MEMBRANE POTENTIALS 225 



That only the positive root is significant can be seen by setting 

 a = 0, when cli = CI2, and not ch = — CI2. 

 By substitution in (1) we obtain 



Vf 



hi = - ^ + t/'^. + cU" (4) 



Thus we see that cli > hi, i.e., the positively charged colloid ion simul- 

 taneously attracts the negative Cl-ions and repels the positive H-ions 

 outward through the membrane. 



If we assume that in addition to HCl, NaCl is also present in the 

 solution, then in the inner solution the ions are in the concentrations 

 a, hi, cli, ni (where Ui is the Na+-concentration) ; and in the outer 

 solution h2, CI2, n2. And now we have the following relationships. 



,^J. ^ ' T 1 ' ~ ^t' \ From the law of electroneutrality 



(II) no + ho = CI2 J 



From Donnan's law of ionic distribution 



We shall now solve these equations for hi, so that hi appears as a 

 function only of a and other values carrying the subscript 2. This 

 is done by solving (III) for hi, ehminating cli by means of (I) and ni 

 by means of (IV) which results in 



cU • hi 

 hi = - 



a + — • n2 + hi 



h2 

 This in turn yields a quadratic equation for hi 



a • h2 //a • haX^ , „ 



2 CI2 iV 2 CI2/ 



And hence, 



hi 



hi 2 cl 



(5) 



From the last equation and (III) we can at once derive the ratio 

 cli/cl2, and, similarly by using (IV), also ni/n2. Furthermore, we 

 can now express the total ionic distribution and the membrane poten- 



