The steady-state approximation SSA

Obviously, the termination reaction in this case is just the reverse of the initiation reaction. This is not always the case. Also, the initiation and termination reactions may not be elementary as written. As discussed in connection with Table 5-1, an inert molecule (M) may be required to serve as an energy source or sink. It may be more accurate to write the initiation and termination reactions as 

However, this issue does not affect the above discussion. 

Consider two, first-order reactions occurring in series, i.e., 

The parameters k and k2 are the first-order rate constants for the reactions A —B and B — C, respectively. 

Suppose these reactions occur in an ideal, batch reactor. The initial concentration of A at time, t = 0 will be designated Cao, and we will assume that there is no B or C in the reactor at t = 0. 

The essence of the steady-state assumption is that the concentration of a reactive intermediate (X) is assumed to build up during a brief initial induction period, but then, during most of the rest of the reaction, its formation and decomposition are balanced (i.e. d [ X ] /d f = 0) so that its concentration remains essentially constant. Obviously, during the final stages of the reaction, the concentration of the intermediate decreases to zero. 

the conditions for the applicability of the SSA to the mechanism ofEquation 4.7 lead to the prediction of a simple first-order rate law for the disappearance of the reactant A. If the assumptions are sound, the experimental rate law shown in Equation 4.10 will be observed  

Since [B]ss is extremely small, mass conservation allows us to write [A]t = [A]0 - [C]t, so we may now substitute for [A], whereupon we obtain as our prediction, 

if the assumptions are sound, a first-order increase in [C] will be observed experimentally  

Here also, then, we see that application of the steady-state approximation to the concentration of B when (k2 and/or k-1) y k leads to a simple first-order concentration-time relationship for the formation of the product C, i.e. very much simpler than that in Equations 4.8. 

3 being formed on a pair of neighboring active sites, Sp, and the reaction rate from TST per unit surface area is  

it has been mentioned before that the site density L (or Lp in this case) is typically near 10 cm , and a reasonable value for Qt from Table 6.1 is 

It is clear that a catalyst should lower the apparent activation energy if it is to enhance the reaction rate, even if the catalyst has a high specific surface area. If a catalyst has a relatively high surface area of 100m g and its density is near unity, then there is about 10 cm per cm, and if Ehom = Eiiet- 

It has been stated by Boudart that the steady-state approximation (SSA) can be considered as the most important general technique of applied chemical kinetics [9]. A formal proof of this hypothesis that is applicable to all reaction mechanisms is not available because the rate equations for complex systems are often impossible to solve analytically. However, the derivation for a simple reaction system of two first-order reactions in series demonstrates the principle very nicely and leads to the important general conclusion that, to a good approximation, the rate of change in the concentration of a reactive intermediate, X, is zero whenever such an intermediate is slowly formed and rapidly disappears. 

Integration of these differential equations with the stated boundary conditions gives  

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Applying the steady-state approximation (SSA) to the transient enolate intermediate in Scheme 4.7, its concentration is given by... 

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 steady-state approximation (SSA) can be used to derive die form of a rate equation for a reaction whose mechanism is constmeted of elementary reactions. 

In dealing with complicated reaction mechanisms, a simplification can often be introduced that when the reaction has reached some kind of steady state (akin to an equilibrium, except that further reactions are possible beyond this equilibrium hence the term steady-state approximation (SSA) is used. Mathematically, after the reaction has started, some intermediate product B has the condition d [B]/dt = 0. This is best illustrated by an example. 

Reaction mechanisms may be developed to explain rate laws, but often they are developed in parallel, one helping the other. In translating a mechanism into a rate law a useful tool is Bodenstein s steady state approximation (SSA) or stationary state hypothesis. This approximation assumes that after a very short interval of time any reactive intermediate, which because of its reactivity will only be present in negligible proportions, will have its rate of decay equal to its rate of production, i.e., it will reach a steady concentration on a vanishingly small time scale. If this did not happen the amount of the intermediate would build up to measureable proportions and it would become an intermediate product. It is assumed at any instant that dcj /dt = 0, where R is the reactive intermediate. 

The equality of the three net rates is a direct consequence of the steady-state approximation. If the SSA is valid, these rates are necessarily equal. 

This is a closed sequence representing a maximum of two reactants and two products, with Si representing an empty active site and S2 representing a filled active site. Application of the SSA (steady-state approximation) to Si and S2 yields for the net rate on these sites ... 

The second approach to speeding up the SSA involves separating the system into slow and fast subsets of reactions. In these methods, analytical or numerical approximations to the dynamics of the fast subset are computed while the slow subset is stochastically simulated. In one of the first such methods, Rao and Arkin (see Further reading) applied a quasi-steady-state assumption to the fast reactions and treated the remaining slow reactions as stochastic events. 

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