Different Kinetic Situations

The overall kinetics vary considerably depending largely on the mode of termination in a particular system. Consider the case of termination exclusively by combination of the propagating center with the counterion (Eq. 5-21). The kinetic scheme of initiation, propagation, and termination consists of Eqs. 5-3, 5-4, 5-16, and 5-21, respectively. The derivation of the 

The number-average degree of polymerization is obtained as the propagation rate divided by the termination rate  

The rates of spontaneous termination and the two transfer reactions are given by 

For the case where chain transfer to S (Eq. 5-29) terminates the kinetic chain, the polymerization rate is decreased and is given by 

The various rate expressions were derived on the assumption that the rate-determining step in the initiation process is Reaction 5-4. If this is not the situation, the forward reaction in Eq. 5-3 is rate-determining. The initiation rate becomes independent of monomer concentration and is expressed by 

Rt -H Rtl -H RtrM + Rtr,S The rates of spontaneous termination and the two transfer reactions are given by 

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Table 5.4-43 Advantageous modes of operation for different kinetic situations ... 

Two quite different kinetic situations arise from Eq. 2-17 depending on the identity of HA, that is, on whether a strong acid such as sulfuric acid orp-toluenesulfonic acid is added as an external catalyst. 

A detailed study of the different kinetic situations that may arise and of the dependence of the peak parameters on the rate constants of the two steps can be found in [6, 19, 50, 52]. 

Figure 50 shows three examples for (different) kinetic situations in which bulk diffusion, surface reaction, and transport across a grain boundary are the sluggish steps. Nonetheless, the other parameters can also be evaluated. This becomes especially clear from the top figure, where the nonunity intercepts reveal surface effects. Similarly, the nonzero bending of the profiles in the other two figures indicates transport resistances. 

Even if the peak behavior fits well for a given apparent desorption order, the real kinetic situation may be a different one. As a rate controlling step in a second-order desorption, random recombination of two particles is assumed most frequently. However, should the desorption proceed via a nonrandom recombination of neighboring particle pairs into an ordered structure, the resulting apparent first-order desorption kinetics is claimed to be possible (36). The term pseudo-first-order kinetics is used in this instance. Vice versa, second-order kinetics of desorption can appear for a nondissociative adsorption, if the existence of a dimer complex is necessary before the actual desorption step can take place (99). A possibility of switching between the apparent second-order and first-order kinetics by changing the surface coverage has also been claimed (60, 99, 100). 

In 18.4-1, kj and kr are first-order rate constants with known Arrhenius parameters (Af, EAj) and (A, EAr), respectively. This situation, with different kinetics, arises in cases of gas-phase catalytic reactions and is further treated from this point of view in Chapter 21. Here, we consider four cases, and assume the reaction is noncatalytic. 

Extrapolation of G to low energies yields G = 1, at least for CFjBr, CF3CI, and CH3Br. This implies that at the threshold one orientation is unreactive and is consistent with our early suspicion that different orientations have different kinetic energy thresholds. This is indeed the situation, as illustrated by the example shown in Figure 11. 

Following the introduction of basic kinetic concepts, some common kinetic situations will be discussed. These will be referred to repeatedly in later chapters and include 1) diffusion, particularly chemical diffusion in different solids (metals, semiconductors, mixed conductors, ionic crystals), 2) electrical conduction in solids (giving special attention to inhomogeneous systems), 3) matter transport across phase boundaries, in particular in electrochemical systems (solid electrode/solicl electrolyte), and 4) relaxation of structure elements. 

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. 

Reduction of C(5)-substituted 2-hydroxychromans selectively provides 2,4-cis-chromans using large silane RsSiH reductants and 2,4-frans-chromans using the smaller silane PhSiH3. The stereochemical outcome has been rationalized on the basis of a Curtin-Hammett kinetic situation arising from hydride delivery to two different conformations of an intermediate oxocarbenium ion.363... 

The contents of the present contribution may be outlined as follows. Section 6.2.2 introduces the basic principles of coupled heat and mass transfer and chemical reaction. Section 6.2.3 covers the classical mathematical treatment of the problem by example of simple reactions and some of the analytical solutions which can be derived for different experimental situations. Section 6.2.4 is devoted to the point that heat and mass transfer may alter the characteristic dependence of the overall reaction rate on the operating conditions. Section 6.2.S contains a collection of useful diagnostic criteria available to estimate the influence of transport effects on the apparent kinetics of single reactions. Section 6.2.6 deals with the effects of heat and mass transfer on the selectivity of basic types of multiple reactions. Finally, Section 6.2.7 focuses on a practical example, namely the control of selectivity by utilizing mass transfer effects in zeolite catalyzed reactions. 

Sophisticated mathematical models based on the numerical simulation of the chromatographic process consider different kinetic and thermodynamic mechanisms [19], The theoretical approaches describe the biospecific adsorption of monovalent and multivalent adsorbates. They also account for the film mass transfer and pore diffusion contributions to the adsorption process and can be applied to analyze various complex experimental situations. Thus, ideally, the appropriate model will have to be selected to describe the actual chromatographic system. 

Numerous acidity and basicity scales have been elaborated for water and other solvents. However, there is no one single scale of acidity and basicity, equally valid and useful for all types of solvents and applicable to both equilibrium and kinetic situations. Excellent reviews on different acidity functions are given by Boyd [60] and Bates [50]. 

At this point, we can only refer to the various acidity scales for different series of solvents, such as those of Hammett [15] and Grunwald [16]. The acidity functions are introduced in order to obtain expressions that are not affected by the relative permittivity and that allow a quantitative comparison of acidity in different solvents. It should be stated, however, that there exists no single scale of acidity or basicity that is universally valid in all types of solvents and appHcable to both equilibrium and kinetic situations [17, 109]. 

Next, let us examine the situation where the reversible RR dimerization (ii) results in a dimer C that is stable on the time scale of the voltammetric experiment. The voltammo-grams that ideally may be obtained in the different kinetic regimes are shown in Fig. 15(a). At low sweep rates (full line), reaction (ii) behaves like a thermodynamic equilibrium and the response observed is close to that shown in Fig. 14(a). At intermediate sweep rates the dissociation of C is too slow to allow for the observation of oxidation current for B, and at high sweep rates it is possible even to outrun forward reaction (ii) and the response is essentially that for a reversible electron transfer process. 

Sometimes, however, this procedure may lead to incorrect conclusions. Suppose, for example, that one reactant is present in large excess, so thdt its concentration does not change appreciably as the reaction proceeds moreover (for example, if it is the solvent) its concentration may be the same in different kinetic runs. If this is so the kinetic investigation will not reveal any dependence of the rate on the concentration of this substance, which would therefore not be considered to be entering into reaction. This situation is frequently found in reactions in solution where the solvent may be a reactant. For example, in hydrolysis reactions in aqueous solution, a water molecule may undergo reaction with a solute molecule. Unless special procedures are employed the kinetic results will not reveal the participation of the solvent. However, its participation is indicated if it appears in the stoichiometric equation. 

Figure 1.4. Flow chart sheet of the strategy of bioprocess kinetic analysis for different process situations stationary/instationary, homogeneous/heterogeneous, differential/ integral, and true dynamic/balanced (frozen) reactor operation, rds, rate-determining step qss, quasi-steady-state, (From Moser, A. 1983.)...

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