Position-dependent rate

The remote transfer in condensed matter is characterized by the position-dependent rate W(r), which is the input data for encounter theory. In its differential version (DET), the main kinetic equation (3.2) remains unchanged, but the rate constant acquires the definition relating it to W(r) ... 

It is assumed throughout this chapter that the relative positions of the donor and acceptor molecules between which an electron is transferred are fixed either within a large molecule or by a molecular bridge in solvents. In this situation, electron-transfer (ET) reactions are governed by first-order kinetics. In second-order reactions [1], the donor and acceptor first diffuse to mutually appropriate positions where the reaction takes place with a position-dependent rate constant. In that case, this is the rate constant discussed in the present article. 

Usually electrons in interband defect states are considered to be immobile the rate of change of the concentration of ionized interband states is equal to their (position-dependent) rate of production, Gj. 

Thus far we have a position-dependent rate T(R ) at which the upper state is populated during application of the two writing beams. If we take the effective time for which the beams are applied as At, the probability that molecule E, is excited immediately after the pulses have passed is P(R ) = T(Rt)Af. The probe pulse arrives after a delay of i(> At) seconds, during which time the molecule, if excited, may relax. We suppose that it relaxes to the ground state via a simple exponential decay. At time x the probability that the molecule is excited is hence... 

The local (position-dependent) rate of consumption of gaseous reactant in a unit volume of pellet is equal to the local rate of consumption of solid reactant per unit volume of pellet. 

The data plotted in the figure clearly support the predicted positive dependence of crystal size on agitation rate. Precipitation in the crystal film both enhances mass transfer and depletes bulk solute concentration. Thus, in the clear film model plotted by broken lines, bulk crystal sizes are initially slightly smaller than those predicted by the crystal film model but quickly become much larger due to increased yield. Taken together, these data imply that while the initial mean crystal growth rate and mixing rate dependence of size are... 

Finally the success of the model can be judged from Figures 6.25a and b which show the experimental and model-predicted rate dependence on pCo and work function during CO oxidation on Pt/pM-Al203.71 Note the transition from a classical Langmuir-Hinshelwood to a positive order rate dependence on pco with decreasing work function. Also notice that on every point of the experimental or model predicted rate dependence, the basic promotional mle ... 

The solvent dependence of the reaction rate is also consistent with this mechanistic scheme. Comparison of the rate constants for isomerizations of PCMT in chloroform and in nitrobenzene shows a small (ca. 40%) rate enhancement in the latter solvent. Simple electrostatic theory predicts that nucleophilic substitutions in which neutral reactants are converted to ionic products should be accelerated in polar solvents (23), so that a rate increase in nitrobenzene is to be expected. In fact, this effect is often very small (24). For example, Parker and co-workers (25) report that the S 2 reaction of methyl bromide and dimethyl sulfide is accelerated by only 50% on changing the solvent from 88% (w/w) methanol-water to N,N-dimethylacetamide (DMAc) at low ionic strength this is a far greater change in solvent properties than that investigated in the present work. Thus a small, positive dependence of reaction rate on solvent polarity is implicit in the sulfonium ion mechanism. 

B. Carmeli and A. Nitzan, Theory of activated rate processes position dependent friction, Chem. Phys. Lett. 102, 517 (1983). 

In Equation 6, the dlffuslvlty and mobility are second rank tensors whose positional dependence is a consequence of the hydrodynamic wall effect and F represents the probabllllty that the Brownian particle, initially at some fixed point, will be at some position in space R at a later time t. At low concentrations, P is replaced by the number concentration, C (25). Conceptually the approach followed is similar to that developed by Brenner and Gaydos (25), however, one needs to include an expression for the flux of particles at the wall due to exchange with the pores. Upon averaging over the interstitial tube cross section of Figure 2, one arrives at the following expression (29) for the area averaged rate equation for the mobile phase transport. 

The presence of 4e as the predominant species during the catalysis is also in accord with the observed kinetic behavior of this catalyst with 1-octene and styrene as the substrates. The observation of this saturated acyl rhodium complex is in line with the positive dependence of the reaction rate on the hydrogen concentration and the zero order in alkene concentration. It was concluded previously that this saturated acyl complex is an unreactive resting state [18]. Before the final hydro-genolysis reaction step can occur, a CO molecule has to dissociate in order to form... 

The first is a variational approach that maps the position-dependent problem to an effective parabolic barrier transfer problem, with an effective friction that is position-independent. This approach leads to a result for the rate that can be interpreted as a Grote-Hynes coefficient with a position-dependent friction. 

In an earlier work we performed a quantum calculation using the exponential resummation technique and found results that agreed qualitatively with those of Azzouz and Borgis. When we allowed for a position-dependent friction, we obtained a function g(s) that is plotted in Fig. 2. The results for the quantum rate are presented in Tables II and 111. The column g(s) = s refers to the position-independent case, as calculated in our earlier work on this system. 

The spots are rated according to the intensity of their color. The positive control should be taken as an orientation point. Blue specimens are rated as negative, distinctly colored samples are considered positive (Fig. 4.1.5), while intensely pink specimens are rated strongly positive. The latter are highly suspicious of MPSs and follow-up examinations are recommended. Positive specimens (rating 2) should be discussed with the patient s physician. Depending on the clinical symptoms, further evaluations may be indicated. 

Figure 2 Model calculation according to Ref. 37 of the photocatalytic rate as a function of parameters F = 02 and Z = 03 (see text), given 4) = 0.001 and ai = 0.01. Note that the rate shows a maximum which position depends on the photon flux. At low Z, the rate shape is comparable to those observed in Fig. 1. At high Z, a Langmuirian shape is produced.

Since the rate of reaction r and the volumetric flow rate V at each position depend on T, P, and local molal flow rate n of the key component of the reacting mixture, finding the true residence time is an involved process requiring many data. The easily evaluated apparent residence time usually is taken as adequate for rating sizes of reactors and for making comparisons. 

---