LAWS OF ELECTROLYTIC DISSOCIATION 53 



where log is the decimal and In is the natural logarithm, therefore, 



^" = - AXi^ . I, 10 (2) 



d log [H+] (k + [H+]2) 



and differentiating once more with respect to log [H+] we obtain 



d at 



d^ a d d log [H+] d [H+] d In [H+] 



d log [H+]2 d [H+] d In [H+] d log [H+1 



k . [H+] • (k - [H+]) 



(k + [H+]3 



(In 10)2 (3) 



Letting this expression = we obtain the conditions for the 

 transformation point of the a-or log [H+]-curve, This condition 

 obtains, as may be readily seen when [H+] = k, or pH = pK. 



By substituting in (1) [H+] = k, we have a = \. 



Thus at the transformation point, 



1. pH = pK 



2. a = i 



which was to be demonstrated. 



The same holds true for the residue curve which at the transfor- 

 mation point ordinate is sjTnmetrical with the dissociation curve. 



The angle at which the curve intersects the ordinate in the vicin- 

 ity of the transformation point is of interest. According to the 

 fundamental laws of differential calculus the differential quotient 

 represents the tangent of the unknown angle ip. Since at the point 

 concerned [H+] = k, it follows by substitution in (2) above that 



d « In 10 2.303 ^ ^„„ 

 = - — — = = 0.576 



d log [H^ 



and arc tan ^ = almost exactly 30°. In order to obtain the correct 

 angle by this method one must take care that in the graphic repre- 

 sentation the ordinates are drawn to the same scale as the abscissa. 

 If the scale of the ordinates is n times that of the abscissa then the 

 tangent of the angle of intersection is n times greater. In the 

 curves of figures 3 and 4, n = 5, and therefore, the tangent = 2.88 

 and the angle of inclination is approximately 71°. 



