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in essence they are the same phenomenon, just with different materials. Diffusion is the process whereby in a freely mobile solution (i.e. liquid), a substance will have a net spread from areas of high concentraion to areas of low concentration. There is still movement in the opposite direction but it is outweighed by the above movement so that eventually, the solution will be of equal concentraion throughout. This is seen if you have a glass of water and pour orange juice concentrate in it. The orange juice will swirl around and slowly spread out, given enough time will produce a glass of equal concentration throughout.
Osmosis is diffusion, but referring to the specific movement of water. So in the above example, yes the orange juice will move but so will the water itself. As you see in the below example, it will tend to move in the opposite direction though.
If you have two compartments with water and another substance (call it X), separated by a semi-permeable membrane which allows only water to pass through it, and you put some more substance (call it X) in one side, that side will become more concentrated. Normally X itself will diffuse throughout both compartments so the concentration becomes equal, however because of the membrane it can't pass between compartments. The water itself can though, so the water moves from one to the other. However, to make both compartments the same concentration, unlike X which would go from high conc to low conc, the water must go from areas of low conc (i.e. not much X) to areas of high conc, in order to dilute the high conc part and make it lower conc, which making the low conc part become a bit higher conc as water leaves.
Eventualy both compartments will have the same concentration of X, but it will be higher than what you started with as X has been increased but water hasn't. Furthermore, you would find that because there was a net movement of water from one compartment to the other, one side would be higher than the other. After equilibrium, when both sides have the same conc, there will still be movement of water between compartments but there will be no net movement, i.e. movement from side A to side B equals movement from side B to side A.
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