Different kinetic laws

This will lead to a different kinetic law. In presence of 02, the following chain termination processes take place ... 

It should be stressed that Eq. (66) is not the sole representation of kinetics of nucleophile-activated racemization of optically active silanes. Silylonium complexes often appear to be thermodynamically more stable than the substrate in the reaction systems, i.e.,, [Nu] > -l. The steady-state conditions are not met, which usually leads to complex kinetics. On the other hand, the different kinetic laws observed in these processes (287,288) are related to other types of mechanisms, which are discussed later. 

The reason for the observation of different kinetic laws of radical decay in the above mentioned papers is probably due to a difference of poyethylene samples. Thus Charlesby et al. (19) have found that the rate of alkyl radicals decay in samples of low pressure polyethylene is much higher than in high pressure polyethylene. Loy (22) has investigated radical decay in irradiated cellulose and found that at low irradia-tiori doses (up to 10 Mrad) the radical decay kinetics can be represented by superposition of two processes. At high radical concentrations the kinetics corresponds to a monomolecular law and then a transition to... 

There are, in principle, three zones in which different kinetic laws may operate following Albery and Bartlett [131] we denote these by I—III as in Fig. 60 where... 

Reactions of the three halide ions with iodate are reported as obeying different kinetic laws (Table 28). For the chloride and bromide reactions, the evidence is not extensive, and even the great attention given to the iodide reaction has not produced complete agreement about the reaction orders. Earlier papers refer to this last as the Dushman reaction, on account of a kinetic study which established the reaction as close to fifth order overall (rate = k[H ] [IOJ][I ] ). Kubina examined the same reaction in the presence of arsenite (which did not affect the rates) and claimed that the rate expression was different, viz. 

It is clear that a detailed reaction mechanism cannot be deduced from kinetic measurements only. However, since different kinetic laws have been deduced from the results of the kinetic study of the reaction on fresh, regenerated and oxygenated samples, the kinetic results demonstrate that the surface of NiO(250°) is modified when it contacts nitrous oxide or oxygen at 250° and that each gas produces a specific surface modification. 

With H-Y as catalyst, the vinyl ethers polymerize following a different kinetic law in which Qt is a linear function of In t,... 

Various rate expressions resulting from different kinetic laws have been treated in model <42> and <43> by Mashkar [39] and Juvekar and Sharma [40]. Closed solutions for a number of cases were developed [40]. In what follows, only the more important case of first order kinetics with respect to A will be considered. For model <43> with consideration of gas flow and pressure variations the governing balance equation follows from eq. (84) by setting = o and Stg =... 

Although heterogeneous reactions have the same basic equations as homogeneous reactions, they featnre very different properties - mainly related to their multizone nature - and lead to very different kinetic laws and mathematical formulations of speed. Moreover, two new processes are involved ... 

The reaction of MeO /MeOH with 2-Cl-5(4)-X-thiazoles (122) follows a second-order kinetic law, first order with respect to each reactant (Scheme 62) (297, 301). A remark can be made about the reactivity of the dichloro derivatives it has been pointed out that for reactions with sodium methoxide, the sequence 5>2>4 was observed for monochlorothiazole compounds (302), For 2.5-dichlorothiazole, on the contrary, the experimental data show that the 2-methoxy dehalogenation is always favored. This fact has been related to the different activation due to a substituent effect, less important from position 2 to 5 than from... 

The dissimilarity in the kinetic laws for chain fracture observed under different flow conditions reflect the deficiencies of the present theories which should be able to incorporate both dependences (M 1 and M 2) into its structure, with either... 

Now we shall apply this to different rate laws. For first-order kinetics either of two forms can be used ... 

The most significant difference between brominations in protic and non-protic solvents concerns the kinetic law. Whereas in protic media the reaction is first-order in bromine, in halogenated media it is second-order (Bellucci et ai, 1980). CTC ionization is electrophilically assisted by hydrogen bonding by a protic solvent to the leaving bromide and leads to a bromonium-bromide ion pair. In non-protic media, assistance to the bromination step is provided by a second bromine molecule, leading to a bromonium-tribromide ion pair. In other words, in protic media bromination is solvent-assisted (56) while in halogenated media it is bromine-catalysed (57). 

The kinetic models just described are only a few of those that have been found to represent reactions in solids. Moreover, it is sometimes observed that a reaction may follow one rate law in the early stages of the reaction, but a different rate law may apply in the later stages. Because many of the rate laws that apply to reactions in solids are quite different from those encountered in the study of reactions... 

For simplicity, up to now, first-order kinetics have been assumed, but obviously other rate laws may apply. Further complications can be generated by the presence of multiple paths for M on a variety of sites exhibiting different kinetics [5,11] or sequential enzymatic processes [100], Some complexes, labelled as lipophilics , have been shown to cross the membrane without the need for specific pre-adsorption sites [5,11,18,19,50] see also Chapters 5, 6 and 10 in this volume. Fortin and Campbell [76] have recently reported the accidental uptake of Ag+ induced by thiosulfate ligand. 

The rate-law (2) can usually be established without too much difficulty by appropriate kinetic experiments, but it must be remembered that the same system may follow different rate-laws under different conditions, and it is evident that such kinetically complicated systems are generally unsuitable for attempts to determine the fundamental rate constants. However, a sufficient number of kinetically simple systems is now known which are much more useful for such studies. 

Analysis of the cyclic voltammetric responses is also possible if a kinetic law different from Butler-Volmers governs the electrode electron transfer. Derivation of the kinetic law from the cyclic voltammetric responses may benefit from a convolution approach similar to that described in the preceding section. 

In the preceding sections throughout this chapter, several aspects of the influence of the nucleophile on the rates of the different reaction steps and/or mechanisms involved in ANS with amines have been discussed. One of the most outstanding features and most widely studied phenomena is the observation or the absence of base catalysis and, somewhat related with this subject, is the occurrence of a first, second or third order in amine kinetic law. 

Kinetic isotope fractionation obeys a different fractionation law. Young et al. (2002b) showed recently that kinetic processes, again written here in terms of the two Mg isotope fractionation factors, obey the relation... 

The high precision with which Mg isotope ratios can be measured using MC-ICPMS opens up new opportunities for using Mg as a tracer in both terrestrial and extraterrestrial materials. A key advance is the ability to resolve kinetic from equilibrium mass-dependent fractionation processes. From these new data it appears that Mg in waters is related to mantle and crustal reservoirs of Mg by kinetic fractionation while Mg in carbonates is related in turn to the waters by equilibrium processes. Resolution of different fractionation laws is only possible for measurements of Mg in solution at present laser ablation combined with MC-ICPMS (LA-MC-ICPMS) is not yet sufficiently precise to measure different fractionation laws. 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

Many high-pressure reactions consist of a diffusion-controlled growth where also the nucleation rate must be taken into account. Assuming a diffusion-controlled growth of the product phase from randomly distributed nuclei within reactant phase A, various mathematical models have been developed and the dependence of the nucleation rate / on time formulated. Usually a first-order kinetic law I =fNoe fi is assumed for the nucleation from an active site, where N t) = is the number of active sites at time t. Different shapes of the... 

We can apply classical germination laws to this supersaturated system thus, the Avrami-Mempel laws confirm the unidimensional growth of the solid-like gel network. Induction times can also be studied in this framework 11). Here, we are interested first by the different kinetic behaviors which are dependent upon the location in the phase diagram of the initial solution defined by its supersaturation degree. 

The implication of this behavior suggests that there will be a quantitative difference in the kind of step fluctuation dynamics observed for each kinetic law system. For i-kinetics, we expect steps to fluctuate by variations in the flux of adatoms hitting the step from a uniform, quickly moving sea of equilibrated adatoms on the adjoining terrace. In this case, the time to create a step fluctuation of amplitude y (perpendicular to both the surface normal and the average step direction) will be given by... 

It becomes very reasonable, therefore, to attribute the different kinetic behaviors for series al, a3, and a5, or for the new sets of series, to a specific role of hydrogen ion in the reaction. A more detailed form of the experimental rate law (Equation 9), which predicts quite well the relative values of the slopes and intercepts in the — [Ce(IV)]-... 

An investigation of the effective kinetic law of the anodic hydrogen oxidation in the cell shows a kinetic law that differs from Eq. (29). The latter law had been obtained by procedures that eliminate mass-transfer limitations. 

The consecutive reactions have been investigated by the method of preparative voltammetry, namely by analyzing the product mixture, which is essentially composed of the anodic monomer and dimer independence of working potential or current density resp. Correlating the respective current and mass yields of monomers and dimers with current density allows the kinetic laws governing the selectivities at different anode materials to be obtained. 

In Section 4.4.2 some concepts were developed which allow us to quantitatively treat transport in ionic crystals. Quite different kinetic processes and rate laws exist for ionic crystals exposed to chemical potential gradients with different electrical boundary conditions. In a closed system (Fig. 4-3a), the coupled fluxes are determined by the species with the smaller transport coefficient (c,6,), and the crystal as a whole may suffer a shift. If the external electrical circuit is closed, inert (polarized) electrodes will only allow the electronic (minority) carriers to flow across AX, whereas ions are blocked. Further transport situations will be treated in due course. 

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