10.3.7 Temperature Dependence of the Rate Law

Consider a surface-reaction-limited irreversible isomerization

in which both A and B are adsorbed on the surface, the rate law is

10-55

The specific reaction rate, k, will usually follow an Arrhenius temperature dependence and increase exponentially with temperature. However, the adsorption of all species on the surface is exothermic. Consequently, the higher the temperature, the smaller the adsorption equilibrium constant. That is, as the temperature increases, K_A and K_B decrease resulting in less coverage of the surface by A and B. Therefore, at high temperatures, the denominator of catalytic rate laws approaches 1. That is, at high temperatures (low coverage)

The rate law could then be approximated as

10-56

or for a reversible isomerization we would have

10-57

The algorithm we can use as a start in postulating a reaction mechanism and rate-limiting step is shown in Table 10-4. Again, we can never really prove a mechanism to be correct by comparing the derived rate law with experimental data. Independent spectroscopic experiments are usually needed to confirm the mechanism. We can, however, prove that a proposed mechanism is inconsistent with the experimental data by following the algorithm in Table 10-4. Rather than taking all the experimental data and then trying to build a model from the data, Box et al.¹⁴ describe techniques of sequential data collection and model building.

10.4 Heterogeneous Data Analysis for Reactor Design

In this section we focus on four operations that reaction engineers need to be able to accomplish:

1. Developing an algebraic rate law consistent with experimental observations,
2. Analyzing the rate law in such a manner that the rate law parameters (e.g., k, K_A) can readily be determined from the experimental data,
   

   

3. Finding a mechanism and rate-limiting step consistent with the experimental data
4. Designing a catalytic reactor to achieve a specified conversion

We shall use the hydrodemethylation of toluene to illustrate these four operations.

Hydrogen and toluene are reacted over a solid mineral catalyst containing clinoptilolite (a crystalline silica-alumina) to form methane and benzene¹⁵:

We wish to design a packed-bed reactor and a fluidized CSTR to process a feed consisting of 30% toluene, 45% hydrogen, and 25% inerts. Toluene is fed at a rate of 50 mol/min at a temperature of 640° C and a pressure of 40 atm (4052 kPa). To design the PBR, we must first determine the rate law from the differential reactor data presented in Table 10-6. In this table we are given the rate of reaction of toluene as a function of the partial pressures of hydrogen (H₂), toluene (T), benzene (B), and methane (M). In the first two runs, methane was introduced into the feed together with hydrogen and toluene, while the other product, benzene, was fed to the reactor together with the reactants only in runs 3, 4, and 6. In runs 5 and 16, both methane and benzene were introduced in the feed. In the remaining runs, none of the products was present in the feedstream. Because the conversion was less than 1% in the differential reactor, the partial pressures of the products, methane and benzene, formed in these runs were essentially zero, and the reaction rates were equivalent to initial rates of reaction.

Table 10-6. Data from a Differential Reactor