84 PROPERTIES OF ELECTRICALLY CONDUCTING SYSTEMS 



and the curve is everywhere convex toward the Cy-axis. In the limiting 

 case, m = 1, and 



(30) 



and K is a linear function of Cy. 



From Equation 24, it follows that when m < 1, 



dK' 



< 31 > c !L m o d(CA) = 



and when m > 1, 



In the first case the K', Cy-curve approaches the limit K asymptotic 

 to the axis of K', while in the second case it approaches the limit 

 asymptotic to a line parallel to the axis of Cy. 



The curvature of both curves increases as the concentration de- 

 creases. 1 * For the radius of curvature of the K', Cy-curve we have the 

 equation: 



2(2-w) 



, mV*JH/ 3 (C Y ) 

 " 



[Dm(m I)] 3 / 2 (m I) 2 / 3 



The exponent 2 m of the first term of the right-hand member of this 

 equation is positive for all values of m less than 2. Since no solutions 

 are known for which m is greater than 2, we need not consider greater 

 values of m. It is evident, therefore, that, due to the first term, the 

 radius of curvature increases with Cy for all values of m. For m > 1, the 

 coefficient 2m 1 is positive and the radius of curvature increases with 

 Cy due to this term also. When m < 1, 2 m > 2m 1 so long as m is 

 greater than zero. It follows, therefore, that the first term overbalances 

 the second and that the curvature, for all values of m between zero and 2, 

 decreases with increasing concentration, becoming infinite in the limit. 

 For m = 1, R = oo, and the curvature is zero. For m < 1, the curvature 

 is negative; that is, the curve is concave toward the Cy-axis. While for 

 m > 1, the curvature is positive, and the curve is convex toward this axis. 

 For given values of D and K and for different values of m we have a 

 family of curves passing through the points Cy = 0, K' = K and Cy = 1, 

 K' D -}- K. Such a system of curves is shown in Figure 13. The con- 



14 Kraus, J. Am. Chem. Soc. 2, 6 (1920). 



