Stationary state hypothesis,

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

The derivation of a rate law from a postulated mechanism is a useful application of reaction mechanisms. It shows how the kinetics of the elementary reaction steps are reflected in the kinetics of the overall reaction. The following example illustrates this for a simple, gas-phase reaction involving an open sequence. The derivations typically employ the stationary-state hypothesis (SSH) to eliminate unknown concentrations of reactive intermediates. 

We eliminate cNC>3 (not allowed in the final rate law) by applying the stationary-state hypothesis to NO, rNOj = 0 (and subsequently to NO) ... 

Apply the stationary-state hypothesis to the free radicals CH and C2H50 to derive the rate law for this mechanism. 

For a constant-volume batch reactor operated at constant T and pH, an exact solution can be obtained numerically (but not analytically) from the two-step mechanism in Section 10.2.1 for the concentrations of the four species S, E, ES, and P as functions of time t, without the assumptions of fast and slow steps. An approximate analytical solution, in the form of a rate law, can be obtained, applicable to this and other reactor types, by use of the stationary-state hypothesis (SSH). We consider these in turn. 

We will regard a and 6 as functions of the reactant concentration as expressed in fi and assume that they change on a fast timescale compared with reactant consumption (small e), i.e. we apply the pseudo-stationary-state hypothesis. The pseudo-stationary-state condition da/dt = dO/dz = 0 yields the following simultaneous equations ... 

The necessary conditions for the application of the stationary-state hypothesis, that Mi and M2 be small compared to A or C, can be inspected in a quantitative fashion with Eq. (III.9.8). On rewriting these equations in terms of concentration ratios, we have... 

The philosophy of a rate-determining step is that one of these times is much larger than any of the others following it. Note that, for the stationary-state hypothesis to apply, all times involving formation of intermediates must be successively decreasing. 

If we apply the stationary-state hypothesis to Eq. (XII.15.1), we have for the rate of formation of products... 

If we ignore the pre-stationary-state period, which can be shown to be negligible at all temperatures below lOOO K, then we can make use of the stationary-state hypothesis and set d(Br)/d< = d(H)/d = 0 and solve Eq. (XIII.2.4) for the stationary-state concentrations (H) and (Br). ... 

Much has been made of the kinetics of the hydrocarbon pyrolyses despite the fact that they fit no simple kinetic expression and seem very complex. The schemes given in this section for the decompositions of w- and isobutanes [Eqs. (XIII.9.1) and (XIII.9.2)] cannot be solved explicitly for the over-all reaction rates even by using the stationary-state hypothesis. 

If we apply to the above system of eciuations the stationary-state hypothesis, then we can write an equation for the rates of initiation and termination of radicals in the system... 

The appropriateness of the stationary-state assumption is open to question in such a system when the amount of enzyme-substrate complex is large compared to free enzyme and (Eo) is not very much smaller than (S) (Sec. IIL9). When, however, (S) (Eo) or (E-S) (Eo) then the stationary-state hypothesis is valid for amounts of reacti (So) - (S) > (E-S). 

Spin system, 157, 159 Splitting, Zeeman, 154 Stability study, accelerated, 260 Standard potential, 254, 359 Standard reaction, 315 Standard state, 253 selection of, 208 Stationary-state hypothesis, 101 Statistical analysis, 40 Statistical analysis of solvent effects,... 

Consider a monodispersed latex, where water-phase termination is negligible and termination is instantaneous when a radical enters a polymer particle containing one radical. By definition, IV2 = IV3 = — = 0 and the total radical entry rate per liter of latex equals p. Application of the stationary-state hypothesis gives... 

This statement is termed the stationary-state hypothesis—a concept widely used for reactions involving species of a transitory nature, such as free radicals and atoms. See Sec. 2-11 for a discussion of its use in photochemical kinetics. It is equally applicable here to the high-pressure case and also leads to Eq. (2-46), but by a more complicated route. 

Suppose that the concentrations of intermediates rapidly attain constant low values that is, the stationary-state hypothesis is valid Then a steady state will quickly be reached, after which the production and destruction rates of Pr and Cl will balance each other, or... 

Note that the requirement for the stationary-state hypothesis to be valid is that the concentrations of intermediates be small with respect to those of the reactants and products. In writing these expressions only the one termination step involving k-j is needed, since it is assumed to control the termination process. 

The concentration of [Pr] is small but unknown. However, the stationary-state hypothesis has allowed us to develop Eq. (C) for [Pr] in terms of known quantities. Using Eq. (C) in Eq. (D) gives... 

S. W. Benson, The Foundations of Chemical Kinetics, McGraw-Hill Book Company, New York, 1960. The stationary-state hypothesis mentioned in Secs. 2-6 and 2-11 is defined and illustrated on pp. 50-53. In chap. XII the collision theory is considered in detail, complications related to the energy distribution of molecules and the steric factor are discussed, and the results are compared in depth with those from the transition-state theory. 

Michel Boudart, Kinetics of Chemical Processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. A concise presentation of the fundamental concepts of kinetics for homogeneous and heterogeneous reactions, including a chapter on the application and validity of the stationary-state hypothesis. 

Use the stationary-state hypothesis to derive an expression for the overall rate of decomposition. Do order and stoichiometry agree in this case ... 

In solving the problem for this case, use the stationary-state hypothesis for the intermediates (free radicals Cl, RH, RHCl ) to obtain rate equations for Trhj and ErhiCi analogous to the rate equations for the first-order and second-order cases. Note that the rate constant for the termination steps is usually much less than that for the propagation steps. 

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