Crude collision theory

I). The colliding molecules are supposed to behave as rigid spheres, with no internal degrees of freedom. This is the well-known approximation called the crude collision theory. ... 

As a conclusion of this section we may say that we know for sure that the crude collision theory can be much poorer than we thought, but improvements of the molecular models may improve it considerably. However this is obviously difficult to do for complex systems, where the activated state method may be sometimes easier to apply. 

The collision theory of reaction rates dealt with earlier gives a useful, even if a crude, picture of reaction rates and permits us to calculate the rates of reactions between simple molecules when the activation energies are known. However, this theory leaves much to be desired. It does not furnish a method of calculating activation energies theoretically. It provides no information on the details of reactive collisions. It also does not account for the role that the internal energy might play in the reaction. 

If we use the crude method of estimating partition functions given in Sec. XII.5, we find an expected value of about 10 (liter/mole) for the ratio of the partition functions, and by combining this with a mean value of 10 sec for cither V or kT/hy we see that the preexponential factors are about 10 liters/mole -sec for either of these latter theories, in good agreement with the collision theory and the actual data. 

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. 

We have used here Eq. (10.131) for Pe., and Mr Vrpy When these expressions are compared with (10.38) and (10.39), it is seen that the NR approximation for leads rather easily to a statement in the standard form. Note, however, that because of the flat flux assumption the present derivation does not contain the resonance disadvantage factor fr. This quantity is customarily computed using a one-velocity model to represent the entire fast-neutron population. It is well recognized that this point of view is crude and somewhat unclear. On the one hand, when used with the NR approximation, it may be observed that only one collision is required to remove a resonance neutron from the vicinity of a resonance thus the spatial distribution would be very nearly uniform and isotropic. On the other hand, if the NRIA approximation is valid, then the absorptions are necessarily very strong and the use of a disadvantage factor based on diffusion theory is not well justified. For these reasons it has been omitted in this treatment as an unwarranted refinement not in keeping with the precision of the over-all model. ... 

Equation 5 represents a good approximation for situations in which momentum relaxation takes place considerably faster than nonthermal reaction. The local equilibrium model becomes increasingly inadequate as these rates approach one another, so that the present form of the steady state theory will be least accurate for systems that involve very rapid reactions. Higher order Chapman-Enskog solutions of the Boltzmann equation, which provide successive degrees of refinement, could be incorporated into the theory. Such modifications would introduce additional mathematical structure in Eq. 5, which is probably not needed except for the description of systems that closely approach true steady state behavior. This does not occur for any of the cases of present Interest (vide infra) or. Indeed, for any known nuclear recoil reaction system. For this fundamental reason and also because of the crude level of approximation Involved in our treatment of nonreactive collisions, the further refinement of Eq, 3 has not yet been considered to be worthwhile. 

The Drude model is a crude model, but it contains the accepted mechanism for electrical resistance in solids, which is the effect of collisions with the cores of the crystal. There are a number of more sophisticated theories than the Drude theory. However, the results of these theories are similar in their general form to Eq. (28.4-9). The major differences are in the interpretation of the quantities r, and m One problem with the Drude theory is that the conductivities of most common metals are found experimentally to be approximately inversely proportional to the temperature, instead of being inversely proportional to the square root of the temperature, as in Eq. (28.4-11). One can rationalize this by arguing that the mean free path should decrease as the temperature rises, because of the increased vibrational amplitude of the cores, making them into targets with larger effective sizes at higher temperature. 

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