14.5 Polyprotic Acids

Polyprotic Acids

Polyprotic acids can donate more than one proton ($H^+$) per molecule. They ionize in steps, each with its own equilibrium constant, and each step is weaker than the last. Understanding this stepwise behavior is essential for pH calculations, buffer design, and interpreting titration curves that have multiple equivalence points.

Stepwise Ionization of Polyprotic Acids

A polyprotic acid loses its protons one at a time, not all at once. Each step produces a conjugate base that then acts as the acid in the next step, and each step has its own $K_a$ value.

Sulfuric acid ($H_2SO_4$) is diprotic (two protons):

1. $H_2SO_4 + H_2O \rightleftharpoons HSO_4^- + H_3O^+$ ($K_{a1}$ = very large; strong acid, essentially complete)

2. $HSO_4^- + H_2O \rightleftharpoons SO_4^{2-} + H_3O^+$ ($K_{a2} = 1.2 \times 10^{-2}$)

Notice that the first step is a strong acid dissociation, but the second step is weak. This makes sulfuric acid unusual among polyprotic acids.

Phosphoric acid ($H_3PO_4$) is triprotic (three protons):

1. $H_3PO_4 + H_2O \rightleftharpoons H_2PO_4^- + H_3O^+$ ($K_{a1} = 7.5 \times 10^{-3}$)

2. $H_2PO_4^- + H_2O \rightleftharpoons HPO_4^{2-} + H_3O^+$ ($K_{a2} = 6.2 \times 10^{-8}$)

3. $HPO_4^{2-} + H_2O \rightleftharpoons PO_4^{3-} + H_3O^+$ ($K_{a3} = 4.8 \times 10^{-13}$)

Each product on the right is the conjugate base of the acid on the left, and it becomes the acid in the next step.

Relative Strengths of Successive Ionizations

Each successive ionization is weaker than the one before it: $K_{a1} > K_{a2} > K_{a3}$. The reason is electrostatic: once the molecule has already lost a proton and carries a negative charge, it's harder to pull away another positive $H^+$ from a species that's now more negatively charged.

You can also compare strengths using $pK_a$ values ($pK_a = -\log K_a$). A smaller $pK_a$ means a stronger acid.

• Sulfuric acid: $pK_{a1} < 0$, $pK_{a2} = 1.92$
• Phosphoric acid: $pK_{a1} = 2.12$, $pK_{a2} = 7.21$, $pK_{a3} = 12.32$

For phosphoric acid, each $pK_a$ jumps by roughly 5 units, reflecting how dramatically weaker each step becomes. This same trend applies to polyprotic bases in reverse: successive protonation steps become progressively weaker ($K_{b1} > K_{b2} > K_{b3}$).

Equilibrium Calculations for Polyprotic Acids

The large gap between successive $K_a$ values is what makes these calculations manageable. The key simplifying assumption: if $K_{a1}$ is much larger than $K_{a2}$, you can treat the first ionization separately and ignore the contribution of later steps to $[H_3O^+]$.

For a weak polyprotic acid like $H_3PO_4$:

1. Set up an ICE table using only the first ionization and $K_{a1}$.
2. Solve for $[H_3O^+]$ and $[H_2PO_4^-]$ from that step. These two concentrations will be approximately equal.
3. If you need $[HPO_4^{2-}]$, plug your results into the $K_{a2}$ expression. You'll find that $[HPO_4^{2-}] \approx K_{a2}$ (a useful shortcut when $K_{a1} \gg K_{a2}$).
4. For $[PO_4^{3-}]$, use the $K_{a3}$ expression with the values from step 3. This concentration will be extremely small.

For $H_2SO_4$ the approach differs because the first step is strong:

1. Assume the first dissociation goes to completion. If you start with 0.10 M $H_2SO_4$, you immediately have 0.10 M $HSO_4^-$ and 0.10 M $H_3O^+$.
2. Use the $K_{a2}$ expression with an ICE table to find how much additional $H_3O^+$ comes from the second step, and solve for $[SO_4^{2-}]$.

The pH comes from the total $[H_3O^+]$ across all relevant steps.

Acid-Base Behavior and Applications

• Buffers: Polyprotic acids and their conjugate bases are excellent buffer components. For example, the $H_2PO_4^- / HPO_4^{2-}$ pair buffers near pH 7.21 (its $pK_{a2}$), which is why phosphate buffers are widely used in biological systems.
• Titration curves: When you titrate a polyprotic acid with a strong base, you'll see multiple equivalence points, one for each ionization step. A diprotic acid gives two equivalence points; a triprotic acid gives three. The $pK_a$ values appear as the midpoints of each buffering region on the curve.