22 RADIATION BIOLOGY 



directions of approach. In collision processes the successful trajectories 

 for energy transfer will, in general, cluster together at the place where 

 they reach their highest potential energy. There may be more than one 

 such region or gateway, each with its own highest potential energy £"0. 

 The rate process will count all successful trajectories just as in chemical 

 reactions. Where more than one saddle point exists for the completion 

 of a chemical reaction, the sum of rates for each saddle yields the over-all 

 rate. In energy-transfer processes, besides the requirement that success- 

 ful trajectories must fall within the gateway, there will usually be addi- 

 tional conditions that the velocities lie within certain definite limits, 

 whereas for chemical reactions the velocities normal to the barrier may 

 have any value between zero and infinity. In spite of this, we may 

 write for the specific rate constant k of any process that can be specified 

 as passage through a gateway the eciuation 



k = '-^-^e-^o/^^, (1-11) 



where t = transmission coefficient that measures any special properties 

 of the potential barrier which might restrict the passage of a 

 configurational point; 

 k = Boltzmann's constant; 

 T = absolute temperature; 

 h = Planck's constant; 



/ = single-molecule partition function for the colliding partners in 

 their average states; 

 ft = partition function of the colHsion complex at the gateway; and 

 Eo = highest potential energy per mole reached for the successful 

 trajectory most economical of energy. 

 The expression /Jp-^»/«r is the partition function for all trajectories 

 passing through the gateway reduced to an energy zero, which is the 

 zero-point energy of the reactants. Multiplying through by the frac- 

 tion t, the transmission coefficient, restricts the trajectories to the suc- 

 cessful ones. Clearly such a formalism is necessarily correct, since t is 

 expressly defined to give the correct value. However, in each particular 

 case the procedure for a priori calculations will be to construct enough 

 trajectories, successful and otherwise, to define the gateway or gateways 

 and then to use quantum mechanics to calculate t for the established 

 values of the collision energy Eo. For purposes of discussion and sys- 

 tematization of data, it is convenient to replace Eq. (l-U) by the two- 

 parameter equation 



hT 

 jj' ^ !li_ gASt/RQ-AHt/RT (1-12) 



where ASt and A//| are the apparent entropy and heat for the process in 

 question. If only a small fraction of paths through a gateway are sue- 



