Collisions model

Let us continue with the atom-diatom collinear collision model, this time allowing for the possibility of the reaction A -r BC —> AB -i- C. We first introduce mass-scaled coordinates, as these are especially convenient to describe rearrangements, using... 

It is clear by comparing the uansition state theory with the collision model that the conesponding entropy of activation can be calculated from the value... 

In order to describe the incoherent exchange, in the pulse collision model [Lynden-Bell 1964 Johnson 1967 Stunzes and Benderskii 1971] an effective exchange term, whose real part is... 

The collision model of reaction rates just developed can be made quantitative. We can say that the rate constant for a reaction, k, is a product of three factors ... 

Substituting this expression for f into the equation written above for k, we obtain the basic equation for the collision model ... 

The transition-state model is generally somewhat more accurate than the collision model (at least with p = 1). Another advantage is that it explains why the activation energy is ordinarily much smaller than the bond enthalpies in the reactant molecules. Consider, for example, the reaction... 

You will recall from Section 11.4 that the collision model yields the following expression for the rate constant ... 

Reaction quotient (Q) An expression with the same form as Kbut involving arbitrary rather than equilibrium partial pressures, 333-334 Reaction rate The ratio of the change in concentration of a species divided by the time interval over which the change occurs, 285 catalysis for, 305-307 collision model, 298-300 concentration and, 287-292,314q constant, 288 enzymes, 306-307 egression, 288... 

At higher pressures only Raman spectroscopy data are available. Because the rotational structure is smoothed, either quantum theory or classical theory may be used. At a mixture pressure above 10 atm the spectra of CO and N2 obtained in [230] were well described classically (Fig. 5.11). For the lowest densities (10-15 amagat) the band contours have a characteristic asymmetric shape. The asymmetry disappears at higher pressures when the contour is sufficiently narrowed. The decrease of width with 1/tj measured in [230] by NMR is closer to the strong collision model in the case of CO and to the weak collision model in the case of N2. This conclusion was confirmed in [215] by presenting the results in universal coordinates of Fig. 5.12. It is also seen that both systems are still far away from the fast modulation (perturbation theory) limit where the upper and lower borders established by alternative models merge into a universal curve independent of collision strength. 

The simple fitting procedure is especially useful in the case of sophisticated nonlinear spectroscopy such as time domain CARS [238]. The very rough though popular strong collision model is often used in an attempt to reproduce the shape of pulse response in CARS [239]. Even if it is successful, information obtained in this way is not useful. When the fitting law is used instead, both the finite strength of collisions and their adiabaticity are properly taken into account. A comparison of... 

Fig. 6.7. The first-order (curve 1), second-order (curve 2) and third-order (curve 3) approximations to the exact dependence x x ) in the strong collision model (curve 4).

Petrunina E. B., Romanov V. P., Soloviov V. A. The computation of the relaxation times in liquid in bimolecular collisions model, Acoustic Journal, 21, 782-8 (1975) [in Russian]. 

Rahn L. A., Palmer R. E., Koszykowski M. L., Greenhalgh D. A. Comparison of rotationally inelastic collision models for Q-branch Raman spectra of N2, Chem. Phys. Lett. 133, 513-6 (1987). 

Bulgakov Yu. I., Storozhev A. V., Strekalov M. L. Comparison and analysis of rotationally inelastic collision models describing the Q-branch collapse at high density, Chem. Phys. 177, 145-55 (1993). 

Q-branch rotational structure 179-82 spectra of nitrogen in argon 180 spectral collapse theory 150 spectral width 107 strong collision model 188 cumulant expansions 85-91... 

Maxwellian distribution 129 infinite-order sudden (IOS) approximation 155-6 semi-classical calculation 136-7 Sack s model rotational relaxation 19 strong collision model 219 scattering see isotropic scattering spectra ... 

Now that we have a model, we must check its consistency with various experiments. Sometimes such inconsistencies result in the complete rejection of a model. More often, they indicate that we need to refine the model. In the present case, the results of careful experiments show that the collision model of reactions is not complete, because the experimental rate constant is normally smaller than predicted by collision theory. We can improve the model by realizing that the relative direction in which the molecules are moving when they collide also might matter. That is, they need to be oriented a certain way relative to each other. For example, the results of experiments of the kind described in Box 13.2 have shown that, in the gas-phase reaction of chlorine atoms with HI molecules, HI + Cl — HC1 I, the Cl atom reacts with the HI molecule only if it approaches from a favorable direction (Fig. 13.28). A dependence on direction is called the steric requirement of the reaction. It is normally taken into account by introducing an empirical factor, P, called the steric factor, and changing Eq. 17 to... 

Here va and va are the stoichiometric coefficients for the reaction. The formulation is easily extended to treat a set of coupled chemical reactions. Reactive MPC dynamics again consists of free streaming and collisions, which take place at discrete times x. We partition the system into cells in order to carry out the reactive multiparticle collisions. The partition of the multicomponent system into collision cells is shown schematically in Fig. 7. In each cell, independently of the other cells, reactive and nonreactive collisions occur at times x. The nonreactive collisions can be carried out as described earlier for multi-component systems. The reactive collisions occur by birth-death stochastic rules. Such rules can be constructed to conserve mass, momentum, and energy. This is especially useful for coupling reactions to fluid flow. The reactive collision model can also be applied to far-from-equilibrium situations, where certain species are held fixed by constraints. In this case conservation laws... 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

A. Malevanets and R. Kapral, Mesoscopic multi-particle collision model for fluid flow and molecular dynamics, in Novel Methods in Soft Matter Simulations, M. Karttunen, I. Vattulainen, and A. Lukkarinen (eds.), Springer-Verlag, Berlin, 2003, p. 113. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

Relationships having the same form as eq 14 can also be written for the enthalpic and entropic contributions to the intrinsic free energy barriers (10). Provided that the reactions are adiabatic and the conventional collision model applies, eq 14 can be written in the familiar form relating the rate constants of electrochemical exchange and homogeneous self-exchange reactions (13) ... 

Bashford D Weaver D. L. and Karplus M. Diffusion-collision model for the folding kinetics of the lambda-repressor operatorbinding domain. J. Biomol. Str. Dyn. (1984) 1 1243-1255. 

Karplus M. and Weaver D. L. Protein-folding dynamics - the diffusion-collision model and experimental-data. Prot. Sri. (1994) 3(4) 650-668. 

CPTs kill cells in S-phase by the mechanism involving the replication fork collision model. It is known that reversible TOPO-1 CPT DNA cleavable complexes are in themselves nonlethal and that collisions with an advancing replication fork cause cell death (8). Several biochemical events follow collision with the replication fork the formation of double-strand breaks, irreversible arrest of the replication fork, and the formation of TOPO-1-linked DNA break at the site of collision. These mechanisms account for the S-phase-specific cytotoxicity seen at relatively low doses of camptothecin. 

Billiard-ball model, see also Elastic-collision model... 

Elastic-inelastic collision model, Szilard-Chalmers reaction and, 1 269 Electrical conduction, in organic superconductors, 29 278-286 Electrical conductivity of chalcogenide halide compounds, 23 331 of Group IB, 23 337-339, 342, 346-349 photoelectric effects, 23 368, 410 semiconductors, 23 368, 390, 395-396, 400-402, 410-412 superconductors, 23 375-377 of graphite intercalation compounds, 23 290, 294, 309-310, 312, 317-318 Electric discharges arc type, 6 146-147 chemical reactions in, 6 189-191 chemical reactions in, 6 143-206... 

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