In his discourse on the theory of unimolecular reactions, Bunker associated the onset of weak collision effects with the breakdown of either the strong collision or the randomisation assumptions [66.B2]. We have already constructed (in Chapter 7) an apparatus for examining the latter problem, and we will now attempt to inspect the problem of departures from the strong collision transition rates. By introducing the concept of a generalised strong collision, Nordholm and I have rendered the problem of detecting such departures a little more difficult. In the past, an internal relaxation rate lower than the hard sphere collision rate could be taken as, more-or-less, prima facie evidence for the weakness of the collisions, but as we have seen in the two preceding chapters, both cyclopropane (which conforms exactly to strong collision fall-off behaviour) and methyl isocyanide (which departs only very slightly from such behaviour) both appear to have internal relaxation rates of about one-tenth of the hard sphere collision rate. I have identified this so-called strong collision behaviour with an absence of dispersion in the rates of the normal-mode processes which are important in feeding the reactive states of the molecule. By this definition, collisions between helium and methyl isocyanide are probably strong, as we saw in Section 7.3, and so are collisions between argon and cyclopropane, as I will demonstrate in Section 9.4.

The textbook by Forst [73.F] provides a number of interesting exercises which I do not propose to duplicate. Instead, I will suggest below a selection of computational experiments, all of which can be undertaken once the basic algorithms in Appendix 3 have been assembled together in an available machine and the inevitable organisational conflicts have been resolved. I would suggest that, initially, the student should practise with a few computer runs in which only the input parameters are varied, in order to find out the general behaviour of the method; several of these are listed first. Then follow a number of fairly extensive numerical experiments, each marked *, which are designed to illustrate relevant points of the theory; most students will not have the time to venture beyond these. Finally, for the student whose concern is primarily with the unimolecular kinetic problem, there are three marked ** and ***, which are more involved: the first two can be completed either by a small amount of reprogramming of the routines MAIN and UNIRAT, or by the construction of small routines specially designed for the problem, but the last one should not be attempted without a thorough understanding of the procedures used to calculate sums and densities of states.

Any reaction may be chosen as the subject for each calculation, but the methyl isocyanide isomerisation provides a fairly realistic example of a strong collision reaction without consuming too much computer time; data sets are given for both methyl isocyanide and cyclopropane.

This appendix provides a set of self-contained computer programs for the calculation of rate constants in thermal unimolecular systems. Most of them were written by Andrew Yau [78.Y2], and have now stood up to five years, or so, of continual use. They may run into trouble occasionally if anharmonicity values in the range 0.005–0.006 are used for several oscillators at the same time, or if other unusual molecular parameters are tried but, by and large, they perform acceptably in normal circumstances. Some effort has been expended in trying to make the repetitive segments of the calculations efficient, but there is no pretence at elegance – a contradiction in terms when one is constrained to use FORTRAN [72.D3].

The organisation of the routines is as follows. The MAIN routine handles the flow of the calculation and the assembly of the input parameters. Then there is a UTILITY package which calculates the required sums and densities of states with appropriate precision; this is sufficiently efficient that it is no longer necessary to group the molecular frequencies in most cases. It also divides the reactive energy range into grains of the specified size, computes the equilibrium grain populations βr, and their respective specific rate constants dr = k(E) by the inverse Laplace transform method, equation (4.9). Finally, UNIRAT calculates the rate constant at the requested values of the pressure.

The concept of a strong collision has been bound intimately with the development of the theory of unimolecular reactions ever since its inception in the 1920s. In 1927, Rice & Ramsperger introduced the approximation that an activated molecule would be deactivated whenever it suffered a collision, although they were well aware of the limitations in making such an assumption [27.R]; they supposed that it would be possible at a later stage to determine the extent to which this approximation was untrue, but I doubt if it was envisaged that it would take more than 50 years for us to achieve a realistic understanding of the problem. The principal reason for this delay has been the failure of kineticists to formulate a proper definition of a strong collision; the usual description has been a rather intuitive one [72.R; 73.F], generally adequate within the framework of a steady state treatment of the reaction process, but not always consistent with the principle of detailed balancing. Two useful (and equivalent) verbal definitions of a strong collision taken from the literature are as follows: a strong collision is one in which so much energy is transferred that the subsequent condition of the molecule may be chosen at random (with appropriate weighting factors for energy) from all its possible states [66.B2]; strong collisions are Markovian events whose outcome follows the probabilities given by the Boltzmann distribution, without reference to the initial conditions [77.T1].

Because of an editorial policy discouraging the presentation of experimental results in both graphical and tabular form, the primary rate constant data for the thermal isomerisation of cyclopropane in the fall-off region [53.P2] are only available in thesis form. One might have expected these results to have been superseded by now, but that has not happened, and Sowden's rather inaccessible thesis [54. S] remains the only source of these key data. In view of their continuing importance in the testing of unimolecular reaction theories, I am reproducing those results here (and also those of the cyclobutane reaction) for the convenience of future users.

The general perception, all these years, of the course of a unimolecular reaction has been that the reactant molecule receives the requisite amount of energy by collision, but that it can only react to form products after a time delay during which it has to rearrange that energy and reassemble it into some more appropriate fashion. This reorganisation of the internal energy has almost always been regarded as an intramolecular process [30. R], with a rate which is independent of all external conditions; except for this, the nature of the reorganisation process has been rather poorly defined. Broadly speaking, older treatments of the problem tend to visualise the molecule comprising two domains, one where the energy is collected from external sources by collision, and the other where it is needed to bring about the reaction, with a time delay for communication of internal energy between the two. On the other hand, many modern treatments use the language of radiationless processes: as such, they tend to be unnecessarily complicated from the point of view of treating bulk unimolecular reactions, and they also tend to neglect other possible causes which could have the same end result. The treatment given below is an elaboration of one which I have evolved [80.P2] and which I believe is helpful in clarifying the nature of the unimolecular reaction process.

The reader who has followed the development of this book from the beginning will recognise a progressive increase in the degree of speculation, chapter upon chapter; after describing the basic experimental phenomena, we began with a treatment of relaxation in simple molecules, which is virtually irrefutable, and ended up with an attempt to treat in a semiquantitative manner such concepts as those of randomisation and of activation bottlenecks, about which we still know really very little. For the sake of brevity, I have tried to keep the speculation to the minimum required to form a consistent foundation for the treatment of unimolecular reaction manifestations, wherever they may occur. My discussion has concentrated almost solely on the shape of the fall-off curve for a few thermal unimolecular reactions: this, despite the fact that I enumerated other interesting properties of simple thermal reactions in Chapter 1, and omitted entirely to mention the wide range of other experimental properties usually encompassed within the general topic of unimolecular reaction theory [72.R; 73.F]. In conclusion, therefore, I would now like to hold a brief inspection of each of those topics raised in Chapter 1, together with a few others so far not mentioned, to see how successful the present theory appears to be.