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In general this refers to acid-base theories. Specifically this refers to the Bronsted-Lowry definition of an acid. In general, according to this theory, a reaction occurs and there is a proton transfer. An acid will "donate" a proton and a base will "accept" a proton in reaction.

This theory also stipulates that there is a conjugate acid and conjugate base. This means that in the forward reaction you have:

A1 (acid) + B1(-) (base) --> A2(-) + B2

This reaction is usually reversible because A2(-) is now able to accept a proton, thus is a BL (Bronsted-Lowry) base.

Therefore, according to BL theory, acids are proton donors and base are proton acceptors. The reactions are usually reversible, and A1 is the acid and A2 is the conjugate base, so it follows that B1 is a base and B2 is a conjugate acid.

more examples:

H2O is the acid and NH3 is the base. Do you see why?

NH3 --> NH4 thus is accepts a proton so NH3 is the acid and NH4 is the conjugate base.

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