646 PRINCIPLES OF GENERAL PHYSIOLOGY 



for the electromotive force of concentration batteries (see Nernst's book, 1911, 

 p. 752). It is 



1^1 RT log, -i 

 u + v c 2 



which gives the electromotive force at the contact of two solutions of concentra- 

 tions PJ and c.y, u and v being the velocities of the cation and anion respectively. 

 When u and v are very nearly equal, as in potassium chloride, the potential 

 difference due to this factor is very small. If u is hydrogen, with a molecular 

 conductivity of 318, and v is carboxyl, as in formic acid, with a molecular 



conductivity of 33*7, the fraction - - becomes 



318- 33-7 _ 284-3 _ Q . gl 

 318 + 33-7 351-7 



R is 0-861 x 10~ 4 and if we take ordinary logarithms and a ratio of concentration 

 of 1 to 10, the expression becomes about 0'05 volt, and even this could only last an 

 infinitesimally short time owing to rapid diffusion. 



We see that, to account for the values actually obtained experimentally in 

 animal tissues, another source of potential difference must be found. The 

 potentials of metallic electrodes naturally suggest themselves, so that one some- 

 times finds it stated that the electromotive activity of tissues is that of a 

 concentration battery. But it is plain that there is nothing in living cells that 

 could be taken as a metallic electrode. On the other hand, we have already seen 

 (page 161) how we can obtain a permanent potential difference of fairly considerable 

 amount when a membrane is present permeable to one only of the two ions of a 

 binary electrolyte. There is formed a Helmholtz double layer, and the electro- 

 motive force is expressed by a formula similar to that of the concentration battery. 

 This point of view was token by Bernstein (1902), on the basis of Ostwald's 

 (1890) considerations regarding semiperrneable membranes, and is developed 

 further by Bernstein in a paper of 1913. 



It may enable this important conception to be grasped more easily if a simple illustration 

 be given. Imagine two large pastures separated by a fence and that the spaces between the 

 liars of the fence are wide enough to allow lambs to pass through, but too narrow for ewes. 

 Introduce into one of these pastures a flock of sheep, each ewe with one lamb. In the course 

 of their wanderings they will arrive at the fence. The propensity of the lambs to wander 

 further will take them through the fence, while the ewes will be left behind. But the attrac- 

 tive forces, particularly that of food, will prevent the lambs from departing' from their 

 mothers for any considerable distance. Similarly, the presence of the lambs in the adjoining 

 field will prevent the ewes from wandering far from the fence. Regarding wool as electric 

 charge, we see that the potential will be higher on the side of the fence occupied by the ewes. 

 It may be said that the thickness of the layer would be considerable, but if we imagine 

 molecules magnified to the size of sheep, the arrangement would not greatly differ from the 

 molecular one. 



Since the question is somewhat fundamental, it is well to consider the 

 mathematical proof that a membrane of the kind postulated gives rise to a 

 potential difference expressed by a formula similar to that of Nernst's concentration 

 battery with metallic electrodes. 



There are two distinct ways in which the calculation can be made. We may 

 take the work done in moving electricity from one solution to the other against 

 electrostatic forces, as in the method used by Nernst, based on Helmholtz's theory 

 of contact potential, and similar to that used on page 33 in calculating the work 

 done in compressing a gas. But for the present purpose, in which we are 

 regarding the phenomenon from the point of view of the potential difference in 

 equilibrium, the following method is more appropriate, besides giving an 

 opportunity for a new aspect of the case. The details of the treatment I owe to 

 Mr W. B. Hardy. 



For simplicity, we will consider the membrane as being infinitely thin, which 

 is very nearly true for the cell membrane. Let it be situated, to begin with, 

 between water and a solution containing a salt which is electrolytically dissociated 



