278 PROPERTIES OF ELECTRICALLY CONDUCTING SYSTEMS 



It is evident that: 



(77) K = K 1 K 2 . 



An inspection of the above equation shows that, owing to the formation 

 of the intermediate ion MX + , the value of whose concentration is given 

 by the second term of the right-hand member, the solubility is increased 

 due to the formation of the intermediate ion. With increasing value of 

 M ++ , this term may become sufficiently great to overbalance the effect 

 of the last term of the right-hand member. This is more readily seen on 

 differentiating Equation 76 with respect to the concentration of the 

 common ion M ++ , which leads to the equation : 



tm dS _ y *MX 2 W 1 1 \ 



dM ++ " M ++ % \2K 2 2M+V* 



The solubility will be a minimum when: 

 (79) 1 



Obviously, the concentration of the common divalent ion M ++ at the 

 minimum point of the solubility curve is equal to the equilibrium con- 

 stant K 2 . If this constant is small, then the minimum point will lie at a 

 low concentration; whereas, when this constant is large, the minimum 

 point will lie at high concentrations. In other words, when K 2 is large 

 the fraction of salt present in the form of intermediate ions MX + is 

 relatively small; whereas when K z is small this fraction is relatively 

 large and the minimum point accordingly appears at low concentrations. 

 It may be noted, in this connection, that the solubility curves of lead salts 

 exhibit a pronounced minimum at relatively low concentrations. That 

 for lead iodate in the presence of lead nitrate is in the neighborhood of 

 0.04 N ; that for lead chloride in the presence of lead nitrate is at approxi- 

 mately the same concentration. Silver sulphate, in the presence of potas- 

 sium sulphate, exhibits a minimum in the neighborhood of 0.1 N. Cal- 

 cium sulphate exhibits minima in the neighborhood of 0.15 N in the 

 presence of salts with a common S0 4 ~~ ion. In the case of salts with a 

 common Ca + * ion, this minimum does not appear. The difference in the 

 behavior of calcium sulphate in the presence of a common positive or 

 negative divalent ion may be due to various causes, since in this case 

 there is involved the formation of two different types of complexes. 

 Considering the behavior of uni-divalent salts, it is evident that those 

 salts which exhibit a pronounced tendency to form complexes, such as 

 lead salts for example, likewise exhibit a pronounced minimum in the 

 solubility curve in the presence of a common divalent ion. 



