32 RADIATION BIOLOGY 



gateways increases as the number of vibrations. When there are man.y 

 gateways, perhaps every vibration of each mode results in some energy 

 transfer. Teller (1941) has described the quantum-mechanical mecha- 

 nism in the following way: Certain wave functions of each mode will 

 correspond in energy eigenvalues to certain wave functions of other nor- 

 mal modes. When pairs of wave functions correspond to the same 

 energy, they recombine to form a new independent pair, each mode of 

 which contains elements of the original pair. The energy is redistributed 

 in the time allowed before the degeneracy is destroyed by further cou- 

 pling with other modes. 



In terms of absolute collision theory, it can be seen that the rate con- 

 stants for energy transfer in internal collisions may be of considerable 

 magnitude. The rate constant after cancellation of single-coordinate 

 partition functions, which are identical for the average and activated 

 forms of the molecule, looks like this: 



g-Eo/RT 



kT } — p-hrt/kT 



^-'T 1 <'-23) 



Y Q—hvi/kT J g—hvi/kT 



One vibrational partition function of the average molecule approximately 

 cancels the single vibrational partition function of the activated mole- 

 cule, and since the remaining term in the denominator is nearly unity, 

 the rate constant will take on a value limited only by the collision energy, 

 which is generally low, and the transmission coefficient, which can be 

 relatively large because of the types of potential surfaces involved. The 

 average time for this exchange will vary from mode to mode and mole- 

 cule to molecule, but it is undoubtedly no faster than a fast vibrational 

 time, i.e., about 10"^^ sec, and this only when the molecule is large and 

 the collision energy nearly zero. Arrhenius frequency factors for these 

 processes, or indeed for any processes that depend on the motions of 

 nuclei, must therefore seldom exceed 10^^ though some increase can be 

 expected as a result of coincident transfers taking place at other parts 

 of the molecule. 



Lack of precise knowledge of an average time for a single transfer 

 seriously limits the applicability of the Kassel and Rice theory for uni- 

 molecular reactions, though this is the only widely used theory for these 

 reactions. Even for small molecules the details of quantum size and the 

 strength of coupling between modes are rarely known, so that the theory 

 of Kassel (1932, p. 93) and Rice assumes an Einstein-like single vibra- 

 tional frequency applicable to all modes of motion. The procedure is to 

 calculate the probability p that m of a total of ./ uniform quanta will be 

 found in a given one of s vibrational modes. Two forms of this expres- 



