Second-order kinetics effectiveness factors

Saddle point. 170 Salt effects. 206-214 Scavenging (see Reactions, trapping) Second-order kinetics. 18-22, 24 in one component, 18-19 in two components (mixed), 19-22 Selectivity. 112 Sensitivity analysis. 118 Sensitivity factor, 239-240 Sequential reactions (see Consecutive reactions)... 

Curve B of Figure 12.3 [adopted from Wheeler (38)] represents the dependence of the effectiveness factor on Thiele modulus for second-order kinetics. Values of r for first-order and zero-order kinetics in straight cylindrical pores are shown as curves A and C, respectively. Each curve is plotted versus its appropriate modulus. 

Plots of effectiveness factors versus corresponding Thiele moduli for zero-, first-, and second-order kinetics based on straight cylindrical pore model. For large hr, values of r are as follows ... 

Notice that in the region of fast chemical reaction, the effectiveness factor becomes inversely proportional to the modulus h2. Since h2 is proportional to the square root of the external surface concentration, these two fundamental relations require that for second-order kinetics, the fraction of the catalyst surface that is effective will increase as one moves downstream in an isothermal packed bed reactor. 

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... 

Reactant concentrations and effectiveness factors are calculated for diffusion and second-order kinetics in isothermal catalytic pellets via the methodology described above. In each case, convergence is achieved when the dimensionless molar density of reactant A at the external surface is 1 5, where S is on the order of 10 or less. 

Notice that the molar density of key-limiting reactant A on the external surface of the catalytic pellet is always used as the characteristic quantity to make the molar density of component i dimensionless in all the component mass balances. This chapter focuses on explicit numerical calculations for the effective diffusion coefficient of species i within the internal pores of a catalytic pellet. This information is required before one can evaluate the intrapellet Damkohler number and calculate a numerical value for the effectiveness factor. Hence, 50, effective is called the effective intrapellet diffusion coefficient for species i. When 50, effective appears in the denominator of Ajj, the dimensionless scaling factor is called the intrapellet Damkohler number for species i in reaction j. When the reactor design focuses on the entire packed catalytic tubular reactor in Chapter 22, it will be necessary to calcnlate interpellet axial dispersion coefficients and interpellet Damkohler nnmbers. When there is only one chemical reaction that is characterized by nth-order irreversible kinetics and subscript j is not required, the rate constant in the nnmerator of equation (21-2) is written as instead of kj, which signifies that k has nnits of (volume/mole)"" per time for pseudo-volumetric kinetics. Recall from equation (19-6) on page 493 that second-order kinetic rate constants for a volnmetric rate law based on molar densities in the gas phase adjacent to the internal catalytic surface can be written as... 

The data in the literature is based on functional polymers prepared from vinyl monomers, namely styrene and vinylpyridine isomers. These polymers have flexible hydrocarbon backbones which provide a non-polar environment for the active sites. The major factors causing negative or positive deviation from normal second order kinetics in quaterniztion in polymeric systems are limited to steric, electrostatic field and hydrophilic effects. The influence of backbone rigidity or polarity could not be ascertained. 

Chloromethylated polysulfone indeed exhibits different kinetic behavior in the quaternization with TEA than its corresponding model compound. From experimental results, it is clear that the rate retardation is not due to steric hindrance, the degree of chloro-methylation on the polymer chain, or a salt effect. Stereoisomeric effects are not a potential factor, since chloromethylated polysulfone consists of only one detectable isomer. In spite of these results, we knew that the polymer backbone must play an important role in this reaction. Under the same experimental conditions used to quaternize chloromethylated polysulfone, poly(vinylbenzyl chloride) exhibited normal second-order kinetics with an Ea of 10.4 kcal/mole, as shown in Figure 6. Noda and Kagawa also observed the same phenomenon in the quaternization of chloromethylated polystyrene with TEA in DMF (Ea = 10.5 kcal/mole)The major difference between these two systems is the polymer backbone polysulfone... 

With 77 % aqueous acetic acid, the rates were found to be more affected by added perchloric acid than by sodium perchlorate (but only at higher concentrations than those used by Stanley and Shorter207, which accounts for the failure of these workers to observe acid catalysis, but their observation of kinetic orders in hypochlorous acid of less than one remains unaccounted for). The difference in the effect of the added electrolyte increased with concentration, and the rates of the acid-catalysed reaction reached a maximum in ca. 50 % aqueous acetic acid, passed through a minimum at ca. 90 % aqueous acetic acid and rose very rapidly thereafter. The faster chlorination in 50% acid than in water was, therefore, considered consistent with chlorination by AcOHCl+, which is subject to an increasing solvent effect in the direction of less aqueous media (hence the minimum in 90 % acid), and a third factor operates, viz. that in pure acetic acid the bulk source of chlorine ischlorineacetate rather than HOC1 and causes the rapid rise in rate towards the anhydrous medium. The relative rates of the acid-catalysed (acidity > 0.49 M) chlorination of some aromatics in 76 % aqueous acetic acid at 25 °C were found to be toluene, 69 benzene, 1 chlorobenzene, 0.097 benzoic acid, 0.004. Some of these kinetic observations were confirmed in a study of the chlorination of diphenylmethane in the presence of 0.030 M perchloric acid, second-order rate coefficients were obtained at 25 °C as follows209 0.161 (98 vol. % aqueous acetic acid) ca. 0.078 (75 vol. % acid), and, in the latter solvent in the presence of 0.50 M perchloric acid, diphenylmethane was approximately 30 times more reactive than benzene. 

The kinetics of the chlorination of some alkylbenzenes in a range of solvents has been studied by Stock and Himoe239, who again found second-order rate coefficients as given in Table 57. Although the range of rates varies by a factor of 104, there was no marked change in the toluene f-butylbenzene reactivity ratio, and it was, therefore, concluded that the Baker-Nathan order is produced by a polar rather. than a solvent effect. 

At this point it might be helpful to summarize what has been done so far in terms of effective potentials. To obtain the QFH correction, we started with an exact path integral expression and obtained the effective potential by making a first-order cumulant expansion of the Boltzmann factor and analytically performing all of the Gaussian kinetic energy integrals. Once the first-order cumulant approximation is made, the rest of the derivation is exact up to (11.26). A second-order expansion of the potential then leads to the QFH approximation. 

According to this kinetic model the collision efficiency factor p can be evaluated from experimentally determined coagulation rate constants (Equation 2) when the transport parameters, KBT, rj are known (Equation 3). It has been shown recently that more complex rate laws, similarly corresponding to second order reactions, can be derived for the coagulation rate of polydisperse suspensions. When used to describe only the effects in the total number of particles of a heterodisperse suspension, Equations 2 and 3 are valid approximations (4). 

This is illustrated in Figure 7.4 where the effectiveness factor is plotted versus the low ij Aris number An0 for a bimolecular reaction with (1,1) kinetics, and for several values of/ . P lies between 0 and 1, calculations were made with a numerical method. Again all curves coincide in the low tj region, because rj is plotted versus An0. For p = 0, the excess of component B is very large and the reaction becomes first order in component A. For p = 1, A and B match stoichiometrically and the reaction becomes pseudosecond order in component A (and B for that matter). Hence the rj-An0 graphs for simple first- and second-order reactions are the boundaries when varying p. 

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