Diffusion is a spontaneous process in which molecules move with their concentration gradient. For example, if you place food coloring in water, the food coloring will slowly diffuse through the water until the entire solution has been balanced.
Osmosis is specifically the movement of water through a semipermeable membrane - meaning a membrane that can let some substances in but keep others out - and, similar to diffusion, it moves with its concentration gradient. For example, if you place a glucose solution sealed in plastic in water, water will move into the plastic to even out the concentration of glucose in the entire solution because glucose is too large to diffuse freely.
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