Pseudo-steady-state example

Many chemical reactions involve very reactive intermediate species such as free radicals, which as a result 

It is perhaps easiest to explain the pseudo-steady-state approximation by way of an example. Consider the simple reaction A — B + C, whose elementary steps consist of the activation of A by collision with a background molecule M (in the atmosphere M is typically N2 and 02) to produce an energetic A molecule denoted by A, followed by the decomposition of A to give B and C. Thus, we write the mechanism as 

Note that each reaction has a rate constant and that the second reaction is the reverse of the first (i.e., A may be deactivated by collision with M). Assuming the Earth s atmosphere as one big compartment, we have one way of losing A and one way of forming A hence, its rate of change in the atmosphere is 

In this notation, the square brackets mean the concentration of the atom or compound in question for example, [02] is 1017 molecules/cm3 at 30 km altitude. 

In this equation, the derivative is positive and means the rate at which A is formed. The first term on the right is negative because A is being lost by the first reaction. The second term is positive because A is being formed by the second reaction. Both reactions are of second order,7so both terms have two concentrations and a rate constant. 

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The first example we wiU use as an application of the equilibrium step approximation and pseudo-steady-state approximation is the reaction... 

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. 

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. 

In all four cases, tlie initial reaction rates at the start of illumination in the continuous-feed photoreactor were higher than the pseudo-steady-state reaction rates the reaction rates declined over time until pseudo-steady-state operation was achieved. Tliis apparent deactivation phenomenon, often observed with aromatic contaminants, is discussed in Sec. III.E. In a transient reaction system, the time required to reach pseudo-steady-state operation also appears to increase in the same order as the reaction rates. For example, for the continuous photocatalytic oxidation of aromatic contaminants at 50 mg/m in a powder-layer photoreactor, the time required for pseudo-steady-state operation to be achieved was reported to be approximately 90 min for benzene, 120 min for toluene, and as long as 6 hr for wz-xylene [50,51]. Under such conditions, the difference in reaction rates between the aromatic contaminants is magnified by the fact that the more reactive aromatics retain their higher transient reaction rates for longer periods (Fig. 7). 

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]. 

Fairly recently it has been established that a set of pseudo-steady-state equations for complex catalytic reactions can have several solutions only when their detailed mechanisms involve as one step an interaction between various intermediates [22], The simplest catalytic mechanism possessing this property is an adsorption mechanism. For example... 

Example 13.1 Flow in an Idealized Runner System We consider a straight tubular runner of length L. A melt following the Power Law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. We wish to calculate the melt front position and the instantaneous flow rate as a function of time. We assume an incompressible fluid in isothermal and fully developed flow, and make use of the pseudo-steady state approximation. 

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. 

Equation (11.2) provides the basis for studying transient nucleation. For example, if the monomer concentration is abruptly increased at t = 0, what is the time-dependent development of the cluster distribution Physically, in such a case there is a transient period over which the cluster concentrations adjust to the perturbation in monomer concentration, followed eventually by the establishment of a pseudo-steady-state cluster distribution. Since the characteristic time needed to establish the steady-state cluster distribution is generally short compared to the timescale over which typical monomer concentrations might be changing in the atmosphere, we can assume that the distribution of clusters is always at a steady state corresponding to the instantaneous monomer concentration. There are instances, generally in liquid-to-solid phase transitions, where transient nucleation can be quite important (Shi et al. 1990), although we do not pursue this aspect here. 

One approach commonly used in the integration of such chemical kinetics problems is the pseudo-steady-state approximation (PSSA) (see Chapter 3). For example, instead of solving a differential equation for short-lived species like O, OH, and N03, one calculates and solves the corresponding PSSA algebraic equations. For example, McRae et al. (1982a) estimated that nine species (O, RO, OH, R02, N03, RCO, H02, HN04, and N2Os) with characteristic lifetimes less than 0.1 min in the environment of interest could be... 

However, if we were to evaluate the right-hand side numerically we would find that it is very close to zero. Physically, this means that the oxygen atom is so reactive that it disappears by reaction 5.2 virtually as fast as it is formed by reaction 5.1. In dealing with highly reactive species such as the oxygen atom, it is customary, as noted in Chapter 3, to invoke the pseudo-steady-state approximation (PSSA) and thereby assume that the rate of formation is exactly equal to the rate of disappearance, for example. 

Bl. In Example 15-1 operation is at a pseudo-steady state. Brainstorm alternative designs for this diffusion measurement. 

If the sphere in Fig. 6.2-3a is evaporating, the radius r of the sphere decreases slowly with time. The equation for the time for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation (6.2-32), where r is now a variable, to the moles of solid A evaporated per dt time and per unit area as calculated from a material balance. (See Problem 6.2-9 for this case.) The material-balance method is similar to Example 6.2-3. The final equation is... 

It is not possible to cover all of the history or the theory of the chemical kinetics in the context of this chapter. However, the authors intention is to give the student an essential minimum in the theory of chemical kinetics to be able to follow the literature and to incorporate in the design of the chemical reaction units. This chapter is divided into two sections in the first part, the homogeneous kinetics will be covered in detail, covering the collision theory and the transition state theory for the determination of the rate constants and reaction rate expressions. Old but still valid approximations of pseudo-steady-state and pseudoequilibrium concepts will be given with examples. In the second part, the heterogeneous reaction kinetics will be discussed from a mechauis-tic point of view. 

For (1.4.1-l)-(1.4.1-3) to be a useful approximation of the complete scheme, the induction time should be very short, meaning that the concentration of the intermediate must be very small. From (1.3.2-6) it is seen how the maximum in the curve Cq versus t moves towards t = 0 as k2 k. Quite frequently the existence of an intermediate is chemically logical, but it is difficult or impossible to measure its concentration. The pseudo steady state approximation is then a very useful tool. Examples will be encountered in Section 1.5 on bio-kinetics and Section 1.6 on complex reactions. 

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