SOLUTIONS. MACGREGOR. 69 



The first equation is given by the law of kinetic equili- 

 brium. It may be expressed as follows : The regional ionic 

 concentrations of the two electrolytes, i. e , the numbers of their 

 free gramme-ions per unit volume of their respective regions, are 

 equal. The second states that the volume of the solution is 

 equal to the sum of the volumes of the regions of the respective 

 electrolytes. The third and fourth assert that the regional 

 ionic concentrations are functions of the respective regional 

 dilutions. 



As f and f are very complex functions, these equations 

 could not be solved algebraically even if the functions were 

 known. They can be solved graphically, however, without 

 actually determining what the functions are. 



For this purpose we first find, from conductivity obser- 

 vations made on simple solutions of 1 and 2 respectively, 

 corresponding values of dilution and ionic concentration for a 

 sufficient number of solutions of each, and plot dilution-ionic- 

 concentration curves, i. e., curves with dilutions as ordinates 

 and ionic-concentrations as abscissse. To get precise values of 

 the ionization coefficients for the complex solutions, these curves 

 must be accurately drawn. They have, very roughly speaking, 

 the shape of rectangular hyperbolas, and thus, both at great 

 dilution and at great concentration, have but slight curvature, 

 while at moderate dilution they have very rapid curvature. In 

 working with solutions at moderate dilution therefore, it is 

 necessary to have a considerable number of corresponding 

 values of dilution and ionic concentration, in order to plot the 

 curves accurately. When but few are available, it is helpful to 

 plot first a concentration-ionic-concentration curve, i. e., one 

 having concentrations of solutions as ordinates and ionic-con- 

 centrations as abscissae. As the dilution- ionic-concentration 

 curves are something like rectangular hyperbolas, the concen- 

 tration-ionic-concentration curves have comparatively slight 

 curvature, and thus lend themselves readily to interpolation. 

 Corresponding values of concentration and ionic concentration 

 obtained from these curves, when the concentrations are trans- 

 formed into dilutions, may be used to eke out the values 



