Number of Three-Body Collisions

The rate of a termolecular reaction is controlled by the three-body collision frequency Zabc- This frequency is defined as the number of two-body collisions Zx.YZ for each particle X (X = A, B, C) with an unstable complex YZ formed from other particles (YZ = BC, AC, AB) [443]. 

On the other hand, nyz is related to ny, nz, to the two-body collision frequency Z z and the mean lifetime Tyz of the YZ complex by 

With collision frequencies calculated for rigid spheres (Eq. 13.1) we get 

To estimate roughly the order of magnitude of Zg let all masses, lifetimes and collision radii be equal (mx = M, rx = ryz = d, Txy = t). Then 

In the simple case of atomic species the lifetime of a collision complex must be equal in order of magnitude to the time of one species residence near the other in the process of 

---

You can see that Yp 1 if p > 3. This indicates that many-body collisions are really rare. Even the number of three-body collisions in a swollen coil is of order 1. So they cannot seriously affect the conformation of the coil. In contrast, the number of simultaneous pair collisions is about This... 

Fig. 4.18 Vapor pressure ps of argon and local total pressure pioc as a function of normalized distance z =zld from the nozzle in units of the nozzle diameter d for different stagnation pressures po in the reservoir. Condensation can take place in the hatched areas. The numbers of three-body collisions at the points where Ps = Pioc are also given [419]...

The number of three-body collisions is far smaller than the number of two-body collisions in a dilute gas. Consider three-body collisions in a sample of pure argon at 1.000 bar and 300 K. Assume that a three-body collision occurs when a third body collides with a pair of molecules in the act of colliding,... 

In all of these expressions the order appears to be related to the number of molecules involved in the original collision which brings about the chemical change. For instance, it is clear that the bimolecular reaction involves the collision between two reactant molecules, which leads to the formation of product species, but the interpretation of the first and third-order reactions cannot be so simple, since the absence of the role of collisions in the first order, and the rare occurrence of three-body collisions are implied. 

Unfortunately, we cannot do this, so we have to use another approach. There are N particles all together. The probability that each particle has a close partner is (f>. This is why the number of pair collisions is of order N4>- In the same way, the number of three-body colhsions is roughly AT, and so on. In general, the munber Yp of p-body collisions can be estimated... 

Thus, at atmospheric pressure and usual temperatures, the number of gas-kinetic three-body collisions is by two orders of magnitude lower than the number of two-body collisions [443]. However, it will be borne in mind that the above number of gas-kinetic three-body collisions was calculated under the assumption that T is of the order of 10" s as is usual for quasimolecules formed from atoms or simplest molecules. However, as a result of the increase in internal degrees of freedom with increasing number of atoms and molecules, the lifetime of a polyatomic quasimolecule can be by many orders higher than 10" s. Obviously, both Z3 and the ratio of three-body collisions will increase by the same factor. 

A termolecular elementary process in the gas phase involves a three-body collision, which we picture as a collision of a third particle with a pair of molecules that is undergoing a two-body collision. The number of three-body colUsions is proportional to the number of such pairs and is also proportional to the number of third particles. The rate of three-body collisions of a single substance is therefore proportional to the third power of the number density. The rate of three-body collisions of two particles of type 1 and a particle of type 2 is proportional to and so on. If we again... 

While two-body collisions are common in the gas phase, three-body collisions are much less probable and four-body collisions can essentially be ignored because of their low probability. Thus the majority of the reactions we deal with in the atmosphere are bimolecular, with a lesser number being termolecular or unimolecular. 

Inputting solid particles at fixed positions, of different sizes simulates a solid phase in the fluid lattice (Fig. 4). The number of fluid particles per node and their interaction law (collisions) affect the physical properties of real fluid such as viscosity. Particle movements are divided into the so called propagation step (spatial shift) and collisions. Not all particles take part in the collisions. It strongly depends on their current positions on the lattice in a certain LGA time step. In order to avoid an additional spurious conservation law [13], a minimum of two- and three-body collisions (FHP1 rule) is necessary to conserve mass and momentum along each lattice line. Collision rules FHP2 (22 collisions) and FHP5 (12 collisions) have been used for most of the previous analyses [1],[2],[14], since the reproduction of moisture flow in capillaries, in comparison to the results from NMR tests [3], is then the most realistic. 

ZAA and ZAB are the total number of A A or AB collisions per time and volume in a system containing only A molecules, or containing two types of molecules A and B. Three-body collisions can be treated in a similar way. 

The CREAM theory is complicated because of the number of states involved, but the physical principles are straightforward. First it is assumed that the Langevin limit applies such that atoms A and B are more massive than those of bath M. Thus a collision of M with, say, A will not change the momentum of A significantly, but may result in a reorientation of the angular momentum such that its orientation relative to the AB axis is changed, that is, the electronic state is altered. Second it is assumed that a three-body collision involved the interaction of M with only one of the atoms, say A. The... 

---