250 BIOCHEMISTRY OF BACTERIAL LUMINESCENCE 



vessel, which will be equal to that in the black body surroundings: 



U{vo) = {8Trhvo'/c')[e''-'^''^T* - 1]-' (4) 



just such as to balance the forward and backward rate of reaction 



We thus have a reversible machine by which the maximum useful 

 work obtainable from the chemical reaction ( 1 ) can be evaluated 

 through the light emitted. We have 



_AF(°) = -A.4° - APV > NohvoiT* - rj/T*-^^'' 



^ No hMT* - T)/T* (5) 



since we can assume the APV of the reaction to be negligible. 



The temperature T* of the surrounding black body radiation bath, 

 which would just reverse reaction (1), is related to j'o and the rate 

 Ro of the emission of photons in the absence of the reversing bath. 



Let o- be the absorption cross section of molecules C for light of 

 frequency lo to reverse Eq. ( 1 ) . The reverse reaction will go at a rate 



Rr = [U{vo),hpo]CaNo/V (6) 



with No = Avogadro's number, and V the volume (in cubic centime- 

 ters) of one mole of material at the unit concentration used in com- 

 puting - AF<°' of Eq. (2). We use (4) for U(vo) in (6) setting Rr 

 equal to Ro to compute T* as 



with 



\ = C 27ri'o (8) 



The dimensionless quantity, 



Xo = a/Xo- = 4xVoV/c^ (9) 



has a maximum value of unity at resonance, but would normally be 

 expected to be very much smaller for visible light and ordinary 

 molecules, say of the order of Xo = 10 ~- to 10 ~^. If the concentration 

 units of Eq. (2) are moles per liter, then Nn/V — 6 X 10-'\ The rates 

 Ro, photons per cubic centimeter per second per unit frequency range 

 may be of order 10''\ The quantity Xo(No/V)/Ro is of order 10^^ to 

 10'^»1. We can neglect the unity under the logarithm of Eq. (7). 

 Setting 



-AF^°^/Nohvo > 1 - {T/T*) (10) 



