Collision cross-sections

Instead of assuming hard sphere behaviour, we can refine the collision theory by using cross sections calculated on the basis of microscopic interaction potentials of species A and B during their collision, Before we develop the problem for reactive collisions, we will start by treating the simpler situation of an elastic collision between two bodies subjected to a central force, which only leads to scattering. 

The sign must be chosen to agree with that of the radial velocity that is positive for the species leaving and negative for those approaching. 

For any collision, there will always be a maximum approach distance r for which r(t) has its minimum value. In this classical turning point all the initial kinetic energy is converted into potential energy. It is convenient to define f = 0 for rif) = and The complete classical trajectory can be determined by integrating eq. (5.26) from infinity to r and then from r forward. However, it is unnecessary to follow the trajectory 

For us to analyse the behaviour of the angle of deflection as a function of the energy and of the impact parameter, we have to define the potential. In the simplest case of the potential of hard spheres 

A potential energy function that represents the interaction between two bodies in a more realistic form is the Lennard-Jones potential 

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This is the desired result. It shows that the mean free path is mversely proportional to the density and the collision cross section. This is a physically sensible result, and could have been obtained by dimensional... 

We start from a model in which collision cross sections or rate constants for energy transfer are compared with a reference quantity such as average Leimard-Jones collision cross sections or the usually cited Leimard-Jones collision frequencies [54]... 

Equation B2.5.48 introduces the effective average collision cross section Here, the lifetime broadening results from the (coiiisionai) perttirbation of A by collisions with M. 

This expression corresponds to the Arrhenius equation with an exponential dependence on the tlireshold energy and the temperature T. The factor in front of the exponential function contains the collision cross section and implicitly also the mean velocity of the electrons. 

Collision cross section a Debye circular wave-number ... 

Figure 10. Comparison of the velocity dependence of the disappearance cross-section of CHa+, formation cross-section of CH0 +, and Langevin orbiting collision cross-section, all as a function of reciprocal average kinetic energy of ions in the mass spectrometer source...

Top typical saturation curve and variation of mean electron energy with applied field. Middle fraction of the electron swarm exceeding the specific energy at each field strength. Calculated assuming constant collision cross-section and Maxwell-Boltzman distribution. Bottom variation of products typical of involvement of ionic precursors (methane) and excited intermediates (ethane) with applied field strength... 

A low ion pair yield of products resulting from hydride transfer reactions is also noted when the additive molecules are unsaturated. Table I indicates, however, that hydride transfer reactions between alkyl ions and olefins do occur to some extent. The reduced yield can be accounted for by the occurrence of two additional reactions between alkyl ions and unsaturated hydrocarbon molecules—namely, proton transfer and condensation reactions, both of which will be discussed later. The total reaction rate of an ion with an olefin is much higher than reaction with a saturated molecule of comparable size. For example, the propyl ion reacts with cyclopentene and cyclohexene at rates which are, respectively, 3.05 and 3.07 times greater than the rate of hydride transfer with cyclobutane. This observation can probably be accounted for by a higher collision cross-section and /or a transmission coefficient for reaction which is close to unity. 

Some of the rate constants discussed above are summarized in Table VI. The uncertainties (often very large) in these rate constants have already been indicated. Most of the rate constants have preexponential factors somewhat greater than the corresponding factors for neutral species reactions, which agrees with theory. At 2000°K. for two molecules each of mass 20 atomic units and a collision cross-section of 15 A2, simple bimolecular collision theory gives a pre-exponential factor of 3 X 10-10 cm.3 molecule-1 sec.-1... 

However, other data such as the small difference observed in the Si NMR chemical shift (0.9 ppm upheld from TgPhg) and the absence of any measurable Si-F coupling show that the interaction between the huoride ion and the silicon atoms is small. Studies to evaluate the collision cross section of TgPhg using Na show that the cation attaches itself to the outside of the POSS cage and does not significantly distort the structure. 

Based on the molecular collision cross-section, a particle might undergo a collision with another particle in the same cell. In a probabilistic process collision partners are determined and velocity vectors are updated according to the collision cross-section. Typically, simple parametrizations of the cross-section such as the hard-sphere model for monoatomic gases are used. 

Figure 2.2 Non-dimensionalized velocity distribution across a channel for Kn = 0.1 (left) and Kn = 2.0 (right), taken from [2]. The results were obtained by DSCM using two different collision cross-sections and by solution of the linearized Boltzmann equation.

The relaxation of gaseous methane, ethane and propane is by the spin-rotation mechanism and each pure component can be correlated with density and temperature [15]. However, the relaxation rate is also a function of the collision cross section of each component and this must be taken into account for mixtures [16]. This is in contrast to the liquid hydrocarbons and their mixtures that relax by dipole-dipole interactions and thus correlate with the viscosity/temperature ratio. 

If the reverse of Reaction 1 is slow compared to 2 ( the colli sional stabilization step) then overall cluster growth will not depend strongly upon the total helium pressure. This is found to be the case using RRK estimates for k n and hard sphere collision cross sections for ksn for all clusters larger than the tetramer. The absence of a dependence on the total pressure implies that the product of [M] and residence time should govern cluster growth. Therefore, a lower pressure can be compensated for by increasing the residence time (slower flow velocities). 

The kinetic theory of radon progeny attachment to aerosol particles assumes that unattached atoms and aerosol particles undergo random collisions with the gas molecules and with each other. The attachment coefficient, 3(d), is proportional to the mean relative velocities between progeny atoms and particles and to the collision cross section (Raabe, 1968a) ... 

FIGURE 4.5 Inelastic collision cross section of water vapor versus electron energy (LaVerne and Mozumder, 1992). Circles compilation of Hayashi (1989) dashed line unmodified theoretical formula (Pimblott et al., 1990) dot-dashed line theoretical formula scaled to match compilation full curve theoretical formula scaled to match experimental W values. 

According to H. Eyring et al. (1936), the relative rate of the hydride transfer reaction, which is proportional to the collision cross section, should increase monotonically with a my, where a is the polarizability and my is the reduced mass. Using C2D5+, C3D7+, and C4D9+ as ions, Ausloos et al. (1966) have found confirmation of the theory for a number of alkanes and cycloalkanes. 

For the potential given by (5.3), it is easy to show that when b > bc the distance of closest approach is bc /21/2, whereas for b < b, the only thing preventing interpenetration is a repulsive core potential, which is not explicitly considered here. Equation (5.4) is actually the classical collision cross section for the problem. To translate this into a reaction cross section, we may assume that there is another critical separation r0 such that when r < rg chemical forces complete the reaction and no reaction takes place if r > rg. If rg is less than b /2m, then Eq. (5.4) is also the reaction cross section, since reaction definitely takes place if b < b. and it definitely does not take place if b > b.. According to this modification, the high-energy limit of the reaction cross section is nr2 rather than zero as given by (5.4). One therefore has... 

Here fi2 = 1/Lv(t + r), L is the mean free path of radicals at thermal velocity v, and the initial spur radius r0 and the fictitious time T are related by r2 = Lvr. On random scattering, the probability per unit time of any two radicals colliding in volume dv will be ov/dv, where <7 is the collision cross section. The probability of finding these radicals in dv at the same time t is N(N - 1 )p2 dv2, giving the rate of reaction in that volume as crvN(n - 1 )p2 dv. Thus,... 

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