Complex kinetic schemes

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

The kinetic information is obtained by monitoring over time a property, such as absorbance or conductivity, that can be related to the incremental change in concentration. The experiment is designed so that the shift from one equilibrium position to another is not very large. On the one hand, the small size of the concentration adjustment requires sensitive detection. On the other, it produces a significant simplification in the mathematics, in that the re-equilibration of a single-step reaction will follow first-order kinetics regardless of the form of the kinetic equation. We shall shortly examine the data workup for this and for more complex kinetic schemes. 

A reader familiar with the first edition will be able to see that the second derives from it. The objective of this edition remains the same to present those aspects of chemical kinetics that will aid scientists who are interested in characterizing the mechanisms of chemical reactions. The additions and changes have been quite substantial. The differences lie in the extent and thoroughness of the treatments given, the expansion to include new reaction schemes, the more detailed treatment of complex kinetic schemes, the analysis of steady-state and other approximations, the study of reaction intermediates, and the introduction of numerical solutions for complex patterns. 

Neither method will achieve a bumpless startup for complex kinetic schemes such as fermentations. There is a general method, known as constant RTD control, that can minimize the amount of off-specification material produced during the startup of a complex reaction (e.g., a fermentation or polymerization) in a CSTR. It does not require a process model or even a realtime analyzer. We first analyze shutdown strategies, to which it is also applicable. 

For a more complex kinetics scheme, a combination of the explicit and recursion-formula approaches may be required. 

A chemical reaction is a complex process. Besides thermodynamic factors, the process has two other distinct aspects kinetic and molecular mechanistic ones. With the development of modem technology, more and more complex kinetic schemes can be determined by using sufficient experimental information and fairly general computer programs [155]. In order to proceed, it is useful to define what we mean by a theoiy of chemical reactions in the first place. 

It is important to note that Eqs. 5, 8, and 9 were derived entirely from a silicon material balance and the assumption that physical sputtering is the only silicon loss mechanism thus these equations are independent of the kinetic assumptions incorporated into Eqs. 1, 2, and 7. This is an important point because several of these kinetic assumptions are questionable for example, Eq. 2 assumes a radical dominated mechanism for X= 0, but bombardment-induced processes may dominate for small oxide thickness. Moreover, ballistic transport is not included in Eq. 1, but this may be the dominant transport mechanism through the first 40 A of oxide. Finally, the first 40 A of oxide may be annealed by the bombarding ions, so the diffusion coefficient may not be a constant throughout the oxide layer. In spite of these objections, Eq. 2 is a three parameter kinetic model (k, Cs, and D), and it should not be rejected until clear experimental evidence shows that a more complex kinetic scheme is required. 

The detection by epr spectroscopy of a particular spin adduct is generally dependent on a complex kinetic scheme (Scheme 1), which can be divided into three parts. 

Lower), as reported earlier, but also rises more rapidly than kp (Fig. 13 Upper). This result immediately requires a more complex kinetic scheme than that of Scheme I. Excellent self-consistent fits to the time evolution of [1] (t) are obtained with an expression that is the sum of three kinetic phases, all having a common rate constant for triplet decay, kp, but with differing values of the rate constants for the decay of 1(3000 s , 40s , 5s ). We have further seen that complexes with different Cc show similar behavior, but with the fractional contribution of these multiple phases varying with species. 

In previous sections we treated electrochemical processes that proceed according to a single electrochemical reaction [Eq. (6.6)]. However, many electrochemical processes proceed by a multitude of processes characterized by complex kinetic schemes. There are various types of complex mechanisms. 

It is relatively straightforward to solve the differential equations for the time dependence of the transients in simple cases. However, it is important to understand the physical meaning of why a particular case gives rise to a particular form of solution. In this section we will concentrate on an intuitive approach to this understanding. Once a feel for the subject has been developed, algebraic mistakes will not be made and some complex kinetic schemes may be solved by inspection. 

These mathematically complex kinetic schemes should be compared with the mathematically straightforward schemes ... 

The selectivity of a productive reaction refers to the relative amounts of P, P at the time of observation. The ratio of the amounts of P and P which are formed is the ratio of the corresponding rate constants, if the stereoselective is a pair of corresponding reactions53. If, however, the productive stereoselective reaction is a more complex kinetic scheme, then the ratio of the amounts of any two stereoisomeric products, P and P , which depends on time and pairs of the appropriate kinetic constants, has a positive lower bound and a finite upper bound. Both of these bounds are the ratios of two rate constants54. However, since the free enthalpy difference of stereoisomeric transition states is due to different non-bonded interaction and does not, as a rule, exceed 3 kcal/mole, and since the rate constant ratio depends on the free enthalpy difference, this ratio has a rather low upper bound. Accordingly, the stereoselectivity of productive reactions is generally low (50—90% relative yield of the preferred product in most cases). 

If two different substrates bind simultaneously to the active site, then the standard Michaelis-Menten equations and competitive inhibition kinetics do not apply. Instead it is necessary to base the kinetic analyses on a more complex kinetic scheme. The scheme in Figure 6 is a simplified representation of a substrate and an effector binding to an enzyme, with the assumption that product release is fast. In Figure 6, S is the substrate and B is the effector molecule. Product can be formed from both the ES and ESB complexes. If the rates of product formation are slow relative to the binding equilibrium, we can consider each substrate independently (i.e., we do not include the formation of the effector metabolites from EB and ESB in the kinetic derivations). This results in the following relatively simple equation for the velocity ... 

For some complex kinetic schemes, you must calculate the time dependence of the concentrations of reactants that are involved in the chemical transformations. Can the tools of thermodynamics of nonequilibrium processes be helpful in doing this Why or why... 

Chemical kinetics resembles (in some sense) classical mechanics, which allows the final state of the dynamic systems to be accurately predicted when dynamic equations are known for describing the time behavior of the system and exact starting conditions are given. On the other hand, in complex kinetic schemes, the final result of the transformations under consideration is expected to change considerably depending on the partic ular assumptions about elementary steps of the process and the choice of the kinetic scheme, as well as on the assumptions, rather arbitrary some times, about reversibility of each of these steps. This makes it difficult to describe the time evolution of the systems with poorly understood or complex mechanisms of chemical transformations. 

COCH3 + CH2COCH3 (CH3CO)oCH2 This is an imposingly complex kinetic scheme that has three radical intermediates and possibly one species that may reach a stationary state, namely, ketene. Because of the second-order character of the reactions proposed for the destruction of radicals, an explicit etiuation for the stationary-state concentrations is impossible. If we set the rate of reaction 1 = 0/a(l — x) and that of reaction 2 = IaX, where 0 = fraction of excited acetone molecules decomposing, = average number of quanta absorbed per cc-sec, x = fraction of excited acetone molecules which split by the second path, then we can equate the ratio of radical production and destruction in the system ... 

Nowadays, improved computing facilities and, more importantly, the availability of the Chemkin package (Kee and Rupley, 1990) and similar kinetic compilers and processors have made these complex kinetic schemes more user-friendly and allows the study of process alternatives as well as the design and optimization of pyrolysis coils and furnaces. In spite of their rigorous, theoretical approach, these kinetic models of pyrolysis have always been designed and used for practical applications, such as process simulation, feedstock evaluations, process alternative analysis, reactor design and optimization, process control and so on. Despite criticisms raised recently by Miller et al. (2005), these detailed chemical kinetic models constitute an excellent tool for the analysis and understanding of the chemistry of such systems. 

Unfortunately, even in this case we are facing several difficulties. First, different reactions included into a complex kinetic scheme cannot be preliminarily studied to the same extent using independent methods. So, if for some reactions numerous data are available in corresponding literature and databases, in other cases not a single reliable value could be found. This means that in the framework of one kinetic model we have to use parameters obtained using very different experimental methods (direct and indirect) and also by more or less well-grounded evaluations. 

It is clear that a straightforward physical meaning cannot be attached to the monomer amplitudes nor to the excimer amplitudes A2. For a more complex kinetic scheme such as Scheme (II), the expressions for the amplitudes in the triple-exponential decays similarly are functions of all the rate constants involved (25). Therefore, also in this case, a simple physical meaning cannot be attributed to these amplitudes, see Section 4.3.3. 

We now consider the somewhat more complex kinetic scheme of the establishment of a monomer/ dimer equilibrium... 

Analytical solutions are desirable because they explicitly show the functional dependence of the solution on the operating variables. Unfortunately, they are difficult or impossible for complex kinetic schemes and are almost always impossible for the nonisothermal reactors considered in Chapter 5. All numerical solutions have the disadvantage of being case specific, but this disadvantage can be alleviated... 

In its most general form, the fullerene synthesis could be treated as a complex kinetic scheme described by a huge number of kinetic differential equations. The equilibrium composition comes as the limiting case for infinite time. If we treat the problem from a thermodynamic point of view, we should realize that the conventional standard pressure of 1 atm is considerably different from the actual fullerene synthesis conditions. We should expect lower cluster pressures in the carbon-arc synthesis. The actual entropy and Gibbs free energy change with pressure as can be demonstrated [208-212] on the Cgo and C70 cases based on computed or observed [213] data. For example, the equilibrium constant Xgo/yo for an interconversion between the two clusters, expressed in partial pressures p, offers a deeper insight into the problem [208-212] ... 

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