Macroscopic rate equations

Fig. 34. The macroscopic rate equation for a bistable system and the domains of attraction.

Now consider the evolution in a situation as in fig. 34. There are three stationary macrostates 0fl, 0j>, c, of which 0a and 0C are locally stable and (j)b is unstable. Of course, even the pure macroscopist would not regard (j)b as a realizable state, on the ground that a system in (j>b would be caused to move into either 0a or 0C by the smallest perturbation. Systems having a macroscopic characteristic as in fig. 34 are called bistable . There are numerous examples the ones that occur most often in the literature are the laser (section 9 below), the tunnel diode 0, and the Schlogl reaction (X.3.6). The macroscopic rate equation for this reaction is... 

The relation between the master equation and macroscopic rate equations. 

The kinetics of chemical reactions on surfaces is normally described using macroscopic rate equations. The master equation can be used to derive such macroscopic rate equations. Sometimes this derivation is exact, but we often will have to make approximations, which may or may not be appropriate. This will depend on the system. If the approximation to derive the macroscopic rate equations are too crude, the master equation shows, however, how to add corrections to rate equations. It is in general necessary to make approximations even with these corrections, but one has the choice what approximations to make. Of course, in practice one may... 

The desorption of an adatom that does not feel neighboring adatoms is the simplest case to derive macroscopic rate equations for. The derivation in this section will be exact. This is due to the fact that there is no interaction between adatoms. 

It describes correlation between the fluctuations of the numbers of As and Bs. It can be shown that in the thermodynamic limit (i.e., S oo) the summation scales as S, so that the last term vanishes in that limit. [21] The resulting equation is the familiar macroscopic rate equation. In statistical physics this is usually called the mean-fleld approximation. 

We also have Aa = Ap — 2. The macroscopic rate equation becomes... 

A further point worth noting is the reaction rate constant in the macroscopic rate equation. We see that it differs from the transition probability by a factor Z. This means, all other things being equal, that surfaces with high Z are more reactive than those with low Z. It is also important not to forget this factor when one want to derive the reaction rate constant using quantum chemical methods. 

From equations (58)-(60) we can write down the macroscopic rate equations for the ZGB model in a straightforward manner ... 

We have derived the master equation from first principles and shown the relation between the master equation and macroscopic rate equations. After that, several Monte Carlo methods to solve the master equation were discussed. We have thus laid a firm basis to discuss some applications of Monte Carlo techniques in catalysis. 

A simple example—the quantum mechanical basis for macroscopic rate equations... 

However, correlation of these reaction mechanisms (suggested by inspection of the macroscopic rate equations) with molecular-level studies of the elementary surface reactions remains one of the future challenges of catalysis. 

Starred species are held constant by buffering, or reservoirs, or flows. A biological example will be given shortly. The macroscopic rate equations are given by... 

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