## Titration Curves Of Diprotic Acids

A diprotic acid is an acid with two ionizable hydrogens such that it yields two H

^{+}ions for each acid molecule. Examples of diprotic acids are H_{2}CO_{3}(carbonic acid) , H_{2}C_{2}O_{4}(oxalic acid), and H_{2}SO_{3}(sulfurous acid). The ionization of a general diprotic weak acid ionizes in water in two steps:H

_{2}A_{}(aq) H^{+}(aq) + HA_{}^{-}(aq) K_{a1}= [H^{+}][HA_{}^{-}]/[H_{2}A_{}]HA

^{-}(aq) H^{+}(aq) + A^{2-}(aq) K_{a2}= [H^{+}][A_{}^{2-}]/[HA_{}^{-}]Because H

_{2}A has two ionizable hydrogens, its titration curve has two equivalence points, as shown below.

The equations for the acid-base reaction occurring between H

_{2}A_{}and NaOH are:from the beginning to the first equivalence point:

H

_{2}A_{}(aq) + NaOH(aq) NaHA_{}(aq) + H_{2}O(l)from the first to the second equivalence point:

NaHA

_{}(aq) + NaOH(aq) Na_{2}A_{}(aq) + H_{2}O(l)and for the overall reaction:

H

_{2}A_{}(aq) + 2NaOH(aq) Na_{2}A_{}(aq) + 2H_{2}O(l)At the first equivalence point, all the H

^{+}(aq) from the first ionization of H_{2}A_{}have reacted with the NaOH. At the second equivalence point, all the H^{+}(aq) from the second ionization have reacted with the NaOH. The volume of NaOH added at the second equivalence point is twice that of the first equivalence point.It is possible to determine the H

_{2}A acid dissociation constants, K_{a1}and K_{a2}, using the graph below.K

_{a1}= [H^{+}][HA_{}^{-}]/[H_{2}A_{}] (1)K

_{a2}= [H^{+}][A_{}^{2-}]/[HA_{}^{-}] (2)

On the graph, the red vertical line at 5.12 mL is the first half-titration point which represents when one-half of the H

^{+}(aq) ions in the first ionization have been titrated with NaOH. The first half-titration point volume is one-half the volume of the first equivalence point. At the point, 5.12 mL, [H_{2}A_{}] = [HA^{-}] which can be substituted in Equation (1) above to yield:K

_{a1}= [H^{+}]~~[HA~~/_{}^{-}]~~[HA~~= [H_{}^{-}]^{+}]Taking the log of both sides yields:

log K

_{a1}= log [H^{+}]-log K

_{a1}= -log [H^{+}]pK

_{a1}= pHThis means that the pH at the first half-titration point, 5.12 mL, equals the pK

_{a1}of H_{2}A_{}. After pK_{a1 }is determined from the graph, the equation K_{a1 }= 10^{-pKa1}can be used to determine the first dissociation constant for the acid H_{2}A_{}.Using the same steps as above, it can be shown that pK

_{a2}= pH at 15.39 mL of NaOH.The same analysis can be used to determine the K

_{a}'s of triprotic acids such as phosphoric acid, H_{3}PO_{4}.