Average reaction probability

To find the conversion for the reactor, we need the average reaction probability for a great many molecules that have flowed through the system. The averaging is done with respect to residence time since residence time is what determines the individual reaction probabilities ... 

Thus, the energy E has the meaning of effective activation energy. With decreasing temperature, the effective activation energy and the preexponential factor /(E ) diminish. The reaction rate constant is proportional to the averaged reaction probability. The characteristic temperature dependence of the reaction rate constant is shown in Fig. 24. 

Thus, rather than the average reaction probability, Pq), the weighted average (jD(jPqj is required in this case. As in the case of isotropic translational diffusion, the magnitude of the reaction probability is small for small 9, and thus the expression for dimensionless rate constant may be simplified as... 

The functions actually plotted in Figs. 2 and 3 are the ratio of N(E), the reactive flux, to the incident flux (or flux per unit area for the three-dimensional case). These ratios have the more physically meaningful interpretation of an average reaction probability in the collinear case (Fig. 2) and an average reaction cross section in the three-dimensional case (Fig. 3). 

The average reaction probability is evaluated from trajectories with b chosen randomly according to the distribution function... 

With b chosen randomly between 0 and bmax, the average reaction probability is = NJN, where N is the total number of trajectories and Nr, the number of trajectories that are reactive. Thus, the reaction cross... 

We have derived the ABC formulation of quantum reactive scattering theory, and applied it to the calculation of the initial state selected reaction probability. By exploiting the highly localized nature of forces in reactive scattering, the ABC formulation facilitates the direct calculation of detailed or averaged reaction probabilities while sampling only a finite region of space. 

The methodology presented so far allows the calculations of state-to-state. S -matrix elements. However, often one is not interested in this high-level of detail but prefers instead to find more average infomiation, such as the initial-state selected reaction probability, i.e. the probability of rearrangement given an initial state Uq. In general, this probability is... 

Soil water flow is decidedly episodic. During dry times the water solutions in the soil are probably fairly concentrated and not very reactive. Time-averaged reaction rates should be roughly proportional to the fraction of time reacting minerals are in contact with thermodynamically imdersaturated (and reactive) water. In a study of the relationship between denudation rate and runoff for rivers draining igneous and metamorphic rock in Kenya, Dunne (1978) obtains the relationship of (denudation rate in tons/km per year) = 0.28 (runoff in mm/ year)°. ... 

The results for the 2-slte (B/B) model are satisfactory. The weighted sums (Table IX) indicate that there are two sites of roughly equal weights (41% w j for hexane set, and 51% W ] for ether set). The average Bemoullian reaction probabilities for the two sites are = 0.58, Pp,2 = 0.20. These results... 

Change the reaction probability Pr(AB) to 1.0, and let the simulation run for 1000 iterations. At what time (what iteration) does reaction occur Repeat this simulation nine more times and tabulate the results. Find the average time and its standard deviation for your results, as well as the median time. Next change Pr(AB) to 0.05, increase the number of iterations for each run to 5000, and tabulate the results for 10 trial runs. Repeat the averaging process above. This study reveals the influence of the reaction probability on the course of the reaction. 

For determination of reaction probability and reaction cross section, a large number of collision trajectories have to be considered and appropriate averages over the initial conditions performed. The reaction probability is calculated for a specified initial relative velocity vR (i.e. initial relative kinetic energy), rotational state /, and impact parameter b. The reaction probability is the ratio of number of reactive trajectories to the total number trajectories, i.e. 

The choice of other variables R, r, h, 0, and r appropriate for Monte-Carlo averaging is made by pseudo random numbers generated on computer. The reactive cross section can be found by averaging the reaction probability over the impact parameter and rotational state... 

By using reasoning based on reaction probability, Flory [51] has shown that, when Ca = the number-average degree of poly-... 

The total reaction probability for a particular value of the total angular momentum J, averaged over all values of ntj and M for the reactants and summed over aU values of my and M for the products, is given by... 

W. H. Miller I would like to ask Prof. Schinke the following question. Regarding the state-specific unimolecular decay rates for HO2 — H + O2, you observe that the average rate (as a function of energy) is well-described by standard statistical theory (as one expects). My question has to do with the distribution of the individual rates about die average since there is no tunneling involved in this reaction, the TST/Random Matrix Model used by Polik, Moore and me predicts this distribution to be x-square, with the number of decay channels being the cumulative reaction probability [the numerator of the TST expression for k(E)] how well does this model fit the results of your calculations ... 

This partial differential equation is made ordinary by taking its Laplace transform. It is then solved for the transform of u and the transform of the total reaction rate for the particle calculated. There is no need to invert this transform, for what we want is the average reaction rate in the bed and this is the integral of the product of the reaction rate after a time a and the probability of the particle having age a. Because the ages are exponentially distributed, this is none other than the Laplace transform with the dimensionless transform variable set equal to 1. The end result is that the Damkohler number is... 

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