Heterogeneous rate equations

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,... 

Figure 2.3.d.2 I Confidence region heterogeneous rate equation with three parameters after Kittrell [67]). 

Solving equations (11) and (12) for C and <1 substituting these values into the heterogeneous rate equation results in the following expression for the flux ... 

The design of a fixed hed reactor is similar in principle to that of an empty tubular reactors discussed in Chapter 9. The principle differences are that in a fixed bed unit, temperature and concentration gradients occur radially as well as longitudinally whereas radial gradients are uncommon in fluid bed reactors. These variations adversely affect the fluidized reactors and these possibilities make the transition from pilot to commercial scale problematic. However, the calculational procedure for a fixed bed unit is similar to that presented earlier, with a heterogeneous rate equation employed. 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

Information on the composition and temperature changes is obtained from the rate equation, while the mixing patterns are related to the intensity of mixing and reactor geometry. Heat transfer is referred to as the exothermic or endothermic nature of the reactions and the mass transfer to the heterogeneous systems. 

After the rates have been determined at a series of reactant concentrations, the differential method of testing rate equations is applied. Smith [3] and Carberry [4] have adequately reviewed the designs of heterogeneous catalytic reactors. The following examples review design problems in a plug flow reactor with a homogeneous phase. 

Chemical reactions obey the rules of chemical kinetics (see Chapter 2) and chemical thermodynamics, if they occur slowly and do not exhibit a significant heat of reaction in the homogeneous system (microkinetics). Thermodynamics, as reviewed in Chapter 3, has an essential role in the scale-up of reactors. It shows the form that rate equations must take in the limiting case where a reaction has attained equilibrium. Consistency is required thermodynamically before a rate equation achieves success over tlie entire range of conversion. Generally, chemical reactions do not depend on the theory of similarity rules. However, most industrial reactions occur under heterogeneous systems (e.g., liquid/solid, gas/solid, liquid/gas, and liquid/liquid), thereby generating enormous heat of reaction. Therefore, mass and heat transfer processes (macrokinetics) that are scale-dependent often accompany the chemical reaction. The path of such chemical reactions will be... 

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 ... 

An interesting method, which also makes use of the concentration data of reaction components measured in the course of a complex reaction and which yields the values of relative rate constants, was worked out by Wei and Prater (28). It is an elegant procedure for solving the kinetics of systems with an arbitrary number of reversible first-order reactions the cases with some irreversible steps can be solved as well (28-30). Despite its sophisticated mathematical procedure, it does not require excessive experimental measurements. The use of this method in heterogeneous catalysis is restricted to the cases which can be transformed to a system of first-order reactions, e.g. when from the rate equations it is possible to factor out a function which is common to all the equations, so that first-order kinetics results. 

In particular, reactions in heterogeneous catalysis are always a series of steps, including adsorption on the surface, reaction, and desorption back into the gas phase. In the course of this chapter we will see how the rate equations of overall reactions can be constructed from those of the elementary steps. 

Before deriving the rate equations, we first need to think about the dimensions of the rates. As heterogeneous catalysis involves reactants and products in the three-dimensional space of gases or liquids, but with intermediates on a two-dimensional surface we cannot simply use concentrations as in the case of uncatalyzed reactions. Our choice throughout this book will be to express the macroscopic rate of a catalytic reaction in moles per unit of time. In addition, we will use the microscopic concept of turnover frequency, defined as the number of molecules converted per active site and per unit of time. The macroscopic rate can be seen as a characteristic activity per weight or per volume unit of catalyst in all its complexity with regard to shape, composition, etc., whereas the turnover frequency is a measure of the intrinsic activity of a catalytic site. 

Three types of rate equations are shown here. These rate equations ean be used for quite complieated reactions, but a specific method or measurement approach is needed. How we do this is critical to determining accurate estimation of the progress of a solid state reaction. We will discuss suitable methods in another chapter. We now return to the subject of nucleation so that we can apply the rate equations given above to specific cases. First, we examine heterogeneous processes. 

The first of these factors pertains to the complications introduced in the rate equation. Since more than one phase is involved, the movement of material from phase to phase must be considered in the rate equation. Thus the rate expression, in general, will incorporate mass transfer terms in addition to the usual chemical kinetics terms. These mass transfer terms are different in type and number in different kinds of heterogeneous systems. This implies that no single rate expression has a general applicability. 

In general the rate equation for a heterogeneous reaction accounts for more than one process. The present consideration is directed to the general problem of combining the rates for processes of different kinds. Let r1( r2,..., rn be the rates of changes for the individual processes that are to be accounted for by an overall rate. If the changes occur by parallel paths, then the overall rate will be greater than the rate for any individual path. In fact, if the different parallel paths are independent of each other, the overall rate will be simply the sum of all the individual rates, or... 

Discuss the various steps involved in heterogeneous catalysis. Derive an expression for the rate constant and discuss limiting cases of rate equation. 

If the interface reaction rate is extremely small so that mass/heat transfer is rapid enough to transport nutrients to the interface, then interface reaction rate (Equation 4-33) is the overall heterogeneous reaction rate (Figure 1-lla). If the interface reaction is relatively rapid and if the crystal composition is different from the melt composition, the heterogeneous reaction rate may be limited or slowed down by the mass transfer rate because nutrients must be transported to the interface and extra junk must be transported away from the interface (Figures 1-llb and 1-llc). If the crystal composition is the same as the melt composition, then mass transfer is not necessary. When interface reaction rate and mass transfer rate are comparable, both interface reaction and mass transfer would control the overall heterogeneous reaction (Figure 1-lld). 

The I value does not depend on whether or not the reduction is reversible, quasi-reversible, or irreversible. In general, the net current i is the sum of a cathodic and an anodic component. However, when the reduction of O is irreversible, because either the ET is intrinsically slow or as a consequence of a following fast reaction (e.g. bond cleavage), it can be easily shown that the link between i, /(f), and the heterogeneous rate constant A het is equation (25). 

Fig. 6 Representative examples of the steps involved in the convolution analysis approach to obtaining the potential dependence of the heterogeneous rate constant. From top to bottom (a) background-subtracted cyclic voltammograms as a function of scan rate (left to right 0.5, 1, 2, 5, lOVs " ) (b) corresponding convolution curves (c) corresponding potential dependence of logkhet obtained using equation (25). Figures shown are for the reduction of (MeS)2 in DMF/0.1 M TBAP at a glassy carbon electrode.

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. 

Most standard chemical engineering tests on kinetics [see those of Car-berry (50), Smith (57), Froment and Bischoff (19), and Hill (52)], omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (55) from a mathematical standpoint. 

The oxidation of sulphur dioxide to trioxide is one of the oldest heterogeneous catalytic processes. The classic catalyst based on V2Os has therefore been the subject of numerous investigations which are amply reviewed by Weychert and Urbaneck [346]. These authors conclude that none of the 34 rate equations reported is applicable over a wide range of process conditions. Generally, these equations have the form of a power expression, in which the reverse reaction is taken into account within the limits imposed by chemical equilibrium, viz. 

Rate equations for heterogeneous catalytic alkylation of aromatic hydrocarbons... 

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