Titration Curves Explained: Equivalence Points, Half-Equivalence and Choosing an Indicator

What a titration curve shows

In an acid-base titration, a solution of known concentration (the titrant) is added from a burette to a measured volume of the analyte. The equivalence point is where the moles of titrant exactly match the moles of analyte. The end point is where the indicator changes colour.

The curve has four parts: the initial pH, the buffer region (for a weak acid or base), the steep jump around the equivalence point and the flat region of excess titrant.

The formulas for each region

For a weak acid HA titrated with NaOH:

• Start: [H⁺] = √(Ka × C).
• Buffer region: pH = pKa + log(moles A⁻ / moles HA).
• Half-equivalence: moles A⁻ = moles HA, so pH = pKa.
• Equivalence: only A⁻ is left. [OH⁻] = √(Kb × C), where Kb = Kw / Ka and C is the salt concentration in the total volume.
• After equivalence: [OH⁻] = moles of excess NaOH / total volume.

The simulator does not switch between these formulas. It solves the full charge balance, [H⁺] + [Na⁺] = [OH⁻] + [A⁻] + 2[A²⁻] + 3[A³⁻] + [Cl⁻], at every volume. This gives correct values for polyprotic acids and very dilute solutions, where the shortcuts break down.

How to use the titration curve simulator

1. Open the titration curve simulator.
2. Choose the Analyte in the flask. Acids are titrated with NaOH and bases with HCl. Pick Custom weak acid or Custom weak base to type your own pKa or pKb.
3. Enter the Analyte concentration (mol/L), the Analyte volume (mL) and the titrant concentration in the burette.
4. Read the results panel: initial pH, each equivalence point and each half-equivalence point.
5. Pick an indicator under Show indicator colour band to see its range on the plot. The indicator table lists the end point error of methyl orange, methyl red, litmus, bromothymol blue and phenolphthalein.
6. Type volumes in pH at chosen volumes to get a table of pH values.
7. Save the plot with Export PNG or the data with Export CSV.

Worked examples

1. 25.00 mL of 0.100 M HCl with 0.100 M NaOH

Start: pH = −log 0.100 = 1.00. After 10.00 mL: 2.50 − 1.00 = 1.50 mmol H⁺ in 35.00 mL, so [H⁺] = 0.0429 M and pH = 1.37.

At 25.00 mL the pH is 7.00. At 30.00 mL there is 0.500 mmol of excess OH⁻ in 55.00 mL, so pOH = 2.04 and pH = 11.96. The pH jumps from 3.7 to 10.3 between 24.90 and 25.10 mL.

2. 25.00 mL of 0.100 M acetic acid with 0.100 M NaOH

Start: [H⁺] = √(1.74 × 10⁻⁵ × 0.100) = 1.32 × 10⁻³ M, pH = 2.88. At 12.50 mL, pH = pKa = 4.76. At 20.00 mL, pH = 4.76 + log(2.00/0.50) = 5.36.

At 25.00 mL the flask holds 0.0500 M acetate. Kb = 5.75 × 10⁻¹⁰, [OH⁻] = 5.36 × 10⁻⁶ M, pH = 8.73. Phenolphthalein is the right indicator.

3. 25.00 mL of 0.100 M NH₃ with 0.100 M HCl

Start: pH = 11.12. At half-equivalence, pH = pKa of NH₄⁺ = 9.25. At equivalence the flask holds 0.0500 M NH₄⁺, so pH = 5.28. Use methyl red.

4. 25.00 mL of 0.100 M H₃PO₄ with 0.100 M NaOH

pKa values are 2.15, 7.20 and 12.38. The first equivalence point is at 25.00 mL, pH 4.70. The second is at 50.00 mL, pH 9.67. At 37.50 mL, pH = pKa₂ = 7.20.

The third equivalence point at 75.00 mL shows no jump, because HPO₄²⁻ is too weak an acid to titrate in water.

5. 25.00 mL of 0.0500 M Na₂CO₃ with 0.100 M HCl

Carbonate takes two protons. The first equivalence point (CO₃²⁻ → HCO₃⁻) is at 12.50 mL, pH 8.34. The second (HCO₃⁻ → H₂CO₃) is at 25.00 mL, pH 3.98. Methyl orange marks the second, sharper end point.

Summary table

Indicator colour ranges: methyl orange 3.1 to 4.4 (red to yellow), methyl red 4.4 to 6.2 (red to yellow), litmus 4.5 to 8.3 (red to blue), bromothymol blue 6.0 to 7.6 (yellow to blue), phenolphthalein 8.2 to 10.0 (colourless to pink).

Common mistakes

• Assuming every equivalence point is at pH 7.
• Forgetting to add the titrant volume to the analyte volume before working out concentrations.
• Using Henderson-Hasselbalch at 0 mL or at the equivalence point, where only one form is present.
• Choosing an indicator because its range includes 7, not because it falls on the steep part.
• Expecting a visible jump for every proton of a polyprotic acid.
• Treating the end point and the equivalence point as the same thing.

Exam tips

• AP Chemistry: you are often asked to sketch the curve and label the half-equivalence point. Say "pH = pKa because [HA] = [A⁻]".
• A-level: learn the four curve shapes and the two indicators. Phenolphthalein for weak acid with strong base; methyl orange for strong acid with weak base.
• JEE and NEET: at the equivalence point of a weak acid with a strong base, pH = 7 + ½(pKa + log C). For 0.05 M acetate: 7 + ½(4.76 − 1.30) = 8.73.
• The volume of titrant at equivalence depends only on moles, not on acid strength. The pH there depends on strength.

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