Complex reactions Rate equations

If the forward and reverse reactions are nonelementary, perhaps involving the formation of chemical intermediates in multiple steps, then the form of the reaction rate equations can be more complex than Equations 5.33 to 5.36. 

Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series... 

The overall rate equation of complex single-route reaction with the linear detailed mechanism was derived and analyzed in detail by many researchers. King and Altman (1956) derived the overall reaction rate equation for single-route enzyme reaction with an arbitrary number of intermediates... 

The reaction system involved in the case studied is a kind of relatively complex van de vusse reaction, nevertheless the reaction system in real manufacturing process may involve more reaction types and, therefore, is more complex than that one. One can, however, simulate the change of environmental indexes within a reactor by combining traditional reactor mathematical model with the PEI balance, and may also discover the effects of reaction conditions and engineering factors on environmental performance by PEI rate-law expression and/or combinations it with other reaction rate equations as well as other related equations in reactor mathematical models. 

Steady-state reaction theory gives an answer to an important question, namely that if one knows a kinetic law for some elementary reaction, in what way can an equation for the complex reaction rate be derived ... 

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

Table 9.11 Rate parameters for the surface complexation reactions in equation (9.11) (goethite adsorbent) at pH 5...

In this respect, the overall mineralization rate of phenol has often been approximated with a zero-order reaction rate (Salaices et al., 2004) as it follows a fairly straight line. For the Fe-assisted PC reaction, however, this approximation cannot be applied given the sharp change of slope in the last part of the photoconversion reaction. Thus, a more complex kinetic rate equation needs to be developed to account for this behavior. 

In order to understand how the constant k depends on temperature, it was assumed that the chemical reactions may take place only when the molecules collide. Following this collision, an intermediate state called an activated complex is formed. The reaction rate will depend on the difference between the energy of the reactants and the energy of the activated complex. This energy E is called activation energy (other notation E ). The reaction rate will also depend on the frequency of collisions. Based on these assumptions it was shown (e.g. [3]) that k has the following expression (Arrhenius reaction rate equation) ... 

In this model, the first step is the dissociation of C02 at a carbon free active site (Cfas), releasing CO and forming an oxidized surface complex [C(O)]. In the second step, the carbon-oxygen complex subsequently produces CO and a new free active site. The reverse reaction is relatively slow compared with the forward reaction, so the second reaction can be treated as an irreversible reaction. In this model, desorption of the carbon-oxygen surface complex is the rate-limiting step. The rate for this mechanism can be described by the Lang-muir-Hinshelwood rate equation. Furthermore, the C/C02 reaction rate is dependent on the CO and C02 partial pressures and is inhibited by the presence of carbon monoxide. A widely utilized reaction rate equation based on this mechanism is... 

The application of any interpretation of kinetic data to reaction rate equations containing more than one concentration term can be somewhat complex. Consider the reaction... 

In keeping with this complex mechanistic picture, the investigation of the reaction rate equation for the copolymerization of ethene, using [Pd(dppp)(O2CCF3)2],... 

A steady-state approximation is often used in order to simplify the mathematical description of complicated reaction mechanisms. Below we will use simulation to illustrate when such a simplifying assumption is appropriate, and when it is not. We will use the reaction sequence A + B—C —> products. C can either be a chemically identifiable species, or a presumed or hypothetical intermediate, such as a transition state or activated complex. The rate equations are... 

To summarize, the proton transfer reaction can be broken into three distinct parts Diffusion of the reactants to within the radius of the ionic atmosphere accelerated diffusion to within the encounter distance and subsequent conversion of the encoimter complex to products. For reactions in which the equilibrium is rapidly established within the encounter complex, the rate equations are dominated by the diffusion process. This results in the loss of information about the dynamics of the encounter complex. For such a reaction some information can be obtained about the ionic radius by varying the ionic strength and using an electrostatic theory (such as is done for Deby-Hiickel activity coefficients) to calculate the effect of shielding by the ions. ... 

According to the main postulate of the chemical kinetics, correlation of complex reaction rate vs. concentration and order is subjected to the following equation... 

In the foregoing examples, rate equations were developed on the basis of a single rate-determining step. It is possible that many steps of a cycle are simultaneously controlling, as in the Wacker process. The rate equation for such a reaction tends to be more complicated but can be developed by the methods discussed in Chapter 7. Thus for the oxidation of triphenylphosphine with a Pt complex, a rate equation can be developed based on the catalytic cycle shown in Figure 8.9 (Halpern and Pickard, 1970 Birk et al., 1968a,b) ... 

The individual steps in the mechanism proposed for the ruthenium catalyzed oxidation of triphenylphosphine are similar to those of the platinum catalyzed reaction. However, the stoichiometry of one step and the relative rates of several other steps appear to be somewhat different. The observed rate laws are quite different from those of the platimun complex. The rate equations for the proposed mechanism are shovm in equation (71) for the dioxygen complex and equation (72) for the carbonyl complex (RuOj = [Ru(NCSXN0XPPh3)j(02)] RuCO = [Ru(NCSXCOXNOXPPh3)2]). 

This is quite a complex integrated rate equation. However, if we study the kinetics of the reaction at points in time near the establishment of equilibrium, we make the assumption that the forward and reverse rates are becoming equal (as when equilibrium is really established). At equilibrium we define [x] as [x]e, where the extent of reaction is as far as it is going to go, which leads to W[ A]o - [x]c) = fcr([B]o + [x]e). Solving this equality for fcf[ A] - A r[B] , and substituting the result into Eq. 7.41, leads to Eq. 7.42. This tells us that as one approaches equilibrium, the rate appears first order with an effective rate constant that is the sum of the forward and reverse rate constants. This is an approximation because we defined [.v] as [. ]e to obtain this answer, but it is a very common way to analyze equilibrium kinetics. Chemists qualitatively estimate that the rate to equilibrium is the sum of the rates of the forward and reverse reactions. 

Mathematical representation of simple and complex reactions Independent reactions Rate equations... 

---