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Alkali salts or basic salts are salts that are the product of incomplete neutralization of a strong base and a weak acid.
Rather than being neutral , alkali salts are bases as their name suggests. What makes these compounds basic is that the conjugate base from the weak acid hydrolyzes to form a basic solution. In sodium carbonate, for example, the carbonate from the carbonic acid hydrolyzes to form a basic solution. The chloride from the hydrochloric acid in sodium chloride does not hydrolyze, though, so sodium chloride is not basic.
The difference between a basic salt and an alkali is that an alkali is the soluble hydroxide compound of an alkali metal or an alkaline earth metal. A basic salt is any salt that hydrolyzes to form a basic solution.
Another definition of a basic salt would be a salt that contains amounts of both hydroxide and other anions. White lead is an example. It is basic lead carbonate, or lead carbonate hydroxide.