1864 SPECTROSCOPY AND FLUORESCENCE OF PIGMENTS CHAP. 37C 



The second method of calculation, also devised by Duysens and Huis- 

 kamp (1953), is based on the apparent depression of absorption coefficients 

 by accumulation of colored molecules in small particles (as compared to 

 the absorption caused by the same number of molecules in true molecular 

 solution; c/. chapter 22, p. 714). The following derivation was made for 

 the idealized case of cubic particles with one edge parallel to the incident 

 beam. For a given wavelength, the optical density of a single particle 

 (subscript p) is: 



(37C.10) d^ = In {l/T)p = In (///o)p = aCr>d 



where a is the molar absorption coefficient of the pigment, Cp its molar 

 concentration (within the particles), and d the edge of the (cubic) particle. 

 If the area density of the particles in suspension is A'' (per cm. 2), dis- 

 persing the pigment uniformly would produce a solution (subscript S) 

 with a molar area concentration c^Nd'^ (per cm. 2) and an optical density: 



(37C.il) Ds = In {h/I)s = aCj>Nd^ 



Assuming the molecular absorption coefficient to be the same in (37C.10) 

 and (37C.11), we have: 



(37C.12) Ds = In {l/T\Nd'' 



Since d"^ is the cross section of a single particle, the average number p of 

 particles the light beam crossing the suspension traverses is Nd^, and we 

 can therefore wi'iie, instead of (37C.12): 



(37C.13) Ds = p\n{l/T\ 



When the molecules are bunched together in particles, the average optical 

 density of the suspension can be calculated by using Poisson's formula for 

 the probability of a beam encountering a certain number of particles, k, 

 on 1 cm. of its path through the suspension. This probability is: 



(37C.14) Pk = e-^p^/k\ 



where p is, as above, the average number of particles encountered. The 

 beam that traverses k particles is weakened by the factor Tp/ the average 

 transmission of the suspension is therefore obtained by summation of: 



over all values of k from to oo . This gives : 



(37C.15) f = e-pd-^p) 



or an average optical density of suspension (subscript P) : 



(37C.16) Dp = p(l - Tp) 



as compared with the optical density (37C.13) of the same pigment in 



