Rate single site model

Figure 5. Calculated vs. observed rate—single site model...

At low 7t, the denominator simplifies to unity in each case and both models are linear in n. For sufficiently high n, the parenthesis in the denominator approaches Ketv, the initial rate for the dual-site model then approaches zero, and that of the single-site model approaches a constant value. Thus the plot of the experimental data will indicate that the dual-site model is preferable if a maximum exists in the data, or that the single-site model is preferable if a horizontal high-pressure asymptote exists. Hence, for the data of Franckaerts and Froment (FI) shown in Fig. 2, the dual-site model is preferred over the single-site model. 

By using only simple hand calculations, the single-site model has been rejected and the dual-site model has been shown to represent adequately both the initial-rate and the high-conversion data. No replicate runs were available to allow a lack-of-fit test. In fact this entire analysis has been conducted using only 18 conversion-space-time points. Additional discussion of the method and parameter estimates for the proposed dual-site model are presented elsewhere (K5). Note that we have obtained the same result as available through the use of nonintrinsic parameters. 

TCE degradation on metallic iron single site model with competitive adsorbate // Based on adsorption equilibrium and assume that the reaction rate is directly... 

The model fit variances are given in Table IV and calculated rate parameters for the two reactions in Table V. In nearly all cases an improvement in model fit was observed. This is typified by the comparison between the fit gg a single first-order and double first-order (DFO) model to the 0 Co data, as shown in Figure 5. Although the DFO model shows considerable improvement over single-site models, residual plots indicate a small systematic deviation at higher concentrations. This is discussed further in the following section. 

In many reactions it has been demonstrated that more than one site is involved in the catalytic process. This is particularly often the case for dissociation reactions. The same procedure as depicted above for a single site model can be used for the derivation of the rate expression for a dual site model, but the result is somewhat different. This is exemplified for the following dissociation reaction A 2B, which is thought to proceed according to the three step sequence ... 

A further simplification of the rate expressions is obtained if the product concentrations or partial pressures are negligibly small. For the single site model this results in the following expressions ... 

In Sections 3.2 and 3.4, the approach based on one rate-determining step with the other steps in quasi-equilibrium was applied to simplify the derivation of rate expressions. A further simplification of the rate expressions is obtained when the product concentrations or partial pressures are negligibly small, i.e., at low conversions of a pure reactant feed stream. Similarly, at high conversions, when nearly only product is present, the rate expressions can also be simplified. Applying e single-site model, Eqns. (3.17)-(3.19), for the reaction A B at low conversion, the following expressions (often called "initial rate expressions") are obtained, provided that the feed contains pure A. 

A kinetic model for n-butane isomerization over sulfated zirconia catalysts is proposed involving two types of active sites on the catalyst surface and a bi-molecular reaction mechanism. Deactivation kinetics are included in which the two different active sites deactivate at different rates. The proposed model more accurately captures the activity trends observed experimentally with respect to time on stream behavior compared to a single site model with deactivation. 

The kinetic data of oleic acid esterification in Problem 2.1 is next evaluated against the single-site model in which the adsorption of B is controlling. The rate expression can be easily determined to be... 

For a surface-reaction-controlling, single-site model, the rate of reaction can be derived as... 

For the estimation of the contaminants uptake rate and consequently the time required for the phytoremediation of a contaminated site, single mathematical models may be applied. In the following paragraphs, analysis as well as the examples given in Groundwater Remediation Technologies Analysis Center Technology Evaluation Report on Phytoremediation, by Schnoor, is presented without modification.8... 

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. 

This method has been applied (M5) for modeling the vapor-phase rate of dehydration of secondary butyl alcohol to the olefin over a commercial silica-alumina cracking catalyst. Integral reactor data are available at 400, 450, and 500°F. Two models considered for describing this reaction are the single site... 

The Q (initial reaction rate) plots were presented in Fig. 19. Note again that the data may be correlated well by straight lines for both models. The C2 values are correlated by the solid lines of Fig 20. Note that the dual-site values can again be correlated by a straight line, but that the single-site values of C2 show a definite curvature. Alternatively, the 0.975 atm value of the single-site C2 could be rejected, and the three high-pressure points... 

Only in one case (the hydration of 2-methylpropene over a H2S04—Si02 catalyst [278]) was the so-called Rideal mechanism proposed as a preferable model and expressed by the rate equation for single-site adsorption with retardation by the product alcohol, viz. 

Equation (24) is, in fact, a Langmuir—Hinshelwood-type equation. Similar models with a single site surface reaction as the rate-determining step were used for other liquid phase esterifications [448,451]. Experimental data for the l-butanol-x>leic acid system were best fitted by eqn. (24) [452] or eqn. (25) [451]... 

Careful analysis of the kinetic data for the reduction of CN by Mo nitrogenase from A. vinelandii shows that at high concentrations of CN a constant rate of reduction is observed, that rate depends only on the pH of the solution [128], This observation is inconsistent with the earlier proposal that CN and HCN bind independently, and the revised model is one in which the CN binds to a single site (Figure 21). 

Within this model all mutation rates can be expressed in terms of only three quantities the chain length of the polymer, X, the single-digit accuracy of replication, q, often expressed as mutation rate per site and replication, p = 1 - q, and the Hamming distance, d(Ij,Ii). Finally, the (dependent) parameter, s = (1 - q)/q is the ratio between single digit mutation rate and accuracy. 

The electron transfer from cytochrome c to O2 catalyzed by cytochrome c oxidase was studied with initial steady state kinetics, following the absorbance decrease at 550 nm due to the oxidation of ferrocyto-chrome c in the presence of catalytic amounts of cytochrome c oxidase (Minnart, 1961 Errede ci a/., 1976 Ferguson-Miller ei a/., 1976). Oxidation of cytochrome c oxidase is a first-order reaction with respect to ferrocytochrome c concentration. Thus initial velocity can be determined quite accurately from the first-order rate constant multiplied by the initial concentration of ferrocytochrome c. The initial velocity depends on the substrate (ferrocytochrome c) concentration following the Michaelis-Menten equation (Minnart, 1961). Furthermore, a second catalytic site was found by careful examination of the enzyme reaction at low substrate concentration (Ferguson-Miller et al., 1976). The Km value was about two orders of magnitude smaller than that of the enzyme reaction previously found. The multiphasic enzyme kinetic behavior could be interpreted by a single catalytic site model (Speck et al., 1984). However, this model also requires two cytochrome c sites, catalytic and noncatalytic. 

---