Cross section, hard sphere

Hard sphere cross section) X (Mean velocity) X (Arrhenius factor)... 

The probability of transferring energy during the collision process is determined by a hard sphere cross-section, i.e. ... 

The cross-section for collision-induced rotational energy transfer is very large, of the order of magnitude of the hard sphere cross-section or larger. 

Fig. 1. The variation of product cross sections with translational energy in the laboratory frame (upper scale) and the center-of-mass frame (lower scale) for the reaction of He" ( Siy2) with SiH4. The solid line shows the total cross section. The dashed line shows the collision cross section given by the maximum of either the ion-induced dipole (LGS) or the hard sphere cross section. Reprinted with permission from Fisher and Armentrout (1990b). Copyright 1990, American Institute of Physics.

In Equation 7 /ab, Xr, and mixture component (R ), the mole hraction of Ri, and the hard sphere cross section for A vs. R elastic collisions. 

In order to understand the meaning of the reaction cross section, let us remember that the collision number is proportional to a, the mean cross section of the reactants. Just as, by Eq. 49, the total collision number is equal to relative velocity, so the number of collisions between molecules approaching each other with some velocity v is equal to (98a) av. It is now assumed that a differs from the hard-sphere cross section depending, in the simplest case, on the relative velocity v. Thus a becomes a (v) and is called the reaction cross section. This must now be multiplied by the fraction of colliding pairs which have relative velocity in the range v and v -f dv, say/(v), and integrated, to yield k. Thus (98a)... 

Figure 8.12 shows evaluated (ADO) cross section of collisions of Ar+ with H2, CH4 and NH3 as a function of kinetic energy. At low energy, the interaction is defined by charge-dipole interaction and the cross section is many times larger than hard-sphere cross section. Under these conditions, the difference in collision cross sections of atomic and polyatomic ions of the same charge and mass is diminished. 

A simple starting assumption is to view the colliding particles as structureless hard spheres and to assume that reaction occurs when the total collision energy exceeds a critical value E. This hard-sphere cross section is the same for all relative energies greater than E ... 

This line-of-centers cross section increases linearly with E above E and approaches the hard-sphere cross section at high collision energies (Fig. 3). When this function is used in Eq. (3.7) we obtain for the thermal rate coefficient... 

An important refinement of the hard-sphere cross-section formula [Eq. (3.14)] arises when there are long-range attractive interactions. For reaction without barrier this interaction can be described by a spherically symmetric potential of the form... 

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