The Chemical Kinetics Approach

This chapter was devoted to introducing the thermodynamic concepts and formalism essential to understand chemical reactions. Thus, we reviewed the first and second laws of thermodynamics, introduced the concept of thermodynamic equilibrium, defined the free energy change, and used it to prove that thermodynamic and chemical equilibrium are equivalent concepts. Interestingly, we were able to obtain the las of mass action Ifom purely thermodynamic considerations, suggesting that the thermodynamic and the chemical kinetics approaches are closely related. This connection is explored in detail in the next chapter. Finally, the last section of the chapter was dedicated to understanding the concept of chemical potential from the perspective of statistical mechanics. In later chapters we tackle this same question from different angles. 

Different Approaches to Analyzing a Simple Chemical Reaction 

Let us begin by introducing a simple chemical reaction (perhaps the simplest possible one)  

As we discussed in Chap. 1, this reaction (like all other ones) is reversible. Therefore (3.1) denotes two complementary chemical reactions. The first one, in which a molecule of the chemical species A turns into a molecule of the species B, is represented by the right harpoon. The second reaction, corresponding to a molecule of species B turning into a molecule of species A, is represented by the left harpoon. Despite its simplicity, this reversible reaction-set is actually a good model for some essential biochemical processes like the gating of an ion channel between the close and open states, a protein dipping between two different conformational states, and the switching of a promoter between the active and repressed states. 

Although the following assertions are valid for all the chemical reactions that can be depicted by Eq. (3.1), I believe it is easier if one has an specific example 

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The chemical kinetics approach provides us with more insight with respect to the range of validity. We see from Table 4 that Fick s law indeed results without approximations (l.h.s.). For the pure electrical conduction (r.h.s.) we have to linearize the exponentials (Eqs. 98 and 99), i.e., to assume Ifa I RT which is definitely a good approximation for transport in usual samples it fails at boundaries or for ultrathin samples. Hence the application of Eq. (103) has to presuppose sufficiently thick samples and not too high fields. Table 4 also reveals the connection between Dk, uk and the transport rate constant kk and hence their microscopic meaning ... 

The usual chemical kinetics approach to solving this problem is to set up the time-dependent changes in the reacting species in terms of a set of coupled differential rate equations [5,6]. 

In hindsight, the primary factor in determining which approach is most applicable to a particular reacting flow is the characteristic time scales of the chemical reactions relative to the turbulence time scales. In the early applications of the CRE approach, the chemical time scales were larger than the turbulence time scales. In this case, one can safely ignore the details of the flow. Likewise, in early applications of the FM approach to combustion, all chemical time scales were assumed to be much smaller than the turbulence time scales. In this case, the details of the chemical kinetics are of no importance, and one is free to concentrate on how the heat released by the reactions interacts with the turbulent flow. More recently, the shortcomings of each of these approaches have become apparent when applied to systems wherein some of the chemical time scales overlap with the turbulence time scales. In this case, an accurate description of both the turbulent flow and the chemistry is required to predict product yields and selectivities accurately. 

The mechanism for cross-linking of thermosetting resins is very complex because of the relative interaction between the chemical kinetics and the changing of the physical properties [49], and it is still not perfectly understood. The literature is ubiquitous with respect to studies of cure kinetic models for these resins. Two distinct approaches are used phenomenological (macroscopic level) [2,5,50-72] and mechanistic (microscopic level) [3,73-85]. The former is related to an overall reaction (only one reaction representing the whole process), the latter to a kinetic mechanism for each elementary reaction occurring during the process. 

Reduced mechanisms are used increasingly to describe chemical reactions in computational fluid mechanics. However, the development of a reduced mechanism often requires a thorough knowledge of the chemical kinetics of the system of interest, and the results obtained with the reduced mechanism are only valid in a limited domain of initial and operating conditions. Methods to automate the reduction procedure are currently being developed to facilitate the use of this modeling approach, for example, as discussed in Ref. [314],... 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

As it was mentioned above, up to now only the dynamic interaction of dissimilar particles was treated regularly in terms of the standard approach of the chemical kinetics. However, our generalized approach discussed above allow us for the first time to compare effects of dynamic interactions between similar and dissimilar particles. Let us assume that particles A and B attract each other according to the law U v(r) = — Ar-3, which is characterized by the elastic reaction radius re = (/3A)1/3. The attraction potential for BB pairs is the same at r > ro but as earlier it is cut-off, as r ro. Finally, pairs AA do not interact dynamically. Let us consider now again the symmetric and asymmetric cases. In the standard approach the relative diffusion coefficient D /D and the potential 1/bb (r) do not affect the reaction kinetics besides at long times the reaction rate tends to the steady-state value of K(oo) oc re. 

Cyclic voltammetry is one of the most reliable electrochemical approaches to elucidate the nature of electrochemical processes, and to provide insights into the nature of processes beyond the electron-transfer reaction. Several investigations27-29 have extended this method to the study of the chemical kinetics for chemical processes that precede or follow the electron-transfer process, as well as for the study of various adsorption effects that occur at the electrode surface. However, these are sufficiently complicated that those interested should consult the original papers or recent reviews.13,14 30"38 Some simple, general cases are discussed in this chapter, and other examples are included in later chapters. 

There are a number of possible approaches to the calculation of influences of finite-rate chemistry on diffusion flames. Known rates of elementary reaction steps may be employed in the full set of conservation equations, with solutions sought by numerical integration (for example, [171]). Complexities of diffusion-flame problems cause this approach to be difficult to pursue and motivate searches for simplifications of the chemical kinetics [172]. Numerical integrations that have been performed mainly employ one-step (first in [107]) or two-step [173] approximations to the kinetics. Appropriate one-step approximations are realistic for limited purposes over restricted ranges of conditions. However, there are important aspects of flame structure (for example, soot-concentration profiles) that cannot be described by one-step, overall, kinetic schemes, and one of the major currently outstanding diffusion-flame problems is to develop better simplified kinetic models for hydrocarbon diffusion flames that are capable of predicting results such as observed correlations [172] for concentration profiles of nonequilibrium species. 

The information needed about the chemical kinetics of a reaction system is best determined in terms of the structure of general classes of such systems. By structure we mean quahtative and quantitative features that are common to large well-defined classes of systems. For the classes of complex reaction systems to be discussed in detail in this article, the structural approach leads to two related but independent results. First, descriptive models and analyses are developed that create a sound basis for understanding the macroscopic behavior of complex as well as simple dynamic systems. Second, these descriptive models and the procedures obtained from them lead to a new and powerful method for determining the rate parameters from experimental data. The structural analysis is best approached by a geometrical interpretation of the behavior of the reaction system. Such a description can be readily visualized. 

Szpakowska M, Nagy OB, Chemical kinetic approach to the mechanism of coupled transport of Cu(II) ions through bulk liquid membranes. J. Phys. Chem. A 1999 103 1553-1559. 

The chemical kinetics are usually not known for many industrial reactions and are often quite difficult to determine. While CFD holds the most promising approach for homogeneous systems when the kinetics are fully known, some experimental work with verification of CFD predictions is required for a successful scale-up. Most frequently, there is no luxury of determining detailed kinetics and doing CFD computations and verification. Hence, the following procedure (modified from Fasano [31]) is recommended ... 

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