van't Hof's equation for osmotic pressure is derived with the assumption, among others, that the particle size of the solute is comparable to the particles size of the solvent. It holds well so long as they are at least within an order of magnitude or two of each other -- but clearly it breaks down eventually, because you don't get much in the way of osmosis with a suspension of sand, or (to be silly) a handful of ball bearings.

I'm interested in the middle range, here -- in particular, I'm assuming there is a middle range, where particles contribute somewhat to osmolarity but at a lower rate than unity. Is there a decent derivation -- or a formula, I'd accept a formula but I'd prefer a derivation -- for the dependence of osmotic pressure (or potential, or otherwise closely related property) on particle size?


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