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Two Excluded volume paradoxes, and a discussion of osmotic pressure

images/piston0.50

The Big solute paradox

Yes, a particle behind a piston will exert a force on that piston, if it's alone. But what if there are other particles around?

images/piston1.50 These other particles (water, say), will normally exert balancing forces on both sides of the piston.

When the big particle comes close to the piston, in order to bump it, and thus exert an excess force to the right, surely images/piston2.50 it will get in the way of the little guys, who will thus become unable to exert their usual force on the left hand side? So while the big guy gives the piston an extra bump, the little guys can't give it any bumps at all. Since the big guy moves so slow, surely the net effect will be that there is a net LEFTward force, not rightward?

Paradox! (It's a paradox, because this kinetic view appears to conflict with what we know happens on entropic grounds, namely, the accessible state space is bigger the further right the piston moves, so there is an entropic force to the right, little guys or no little guys.)

Small solute paradox

images/piston3.50

Having resolved the above paradox, we might worry about a second one.

What if the special solute molecule has the property that it cannot get past the piston but it is otherwise identical in size to the water molecules? In this situation, surely the force exerted by the lefthand red sea with one blue dude will be equal to the force exerted by the righthand red sea with no blue dude? But according to entropy (and osmotic pressure), the piston moves to the right and water floods into the left-hand compartment to dilute the blue guy down.

More in due course...


Site last modified Tue Jul 5 12:32:01 BST 2005