Slow kinetics equations

Kinetic Equations 3-143 and 3-153 are obeyed by nucleides undergoing radioactive decay, where the rate constant kj is large and kj is small. The reactant A is converted rapidly into the intermediate B, which slowly forms C. Figure 3-13b shows plots of the exponentials g-kit and and of tlieu difference. Since kj is small, tlie exponential g-kit shows a slow decay while e d shows a rapid decline. The... 

In 1930, Max Volmer and Tibor Erdey-Griiz used the concept of a slow discharge step for cathodic hydrogen evolntion (slow discharge theory). According to these ideas, the potential dependence of electrochemical reaction rate constants is described by Eq. (6.5). Since hydrogen ions are involved in the slow step A, the reaction rate will be proportional to their concentration. Thus, the overall kinetic equation can be written as... 

Usually, proton transfer in acid-base equilibria is a fast process, but very hindered systems show slow kinetics. The equilibrium (equation 3) regarding 2,4-dialkylpyridines proceeds through locked-rotor and with intermediates of low entropy44. 

The kinetics of the reaction have been studied [2], A first order has been found in the concentrations of the titanium catalyst, t-butylhydroperoxide, and the allylic alcohol. The reaction is slowed down by 2-propanol, or more accurately, it has an inverse second order in 2-propanol. Thus, the kinetic equation reads ... 

If 8L/L0 is small, as observed experimentally, perhaps due to very slow kinetics of chain rearrangements, then the relationship between Lq and Tc is the same as the Gibbs-Thompson equation ... 

The kinetic model may be formulated using kinetic equations for all steps or using equilibrium equations for all but the slowest steps. The latter approach reduces the computational effort and leads to a kinetic expression, which is far easier to analyze. However, if a step, which is slow in reality, is modeled by an equilibrium equation, the micro-kinetic model becomes unrealistic and it may in some cases be the simplest to treat a problematic step by a kinetic equation. 

The model can be further tested for internal consistency. Steps treated through kinetic equations should be slow at least under some conditions or the model may obviously be... 

Constant pattern condition This condition reduces the mass balance equation (4.128) to the simple relation C/C0=q/qm lx (see the section A look into the constant pattern condition). Practically, the constant pattern assumption holds if the equilibrium is favorable, and at high residence times (Perry and Green, 1999 Wevers, 1959 Michaels, 1952 Hashimoto et al., 1977). However, the constant pattern assumption is weak if the system exhibits very slow kinetics (Wevers, 1959). 

Exercise 17-16 When the aldol reaction of ethanal is carried on in D20 containing OD , using moderate concentrations of undeuterated ethanal, the product formed in the early stages of the reaction contains no deuterium bound to carbon. Assuming the mechanism discussed in this section to be correct, what can you conclude as to which step in the reaction is the slow step What then would be the kinetic equation for the reaction What would you expect to happen to the kinetics and the nature of the product formed in D20 at very low concentrations of ethanal ... 

Since the concentration of these active centres increases in the initial stage of copolymerization where only small changes in the concentration of the components occur, initial monomer concentrations can be used in the kinetic equation for the formation of the active centres. The concentration of active centres is usually much lower than the concentration of any other component participating in the equilibrium. Since reaction (55) has not been confirmed experimentally and is assumed to be rather slow under the given experimental conditions, the existence of the induction period followed by a steady state is compatible with the scheme described by Eqs. (69)-(72). For initiation without proton-donor compounds it holds 45) ... 

The occurrence of self-acceleration during curing of epoxy resins and epoxy-based compounds was proven by rheokinetic and calorimetric methods.53 This phenomenon can be treated formally in terms of an induction period (when the reaction is very slow in the initial stage of a process), followed by a constant rate. However, it seems preferable to use a single kinetic equation incorporating the self-acceleration effect to describe reaction as a whole. Such a kinetic equation contains only a limited number of constants (K and co in Eq. (2.33)) and allows easy and unambiguous interpretation of their dependencies on process factors. 

The results of the numerical experiment for system (20) necessitated a general mathematical investigation of slow relaxations in chemical kinetic equations. This study was performed by Gorban et al. [226-228] who obtained several theorems permitting them to associate the existence of slow relaxations in a system of chemical kinetic equations (and, in general, in dynamic systems) with the qualitative changes in the phase portrait with its parameters (see Chap. 7). 

Finally, we can suggest a third explanation fast steps can compose a mechanism with slow relaxations. Indeed, nothing suggests that the relaxation time for a set of chemical kinetic equations is directly dependent on the characteristic times of the individual steps. But it cannot be treated as a reason for slow relaxations. It is only a simple indication for the possibility of finding such reasons here. Let us now indicate the reasons according to which fast steps can compose a mechanism with slow relaxations. 

The rate of elimination is an important characteristic of a drug. Too rapid an elimination necessitates frequent repeated administration of the drug if its concentration is to reach its therapeutic window. Conversely, too slow an elimination could result in the accumulation of the drug in the patient, which might give an increased risk of toxic effects. Most drug eliminations follow first order kinetics (equations (8.1) and (8.2)), no matter how the drug is administered, but there are some notable exceptions, such as ethanol which exhibits zero order kinetics where ... 

To develop the kinetic equations in condensed phases the master equation must be formulated. In Section 3 the master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The latter set of equations permits consideration of history of formation of the local solid structure as well as its influence on the subsequent elementary stages. The many-body problem and closing procedure for kinetic equations are discussed. The influence of fast and slow stages on a closed system of equations is demonstrated. The multistage character of the kinetic processes in condensed phase creates a problem of self-consistency in describing the dynamics of elementary stages and the equilibrium state of the condensed system. This problem is solved within the framework of a lattice-gas model description of the condensed phases. 

To obtain the closed system of kinetic equations one should divide all variables on slow (with the characteristics time xm) and rapid (with the characteristics time xr) variables with respect to the chosen time interval x, which is presented in the left sides of the differential Eqs. (31) and (32). As rule, in condensed reaction systems inner degrees of freedoms for reactants can reach the local relaxation state during less time by comparison with rapid variables. In opposite cases, various inner states of reactants should be considered as independent kinds of species. 

In the second case, if the species mobilities differ greatly, the dimensionality of the system of kinetic equations decreases [103], Let all the components be divided into two groups of species a slow (5) and a rapid (r) one. This yields three types of pair functions. For the rapid species the condition of the equilibrium distribution can be considered as satisfied. Then, for the pair functions of types sr and rr instead of the kinetic Eqs. (32) algebraic relations in Appendix A apply, whose dimensionality can be lowered using the method of substitution variables according to Appendix B. In this case the kinetic Eqs (31) for the local concentrations and Eq. (32) for the pair functions type ss do not change. A similar situation remains in passing to the one-dimensional discrete and point-like models. 

Kinetic Equations 3-143 and 3-153 are obeyed by nucleides undergoing radioactive decay, where the rate constant k, is large and k2 is small. The reactant A is converted rapidly into the intermediate B, which slowly forms C. Figure 3-13b shows plots of the exponentials C-M and e-M and of their difference. Since k2 is small, the exponential e-k2t shows a slow decay while e klt shows a rapid decline. The difference of e-k2t-e-kl is shown by the dashed line in Figure 3-13b. The concentration of B is (Equation 3-143) equal to this difference multiplied by CAO (since kt k2). Therefore, the concentration of B rapidly rises to the value of CAO and then slowly declines. The rise in concentration C then approximately follows the simple first-order law. Conversely, when k, is small and k2 is large (k2 kj), the concentration of B is given by Equation 3-143 ... 

The paraffins dehydrogenation on platinum-alumina catalysts proceeds with constant rate up to some time-on-stream after which a slow deactivation of the catalysts takes place Since relative changes of the catalyst activity ( characterized by reaction rate) are proportional to relative amounts of the deposited coke it can suppose that coke formation is the main reason of deactivation. Deactivation can be related with an attainment of a threshold in coke concentration (Co) on catalysts. The threshold amounts are 1.8 wt.% for A-I, 6,8% and 2.2% for A-II and A-IXI catalysts respectively. The isobutane dehydrogenation in non-stationary region (C > Co) is described by the following kinetic equation ... 

This kinetic equation was originally proposed by Crank (1953), who referred to the first term as the instantaneous part and the second term as the slow part of the time dependence of D. He considered that these parts are associated with the instantaneous and retarded deformations of a polymer molecule occurring when it is subject to an external force. In accordance with Crank and Park (1951), the diffusion process governed by a diffusion coefficient depending explicitly on time is generally termed the time-dependent diffusion or the history-dependent diffusion. 

Together with acid-base reactions, where a proton transfer occurs (pH-dependent dissolution/ precipitation, sorption, complexation) redox reactions play an important role for all interaction processes in aqueous systems. Redox reactions consist of two partial reactions, oxidation and reduction, and can be characterized by oxygen or electron transfer. Many redox reactions in natural aqueous systems can actually not be described by thermodynamic equilibrium equations, since they have slow kinetics. If a redox reaction is considered as a transfer of electrons, the following general reaction can be derived ... 

As seen in the figure, the major portion of cyanide is present as hydrogen cyanide at the pH range of the rivers and soil solutions. (It is usually neutral or weakly alkaline pH 6-8.) The rate of evaporation decreases as pH increases. When studying these systems, it is important to decrease the rate of evaporation as much as possible, or else the evaporated quantity must be taken into consideration. The evaporation of cyanide, however, is slow the rate-determining step is diffusion, and it can be described with an exponential relation similar to a formally first-order kinetic equation. The evaporation of cyanide slightly increases as temperature increases. 

The steady-state V or P values do not depend upon the micro-wave power level. The lack of power dependence and the requirements of ti/2 and Ti corrections in the slow-response detection mode can be demonstrated together by considering an ensemble of radicals whose two spin states, a and 3, are populated at a rate <(>a and respectively. With this assumption the radicals can only decay by reaction between radicals of opposite spin states. The kinetic equations for these two states can then be written as... 

Assuming that carbonium ion isomerization is a slow stage (stage 2) while the other intermediate stages are fast and are in equilibrium, one may derive the following kinetic equation ... 

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