152 ATOMS, IONS, SALTS, AND SURFACES 



THE MEMBRANE EQUILIBRIUM 



Let the solid salt be replaced by a membrane, and let one of the salts in the left- 

 hand solution consist of one ion {Na'^), which is, and one (i?~), which is not, diffusible 

 through the membrane. The equilibrium relations are represented below: 



Equilibrium 



Na+ Cl+ 



a" a" 



Na+ Cl- 



R- 



QR- 



If both solutions are considered infinite in volume, then the transfer of i mol of 

 sodium chloride from left to right will not disturb the equihbrium. If i mol is thus 

 transferred, the decrease of free energy ( — A/^) is 



aifa+ aci~ 



But the characteristic of an equilibrium process is that the change of free energy 

 (— AF) is zero, so 



aNa+ aci~ 

 or 



aNa+ aci- 



Ijut this is identical with (3), the relation which expresses ratios obtained from the 

 solubility product. Thus, to this extent the membrane equilibrium is identical with 

 the equilibrium with a solid. Equation (6) may be written 



iiNa + X aci - = axa + X aci , (7) 



which is the same as the solubility-product relation between the two solutions (2) but 

 differs in that no solid of constant activity is concerned in the equilibrium. In the 

 solubility product the product of the activities of the ions of the saturating salt, in 

 either of the solutions on the two sides of the solid salt, equals the activity of the solid 

 salt which is constant. In the membrane equilibrium the same product is equal to the 

 activity of the salt in the membrane between the two solutions, but this activity is 

 variable. 



In order to calculate the cciuilibrium in any special case, the activities under the 

 conditions of the membrane equilibrium must be known. An attempt was made by 

 Donnan to solve this problem without a knowledge of these activities, but there is no 

 conclusive evidence to indicate that such a solution corresponds with the actual equi- 



