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If we have two compounds in aqueous solution, in order to separate one from the other the solution is placed into an ice bath. For example the formation of $\ce{KClO3}$ from a combination of $\ce{NaClO3(aq)}$ and $\ce{KCl(aq)}$.

I am trying to understand why potassium chlorate precipitates out of solution first as the temperature is lowered. My hypothesis is that the melting points for potassium chlorate, which is $\pu{356 ^\circ C}$, is lower than that of sodium chloride, which is $\pu{801 ^\circ C}$.

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Potassium chlorate precipitates because it's only slightly soluble is water compared to Sodium chlorate and Potassium chloride. The ice-bath only helps to speed up the process as solubility is temperature dependent.

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The reason for cooling is related to solubility at different temperatures not melting point

The reason many precipitation reactions cool the solution to generate a precipitate (or to maximise the yield of it) is because the solubility of most substances in most solvents decrease with temperature. For those substances where the solubility is much lower at low temperatures cooling make a big difference to the amount of precipitate you get.

But some substances are so insoluble cooling won't make much difference and they precipitate immediately as soon as the reaction happens. For example: Sodium sulphate is much less soluble in cold water; temperature makes little difference for sodium chloride; and cerium sulphate is more soluble in cold water. So the general idea of cooling to precipitate a solid won't always work.

None of this has any relationship to the melting points of the solid reactants or products.

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