The macroscopic rate equation

Equation (219) is a complicated equation and difficult to solve without a considerable simplification. Wilemski and Fixman [51] suggested the equation could be simplified considerably if the rate of the quenching reaction was slow compared with diffusion. The last term on the right-hand side perturbs the density, n, slightly under such circumstances and the equation can be solved by standard Green s function techniques (Appendix A) to give 

The source term describes the formation of the m + 1 reactants at time t — t0 with these initial positions at rA° and r ls r 2... etc. The integral in eqn. (220) describe the reduction in the density of the quencher and fluorophor distribution if the quenching process is very slow. Unfortunately, within the integral is n( r0, f°), which is the very density that is sought. As a first approximation, ne 1 could be used. Wilemski and Fixman suggested a more satisfactory (closure) approximation 

Now let us consider the simplest case where this approximation for n( r, t) = neq v(t) is adequate. Take the volume integral of eqn. (219) and utilise the boundary conditions on re (Vn = 0 on all surface) and also eqn. (218) 

This has utilised the definition of y( r ) as the overlap of all m quenchers with the fluorophor and the fact that the integral is really /drA /drQ j. .. /drQ , an integral over all 3(m + 1) coordinates of the quenchers and fluorophor. Note that using eqn. (218) 

This is a familiar first-order equation for the decay of the excited fluorophor and has a second-order rate coefficient, k, and the quencher concentration, [Q], = m/V in the rate expression just as expected. It is only valid when the rate of quenching is very slow compared with diffusion, so that a homogeneous distribution of reactants is developed and maintained. 

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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 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... 

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 ... 

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... 

Mechanisms. Mechanism is a technical term, referring to a detailed, microscopic description of a chemical transformation. Although it falls far short of a complete dynamical description of a reaction at the atomic level, a mechanism has been the most information available. In particular, a mechanism for a reaction is sufficient to predict the macroscopic rate law of the reaction. This deductive process is vaUd only in one direction, ie, an unlimited number of mechanisms are consistent with any measured rate law. A successful kinetic study, therefore, postulates a mechanism, derives the rate law, and demonstrates that the rate law is sufficient to explain experimental data over some range of conditions. New data may be discovered later that prove inconsistent with the assumed rate law and require that a new mechanism be postulated. Mechanisms state, in particular, what molecules actually react in an elementary step and what products these produce. An overall chemical equation may involve a variety of intermediates, and the mechanism specifies those intermediates. For the overall equation... 

Before deriving the rate equations, we first need to think about the dimensions of the rates. As heterogeneous catalysis involves reactants and products in the three-dimensional space of gases or liquids, but with intermediates on a two-dimensional surface we cannot simply use concentrations as in the case of uncatalyzed reactions. Our choice throughout this book will be to express the macroscopic rate of a catalytic reaction in moles per unit of time. In addition, we will use the microscopic concept of turnover frequency, defined as the number of molecules converted per active site and per unit of time. The macroscopic rate can be seen as a characteristic activity per weight or per volume unit of catalyst in all its complexity with regard to shape, composition, etc., whereas the turnover frequency is a measure of the intrinsic activity of a catalytic site. 

With these definitions the equations for the macroscopic rate become... 

Exercise. Find both versions (3.1) and (V.8.6) of the macroscopic equation belonging to the equation (VI.9.5). Check that the former is the familiar rate equation for the reaction. 

We have taken a different route. Similar to the way we found the relationship between q+ and q by comparing equation (1) with the thermodynamics, we shall find the absolute value of D by comparing (5) with the macroscopic rate of change of the system, in the spirit of the general principle of correspondence of theoretical physics. 

Detailed balance provides a relation between the macroscopic rate constants kf and kr for the forward and reverse reactions, respectively. On a macroscopic level, the relation is derived by equating the rates of the forward and reverse reactions at equilibrium. Here it will be shown that the principle of detailed balance can be readily obtained as a direct consequence of the microscopic reversibility of the fundamental equations of motion. 

The biexponential rate equation associated with this model was fitted to the experimental data using a nonlinear least squares procedure. Pharmacokinetic constants for the two-compartment model were calculated by standard methods. The fraction amount absorbed as a function of time was estimated by the Loo-Riegelman method using the macroscopic rate constants calculated from the intravenous data. The slope of the linear part of the Loo-Riegelman plot combined with the total amount absorbed (quantitated by depletion analysis of the saturated donor solution) was used to calculate the zero-order rate constant for buccal permeability. 

The relation between the master equation and macroscopic rate equations. 

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. 

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. 

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 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. 

The crucial assumption is to relate the macroscopic rate of dissipation (which appears on the left-hand side of the following equation) with the microscopic rate of dissipation (appearing on the right-hand side) ... 

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

The flow behavior of fluids is governed by the basic laws for conservation of mass, energy, and momentum coupled with appropriate expressions for the irreversible rate processes (e.g., friction loss) as a function of fluid properties, flow conditions, geometry, etc. These conservation laws can be expressed in terms of microscopic or point values of the variables, or in terms of macroscopic or integrated average values of these quantities. In principle, the macroscopic balances can be derived by integration of the microscopic balances. However, unless the local microscopic details of the flow field are required, it is often easier and more convenient to start with the macroscopic balance equations. 

The function is a sum of two terms, one is characterized by a fast (72) and the other by slow (71) macroscopic rate constants. The macroscopic rate constants 71 and 72 are the roots of the Laplace polynomial and are given by Equation (19), which relates the macroscopic with the microscopic rate constants... 

Figures 23-25 demonstrate that the simulation technique is suitable for reproducing the outcome of the experiment, thus these calculations may be convenient for understanding the complex relationship between the various parameters appearing in the differential rate equations and the observed macroscopic parameters.

Equation (5-9) is identical with the relation obtained from collision theory, Eq. (2-33) for k = 1 (unlike reactants), and for a reaction whose steric factor p is unity. It is also worth noting that the synthesis of the macroscopic rate constant /f(T) in Eq. (5-6) from the microscopic quantities in Eqs. (5-2) to (5-5) strongly parallels portions of the rigorous derivation of the macroscopic transport coefficients, e.g., those of viscosity, thermal conductivity, or diffusion [4, 5]. 

Analysis of the transport and kinetics must be approached on two levels the first is essentially macroscopic. The steady-state Ficksian diffusion/reaction equation must be solved for the substrate in the bounded diffusion space of the film of extent L. This type of analysis has been discussed in previous sections of Chapter 2. From this analysis the pseudo first-order rate constant k for substrate reaction can be derived however the analysis must be taken a step further. We must also adopt a microscopic approach. In this case the spherical geometry of the microparticle must be considered, and the steady-state spherical diffusion of the substrate to the microparticle must be examined. We must then relate the macroscopic rate constant k for substrate reaction to the spherical diffusion and reaction at each microparticle. 

In deriving the latter expression we assumed that R R- Equation 196 is very fundamental, and it provides a relationship between the macroscopic rate constant k and the macroscopic processes of substrate electrode kinetics and spherical diffusion at the catalytic particle. On the rhs of Eqn. 196 the first term describes the effect of heterogeneous kinetics of the substrate at the particle surface. This rate constant k is a strong function of electrode potential. The second term describes the spherical diffusion of substrate to the catalytic particle. When ks DsIR the mass transport of S to the particle surface is rate-determining. Alternatively when k E DsiR then the electron transfer process at the particle surface is rate-determining. 

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