General Approach—Single Reaction

Concentration and temperature gradients inside a catalyst particle can influence the rate of reaction, i.e., the apparent catalyst activity. They can also influence the product distribution, i.e., the apparent catalyst selectivity. First, let s deal with the reaction rate. 

The most common approach to quantifying the effect of intranal concentration and temperature gradients on the reaction rate is to apply a correction factor. 

In this equation, the actual reaction rate is the rate that is obs ed, i.e., measured, in a catalyst particle in which gradients are present. The rate with no gradients is die rate that would be observed if the concentrations and the temperature throughout the particle were equal to their respective values at the external surface. Both of these rates are intensive variables, i.e., rate per unit weight or per unit volume of catalyst. For example, if the reaction was irreversible and first order in A, the rate with no internal gradients would be (7s)CA S — Aexp( /f 7s)CA,s- 

The parameter r is the correction factor that accounts for the effect of intranal transport on the reaction rate. This parameter is known as the int al effectiveness factor, or simply the effectiveness factor. According to Eqn. (9-2), the actual reaction rate can be obtained by multiplying the rate with no internal gradients by rj. The problem of accounting for internal temperature and concentration gradients then boils down to predicting the value of t.  

The effectiveness factor can be related to the system parameters by solving the differential equations that describe mass and energy transport inside the catalyst particle. To illustrate, consider a control volume that consists of a spherical shell of thickness dr within a spherical catalyst particle, as shown below. 

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The examples in this section have treated a single, second-order reaction, although the approach can be generalized to multiple reactions with arbitrary... 

In the range of temperatures and pressures where the reaction is substantially reversible, the kinetics is much more complicated. There is no grounds to consider chemical changes described by (272) and (273) as independent, not interconnected, reactions. Conversely, if processes (272) and (273) occur on the same surface sites, then free sites will act as intermediates of both processes. Thus one must use the general approach, treating (272) and (273) as overall equations of a certain single reaction mechanism. But if a reaction is described by two overall equations, its mechanism should include at least two basic routes hence, the concept of reaction rate in the forward and reverse directions can be inapplicable in this case. However, experiments show that water-gas equilibrium (273) is maintained with sufficient accuracy in the course of the reaction. Let us suppose that the number of basic routes of the reaction is 2 then, as it has been explained in Section VIII, since one of the routes is at equilibrium, the other route, viz., the route with (272) as overall equation, can be described in terms of forward, r+, and reverse, r, reaction rates. The observed reaction rate is then the difference of these... 

So far the generalized modulus of Bischoff has only been applied for a single reaction. Since two reactions are occurring simultaneously, each with their effectiveness factor, (18) can only be conveniently applied when a simple relationship exists between dCj and dC2. The experimental program led to a fairly constant ratio of CO/COj over a rather broad range of total conversion, confirming previous observations by Akers and Camp (18). Therefore, the ratio of dCj/dC2 was approximated by (cf-cf T/ (cl-C ), which is the more accurate the more equilibrium is approached. 

The postulate of quasi-equilibrium of all steps except a single one that controls the rate is very powerful. It reduces the mathematical complexity of kinetics even of large networks to quite simple rate equations and has become a favorite tool, employed today in a great majority of publications on kinetics of multistep homogeneous reactions, sometimes uncritically. In many cases, a sharp distinction between fast and slow steps cannot be justified. A more general approach that avoids the postulate of a single rate-controlling step and contains the results obtained with it as special cases will be described in Sections 4.3 and 6.3 and widely used in later Chapters. 

The following describes results of three, relatively simple chemical reactions involving hydrocarbons on model single crystal metal catalysts that illustrate this general approach, namely, acetylene cyclotrimerization and the hydrogenation of acetylene and ethylene, all catalyzed by palladium. The selected reactions fulfdl the above conditions since they occur in ultrahigh vacuum, while the measured catalytic reaction kinetics on single crystal surfaces mimic those on reahstic supported catalysts. While these are all chemically relatively simple reactions, their apparent simplicity belies rather complex surface chemistry. 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

However, we need to go beyond the simpler ideas. Chemical reactions are more complex than the first few sections of this article would suggest. For example, there may be multiple reaction paths (or mechanisms, if you like) for a single reaction there may be multiple symmetry related paths (versions of the same mechanism) for a reaction there may be different reaction paths for different reactions in close proximity on the PES (competing reactions). These macroscopic difficulties for the reaction path concept have echoes even in the simple case of a single reaction path when we consider in detail the shape of the valley walls. In later sections we explore a way in which the reaction path approach forms the nucleus of a more general approach to interfacing ab initio quantum chemistry and reaction dynamics in these more complex scenarios. 

As mentioned, complexity can also arise by a combination of homogeneous and catalytic steps in a single reaction. The approach to such a reaction need not follow the formal procedure outlined for complex reactions in general. It tends to be system specific, and we illustrate in Example 5.S an approach that can be modified as required to suit individual systems. 

In this section we use two types of complex charge-transfer reaction to illustrate the general approach to the elucidation of reaction mechanisms at single crystal electrodes. These reactions are photocurrent doubling at -type and p-type semiconductors and the (photo)anodic oxidation of the semiconductor itself. 

Dabes, Finn, and Wilke (1973) in a more general approach suggested that (a) only the upper limit of growth rate is fixed by a single enzymatic step (rds concept proposed by Blackman in 1905), and (b) at low substrate concentrations, more than one step in a series of enzymatic reactions influences growth rate. These authors analyzed the linear sequence of enzymatic reactions... 

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