Simple kinetic models

In this section we will see how all these rules can be described mathematically by a single and simple kinetic model based on fundamental thermodynamic and catalytic principles."... 

We wish to conclude with a simple kinetic model that is compatible with the above picture and is based on the theory of continu um Zip-reactions (46,47). Let us assume random initiation at sites distributed at random along a chain followed by zip in both directions with speed v. If we denote by p the linear density of such... 

In the case of radiative transfer between identical molecules, the fluorescence decays more slowly as a result of successive re-absorptions and re-emissions. A simple kinetic model has been proposed by Birks (1970). It is based on the assumption of a unique value for the average probability a that an emitted photon is absorbed, i.e. without distinction between the generations of photons (a photon of generation n is emitted after n successive re-absorptions). This model leads to the following expressions for the effective lifetime and the macroscopic fluorescence quantum yield ... 

In this case, a simple kinetic model could not be derived to explain all the experimental observations. Independently prepared hexadecyl trimethyl ammoniurn persulfate was found to be soluble in toluene/AN mixtures and to catalyze polymerization in this homogeneous system at approximately the same rate as that observed in the two phase system. This result implies that anion exchange at the interface (see below) must be essentially complete under these conditions. Factors which complicated further analysis of the mechanism included a) precipitation of poly(acrylonitri 1 e) during the polymerization ... 

Using a simple kinetic model, Solomon demonstrated that the spin-lattice relaxation of the I and S spins was described by a system of coupled differential equations, with bi-exponential functions as general solutions. A single exponential relaxation for the I spin, corresponding to a well-defined Tu, could only be obtained in certain limiting situations, e.g., if the other spin, S, was different from I and had an independent and highly efficient relaxation pathway. This limit is normally fulfilled if S represents an electron spin. The spin-lattice relaxation rate, for the nuclear spin, I, is in such a situation given by ... 

Data from these studies were analyzed by a computer using equations 8 based on our simple kinetic model for the sediment/water systems (eqn. 7). The computer program (23) uses concentrations of chlorpyrifos in the water and sediment phases and product concentrations (obtained by difference) as a... 

This is well evidenced by simple kinetic models in organometallic catalysis. We first developed the now so-called ML2 model, where M is a metalllic center and L a chiral hgand (Figure 7.2). In this model we assumed that the catalytic reaction... 

When a simple kinetic model is not enough, two separate first-order kinetic models for fast and slow degradation, respectively, have been used. But, it is difficult to define the time when the fast mechanism ends and the slow mechanism begins during cooking (Lee et al., 2007). However, the fractional conversion provides improved accuracy and reliability in determining... 

There is computational and/or experimental evidence that a structurally diverse range of singlet biradicals occupy plateau-like regions of their PESs. A selection of them, including the reactions that they mediate, are shown in Scheme 21.2. The occurrence of a plateau at a mechanistically crucial region of the PE hypersurface guarantees that there can be no clear distinction between concerted and stepwise mechanisms, and also strongly indicates that the reaction may be one whose course will not be describable by any simple kinetic model. 

In the present chapter, we have attempted to illustrate how surface bonding and catalytic activity are closely related. One of the main conclusions is that adsorption energies of the main intermediates in a surface catalyzed reaction is often a very good descriptor of the catalytic activity. The underlying reason is that we find correlations, Brpnsted-Evans-Polanyi relations, between activation barriers and reaction energies for a number of surface reactions. When combined with simple kinetic models such correlations lead to volcano-shaped relationships between catalytic activity and adsorption energies. 

Lewandowski and Ollis have proposed a simple kinetic model describing the transient photocatalytic oxidation of aromatic contaminants [50]. The model considered three chemical species an aromatic contaminant, preadsorbed onto the catalyst in the dark and refreshed continuously from the gas phase a strongly bound, recalcitrant reaction intermediate and final reaction products (CO or CO2), assumed for simplicity to be strictly gas-phase species. The model also assumed that two types of catalyst site were present on the photocatalyst surface, with the first suitable for the adsorption of aromatic contaminants, as well as reaction intermediates, and the second type considered to be more polar in nature, suitable only for adsorption of partially oxidized reaction intermediates. 

Chapter 2 will discuss in some detail a simple isothermal autocatalytic model whereby a reactant P is converted to a product C through two intermediates A and B. The simple kinetic model can be written as... 

In addition to the general aims set out at the beginning of this chapter we have discovered a wealth of specific detail about the behaviour of the simple kinetic model introduced here. Most results have been obtained analytically, despite the non-linear equations involved, with numerical computation reserved for confirmation, rather than extension, of our predictions. Much of this information has been obtained using the idea of a pseudo-stationary state, and regarding this as not just a function of time but also as a function of the reactant concentration. Stationary states can be stable or unstable. 

However, simple kinetic models, especially of the Langmuir—Hinshel-wood type, can serve with advantage for correlation of experimental data in spite of simplifying assumptions which are necessary for their derivation. Experience shows that heterogeneous acid—base catalysis is the very field where they fit best. Their most frequent general form... 

Equations 3.71 and 3.72 can be further developed in terms of the self-diffusivity using the atomistic models for diffusion described in Chapters 7 and 8. The resulting formulation allows for simple kinetic models of processes such as dislocation climb, surface smoothing, and diffusional creep that include the operation of vacancy sources and sinks (see Eqs. 13.3, 14.48, and 16.31). 

In order to understand the reaction mechanism of photocatalysis, a simple kinetic model can be presented as follows. 

Simple kinetic models for the direct and homogeneous electrocarboxylation of benzyl halides were recently developed based on competitions at two successive stages involving R and R, respectively, and on the competition between reduction processes involving C02 and RX. Interestingly, the experimental data obtained were in satisfactory agreement with theoretical predictions, both in the case of the direct process [33] (in spite of the fact that no adjustable parameters were used in the model) and of the process mediated by a HCTC [21]. 

To obtain a simple kinetic model for this heterogeneous reaction, it has to be assumed that the concentration of hydrogen as a reactant, in terms of partial pressure, is constant, and that the hydrogen is thoroughly mixed with the reaction mass to avoid external mass-transfer limitations. Consequently, experiments were carried out in which the speed of the stirrer was varied from 600 to 1800 rpm. For stirrer speeds up to 1200 rpm, a strong dependence of the rate of reaction on the stirrer speed was found. For stirrer speeds above 1200 rpm, no significant increase in the reaction rate was found therefore, a speed of 1200 rpm was... 

However, for several epoxy-amine systems, the simple kinetic model expressed by the set of Eqs (5.18) (5.21) does not provide a good fitting with experimental results. Reaction mechanisms, including the formation of different kinds of complexes, have been postulated to improve the kinetic description (Flammersheim, 1998). Also, a more general treatment of the kinetics of epoxy-amine reactions would have to include the possibility of the homopolymerization of epoxy groups in the reaction path. Sets of kinetic equations including this reaction have been reported (Riccardi and Williams, 1986 Chiao, 1990 Cole, 1991). 

This simple kinetic model predicts the presence of an induction period in isothermal runs, where (3 gets close to (3eq, followed by a first-order kinetics, as experimentally observed. 

Some very important features of thermal aging cannot be easily understood without a kinetic reasoning. In the following section, a simple kinetic model is postulated, which, in spite of its oversimplification, allows us to understand most of the general properties of thermal aging. The structure-thermal stability relationships are briefly examined and then some practical aspects are presented. 

A number of authors (e.g., Bruggeman et al., 1981 and 1984 Opperhuizen et al., 1985 Gobas et al., 1989) have applied simple kinetic models (i.e., equations (13)-(15), (25)-(27), (29, (30))to fish. In most cases, the models are used in a descriptive sense to describe empirical data derived from bioconcentration or bioaccumulation tests. These models can play an important role in the analysis of the results of bioconcentration tests, but they are generally inapplicable to bioaccumulation under field conditions. 

In terms of the law of acting surfaces and without any additional assumptions, we will consider a simple kinetic model characterized by rate selfoscillations. 

Let us state the conclusions of this section. We have shown that, in terms of the law of acting surfaces (without any additional assumptions), it is possible to construct sufficiently simple kinetic models for the qualitative interpretation of self-oscillations in the rates of heterogeneous catalytic reactions. 

In summary, it was confirmed that simple kinetic models derived from the classical Frank scheme can reproduce the variety of unprecedented effects of... 

Such an expression is based on the simple kinetic model of olefin polymerisation and is valid for systems with non-supported Ziegler-Natta catalysts (of moderate activity). Some important limiting expressions can be deduced when [M] is high and when k kp km, and Cp = C, where C denotes the total concentration of active sites, and Pn denotes the number-average degree of polymerisation ... 

For excited state racemization that is fast enough to compete with emission, a simple kinetic model shows that the time-dependent difference in the excited... 

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