Polyprotic Acids: Stepwise Dissociation

Sulfuric acid has two acidic hydrogens. So a 0.010 mol/dm³ solution should contain 0.020 mol/dm³ of H⁺ and have a pH of 1.70, right? Measure it, and you’ll find the pH is closer to 1.84. Somewhere along the way, a chunk of the second proton has gone missing.

It hasn’t really gone missing. It just hasn’t left. Acids with more than one acidic proton, called polyprotic acids, give them up one at a time, and each step is weaker than the one before. Understanding this stepwise behaviour explains the pH of sulfuric acid, how phosphate buffers your cells, and how carbon dioxide dissolving in the ocean changes its chemistry.

Types of acids by number of protons

Type	Acidic protons	Examples
Monoprotic	1	HCl, HNO₃, CH₃COOH, HF
Diprotic	2	H₂SO₄, H₂CO₃, H₂S, oxalic acid (H₂C₂O₄)
Triprotic	3	H₃PO₄, citric acid (C₆H₈O₇)

Only hydrogens bonded to highly electronegative atoms (usually oxygen) are acidic. Phosphorous acid, H₃PO₃, has three hydrogens but only two are acidic; the third is bonded directly to phosphorus and stays put. Citric acid has eight hydrogens but only three acidic ones, one on each –COOH group.

Stepwise dissociation

A polyprotic acid ionises in separate equilibria, each with its own constant. For phosphoric acid:

H₃PO₄ ⇌ H⁺ + H₂PO₄⁻ Ka1 = 7.1 × 10⁻³ (pKa1 = 2.15) H₂PO₄⁻ ⇌ H⁺ + HPO₄²⁻ Ka2 = 6.3 × 10⁻⁸ (pKa2 = 7.20) HPO₄²⁻ ⇌ H⁺ + PO₄³⁻ Ka3 = 4.5 × 10⁻¹³ (pKa3 = 12.35)

Notice the pattern: each Ka is roughly 100,000 times smaller than the last. That’s typical.

Why each step gets weaker

Two reasons, both about electric charge:

1. Pulling a proton off a negative ion is harder. The first proton leaves a neutral molecule. The second has to escape from an ion that’s already −1, which attracts the positive proton back. The third must leave a −2 ion.
2. The ion left behind carries more charge. Concentrating extra negative charge on a smaller number of atoms is less stable.

So the first dissociation always dominates, which makes calculations much simpler than they look.

Acid constants of common polyprotic acids

Acid	pKa1	pKa2	pKa3
Sulfuric acid, H₂SO₄	strong (below 0)	1.99	—
Oxalic acid, H₂C₂O₄	1.25	4.27	—
Sulfurous acid, H₂SO₃	1.86	7.2	—
Phosphoric acid, H₃PO₄	2.15	7.20	12.35
Citric acid	3.13	4.76	6.40
Carbonic acid, H₂CO₃*	6.35	10.33	—
Hydrogen sulfide, H₂S	7.0	very high (above 12; exact value disputed)	—

*The carbonic acid value is for all dissolved CO₂ treated as H₂CO₃, the convention used in most data books.

Calculating pH: usually only the first step matters

Because Ka2 is so much smaller than Ka1, the H⁺ from the second step is tiny by comparison. For most polyprotic acids you can calculate pH from Ka1 alone, treating the acid as if it were monoprotic.

Example: 0.10 mol/dm³ phosphoric acid.

The approximation √(Ka1 × c) gives 0.027 mol/dm³, which is 27% of c, far too big for the shortcut. Solve the quadratic instead:

x² + (7.1 × 10⁻³)x − 7.1 × 10⁻⁴ = 0 → x = 0.0233 mol/dm³

pH = 1.63

The second step adds only about 6 × 10⁻⁸ mol/dm³ of H⁺, which is completely negligible. A neat result falls out: in a solution of a diprotic or triprotic acid, the concentration of the doubly-deprotonated ion is roughly equal to Ka2. Here, [HPO₄²⁻] ≈ 6.3 × 10⁻⁸ mol/dm³.

The special case of sulfuric acid

Sulfuric acid breaks the “only the first step matters” rule, because its second step, Ka2 = 1.0 × 10⁻² to 1.2 × 10⁻², is still quite large.

Example: pH of 0.010 mol/dm³ H₂SO₄ (Ka2 = 1.2 × 10⁻²).

Step 1 is complete: [H⁺] = 0.010, [HSO₄⁻] = 0.010.

Step 2: let x of the HSO₄⁻ dissociate.

	HSO₄⁻	H⁺	SO₄²⁻
Start	0.010	0.010	0
Change	−x	+x	+x
Equilibrium	0.010 − x	0.010 + x	x

Ka2 = (0.010 + x)(x) ÷ (0.010 − x) = 0.012

This rearranges to x² + 0.022x − 1.2 × 10⁻⁴ = 0, giving x = 0.0045 mol/dm³.

Total [H⁺] = 0.010 + 0.0045 = 0.0145 mol/dm³ → pH = 1.84

So the second proton is only about 45% released at this concentration. Many school courses simplify by treating sulfuric acid as fully diprotic (pH 1.70). That’s a convenient approximation, and you should use whatever your exam specifies, but it’s worth knowing it’s an approximation. In more concentrated solutions the second step is suppressed even more, because the large amount of H⁺ from step one pushes the second equilibrium back to the left.

Which species dominates at which pH?

For each step, the two species are present in equal amounts when pH = pKa. That gives a simple map for phosphoric acid:

pH range	Main species
below 2.15	H₃PO₄
2.15 to 7.20	H₂PO₄⁻
7.20 to 12.35	HPO₄²⁻
above 12.35	PO₄³⁻

At pH 7.4, the pH of blood, phosphate exists mainly as a mixture of H₂PO₄⁻ and HPO₄²⁻, in a ratio of about 1 : 1.6. That pair makes an excellent buffer right where your body needs it. Read more in buffers explained.

Carbonic acid and the oceans

When CO₂ dissolves in water, it forms carbonic acid, which dissociates in two steps:

CO₂ + H₂O ⇌ H₂CO₃ ⇌ H⁺ + HCO₃⁻ ⇌ 2H⁺ + CO₃²⁻

Seawater has a pH of about 8.1, between pKa1 (6.35) and pKa2 (10.33), so most dissolved carbon is present as hydrogencarbonate, HCO₃⁻. As the ocean absorbs more CO₂ from the atmosphere, the balance shifts: more H⁺ forms, pH drops, and the carbonate ion concentration falls. Corals, shellfish and plankton need carbonate ions to build calcium carbonate shells, which is why ocean acidification worries marine biologists.

Titrating a polyprotic acid

When you titrate a polyprotic acid with sodium hydroxide, each proton is removed in turn. If the pKa values are far enough apart (roughly 4 units or more), the titration curve shows two or three separate steep jumps, one for each equivalence point. Phosphoric acid shows clear first and second equivalence points; the third is too weak to show in water. Each equivalence point needs the same volume of alkali as the first.

Quick check

Oxalic acid has pKa1 = 1.25 and pKa2 = 4.27. Which species dominates at pH 3? Answer: HC₂O₄⁻, because pH 3 lies between the two pKa values.

Common mistakes

• Doubling or tripling [H⁺] automatically. Only valid if every step is essentially complete, which is rarely true.
• Adding pKa values or averaging them to get a single “acid strength”.
• Counting every H as acidic. Check which hydrogens are attached to oxygen.
• Forgetting that the middle species are amphiprotic. H₂PO₄⁻ and HCO₃⁻ can act as acids or bases.

Key takeaways

• Polyprotic acids lose protons one at a time, with Ka1 > Ka2 > Ka3, often by a factor of about 10⁵ each.
• For most, the pH depends almost entirely on Ka1.
• Sulfuric acid’s second step is large enough to matter: 0.010 mol/dm³ H₂SO₄ has pH ≈ 1.84, not 1.70.
• Each conjugate pair is 50 : 50 at pH = pKa, which maps out which species dominates.
• For single-step calculations, see pH of weak acids and Ka and pKa.