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Acids and Bases
Chemists use several definitions to determine which compounds are acids or bases, but the one most people are familiar with is the Lowry-Bronsted version. This says that an acid is a chemical that can donate a positive hydrogen ion (H+), while a base is a chemical that can accept H+. The strong acids that are traditionally used in chemistry -- such as HCl and HNO3 - all fit this definition because they all contain hydrogen and release H+ in a water solution.
Conjugate Acids and Bases
When an acid dissolves in water and releases H+, the remaining part of the molecules is left with a negative charge. Interestingly, that remaining portion can now act as a base because it can accept an H+ and change back into the original acid. This type of species is known as a conjugate base. In the same manner, when a base such as ammonia (NH3) accepts an H+, it turns into a conjugate acid (NH4+ in the case of ammonia).
Sulfuric Acid and Its Conjugate Bases
Sulfuric acid is slightly unusual for an acid. It meets the standard definition of an H+ donor, but it has two H+ that it can give up. After H2SO4 loses its first H+, it becomes bisulfate (HSO4-), which is the conjugate base of the original acid, but, oddly, is still acidic since it can lose its own H+ to become the sulfate ion. The sulfate ion is thus the conjugate base of the bisulfate ion.
Sulfate and Acid
The sulfate ion is truly a conjugate base, since it can potentially gain an H+ and change back into the bisulfate ion HSO4-. This is possible because bisulfate, although acidic, is a weak acid. When dissolved in water, only some bisulfate molecules lose an H+, so there is an equilibrium between HSO4- and its products H+ and SO4(2-). So, if sulfate is combined with any acidic species, such as HCl, it will react with that acid by accepting its H+ and changing into HSO4-.