Detailed rate constants

Fig. 8. Three contour plots of the detailed rate constants k(V, R, T) for H + Cl2. The left-hand plot is for the room temperature reaction, the upper right-hand plot is for translationally excited reagents and the lower right-hand plot is for vibrationally excited reagents, H + Cl2 (u => 1). (Reproduced from ref. 196 by permission of the authors and the Royal Society of Chemistry.)...

The quasiclassical trajectory method disregards completely the quantum phenomenon of superposition (13,18,19) consequently, the method fails in treating the reaction features connected with the interference effects such as rainbow or Stueckelberg-type oscillations in the state-to-state differential cross sections (13,17,28). When, however, more averaged characteristics are dealt with (then the interference is quenched), the quasiclassical trajectory method turns out to be a relatively universal and powerful theoretical tool. Total cross-sections (detailed rate constants) of a large variety of microscopic systems can be obtained in a semiquantitative agreement with experiment (6). 

As noted above, development of reliable atmospheric models requires the elucidation of detailed rate constants for specific deactivation pathways. Presented below is a discussion of the collisional dynamics of O ( D2) deactivation by molecules of atmospheric interest. Additional species which are not of primary importance to environmental chemistry will also be mentioned in order to illustrate the general behavior of O ( D2) in gas phase encounters with quenching and reactive substrates. 

Ha(HD) + Clzlhv Spectroscopy, investigation of pumping processes, discussion of detailes rate constants Coraeil, Pimentel... 

Detailed rate constants, which describe the rates of reaction in collisions between species in defined internal states but occurring with a thermal spread of relative... 

Figure 1 Triangular contour plots showing the variation of detailed rate constants. Values of vibrational energy (V ) and rotational energy (RO are plotted, ignoring quantization along the rectilinear axes and those of translational energy (T ) are indicated by the dashed diagonal lines. Units are kcal mol (1 kcal mol = 4.18 kJmol ). Panel (a) shows the variation of the detailed rate constants for reaction (22) in the exothermic direction, i.e. kt ( , i) kt ( , J ,T),as determined by i.r. chemiluminescence experiments.

Panels (b) to (d) indicate the iktailed rate constants for reactions (—20) to ( -22), i.e. k, (v, jy,J S ), as obtained from application of equation (27). The horizontal line on each diagram ideates the energy of the actual vibrational states and the value of kt or k, beside these lines indicates the results of summing the detailed rate constants over rotational states (Reproduced by permission from /. Chem. Phys., 1969,51, 5716, 5717)... 

Figure 2 Information-theoretic analysis of detailed rate data foe the reaction O + CS CO( ) + S. Panel (a) compares P(f ), the observed distributUm over CO vibrational states from the exothermic reaction, > with P°(f/) the ittstrStution expected on prior grounds. Panel (b) shows the surprisal associated with these CO(<0 concentrations as a function of f. In panel (c), the partially detailed rate constants for the endothermic reaction from selected CO v rational levels, calculated from detailed balance, are plotted against e >jhTfor T — 300 K...

To consider reaction out of specified mctant states it is necessary to modify slightly the definition of surinisal that was given in equation (28). Kaplan, Levine, and Manz consider the partially detailed rate constants for reaction from specified vibrational levels, the rotationaland translational d rees of freedom having thermal distributions defined by T. Now the (vibrational) surprisal is... 

The Dynamics of ElectronicaUy Adiabatic Collisions.— There are three parts to a detailed rate theory of processes occurring in electronically adiabatic collisions. First, the potential describing the molecular interaction must be calculated or estimated. Secondly, the equations of motion have to be solved for individual, fully specified, collisions. Finally, the results of calculations on single collisions must be averaged correctly to yield the required result for example, a reactive cross-section or a detailed rate constant. The procedures for the third stage were outlined in Section 2. In the forward direction, i.e. from o(n ln 6) to ic(T), this averaging presents no problems, but it is the difficulty of reversing this process which makes it impossible to obtain detailed information about the collision dynamics or potential from experimental measurements of thermal rate constants. 

The data in Table 2 emphasize that the rate constants associated with the reactive and inelastic processes removing molecules from excited vibrational levels may increase rapidly with v if the system is potentially reactive. This has an important consequence. If experimental measurements are made under conditions where V-V energy exchange is much faster than the overall relaxation of vibrational energy, only the total loss of vibrational quanta is observed. In a system, such as H + H2 (v), where chemical reaction caimot be distinguished, the rate of loss of quanta is equal to YT, (v -M, where is the detailed rate constant for " ... 

In the first part of Section 2, it was shown that detailed rate constants for reactions proceeding in opposite directions can be related quantitatively by application of the principle of microscopic rev bility. This provides a powerful method for deriving rate constants for endothermic reactions, detailed as regards reactant states - i.e. Ar( n T) - when the product state distribution from the reverse, exothermic reaction has been measured. In addition, these relationships... 

Halperin and Alexander extended the theory of the Aniansson and Wall approach to calculate the detailed rate constants and the associated activation energies for polymeric materials, i.e., block copolymer micelles. We briefly review the central results in the following section. 

For such reactions no single well-defined transition state exists. In order to estimate a thermal rate constant it is necessary to adopt a microcanonical approach in which detailed rate constants, k(E,J), are caJculated for the formation of complexes of given total energy and total angular momentum. Values of k(E,J) are then averaged... 

TABLE 2. Measured relative vibrational occupations N (v) for the reaction F + CF3I at two collision enet es Eg. The number densities have been converted into detailed rate constants k (v) by multiplying by the corresponding laboratory velocities. 

If one converts the measured number densities into detailed rate constants, assuming forward-backward symmetry, this effect becomes even more pronounced (Table 2). Summarizing the data we find that a total of 42% of the excess reagent translation is channelled into product vibration (Table 3). A more detailed analysis of the measured internal product state distributions will be given elsewhere [15]. 

FIGURE 2. Plot of detailed rate constants, ky, for partitioning of vibrational energy in IF(X) formed from F + CF3I. Data taken from Ref. 30. 

Here Zc u> Ub u ( teui2/6 ) the stoichiometric coefficients Zc>, Yv (Zg, Yi) are the chemicEil symbols of the reactants (products) involved in the chemical reaction firom adsorbed (Z) and gaseous (V) phases index u enumerates reactions resulting in the production of c component and are the detailed rate constants of direct and reverse reactions. Balance equation for the operator X for reactions of the type (6.1.17) reads... 

At last, for the detailed rate constants in (6.1.18), (6.1.19) analogous representations may be used with an eye to the nature of the adsorption cells... 

Figure 7. Contour plot of the detailed rate constants for formation of HF in a given v,J state for the F -f HCl reaction. The 0.5 contour has been omitted for sake of clarity the dotted contour is the 0.05 value. Based upon a sur-prisal analysis, the relative population of v = 0 is 0.10.

Figure 8. Contour plot of the detailed rate constants for formation of HF in given vj states for the F + (CHs)iO reaction, from (32). The failure of HF%3 to acquire energy up to the thermochemical limit is attributed to the presence of the CHgOCHg stabilization energy which is retained in the radical. This stabilization energy displaces the contour plot from the dotted...

In this paper we have presented detailed rate constant calculations based on an lOS treatment of the collision dynamics. The systems considered included the H + H2 and F + H2 reactions. The former was considered because of the availability of accurate converged CC dynamical results for the initial reactant state Vj =jj =0 and the PK2 potential surface. In addition, extensive QCT studies of both systems have been made over the past two decades. Some conclusions based on this study are as follows. 

The potential economic importance of photochemical isotope enrichment has led to it being fully and frequently reviewed in recent years. " Here, the object is simply to illustrate how basic research into state-selected kinetics is closely related to the successful application of photochemical methods of isotope enrichment. Not only can results from fundamental studies assist in assessing the viability of any particular enrichment scheme, but also both kinds of experiment face similar difficulties in regard to processes which interfere with their primary objective, whether it be the measurement of a detailed rate constant or the attainment of a high enrichment factor. 

Detailed rate constants for reactions between species in defined v, J states, but occurring in collisions with a thermal spread of energies (or relative translational velocities), are derived from detailed rate coefficients by carrying out an integration such as... 

In many experiments only partially detailed information is obtained. For example, most experimental studies of energy disposal (see Chapter 2) measure the distribution of reaction products over v" or v J but the reagents are thermally equilibrated there is no initial-state selection. If rovibrational distributions are determined in such experiments, the appropriate partially detailed rate constants are defined by... 

Examples of the application of equations (1.20), (1.23), and (1.24) are given later. However, it may be useful at this stage to consider what these equations predict for the ratio of two detailed rate constants. For reaction out of two neighboring vibrational levels, i 4- 1 and v ... 

Relative values of k(v T)—or of k(v J T)—can only be determined for those product levels for which < ev=o- To discover the detailed rate constants for values of v or v with energies greater than ,=0 requires direct experiments involving selective excitation. 

Recently, Pollak has demonstrated that the linearity of vibrational surprisal plots is consistent with an exponential gap law for the detailed rate constants, k v v T), of the form ... 

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