procedure, since it considerably reduces the reliability of the solution. Experimental results can in some, not too complex cases be described well by means of several different sets of equations or of constants. An example would be the study of Wajc et al. (14) who worked up the data of Germain and Blanchard (15) on the isomerization of cyclohexene to methylcyclopentenes under the assumption of a very simple mechanism, or the simulation of the course of the simplest consecutive catalytic reaction A — B —> C, performed by Thomas et al. (16) (Fig. 1). If one studies the kinetics of the coupled system as a whole, one cannot, as a rule, follow and express quantitatively mutually influencing single reactions. Furthermore, a reaction path which at first sight is less probable and has not been therefore considered in the original reaction network can be easily overlooked. 

The kinetics of a coupled catalytic reaction can be well described by equations of the Langmuir-Hinshelwood type, since these are able to express mutual influencing of single reactions. Power-law equations are not suitable for this purpose. 

Kinetic data fitting the rate equation for catalytic reactions that follow the Michaelis-Menten equation, v = k A]/(x + [A]), with[A]0 = 1.00 X 10 J M, k = 1.00 x 10 6 s 1, and k = 2.00 X 10-J molL1. The left panel displays the concentration-time profile on the right is the time lag approach. 

The pseudohomogeneous reaction term in Equation (11.42) is analogous to that in Equation (9.1). We have explicitly included the effectiveness factor rj to emphasis the heterogeneous nature of the catalytic reaction. The discussion in Section 10.5 on the measurement of intrinsic kinetics remains applicable, but it is now necessary to ensure that the liquid phase is saturated with the gas when the measurements are made. The void fraction s is based on relative areas occupied by the liquid and soUd phases. Thus,... 

Catalytic reactions (as well as the related class of chain reactions described below) are coupled reactions, and their kinetic description requires methods to solve the associated set of differential equations that describe the constituent steps. This stimulated Chapman in 1913 to formulate the steady state approximation which, as we will see, plays a central role in solving kinetic schemes. 

In electrocatalysis, in contrast to electrochemical kinetics, the rate of an electrochemical reaction is examined at constant external control parameters so as to reveal the influence of the catalytic electrode (its nature, its surface state) on the rate constants in the kinetic equations. 

Quantitative information can be drawn from such plots. For the a-th order kinetics the slope is the reaction order a and the intercept is In k. For the catalytic reaction considered above with the surface reaction as the rate-limiting process, linearization of the rate equation (5.4-112) leads to ... 

A kinetic model describing the HRP-catalyzed oxidation of PCP by H202 should account for the effects of the concentrations of HRP, PCP, and H202 on the reaction rate. To derive such an equation, a reaction mechanism involving saturation kinetics is proposed. Based on the reaction scheme described in Section 17.3.1, which implies that the catalytic cycle is irreversible, the three distinct reactions steps (Equations 17.2 to 17.4) are modified to include the formation of Michaelis-Menten complexes ... 

Equation 8.3.4 may also be used in the analysis of kinetic data taken in laboratory scale stirred tank reactors. One may directly determine the reaction rate from a knowledge of the reactor volume, flow rate through the reactor, and stream compositions. The fact that one may determine the rate directly and without integration makes stirred tank reactors particularly attractive for use in studies of reactions with complex rate expressions (e.g., enzymatic or heterogeneous catalytic reactions) or of systems in which multiple reactions take place. 

As a result of the kinetics and the equilibria mentioned above, all iodide in the system occurs as methyl iodide. The reaction in Equation (2) makes the rate of the catalytic process independent of the methanol concentration. Within the operation window of the process, the reaction rate is independent of the carbon monoxide pressure. The selectivity in methanol is in the high 90s but the selectivity in carbon monoxide may be as low as 90%. This is due to the water-gas shift reaction ... 

Attempts to employ allenes in palladium-catalyzed oxidations have so far given dimeric products via jr al lyI complexes of type 7i62.63. The fact that only very little 1,2-addition product is formed via nucleophilic attack on jral ly I complex 69 indicates that the kinetic chloropalladation intermediate is 70. Although formation of 70 is reversible, it is trapped by the excess of allene present in the catalytic reaction to give dimeric products. The only reported example of a selective intermolecular 1,2-addition to allenes is the carbonylation given in equation 31, which is a stoichiometric oxidation64. 

In previous chapters, we deal with simple systems in which the stoichiometry and kinetics can each be represented by a single equation. In this chapter we deal with complex systems, which require more than one equation, and this introduces the additional features of product distribution and reaction network. Product distribution is not uniquely determined by a single stoichiometric equation, but depends on the reactor type, as well as on the relative rates of two or more simultaneous processes, which form a reaction network. From the point of view of kinetics, we must follow the course of reaction with respect to more than one species in order to determine values of more than one rate constant. We continue to consider only systems in which reaction occurs in a single phase. This includes some catalytic reactions, which, for our purpose in this chapter, may be treated as pseudohomogeneous. Some development is done with those famous fictitious species A, B, C, etc. to illustrate some features as simply as possible, but real systems are introduced to explore details of product distribution and reaction networks involving more than one reaction step. 

Equation 8.5-11 applies to a first-order surface reaction for a particle of flat-plate geometry with one face permeable. In the next two sections, the effects of shape and reaction order on p are described. A general form independent of kinetics and of shape is given in Section 8.5.4.5. The units of are such that is dimensionless. For catalytic reactions, the rate constant may be expressed per unit mass of catalyst (k )m. To convert to kA for use in equation 8.5-11 or other equations for d>, kA)m is multiplied by pp, the particle density. 

Through such chemisorption studies, the values of >, have been determined not only by geometric accessibility, but also by the chemical heterogeneity of the surface. This can result in abnormal values of D, and demonstrates the scale effect on the kinetics and selectivity of catalytic reactions. For such studies, Farin and Anvir [213] derived the equations that can be applied for characterization of supported catalysts ... 

As compared to the Nemstian case, the plateau is the same but the wave is shifted toward more negative potentials, the more so the slower the electrode electron transfer. An illustration is given in Figure 4.13 for a value of the kinetic parameter where the catalytic plateau is under mixed kinetic control, in between catalytic reaction and substrate diffusion control. For the kjet(E) function, rather than the classical Butler-Volmer law [equation (1.26)], we have chosen the nonlinear MHL law [equation (1.37)]. 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

Catalyst decomposition ( die-out ) during the catalytic reaction is a common phenomenon also distorting the kinetic measurements. If the decomposition reaction obeys a rate equation in a well-behaved manner, one can include the decomposition reaction in the kinetics, but usually one will prefer the use of a stable catalyst. Catalyst decomposition is an import issue in applied catalysis although it has received relatively little attention in literature as far as homogeneous catalysis is concerned [5],... 

Another way of representing catalytic cycles is shown in Figure 4.1. Note that the reactions have been drawn as irreversible reactions while most of them are actually equilibria. If one wants to derive kinetic equations, we strongly recommend the use of reaction sequences as shown on the previous page, because cycles such as 4.1 often lead to mistakes. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Because of the great industrial importance of catalytic reactions, considerable effort has been spent in developing theories from which kinetic equations can rationally be developed. The most useful for our purposes supposes that the reaction takes place on an active site on the surface of the catalyst. Thus three steps are viewed to occur successively at the surface. 

Any type of reactor with known contacting pattern may be used to explore the kinetics of catalytic reactions. Since only one fluid phase is present in these reactions, the rates can be found as with homogeneous reactions. The only special precaution to observe is to make sure that the performance equation used is dimensionally correct and that its terms are carefully and precisely defined. 

A closed loop batch G/S reactor (see Fig. P18.19) is used for catalytic rate studies. For this purpose feed gas with reactant is introduced into the system and is rapidly circulated through the catalyst loop. From the following composition-time data find a kinetic equation in units of mol/ gm min to represent this reaction. 

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