Heterogeneous catalysis rate expressions

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

All these steps can influence the overall reaction rate. The reactor models of Chapter 9 are used to predict the bulk, gas-phase concentrations of reactants and products at point (r, z) in the reactor. They directly model only Steps 1 and 9, and the effects of Steps 2 through 8 are lumped into the pseudohomoge-neous rate expression, a, b,. ..), where a,b,. .. are the bulk, gas-phase concentrations. The overall reaction mechanism is complex, and the rate expression is necessarily empirical. Heterogeneous catalysis remains an experimental science. The techniques of this chapter are useful to interpret experimental results. Their predictive value is limited. 

Steps 1 through 9 constitute a model for heterogeneous catalysis in a fixed-bed reactor. There are many variations, particularly for Steps 4 through 6. For example, the Eley-Rideal mechanism described in Problem 10.4 envisions an adsorbed molecule reacting directly with a molecule in the gas phase. Other models contemplate a mixture of surface sites that can have different catalytic activity. For example, the platinum and the alumina used for hydrocarbon reforming may catalyze different reactions. Alternative models lead to rate expressions that differ in the details, but the functional forms for the rate expressions are usually similar. 

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. 

Using the concept of the rate-determining step significantly simplifies the overall rate expression. Therefore, it is widely used in the analysis of kinetie data, especially in the field of heterogeneous catalysis. 

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 ... 

In the case of gel entrapped biocatalysts, or where the biocatalyst has been immobilised in the pores of the carrier, then the reaction is unlikely to occur solely at the surface. Similarly, the consumption of substrate by a microbial film or floe would be expected to occur at some depth into the microbial mass. The situation is more complex than in the case of surface immobilisation since, in this case, transport and reaction occur in parallel. By analogy with the case of heterogeneous catalysis, which is discussed in Chapter 3, the flux of substrate is related to the rate of reaction by the use of an effectiveness factor rj. The rate of reaction is itself expressed in terms of the surface substrate concentration which in many instances will be very close to the bulk substrate concentration. In general, the flux of substrate will be given by ... 

In some cases of heterogeneous catalysis it is the mere fact of adsorption which, by increasing the concentration of the reacting molecules, accelerates the reaction. The expression for the rate v of a chemical reaction can be given as... 

Complex formation in homogeneous catalysis and adsorption on a catalyst surface share the same principle the total number of sites is constant. The rate expressions for homogeneous and heterogeneous catalysts therefore have a similar form. Usually, for gas phase reactions, partial pressures are used, whereas concentrations are employed for liquid phase reactions. This only has consequences for the dimensions of the constants in the rate expressions. The following approach to the derivation of these rate expressions can be applied to homogeneous as well as to heterogeneous catalyst systems. 

This Langmuir-Hinshelwood mechanism is the one most commonly encountered in the heterogeneous catalysis of gas reactions and the appropriate rate expressions for various special cases are well known [9, 31, 42], In general, we may write... 

These kinetic expressions can be useful in many situations, since they capture two key aspects of heterogeneous catalysis the rate of the reaction, and the saturation of the surface by the reactants. The values assigned to the various kinetic and adsorption parameters in this work produce rates that agree well with those reported in the literature. The liquid-phase components were considered nonvolatile. The saturation concentration of H2 was evaluated using Henry s law. All physical parameters were treated as constants. The catalyst properties were representable for a supported noble metal hydrogenation catalyst. 

Therefore, it is necessary to determine the influence of mass transfer to or from the above-mentioned interfaces on the conversion, which leads to expressions for the flux of a reactant across the interface and for the overall reaction rate. After balancing the disappearance of the components Ai and A2, e. g., at the gas/liquid interface, by analogy with the treatment of the rate of chemical reaction and pore diffusion in heterogeneous catalysis, the overall reaction rate is given by eq. (1) [2] ... 

Adsorption on a solid catalyst surface, complex formation in homogeneous catalysis with metallo-organic complexes and in biocatalysis with enzymes share the same principle, i.e. the total number of sites is constant. Therefore, the rate expressions for reactions on heterogeneous, homogeneous and biocatalysts have a similar form. The constant number of active sites results in rate expressions that differ from homogeneous gas phase kinetics. Partial pressures are usually used in rate expressions for gas-phase reactions, while concentrations are used when the reactions take place in the liquid phase. It appears that definitions and nomenclature of particular kinetics constants in the different sub-communities differ sometimes. In the following sections the expressions used by the different subdisciplines will be compared and their conceptual basis outlined. 

The form of the resulting expression differs from the gas-phase reaction rate expressions due to the presence of a denominator representing the reduction in rate due to adsorption phenomena. The individual terms of this denominator respresent the distribution of the active sites among the possible surface complexes and vacancies. Expressions of this type are termed the Langmuir-Hinshel-wood-Hougen-Watson (LHHW) rate expressions in heterogeneous catalysis and Michaelis-Menten expressions in biocatalysis. 

As in most sections of this chapter we are dealing with heterogeneous catalysis, the production rate will generally be expressed per unit mass of catalyst. The rate of production is negative for reactants and positive for products ... 

---