86 CHEMICAL REACTIONS IN THE GAS PHASE 



continuous one. Even without this distribution function in explicit 

 form it is possible to formulate a fairly simple theory of the decompo- 

 sition of these molecules which, although it oversimplifies the situation, 

 serves to provide a reasonable picture of the processes involved. 



As has already been done by Kassel (36), we shall use as a model to 

 represent the molecule a collection of harmonic oscillators coupled by 

 forces which are sufficiently large to allow the energy to pass freely from 

 one to the other but small enough so that the energy of a group of such 

 oscillators may be expressed as a sum of squares of their coordinates and 

 momenta. Also to further simplify the problem we shall assume that 

 the harmonic oscillators all have the same frequency, v. 



We shall let P{Ei) be the probability that after ionization the molecule 

 has acquired, in the manner discussed above, an amount of vibrational 

 energy Ei and shall assume that this energy is randomly distributed 

 among the vibrational degrees of freedom of the molecule. It now 

 becomes necessaiy to formulate an expression for the specific rate of a 

 particular reaction in which molecules, each with energy Ei, decompose. 



A molecule with total vibrational energy Ei will contain 



Ei 

 n = ~ (1) 



hv 



quanta. The molecule, as already mentioned, will be represented by 

 harmonic oscillators corresponding to the vibrational degrees of freedom 

 of the molecule. 



Consider a single possible way of arranging these n quanta in the s 

 oscillators such that there are rii quanta in the first oscillator, 712 in the 

 second, and so on up to Ug quanta in the last oscillator. This set of n's 

 obviously must conserve energy, that is, 



J^^r = n (2) 



r 



We shall consider that the slow step in the decomposition process is the 

 transfer of energy to the oscillator corresponding to the reaction co- 

 ordinate which is to rupture in a particular reaction. In the following 

 formulation we shall consider the reaction to be governed by accumula- 

 tion of a critical number of quanta, n*, in a single oscillator. The number 

 of quanta, Ikj, which when transferred from the kth. to the jth oscillator 

 will cause a break must satisfy the relation Ikj > fij* — rij, where rij is 

 the number of quanta in the reacting oscillator at the start of the re- 

 action. We now define a transmission coefficient, 



7(n, 7li, 712, • " Ur, " ■ W«-b hj) 



