pT(E) = rho(E) e-E/kBT / Q(T) ,
where rho(E) is the internal density of states of the reactant molecule and Q(T) its corresponding partition function ensuring that pT(E) is properly normalized. An important restriction on the possible form of P(E,E') is that it must satisfy normalization and detailed balance.
Thus, the simplest representation of the energy transfer mechanism is then to set
P(E,E') = pT(E) .
This is the mathematical formulation of the strong collision assumption (SCA) or the canonical SCA in our terminology. Note that it would, interpreted in a physically literal sense, imply that a single collision would thermally equilibrate the reactant molecule irrespective of its initial energy E'. This is at the same time a beautifully simple (an Alexandrian swipe at a Gordonian knot!) and physically implausible assumption for small medium molecules.
There are, however, many problems attached to the canonical SCA in the RRKM theory:
i) How should we obtain the corresponding collision frequency [[omega]]?
ii) Why should we use a canonical equilibrium assumption in the representation of the energy transfer mechanism when the reaction mechanism is described at a more microcanonical level?
iii) How can we ignore the clear evidence that typical collisions, at least if the medium molecule is small, are far from strong?
A. The microcanonical strong collision assumption
In its literal interpretation the canonical strong collision assumption is unphysical in most cases. Its most obvious flaw is that it ignores the contradiction posed by energy conservation and the assumption of complete thermal relaxation of the reactant molecule. By logical equivalence, one would have to assume thermal equilibrium also for the medium molecule after the collision but for a highly excited reactant molecule and a small thermal medium molecule this outcome violates energy conservation. Given the nature of the RRKM theory the logical treatment of the activation-deactivation mechanism is based instead on a microcanonical strong collision assumption, i.e., on the assumption that the energy distribution over the two molecules and their relative motion reflects a relaxation to microcanonical equilibrium at the total energy available to the collision complex. The implementation of this idea will be referred to as the ergodic collision theory (ECT). The energy transfer kernel can then be written as a convolution over the two molecular densities of states (the densities of states of the reactant, medium and uncoupled complex, respectively). It is assumed that the medium is in thermal equilibrium at the temperature T.
The implementation of the ECT is particularly simple in the classical case if we use the usual assumption that the vibrations are harmonic and the rotations separable. The moments of the energy transfer kernel can then be obtained analytically and we find
<[[Delta]]E>E = - F(nM + 2,n + nM + 4) (E - <E>T) ,
<([[Delta]]E)2>E = F((nM + 4)(nM + 2),(n + nM + 6)(n + nM + 4)) (E2 - 2 F(nM + 2,nM + 4) E<E>T - <E2>T) .
Here n and nM are doubled exponents in the power law form of the densities of states, e.g.,
rho(E) = c En/2 .
They can be obtained from the number of atoms N and NM in the reactant and medium molecule, respectively. The result for the first moment tells us that the two molecules divide up the energy in proportion to their molecular complexity which is nearly the same as the number of atoms in each of the molecules. This gives us the insight that energy transfer efficiency is related to the relative molecular sizes. Monatomic medium molecules are inefficient while large medium molecules may even approach the canonical strong collision prediction.
As it turns out, the effective molecular "size" in this context is strongly influenced by quantum effects on the vibrational contributions to the densities of states. If insufficient energy is available to simultaneously excite all vibrations from their ground states then some of the vibrations of the highest frequency will be "frozen", i.e., unable to leave the ground state. If both molecules are "quantum frozen" to a similar extent then the effect on the energy transfer efficiency within the ECT is small. The effect is largest when one of the two molecules is completely classical ( e.g., a monatomic collider) and the other heavily quantized ( e.g., CH4). Although the simple results for <[[Delta]]E> and <([[Delta]]E)2> above are of a classical origin they can be used for real molecules. If the freezing effect is reasonably symmetrical between reactant and collider it is only necessary to evaluate <E>T properly quantum mechanically to get a reasonable estimate of the ECT limit. To get a more accurate estimate one can use the thermodynamic method for the estimation of quantized vibrational densities of states which yields effective molecular sizes and thermal energies (n and <E>T therefore).
The ECT has great advantages over the canonical strong collision assumption but it also has disadvantages. The main ones are that the simple solution for the unimolecular rate constant is lost and the ECT is still an upper bound but not an accurate estimate of energy transfer efficiency. The former problem we shall leave aside here. With respect to accuracy the ECT is about half way between the canonical strong collision assumption and reality. We shall consider now the reasons for remaining errors in a general way returning to them in greater detail in the next section. Recall first our discussion of the limitations of the "hit or miss" model of collisional transfer. This limitation applies also to the ECT. Even if head-on collisions were microcanonically strong there must be glancing collisions which contribute less efficiently to the energy transfer. Thus the very fact that reality produces a range of intermediates between hits and misses would force us to consider weak collision models of energy transfer.
B. Angular momentum conservation
Head on or nearly head on collisions may not be strong (in the ECT sense) either. A particularly obvious reason for this is the conservation of angular momentum. In principle, we should therefore develop an EJ-ECT theory based on an equilibration of products subject to the conservation of both energy and angular momentum. Although this is more difficult than the development of the ECT it could be done. However, the gain in accuracy may not justify the added complication. The effects of angular momentum conservation are not very large. They are largest for small molecules but even for Br2 + Ar and Br2 + Br2 they decrease the magnitude of <[[Delta]]E> by no more than about 20 and 10 percent, respectively, according to our recent investigations. For larger molecules they should be even less. Thus, the use of a model of E,J-strong collisions may not, in practice, constitute a major improvement.
C. Impulsive collisions
The major source of collision inefficiency is likely to be dynamical rather statistical in nature. Note that a strong collision would imply a long lifetime of the collision complex so that energy in the two molecular parts have time to redistribute completely. Little is known as to how long the lifetime must be, but it is clear that real collisions often do not display the long lived complexes required and therefore show weak collision character. An interesting limit to consider is the case when the complex lifetime is zero. In that limit potential energy can't be transferred since the spatial configuration must be conserved in a zero-lifetime collision. Thus, only kinetic energy can be transferred. In the impulsive ECT (or IECT) we furthermore assume that the kinetic energy is complete equilibrated in the products. We can then work out another form of P(E,E') and obtain a slightly modified expression for <[[Delta]]E>,
<[[Delta]]E>E = - F(nKM + 2,nK + nKM + 4) (K - <K>T) , (3-7)
where nK is 3N-5 and nKM is 3N-2. K is the reactant kinetic energy before collision and <K>T its thermal average. The branching ratio for the redistribution of the thermal excess kinetic energy of the reactant generally does not differ greatly from that which applies in the ECT but the amount of energy to be redistributed is down by 50%. Thus, the magnitude of <[[Delta]]E> drops by ~50% when we go from the ECT to the IECT estimate. This is a significant drop bringing us closer, for small molecules, to experimental realities.
Some idea of how well the statistical ECT and IECT theories compare with experiment can be gained from an examination of the collision efficiency [[beta]]c(R|M) of a medium molecule M relative to the reactant molecule R. This quantity is insensitive to the definition of the collision frequency and is readily accessible experimentally. For the ECT
[[beta]]c(R|M) = F(nM + 2,n + nM + 4) F(2nM + 7,n + 5) , (3-8)
and the same formula applies for the IECT with the exponents n, nM replaced by nK, nKM. The dependence of the relative collision efficiency on the size of the medium molecule for methyl isocyanide in the presence of various medium molecules Whereas the strong collision assumption would predict [[beta]]c(R|M) = 1 for all medium molecules regardless of size, it is clear that the ECT and IECT predictions are much closer to the truth. For medium molecules smaller than the reactant the deviation between the theoretical predictions and experimental values is ~25 % . The experimental relative collision efficiencies appear to reach a maximum relative collision efficiency of roughly unity for medium molecules of the size of the reactant molecule or greater whereas the statistical theories predict a value of [[beta]]c(R|M) = 2 in the limit of large medium molecules. It is clear that there are non-statistical effects related to the dynamics of the collision that are not being taken into account.
The IECT also serves to remind us that kinetic energy is easier to transfer than potential energy. Thus we would expect rotational energy to transfer more readily than vibrational energy (despite the constraint that angular momentum conservation may impose) which is generally borne out by experiment and simulation; e.g.. However, we must note that the IECT is not an upper bound on the energy transfer efficiency.
If the collision lifetime is zero one is lead to consider the notion of hard sphere atoms colliding pairwise. If we thus restrict the kinetic energy transfer to occur between two atoms, one on each molecule, then the energy transfer efficiency will be further reduced as has been illustrated for Br2 in an inert gas medium. In the corresponding impulsive collision theory (ICT) the dynamical constraints on energy exchange between hard spheres is included, i.e., the conservation of linear momentum and the conservation of momenta parallel to the hard sphere surface at the point of contact. As a result, we get an estimate of the glancing collision effect and the dependence on the mass ratio of the atoms. A very major reduction of the energy transfer efficiency is found. The results are now close to those observed in molecular dynamics simulations. The ICT may yield an underestimate, e.g., due to the neglect of multiple atom-atom encounters. It may also yield an overestimate since hard encounters are known to be more efficient than soft encounters in many cases.
The impact of impulsive collisions on diatomic dissociation rates in the low pressure limit has also been considered in an extended form of RRKM theory. The effects of angular momentum conservation and the merits of a local impulsive strong collision assumption (ISCA) versus the traditional global strong collision assumption (SCA) were considered. The SCA predictions are not well-defined as they depend on the spatial location of the transition state whereas the ISCA predictions are always well defined.
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