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The dye will diffuse fastest in the hot water, and most slowly in the ice cold water. This happens because temperature is actually an indicator of the kinetic energy, or motion, of the molecules. The higher the temperature, the higher the average speed of the molecules is. Since molecules move about randomly and collide with each other and the walls of the container, they change position and mix, which is what we refer to as diffusion. The hotter, faster moving molecules will experience more collisions per second and so will mix faster, resulting in faster diffusion of the dye in the hottest water. By the same logic, the ice cold water molecules are moving very slowly, and they will have fewer collisions. As a result the dye will tend to stay in one place much longer in the ice water, and diffusion will be very slow.
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