Catalytic reaction steady-state approximation

Figure 3.2. A simple two-step catalytic reaction steady-state approximation...

Catalytic reactions (as well as the related class of chain reactions described below) are coupled reactions, and their kinetic description requires methods to solve the associated set of differential equations that describe the constituent steps. This stimulated Chapman in 1913 to formulate the steady state approximation which, as we will see, plays a central role in solving kinetic schemes. 

Historically, the steady state approximation has played an important role in unraveling mechanisms of apparently simple reactions such as H2 + CI2 = 2HC1, which involve radicals and chain mechanisms. We discuss here the formation of NO from N2 and O2, responsible for NO formation in the engines of cars. In Chapter 10 we will describe how NO is removed catalytically from automotive exhausts. 

Assuming that the catalytic reaction takes place in a flow reactor under stationary conditions, we may use the steady state approximation to eliminate the fraction of adsorbed intermediate from the rate expressions to yield ... 

The last equation is not independent of the others due to the site balance of Eq. (141) hence, in general, we have n-1 equations for a reaction containing n elementary steps. Note that steady state does not imply that surface concentrations are low. They just do not change with time. Hence, in the steady state approximation we can not describe time-dependent phenomena, but the approximation is sufficient to describe many important catalytic processes. 

In solving the kinetics of a catalytic reaction, what is the difference between the complete solution, the steady-state approximation, and the quasi-equilibrium approximation What is the MARI (most abundant reaction intermediate species) approximation ... 

A standard kinetic analysis of the mechanism 4a-4e using the steady state approximation yields a rate equation consistent with the experimental observations. Thus since equations 4a to 4e form a catalytic cycle their reaction rates must be equal for the catalytic system to be balanced. The rate of H2 production... 

Equilibrium studies under anaerobic conditions confirmed that [Cu(HA)]+ is the major species in the Cu(II)-ascorbic acid system. However, the existence of minor polymeric, presumably dimeric, species could also be proven. This lends support to the above kinetic model. Provided that the catalytically active complex is the dimer produced in reaction (26), the chain reaction is initiated by the formation and subsequent decomposition of [Cu2(HA)2(02)]2+ into [CuA(02H)] and A -. The chain carrier is the semi-quinone radical which is consumed and regenerated in the propagation steps, Eqs. (29) and (30). The chain is terminated in Eq. (31). Applying the steady-state approximation to the concentrations of the radicals, yields a rate law which is fully consistent with the experimental observations ... 

Two-Electron Catalytic Reactions In a number of circumstances, the intermediate C formed upon transformation of the transient species B is easily reduced (for a reductive process, and vice versa for an oxidative process) by the active form of the mediator, Q. This mechanism is the exact counterpart of the ECE mechanism (Section 2.2.2) changing electron transfers at the electrode into homogeneous electron transfers from Q, as depicted in Scheme 2.9. In most practical circumstances both intermediates B and C obey the steady-state approximation. It follows that the current is equal to what it would be for the corresponding EC mechanism with a... 

Insofar as the reactions in the catalytic cycle are fast, we may assume that the various forms of the enzyme obey the steady-state approximation. 

Two-Electron Catalytic Reactions The steady-state approximation may be applied to both transient intermediates B and C. It follows that not only does... 

As for the quasi (pseudo)-steady-state case, the basic assumption in deriving kinetic equations is the well-known Bodenshtein hypothesis according to which the rates of formation and consumption of intermediates are equal. In fact. Chapman was first who proposed this hypothesis (see in more detail in the book by Yablonskii et al., 1991). The approach based on this idea, the Quasi-Steady-State Approximation (QSSA), is a common method for eliminating intermediates from the kinetic models of complex catalytic reactions and corresponding transformation of these models. As well known, in the literature on chemical problems, another name of this approach, the Pseudo-Steady-State Approximation (PSSA) is used. However, the term "Quasi-Steady-State Approximation" is more popular. According to the Internet, the number of references on the QSSA is more than 70,000 in comparison with about 22,000, number of references on PSSA. 

Although all reactions showing a closed sequence could be considered to be catalytic, there is a difference between those in which the entity of the active site is preserved by a catalyst and those in which it survives for only a limited number of cycles. In the first category are the truly catalytic reactions, whereas the second comprises the chain reactions. Both types can be considered by means of the steady-state approximation, as in Christiansen s treatment. This important development dates to 1919 (reaction between hydrogen and bromine reported earlier by Bodenstein and Lind. 

In this section, the different behavior of processes with coupled noncatalytic homogeneous reactions (CE and EC mechanisms) is discussed in comparison with a catalytic process. We will consider that the chemical kinetics is fast enough and in the case of CE and EC mechanisms K (- c /cf) fulfills K 1 so that the kinetic steady-state and even diffusive-kinetic steady-state approximation can be applied. 

Appropriate expressions for the fluxes of each of the reactions in the system must be determined. Typically, biochemical reactions proceed through multiple-step catalytic mechanisms, as described in Chapter 4, and simulations are based on the quasi-steady state approximations for the fluxes through enzyme-catalyzed reactions. (See Section 3.1.3.2 and Chapter 4 for treatments on the kinetics of enzyme catalyzed reactions.)... 

If the number of catalytic species is three, then the reaction rate becomes more complicated, but still managable to derive using a steady-state approximation. However, this laborous excercise is not needed, as we can directly use the derivation presented for the three-step catalytic sequence with linear steps, i.e. equation (4.114). Examples of direct and indirect hydrogenation mechanisms when the reaction mechanism can be expressed by a cycle with three intermediates are presented in Figure 5.10 and Figure 5.11 respectively. 

In modeling reactions, in general, and catalytic reactions, in particular, the kineticist must draw on as many tools at his disposal as possible. Some of the most important concepts that are routinely used to derive, simplify and evaluate complicated rate expressions are 1) Transition-state theory 2) The steady-state approximation 3) Bond-order conservation calculations for surface species 4) A rate determining step 5) A most abundant reaction intermediate and 6) Criteria to evaluate parameters in derived rate expressions. Let us examine these topics prior to their utilization in deriving and evaluating reaction models and rate equations. 

ABSTRACT. A fundamental approach is outlined for the kinetic modeling of complex processes like thermal cracking or catalytic hydrocracking of mixtures of hydrocarbons. The reaction networks are written in terms of radical mechanisms in the first case and of carbenium ion mechanisms in the second case. Since the elementary steps of the networks pertain to a relatively small number of classes, the number of rate coefficients is kept within tractable limits. The reaction networks are generated by computer through Boolean relation matrices. The number of continuity equations is limited by the elimination of radicals or carbenium ions through the pseudo-steady-state approximation. 

Especially in catalytic processes, identification of the rate-limiting step is essential. It determines the form of the expression to be used as reaction-rate constant. From this expression, information can be deduced as to which reaction steps or surface equilibria the catalyst has to modify. We will return to this point after having analyzed the steady-state approximation further. 

After reduction and surface characterization, the iron sample was moved to the reactor and brought to the reaction conditions (7 atm, 3 1 H2 C0, 540 K). Once the reactor temperature, gas flow and pressure were stabilized ( 10 min.) the catalytic activity and selectivity were monitored by on-line gas chromatography. As previously reported, the iron powder exhibited an induction period in which the catalytic activity increased with time. The catalyst reached steady state activity after approximately 4 hours on line. This induction period is believed to be the result of a competition for surface carbon between bulk carbide formation and hydrocarbon synthesis.(6,9) Steady state synthesis is reached only after the surface region of the catalyst is fully carbided. 

To verify that steady state catalytic activity had been achieved, the catalyst was allowed to operate uninterrupted for approximately 8 hours. The catalyst was then removed from the reactor and the surface investigated by XPS. The results are shown in Figure 2c. The two major changes in the XPS spectrun were a shift in the iron 2p line to 706.9 eV and a new carbon Is line centered at 283.3 eV. This combination of iron and carbon lines indicates the formation of an iron carbide phase within the XPS sampling volume.(J) In fact after extended operation, XRD of the iron sample indicated that the bulk had been converted to FecC2 commonly referred to as the Hagg carbide.(2) It appears that the bulk and surface are fully carbided under differential reaction conditions. 

The steady state experiments showed that the two separate phases and the mixture are not very different in activity, give approximately the same product distributions, and have similar kinetic parameters. The reaction is about. 5 order in methanol, nearly zero order in oxygen, and has an apparent activation energy of 18-20 kcal/mol. These kinetic parameters are similar to those previously reported (9,10), but often ferric molybdate was regcirded to be the major catalytically active phase, with the excess molybdenum trioxide serving for mechanical properties and increased surface area (10,11,12). 

For the analysis of nonlinear cycles the new concept of kinetic polynomial was developed (Lazman and Yablonskii, 1991 Yablonskii et al., 1982). It was proven that the stationary state of the single-route reaction mechanism of catalytic reaction can be described by a single polynomial equation for the reaction rate. The roots of the kinetic polynomial are the values of the reaction rate in the steady state. For a system with limiting step the kinetic polynomial can be approximately solved and the reaction rate found in the form of a series in powers of the limiting-step constant (Lazman and Yablonskii, 1988). 

Table 15 shows the rate constants of each elementary step as determined by the above-mentioned methods153,15S. A comparison of the approximate magnitudes of each rate constant indicates that the electron-transfer step is the slowest one in the catalytic cycle. From the fractions of the catalyst species in the steady state of the reaction, it is possible to construct a profile of the Cu complex on a QPVP ligand, which is acting as a catalyst for the oxidative polymerization of XOH, as illustrated in Fig. 28. 

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