Effective collision time

In the derivation of the Boltzmann equation it is assumed that the distribution function changes only in consequence of completed collisions, i.e., the effect of partial collisions is neglected. We shall, therefore, consider the single-particle distribution function averaged23 over a time r, which will (later) be taken large compared with a collision time ... 

The collisions that take place at the times x represent the effects of many real collisions in the system.1 These effective collisions are carried out as follows.2 The volume V is divided into Nc cells labeled by cell indices Each cell is assigned at random a rotation operator 6v chosen from a set Q of rotation operators. The center of mass velocity of the particles in cell , is Vj = AT1 JTJj v where is the instantaneous number of particles in the cell. The postcollision velocities of the particles in the cell are then given by... 

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. 

Cross-sections for reactive scattering may exhibit a structure due to resonance or to other dynamical effects such as interference or threshold phenomenon. It is useful to have techniques that can identify resonance behavior in a system and distinguish it from other sorts of dynamics. Since resonance is associated with dynamical trapping, the concept of the collision time delay proves quite useful in this regard. Of course since collision time delay for chemical reactions is typically in the sub-picosecond domain, this approach is, at present, only useful in analyzing theoretical scattering results. Nevertheless, time delay is a valuable tool for the theoretical identification of reactive resonances. 

Twenty years ago, Bogolubov3 developed a method of generalizing the Boltzmann equation for moderately dense gases. His idea was that if one starts with a gas in a given initial state, its evolution is at first determined by the initial conditions. After a lapse of time—of the order of several collision times—the system reaches a state of quasi-equilibrium which does not depend on the initial conditions and in which the w-particle distribution functions (n > 2) depend on the time only through the one-particle distribution function. With these simple statements Bogolubov derived a Boltzmann equation taking into account delocalization effects due to the finite radius of the particles, and he also established the formal relations that the n-particle distribution function has to obey. 

Filippi and coworkers discovery of tightly-bound, isomeric rj -type complexes on the 2-butyl ion/toluene PES put into question this widely accepted model." Indeed, their occurrence provide a rationale for the significant pressure effect on the isomeric product pattern observed in the gas-phase sec-butylation of toluene T = 24 °C). ° This pressure effect can be explained only by acknowledging the intermediacy of isomeric 17 -type complexes with lifetimes comparable with their collision time with the bulk gas at 70-710 torr (5 X 10 ° 5 X 10 s). 

A higher concentration of reactants leads to more effective collisions per unit time, which leads to an increasing reaction rate (except for zero order reactions). Similarly, a higher concentration of products tends to be associated with a lower reaction rate. ... 

The rate of a reaction is proportional to the effective collisions in a unit of time. If the number of effective collisions increases, so does the rate of reaction. 

The loss term contains contributions from all possible collisions that can deflect molecule i during the time dt. With reference to Fig. 12.13, any molecule j that arrives within the effective collision distance A is assumed to contribute to the loss term. Molecules j approaching i with relative velocity... 

Concentration. In most reactions, the rate increases when the concentration of either or both reactants is increased. This is understandable on the basis of the collision theory. If we double the concentration of one reactant, it will collide in each second twice as many times with the second reactant as before. Since the rate of reaction depends on the number of effective collisions per second, the rate is doubled (Fig. 20.2). 

The exchange of energy between an oscillator and a simple molecule was first analyzed from a classical viewpoint by I andau and Teller, who showed that, for a very slow collision, the net inelastic transfer is zero. This can be seen intuitively by considering the behavior of an infinitesimal and nearly constant force applied to one atom of a vibrating molecule. On one half cycle when the force and motion are in phase there will be an increase in momentum and kinetic energy of this atom which will be almost precisely compensated in the next half cycle by a decrease in momentum and kinetic energy. Closer analysis shows that the net effect of such a force over a cycle is to slowly accelerate the entire oscillator but not to excite it. The probability of inelastic transfer increases with the hard-ness of the collision. This latter is measured by the ratio of the time of a vibration to the collision time, rtr/rcoii = Vnl Tva, where intermolecular forces/ v is the oscillator frequency, and Vr is the relative collision velocity. 

If the attacking species X has a partial pressure [X] the number of effective collisions with the particle surface in unit time will be kA[X],... 

Breckenridge and Umemoto provided sample calculations based on the reaction of Mg( Pi) with H2 to demonstrate that the effective delay time for secondary collisions of the MgH product is different from the delay time... 

The process of ion scattering is illustrated schematically in Figure 5. Because collision times are very short (10 to 10 s), the interactions can be approximated as elastic binary collisions [28] between the incident ion and a single surface atom (i.e., with an effective mass equal to the atomic mass). Diffraction effects are negligible. The basic equation in ISS, using energy and momentum conservation, is... 

Under their conditions more than 80% of the decomposition was effected by wavelengths between 170 and 140 nm. Absorption by ground state C2O was only observed after addition of CO to the photolysis mixture. This is presumably due to formation of C2O in a three-body recombination process. Through spectroscopic comparison of the yield of CO and the consumption of C3O2 at times shorter than collision times, Braun et al. found that 2 molecules of CO were formed for each C3O2 removed. This indicates that primary process 2 dominates process 1 in this wavelength interval. An upper limit of 25% is placed on the yield of C2O. The dependence of carbon atom yields on flash intensity suggest that C( P) and C( D) are formed in a primary photolytic process, while C( S) is the result of some secondary reaction. The relative yields of C( P), C( D) and C( S) were determined to be 1.00, 0.25, <0.025, respectively. 

In the experiments of our concern, the otherwise inert carrier gas usually contains a minor chemical active component. To estimate the rate of chemical interaction, we should take the concentration of the reagent for m, which is still by many orders of magnitude larger than n. Due to this, the reaction is formally of the first kinetic order, and a useful quantity is the mean time of interaction rr. It is obviously the reciprocal of the rate of effective collisions, the above ncz, multiplied by the Boltzmann factor due to eea - the energy of activation. It yields ... 

The fundamental notion of the collision theory of reaction rates is that for a reaction to occur, molecules, atoms, or ions must first collide. Increased concentrations of reacting species result in greater numbers of collisions per unit time. However, not all collisions result in reaction that is, not all collisions are effective collisions. For a collision to be effective, the reacting species must (1) possess at least a certain minimum energy necessary to rearrange outer electrons in breaking bonds and forming new ones and (2) have the proper orientations toward one another at the time of collision. 

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