Chemical reactions finite-rate

Every chemical reaction occurs at a finite rate and, therefore, can potentially serve as the basis for a chemical kinetic method of analysis. To be effective, however, the chemical reaction must meet three conditions. First, the rate of the chemical reaction must be fast enough that the analysis can be conducted in a reasonable time, but slow enough that the reaction does not approach its equilibrium position while the reagents are mixing. As a practical limit, reactions reaching equilibrium within 1 s are not easily studied without the aid of specialized equipment allowing for the rapid mixing of reactants. 

There is no sharp dividing hne between pure physical absorption and absorption controlled by the rate of a chemic reaction. Most cases fall in an intermediate range in which the rate of absoration is limited both by the resistance to diffusion and by the finite velocity of the reaction. Even in these intermediate cases the equihbria between the various diffusing species involved in the reaction may affect the rate of absorption. 

Imagine a closed reaction vessel in which an exothermic reaction proceeds at room temperature at a finite rate. Although the temperature in the reaction vessel is initially the same as room temperature, it rises gradually until the rate of heat generation due to the exothermic chemical reaction is equal to the rate of heat escape from the reaction vessel surface. However, if a thermal balance is not established for such a chemical reaction, the reaction rate is accelerated by self-heating as the temperature rises, leading to thermal runaway. The temperature change in a reaction vessel is represented by Eq. (1),... 

The focus of the remainder of this chapter is on interstitial flow simulation by finite volume or finite element methods. These allow simulations at higher flow rates through turbulence models, and the inclusion of chemical reactions and heat transfer. In particular, the conjugate heat transfer problem of conduction inside the catalyst particles can be addressed with this method. 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

The FM approach to modeling turbulent reacting flows had as its initial focus the description of turbulent combustion processes (e.g., Chung 1969 Chung 1970 Flagan and Appleton 1974 Bilger 1989). In many of the early applications, the details of the chemical reactions were effectively ignored because the reactions could be assumed to be in local chemical equilibrium.26 Thus, unlike the early emphasis on slow and finite-rate reactions... 

In (5.297), the interpolation parameter is defined separately for each component. Note, however, that unlike the earlier examples, there is no guarantee that the interpolation parameters will be bounded between zero and one. For example, the equilibrium concentration of intermediate species may be negligible despite the fact that these species can be abundant in flows dominated by finite-rate chemistry. Thus, although (5.297) provides a convenient closure for the chemical source term, it is by no means guaranteed to produce accurate predictions A more reliable method for determining the conditional moments is the formulation of a transport equation that depends explicitly on turbulent transport and chemical reactions. We will look at this method for both homogeneous and inhomogeneous flows below. 

The development of the mass burning rate [Eq. (6.118)] in terms of the transfer number B [Eq. (6.120)] was made with the assumption that no oxygen reaches the fuel surface and no fuel reaches °°, the ambient atmosphere. In essence, the only assumption made was that the chemical reactions in the gas-phase flame zone were fast enough so that the conditions mos = 0 = m[m could be met. The beauty of the transfer number approach, given that the kinetics are finite but faster than diffusion and the Lewis number is equal to 1, is its great simplicity compared to other endeavors [20, 21],... 

If finite chemical reaction times are put into the columnar diffusion flame theory (76), burning rates are predicted to be linearly proportional to pressure at low pressure and independent of pressure (plateau burning) at high pressure. Based on this model, von Elbe et al. (97) proposed the simple equation ... 

The above models describe a simplified situation of stationary fixed chain ends. On the other hand, the characteristic rearrangement times of the chain carrying functional groups are smaller than the duration of the chemical reaction. Actually, in the rubbery state the network sites are characterized by a low but finite molecular mobility, i.e. R in Eq. (20) and, hence, the effective bimolecular rate constant is a function of the relaxation time of the network sites. On the other hand, the movement of the free chain end is limited and depends on the crosslinking density 82 84). An approach to the solution of this problem has been outlined elsewhere by use of computer-assisted modelling 851 Analytical estimation of the diffusion factor contribution to the reaction rate constant of the functional groups indicates that K 1/x, where t is the characteristic diffusion time of the terminal functional groups 86. ... 

This is a book about chemical kinetics—not necessarily the most familiar aspects of that subject, but nevertheless the various phenomena to be described arise primarily because reactions occur at finite rates, and different reactions may occur at different rates. Before proceeding along our kinetics course, however, it is worth while examining what information we can gain from thermodynamics. For most of us, the familiar aspects of thermodynamics are those dealing with systems at chemical equilibrium. Then we can use concepts such as enthalpy and entropy to place strong restrictions on the final equilibrium composition attained from a given set of initial reactant concentrations. 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

These approaches can be divided into two groups. In the first group, fast chemistry (approaches 1 and 2), it is assumed that the rate of chemical conversion is not kinetically controlled. The second group,finite rate chemistry (approaches 3-5), allows for kinetically controlled processes, in that restrictions are put on the chemical reaction rate. Below we discuss these different approaches in more detail. 

Global Reactions The use of global or multi-step reactions to represent the chemistry in a reacting flow system may be a significant improvement compared to assumptions of fast reaction or chemical equilibrium. The use of global reactions such as in Eq. 13.2 is the simplest way to introduce finite rate chemistry. 

The chemical kinetics occur at a finite rate, with a certain time required for reactions to proceed. As the frequency decreases, providing more time at relatively higher temperature and pressure within each cycle, there is time for the chain-branching free-radical species to build up to levels that trigger an ignition. As the frequency increases, the time available... 

Since the invention of d.c. polarography [10, 11], numerous inorganic and organic compounds have been studied by means of this method in Heyrovsky s school and extensive knowledge gathered about the electrochemical properties of these compounds. Among them, many cases were discovered where the polarographic wave appeared to be influenced by the existence of chemical equilibria between the electroactive substance and other, in most cases electroinactive, species in the electrolyte solution, more particularly by the finite rate at which these equilibria relax after the electrochemical perturbation. In fact, the chemical reaction serves as either a source or a sink to deliver or to consume the electroactive reactants and products, in addition to diffusion. 

However, in this paper Ya.B. went further and considered the chemical kinetics. He determined the limit of intensification of diffusion combustion, which is related to the finite chemical reaction rate and the cooling of the reaction zone, for an excessive increase of the supply of fuel and oxidizer. If the temperature in the reaction zone decreases in comparison with the maximum possible value by an amount approximately equal to the characteristic temperature interval (calculated from the activation energy of the reaction), then the diffusion flame is extinguished. The maximum intensity of diffusion combustion, as Ya.B. showed, corresponds to the combustion intensity in a laminar flame of a premixed stoichiometric combustible mixture. 

Finally, comparatively recently, Ya.B., A. P. Aldushin, and S. I. Khudyaev (23) completed a theory of flame propagation which considers the most general case of a mixture in which the chemical reaction occurs at a finite rate at the initial temperature as well. In this work the basic idea is followed through with extraordinary clarity flame propagation represents an intermediate asymptote of the general problem of a chemical reaction occurring in space and time. At the same time, the relation between the two types of solutions (KPP and ZFK) is completely clarified. 

Let us introduce a finite chemical reaction rate. One might think that the type of the chemical kinetics (autocatalysis or a classical reaction of some order) is in a certain sense not in itself essential autocatalysis changes the absolute value of the induction period and places it in dependence on small admixtures in the original mixture, but the form of the kinetic curve itself hardly changes since, in a classical reaction as well, with any significant activation heat one observes significant self-acceleration related to the increase in temperature. 

Rivin and Sokolik [1] were the first to notice that in a detonation wave the chemical reaction zone, in which the transformation of the original mixture into the reaction products occurs, must have a finite width which depends on the reaction rate. 

Important classes of chemical reactions in the ground electronic state have equal parity for the in- and out-going channels, e.g., proton transfer and hydride transfer [47, 48], To achieve finite rates, such processes require accessible electronic states with correct parity that play the role of transition structures. These latter acquire here the quality of true molecular species which, due to quantum mechanical couplings with asymptotic channel systems, will be endowed with finite life times. The elementary interconversion step in a chemical reaction is not a nuclear rearrangement associated with a smooth change in electronic structure, it is aFranck-Condon electronic process with timescales in the (sub)femto-second range characteristic of femtochemistry [49],... 

Turning our attention to surface phenomena rather than diffusion, we recognize that species will transit across the boundary layer and may be created or destroyed in this passage due to chemical reactions which will proceed at finite rates (homogeneous gas phase reactions). Upon impacting the surface, they may adsorb and then decompose, leaving a solid thin film. This will be a heterogeneous surface reaction which will have a characteristic chemical reaction rate. One way to describe this phenomena is in terms of a mass transfer" coefficient. The mass flux can be expressed in terms of this coefficient, as follows ... 

In the first chapter, we consider the fundamental nature of the thermally-induced CVD. Initially, we consider the behavior of CVD reactions under the assumption of chemical equilibrium. Much useful information can be derived by this technique, especially for very complex chemical systems where several different solid phases can be deposited. In order to extend our understanding of CVD, it is necessary to consider reacting gas flows where the rates of chemical reactions are finite. Therefore, the next subject considered is the modeling of CVD flows, including chemical kinetics. Depending on processing conditions, the film being deposited may be amorphous, polycrystalline, or epitaxial. 

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