Langevin rate coefficient

Ion-Molecular Reactions, Ion-Molecular Polarization Collisions, and the Langevin Rate Coefficient... 

The numerical value of the Langevin rate coefficient for an electron-neutral collision is about 10 cm /s and also does not depend on temperature. 

Incorporation of this attractive potential into a simple collision model, which we shall not present here, allows the prediction of the reaction cross section and, ultimately, the reaction rate coefficient. The Langevin rate coefficient, as it is often known [21], is given by... 

In order to attempt the calculation, the polarizability and permanent dipole moment of acetone are required. Literature values of these quantities are a = 6.4 x 10 m and = 2.88 D [16]. The reduced mass for this system is 2.38 x 10 kg and other quantities required are = e = 1.602 x 10 Cand 0 = 8.854 x 10 C m Thereisnow sufficient information to calculate the Langevin rate coefficient, which yields... 

The experimental rate coelHcient for the reaction of H3O+ with acetone has been determined to be (3.9 1.0) x 10 cm s at 300 K. Clearly the Langevin rate coefficient lies significantly below the experimental rate coefficient, which is expected given the neglect of the ion-permanent dipole interaction. Both the ADO and the trajectory theories do significantly better, with the latter in particular yielding excellent agreement with experiment. 

The simulation of motion of the solvent as a Langevin-type Brownian motion is a most useful device to reduce computing time. Nevertheless, the essential features of the solvent cage are preserved. Allen was particularly interested in the reaction occurring in the low coupling limit, that is where the solvent does not interact with the particle which is transfered, B. However, when there is some interaction, the rate coefficient becomes diffusion-limited and can be described by a Kramers expression quite well (see Chap. 8, Sect. 2.4) [67]. 

Some of the initial work dealt with the formation of proton-bound dimers in simple amines. Those systems were chosen because the only reaction that occurs is clustering. A simple energy transfer mechanism was proposed by Moet-Ner and Field (1975), and RRKM calculations performed by Olmstead et al. (1977) and Jasinski et al. (1979) seemed to fit the data well. Later, phase space theory was applied (Bass et al. 1979). In applying phase space theory, it is usually assumed that the energy transfer mechanism of reaction (2 ) is valid and that the collisional rate coefficients kx and fc can be calculated from Langevin or ADO theory and equilibrium constants. 

Sunner et al. (1989) used a semiempirical treatment to theoretically evaluate the rate coefficients of hydride transfer reaction sec-C3H7 + iso-C4H10 — C3F18 + tert-C4Hg. Their kinetic scheme is based on a loose and excited chemically activated complex (C3H7 C4H10) formed at the Langevin rate. The complex can decompose back to reactants or form the products of the hydride transfer... 

A similar treatment in which the rates themselves can all be treated as random variables has been given by Shore. Here the treatment by Bartlett is used, which allows for the evolution of the coupled system trader the explicit assumption that g = 1 — s. This will not be true in the case of infall but well approximates the evolution of the closed system. The Fokker-Planck equation in this case results from the interpretation of the coupled one-zone evolution equations as both being Langevin equations. The infall characteristics are not specified, nor is it necessary to state at the outset what the variance of the rate coefficients is like. The equation is of the form... 

Fig. 1.82. Relative reaction efficiencies fc/fcc for the conversion of CO to CO2 by PtnOm cluster of various compositions (fc is the measnred reaction rate coefficient and fee represents the Langevin ion-molecnle collision rate coefficient) [22]...

Numerically, the Langevin capture rate coefficient can be presented as... 

The typical valne of the rate coefficient for ion-molecular reactions is 10 cm /s, which is 10 times higher than the gas-kinetic value for neutral particles ko 10 cm /s. The preceding relations describe the interaction of a charge with an induced dipole. If an ion interacts with a molecnle having a permanent dipole moment /xd, the Langevin capture cross section becomes larger (Sn Bowers, 1973,1975) ... 

Most of the exothermic ion—molecular reactions have no activation energy (Talrose, 1952). Quantum-mechanical repulsion between molecules, which provides the activation barrier even in the exothermic reactions of nentrals, can be suppressed by the charge-dipole attraction in the case of ion-molecular reactions. Thus, rate coefficients of the reactions are very high and often correspond to the Langevin relations (2 8)-(2-50), sometimes partially hmitedby quantum-mechanical factors (Su Bowers, 1975 Virin et al., 1978). The efiect obviously can be apphed to both positive and negative ions. 

Comparison of the Thomson and Langevin models shows that recombination rate coefficients grow with pressme at moderate pressures and begin to decrease as... 

Numerically, the maximum recombination rate coefficient is about 1-3 10 cm /s. This coefficient decreases both with growth and with reduction of pressure from the near-atmospheric value (see Fig. 2-13). A generalized ion-ion recombination model, combining the Thomson and Langevin approaches for moderate and high pressures, was developed by Natanson(1959). 

Thomson and Langevin Models of Three-Body Ion-Ion Recombination. Comparing Thomson s and Langevin s models for the three-body ion-ion recombination, (2-93) and (2-97), find a typical value of pressure when the recombination rate coefficient is the highest. Specifically consider oxygen plasma at room gas temperature. 

Note that /cl is independent of the kinetic energy of the reactants. For ion-molecule reactions without an energy barrier, and in the absence of a permanent dipole moment, the Langevin model gives reasonably good estimates of rate coefficients. For example, the rate coefficient for the reaction of Ar ions with H2 is predicted to be 1.5 x 10 cm mole-and the experimental value is reported to be... 

During the ion-dipole interaction, polarization of the neutral must also occur. The reaction rate coefficient may be considered to be composed of contributions from both the ion-dipole and the ion-induced dipole interactions. An estimate of the total rate coefficient may be made as the sum of the Langevin and the ion-dipole rate coefficients. 

FIG U RE 1.7 Total inelastic rate coefficients for collisions of Li with Li2 (n = 1 and 2, with j = 0 for boson dimers and / = 1 for fermion dimers). The dotted line shows the result of Langevin capture theory. (From Critas, M.T. et ah, Phys. Rev. Lett., 94, 033201, 2005. With permission.)... 

FIGURE 3.18 Rate coefficients as functions of the temperature predicted by the Langevin model compared with experimental data for different coUisional processes involving Rb or Cs atoms. The curves correspond to the Langevin results and the symbols denote the experimental results. 

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