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A base is a type of chemical, one that can be said to receive a proton, or make a -OH ion in water. By the Lewis theory, a base is an electron-pair donor.
Strong vs. weak
A strong base is one that disassociates completely in the solvent. A weak base is one that does not associate completely, but achieves an equilibrium between conjugate acids and bases.
Superbases are very strong bases which react with water, like if it was an acid, to completion. Some examples of superbases are alkoxides and alkyl compounds of alkali metals.
Finally, there are compounds that do not behave as bases in aqueous solution but can be forcibly protonated and forced to act as bases. Such compounds are said to be weaker bases than water. The conjugate acids of these "bases" - usually exotic oniums like phosphonium or methanium - are superacids, and it takes another superacid to synthesize one of these ions. Examples of such extra-weak bases are phosphine and methane.
Arrhenius ran into problems when trying to describe the basic nature of ammonia, because ammonia does not contain a hydroxide group. Instead, Brønsted and Lowry independently described bases as being proton acceptors rather than hydroxide donors. This allowed them to describe the action of ammonia as creating hydroxide ions by removing a proton from the solvent (in most cases, water), forming the conjugate acid, and leaving hydroxide as the base:
NH3 + H2O ⇌ NH4+ + OH-
Ammonia is a weak base, so the reaction does not go to completion.
By this definition, sodium hydroxide is not a base in itself, but it is a source of a base, hydroxide. The hydroxide ion can accept a proton to become water.
Lewis bases extend the theory of proton acceptance to exactly what does the accepting: electron pairs. Lewis bases donate free electron pairs to species which lack them. All Brønsted-Lowry bases are Lewis bases because they donate electron pairs to at least one species (in this case, a proton). However, they can react with any sort of Lewis acid, one common example being metal ions. When a base such as ammonia or ethylenediamine interacts with a sufficiently acidic metal ion such as nickel, a complex ion forms that is an adduct between the metal ion and the base.
[Ni(H2O)6]2+ + 3C2H4(NH2)2 → [Ni(C2H4(NH2)2)3]2+ + 6H2O
Complexation reactions are in equilibrium, but the equilibrium normally shifts so far to the right that it is convenient to say that it goes to completion.
A Usanovich base is any donor of a positive species or an acceptor of a negative one. These species can be individual electrons, thus allowing redox reactions to be described as a reaction between an acid (the oxidizer) and a base (the reducer).
Hard-soft acid-base theory
The most common strong base in the lab is sodium hydroxide. Potassium hydroxide, rubidium hydroxide, and cesium hydroxide are all useful as well, but are harder to find and are generally more expensive. Lithium hydroxide, calcium hydroxide, strontium hydroxide, and barium hydroxide are not very soluble in water, but count as strong bases. Of these, Calcium hydroxide is used most, and the least expensive of all hydroxides. Potassium hydroxide is most useful for energetic materials, as the potassium emission lines are weak and do not interfere with any desired colors.
Ammonia is the most common weak base in the lab.