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What is the conjugate base of H2SO3?
The bisulfite anion.
As you know, the conjugate base of an acid is determined by looking at the compound that's left behind after the acid donates one of its acidic hydrogen atoms.
The compound to which the acid donates a proton acts as a base. The conjugate base of the acid will be the compound that reforms the acid by accepting a proton.
In this case, sulfurous acid has two protons to donate. However, the conjugate base of sulfurous acid will be the compound left behind after the first hydrogen ion is donated.
##"H"_2"SO"_text(3(aq]) rightleftharpoons "H"_text((aq])^(+) + color(blue)("HSO"_text(3(aq])^(-))##
The compound left behind after sulfurous acid donates its first acidic hydrogen is called the bisulfite anion, ##"HSO"_3^(-)##.
As you can see, the bisulfite anion can reform the sulfurous acid by accepting a proton.
SIDE NOTE Sulfurous acid molecules are actually represented as sulfur dioxide and water. The first dissociation for sulfurous acid would look like this
##"SO"_text(2(aq]) + "H"_2"O"_text((l]) rightleftharpoons "H"_text((aq])^(+) + "HSO"_text(3(aq])^(-)##