Apparent rate

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

The course of a surface reaction can in principle be followed directly with the use of various surface spectroscopic techniques plus equipment allowing the rapid transfer of the surface from reaction to high-vacuum conditions see Campbell [232]. More often, however, the experimental observables are the changes with time of the concentrations of reactants and products in the gas phase. The rate law in terms of surface concentrations might be called the true rate law and the one analogous to that for a homogeneous system. What is observed, however, is an apparent rate law giving the dependence of the rate on the various gas pressures. The true and the apparent rate laws can be related if one assumes that adsorption equilibrium is rapid compared to the surface reaction. 

Just as the surface and apparent kinetics are related through the adsorption isotherm, the surface or true activation energy and the apparent activation energy are related through the heat of adsorption. The apparent rate constant k in these equations contains two temperature-dependent quantities, the true rate constant k and the parameter b. Thus... 

The rate of the uncatalysed reaction in all four solvents is rather slow. (The half-life at [2.5] = 1.00 mM is at least 28 hours). However, upon complexation of Cu ion to 2.4a-g the rate of the Diels-Alder reaction between these compounds and 2.5 increases dramatically. Figure 2.2 shows the apparent rate of the Diels-Alder reaction of 2.4a with 2.5 in water as a lunction of the concentration of copper(II)nitrate. At higher catalyst concentrations the rate of the reaction clearly levels off, most likely due to complete binding of the dienophile to the catalyst. Note that in the kinetic experiments... 

In the kinetic runs always a large excess of catalyst was used. Under these conditions IQ does not influence the apparent rate of the Diels-Alder reaction. Kinetic studies by UV-vis spectroscopy require a low concentration of the dienophile( 10" M). The use of only a catalytic amount of Lewis-acid will seriously hamper complexation of the dienophile because of the very low concentrations of both reaction partners under these conditions. The contributions of and to the observed apparent rate constant have been determined by measuring k pp and Ka separately. ... 

From the equilibrium constant and the apparent rate constant, the rate constant for reaction of the metal-ion coordinated dienophile can be calculated using equation 2.2 (derived in... 

Figure 4.1. The apparent rate constant of the Diels-Alder reaction of 4.8 with 4.6 versus the concentration of MeReO-j catalyst according to reference 7.

Berezin and co-workers have analysed in detail the kinetics of bimolecular micelle-catalysed reactions ". They have derived the following equation, relating the apparent rate constant for the reaction of A with B to the concentration of surfactant ... 

Studies of micellar catalysis of himolecular reactions of uncharged substrates have not been frequent" ". Dougherty and Berg performed a detailed analysis of the kinetics of the reaction of 1-fluoro-2,4-dinitrobenzene with aniline in the presence of anionic and nonionic surfactants. Micelles induce increases in the apparent rate constant of this reaction. In contrast, the second-order rate constant for reaction in the micellar pseudophase was observed to be roughly equal to, or even lower than the rate constant in water. 

In summary, all studies on the influence of micelles on himolecular Diels-Alder reactions indicate that the apparent rate constants in these media are strikingly similar to the rate constants in water. 

The effect of micelles of SDS, CTAB and C12E7 on the apparent second-order rate constants of the Diels-Alder reaction between nonionic 5.1a, anionic 5.1 f and cationic 5.1g with 5.2 is reported in Table 5.1. These apparent rate constants are calculated from the observed pseudo-first-order rate constants by dividing the latter by the overall concentration of 5.2. 

Further evidence for an increased efficiency of complexation in the presence of micellar aggregates with bivalent metal counterions is presented in Table 5.4. The apparent rate constants of the reaction of 5.1c with 5.2 in the presence of micelles of Co(DS)2, Ni(DS)2, Cu(DS)2 and Zn(DS)2 are compared to the rate constants for the corresponding bivalent metal ion - dienophile complexes in the absence of micelles. The latter data are not dependent on the efficiency of the formation of the catalyst - dienophile complex whereas possible incomplete binding will certainly be reflected in the former. The good correlations between 1 and and the absence of a correlation between and... 

The apparent rate constant in Eq. (6.26) follows the Arrhenius equation and yields an apparent activation energy ... 

In a neutral azole, the apparent rate of formation of an A-substituted derivative depends on the rate of reaction of a given tautomer and on the tautomeric equilibrium constant. For example, with a 3(5)-substituted pyrazole such as (199), which exists as a mixture of two tautomers (199a) and (199b) in equilibrium, the product composition [(200)]/[(201)] is a function of the rate constants Ha and fcs, as well as of the composition of the tautomeric mixture (Scheme 16) <76AHC(Si)l). 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

Here AX is the acetyl compound (acetyl chloride or acetic anhydride), N is N-methylimidazole, I is the intermediate (presumably A -acetyl-A -methylimidazo-lium ion), X is the counterion (chloride or acetate), and ROH is the acetyl acceptor (alcohol or water). A general treatment of Scheme XXIII requires specification of the detailed nature of and k[ and is probably too complicated to be of practical use. However, several important special cases may arise from the operation of the ratio kxlk x, the behavior of apparent rate constants k /. and k, the relative magnitudes of k / and k, the relative concentrations of the reactants, the method of observation, and the nature of ROH. These cases are outlined in Scheme XXIV. 

Qualitative cotLdation of tlie apparent rate of 1,4-addition witli tlie reduction potential of tlie enone was later proven to be only superficial, tlirougb quantitative kinetic studies by Ktauss and Sniitli [60]. 

In non-competitive inhibition, the substrate (S) and inhibitor (I) have equal potential to bind to the free enzyme (E). The inhibitor forms a ternary complex with enzyme-substrate (ES) whereas the substrate will form another ternary complex with enzyme-inhibitor (El). Since the non-competitive inhibitor had no effect on the binding of substrate to the enzyme, the Km value remained consistent (or unchanged). There are two different ways for the formation of ESI ternary complex this complex would not form the product and therefore was decreased. Non-competitive inhibitor had no effect on substrate binding or the enzyme-substrate affinity, therefore the apparent rate constant (K ) was unchanged.5 A possible reason for product inhibition was because of the nature of 2-ethoxyethanol,... 

When the logarithm of c(t) is plotted against t, the slope which represents an apparent rate parameter (k) will gradually decline... 

De Tar and Day [498] considered problems which arise in applying the first-order expression [eqn. (15)] to kinetic data of limited accuracy. If a is known to a high degree of accuracy (e.g. 0.1%), then small components (5—10%) of higher or lower order behaviour are detectable. When the accuracy of a is reduced to 1%, a second-order component of 25% could escape detection, and perhaps an even larger contribution might be missed within a limited a interval. Using exact values of a from known kinetic behaviour, the apparent rate coefficient, ft, was found to depend on both the proportion of the non-first-order component and on the extent of reaction considered. 

Steady-state mechanism. Consider the oxidation of RufNHj) by CL, which is believed to occur by the scheme shown below at constant pH. Imagine that one does a series of experiments with [Ru(NHs)g+ ] [O2]. Derive the steady-state rate law. Could these experiments equally well have had the reverse inequality of concentrations Should [RulNH.O ] also be adjusted (how and why) What apparent rate constant could be obtained from the concentration conditions that you consider optimum How would you design a longer series of experiments, and what rate constants could be obtained from the data If the data were examined graphically, what quantities would be displayed on the axes to obtain linear plots, and how would the rate constants be obtained from them ... 

As an example of the problem that arises when the mechanism changes, consider the data presented in Figure 10-3. Part (a) depicts the apparent rate constants for semi-carbazone formation measured at an intermediate pH of 3.9. The rate-controlling step is not clearly defined under such conditions, and a curved plot is obtained. 

Many reactions are accelerated by the presence of H+ or OH-, or both. If both are effective catalysts, then the apparent rate constant over a wide pH range may be expressed by... 

Thus the apparent rate constant (kcat/KM) is determined by the apparent activation barrier Ag. In fact, both Ag and Agjfat should have been written as AG and AGaat, respectively [see eq. (2.11)], but as long as we do not have large entropic effects (see Chapter 9), the approximation given above is reasonable. 

A low energy drop-off in the cross-section is also consistent with recent afterglow measurements of the apparent rate constant (4), 3 X 10-12 cc. sec.-1, which is well below the predicted (9, 13) ion-dipole rate, 9.7 X 10-10 cc. sec.-1... 

The latter seem to be the only species contributing to propagation, and hence the apparent rate constant of propagation is given by... 

If no concentration of the educt is given the standard exchange current density y oo is stated. Values of)t are printed in italics values of the apparent rate constant k pp are printed in parentheses in italics. For electrode potentials where the latter rate constant was actually determined the reader is referred to the original literature. 

Data given in Tables 1-6 clearly show a significant dependence of P2 and p4 on amine concentration, that is, at least one of the apparent rate constants kj contains a concentration factor. Thus, according to the mathematical considerations outlined in the Analysis of Data Paragraph, both p2, P4 exponents and the derived variables -(P2 + p)4> P2 P4 ind Z (see Eqns. 8-12) are the combinations of the apparent rate constants (kj). To characterize these dependences, derived variables -(p2+p)4, P2 P4 and Z (Eqns. 8,11 and 12) were correlated with the amine concentration using a non-linear regression program to find the best fit. Computation resulted in a linear dependence for -(p2 + p)4 and Z, that is... 

The values of the apparent rate constants kj for each temperature and the activation enthalpies calculated using the Eyring equation (ref. 21) are summarized in Table 10. However, these values of activation enthalpies are only approximative ones because of the applied simplification and the great range of experimental errors. Activation entropies were not calculated in the lack of absolute rate constants. Presuming the likely first order with respect to 3-bromoflavanones, as well, approximative activation entropies would be between -24 and -30 e.u. for la -> Ih reaction, between -40 and - 45 e.u. for the Ih la reaction and between -33 and -38 e.u. for the elimination step. These activation parameters are in accordance with the mechanisms proposed above. 

Table 10. Apparent rate constants kj and approximative activation enthalpies ... 

Coefficients ai, c and dj of Ih computed for N=6 (see Tables 7-9) were used to calculate the apparent rate constants. 

Proteolysis does not affect transcription and translation bnt by degrading the desired product it influences the apparent rate of gene expression. Although proteolysis can be reduced it is difficult to eliminate it completely. One approach is to nse protease-defident... 

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