Transfer probability distribution function

C3.3.5.2 EXTRACTING THE ENERGY TRANSFER PROBABILITY DISTRIBUTION FUNCTION P(E, E)... 

Figure C3.3.11. The energy transfer probability distribution function P(E, E ) (see figure C3.3.2) for two molecules, pyrazine and hexafluorobenzene, excited at 248 nm, arising from collisions with carbon dioxide...

Figure C3.3.12. The energy-transfer-probability-distribution function P(E, E ) (see figure C3.3.2 and figure C3.3.11) for two molecules, pyrazine and hexafluorobenzene, excited at 248 nm, arising from collisions with carbon dioxide molecules. Both collisions that leave the carbon dioxide bath molecule in its ground vibrationless state, OO O, and those that excite the 00 1 vibrational state (2349 cm ), have been included in computing this probability. The spikes in the distribution arise from excitation of the carbon dioxide bath 00 1 vibrational mode.

For the purposes of modeling, it is convenient to factor the rate coefficient for collisional energy and angular momentum transfer into an overall rate coefficient for collisions between the molecule and bath gas (irrespective of the amount of energy transferred), z, and a collisional energy and angular momentum transfer probability distribution function (TPDF) P E,J E, J ),... 

The quantity 17(f) is the time-dependent friction kernel. It characterizes the dissipation effects of the solvent motion along the reaction coordinate. The dynamic solute-solvent interactions in the case of charge transfer are analogous to the transient solvation effects manifested in C(t) (see Section II). We assume that the underlying dynamics of the dielectric function for BA and other molecules are similar to the dynamics for the coumarins. Thus we quantify t](t) from the experimental C(t) values using the relationship discussed elsewhere [139], The solution to the GLE is in the form of p(z, t), the probability distribution function. 

Figure 27. The time-dependent probability distribution function p(z, f) for the excited state charge transfer of BA from a GLE simulation (See Refs. 132 and 133). The 5, potential employed in the simulation is shown in Figure 24.

Danckwerts assumed a random surface renewal process in which the probability of surface renewal is independent of its age. If s is the fraction of the total surface renewed per unit time, obtain the age distribution function for the surface and show that the mean mass transfer rate Na over the whole surface is ... 

In this equation g(r) is the equilibrium radial distribution function for a pair of reactants (14), g(r)4irr2dr is the probability that the centers of the pair of reactants are separated by a distance between r and r + dr, and (r) is the (first-order) rate constant for electron transfer at the separation distance r. Intramolecular electron transfer reactions involving "floppy" bridging groups can, of course, also occur over a range of separation distances in this case a different normalizing factor is used. 

Due to the rapid decrease in the process probability with increase of the distance between the reagents, it should be expected that reaction (13) will result in electron transfer primarily to the particle A which is nearest to the excited donor particle D. In this case, the condition n < N is satisfied for reaction (13), where n is the concentration of the particles D and N is that of the particles A, and with the random initial distribution of the particles, A, the distribution function over the distances in the pairs D A formed, will have the same form [see Chap. 4, eqn. (13)] as with the non-paired random distribution under the conditions when n IV. In such a situation the kinetics of backward recombination of the particles in the pairs D A [reaction (12)] will be described by eqn. (24) of Chap.4 which coincides with eqn. (35) of Chap. 4 for electron tunneling reactions under a non-paired random distribution of the acceptor particles. Therefore, in the case of the pairwise recombination via electron tunneling considered here, the same methods of determining the parameters ve and ae can be applied as those described in the previous section for the case of the non-pair distribution. However, examples of the reliable determination of the parameters ve and ae for the case of the pairwise recombination using this method are still unknown to us. 

The desorption data collected during this investigation were analyzed with a distributed-rate model that has been described in detail elsewhere (Culver et al. 1997 Deitsch et al. 1998 Deitsch and Smith 1999). Briefly, to account for soil heterogeneity, the distributed-rate model replaces a single mass-transfer rate coefficient with a continuous distribution of rate coefficients. In this study, a T-probability density function (r-PDF) was used to generate the distribution of rate coefficients. The T-PDFis given by... 

The movement of the particles in this stage is very complex and extremely random, so that to determine accurately the residence time distribution and the mean residence time is difficult, whether by theoretical analysis or experimental measurement. On the other hand, the residence time distribution in this stage is unimportant because this subspace is essentially inert for heat and mass transfer. Considering the presence of significant back-mixing, the flow of the particles in this stage is assumed also to be in perfect mixing, as a first-order approximation, and thus the residence time distribution probability density function is of the form below ... 

Before the transfer starts, the energy distribution of electrons takes the form of a Fermi-Dirac distribution function. While the number of electrons is decreasing steadily with time, the distribution of electrons keep the form of a Fermi-Dirac distribution function. This constancy of the distribution is due to the fact that the capture rate of free electrons by the localized states is much faster than the loss of free electrons caused by the transfer when the occupation probability of localized states is not approximately one. Therefore, electrons are considered to be in their quasi-thermal equilibrium condition i.e., the energy distribution of electrons is described by quasi-Fermi energy EF. Then the total density t of electrons captured by the localized states per unit volume can be written as... 

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