Reactions steady-state approximation

The most common elementary processes are unimo-lecular (one molecule dissociates) and bimolecular (two molecules collide). Some of the species in elementary processes may be reaction intermediates, species produced in one elementary process and consumed in another. One of the elementary processes may be the rate-determining step. When a single rate-determining step cannot be identified, the mechanism can often be established by the steady-state approximation. Reaction mechanisms can also be depicted through reaction profiles (Fig. 20-14). 

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). 

For a sequenee of reaetion steps two more eoneepts will be used in kinetics, besides the previous rules for single reaetions. One is the steady-state approximation and the seeond is the rate limiting step eoneept. These two are in strict sense incompatible, yet assumption of both causes little error. Both were explained on Figure 6.1.1 Boudart (1968) credits Kenzi Tamaru with the graphical representation of reaction sequences. Here this will be used quantitatively on a logarithmic scale. 

A useful approach that is often used in analysis and simplification of kinetic expressions is the steady-state approximation. It can be illustrated with a hypothetical reaction scheme ... 

The overall rate of a chain process is determined by the rates of initiation, propagation, and termination reactions. Analysis of the kinetics of chain reactions normally depends on application of the steady-state approximation (see Section 4.2) to the radical intermediates. Such intermediates are highly reactive, and their concentrations are low and nearly constant throughout the course of the reaction ... 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

The concentration of the reaction intermediate AB may be determined by using the steady state approximation for intermediates,... 

This also accounts for the production of the small amount of butane. If the reaction mechanism were steps 1, 2, 3, 4, 5a, and 5b, then applying the steady state approximations would give the overall order of reaction as 1/2. 

Consider further Scheme XIV and rate equations (3-139) to (3-141). Evidently Cb will be small relative to (Ca + Cc) if ( -i + 2) i- Then B plays the role of a reactive intermediate in the overall reaction A— C. This is the usual condition that is taken as a warrant for the application of the steady-state approximation. If Cb is small, it is reasonable that Cb will be small throughout most the reaction, so it is set equal to zero. As Wong (53) has pointed out, however, the condition Cb = 0 is a sufficient but unnecessary condition for Eq. (3-142) to hold. Erom Eq. (3-140) we obtain... 

The sufficient and necessary condition is therefore Cb iCa. As a consequence of imposing the more restrictive condition, which is obviously not correct throughout most of the reaction, it is possible for mathematical inconsistencies to arise in kinetic treatments based on the steady-state approximation. (The condition Cb = 0 is exact only at the moment when Cb passes through an extremum and at equilibrium.)... 

One way to examine the validity of the steady-state approximation is to compare concentration—time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , 2 = 2 s . The period during which the concentration of the intermediate builds up from its initial value of zero to the quasi-steady-state when dcfjdt is vei small is called the pre-steady-state or transient stage in Fig. 3-10 this lasts for about 2 s. For the remainder of the reaction (over 500 s) the steady-state and exact solutions are in excellent agreement. Because the concen-... 

Of course it is also possible for a reaction system not to belong to any of these classes of approximate description.) Only in class III can equilibrium be said to be a special case of the steady-state treatment. Note that, for class III systems, the steady-state concentration of intermediate is very large,whereas for class I it is very small. Zuman and Patel have discussed the equilibrium and steady-state approximations in terms similar to the present treatment. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

If a reaction system consists of more than one elementary reversible reaction, there will be more than one relaxation time in general, the number of relaxation times is equal to the number of states of the system minus one. (However, even for multistep reactions, only a single relaxation time will be observed if all intermediates are present at vanishingly low concentrations, that is, if the steady-state approximation is valid.) The relaxation times are coupled, in that each relaxation time includes contributions from all of the system rate constants. A system of more than... 

Either step could be rate determining. Study of many reactions has shown that most occur with a significant isotopic effect, but for some reactions the isotope effect is absent.If we apply the steady state approximation to the intermediate, this reaction scheme leads to... 

Because the cationic intermediate is unstable, it will be permissible to apply the steady-state approximation, leading to Eq. (8-65) for the reaction rate. 

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

Steady-state. An erroneous rate law is shown below for the reaction scheme believed to represent the reaction between Fe3+ and I-, in that an extraneous denominator term appears. In the scheme shown, I2 and Fel2+ obey the steady-state approximation. Show what the incorrect part of the expression is. Suggest a simple derivation of the correct equation that avoids extensive algebraic manipulations. 

Reactant fluxes. Calculate values of , for the combination of rate constants in Tables 4-1 and 4-2 for those systems for which the steady-state approximation holds. Construct a diagram of the fluxes at the start of the reaction when [A]o = 1. 

We first explore a system that could perhaps be handled by the steady-state approximation. The reactions are ... 

Steady-state approximation product ratios. The cycloaddition reaction between ben-zyne and cis-1,2-dichloroethene proceeds to a mixture of cis and trans products according to the following scheme, in which the two diradicals are steady-state intermediates 37... 

The new pathway, too, is a chain reaction Note that the first term of Eq. (8-31) does not give a meaningful transition state composition. Since the scheme in Eqs. (8-20M8-23) seems valid for the Cu2+-free reaction, we can seek to modify it to accommodate the new result. This approach is surely more logical than inventing an entirely new sequence. To arrive at the needed modification, we simply replace Eq. (8-23) by a new termination step, Eq. (8-30). With that, and the steady-state approximation, the rate law is... 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

The composite of sequences a, f3, and y gives the overall stoichiometry of reaction (8-56). The steady-state approximations, abbreviated as before, are... 

STRATEGY Construct the rate laws for the elementary reactions and combine them into the overall rate law for the decomposition of the reactant. If necessary, use the steady-state approximation for any intermediates and simplify it by using arguments based on rapid pre-equilibria and the existence of a rate-determining step. 

The rate law of an elementary reaction is written from the equation for the reaction. A rate law is often derived from a proposed mechanism by imposing the steady-state approximation or assuming that there is a pre-equilibrium. To be plausible, a mechanism must be consistent with the experimental rate law. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

The reaction rate of CO2 under the condition of steady-state approximation to formation of Cl is presented as follows ... 

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. 

Coupled Reactions in Flow Reactors The Steady-state Approximation 41... 

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. 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

Are situations conceivable in which the steady-state approximation can be applied to the kinetics of a batch reaction ... 

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 ... 

Note that it is assumed that reaction (1) is reversible, while reactions (2) and (3) are irreversible. Show, using steady-state approximations for the intermediates, that the overall reaction rate can be written as... 

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