QUENCHING BY COLLISIONS AND BY RESONANCE 785 



However, deviations from the linear relation required bj^ equation (23.16B) 



do occur, particularly at the higher values of [Q] (fig. 23.8). Livingston 



and Ke obtained a better approximation by using an empirically generalized 



equation : 



Fo 



(^^•^^^^ ^ - 1 + kdQ] +k2lQr- 



The broken lines in figure 23.8 show how this equation can be made to fit 

 the data. The values of /ci are given in Table 23.IIIC. Livingston com- 

 pared the empirical two-constant equation (23.1()C) with a theoretical 

 equation of ^\nvilov and Frank: 



(23.1GD) F 



elQ]p + iAT/v)[Q\ 



Here, the first exponential term in the denominator represents "static 

 ciuenching," i. e., quenching determined by average distance between 

 quencher and fluorescent molecule (p = effective radius of energy ex- 

 change). The second term accounts for "kinetic quenching" {A = const., 

 ij = viscosity of the medium). Quenching by resonance exchange of 

 energy is a possible mechanism of static quenching; chemical reaction 

 by the first, or one of the first, kinetic encounters (the frequency of which 

 is determined by the rate of diffusion, and thus indirectly by viscosity), 

 can be suggested as one mechanism of "kinetic" quenching. 



Developing the exponential in (23.16D) and retaining only the first 

 term, one can obtain an equation of the type (23.16C). Livingston and Ke 

 calculated from their empirical constants (h and fc2) the "action ^radii", p 

 for different quenchers, and obtained values between 23 and 7 A— which 

 they considered as plausible in view of Forster's calculations of the range 

 of energy exchange (chap. 32). However, Forster's calculations were for 

 the case of resonance between the two molecules taking part in the energy 

 exchange, while the molecules listed in Table 23.niC have no absorption 

 bands which could resonate with the red fluorescence band of chlorophyll. 

 Therefore, no exchange of excitation energy over distances wider than a 

 molecular collision diameter appears possible — unless one resorts to theo- 

 retically feasible, but experimentally as yet unsupported hypotheses. 



Resonance between two separately "prohibited"— and therefore spectroscopically 

 unknown — transitions, one in the excited molecule and one in the quencher, which be- 

 come "permitted" by being coupled together, was suggested by Rollefson; the total 

 spin of the system can be preserved if one molecule goes from a triplet into a singlet 

 state, while the other simultaneously undergoes a reverse change. 



Testing equation (23.16D) by comparing quenching eflaciencies in dif- 

 ferent solvents did not give satisfactory results. For example, the h 



