150 PRINCIPLES OF GENERAL PHYSIOLOGY 



solutions, we are allowing for the volume of the molecules of the solute, or taking 

 V-b instead of V in the simple equation. 



But this procedure, as the table shows, is not a complete solution of the 

 question, and we must also take into consideration the remaining constant of van 

 der Waals, viz., a, which refers to the mutual attraction of the molecules, and 

 therefore acts in the opposite way to 6. This mutual attraction of the molecules 

 has been already met with in Chapter III., in the case of liquids, as the internal 

 pressure of Laplace, giving rise to the surface tension. These attracthr t'mvcs are 

 naturally less the further the molecules are from one another. They are in fact 

 inversely proportional to the square of the volume occupied by a given number of 



molecules, i.e., . We must then increase P, in the simple gas equation, by 



this quantity. 



In the application of the van der Waals theory to solutions I propose to follow, 

 in the main, the treatment of Otto Stern (1912), since it is, on the whole, capable 

 of easier explanation than the similar one of Berkeley (1907). For a complete 

 account, however, the original papers must be consulted. 



In the first place, we must not expect even dilute solutions to obey the simple 

 gas law exactly, because the solvent itself is, as regards its molecular state, very 

 concentrated when compared with a gas. In other words, its molecules are closely 

 packed. According to van der Waals, at the boiling point, the volume of the 

 molecules is about one-quarter of the entire space occupied by the liquid. 



That there is space between the molecules of a liquid is shown by the fact, amongst others, 

 that liquids are not altogether incompressible. Parsons and Cook (1911, p. 343) find that 

 water at 4 can be compressed to 87 per cent, of its volume by a pressure of 4, 500 atmospheres, 

 and ether at 35 to 80 per cent, of its volume by 4,000 atmospheres. 



Moreover, the molecules of the solvent affect those of the solute in both the 

 attractive and the repulsive ways of the van der Waals equation ; so that it 

 is, in point of fact, rather unexpected to find, even in the case of dilute solutions, 

 that the osmotic pressure is so nearly equivalent to the gas pressure of the solute. 

 The reason for this, according to Stern, is the presence of the semi-permeable 

 membrane itself, which causes the effects due to both the attractive and repulsive 

 forces to be compensated in dilute solutions in the following way : As regards a, 

 a molecule of the solute which hits against the membrane is surrounded on all 

 sides by the solvent, since the membrane is permeable to these. The attractive 

 forces are therefore equal on all sides, as if the membrane were not present, and 

 play no part in the production of the osmotic pressure, which can only be affected 

 by forces which are unequal on the two sides of the membrane. As regards 6, 

 an increased osmotic pressure must undoubtedly be caused thereby, but a part 

 of the total osmotic pressure, and, in fact, a part which is exactly equal to that 

 due to the volume of the molecules, is taken up, not by the membrane, but by 

 the molecules of the solvent in the act of passing through the membrane. A 

 certain part of the membrane is occupied by molecules of the solvent, instead 

 of membrane substance, so that a certain number of the molecules of the solute 

 hit against these molecules of the solvent, instead of against the membrane, 

 and are therefore inactive osmotically. 



The above considerations apply only to dilute solutions, where the osmotic 

 pressure is given by the simple gas law, and the number of molecules of the 

 solute is so small in comparison with those of the solvent, its "molar fraction," 

 that mutual action may be neglected. 



This mutual action cannot be neglected in more concentrated solutions, and 

 Otto Stern has developed the following modification of the van der Waals 

 formula : 



[ V - &1 + b^x. - x)] = RT, 



where TT is the osmotic pressure, a l and b l are the van der Waals constants of 

 the pure solute, a 1>2 and b rz are constants depending on the attraction and 

 repulsion respectively between the molecules of solvent and solute, and x x 



