Langevin rate constant

Equation (16) represents the locked-dipole capture rate constant. The first term, which at the same time is the high-temperature limit, denotes the well-known Langevin rate constant... 

Fig. 19. Rate constants for the reaction NJ(d) -I- Kr - N2 + Kr as a function of the vibrational quantum number v. Langevin rate constant = 8.1 x 10 " cm s (dashed line). The prediction is based on Franck ondon factors for energy-resonant transitions (solid line), whereas the dotted line is based on Franck-Condon factors assuming a relative shift of 0.02 A of the vibrational wave functions of and N2 (Kato eta/., 1997).

If /C3 and can be taken as Langevin collision rate constants, then to the extent that t (= V3 ) can be taken from (7) (or other, more refined expressions) and p can be assumed constant (or otherwise approximated), one has a theory for three-body association reactions. If and/or deviate from Langevin rate constants, the factor p must absorb this effect also. Variations in third-body efficiency are commonly observed, although not generally very large ones. This establishes that k j) cannot be equal to /cl for the third-body collisions, i.e., either k ki ov p 1. 

As can be seen from Eq. 2.14, classical theoiy predicts the capture cross section should vary inversely as the relative velocity of the coUiding pair and hence the Langevin rate constant, shonld be independent of the relative velocity and the temperature. Equation 2.15 predicts reasonably well the rate constants for ion-molecule reactions involving only ion-indnced dipole interactions. This indicates that a reaction occurs on every collision for many ion/molecule pairs there can be no activation energy for the reaction. The rate constant predicted is of the order of 1X 10 cm molecule s" . 

Predictions of collision rate constants for systems in which a reactive collision of N2H with its partner would result in proton transfer have not been treated in the text. Such collision rate constants obtained using classical theories, such as the Langevin ion-induced dipole theory, the locked-dipole theory, and the average dipole orientation theory, are given, for example, in [1 to 11]. Langevin rate constants for the reactions of N2H with many species, including those for which there are no laboratory data (for example C, S, NH, OH, CH2, NH2, HCO, C2H), are given in [12]. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

We recently received a preprint from Dellago et al. [9] that proposed an algorithm for path sampling, which is based on the Langevin equation (and is therefore in the spirit of approach (A) [8]). They further derive formulas to compute rate constants that are based on correlation functions. Their method of computing rate constants is an alternative approach to the formula for the state conditional probability derived in the present manuscript. 

Hase s trajectory value for the association rate constant, /cp of 1.04 cm- s maybe used in conjunction with the above Langevin value of the collisional stabilization rate constant to yield a unimolecular dissociation rate constant of 3.75 x 10 ° s and a lifetime of 27 ps. In each case, these values are in excellent agreement with the order of magnitude of lifetimes predicted by Hase s calculations for cr/CHjCl collisions at relative translational energies of 1 kcal mor , rotational temperatures of 300 K, and vibrational energies equal to the zero-point energy of the system. 

Pressure dependence analysis. Kofel and McMahon pointed out that if the apparent bimolecular association rate constant is measured as a function of pressure, k and can be obtained from the slope and intercept of the pressure plot, provided that k and k are independently known k is often taken equal to the Langevin or ADO orbiting rate constant k (the strong collision assmnption), and kf is either taken equal to k or is measured independently by high-pressure mass spectrometry. 

In Table 1 (pp. 251-254), IM rate constants for reaction systems that have been measured at both atmospheric pressure and in the HP or LP range are listed. Also provided are the expected IM collision rate constants calculated from either Langevin or ADO theory. (Note that the rate constants of several IM reactions that have been studied at atmospheric pressure" are not included in Table I because these systems have not been studied in the LP or HP ranges.) In general, it is noted that pressure-related differences in these data sets are not usually large. Where significant differences are noted, the suspected causes have been previously discussed in Section IIB. These include the reactions of Hcj and Ne with NO , for which pressure-enhanced reaction rates have been attributed to the onset of a termolecular collision mechanism at atmospheric pressure and the reactions of Atj with NO and Cl with CHjBr , for which pressure-enhanced rate constants have been attributed to the approach of the high-pressure limit of kinetic behavior for these reaction systems. 

In Kramers theory that is based on the Langevin equation with a constant time-independent friction constant, it is found that the rate constant may be written as a product of the result from conventional transition-state theory and a transmission factor. This factor depends on the ratio of the solvent friction (proportional to the solvent viscosity) and the curvature of the potential surface at the transition state. In the high friction limit the transmission factor goes toward zero, and in the low friction limit the transmission factor goes toward one. 

The Langevin equation, Eq. (11.5), that was used in Kramers calculation of the dynamical effects on the rate constant, is only valid in the limit of long times, where an equilibrium situation may be established. The reaction coordinate undergoes many collisions with the atoms in the solvent due to thermal agitation. From the Langevin equation of motion and Eq. (11.9), we obtained an expression for the autocorrelation function of the velocity ... 

The long-time balance between recombination and drift of carriers as expressed by the y/n ratio has been analyzed using a Monte Carlo simulation technique and shown to be independent of disorder [40]. Consequently, the Langevin formalism would be expected to obey recombination in disordered molecular systems as well. However, the time evolution of y is of crucial importance if the ultimate recombination event proceeds on the time scale comparable with that of carrier pair dissociation (Tc/Td l). The recombination rate constant becomes then capture—rather than diffusion-con-trolled, so that Thomson-like model would be more adequate than Langevin-type formalism for the description of the recombination process (cf. Sec. 5.4). 

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