Collisional rates

Here long-time asymptotic (collisional) rate... 

In CO2 gas, the density-normalized electron mobility /ig fe is independent of temperature (2 X 10 molecule/cm V sec [25]), although the apparent mobility steadily decreases with the pressure free electrons are trapped by neutral (C02) clusters ( = 6) with nearly collisional rates, and the electron affinity of these clusters > 0.9 eV. When extrapolated to solvent densities of (2-15) x 10 cm typical for sc CO2, these estimates suggest that the free electron mobility is ca. 1 cm /V sec and its collision-limited lifetime Xg < 30 fsec [18]. If the lifetime were this short, the electrons would negligibly contribute either to the conductivity or the product formation. However, this extrapolation is not supported by experiment [18,20]. 

Evaluate the Lennard-Jones collisional rate constant, ku at this temperature. How large a correction to the hard-sphere value does this make ... 

The rate of collision is thus rate determining, and the slow forward step will be the collisional rate of activation. The rate of collision is rate determining and the slow forward step is activation, step 1, and... 

During the collision, particle 2 gains a quantity (ir 2 — ir2) of if, where primed and unprimed quantities refer to the values after and before the collision, respectively. Considering only particles about to collide (i.e., taking v 2k> 0), we can express the collisional rate of increase for the mean of if per unit volume as... 

Hence, the collisional rate of dissipation and the collisional stress tensor are related to the coefficient of restitution by Eq. (5.284) and Eq. (5.277). 

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. 

Given knowledge of the atomic parameters B- and A21 and the collisional rates, one can directly relate the observed N2 to the total number density of the species. If calibration is possible, only the temperature dependence of the yield, Y, need be known. 

Use of Saturation. Because of the potential for simplification of the population balance equations, much recent work has concentrated on studying saturation phenomena. First proposed by Piepmeier (9), and elaborated on by Daily (10), saturation in atomic species can lead to complete elimination of the need to know any collisional rates, and in molecular species may provide substantial simplification of the balance equation analysis. 

Saturation in molecular species is more difficult due to syphoning of population to other levels. Thus higher laser powers are required. Baronavski and McDonald (15) have studied the approach to saturation of C2 and suggested means to use the saturation curve to extract collisional rate information. 

Excitation Dynamics. The response of atomic and molecular systems to exciting radiation has long been of interest and work has been going on to understand such phenomena for over one hundred years (18). Recent work has involved the use of lasers and modern detection systems to observe and measure individual radiative and collisional rates. 

The field of unimolecular reaction rates had an interesting history beginning around 1920, when chemists attempted to understand how a unimolecular decomposition N2Os could occur thermally and still be first-order, A — products, even though the collisions which cause the reaction are second-order (A + A— products). The explanation, one may recall, was given by Lindemann [59], i.e., that collisions can produce a vibrationally excited molecule A, which has a finite lifetime and can form either products (A — products), or be deactivated by a collision (A + A— A + A). At sufficiently high pressures of A, such a scheme involving a finite lifetime produces a thermal equilibrium population of this A. The reaction rate is proportional to A, which would then be proportional to A and so the reaction would be first-order. At low pressures, the collisions of A to form A are inadequate to maintain an equilibrium population of A, because of the losses due to reaction. Ultimately, the reaction rate at low pressures was predicted to become the bimolecular collisional rate for formation of A and, hence, second-order. 

The two work-corrected rate constants, k and kj., should be distinguished carefully. The latter quantity arises from the use of the preequilibrium model and describes the rate constant (s ) for electron transfer at a given electrode potential taken wnth respect to the precursor intermediate for the preequilibrium model. The former quantity is an intermolecular (cm s ) rate constant, which can be utilized in conjunction with either the preequilibrium or collisional-rate formulation. In view of Eqs. (a), (n) and (o) ... 

To calculate the collisional rate of change for dense suspensions Gidaspow [22] adopted the dense gas collision operator (4.13), but in this case the particles are inelastic so g2i is related to g2i through the empirical relation (2.123). The kinetic fluxes for dilute suspensions were determined adopting the dilute gas collision operator (4.10), valid for elastic particles, instead. For dense suspensions the kinetic fluxes were approximated by those deduced for dilute suspensions. 

The collisional rate of change Z (V )coiiision of any particle property tjj is the integral over all possible binary collisions of the change in 0 in a particular collision multiplied by the probability frequency of such a collision. Hence, particle 1 gains of the microscopic property tjj during the collision... 

A similar expression for the collisional rate of change for particle 2 can be obtained. In this case we utilize the collision s unmetry properties, so this relation is achieved by interchanging the labels 1 and 2 in (4.15) and replacing k by —k. As distinct from the previous analysis, to determine this probability frequency at the instant of a collision between particles labeled 1 and 2 we now take the center of the second particle to be located at position r and the center of particle 1 to be at r — di2k. This approach represents a collision dynamically identical but statistically different from the previous one [31] [49] [32]. The result is ... 

Each of the integrals equals the collisional rate of change (V )coiiision, but a more symmetric expression of /i(V )Collision might be achieved taking one half of the sum of (4.15) and (4.16) [31, 32]. 

To complete the reformulation of the Boltzmann equation replacing the microscopic particle velocity with the peculiar velocity, the collisional rate of change term has to be modified accordingly. Jenkins and Richman [32] proposed the following approximate formula ... 

Since the rate of stimulated emission can be very much faster than the spontaneous emission rate, the time resolution of chemical laser measurements can be quite high. In a very crude way one might say that the rate of emission under lasing conditions can be deliberately increased by a sufficiently intense stimulating field so as to exceed any other collisional rate in... 

In writing rate equations only for it is tacitly assumed that translational relaxation is instantaneous on the time scale of all the other rate processes. Hence, a well-defined temperature, T, characterizes the translational degrees of freedom of the lasing molecules and all degrees of freedom of the nonlasing species. This heat bath temperature appears as a parameter in the collisional rate constants. It also enters the gain coefficients via the linewidth and in the case of rotational equilibrium mainly via the population inversion. Thus (1) and (2) should be supplemented by a rate equation for T. Additional kinetic equations describe the time dependence of the nonlasing species concentrations. 

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