Question

Write out the stepwise \(K_{\mathrm{a}}\) reactions for citric acid \(\left(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}\right)\), a triprotic acid.

Short answer

The stepwise dissociation reactions and corresponding Ka expressions for citric acid are: 1. \( \mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} (aq) + \mathrm{H}_{2} \mathrm{O} (l) \longleftrightarrow \mathrm{H}_{3} \mathrm{O}_{\mathrm{a}} + \mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-} (aq)\) $$K_{a1} = \dfrac{[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}][\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-}]}{[\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}]}$$ 2. \( \mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-} (aq) + \mathrm{H}_{2} \mathrm{O} (l) \longleftrightarrow \mathrm{H}_{3} \mathrm{O}_{\mathrm{a}^{+}} + \mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-} (aq)\) $$K_{a2} = \dfrac{[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}^+][\mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-}]}{[\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-}]}$$ 3. \( \mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-} (aq) + \mathrm{H}_{2} \mathrm{O} (l) \longleftrightarrow \mathrm{H}_{3} \mathrm{O}_{\mathrm{a}}^{+} + \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{3-} (aq)\) $$K_{a3} = \dfrac{[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}^+][\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{3-}]}{[\mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-}]}$$

Step-by-step solution

Identify acidic protons

Citric acid has three acidic protons in its structure. These protons are attached to oxygen atoms in the carboxylic acid (COOH) functional groups, and can dissociate in water to form a hydronium ion (H₃O⁺) and corresponding citrate ion.

First dissociation reaction

The first dissociation reaction involves one acidic proton being removed from the citric acid molecule, forming the hydrogen di-citrate ion. The balanced chemical equation for the first dissociation reaction is: $$ \mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} \left( \mathrm{aq} \right) + \mathrm{H}_{2} \mathrm{O} \left( \mathrm{l} \right) \longleftrightarrow \mathrm{H}_{3} \mathrm{O}_{\mathrm{a}} + \mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-} \left( \mathrm{aq} \right) $$

First Ka expression

The acid dissociation constant (Ka) expression for the first dissociation step is: $$ K_{a1} = \dfrac{[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}][\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-}]}{[\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}]} $$

Second dissociation reaction

The second dissociation reaction involves another acidic proton being removed, this time from the hydrogen di-citrate ion, forming the dihydrogen citrate ion. The balanced chemical equation for the second dissociation reaction is: $$ \mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-} \left( \mathrm{aq} \right) + \mathrm{H}_{2} \mathrm{O} \left( \mathrm{l} \right) \longleftrightarrow \mathrm{H}_{3} \mathrm{O}_{\mathrm{a}^{+}} + \mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-} \left( \mathrm{aq} \right) $$

Second Ka expression

The acid dissociation constant (Ka) expression for the second dissociation step is: $$ K_{a2} = \dfrac{[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}^+][\mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-}]}{[\mathrm{H}_{2} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{-}]} $$

Third dissociation reaction

The third and final dissociation reaction involves the last acidic proton being removed, from the dihydrogen citrate ion, forming the tricitrate ion. The balanced chemical equation for the third dissociation reaction is: $$ \mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-} \left( \mathrm{aq} \right) + \mathrm{H}_{2} \mathrm{O} \left( \mathrm{l} \right) \longleftrightarrow \mathrm{H}_{3} \mathrm{O}_{\mathrm{a}}^{+} + \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{3-} \left( \mathrm{aq} \right) $$

Third Ka expression

The acid dissociation constant (Ka) expression for the third dissociation step is: $$ K_{a3} = \dfrac{[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}^+][\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{3-}]}{[\mathrm{HC}_{6} \mathrm{H}_{5} \mathrm{O}_{7}^{2-}]} $$ These are the final stepwise dissociation reactions and Ka expressions for citric acid.