Theory of Fractional Precipitation. 123 



£->-4[(£-»)w(2-,)4 ■ • ffl 



l=,'p-->[(t-)-(!-^]. • • < 2 > 



by the integration of which the amounts of the precipitates x 

 and y formed at any time could be determined ; results, how- 

 ever, which would be of little practical value. 



Comparing the rates of formation of the precipitates 



art=;( A -*> ! 7( B -'>> • ■ ■ ■ ^ 



or 



(A-xy=F(B-yY ; 



but as A e = FB 6 ', the equation becomes 



(>-£H-# w 



or 



log(l-|) 



which gives the quantities of the precipitates formed, x, y 

 (calculated in terms of the original salts A and B), in terms 

 of the basic powers. Hence by a single experiment it is pos- 

 sible to calculate the ratio — , A being the more basic material, 



and consequently e>e / . 



If e=ne f , then the greater the value of n the greater will 

 be the tendency of B to accumulate in the precipitate ; and 

 considering that fractional precipitation is the only method 

 yet discovered for effecting the separation of several of the 

 rare earths, for pairs of which the value of n evidently does 

 not differ much from unity, the experimental study of the 

 process, employing the commoner metals, with regard to tem- 

 perature in particular, might indicate in which direction to 

 work so as to increase n and render the operations less tedious. 

 If n increases with the temperature, then fractional precipita- 

 tion would be more effectual at high than at low temperatures, 

 and vice versa. 



It is customary, in the separation of the rare earths, to 

 employ the nitrates and ammonia as the precipitant, the solu- 

 tions being made so dilute that no precipitate is perceptible 



