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Osmotic pressure is defined as:

The pressure that would have to be applied to a pure solvent to prevent it from passing into a given solution by osmosis.

And yet water moves from low osmotic pressure to high osmotic pressure. Shouldn't high osmotic pressure prevent osmosis? It would be really helpful is this definition were explained.

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The definition in your first paragraph doesn't match your understanding in the second. If osmotic pressure is high in "A" relative to "B", you would have to apply a physical pressure to "A" to prevent solvent moving from B to A. If there is no such pressure applied, then solvent does move from B to A. The osmotic pressure and physical pressure are separate and opposite forces.

I prefer to think of osmotic pressure as sort of a "vacuum" that "pulls" solvent towards it (of course it isn't really a vacuum so don't take this analogy too far...). The definition still works given this form of thinking: you'd have to apply as much external pressure to equal the "vacuum" in order to have no movement of solute.

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The second paragraph in the question describes two solutions separated by a semipermeable membrane (a membrane that allows water through, but not anything dissolved in the water). As it says, in those conditions, water will move from the section with low osmotic pressure to the solution with high osmotic pressure.

Osmotic pressure, like other colligative properties, is proportional to the concentration of solutes in a solution. So you can think of it is a measure of how much other stuff is in the water. Low osmotic pressure means low solute concentration. High osmotic pressure means high solute concentration. Water moving across a semipermeable membrane from an area of low solute concentration to an area of high solute concentration may be a little more intuitive. Given a barrier to solutes but not water, water will move in a way that will even out the concentration.

Now come back to the definition of osmotic pressure. The definition starts by setting up a very special set of conditions to allow us to measure the pressure -- pure solvent in one compartment, a known concentration in another compartment, and the key difference, pressure to prevent water from moving the way it otherwise would.

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