ELECTRICAL CHANGES IN TISSUES 647 



into the ions B- and S', and be freely permeable to B-, but impermeable to S', in a 

 purely mechanical way, as a filter or sieve, for example. 



The ions B- tend to pass from solution inside to water outside owing to their 

 osmotic pressure-and to this alone. Since ions S' cannot pass through, in order that 

 ions B' shall diffuse into the outer solvent, they must separate from their 

 companions. This they cannot do for more than a minute distance, owing to the 

 enormous electrostatic force between the oppositely charged ions. The amount of 

 this force was calculated by Arrhenius, as we saw on page 179 above. 



These ions B' are, therefore, acted on by two forces in opposite directions, and 

 they will take up a position in which the two forces are equal and opposite. 



The osmotic pressure exerted on a membrane of area A is Ao?P, where P is the 

 pressure per unit area. 



The opposite electrostatic force is obtained thus : 



Let E be the potential difference between the two members S' and B' of the 



Helmholtz double layer. Then - is the potential gradient, or rate of fall of 



potential across the space between the two layers. 



Further, if q is the quantity of electricity carried by one gram-equivalent of 



ion B', then the force acting on this gram-equivalent is q . That this is so 



dx 



will be clear by consideration of the fact that the force is directly proportional 

 to the quantity of electricity producing it, and the fact that the greater the 

 difference of potential between the layers of B' and S', the greater will be the 

 attractive force between them. 



Let c be the concentration of the diffusible ion B' in gram-equivalents per c.c. 

 of solution. 



The volume of the space between the two layers is A&r, if the depth be &r, 

 and the number of gram-equivalents of the ion B' contained in the space is A8xc. 



Then the force acting upon them, due to the potential gradient , is A8xc q. 



dx dx ^ 



This force is equal and opposite to their osmotic pressure, therefore 



fJV 

 A 5 "^ A ^J"Q 



A.OOCC q = A-dr , 



dx 



or, = . . 



dx cq ox 



P is equal to c-RT, since c = -, 







dE RT dP 1 RT 



therefore, = , since - = -TT-ri 



dx Pq 8x cq ctilq 



RT (dx dP^ 

 and, o?E = - 



The treatment will be more general if we suppose that the concentration of 

 the ion B' is a positive quantity on both sides the membrane. In that case, we 

 must integrate between the limits, p l and p v which are the osmotic pressures 

 of the ions B - on the two sides of the membrane. 



jpyi 



- is a constant. Therefore, 



q 



E 



_RT rPi/dx dP 

 9 'J TZ ^ P ' ^ x 



(i) 



9 Pi 



or, if c 2 is the concentration of the stronger solution and c, that of the weaker 

 solution, 



= log, - 2 (2) 



