Bodenstein’s steady state

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. 

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. 

Using Bodenstein s steady-state approximation (Sect 4.3) aftCT a short time the following equations hold ... 

Henri and Michaelis-Menten kinetics assumed that the rate of formation of products was much less than that for the back reaction from ES to yield E + S. Van Slyke assumed the reverse. A more rigorous formulation was offered by Briggs and Haldane (1925) using steady-state assumptions previously applied to chemical kinetics by Bodenstein (1913). 

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 original more formal proof (31) of (44) consisted in the demonstration of its equivalence to the steady-state condition in the Bodenstein s form i.e., the condition that the rates of formation of intermediates are equal to zero. 

The application of Bodenstein s principle (a quasi-steady state with respect to radicals) to nonisothermal processes is shown not always to be correct. 

In general, the coke (or carbon deposit) could be produced from any surface species, not only A. Deactivation rs and self-regeneration r. proceed simultaneously. In order for a chemical reaction to occur, deactivation steps rs and r.s must be essentially slower than the reaction steps. This means in turn, that quasi steady state (Bodenstein) conditions could be applied for the main reaction, but not for deactivation. In terms of surface coverage it follows then that... 

The concentration of the primary radical cr is characterized by the fast establishment of a steady-state value. Applying Bodenstein s quasi-steady-state principle to this concentration yields the following description of the initiating reaction ... 

The steady-state apiHoximation, advocated by Bodenstein in the 1920 s, is now a clasncal tool of chemical kinetics. Its limitations, as sketched in this section, arc still under discusaon. The fnesent treatment is based on the work of J. C. Giddings and H. K. Shin, Trans. Parade Soc., 57, 468 (1961). A more powerM approach has been outlined by J. R. Bowen, A. Acrivos and A. K. Oppenheim, Chem. Eng. Sei. 18, 177 (1963). 

In these conditions, all the other steps will be regarded as constantly in equilibrium and the pseudo-steady state will have the meaning of the Bodenstein s quasi-steady state. 

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