MIXTURES OF ELECTROLYTES MACGREGOR. 109 



times the dilution (V' 3 ) of the latter solution. Then, since 

 V.' -:&>- -d V, -%, = i 



we have 



V ' - V 



* i * 



1 11. 



Hence F E and H G are equal. In order to determine the con- 

 centration of ions after mixing, we must find two points J, K, 

 on the curves A B and C D respectively, having according to 

 equation 1, a common abscissa O L, and having according to 

 equation 2, ordinates L K and L J which together are equal to 

 p times the sum of F E and H G. If p is equal to unity (i- e., if 

 the change of volume on mixing is negligible), the line J K 

 joining the points J and K, when properly selected, will obviously 

 be bisected by the line E G. The points J and K may thus be 

 easily found by inspection. If p is not equal to unity, we must 

 cut off from F E and H G or from these lines produced, portions 

 F N and H P, equal to p. F E and p. H G respectively. Then, 

 as before, the proper points Q, R will be so situated that the line 

 Q R will be bisected by the line N P. Thus in this case also the 

 points Q, R may readily be found by inspection. The points J, 

 K (or Q, R) being thus found, O L (or O S) will represent the 

 common concentration of ions in the mixture, and L K and 

 L J (or S R and S Q) will represent the dilution of electrolyte 1 ? 

 and t njn l times the dilution of electrolyte 2, respectively, in the 

 mixture. , 



If the solutions to be mixed have unequal volumes (y^ and i> 2 ') 

 we must draw the curve c d having as abscissae the concentrations 



of ions, and as ordinates n z v * f times the corresponding dilu- 

 tions of a series of solutions of electrolyte 2. Let Oh and hg 

 represent the concentration of ions, and the dilution (V/) 

 respectively, of the solution of electrolyte 1, which is to be 

 mixed with a solution of electrolyte 2, and let 0/and fe represent 



the concentration of ions, and n 2 - ^ , times the dilution (V 2 ') of 

 the latter solution, respectively. 



