88 CHEMICAL REACTIONS IN THE GAS PHASE 



Knj=C2^ 2^ 2^ 2^ — ~vi{n,nk,nj,lkj) 



k nk = 7iy = lki = nj*-nj (Tl — Uk — Uj) .{S — 6)1 



(8) 

 Actually we should exclude from the terms 



(n — nk — Uj -{- s — 3) ! 



{n — Uk — nj)\{s — 3)! 



those states which correspond to dissociation, that is, n^ > rir*. How- 

 ever, this would only decrease the expression slightly at the price of 

 great complication. 



Expression 8 is apparently quite complicated and in this form would 

 be difficult to evaluate. Actually, however, it is almost certainly true 

 that the sums involved will contain a large number of terms equal to 

 or very near zero. First of all, 7 will be strongly dependent uopn Ikj, 

 the number of quanta which are transferred from oscillator k to oscillator 

 j in a single period of the frequency factor. In fact it seems likely that 

 the probability of transferring more than one quantum in time 1/v may 

 be negligible. This means that all terms are essentially zero unless Ikj = 

 1. If this is true then w^e must have 



Uj* — Tlj — I 



SO that only those molecules having Uj = Uj* — 1 can decompose in time 

 1/v. This reduces the summation over hj to a single term kj = 1. 

 Since in the summation over Uj the lower limit is rij = Uj* — kj, this 

 now becomes ny = rij* — 1, and the summation over rij is thus also 

 reduced to a single term. In the molecule, energy transfer is principally 

 between adjacent bonds; the summation over A; can be taken as zero 

 except for three or four terms. The rate then reduces to 



->^^[n -Uk- (uj* - 1) + s - 3]! _ / * ix n 



knj = C E E 1 r^^ T^TTT ^ n[^> ^k, {uj* - 1), 1] 



k nk^o [n - Uk- (rij* - l)]!(s - 3)! 



For quanta corresponding to j'osc = 1000 cm~\ a reasonable value for a 

 hydrocarbon molecule, the factorial term will decrease rapidly in size. 

 For example, if n - n^ - (wy* - 1) = 12 and s - 3 = 52 (values which 

 are approximately correct for a molecule like hexane), the factorial term 

 is 3.3 X 10^^; and for the next term, where n — Uk - (rij*) = 11, its 

 value is 6.2 X 10^ ^ The ratio of the second term to the first is approxi- 

 mately 0.2. The transmission coefficient 7 will increase as rik becomes 

 larger. There will therefore be a maximum term in the summation over 

 Uk. Assuming this maximum to be strong allows us to discard the 



