Macroscopic reaction rate

Wintterlin J, Vdikening S, Janssens T V W, Zambelli T and ErtI G 1997 Atomic and macroscopic reaction rates of a surface-catalyzed reaction Soienoe 278 1931-4... 

The transition state theory of reaction rates [21] provides the link between macroscopic reaction rates and molecular properties of the reactants, such as translational, vibrational, and rotational degrees of freedom. For an extensive discussion of transition state theory applied to surface reactions we refer to books by Zhdanov [25] and by Van Santen and Niemantsverdriet [27]. The desorption of a molecule M proceeds as follows ... 

Let us now have a closer look at three basic types of the relative probabilities appearing in the model for an isomerization vs. another isomerization, the 1,2-insertion vs. 2,1-insertion, and an isomerization vs. an insertion. The right-hand part of Figure 11 summarizes the equations for the macroscopic reaction rates for the alternative reactive events starting from an alkyl complex p0 let us assume that the secondary carbon atom is attached to the metal, so that two isomerization reactions have to be considered. 

From a kinetic point of view processes (1) and (2) are characterized by equal macroscopic reaction rates for both enantiomers (v = Vj), while for (3) v Vj (in the extreme case, one of the two rates is zero). At the microscopic level, for a stereoselective polymerization... 

Using eqns. (42)-(44) and assuming T > co /2n, i.e. that the temperature is not too low, we have that tunneling does not practicably influence the macroscopic reaction rate constant at high pressures when the molecules have an equilibrium energy distribution. In this case... 

The knowledge of the two-minima energy surface is sufficient theoretically to determine the microscopic and static rate of reaction of a charge transfer in relation to a geometric variation of the molecule. In practice, the experimental study of the charge-transfer reactions in solution leads to a macroscopic reaction rate that characterizes the dynamics of the intramolecular motion of the solute molecule within the environment of the solvent molecules. Stochastic chemical reaction models restricted to the one-dimensional case are commonly used to establish the dynamical description. Therefore, it is of importance to recall (1) the fundamental properties of the stochastic processes under the Markov assumption that found the analysis of the unimolecular reaction dynamics and the Langevin-Fokker-Planck method, (2) the conditions of validity of the well-known Kramers results and their extension to the non-Markovian effects, and (3) the situation of a reaction in the absence of a potential barrier. 

We will see that for bimoleculcir reactions we cannot derive exact macroscopic reaction rate equations. The reason for this is that the reaction rate depends on the environment of each reactant i.e., it depends on how many reactants cire on neighboring sites. The approximation that is implicitly made when the phenomenological rate equation is written down may not always be appropriate. 

The velocity of this reaction v = d[P]/dt) is a function of the bimolecular rate of substrate binding (ki) and the unimolecular rates of chemistry (k2,fe-2) and substrate and product release (k i,k3). The steady-state velocity expression under initial rate conditions (Eq. (10.2)) demonstrates how each microscopic rate constant contributes to the macroscopic reaction rate and the dependence of the velocity upon substrate concentration. 

Note that we cannot write the usual form of the macroscopic reaction rate expression, Eq. (250), before making the equilibrium assumption in Eq. (249). For a detailed discussion of when Eq. (250) is valid, see Reference 86. 

The slow reaction limit corresponds to an inequality opposite that of (3-177). In this case the population of highly reactive states (E > E ) by W exchange is faster than the elementary chemical reaction itself. The vibrational distribution function in this case is almost not perturbed by chemical reaction / E) f ° E), and the total macroscopic reaction rate coefficient can be found as... 

The aforementioned example is characteristic of a situation in which the steady-state (macroscopic) reaction rate is constant, but the reacting adsorbed species are nevertheless not uniformly distributed over the surface. Most common in this respect are spiral waves that are characteristic of excitable media, where the excitation not necessarily requires an external perturbation, but may often be identified with an inherent local variation of kinetic parameters such as a defect zone on the surface. The core of the spiral may then be formed by a local defect to which the spiral is... 

Figure 6.4 Raman-spectroscopic monitoring of a polymerization reaction conducted in a microreactor to form a semiconducting polymer on basis of a polyphenylene-vinylene structure for OLED applications. The C—C double bond at 1580cm that is formed during reaction progress undergoes a strong increase in intensity that allows the determination of the macroscopic reaction rate.

When condition (dai/dt)react (dai/dt)rei is not satisfied, the perturbation of equilibrium distribution is substantial. However, in this case realization of the quasi-steady-state condition is possible, provided the overall rate dai/dt is low compared to partial rates (daj/dt)rei and (dai/dt)i.eact Then, the microscopic kinetic equation can be solved by the quasi-steady-state approximation. The approximation implies that the non-equilibrium distribution functions depend on time implicitly via the total concentration of reactants rather than explicitly. This also means that the macroscopic reaction rates are low compared to microscopic reaction and relaxation rates. Since the distribution functions in this approximation depend on the total concentration only, the reaction rates, according to Eq. (8.50) also depend on the total concentration. Hence, we come to macroscopic kinetic equations that involve only the total concentration of reactants and certain combinations of microscopic rate constants that have the meaning of macroscopic constants. Note that these macroscopic equations need not be consistent with the macroscopic kinetic law as, besides elementary reactive processes, they involve unreactive processes. 

We should finally mention recent theoretical and experimental studies of chemical reactions on a more detailed microscopic level. In these studies the collision dynamics are studied in detail by classical or quantum mechanics in order to obtain the cross section for the reaction. The cross section is a function of relative speed, orientation, and possibly the internal state of the colliding molecules and can be averaged over these variables to give the macroscopic reaction rate. This very intriguing theoretical approaches unfortunately extremely complicated and so far has been applied only in the simplest reactions. References on this work may be found in [1]. 

A macroscopic reaction rate is defined as the derivative of the concentration of some species involved in the reaction with respect to time. 

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