Cumulative reaction rate

N E) is called the cumulative reaction probability. It is directly related to the themial reaction rate k T) by... 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

The plot of residuals versus some measure of the time at which experiments were run can also be informative. If the number of hours on stream or the cumulative volume of feed passed through the reactor is used, nonrandom residuals could indicate improper treatment of catalyst-activity decay. In the same fashion that residuals can indicate variables not taken into account in predicting reaction rates, variables not taken into account as affecting activity decay can thus be ascertained. 

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 ... 

The rate constant for a chemical reaction is conveniently expressed in terms of the cumulative reaction probability [1] (CRP) N(E),... 

One expects to observe a barrier resonance associated with each vibra-tionally adiabatic barrier for a given chemical reaction. Since the adiabatic theory of reactions is closely related to the rate of reaction, it is perhaps not surprising that Truhlar and coworkers [44, 55] have demonstrated that the cumulative reaction probability, NR(E), shows the influence barrier resonances. Specifically, dNR/dE shows peaks at each resonance energy and Nr(E) itself shows a staircase structure with a unit step at each QBS energy. It is a more unexpected result that the properties of the QBS seem to also imprint on other reaction observables such as the state-to-state cross sections [1,56] and even can even influence the helicity states of the products [57-59]. This more general influence of the QBS on scattering observables makes possible the direct verification of the existence of barrier-states based on molecular beam experiments. 

Enhance reaction rates by reducing the activation energy barrier, e.g. by altering cumulations of charge (bind polarities) and thus favouring reactions which... 

Although instantaneous cracking rate is assumed to be directly proportional to oil partial pressure, the net effect of pressure in actual cracking operation is much less, particularly in a fixed bed, because of the increased coke deposition and more rapid activity decline at higher pressures (73). Even in catalyst-circulation processes the cracking rate is less than proportional to pressure for example, the cumulative reaction-velocity constant in fluid-catalyst operation appears to be proportional to about the 0.5 power of pressure. 

Abstract. The Chebyshev operator is a diserete eosine-type propagator that bears many formal similarities with the time propagator. It has some unique and desirable numerical properties that distinguish it as an optimal propagator for a wide variety of quantum mechanical studies of molecular systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calculation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

In studying reaction dynamics, one may only be interested in averaged properties such as cumulative reaction probabilities and thermal rate constants. These quantities can of course be obtained from state-to-state probabilities, but as shown by Miller and coworkers they can be calculated directly and more efficiently without knowledge of the S-matrix elements.[44,45] The cumulative reaction probability, for example, can be computed as follows ... 

Eq. (20.36) is a relation of (cumulative) reaction probability, that is, reaction dynamics, and hot electron dynamics. Because the rate of energy transfer is much higher in DIET than DIMET per single hot electron attachment, the eTST is suitable for a DIET process as a first approximation. 

The results obtained (Table 2) show that the effect of substituents on the reaction rate is comparable to that observed in homogeneous catalysis [2,6,7]. The reaction rate greatly increases in the presence of a strong electron-withdrawing group such as NO2, especially in theorf/io position, due to the destabilization of the N-acyl bond by cumulative resonance and inductive effect. 

Because the total angular momentum and its component are conserved during a collision, we can study the reaction dynamics for each value of 7 and M, independently. Since the results are independent of M, we always set M, - 0, and we will not mention it again (but the existence of the Af, quantum number is the reason for the factor of 27 + 1 in the following sentence). In particular, we can study the 7-specific contributions to the rate constant, k (E) [with k(E) of Eq. (3) being a (27 + l)-weighted sum of individual k (E), to the cumulative reaction probability, N (E), and to the density of reactive states, pJ(E). The influence of quantized transition states on chemical reactivity will be analyzed through studies of k (E). 

The completely rigorous equilibrium rate constant can also be written in the form of Eq. (11), where for a bimolecular reaction the rigorous expression for the cumulative reaction probability is (17). 

Most of the above discussion has concentrated on calculation of the cumulative reaction probability, N(E), from which one obtains the microcanonical rate k(E) via Eq. (8) or the canonical rate k(T) by averaging over total energy as in Eq. (11). If one is primarily interested in the thermal rate, however, it would clearly be desirable to be able to calculate it directly for a given temperature T and not have to calculate N(E) at many values of E. 

For many chemical reactions a sufficient description is provided by the rate constant, either the canonical rate constant characterized by the temperature, k T), or the microcanonical rate constant characterized by the total energy, k E). These rate constants can be obtained using appropriate averages of the state-to-state differential cross sections. This averaging process yields the cumulative reaction probability (CRP) ... 

The thermal rate constant is given in terms of N(E), the cumulative reaction probability—which is approximated in transition-state theory by the minimum of the microcanonical flux—by... 

A hierarchy of reduced dimensionality exact quantum theories of reactive scattering is presented for the vibrational state-to-state cumulative reaction probability and vibrational state-to-state thermal rate constant. The central approximation in these theories is the adiabatic treatment of the bending motion of the reactive species in the strong interaction region of configuration space. Applications of the theories are made to the reactions MU+H2, 0( P)+H2, D2 and HD... 

The reduced dimensionality quantum theory of reactive scattering we have developed focuses on the so-called cumulative reaction probability. In order to motivate and review the importance of this quantity we begin this section with the rigorous collisional expressions for the thermal rate constants for an A+BC -> AB+C reaction. 

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