Solubility is the greatest mass of a solute that will dissolve in 100 g of water. Cooling a solution that was saturated while hot leaves more solute than the water can hold, and the excess separates out as crystals. This finds how much.
Solubility is the greatest mass of a solute that will dissolve in 100 g of water. Cool a solution that was saturated while hot and the water can no longer hold all of it; the excess separates out as crystals. This is recrystallisation, the basic method for purifying a solid.
Here is the mass of water, the solubility when hot, the solubility when cold and the mass recovered. Solubility is quoted per 100 g of water, so it is scaled to the water actually present.
Cooling alone removes no water, so the difference between what the water can hold hot and cold is exactly what comes out.
With a solubility of 110 when hot, 32 when cold, and 100 g of water, 78 g of crystals are recovered.
The hot solution holds 110 g dissolved; cold, it can hold only 32 g. The difference of 78 g separates out. The hot saturated solution itself is 110 ÷ (110 + 100) = 52.4 per cent solute by mass.
How strongly solubility depends on temperature varies enormously between substances. Potassium nitrate changes a great deal with temperature and purifies well this way, while sodium chloride barely changes at all and yields very little on cooling.
If water is boiled off before cooling, this calculation does not apply, because the mass of water itself has changed.