Rate expressions catalysts

Rate Expression Catalyst Activation Energy (kJ/mol) Reference... 

For the synthesis of ammonia, Nj -i- 3H2 —> 2NH3, over an iron catalyst, develop the rate expression for the following mechanism... 

Sometimes the rate expression obtained by the process just described involves a reactive intermediate, that is, a species produced in one step of the mechanism and consumed in a later step. Ordinarily, concentrations of such species are too small to be determined experimentally. Hence they must be eliminated from the rate expression if it is to be compared with experiment. The final rate expression usually includes only those species that appear in the balanced equation for the reaction. Sometimes, the concentration of a catalyst is included, but never that of a reactive intermediate. 

For the reaction of stannic chloride with toluene (this aromatic being used here because of the lower effectiveness of the catalyst), different kinetics were obtained the rate expression being... 

A promoter is, first of all, just another adsorbate on the catalyst surface. In a Langmuirian context it blocks sites. Thus if its coverage on the catalyst surface is 0P, the simplest LHHW rate expression (2.14) becomes ... 

Figure 8.2. Plot of the effect of gaseous composition and of n=( )/kbT during C2H4 oxidation on two Pt catalyst films, labeled R1 and R2, showing that the rate expression given by (Eq. 8.1) is valid both under open-circuit conditions (open symbols) and also under NEMCA conditions (filled symbols).1 Reprinted with permission from Academic Press.

The first step, as we have already seen (12-3), actually consists of two steps. The second step is very similar to the first step in electrophilic addition to double bonds (p. 970). There is a great deal of evidence for this mechanism (1) the rate is first order in substrate (2) bromine does not appear in the rate expression at all, ° a fact consistent with a rate-determining first step (3) the reaction rate is the same for bromination, chlorination, and iodination under the same conditions (4) the reaction shows an isotope effect and (5) the rate of the step 2-step 3 sequence has been independently measured (by starting with the enol) and found to be very fast. With basic catalysts the mechanism may be the same as that given above (since bases also catalyze formation of the enol), or the reaction may go directly through the enolate ion without formation of the enol ... 

Equation (1.20) is frequently used to correlate data from complex reactions. Complex reactions can give rise to rate expressions that have the form of Equation (1.20), but with fractional or even negative exponents. Complex reactions with observed orders of 1/2 or 3/2 can be explained theoretically based on mechanisms discussed in Chapter 2. Negative orders arise when a compound retards a reaction—say, by competing for active sites in a heterogeneously catalyzed reaction—or when the reaction is reversible. Observed reaction orders above 3 are occasionally reported. An example is the reaction of styrene with nitric acid, where an overall order of 4 has been observed. The likely explanation is that the acid serves both as a catalyst and as a reactant. The reaction is far from elementary. 

Strictly gas-phase CSTRs are rare. Two-phase, gas-liquid CSTRs are common and are treated in Chapter 11. Two-phase, gas-solid CSTRs are fairly common. When the solid is a catalyst, the use of pseudohomogeneous kinetics allows these two-phase systems to be treated as though only the fluid phase were present. All concentration measurements are made in the gas phase, and the rate expression is fitted to the gas-phase concentrations. This section outlines the method for fitting pseudo-homogeneous kinetics using measurements made in a CSTR. A more general treatment is given in Chapter 10. 

Studies on similar catalysts have suggested a rate expression of the form... 

More complicated rate expressions are possible. For example, the denominator may be squared or square roots can be inserted here and there based on theoretical considerations. The denominator may include a term k/[I] to account for compounds that are nominally inert and do not appear in Equation (7.1) but that occupy active sites on the catalyst and thus retard the rate. The forward and reverse rate constants will be functions of temperature and are usually modeled using an Arrhenius form. The more complex kinetic models have enough adjustable parameters to fit a stampede of elephants. Careful analysis is needed to avoid being crushed underfoot. 

Let us look at the limiting cases, starting from the complete rate expression in Eq. (111). If the conversions from adsorbed intermediate into either product or reactant are fast, then the denominator approaches 1 and the catalyst surface will be mostly empty, 6r s 0. In this case the rate expression becomes equal to that of the uncatalyzed reaction, and the order in the reactant becomes +1. Under such conditions it is beneficial to increase the reactant concentration since the surface is mainly unoccupied, and the catalyst inefficiently used. 

A full analysis of the rate expression reveals that all data on the Cu(lOO) single crystal are modeled very well, as shown in Fig. 8.10. Even more important is that the model also describes data obtained on a real catalyst measured under considerably different conditions reasonably well, indicating that the micro-kinetic model captures the most important features of the methanol synthesis (Fig. 8.11). 

As the adsorption of hydrogen is rather weak, the corresponding term in the denominator may be omitted. The rate expression shows that the reaction is suppressed by H2S. Hence, the most active catalysts (which appear at the top of the volcano curve of Fig. 9.7)... 

Using the pulse microreactor method( ), the general rate expression for reoxldatlon of bismuth molybdate catalysts was found to be ... 

Greater success in extending kinetic measurements to higher degrees of polymerization has been achieved with polyesterifications catalyzed by a small amount of a strong acid catalyst. The catalyst concentration being constant throughout the process, the second-order rate expression... 

The Cu-, Co- and Fe-ZSM-5 catalysts are active systems for the decomposition of N2O, but their behaviour differs with respect to conditions and gas atmospheres. They all seem to obey a (nearly) first order dependency towards pmo> which can be rationalised by the two step kinetic model given by eqs. (2) and (3). A step like eq. (3) is quite well feasible, since the TM ions in ZSM-5 can be coordinated by several ligands simultaneously [18,22], The resulting rate expression is given by eq. (7). 

The TOF was calculated for the 4-fert-butylaniline hydrogenation over each of the four catalysts. Using the relationship between metal crystallite size and TOF shown in Figure 3, the TOF for each catalyst was normalised to 1 nm and back calculated to a rate expression to remove the metal ciystallite size effect from the rate. With the rate expressed in this way the effect of the pore size is clearly seen (Figure 4). 

According to the proposed mechanism, the hydrogenation of olefin by iridium catalyst should conform to the following rate expression,... 

In contrast to kinetic models reported previously in the literature (18,19) where MO was assumed to adsorb at a single site, our preliminary data based on DRIFT results suggest that MO exists as a diadsorbed species with both the carbonyl and olefin groups being coordinated to the catalyst. This diadsorption mode for a-p unsaturated ketones and aldehydes on palladium have been previously suggested based on quantum chemical predictions (20). A two parameter empirical model (equation 4) where - rA refers to the rate of hydrogenation of MO, CA and PH refer to the concentration of MO and the hydrogen partial pressure respectively was developed. This rate expression will be incorporated in our rate-based three-phase non-equilibrium model to predict the yield and selectivity for the production of MIBK from acetone via CD. 

While in presence of catalyst the reaction is assumed to LHHW type of mechanism and its rate expression is given by ... 

Neyens (4) has studied the bromination of metaxylene at 17 °C. The reaction is carried out by introducing small quantities of iodine and bromine into pure liquid xylene and following the rate of disappearance of bromine by titrating samples removed from the liquid to determine their bromine content. The iodine serves as a catalyst for the reaction. Since the concentrations of xylene and catalyst remain essentially unchanged during the course of the reaction, it may be assumed that the rate expression is of the form... 

Thus mechanism B, which consists solely of bimolecular and unimolecular steps, is also consistent with the information that we have been given. This mechanism is somewhat simpler than the first in that it does not requite a ter-molecular step. This illustration points out that the fact that a mechanism gives rise to the experimentally observed rate expression is by no means an indication that the mechanism is a unique solution to the problem being studied. We may disqualify a mechanism from further consideration on the grounds that it is inconsistent with the observed kinetics, but consistency merely implies that we continue our search for other mechanisms that are consistent and attempt to use some of the techniques discussed in Section 4.1.5 to discriminate between the consistent mechanisms. It is also entirely possible that more than one mechanism may be applicable to a single overall reaction and that parallel paths for the reaction exist. Indeed, many catalysts are believed to function by opening up alternative routes for a reaction. In the case of parallel reaction paths each mechanism proceeds independently, but the vast majority of the reaction will occur via the fastest path. 

In the design of an industrial scale reactor for a new process, or an old one that employs a new catalyst, it is common practice to carry out both bench and pilot plant studies before finalizing the design of the commercial scale reactor. The bench scale studies yield the best information about the intrinsic chemical kinetics and the associated rate expression. However, when taken alone, they force the chemical engineer to rely on standard empirical correlations and prediction methods in order to determine the possible influence of heat and mass transfer processes on the rates that will be observed in industrial scale equipment. The pilot scale studies can provide a test of the applicability of the correlations and an indication of potential limitations that physical processes may place on conversion rates. These pilot plant studies can provide extremely useful information on the temperature distribution in the reactor and on contacting patterns when... 

This section is concerned with analyses of simultaneous reaction and mass transfer within porous catalysts under isothermal conditions. Several factors that influence the final equation for the catalyst effectiveness factor are discussed in the various subsections. The factors considered include different mathematical models of the catalyst pore structure, the gross catalyst geometry (i.e., its apparent shape), and the rate expression for the surface reaction. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities Extension to Reactions Other than First-Order and Various Catalyst Geometries. The analysis developed in Section 12.3.1.3 may be extended in relatively simple straightforward fashion to other integer-order rate expressions and to other catalyst geometries such as flat plates and cylinders. Some of the key results from such extensions are treated briefly below. 

Effectiveness Factors for Hougen-Watson Rate Expressions. The discussion thus far and the vast majority of the literature dealing with effectiveness factors for porous catalysts are based on the assumption of an integer-power reaction rate expression (i.e., zero-, first-, or second-order kinetics). In Chapter 6, however, we stressed the fact that heterogeneous catalytic reactions are more often characterized by more complex rate expressions of the Hougen-Watson type. Over a narrow range of... 

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