254 BIOCHEMISTRY OF BACTERIAL LUMINESCENCE 



radiation ("black body radiation"). Planck showed that this is 



av/ 



^' ^hvJkT „ \ 



(4) 



If C and D are in thermodynamic equilibrium, (D)/(C) = exp(/ii/o/ 

 fcT) and Eq. (3) becomes identical with Planck's formula — as of 

 course it must. (Indeed, this is a simple and often used way of 

 deriving Planck's formula.) If C and D are present in comparable 

 concentrations, as they must be during most of the course of a chemi- 

 luminescent reaction, we can write (C) = (D), giving 



av? 



*'^ ghv,*/kT _ 1 



We see that the radiation density thus obtained is related to the black 

 body radiation density p;* associated with the quantum /ivi* in the 

 following way: 



Pi ~ iVi/v^*Y Pi* 



That is, during a chemiluminescent reaction involving an intermediate 

 which is activated thermally by an amount equivalent to a quantum of 

 energy /ivi*, the radiation density may be greater than the ordinaiy 

 black body radiation of photons of energy hvi* at the same tempera- 

 ture by a factor [(vo + vi*)/n*]^. This factor might be quite large; 

 for instance, if vo corresponds to a wavelength of 5300 A or an energy 

 of 54,000 cal/mole, and vi* corresponds to an energy of 6000 cal/mole 

 (or about lOfcT), the steady state chemiluminescent radiation density 

 at 4750 A ( corresponding to an energy of 60,000 cal/mole ) would be 

 1000 times greater than the black body radiation at 47,500 A (cor- 

 responding to 6000 cal/mole). The physical reason for this somewhat 

 surprising result is that the number of states available to a photon 

 is proportional to the square of its frequency, while its energy is pro- 

 portional to its frequency. By attaching the energy /ir,* to a more 

 energetic photon of energy /jvo we increase the probability of finding 

 the energy /ivi* by a factor of [(vo + vi*)/vi*]^. 



In order to find the numerical value of the limiting intensity of 

 emission, let us assume that the states Cj* form a continuum above 

 a threshold frequency vi* = v*. Let us also assume that there is a small 

 hole in the side of the vessel containing the chemiluminescent reaction, 

 through which radiation may leak out. The rate of emission of energy 

 associated with a frequency in the range from vi to vj + dvi from such 

 a hole is simply (c/2)/3idvi per unit area of the hole per unit time. For 



