State-specific rate coefficients

So far our discussion has focused on the measurement of thermal rate coefficients for a number of different classes of reactions involved with low temperature combustion. A significant proportion of the kinetics community is interested in looking beyond thermally averaged rate coefficients k T)) towards quantum state specific data the rate coefficient for reagents in the quantum state vj. The two quantities are related via equation (2.71) 

The main goals of dynamical studies are to develop theories for chemical reactions and gain an insight into the molecular mechanisms of elementary reactions. The value of a validated and workable theory for reaction kinetics is obvious, rate coefficients would no longer have to be measured but could be calculated for any set of conditions. More specifically, good models of a specific elementary reaction, validated against dynamical data, allows rate coefficients to be calculated at any temperature. 

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In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, u, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j. Once the reactive cross-sections are known the state specific rate coefficient, can be determined from. 

The approach proposed in the previous section makes it possible to develop the most rigorous model of reacting gas mixtures, since it takes into account the detailed state-to-state vibrational and chemical kinetics in a flow. However, practical implementation of this method leads to serious difficulties. The first important problem encountered in the realization of the state-to-state model is its computational cost. Indeed, the solution of the fluid dynamics equations coupled to the equations of the state-to-state vibrational and chemical kinetics requires numerical simulation of a great number of equations for the vibrational level populations of all molecular species. The second fundamental problem is that experimental and theoretical data on the state-specific rate coefficients and espiecially on the cross sections of inelastic processes are rather scanty. Due to the above problems, simpler models based on quasi-stationary vibrational distributions are rather attractive for practical applications. In quasi-stationary approaches, the vibrational level populations are expressed in terms of a... 

In the zero-order approximation, the multi-temperature rate coefficients of exchange and dissociation reactions can be expressed in terms of the state-specific rate coefficients considered in section 3.2 ... 

In order to extract from such laser induced processes quantitative information such as state populations or reliable state specific rate coefficients, it is mandatory to separate relaxation, reactions and laser induced processes. This is possible by combining the flexible trapping method with short gas pulses, chopped or modulated CW lasers, and pulsed effusive or supersonic beams. 

Fig. 6.15. Photodetachment of electrons from OH trapped at 180 K in a 22-pole. The small loss of ions (time constant 133 s) is significantly increased, if a He-Ne laser is switched on (here at 10s). The photon energy (1.96eV) is sufficient to detach the electron from the anion OH (electron affinity 1.8 eV). The solid lines are exponential fits. Measurements performed at various temperatures of the trap allow state specific rate coefficients to be extracted.

So far we have neglected internal quantum states of the reactants and attributed the temperature dependence of the rate coefficient only to the variation of cross section with relative translational energy. In this section we extend the collision theory concept to state-specific rate coefficients and their thermal average behavior. 

Hence the thermal rate coefficient is the sum of the state-specific rate coefficients weighted by their corresponding Boltzmann factors (Eliason and Hirschfelder, 1959). 

In summary, non-Arrhenius temperature dependence of a thermal rate coefficient occurs if a substantial fraction of the total reactive flux at higher temperatures occurs via excited states, provided that the enhancement of the state-specific rate coefficients is largely an effect of enhancement of preexponential factors. 

Arrhenius graph curvature for this reaction has also been interpreted in terms of state-specific rate coefficients in the collision theory framework. The ratio of rate coefficients for the y = 1 and i = 0 states of H2 was measured to be 2600 at 300 K by Light (1978). By assigning to the i = 1 rate coefficient a temperature dependence derived from an LEPS trajectory study (Johnson and Winter, 1977) a major part of the supplementary reaction over 1000 K was predicted to be due to reaction in the i = 1 channel. The amount of supplementary reaction, however, was not sufficient to account for the data of Schott et al... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Aryl substituents which can release electrons to the allylic system increase the ease of isomerization of a-arylallyl alcohols. Aryl substituent effects support the hypothesis that the rate-limiting transition state has considerable car-bonium-ion character. The specific rate coefficients for acid catalyzed isomerization of a-aryl-y-methylallyl alcohols in aqueous 60% dioxane correlate poorly with Hammett s cr-substituent constants (see ref. 182, p. 184) whereas a plot of log versus Brown s cr -substituent constants is linear, with a slope of 2.7. 

The total recombination rate coefficient fcfec c is defined in terms of the state-specific rate... 

Lifetimes for collision complexes and specific rate coefficients for unimolecular decay of metastable states can be derived in several ways in the framework of the adiabatic channel model, resulting in similar fundamental expressions. The major differences between the various derivations of lifetimes are connected to the physical interpretation. 

In this pressure range the actual populations of excited molecular states are close to the equilibrium populations given by f. The rate coefficient /c2 is therefore the thermal equilibrium average of the specific rate coefficient k(E) of the unimolecular reaction. The rate coefficient for the reverse recombination of a simple bond fission reaction... 

In a fourth step the cross section is related to a state-selected specific bimolecular rate coefficient... 

Two types of interac tion, competition, and predation are so important that worthwhile insight comes from considering mathematical formulations. Assuming that specific growth-rate coefficients are different, no steady state can be reached in a well-mixed continuous culture with both types present because, if one were at steady state with [L = D, the other would have [L unequal to D and a rate of change unequal to zero. The net effect is that the faster-growing type takes over while the other dechnes to zero. In real systems—even those that approximate well-mixed continuous cultures—there may be profound... 

That is, the equilibrium constant (K ) for an elementary reaction at any temperature is related to the ratio of the rates for the forward (fcf) and backward (k ) rate coefficients. The can be determined from the knowledge of the standard state AHf and S of all the species participating in the elementary reaction, and the temperature-dependent specific heats. 

Little attention has so far been paid to studying exchange energy-transfer processes in media so viscous that a steady-state is no longer established. Butler and Pilling [200] specifically sought experimental evidence for time-dependent rate coefficients of the form of eqn. (98). They chose to study triplet phenanthrene in methanol—water mixtures and used cupric chloride as the acceptor since it is readily soluble and a very efficient quencher of triplet phenanthrene. To observe even the t 1/2 dependence of the time-dependent rate coefficient, concentrations [A] > 10-2 are required that is with Re 1 nm and [A] > 10 mmol... 

When potential surfaces are available, quasiclassical trajectory calculations (first introduced by Karplus, et al.496) become possible. Such calculations are the theorist s analogue of experiments and have been quite successful in simulating molecular reactive collisions.497 Opacity functions, excitation functions, and thermally averaged rate coefficients may be computed using such treatments. Since initial conditions may be varied in these calculations, state-to-state cross sections can be obtained, and problems such as vibrational specificity in the energy release of an exoergic reaction and vibrational selectivity in the energy requirement of an endo-... 

Thermodynamic activation parameters calculated from second-order rate coefficients, unless specifically indicated to the contrary, all refer to the molar scale (standard state 1 mole/1 and unit activity). 

If redissociation into reactants is faster than stabilization, equations (3.15) and (3.16) simplify into a product of k,/k, and either kr or kcoll. Under these conditions, to obtain a theory for a total association rate coefficient, one must calculate both k,/k i and kr or kco . Three levels of theory have been proposed to calculate k, /k, . In the simplest theory, one assumes (Herbst 1980 a) that k, /k 3 is given by its thermal equilibrium value. In the next most complicated theory, the thermal equilibrium value is modified to incorporate some of the details of the collision. This approach, which has been called the modified thermal or quasi-thermal treatment, is primarily associated with Bates (1979, 1983 see also Herbst 1980 b). Finally, a theory which takes conservation of angular momentum rigorously into account and is capable of treating reactants in specific quantum states has been proposed. This approach, called the phase space theory, is associated mainly with Bowers and co-workers... 

According to transition-state theory it is possible to consider reaction velocities in terms of a hypothetical equilibrium between reactants and transition state. It follows that the influence of the isotopic composition of the medium on reaction velocity can be considered to be the same as its influence on the concentration of transition states. The kinetic formulation of the problem can thus be replaced by one couched in equilibrium terms, and the equilibrium theory of the preceding section can be applied with a minimum of modification (Kresge, 1964). The rate constant, or catalytic coefficient, (k) for a catalysed reaction can be written as the product of three factors, viz. the equilibrium constant (K ) for the process forming the transition state from the reactants, the transmission coefficient, and the specific rate of transition state decomposition (kT/h). We recognize that the third factor is independent of the isotopic nature of the reaction and assume that there is no isotope effect on the transmission coefficient. It follows that... 

The identification of phenomena that explain the behavior of a studied system depends on the analysis of their kinetic data. Normally, this kinetic analysis is performed using characteristic variables calculated from the experimental data. The specific rates and the yield coefficients are the common values used in this task. When cell concentration data are available, cell growth and death rates, as well as cell viability, are the best kinetic variables to characterize the population physiological state. In the absence of this information - as can occur, for example, with immobilized cells - the treatment must be based on substrate consumption or on metabolites production (Miller and Reddy, 1998). 

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