Equilibrium consistency with kinetics

Calibrations performed using an equilibrium model indicated increasing Kd with time, which is consistent with kinetic effects (i.e., gradual approach to equilibrium). When the kinetic model was calibrated, good model fits were observed for all three columns using a calibrated Kd of 1.4 mL/g and first-order sorption rate constant of 0.15 day 1 (Figure 2). 

Multinuclear Li and NMR and quantum-chemical investigations by Muller and coworkers showed that it is possible to determine the structure of active species in methacrylate polymerization in the presence of triethylaluminum (AlEts) in nonpolar solvents, such as toluene. Reaction of eibLi and trialkylaluminum with methyl pivalate (mpiv), in equimolar amounts, leads to an equilibrium of monomer 46 and dimer 47 complexes (eibLi mpiV"AlEt3) (n = 1, 2), consistent with kinetic studies. Methyl pivalate was... 

One factor that complicates the mechanistic understanding of this reaction is the fact that the olefin-Co, alkyl-Co and acyl-Co complexes are in equilibrium. The existence of equilibrium among these intermediate complexes is consistent with the observations of (1) olefin isomerization and (2) virtually statistical isotope scrambling on using ],l,2-trideuterio-l-propene, 6,6,6-trideuterio-l-hexene or l-l CJ-propene. Isomerization and isotope scrambling are suppressed at high carbon monoxide pressures (>100 bars) and temperatures up to 140°. This fact clearly indicates that vacant coordination sites are necessary for the observed isomerization and isotope scrambling. These observations are also consistent with kinetic studies/ ... 

Two lelH redox systems have been attached to SAMs, and full studies of the kinetics as a funchon of electrolyte pH have been performed [248-250]. The data were compared to the predictions of the stepwise mechanism. In this mechanism, electron and proton transfers are separate steps and proton transfer is treated as an equilibrium [251, 252]. With the inclusion of a potential-dependent transfer coefficient [253], two testable predictions can be made. A plot of logfA ) versus pH has a V shape, and a plot of afO) (the transfer coefficient at the formal potential) versus pH oscillates about 0.5. These predictions are consistent with kinetic data collected for a SAM with an attached galvi-nol (a phenol-hke redox molecule) at pHs greater than 8 [248, 249]. However, the data obtained for an osmium complex ([Os "/"(bpy)2(py)(L)], L = OH or H2O) deviate substantially from the predictions. The plot of hi(A ) versus pH is much less dependent on pH than expected, and the... 

A further paper [167] explains the lamellar thickness selection in the row model. The minimum thickness lmin is derived from the similation and found to be consistent with equilibrium results. The thickness deviation 81 = l — lmin is approximately constant with /. It is established that the model fulfills the criteria of a kinetic theory Firstly, a driving force term (proportional to 81) and a barrier term (proportional to /) are indentified. Secondly, the competition between the two terms leads to a maximum in growth rate (see Fig. 2.4) which is located at the average thickness l obtained by simulation. Further, the role of fluctuations becomes apparent when the dependence on the interaction energy e is investigated. Whereas downwards (i.e. decreasing l) fluctuations are approximately independent... 

More recently, Bagal and coworkers (Luchkevich et al., 1991) obtained similar results in a kinetic investigation of the coupling reactions of some substituted benzenediazonium ions with 1,4-naphtholsulfonic acid, and with 1,3,6-, 2,6,8-, and 2,3,6-naphtholdisulfonic acids. The kinetic results are consistent with the transient formation of an intermediate associative product. The maximum concentration of this product reaches up to 94% of the diazonium salt used in the case of the reaction of the 4-nitrobenzenediazonium ion with 1,4-naphtholsulfonic acid (pH 2-4, exact value not given). The authors assume that this intermediate is present in a side equilibrium, i. e., the mechanism of Scheme 12-77 mentioned above rather than that of Scheme 12-76, and that the intermediate is the O-azo ether. 

When a 1 1 mixture of NO and NO2 (i.e., NO2/NOx=0,5) is fed to the SCR reactor at low temperature (200 °C) where the thermodynamic equilibrium between NO and NO2 is severely constrained by kinetics, the NO2 conversion is much greater than (or nearly twice) the NO conversion for all three catalysts. This observation is consistent with the following parallel reactions of the SCR process [6] Reaction (2) is the dominant reaction due to its reaction rate much faster than the others, resulting in an equal conversion of NO and NO2. On the other hand, Reaction (3) is more favorable than Reaction (1), which leads to a greater additional NO2 conversion by Reaction (3) compared with the NO conversion by Reaction... 

In either neat dioxane or THF, carbene-ether ylides are observed as a broad IR absorption band between 1560 and 1610 cm , distinct from the IR bands of the free carbenes. With discrete spectroscopic signatures for the free carbene and its corresponding ether ylides, TRIR spectroscopy was used to confirm that the effects described above with dilute ether in Freon-113 were due to specific solvation of the carbene (Scheme 4.6, Reaction 2) rather than a pre-equilibration with the coordinating solvent (Scheme 4.6, Reaction 3) or reactivity of the ylide itself (Scheme 6, Reaction 4). In Freon-113 containing 0.095M THF simultaneous TRIR observation of both the free carbene (x = ca. 500 ns) and the carbene-THF ylide (x = ca. 5ps) was possible7 The observation that lifetimes of these species were observed to be so different conclusively demonstrates that the free carbene and the carbene-THF ylide are not in rapid equilibrium and that Reaction 3 of Scheme 4.6 is not operative. By examining the kinetics of the carbene 34 at 1635 cm directly in Freon-113 with small amounts of added dioxane, it was observed that the rate of reaction with TME was reduced, consistent with Reaction 2 (and not Reaction 4) of Scheme 4.6. 

The functional form of this rate expression is consistent with the behavior of the iridium system observed throughout the kinetic investigations. The coordination of nitrile to iridium is anticipated to produce more than a simple inhibitory effect. Being the dominant equilibrium in the mechanism, nitrile coordination may produce the observed first order dependence of the reaction rate with respect to hydrogen. Given Kcn[RCN] is the predominant term in the denominator, the rate expression may be reduced to the form of (8) which is first order with respect to both olefin and [H2]. 

Upon addition of a solution of sulfuric acid in D20 the reaction of A-acetoxy-A-alkoxyamides obeys pseudo-unimolecular kinetics consistent with a rapid reversible protonation of the substrate followed by a slow decomposition to acetic acid and products according to Scheme 5. Here k is the unimolecular or pseudo unimolecular rate constant and K the pre-equilibrium constant for protonation of 25c. Since under these conditions water (D20) was in a relatively small excess compared with dilute aqueous solutions, the rate expression could be represented by the following equation ... 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

The kinetics of the ionic hydrogenation of isobutyraldehyde were studied using [CpMo(CO)3H] as the hydride and CF3C02H as the acid [41]. The apparent rate decreases as the reaction proceeds, since the acid is consumed. However, when the acidity is held constant by a buffered solution in the presence of excess metal hydride, the reaction is first-order in acid. The reaction is also first-order in metal hydride concentration. A mechanism consistent with these kinetics results is shown in Scheme 7.8. Pre-equilibrium protonation of the aldehyde is followed by rate-determining hydride transfer. 

The simulated C02 fugacity matches the initial reservoir C02 content and indicates that the pH is buffered by C02-calcite equilibrium. Further modelling was carried out using the Geochemists Workbench React and Tact modules with the thermodynamic database modified to reflect the elevated P conditions and kinetic rate parameters consistent with the Waarre C mineralogy. The Waarre C shows low reactivity and short-term predictive modelling of the system under elevated C02 content changes little with time (Fig. 1). 

Mechanisms of Sorption Processes. Kinetic studies are valuable for hypothesizing mechanisms of reactions in homogeneous solution, but the interpretation of kinetic data for sorption processes is more difficult. Recently it has been shown that the mechanisms of very fast adsorption reactions may be interpreted from the results of chemical relaxation studies (25-27). Yasunaga and Ikeda (Chapter 12) summarize recent studies that have utilized relaxation techniques to examine the adsorption of cations and anions on hydrous oxide and aluminosilicate surfaces. Hayes and Leckie (Chapter 7) present new interpretations for the mechanism of lead ion adsorption by goethite. In both papers it is concluded that the kinetic and equilibrium adsorption data are consistent with the rate relationships derived from an interfacial model in which metal ions are located nearer to the surface than adsorbed counterions. 

A mechanism is determined from these data by choosing one which is consistent with the overall equilibrium behavior and which correctly matches the rate relationships derived for the postulated mechanism e.g., assuming the bimolecular adsorption/desorption reaction mechanism, as given in Equation 1, and using the kinetic model described above, the following relationship between xp and reactant and product concentrations can be derived (see Appendix C) ... 

If this mechanism is consistent with the experimental relaxation data, then a plot of xp versus the expression in the brackets of Equation 35 will give a straight line with a slope of kjnt and an intercept at the origin. As shown in Figure 11, the data fit this proposed mechanism quite well. Values for i i0, reactant and product concentrations, and K nt input into Equation 35 are from the equilibrium modeling results calculated at each pH value for which kinetic runs were made. Normally a variety of different mechanisms are tested against the experimental data. Several other more complex mechanisms were tested, including those postulated for metal ion adsorption onto y-A O (7) however, only the above mechanism was consistent with the experimental data. Hence it was concluded that the bimolecular adsorption/desorption reaction was the most plausible mechanism for Pb2+ ion adsorption onto a-FeOOH. 

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