General Chemistry · Acid–Base Equilibria · POLYPROTIC ACIDS

Stepwise ionization of phosphoric acid and
species fractions

An acid such as phosphoric acid (H₃PO₄) can give away more than one proton. Which species dominates at each pH? Change the pH yourself and calculate the fraction (α) of the four species with exact equations. You will also solve the pH of the acid dissolved in water from the charge balance. Predict first.

Learning objectives After this activity you can …
write the stepwise ionization of a polyprotic acid as equations with Ka1, Ka2 and Ka3.
use data to decide how the dominant species and the fraction of each species change with pH.compare pH with pKa to decide the dominant species and the fraction of each species.
explain why, when the Ka values differ greatly, you may calculate the pH from the first step only.
Progress 0% · Predict → Observe → Check → Complete
Estimated time 20–25 minPrerequisites: Ka of a weak acid · pH = −log[H⁺] · pKa = −log Ka—
Level Basic Standard Advanced
Key summary Three key ideas
  1. A polyprotic acid gives away its protons one at a time, step by step. Each step has its own acid dissociation constant (Ka1, Ka2, Ka3).
  2. The species fraction α is the share of one species in the total acid concentration. The fractions of all species add up to 1.
  3. When the pH changes, the position of each step's equilibrium changes, and so do the ratios between the species.In each step, the concentration ratio of conjugate base to conjugate acid is set by the ratio of that step's Ka to [H⁺] (Ka = [H⁺][conjugate base]/[conjugate acid]).

Solution conditions

phosphoric acid H₃PO₄ · pH 1.00
Ka1 7.5×10⁻³ · Ka2 6.2×10⁻⁸ · Ka3 4.2×10⁻¹³
H₃PO₄ ⇌ H⁺ + H₂PO₄⁻ ⇌ … (stepwise ionization)
The fraction of each species and the dominant species are shown after you submit your prediction.

Dominant species: —

—

Fraction vs pH (distribution curves)

The species distribution curves are shown after you submit your prediction.
Focus the graph. The right or up arrow raises the pH by 0.05; the left or down arrow lowers it by 0.05. The pH slider does the same.
solid → dashed → dotted → dash-dot: species that have lost more and more protons—
Entry check Before you start — prerequisites and key terms

Which statement correctly describes a polyprotic acid?

An acid with several oxygen atomsAn acid with two or more H atoms that can ionizeA strong acid that ionizes completely in water
Key terms

Stepwise ionization: H₃PO₄ ⇌ H⁺ + H₂PO₄⁻ (Ka1), H₂PO₄⁻ ⇌ H⁺ + HPO₄²⁻ (Ka2), HPO₄²⁻ ⇌ H⁺ + PO₄³⁻ (Ka3). · pKa = −log Ka. · Species fraction α: concentration of one species ÷ total acid concentration (0 to 1). · Dominant species: the species with the largest fraction at that pH. · Charge balance: total positive charge concentration = total negative charge concentration.

Predict Think first, before you observe

A phosphoric acid solution is held at pH 7.21 with a buffer (25 °C). In which form does the phosphoric acid mainly exist?

Mostly PO₄³⁻ — the pH is near neutral, so it has lost all its protons H₂PO₄⁻ and HPO₄²⁻ in nearly equal amounts — about half of the second protons have left Mostly H₃PO₄ — it is a weak acid, so it hardly ionizes 25% each of the four species — all three steps happen evenly
ConfidenceMedium
The explanation opens after you lock in a prediction
phosphoric acid H₃PO₄carbonic acid H₂CO₃oxalic acid H₂C₂O₄citric acid H₃C₆H₅O₇
014
10⁻⁴ M1 M

Fractions are calculated exactly: α = (term of that species)/(h³ + Ka1h² + Ka1Ka2h + Ka1Ka2Ka3), with h = [H⁺]. "The pH of this solution" is found by solving the charge balance [H⁺] = [OH⁻] + Σ(charge × concentration) without approximation, for the acid alone dissolved at C M.

At pH = pKa, two neighboring species are half and half

Take the log of Ka2 = [H⁺][HPO₄²⁻]/[H₂PO₄⁻]: pKa2 = pH + log([H₂PO₄⁻]/[HPO₄²⁻]). When pH equals pKa2 (7.21), the log term is 0, so the two species have equal concentrations — two distribution curves cross near 50%. When pH is 1 unit above pKa, there is about 10 times more conjugate base; 2 units above, about 100 times more.

pH = pKa ⟹ [conjugate acid] = [conjugate base]

pH 7.21 is more than 5 units below pKa3 (12.38), so there is almost no PO₄³⁻ (about 10⁻⁵). The idea "neutral means all protons are lost" is wrong. In soil water near pH 5–8, the same curves show that phosphate from fertilizer exists mainly as H₂PO₄⁻ and HPO₄²⁻.

The pH of a polyprotic acid solution is set almost entirely by the first step

When Ka1 and Ka2 differ by a factor of 10³ or more, the H⁺ from the second step is negligible. For example, for 0.0037 M H₂CO₃, Ka1 alone gives [H⁺] = 4.0×10⁻⁵ M, pH 4.40, and [CO₃²⁻] is only 5.6×10⁻¹¹ M, almost equal to Ka2. The "solution pH" in this simulation checks this with the exact charge-balance solution that includes every step and water.

[H⁺] = [OH⁻] + [H₂A⁻] + 2[HA²⁻] + 3[A³⁻]
Limits Limits of this model

The model uses 25 °C, Kw = 1.0×10⁻¹⁴ and activity coefficients of 1 (ideal solution). The fraction curves assume the pH is held by a buffer. Ka values are common reference values for polyprotic acids. Some sources label pKa3 as 12.67, but Ka3 = 4.2×10⁻¹³ gives 12.38, so this simulation uses the Ka values as the reference. Constants that differ by more than 5% from literature values (CRC Handbook, 84th ed. · Wikipedia) are listed in the teacher layer. For carbonic acid, Ka1 is the apparent value that counts all dissolved CO₂ as H₂CO₃.

Check Formative check — pass all items to complete
Q1—
— — —
Hint

—

Q2 · Task—
Now: — · Goal: —
Q3—
Hint

—

Q4 · Fill in the blank—
Hint

—

Q5 · Apply (advanced, not required for completion)In human blood (pH about 7.4), what is the approximate ratio HPO₄²⁻ : H₂PO₄⁻ of the two main phosphate species? (one decimal place; use the reference Ka2 of phosphoric acid)
Reflect Reflect — metacognition · completion

My prediction result: Not predicted yet

Explain in one sentence why the idea "at pH 7.21, phosphoric acid is mostly PO₄³⁻" is wrong. Use a concept word (pKa, fraction, conjugate acid, conjugate base) and a reason word ("because", "so"):

Apply Use the idea in a new context (not required for completion)

In a carbonated soft drink (pH about 3) and in seawater (pH about 8.1), what is the main dissolved carbonic-acid species in each? Compare with pKa1 and pKa2 and explain in one or two sentences.

Next up: Kb and pH of weak bases → buffer solutions (Henderson–Hasselbalch equation) and titration of polyprotic acids.

COMPLETE · Completed — all checks passed.

Worksheet — Stepwise ionization and species fractions of polyprotic acids

Name ________ Class ________ Date ________

1. Predict — What is the main form of phosphoric acid at pH 7.21? (mostly PO₄³⁻ / H₂PO₄⁻ and HPO₄²⁻ about equal / mostly H₃PO₄ / 25% each) Reason for your choice:

2. Stepwise ionization — Write the three ionization steps of phosphoric acid and each pKa (calculated from the reference Ka, two decimal places).

H₃PO₄ ⇌ ______ pKa1 = ____ · H₂PO₄⁻ ⇌ ______ pKa2 = ____ · HPO₄²⁻ ⇌ ______ pKa3 = ____

3. Observe — On the axes below, sketch the fraction curves of the four phosphate species and mark the pH values where the curves cross (solid H₃PO₄, dashed H₂PO₄⁻, dotted HPO₄²⁻, dash-dot PO₄³⁻).

4. Calculate — pH of a 0.0037 M H₂CO₃ solution = ______

5. Fill in the blank — The fraction curves of two neighboring species cross where pH = [＿＿＿].

Curve key (teacher copy): solid H₃PO₄ · dashed H₂PO₄⁻ · dotted HPO₄²⁻ · dash-dot PO₄³⁻. Markers separate the curves.

Answer key (teacher only · not in the student printout) — 1. H₂PO₄⁻ and HPO₄²⁻ about half each (49.9% : 50.1%); PO₄³⁻ is 3.4×10⁻⁶ · 2. H⁺ + H₂PO₄⁻ (2.12) · H⁺ + HPO₄²⁻ (7.21) · H⁺ + PO₄³⁻ (12.38) · 3. Crossing pH 2.12 · 7.21 · 12.38. A label of 12.67 seen in some graphs does not match the Ka3 value · 4. 4.40 · 5. pKa (the pKa of that step)
Teacher notes — open the lesson plan

Lesson plan · 5E phase and time

  • Explore and Explain, 20–25 min. Right after monoprotic weak acids, before buffer solutions. Whole-class demonstration (?mode=teacher) or individual work.

Demonstration in 3 steps

  • ① Phosphoric acid, scan pH automatically from 0 to 14 — the dominant species changes four times.
  • ② Stop at pH 7.21 — two bars are almost the same length; the third anion is near 0.
  • ③ Carbonic acid 0.0037 M, "Go to the pH of this solution" — 4.40; the second step contributes almost nothing.

Three key questions

  • "Can you predict from the table where two curves cross?"
  • "The acid can give three H⁺. Why is [H⁺] of the acid solution not three times larger?"
  • "HPO₄²⁻ dominates over the wide range between pKa2 and pKa3. What does this mean?"

Common misconceptions

  • MC-poly-01 "mostly PO₄³⁻ at pH 7.21" → the belief that neutral = fully deprotonated. pH is 5 below pKa3, so the PO₄³⁻ fraction is about 10⁻⁵.
  • MC-poly-02 "a diprotic acid doubles [H⁺]" → Ka2 is nearly 10⁴ times smaller than Ka1, so H⁺ from the second step is about 10⁻⁶ of the first. Counting it twice gives 4.10 instead of 4.40.

Calculation method (E2 exact equations)

  • Fractions: αᵢ = (h^(n−i)·K1…Kᵢ)/Σ, no approximation. Solution pH: f(h) = h − Kw/h − C·Σ i·αᵢ(h) = 0 solved by bisection in log space (f increases monotonically with h).

Reference values and a labeling inconsistency

  • Phosphoric acid 7.5×10⁻³ / 6.2×10⁻⁸ / 4.2×10⁻¹³ → pKa 2.12 / 7.21 / 12.38. Carbonic acid 4.3×10⁻⁷ / 5.6×10⁻¹¹, oxalic acid 5.9×10⁻² / 6.4×10⁻⁵, citric acid 7.4×10⁻⁴ / 1.7×10⁻⁵ / 4.0×10⁻⁷.
  • Labels pKa1 2.12 and pKa2 7.21 agree with the Ka table, but a pKa3 label of 12.67 seen in some graphs differs from 12.38 calculated from Ka3. Reference = the Ka table.
  • 0.0037 M H₂CO₃: pH 4.40, [CO₃²⁻] = 5.6×10⁻¹¹ M — exact solution 4.4015.

Literature check (2 sources: LibreTexts Table E1 = CRC Handbook 84th ed. · Wikipedia page of each substance, checked 2026-10-03)

  • Phosphoric acid pKa: CRC 2.16 / 7.21 / 12.32, Wikipedia 2.14 / 7.20 / 12.37. The reference Ka1 (7.5×10⁻³) differs from CRC (6.9×10⁻³) by 9%, and Ka3 (4.2×10⁻¹³) differs from CRC (4.8×10⁻¹³) by 13%, but is close to Wikipedia (4.3×10⁻¹³). 12.67 appears in neither source.
  • Carbonic acid pKa: CRC and Wikipedia both 6.35 / 10.33. The reference Ka2 5.6×10⁻¹¹ differs from the literature 4.7×10⁻¹¹ by 19% (listed side by side).
  • Oxalic acid pKa2: CRC 3.81, Wikipedia 4.28 — the sources disagree. The reference 4.19 lies between them. Citric acid: CRC 3.13 / 4.76 / 6.40, Wikipedia 3.128 / 4.761 / 6.396 — agrees with the reference values.
  • Grading uses the reference values only. Literature values are for display.
University general chemistry · weak acidsEngine E2 equation-driven (exact fractions · charge balance)Verified: independent Python (mpmath) multi-point comparison
Teacher mode — class controls