Pseudo-steady-state, reaction kinetics

Concepts of the psendo-steady and quasi-steady states are apparently veiy close to one another. However the difference is fundamental. Whereas the pseudo-steady states define kinetic modes of reaction, the quasi-steady states are simple mathematical approximations. Moreover the difference is palpable in the sense that the first concept can be directly checked by the experiment independently of the kinetic law, the second is checked only by the conformity between the kinetic law speed, concentrations calculated and the experimental corresponding law. But maity other factors must intervene in addition so that these two laws coincide, in particular a correct choice of the mechanism of the reaction. 

Mechanism. The thermal cracking of hydrocarbons proceeds via a free-radical mechanism (20). Siace that discovery, many reaction schemes have been proposed for various hydrocarbon feeds (21—24). Siace radicals are neutral species with a short life, their concentrations under reaction conditions are extremely small. Therefore, the iategration of continuity equations involving radical and molecular species requires special iategration algorithms (25). An approximate method known as pseudo steady-state approximation has been used ia chemical kinetics for many years (26,27). The errors associated with various approximations ia predicting the product distribution have been given (28). 

As illustrated in Fig. 1, the activated carbon displays the highest conversion and selectivity among all the catalysts during the initial reaction period, however, its catalytic activity continues to decrease during the reaction, which is probably caused by coke deposition in the micropores. By contrast, the reaction over the CNF composites treated in air and HN03 can reach a pseudo-steady state after about 200 min. Similiar transient state is also observed on the CNFs and the untreated composite. Table 3 collects the kinetic results after 300 min on stream over catalysts tested for the ODE, in which the activity is referred to the BET surface area. The air-treated composite gives the highest conversion and styrene selectivity at steady state. 

Students may have seen the acetaldehyde decomposition reaction system described as an example of the application of the pseudo steady state (PSS), which is usually covered in courses in chemical kinetics. We dealt with this assumption in Chapter 4 (along with the equilibrium step assumption) in the section on approximate methods for handling multiple reaction systems. In this approximation one tries to approximate a set of reactions by a simpler single reaction by invoking the pseudo steady state on suitable intermediate species. 

When oxygen is removed from a reaction solution of tetrakis-(dimethylamino)ethylene (TMAE), the chemiluminescence decays slowly enough to permit rate studies. The decay rate constant is pseudo-first-order and depends upon TMAE and 1-octanol concentrations. The kinetics of decay fit the mechanism proposed earlier for the steady-state reaction. The elementary rate constant for the dimerization of TMAE with TMAE2+ is obtained. This dimerization catalyzes the decomposition of the autoxidation intermediate. 

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

A single-route complex catalytic reaction, steady state or quasi (pseudo) steady state, is a favorite topic in kinetics of complex chemical reactions. The practical problem is to find and analyze a steady-state or quasi (pseudo)-steady-state kinetic dependence based on the detailed mechanism or/and experimental data. In both mentioned cases, the problem is to determine the concentrations of intermediates and overall reaction rate (i.e. rate of change of reactants and products) as dependences on concentrations of reactants and products as well as temperature. At the same time, the problem posed and analyzed in this chapter is directly related to one of main problems of theoretical chemical kinetics, i.e. search for general law of complex chemical reactions at least for some classes of detailed mechanisms. 

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. 

When a simple, fast and robust model with global kinetics is the aim, the reaction kinetics able to predict correctly the rate of CO, H2 and hydrocarbons oxidation under most conditions met in the DOC consist of semi-empirical, pseudo-steady state kinetic expressions based on Langmuir-Hinshelwood surface reaction mechanism (cf., e.g., Froment and Bischoff, 1990). Such rate laws were proposed for CO and C3H6 oxidation in Pt/y-Al203 catalytic mufflers in the presence of NO already by Voltz et al. (1973) and since then this type of kinetics has been successfully employed in many models of oxidation and three-way catalytic monolith converters... 

A simple example is the so-called Michaelis-Menten kinetics for enzymatic reactions A + E +C->B + E, which, when the pseudo-steady-state hypothesis is invoked, gives for the concentration of A, for instance, a,... 

Reprint F is an example of analyzing a reaction in formal kinetics. Gray and Scott introduced the autocatalytic A + 2B = 3B as a simple model reaction that proved to have a rich behavior, much richer than the Brusselator for example. However, A + 2B smacks of a three-body interaction, which is a sufficiently rare occurrence as to be avoided. I had done a pseudo-steady-state analysis before I visited Leeds at Gray s invitation, and the chance of working with the fons et origo of this reaction, so to speak, was an opportunity to make sure that the limiting behavior was not lost when certain parameters were small, but not actually zero. For another analysis of autocatalytic behavior, see [107]. 

However if c3 and c4 are constant then c2 = (k2 + k3)c4/k1c3 must be constant, and no reaction takes place. There is therefore a basic inconsistency in the attempt to make the mechanism SR account strictly for the reaction Si. In spite of this, such kinetic equations as (28) have been found to be extremely useful and quite accurate in kinetic studies. The chemical kineticist therefore claims that over an important part of the course of reaction c3 and c4 are approximately constant, or often that they are both small and slowly varying. This is called a pseudo-steady-state hypothesis and however pseudo it must appear to the mathematician it is sufficiently important to merit formalization. We shall therefore propound a formal definition and illustrate further how it may be used. 

Briggs-Haldane approach (Briggs and Haldane, 1925) The change of the intermediate concentration with respect to time is assumed to be negligible, that is, d(CES)/dt = 0. This is also known as the pseudo-steady-state (or quasi-steady-state I assumption in chemical kinetics and is often used in developing rate expressions in homogeneous catalytic reactions. 

Prior to gelation, two types of radicals with solid- and liquid-like mobility are present they are possibly located in microgels (solid-like mobility) and in monomers (liquid-like mobility). The concentration of free radicals increases continuously, so that the pseudo steady-state assumption cannot be applied to model the reaction kinetics. 

For several cases, e.g. for linear pseudo-steady-state equations (linear mechanisms), the steady state is certain to be unique. But for non-linear mechanisms and kinetic models (which are quite common in catalysis, e.g. in the case of dissociative adsorption), there may be several solutions. Multiplicity of steady-states is associated with types of reaction mechanisms. 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

Briggs and Haldane [8] proposed a general mathematical description of enzymatic kinetic reaction. Their model is based on the assumption that after a short initial startup period, the concentration of the enzyme-substrate complex is in a pseudo-steady state (PSS). For a constant volume batch reactor operated at constant temperature T, and pH, the rate expressions and material balances on S, E, ES, and P are... 

From the reaction mechanism, we can derive a reaction rate equation (kinetics) for the overall reaction. Below, we will discuss two such methods rate limiting step and quasi-stationary state (pseudo-steady-state). 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

The kinetic parameters for a free enzyme in solution are readily derived using the Michaelis-Menten approach describing pseudo-steady-state conversions. Consider Equation (31.1) representing the conversion of a substrate S into a product P, catalyzed by an enzyme E. The rate of formation of an enzyme/substrate complex, ES, is denoted as ku the reverse reaction by and the rate of subsequent conversion to the free product by k2. 

The same considerations made before are valid for this case and it is very important to have an available validated reaction mechanism. It can be obtained from three main sources (Blelski et al., 1985 Buxton et al., 1988 Stefan and Bolton, 1998) and it is shown in Table 5. With the available information about the constant k2, k, k, fcg, and k-j, it could be possible to solve a system of four differential equations and extract from the experimental data, the missing constants 4> and k (that in real terms is k /Co2)-This method would provide good information about the kinetic constants, but it is not the best result for studying temperature effects if the same information is not available for the pre-exponential factors and the activation energies. Then, it is better to look for an analytical expression even if it is necessary to make some approximations. This is particularly true in this case, where the direct application of the micro steady-state approximation (MSSA) is more difficult due to the existence of a recombination step that includes the two free radicals formed in the reaction. From the available information, it is possible to know that to calculate the pseudo-steady-state... 

Lumping and Mechanism Reduction It is often useful to reduce complex reaction networks to a smaller reaction set which still maintains the key features of the detailed reaction network but with a much smaller number of representative species, reactions, and kinetic parameters. Simple examples were already given above for reducing simple networks into global reactions through assumptions such as pseudo-steady state, rate-limiting step, and equilibrium reactions. 

In photochemical experiments, this very simple approach may be compromised if desorption of the reactants is fast, in that reactant adsorption-desorption equilibrium is not established during the reaction [then equation (13.5) does not hold]. In addition, active center reactivity is continuous because of continuous illumination thus, no equilibrium is established. This may lead to the derivation of a pseudo-steady-state kinetic model [200,201] with a rate expression slightly different from equation (13.4), the discussion of which is, however, out the scope of this work. 

A non-linear theory of steady-state kinetics of complex catalytic reactions is developed. A system of steady-state (or pseudo-steady-state) equations can always be reduced to a so called kinetic polynomial. This polynomial is a function of the steady-state reaction rate and the process parameters (concentrations of the reactants, temperature). 

The catalytic reaction mechanisms and corresponding kinetic models can be classified into linear and non-linear models. These terms were introduced by Temkin [2]. For linear mechanisms every reaction involves the participation of only one molecule of the intermediate substance. Therefore the rate of each step depends linearly on an intermediate concentration. Using the principle of pseudo-steady-state concentrations (see equation (7)) we can easily find the solution of linear algebraic equations that corresponds to the linear mechanism and then obtain the values of the pseudo-steady-state (or steady-state)... 

The analysis of non-linear mechanisms and corresponding kinetic models are much more difficult than that of linear ones. The obvious difficulty in this case is the follows an explicit solution for steady-state reaction rate R can be obtained only for special non-linear algebraic systems of steady-state (or pseudo-steady-state) equations. In general case it is impossible to solve explicitly a system of non-linear steady-state (or pseudo-steady-state) equations. However, in the case of mass-action-law-model it is always possible to apply to this system a method of elimination of variables and reduce it to a polynomial in one variable [4], i.e., a polynomial in terms of the steady-state reaction rate. We refer a polynomial in the steady-state reaction as a kinetic polynomial. The idea of this polynomial was firstly emphasized in [5]. 

When the activity reaches a pseudo-steady state, deactivation models with residual activity (DMRA) [23] should be used. Eti loying Langmuir-Hinshelwood (L-H) kinetics to analy2 the deactivation process, both the main reaction and the deactivation mechanisms are required to obtain the deactivation kinetic parameters. According to the literature [6,7], the following reaction mechanism can be considered ... 

Enzyme kinetics. The kinetics of transformation processes catalysed by a single enzyme are often described using the Micha-elis-Menten equation (1). The derivation of this equation is, however, based on two assumptions. The pseudo steady state hypothesis ( ) with respect to the intermediary enzyme-siibstrate complex is valid and the reverse reaction from product to substrate can be... 

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