Why does crystallization separate mixtures?

Crystallization separates because the components have different solubilities at high and low temperatures.

Crystallization is the slow precipitation of crystals from a saturated solution.
When it is used to purify an impure solid, the process is often called fractional crystallization or recrystallization.
You heat a sample of the impure compound with a suitable , often at its boiling point.
Then you add enough extra solvent until the solid just dissolves. At this point, the solution is saturated.
As the solution gradually cools, the solubility of the compound decreases. Crystals of the pure compound precipitate out.
The impurity, which is present in small amount, remains unsaturated in the solution.
When you filter the cold mixture, crystals of the pure compound remain on the filter paper. The solvent containing the impurity passes through.

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