Bodenstein-principle approximation

In almost every case differential equations for the quantitative description of the time dependence of particular species resulting from a catalytic cycle cannot be solved directly. This requires approximate solutions to be made, such as the equilibrium approximation [15], the Bodenstein principle [16], or the more generally valid steady-state approach [17]. A discussion of differences and similarities of different approximations can be found in [18]. 

Equipped with the Bodenstein principle, let us now continue the derivation of the rate law for SN reactions that take place according to Figure 2.11. The completely inadequate approximation [carbenium ion] = 0 must be replaced by Equation 2.6. Let us now set the left-hand side of Equation 2.6, the change of the carbenium ion concentration with time, equal to the difference between the rate of formation of the carbenium ion and its consumption. Because the formation and consumption of the carbenium ion are elementary reactions, Equation 2.7 can immediately be set up. If we now set the right-hand sides of Equations 2.6 and 2.7 equal and solve for the concentration of the carbenium ion, we get Equation 2.8. With this equation, it is possible to rewrite the previously unusable Equation 2.5 as Equation 2.9. The only concentration term that appears in this equation is the concentration of the alkylating agent. In contrast to the carbenium ion concentration, it can be readily measured. 

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... 

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 ... 

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