Bodenstein steady state approximation

Using the Bodenstein steady state approximation for the intermediate enzyme substrate eomplexes derives reaetion rate expressions for enzymatie reaetions. A possible meehanism of a elosed sequenee reaetion is ... 

Third, it is often useful to assume that the concentration of one or more of the intermediate species is not changing very rapidly with time (i.e., that one has a quasistationary state situation). This approximation is also known as the Bodenstein steady-state approximation for intermediates. It implies that the rates of production and consumption of intermediate species are nearly equal. This approximation is particularly good when the intermediates are highly reactive. 

ILLUSTRATION 4.3 USE OF THE BODENSTEIN STEADY-STATE APPROXIMATION TO DERIVE A RATE EXPRESSION FROM A CHAIN REACTION MECHANISM... 

This expression contains the concentrations of two intermediate free radicals, C102 and C103. These terms may be eliminated by using the Bodenstein steady-state approximation. 

The Bodenstein steady state approximation is widely applied in catalysis. At the same time this approximation is often not valid and the dynamics should be taken into accout. Transient kinetic modelling as well as oscillation reactions will be considered in Chapter 8. 

ILLUSTRATION 4.3 Use of the Bodenstein Steady-State Approximation to Derive a Rate Expression for a Chain Reaction Mechanism... 

In terms of the rate expressions above, the Bodenstein steady-state approximation for free-radical intermediates... 

Reaction rate expressions for enzymatic reactions are usually derived by making the Bodenstein steady-state approximation for the intermediate enzyme-substrate complexes. This assumption is appropriate for use when the substrate concentration greatly exceeds that of the enzyme (the usual laboratory situation) or when there is both a continuous supply of reactant and a continuous removal of products (the usual cellular situation). 

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. 

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. 

It can be solved by the so-called Bodenstein or steady-state approximation. This approximation supposes that the concentration of the reactive intermediate, in this case MS, is always small and constant. For a catalyst, of which the concentration is always small compared to the substrate concentration, it means that the concentration of MS is small compared to the total M concentration. The rate of production of products for the scheme in Figure 3.1 is given by equation (3). Equation (4) expresses the steady state approximation the amounts of MS being formed and reacting are the same. Equation (5) gives [M] and [MS] in measurable quantities, namely the total amount of M (Mt) that we have added. If we don t add this term, the nominator of equation (6) will not contain the term of k and the approximations that follow cannot be carried out. 

Although all reactions showing a closed sequence could be considered to be catalytic, there is a difference between those in which the entity of the active site is preserved by a catalyst and those in which it survives for only a limited number of cycles. In the first category are the truly catalytic reactions, whereas the second comprises the chain reactions. Both types can be considered by means of the steady-state approximation, as in Christiansen s treatment. This important development dates to 1919 (reaction between hydrogen and bromine reported earlier by Bodenstein and Lind. 

The rate law for the halogenation reaction shown above is derived step by step in Equations 1.4-1.8. We will learn to set up derivations of this type in Section 2.4.1. There we will use a much simpler example. We will not discuss Bodenstein s steady-state approximation used in Equations 1.6 and 1.7 in more detail until later (Section 2.5.1). What will be explained there and in the derivation of additional rate laws in this book is sufficient to enable you to follow the derivation of Equations 1.4—1.8 in detail in a second pass through this book (and you should make several passes through the book to ensure you understand the concepts). 

The concentration of an intermediate in a multistep reaction is always very low when it reacts faster than it is produced. If this concentration is set equal to zero in the derivation of the rate law, unreasonable results may be obtained. In such a case, one resorts to a different approximation. One sets the change of the concentration of this intermediate as a function of time equal to zero. This is equivalent to saying that the concentration of the intermediate during the reaction takes a value slightly different from zero. This value can be considered to be invariant with time, i.e., steady. Consequently this approximation is called Bodenstein s steady state approximation. 

The condition expressed by the Bodenstein approximation rx = 0 is often misleadingly called a steady state. It is not. It is not a time-independent state, only a state in which a specific variation with time (or reactor space time) is small compared with the others. In fact, some older textbooks applied what they called the steady-state approximation to batch reactions in order to derive the time dependence of the concentrations, unwittingly leading the incorrect presumption of a steady state ad absurdum. And a continuous stirred-tank or tubular reactor may, and usually does, come to a true steady state, even if the Bodenstein approximation is and remains inapplicable. [The approximation compares process rates r, it is irrelevant for its validity whether or not the reactor comes to a steady state, that is, whether the rates of change, dC /dr, become zero.]... 

Bei Annahme einer Reaktionsfolge nach (37) fiihrt die Anwendung des Bodenstein-Theorems ( steady-state approximation ) zu Gl. (38), die fur k l k2[P(OR)3] mit den experimentellen Werten iibereinstimmt. 

The steady-state approximation applied to chain reactions by Bodenstein cannot describe several time-dependent phenomena, for instance branched chain reactions, leading to explositions. For such cases the semi (or quasi ) steady-state approach, developed by a Nobel prize winner N. Semenov, assumes that concentrations of all radicals except one (the greatest) are considered as steady state. 

The steady-state approximation was first enunciated by Bodenstein.> It states that in a reaction in which transient species, such as atoms or radicals, are involved, a steady state sets in, characterized by an equal rate of formation and disappearance of the species. This principle, applied to the case of a polymerization reaction, means that at a certain reaction stage the amount of active centers formed is equal to the amount of growing chains terminated ... 

Applying the Bodenstein pseudo-steady-state approximation, where it is assumed that the excited molecule M is formed as quickly as it disappears, yields... 

This method for deriving approximate solutions for kinetic equatimis, known as steady-state approximation, was first introduced by the German physicochemist Max Bodenstein. It is very helpful in simplifying complicated kinetic relations. 

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 quasi-steady-state approximation (QSSA) is also called the Bodenstein principle, after one of its first users (Bodenstein 1913). As a first step, species are selected that will be called quasi-steady-state (or QSS) species. The QSS-species are usually highly reactive and low-concentration intermediates, like radicals. The production rates of these species are set to zero in the kinetic system of ODEs. The corresponding right-hand sides form a system of algebraic equations. These... 

Thus for a homogeneous system operating at constant volume, the pseudo-steady mode gives the same result as the Bodenstein quasi-steady state approximation. We can say that we have a quasi-steady state mode. 

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