Diffusion and Osmotic Pressure

In last weeks physic's friday post we made some general remarks about diffusion and Einstein's ideas. Today we want to discuss the experimental setup in more detail.

To get access to the phenomenon of osmotic pressure consider a solution where the solute is dissolved in a concentration c in the solvent in a volume V' enclosed by a membrane. This membrane is only permeable to the solvent and not to the solute. Furthermore it is assumed to be immersed in a surrounding volume of solvent, meaning a free flow of solvent in- and out is possible. Then a pressure p induced by the solute acts on the membrane - the osmotic pressure. Within the ideal gas framework the pressure can be described as


where R is the ideal gas constant. We can justify this ansatz by assuming that in a solution the size of the solute is microscopic (atomic or molecular) resembling the idea of the ideal gas. 

Sketch of the idea of osmotic pressure (picture from http://www.sparknotes.com/chemistry/solutions/colligative/section1.rhtml)

On the other hand in a suspension the particles immersed in the fluid are macroscopic.  Classical thermodynamics would suggest an osmotic pressure equal to 0. Einstein's (and some others) findings  corrected this view with the statistical theory of heat which answers the question which microscopic changes are originated by the addition and removal of heat. Addition and removal of heat simply increases (decreases) the motion of the particles. As a consequence both microscopical as well as macroscopic quantities must follow the same laws of motion and of statistical mechanics. As a consequence osmotic pressure is built up both in solution and suspensions. Furthermore there is a unique expression for the diffusion constant of particles in a liquid which is proportional to 1/r. Here r is the radius of the (assumed to be spherical) particles. It is obvious that between solutions and suspensions a huge difference in the dissuasion constant may be possible, nevertheless there is no qualitative difference between them in statistical mechanics.