Gaseous unimolecular reactions: Theory of the effects of pressure and of vibrational degeneracy

A theory which gave the high-pressure unimolecular reaction rate as K 8 = v exp ( — E 0 /kT) is extended to find the decline of rate with pressure; the gas molecule is again a classical vibrating system which dissociates at a critical extension of an internal co-ordinate. The general rate K is found to be approximately... where n is the effective number of normal modes of vibration; d is proportional to pT~^n, but depends also on the molecular structure and size. For n < 13, this integral is tabulated, and the pressures at which the rate declines from first order are estimated. The pressure tends to decrease as n increases; for E 0 /k T ~ 40, it is estimated that only molecules with six or more atoms should show rates approaching KCX) at normal pressures. The table of K/K;a is not carried as far as the ‘bimolecular’ range, but a precise technique is developed for this region. The theory is compared with Kassel’s classical theory of a molecule of s ‘oscillators’. The lowpressure activation energy, and the shape of the curve of log K against log p, are similar in the two theories if n = 2s — 1; the absolute values of p for a given rate are also roughly comparable. Two results are proved, for the present severely classical model, concerning special cases. (i) A pair or triplet of degenerate modes with equal frequencies counts as one in assessing ‘n’ for the general rate K. (ii) If the dissociation co-ordinate q relates atoms ml, and mx is replaced by an isotope m*, the high-pressure rate changes in the ratio d{m 1 (m*+ m 2 )/m^(m 1 +m 2 )}; for this, the internal potential energy V need not be quadratic, nor need q be isolated in V from other co-ordinates.