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Water solubility, also referred to as Sw, is a measure of the ability of a given substance to dissolve into a given solvent. The material being dissolved is referred to as the solute. Solubility is a measure of the maximum amount of solute that can be clearly dissolved into a solvent at a given temperature and pressure.
The polarity of the solute and how it is matched to the solvent is one key factor in determining solubility. Commonly it is said that "likes dissolve likes." Polar materials dissolve in polar solvents, nonpolar in nonpolar solvents. The solubility can be expressed in percent by weight at a specific temperature.
Some solvents and solutes show little temperature dependence, such as NaCl, which has a solubility in water of 26.3% by weight at 25oC, and 27.7% by weight at 100oC. Still, others like glucose show more temperature dependence--such as glucose, which has a solubility of 47.6% by weight at 25oC and 84% by weight at 100oC. Some materials are infinitely soluble, like water and ethanol; and still others have vanishing small solubility like silver chloride in water. There really is no completely insoluble material since that would require zero dissolution.
Partition coefficient, unlike solubility, measures the differential ability of a solute to dissolve in two insoluble phases. Most commonly the partition coefficient is measured in octanol and water--the so-called KOW. Almost all molecules will dissolve at least marginally in both phases. The use of a separatory funnel with diethyl ether and water is a very well-known example of using partition coefficient to purify an organic compound.