Concentration profile second-order kinetics

The entire Scheme 4 accounts for the second-order kinetics in picolinic acid for lm in terms of Eq. (12) under the assumption that kM]. 0 and setting [>2 = -Klh lh- Eq. (12) rationalizes both the second-order in picolinic acid concentration and the observed pH profile for 0bs in the case of lm. 

Laboratory studies on both model compounds and real residuum feedstocks under commercial conditions have shown that the metal deposit profile on aged catalyst pellets is often M shaped, having a concentration maximum at some internal radial position. The simple first- or second-order kinetics employed by the models discussed cannot predict such a profile. This profile is a consequence of the sequential path for HDM where the hydrogenated intermediate B must first be formed from the original oil compound A prior to forming the deposit C ... 

Time profiles of the formation of fullerene radical anions in polar solvents as well as the decay of 3C o obey pseudo first-order kinetics due to high concentrations of the donor molecule [120,125,127,146,159], By changing to nonpolar solvents the rise kinetics of Go changes to second-order as well as the decay kinetics for 3C o [120,125,133,148], The analysis of the decay kinetics of the fullerene radical anions confirm this suggestion as well. In the case of polar solvents, the decay of the radical ion absorptions obey second-order kinetics, while changing to nonpolar solvents the decay obey first-order kinetics [120,125,127,133,147]. This can be explained by radical ion pairs of the C o and the donor radical cation in less polar and nonpolar solvents, which do not dissociate. The back-electron transfer takes place within the ion pair. This is also the reason for the fast back-electron transfer in comparison to the slower back-electron transfer in polar solvents, where the radical ions are solvated as free ions or solvent-separated ion pairs [120,125,147]. However, back-electron transfer is suppressed when using mixtures of fullerene and borates as donors in o-dichlorobenzene (less polar solvent), since the borate radicals immediately dissociate into Ph3B and Bu /Ph" [Eq. (2)][156],... 

Tanaka et al. studied the decay reactions of PVB radical anions produced by electron pulses in MTHF [47]. At low concentration ( < 0.05 base-mol dm - 3) of polymers the decay reaction followed a simple second-order kinetics. The charge neutralization reaction is responsible for the decay curve as is the case of biphenyl radical anions. However, the rate constant of the polymer anions was only a half or one-third of that of the biphenyl anion, because of the small diffusion coefficient of the polymer ion in solution. At high concentration of the polymer, a spike was observed in the time-profile of the PVB anion this was attributed to the retarded geminate recombinations within micro-domains where the polymers were entangled with each other. 

The insertion with CpFe(CO)2R in organic solvents follows second-order kinetics, first order in both the alkyl complex and the SO2. However, the rate profile, i.e., R = i-Pr > CH2CMe3 > Me 71, 77), is not that expected for a backside approach of SO2. Such an attack would be suggested by the observation of inversion at a carbon in the insertion with CpFe(CO)2CHDCHDCMe3 in various solvents 13S). However, this stereochemical result cannot be unambiguously applied to the mechanistic problem in question because extremely concentrated SO2 solutions were employed. A similar difficulty exists with the insertion... 

Figure 1 Axial profiles of sulfur removal and hydrogen sulfide concentration in the gas for hydrodesulfurization of oil following second-order kinetics in total sulfur.

The VERSE method was extended to describe the consequences of protein de-naturation on breakthrough curves in frontal analysis and on elution band profiles in nonlinear isocratic and gradient elution chromatography [45]. Its authors assumed that a unimolecular and irreversible reaction taking place in the adsorbed phase accormts properly for the denaturation and that the rate of adsorption/desorption is relatively small compared with the rates of the mass transfer kinetics and of the reaction. Thus, the assumption of local equilibrium is no longer valid. Consequently, the solid phase concentration must then be related to the adsorption and the desorption rates, via a kinetic equation. A second-order kinetics very similar to the one in Eq. 15.42 is used. 

The eoneentration profiles are similar to those in Figure 4.7, and the effeetiveness faetors are plotted in Figure 4.8. For a given (p, the effeetive-ness faetor for second-order is lower than for a first-order reaction, since a decrease in average concentration has more effect for the higher-order reactions. The solution for second-order reactions applies only where the rate is proportional to the square of the concentration of a single reactant, which is rarely the case. For a bimolecular reaction with a rate proportional to both reactant concentrations, r = k2C Cs, the solution for second-order kinetics can be used if the initial concentrations are equal and if the diffusivities of A and B are nearly the same. If reactant B is in considerable excess, the reaction could be considered pseudo-first-order with k = k2Cg and Eq. (4.31) used for a reasonable approximation of rj. 

Dissociation of tetramer to dimer occurred with a relaxation time, t, of a few milliseconds, the second relaxation time, T2, corresponding to transformation of dimers into monomers had a magnitude of a few seconds. The longest relaxation time, T3, was 4.8 x 10 sec at 23°C, and was associated with a conformational change of the monomer. Reassembly from 0.8 M GuHCl by 1 to 10 dilution of denaturant yielded about 70-94% of the original activity depending on the protein concentration. The reactivation process follows second-order kinetics (see Fig. 11.6 for the second-order profiles for return of enzymatic activity) (Parr and Hammes, 1975, 1976). 

Chemical kinetics focuses on the rate of a reaction through studying the concentration profile with time. Based on the number of reactants involved in the chemical reaction, the reaction can be classified as zero, first, or second order. Third-order reactions are rare because the probability of three reactants colliding and reacting is low. The following are simplified mathematic descriptions of the chemical kinetics of the various orders. 

Reduction of [IrCls] by a variety of agents has been cited. The reduction of [IrXs] " (Cl, Br) by 2-thiouracil and 2-thiopyrimidine follows second-order rate laws, first-order with respect to both the concentration of the oxidant and the reductant. A pH-rate profile as well as activation parameters have been presented, and a free radical mechanism proposed." Reduction in aqueous alcoholic solution by aquated electrons, hydrogen and alkyl radicals was investigated by pulse radiolysis, and found to result in the production of [IrClg] " A kinetic examination of reaction (123) in aqueous solution shows that this outer-sphere electron transfer reaction proceeds by parallel paths first and second order in substrate (SCN ). The kinetics of the oxidation of hydrazine by [Ir(Cl)4(H20)2], [Ir(Cl)5(H20)], [IrClef , as well as [IrBrsf , in aqueous acidic perchlorate solution have been investigated." ... 

This is a system of stiff, second-order partial differential equations which can be solved numerically to yield both transient, and steady state concentration profiles within the layer. Comparison of the experimental calibration curves and of the time response curves with the calculated ones provides the verification of the proposed model from which it is possible to determine the optimum thickness of the enzyme layer. Because the Thiele modulus is the controlling parameter in the diffusion-reaction equation it is obvious from Eq.6 that the optimum thickness L will depend on the other constants and functions included in the Thiele modulus. Because of this the optimum thickness will vary from one kinetic scheme to another. 

Figure 8.8 Approximate concentration profiles for second order chemical kinetics.

A vast majority of the kinetic measurements involving RO2 radicals has made use of the relatively intense UV absorption spectrum of these species in order to obtain time-concentration profiles. Moreover, for the study of peroxy radical selfreactions, absolute concentrations must be known in order to analyse second order decay profiles, indicating the importance of accurately measured absolute absorption cross sections. The cited peroxy radicals were generated by photolysis... 

Mirvish. later investigated the nitrosation of aminopyrine in considerable detail (12), proposing the alternative pathway shown in Fig. 5b, but presenting evidence that the mechanism is actually a great deal more complex than that. Firstly, they identified both a "fast" initial reaction, which was essentially complete within 2-5 min., and a "slow" reaction, which proceeded at a nearly constant rate for 15 min. Secondly, they found that the pH rate profile had maxima at both pH 2.0 and pH 3.1. Thirdly, they reported an apparent kinetic order for nitrite which varied considerably under some conditions from the value of 2 required by the mechanism of Fig. 5b, ranging from cleanly first order for the "slow" reaction at pH 2 to as high as 3-4 for the initial reaction at low nitrite concentration (l-6mM). 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

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