Independent and Dependent Chemical Reactions

In the preceding section we noted fliat when multiple chemical reactions take place, only a set of independent chemical reactions should be considered to determine the species composition. As indicated, flie summations in Eq. 2.3.3 and Eq. 2.3.11 are taken over a set of independent chemical reactions and not over all the reactions that take place. This point deserves a closer examination. 

The concept of independent reactions, or, more accurately, independent stoichiometric relations, is an important concept in stoichiometry and reactor analysis. The number of independent reactions indicates the smallest number of stoichiometric relations needed to describe the chemical transformations that take place and to determine all the state quantities of a chemical reactor (species composition, temperature, enthalpy, etc.). As will be seen later, the number of independent reactions also indicates the smallest number of design equations needed to describe the reactor operation. Since state quantities are independent of the path, we can select different sets of independent reactions to determine the change from one state to another. Below, we discuss the roles of independent and dependent reactions in describing reactor operations. We also describe a procedure to determine the number of independent reactions and how to identify a set of independent reactions. 

To develop insight into the concept of independent reactions, consider the following reversible chemical reaction  

According to the stoichiometric methodology adopted here, reversible reactions are represented as two distinct reactions, a forward and a reverse. Hence, Reaction 2.4.1 is described by two chemical reactions a forward reaction. 

Although two chemical reactions take place here, both of them provide the same information on the proportions among the individual species. This is because Reaction 2.4.1b is the reverse of Reaction 2.4.1a, and its stoichiometric coefficients have the negative values of those of Reaction 2.4.1a. In mathematical terms, we say that the two reactions are linearly dependent. Hence, only one chemical reaction (stoichiometric relation) is needed to determine the species compositions. 

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Chemical reaction rates, 14 607. See also Kinetic measurements Chemical reactions. See also Chemical processes Reaction entries with absorption, 2 47-48, 71-76 activated carbon for control of, 4 755 on adsorbents, 2 629-630, 650-651 atomic level of, 16 736 contexts of, 22 336 engine knock and, 22 390—391 heterogeneous, 22 331-332, 339 homogeneous, 22 339 independent and dependent, 22 336—337 mass-transfer coefficients with, 20 753-755... 

In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix T into a vector c that has Nr reacting components cr and N conserved components cc.35 In the limit of equilibrium chemistry, the behavior of the Nr reacting scalars will be dominated by the transformed chemical source term S. 36 On the other hand, the behavior of the N conserved scalars will depend on the turbulent flow field and the inlet and initial conditions for the flow domain. However, they will be independent of the chemical reactions, which greatly simplifies the mathematical description. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

Independent composition specifications depend on the relationships among individual species and the chemical reactions taking place in the reactor, but they are invariant of the specific set of independent reactions selected. Since the set of independent reactions generated by the reduced matrix of the Gaussian elimination... 

We consider a complex chemical system and focus on a set of S species M , u = 1,..., 5, which can carry one or more identical molecular fragments that are unchanged during the process in the following we refer to these species as carriers. For simplicity, in this section we limit ourselves to the case of isothermal, well-stirred, homogeneous systems, for which the concentrations c = c (t), u = 1,..., 5, of the chemicals M , m = 1,..., 5, are space independent and depend only on time. Later on we consider the more complicated case of reaction-diffusion systems. The deterministic kinetic equations of the process can be expressed in the following form ... 

Gaussian plume models These models use a time and spatially independent horizontal wind field, time-dependent point source, and no chemical reactions or loss mechanisms. A bell-shaped downwind distribution is assumed. The answer is a function of source strength, average wind velocity, and two diffusion parameters Easy to use Sanctioned by EPA for developing implementation plans for ambient air-quality standards Much experience with use Assumes wind field constant and uniform Limits use to 1 h and 10 km Not useful for reactive pollutants 114... 

It should be emphasized that the independence of S on potential, together with other characteristics of the process (the dependence of the cathodic and the chemical reaction rate on the concentration of HsCl ions, the ij ] -effects, and the Tafel slope b - 120 mV), cannot be explained within the framework of traditional mechanisms of cathodic hydrogen evolution. A detailed analysis of this problem is given in [415]. Here, we shall confine ourselves to the salient features only. 

In the former case, the rate is independent of the diffusion coefficient and is determined by the intrinsic chemical kinetics in the latter case, the rate is independent of the rate constant k and depends on the diffusion coefficient the reaction is then diffusion controlled. This is a different kind of mass transport influence than that characteristic of a reactant from a gas to ahquid phase. 

The development of combustion theory has led to the appearance of several specialized asymptotic concepts and mathematical methods. An extremely strong temperature dependence for the reaction rate is typical of the theory. This makes direct numerical solution of the equations difficult but at the same time accurate. The basic concept of combustion theory, the idea of a flame moving at a constant velocity independent of the ignition conditions and determined solely by the properties and state of the fuel mixture, is the product of the asymptotic approach (18,19). Theoretical understanding of turbulent combustion involves combining the theory of turbulence and the kinetics of chemical reactions (19—23). 

A catalyst speeds up both the forward and the reverse reactions by the same amount. Therefore, the dynamic equilibrium is unaffected. The thermodynamic justification of this observation is based on the fact that the equilibrium constant depends only on the temperature and the value of AGr°. A standard Gibbs free energy of reaction depends only on the identities of the reactants and products and is independent of the rate of the reaction or the presence of any substances that do not appear in the overall chemical equation for the reaction. 

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

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