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The speed of the effect of dyes depends on the temperature of water but that also depends on the quality of the dye itself. The dye dissolve the fastest in hot water due to its kinetic energy and expanded molecules and because it has more space to move and dissolve around the water while it dissolves the slowest in cold water because molecules in cold water contracts and become more compact making it hard for the solubles to move and travel around the water requiring a much longer time to dissolve. 
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