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I'm curious to know how basicity and acidity arises when a salt or covalent compound is dissolved in water.

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Think of the salt in terms of its components: for example, sodium sulfide can be thought of as derived from sodium hydroxide (a strong base) and hydrogen sulfide (a weak acid) - resulting in a net basic character. When you dissolve sodium sulfide in water, the sulfide ion partially reacts with water to form hydrosulfide ion and hydrogen sulfide, plus hydroxide ion. Aluminum chloride, on the other hand, is derived from aluminum hydroxide (a weak base) and hydrogen chloride (a strong acid), so it is an acidic salt. Sodium chloride, from sodium hydroxide and hydrogen chloride, is a neutral salt (fully ionized). Knowing the dissociation constants of the corresponding acid and base of a salt, one can calculate how acidic or basic the salt is.

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Some salts may have basic Brønsted/Lowry anions but are extremely stable in water. This belies the true nature of the salt's anion.

Consider silver sulfide as an example. This salt has a Ksp value of about ${6*10^{-51}}$. Despite the fact that the sulfide anion is a highly basic Brønsted/Lowry base, a solution of silver sulfide will actually exhibit a pH of 7.0.

Note that salts do not have to be exclusively Brønsted-Lowry bases/acids. Some salts may also exhibit Lewis acidity/basicity in water. This is another way in which basicity and acidity may arise when a salt or covalent compound is dissolved in water. Consider boric acid. This isn't a salt, but it is a covalent compound. It is an acid, but some argue it is not an acid in the sense of being a proton donor. The central boron instead utilizes its empty 2p orbital to heterolytically cleave water molecules as to form H+ in solution.

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This question is directly related to the inter relatedness of Lewis and Brønsted acidity.

Typical metal salts (let's say MCl3) undergo hydrolysis (presumably in an aqueous solution) as Lewis basic water (which donates electrons to the acid in this context) coordinates (binds) to the Lewis acidic metal (which accepts electrons from the Lewis base). After, before or during coordination (it's hard to tell which without extensive experiment), water will give up one or two protons (i.e. hydrogen atom minus an electron), thereby generating a metal oxo/hydroxide (in this case, e.g. MO2H) or metal hydroxide (e.g. M(OH)3), either of which will be accompanied by acid in solution in the form of 3 eq of HCl (which would be completely dissociated into H+ and Cl-).

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