Reaction Cross Section Hard-Sphere Model

Reaction Cross Section Hard-Sphere Model 

Consider a particularly simple model of chemical reaction. The molecules are imagined to be hard spheres of mean diameter a and the reaction has an energy barrier of height Cq. For reaction to occur two conditions must be met the molecules must collide and they must have sufficient collision energy to overcome the barrier. 

Case A All collision energy available for surmounting barrier. In these circumstances the energy dependence of the reaction cross section is particularly simple 

Even at this level of simplification the model cannot be correct. Not all the collision energy is available to overcome the barrier. The fraction of 

A good elementary discussion of statistical thermodynamics is given by T. L. Hill, Introduction to Statistical Thermodynamics (Reading, Mass. Addison-Wesley, 1960). See especially Chapter 3. 

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Reaction Cross Section Hard-Sphere Model... 

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. 

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... 

Some of the defects of the hard-sphere model for the reaction cross section are apparent. It assumes that all the line-of-centers kinetic energy can be used for overcoming the threshold. It does not account for the dependence of Q( , /, j) on the internal states of the reactants. It does not consider any effect due to molecular structure or to the details of the collision process. Improvement requires either direct experimental measurement of g( , Uj) or equivalently, a calculation which takes into account the interaction potential between colliding molecules in specific internal states. 

In Section 8.2 we found that the reaction cross section provides the link between the rate constant and collision dynamics. However, Q , i,J) is just a function of the collision energy and the internal states of the reactants. The outcome of a single collision also depends upon the impact parameter, as demonstrated for the simple hard-sphere model of reaction cross section discussed in Section 8.4. The probability of a specific result p(b, e, i, j) is dependent on all these variables. In Section 8.4 we showed that the relative importance of a collision with impact parameter between b and b + db % proportional to the annular area Inb db. The reaction cross section is found by multiplying the weighting factor for a particular b by the probability of a specific result and integrating over all values of the impact parameter... 

If the reaction cross section, Q e, U j is known, the rate constant for the corresponding chemical reaction can be calculated from (8.6) and (8.11). To interpret the details of molecular beam experiments even more information is needed. The hard-sphere model was obviously far too naive to give reliable estimates of the reaction cross section. Improvement, by considering the dynamics of reaction across a realistic potential-energy surface such as that described in Section 9.2, is a formidable quantum mechanical problem. As already mentioned it has not been solved, except for low-energy H + Hg chemical reaction. 

Koura (l ) has investigated the possible formation of high temperature steady states in the nonthermal + Hg system using a Monte Carlo numerical procedure for solving the time dependent Boltzmann equation. Reactive cross section data reported from this laboratory were employed together with an energy dependent hard sphere model. Time dependent momentum relaxation, reaction rate and yield results were obtained for a variety of assumed initial hot atom momentum distributions. 

Therefore the reaction criterion (3.29) is sometimes formulated as reaction occurs if the kinetic energy along the line of centers of two hard spheres exceeds Eo- The criterion and the cross-section (3.30) is then known as the line of centers model. This interpretation of (3.29) is correct but not essential. We can think of Eo as the value of V(R) atR = d. Equation (3.29) is then the criterion that the motion of the reactants under the potential V(R) can reach the separation R = d where V(d) = Eo and Eo is positive. 

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