Collision theory integration

Miller W H 1970 Semiclassical theory of atom-diatom collisions path integrals and the classical S matrix J. Chem. Phys. 53 1949-59... 

In chemical kinetics, it is often important to know the proportion of particles with a velocity that exceeds a selected velocity v. According to collision theories of chemical kinetics, particles with a speed in excess of v are energetic enough to react and those with a speed less than v are not. The probability of finding a particle with a speed from 0 to v is the integral of the distribution function over that interval... 

The exact kinetic theory of dilute gases leads to expressions for transport coefficients in terms of certain quantities called collision integrals, which depend on the dynamics of binary intermolecular collisions. These integrals are defined in this section. 

Time The Integrated Rate Equation 16-5 Collision Theory of Reaction Rates 16-6 Transition State Theory 16-7 Reaction Mechanisms and the Rate-Law Expression... 

As for the rotational amplitude it may be calculated in the framework of the formalism of path integration in action-angle variables, developed in molecule collision theory (Bogdanov et J. 1989). For plane rotor the action-angle variables coincide with the angular momentum J and the orientation angle 0, and the rotational amplitude is... 

Rate constants for chemical reactions can be obtained theoretically by various methods. We shall only mention two here, namely the collision theoretical approach and the approximate transition state theory (TST). In the collision theory the reaction rate constant is obtained by integrating over a Boltzmann-weighted flux of particles hitting a target. In gas-phase dynamics, the target is another molecule in the gas-phase in surface dynamics it is either the surface itself (sticking, dissociation) or an adsorbed molecule. In any case, the rate constant is obtained as... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

Rate constant, rate law, concentration profile, experimental measurement, integrated rate laws, linear plots, half-lifes Theory of the rate constant (activation energy, orientation factor, collision frequency factor, Transition State Theory)... 

Spectral moments. For the analysis of collision-induced spectra and the comparison with theory, certain integrals of the spectra, the spectral moments, are of interest. Specifically, we define the nth moment of the spectral function, g(v), by... 

According to theory, most collision-induced absorption spectra should not only consist of contributions of the free-state to free-state transitions typical of collisional pairs, but also of contributions arising from bound-to-free and bound-to-bound transitions involving van der Waals molecules. In the rotovibrational spectra such dimer bands have been known for some time, but in CIA studies of the rototranslational band, where path lengths have generally been limited to a few meters, dimer features have been seen only recently [268], The dimer spectra are an integral part of the interaction-induced absorption. 

The theory of collision-induced absorption developed by van Kranendonk and coworkers [405] and other authors [288, 289, 81, 126, 125] has emphasized spectral moments (sum formulae) of low order. These are given in closed form by relatively simple expressions which are readily evaluated. Moments can also be obtained from spectroscopic measurements by integrations over the profile so that theory and measurement may be compared. A high degree of understanding of the observations could thus be achieved at a fundamental level. Moments characterize spectral profiles in important ways. The zeroth and first moments, for example, represent in essence total intensity and mean width, the most striking parameters of a spectral profile. 

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

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