Rate constant apparent

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... 

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 ... 

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. 

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,... 

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. 

Figure 6.3 Dependence of the apparent rate constants, determined by fitting the experimental transients with (6.5), on the step fraction. The final potentials are 0.73 V (triangles), 0.755 V (diamonds), 0.78 V (squares), and 0.805 V (circles). The value of the step fraction for Pt(lll) was estimated using a procedure described in [Lebedeva et al., 2002c]. The inset shows the independence of the apparent intrinsic rate constant per step.

Figure 6.4 Dependence of the apparent rate constants and the apparent intrinsic rate constant on the potential ( Tafel plots ), determined by fitting the experimental transients with (6.5).

Figure 10.8 Arrhenius plots for the apparent rate constant for the HOR (CO-free) at Pt (O), Pt5iCo49 (A), and Pt54Ru46 ( ) working electrodes at 0.020 V vs. RHE(/). (From Uchida et al. [2006], reproduced by permission of the American Chemical Society.)...

In this section, we demonstrate the real ORR activities (apparent rate constant per real active surface area, fe pp) and P(H202) at bulk Pt and nanosized Pt catalysts dispersed on carbon black (Pt/CB) with dp,= 1.6 + 0.4, 2.6 + 0.7, and 4.8 1.0 nm in the practical temperature range 30-110 °C [Yano et al., 2006b]. The use of a channel flow double-electrode (CFDE) cell allowed us to evaluate fe pp and P(H202) precisely. 

Arrhenius Plots of Atapp The value of 4 at a given potential E is determined in the same manner as described in Section 10.2.4. Because the eontribution of two-electron reduction to the production of H2O2 (compared with the overall ORR) was very low, we ean evaluate an apparent rate constant at a constant applied potential from the following equation [Wakabayashi et al., 2005a, b] ... 

The theory presented above accounts for the electrostatic effects on the apparent rate constant for ion transfer by relating the observed changes in to changes in c"(0), or equivalently to 0(0). In the following, we present the simulated electrical potential distributions and the corresponding enhancement factors for a cation transferring from the aqueous phase across the water-l,2-DCE interface (s" = 78.39, s° = 10.36). The rela-... 

Electrocatalysis employing Co complexes as catalysts may have the complex in solution, adsorbed onto the electrode surface, or covalently bound to the electrode surface. This is exemplified with some selected examples. Cobalt(I) coordinatively unsaturated complexes of 2,2 -dipyridine promote the electrochemical oxidation of organic halides, the apparent rate constant showing a first order dependence on substrate concentration.1398,1399 Catalytic reduction of dioxygen has been observed on a glassy carbon electrode to which a cobalt(III) macrocycle tetraamine complex has been adsorbed.1400,1401... 

It is noteworthy that the partition coefficient is included in the apparent rate constants and may be ignored only when VcellK is much smaller than VR and VD. Under these conditions, the above equations reduce to Eqs. (8) and (9) for the sink situation. 

As may be seen from Figure 3, first order plots based on the conversions obtained by both methods were linear with slopes in close agreement. The apparent rate constant is 0.57 h"1 based on H2 evolution and 0.54 h"1 based on the formation of glycine salt thereby indicating that it is valid to use the simpler H2 evolution measurements when measuring the effects of process variables on the reaction. 

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