Use of the Steady-State Approximation

This simple analysis leads to an important and useful relationship known as the psuedo-steady-state approximation, or the Bodenstein steady-state approximation, or simply the steady-state approximation (SSA). As an approximation. 

The SSA is a generalization that is supported hy two important features of the behavior of the active center B in the preceding example  

the concentration of B increased to a maximum value very rapidly at the start of a reaction, before any significant quantity of reactant was consumed  

the net rate of formation of B (re) was very small relative to the rate of disappearance of A, and the rate of formation of C. 

Equation (5-1) simply is a mathematical expression of the second point. The first point suggests that Eqn. (5-1) is valid over the complete duration of areaction, not just over some small period of time. 

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The use of the steady-state approximation gives this velocity as a function of [5] by... 

Considering the rate equation v2 = k2[ES] and making use of the steady-state approximation for [ES], we obtain... 

In general, the effective overall rate constant associated with loss of reactants can be expressed in terms of the individual rate constants a-e in eq 3 by use of the steady state approximation. Simpler expressions can be obtained if the species related by diffusion (a,b) and activation (c,d) processes are assumed to be in thermal equilibrium. In such a case one finds straightforwardly that the effective first order rate constant, k (r), for electron transfer at separation r can be written as... 

Some Observations on the Use of the Steady-State Approximation in Complex Reactions... 

The general rate expressions for the hydrolysis and formation of an ester by this mechanism are complicated, and even the mechanism as written is a simplification, since ROH and ROH2+ will act as general base and general acid, respectively, just as H20 and H30+ do. In a solution containing only the reactants the initial rate of disappearance of ester, E, can be shown, by the use of the steady-state approximation for the tetrahedral intermediate, to be130... 

The use of the steady-state approximation is justified on the basis of two separate, independent observations. Firstly, after sufficient polymer has accumulated, the rate remains constant over an extended range of conversion (Figure 4). Secondly, Bengoughs measurements (10) of the non-steady-state kinetics of acrylonitrile polymerization show that a steady state is established within minutes, whereas the polymerization continues for hours. 

The same type of difficulty that is resolved by use of equation (35) for the partial-equilibrium approximation may also arise in connection with the steady-state approximation. For example, part of the sum of terms that contribute to the production rate of a primary species, to which the steady-state approximation is not applied, may be a constant multiple of cz . for an intermediary that is subject to the steady-state approximation, and the remaining terms in the production rate may be smaller than (U- even though (u. is small compared with. Under this condition, inaccurate results for the concentration history of the primary species will be obtained by use of the steady-state approximation for the intermediary unless a substitution... 

Since it has been seen that dc /dt decreases with t if k > (a - 1)/c/r(0) but may exhibit an explosionlike growth if k < (a — l)k P/ (0), it may be reasonable to identify the equality k = (a — l)kpCj (0) with the boundary of explosion. Alternative reasoning that leads to this same result makes use of the steady-state approximation for the intermediary C. Vanishing of the right-hand side of equation (45) yields... 

These simplifying assumptions must be adapted to some extent to explain the nature of some reactions on catalyst surfaces. The case of ammonia synthesis on supported ruthenium described in Example 5.3.1 presents a situation that is similar to rule 1, except the rate-determining step does not involve the mari. Nevertheless, the solution of the problem was possible. Example 5.3.2 involves a similar scenario. If a mari cannot be assumed, then a rate expression can be derived through repeated use of the steady-state approximation to eliminate the concentrations of reactive intermediates. 

The use of the steady-state approximation instead of assuming substrate adsorption-desorption equilibrium a key feature of Halpem s studies (9) of pressure effects on enantioselectivity, and its implications for enantioselectivity in the current system will be discussed later in the text. 

By making use of the steady state approximation with regard to the concentrations of the intermediates NO3 and NO, derive the rate law for this process. 

The flux coefficients are not uniquely defined. If the boundary is put between A and B, then r = / i and / = / i(B/C). The flux coefficients must satisfy two conditions firstly, rA — r C must be equal to — ifl which is the overall rate of the reaction, —dkldt secondly, at equilibrium r/r must be equal to the equilibrium constant. For this simple example, it can be readily shown that the two conditions are satisfied. Use of the steady-state approximation for B gives... 

An analysis of this kinetic scheme was proposed by Michaelis and Menten and later modified by Briggs and Haldane [2, p. 367-370]. The latter full treatment makes use of the steady-state approximation, which states that very soon after the start of the reaction the rate of formation of ES will be equal to its rate of loss, so that ... 

We will illustrate the use of the steady-state approximation for solving the rate expressions in the reaction between hydrogen and bromine, which proceeds according to a scheme that is slightly more complicated than that for the H2 + CI2 reaction... 

The parameter is called the Michaelis-Menten constant or the Michaelis constant. It is equal to a constant only if the temperature is constant. A version of the Michaelis-Menten equation can be derived with inclusion of a reverse reaction for step 2. (See Problem 13.11.) The Michaelis-Menten mechanism has also been studied without use of the steady-state approximation. ... 

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