382 HANDBOOK OF PHOTOGRAPHY 



rather than an arithmetic one. Since a certain fraction rather than a certain amount 

 of the silver bromide is dissolved in each time interval, it is theoretically impossible 

 for all the bromide to become dissolved, although the amount remaining undissolved 

 may be made as small as may be desired by leaving the photographic material 

 immersed in the fixing bath for a sufEcientlj' long period of time. 



Quantitativel.y, the amount of silver bromide left 'in the negative at any time can 

 be computed in terms of the original amount of silver bromide when the law expressing 

 the rate of dissolution is known. If the temperature and concentration of the fixing 

 bath are maintained constant, the rate at which the silver bromide dissolves is (at 

 least for a first approximation) proportional to the original amoimt of the salt and is 

 also proportional to (1 — k) where k is the fraction of the bromide dissolved in unit 

 time. If aS is the amount of the salt (silver bromide) remaining after any time t, the 

 rate at which the silver bromide dissolves is expressed by the equation 



^ = ~AS[l - k) (4) 



where A is a constant which depends upon the units used and is to be determined 

 experimentally. By separating the variables, integrating, and transforming from the 

 logarithmic to the exponential form, the amount of undissolved salt S at any time t 

 is given by the expression 



S = e-^ <!-*>' • (5) 



The factor k is termed the velocity constant of fixation and depends upon the 

 temperature and concentration of the fixing bath. It is dependent upon the amount 

 of silver bromide and the condition of the gelatin, although, for a given set of condi- 

 tions, k is larger for chloride than for bromide emulsions, so that the latter would 

 require a somewhat longer time for the same proportion of bromide to become dissolved. 



From a theoretical point of view, an infinitely long time would be required to 

 dissolve out completely all the bromide in a negative or plate. However, it is desirable 

 to know at least approximately what length of time will be required to fix the photo- 

 graphic materials satisfactorily from a practical point of view. From the formulas 

 which have alreadj^ been developed, it can be shown that the time of fixing is 



t 



Ail - k) 



h '^^' (I) ^'^ 



where 52 = the amount of bromide left undissolved at the end of the fixing time; 



Si ,= the amount of bromide originally present. 

 Since this equation involves the ratio of S2 to S\, we may determine the time of fixing 

 for the bromide to be reduced to any fraction of its original value. A common prac- 

 tice in photography is to determine the "time of clearing" of a negative and to use this 

 as the minimum time of fixation. The clearing time is that required for the negative 

 to become free from its opalescence. It is claimed by Warwick ^ that from 5 to 12.5 per 

 cent of the bromide remains imdissolved when the negatiA^e is just cleared of its 

 opalescence. If we use a value of 10 per cent as representing an average value, then 

 S-z./Si = 0.1 and log, (So/Si) = -2.3026. Thus the time of clearing is 



= 2-30^6 (7) 



.1(1 - k) ^" 



The equations developed may be regarded as giving some insight into the mecha- 

 nism of operation of the fixing bath. As a general rule, the temperature and concen- 



1 WAR'mcK, W. A., The Laws of Fixation, Am. Phot., November, 1918. 



