Cross section, hard sphere reaction

Reaction Cross Section Hard-Sphere Model... 

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 pre-exponential factor of a bimolecular reaction is related to the reaction cross-section (see Problem 2.3). A relation that is fairly easy to interpret can be obtained within the framework of transition-state theory. Combining Eqs (6.9) and (6.54), we can write the expression for the rate constant in a form that gives the relation to the (hard-sphere) collision frequency ... 

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.

Although the scattering cross sections that underlie these and values are subject to uncertainty, the above conclusions are believed to be valid for many atom transfer reactions. The replacement of hard sphere by mean elastic intercollision lifetimes corresponding to a realistic potential description of the intermolecular forces, for example, would reinforce the present arguments (4). 

The picture to this point then is the collision of hard spheres given in Figure 2.7, with energies as defined in equation (2-14) and probabilities as given in equation (2-15). How do we turn this into a reaction rate expression that is compatable with all the work done previously to get to equation (2-12) This has been done over the years, following the terminology of the theorists, in terms of yet a new quantity that is generally called the reactive cross-section. For now, let us write down the definition and worry about the physical interpretation later. Thus, we define a reactive cross-section as... 

The simplified-kinetic-theory treatment of reaction rates must be regarded as relatively crude for several reasons. Numerical calculations are usually made in terms of either elastic hard spheres or hard spheres with superposed central attractions or repulsions, although such models of molecular interaction are better known for their mathematical tractability than for their realism. No account is taken of the internal motions of the reactants. The fact that every combination of initial and final states must be characterized by a different reaction cross section is not considered. In fact, the simplified-kinetic-theory treatment is based entirely on classical mechanics. Finally, although reaction cross sections are complicated averages of many inelastic cross sections associated with all possible processes by which reactants in a wide variety of initial states are converted to products in a wide variety of final states, the simplified kinetic theory approximates such cross sections by elastic cross sections appropriate to various transport properties, by cross sections deduced from crystal spacings or thermodynamic properties, or by order-of-magnitude estimates based on scientific experience and intuition. It is apparent, therefore, that the usual collision theory of reaction rates must be considered at best an order-of-magnitude approximation at worst it is an oversimplification that may be in error in principle as well as in detail. 

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)... 

Here, a is the collision cross section, which may be estimated using a simple hard sphere model for colliding particles (Fig. 1). Two particles collide with a relative velocity vector, g, the magnitude of which is denoted by g, and impact parameter b, also known as the aiming error of the collision. A hard sphere collision will occur provided 0 < ft < (r -b fg). The collision cross section is therefore the area of a circle of radius t/as = fa + fn, i.e., Oh.s. = rrd g. The incident flux, 1a = [A]ga, may then be substituted into Eq. (6). If the rate of reaction between A and B is simply the collision rate, then... 

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