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Conjugate bases are what remain when an acid donates a proton. In a reaction, you can spot them because they are often negatively charged because they have donated a proton during the reaction. For example, formic acid, HCOOH, has a conjugate base of HCO2- (the H+ is removed from the carboxyl group in carboxylic acids).
If the reaction is in water, then whatever molecule that isn't the hydronium or hydroxide ion is the conjugate acid/base, depending on whether the molecule before the reaction is an acid or a base. Acids have conjugate bases and bases have conjugate acids.
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