Heterogeneous catalysis rate equation

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. 

Equation (10.12) is the simplest—and most generally useful—model that reflects heterogeneous catalysis. The active sites S are fixed in number, and the gas-phase molecules of component A compete for them. When the gas-phase concentration of component A is low, the k a term in Equation (10.12) is small, and the reaction is first order in a. When a is large, all the active sites are occupied, and the reaction rate reaches a saturation value of kjkd-The constant in the denominator, is formed from ratios of rate constants. This makes it less sensitive to temperature than k, which is a normal rate constant. 

This equation gives (0) = 0, a maximum at =. /Km/K2, and (oo) = 0. The assumed mechanism involves a first-order surface reaction with inhibition of the reaction if a second substrate molecule is adsorbed. A similar functional form for (s) can be obtained by assuming a second-order, dual-site model. As in the case of gas-solid heterogeneous catalysis, it is not possible to verify reaction mechanisms simply by steady-state rate measurements. 

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. 

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

In general, the rate (A ) of heterogeneous catalysis in a gas-catalyst (and liquid-catalyst) reaction may be expressed as the product of the rate coefficient A o and a function of pressure (or concentration), p, i.e. k = kof(p) where p is the partial pressure of the reactant, ko depends on the reaction conditions and may involve reaction steps prior to the first rate-determining step of the reaction. A convenient method for determining A o is to use the Arrhenius equation ... 

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. 

Case 3 will be of special interest in this paper. It is encountered in all the examples listed above for Equations (1) and (2). Especially in heterogeneous catalysis it shows what has been called the compensation effect (C.E.). This term indicates that an increase in the enthalpy of activation AH frequently has not the expected result of a considerable decrease in the rate constant, because there occurs a simultaneous increase in the entropy of activation AS or of the frequency factor A, which compensates partly or entirely for the change in the exponent (AH /Er, or AE/ET). 

The statement of this equation is commonly called the rate equation or the rate law. Frequently, in heterogeneous catalysis, the function / is of the form... 

If a chemical reaction is operated in a flow reactor under fixed external conditions (temperature, partial pressures, flow rate etc.), usually also a steady-state (i.e., time-independent) rate of reaction will result. Quite frequently, however, a different response may result The rate varies more or less periodically with time. Oscillatory kinetics have been reported for quite different types of reactions, such as with the famous Belousov-Zha-botinsky reaction in homogeneous solutions (/) or with a series of electrochemical reactions (2). In heterogeneous catalysis, phenomena of this type were observed for the first time about 20 years ago by Wicke and coworkers (3, 4) with the oxidation of carbon monoxide at supported platinum catalysts, and have since then been investigated quite extensively with various reactions and catalysts (5-7). Parallel to these experimental studies, a number of mathematical models were also developed these were intended to describe the kinetics of the underlying elementary processes and their solutions revealed indeed quite often oscillatory behavior. In view of the fact that these models usually consist of a set of coupled nonlinear differential equations, this result is, however, by no means surprising, as will become evident later, and in particular it cannot be considered as a proof for the assumed underlying reaction mechanism. 

As this chapter has shown, rate equations of multistep homogeneous catalysis are still relatively simple if the catalyst-containing intermediates are at trace level, but the free catalyst is not. In heterogeneous catalysis this corresponds to an almost entirely unoccupied catalyst surface. Since adsorption is prerequisite for reaction, low surface coverage results in low rates and therefore is of practical interest only in exceptional situations. Heterogeneous catalysis cannot avoid dealing with substantially covered... 

In essence, the procedures for reduction of complexity described in this book are applicable in principle to reactions on catalyst surfaces, provided the latter are uniform and no segregation of adsorbed species occurs. However, in view of the wealth of other complicating factors, the effort may well be beyond a point of diminishing returns unless a very simple rate equation results. While the strictly kinetic problems are largely analogous in both fields, the much greater complexity of the peripheral conditions in heterogeneous catalysis leaves less room for inclusion of finer reaction-kinetic detail. 

The rates of product formation (and reactant consumption) are seen to be of order one half in the initiator or, if the reaction is initiated by a reactant converted in the propagation cycle, the rate equation involves exponents of one half or integer multiples of one half. For an example, see the hydrogen-bromide reaction below. This is one of the exceptions to the rule that reasonably simple mechanisms do not yield rate equations with fractional exponents. [The other exceptions are reactions with fast pre-dissociation (see Section 5.6) and of heterogeneous catalysis with a reactant that dissociates upon adsorption.]... 

The preceeding four sections conclude our discussion of elementary heterogeneous catalysis mechanisms and design equations. In the following section we shall work through an example problem using experimental data to (1) deduce a rate law, (2) determine a mechanism consistent with experimental data, (3) evaluate the rate law parameters, and (4) design a CSTR and packed-bed reactor. 

In Chap. 8 heterogeneous catalysis was explained by postulating a three-step process (1) chemisorption of at least one reactant on the solid, (2) surface reaction of the chemisorbed substance, and (3) desorption of the product from the catalytic surface. Now our objective is to formulate rate and equilibrium equations for these steps. We shall consider the kinetics and equilibrium of adsorption and then examine rate equations for the overall reaction. 

See also J. M. Thomas and W. J. Thomas, in Introduction to the Principles of Heterogeneous Catalysis, pp. 458-459, Academic Press Inc., New York, 1967, for rate equations assembled in tabular form for various controlling mechanisms for the two reactions A B and A + B C. 

There are important exceptions Most chain reactions and some reactions of heterogeneous catalysis or polymerization or involving pre-dissociation produce exponents of one half or integer multiples of one half in power-law or one-plus rate equations (see Sections 5.6, 9.2, 10.3, and 11.3.1). Such exponents should be accepted if found not to vary with conversion and if there is good reason to believe that a mechanism of this kind may be operative. 

Different mechanisms and rate-controlling steps may produce rate equations of the same algebraic form, making it impossible to identify mechanism and rate control conclusively on the basis of an empirical rate equation alone. This happens more often in heterogeneous catalysis than in homogeneous reactions. A clearer indication may be gained from studies of the temperature dependence of the coefficients, concentration dependence of initial rates, and tests of model predictions. 

The above analysis treats the surface in an ideal manner and does not include phenomena such as surface heterogeneity and surface diffusion (Kiperman et al., 1989 Boudart, 1989). These phenomena and others are usually included implicitly into the rate constants and are rarely treated explicitly in work related to the development of kinetic rate equations. However, in catalysis research, all surface phenomena are of very high importance. 

If there are more than two steps and the mechanism of a heterogeneous catalytic reaction follows the linear Christiansen sequences, then the general kinetic expressions are given by eq. (5.94-5.95). Some particular cases of Christiansen sequences with 3, 4, 6 and 8 steps were presented for homogeneous catalysis by metal complexes, e.g., equations (5.72, 5.76, 5.84) and (5.88) respectively. It should be stressed that in the case of heterogeneous catalysis equations for the reaction rates are exactly the same, which is not surprising as similar kinetic steps describe the reaction mechanisms. 

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