Causes of Osmotic Pressure. 4 ( J5 



permeable membrane in contact with the dilute sugar solution, 

 each molecule of sugar in being turned back receives momen- 

 tum 2mv, so that the whole pressure taken by unit area of the 

 meshes of the membrane is nmv*/3, where n is the number of 

 molecules of sugar per unit volume. Thus for the osmotic 

 pressure we have arrived at exactly the same expression as 

 for the gas pressure of the sugar molecules if the water were 

 annihilated and the sugar remained as a perfect gas. By the 

 nature of the argument, we are not involved in any difficulties 

 in the way of its leading us to assert Boyle's law as holding 

 down to and in the liquid state. We assert that what holds 

 down to and in the liquid state for any fluid is that the 

 number of times a unit plane is crossed in the same direction 

 by molecules per second is nv/6, and as the result of this and 

 of the property of semipermeability the identity of the laws 

 of osmotic pressure with those of perfect gases comes to pass, 

 notwithstanding that the complete equation for the pressure 

 of the pure water of the dilute sugar solution involves all the 

 complications of collisions and powerful attractions with 

 absolute departure from Boyle's law and Charles' law. 



The reason why we cannot assert that a liquid having n mole- 

 cules to unit volume exerts a pressure nmiP/3 on the solid wall 

 containing it because of collisions, is that a unit area of the wall 

 cannot be regarded as a random unit area, but a very special 

 one with the special condition that no molecule can pass it. 

 Now the meshes of our semipermeable membrane in turning 

 back the molecules of sugar give the unit plane containing 

 them a special property, but when the solution is dilute and 

 the numerous water molecules pass freely, the meshes cause 

 what is only a slight derangement from the conditions of a 

 random plane, and in our first approximation we have 

 neglected this derangement on account of its slightness. 



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Molecules of sugar can be turned back by molecules of water 

 which project beyond the plane of the mesh by any fraction of 

 their diameter. 



As the surface of a solid wall bounding liquid cannot be 

 treated as an average plane we have to take a special method 

 of calculating the total eollisional pressure on it, as for instance 

 that in " The Kinetic Theory of Solids " (Phil. Mag. [5] 

 xxxii.). The essential distinction then between a semiperme- 

 able membrane and a solid wall in relation to a dilute solution 

 is that while the membrane has special relation to only the 

 few dissolved molecules, the solid wall has a special relation 

 to all the solvent as well as the dissolved molecules. 



A derivative result of some importance from our argument 

 is the identity of the mean kinetic energy of the molecules of 



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