Why does recrystallization work

To purify a freshly prepared sample, the preparative chemist will crystallize then recrystallize the compound until convinced it is pure. To recrystallize, we first dissolve the compound in hot solvent. The solubility s of the compound depends on the temperature T. The value of s is high at high temperature, but it decreases at lower temperatures until the solubility limit is first reached and then surpassed, and solute precipitates from solution (hopefully) to yield crystals. 

The solubility s relates to a special equilibrium constant we call the solubility product Ks, defined by 

The [solute] term may, in fact, comprise several component parts if the solute is ionic, or precipitation involves agglomeration. This equilibrium constant is not written as a fraction because the effective concentration of the undissolved solute [solute] (S) can be taken to be unity. 

Like all equilibrium constants, the magnitude of the equilibrium constant Ks depends quite strongly on temperature, according to 

We say the value of [solute](s) = 1 because its activity is unity see Section 7.3. 

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