Two Excluded volume paradoxes, and a discussion of osmotic pressure
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
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
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?
(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
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
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...