Lesson 3 · 30 min

Complexometric titration: EDTA and water hardness

One molecule grabs almost every metal ion, yet is fussy about pH. Conditional formation constants, the wine red and pure blue of Eriochrome Black T, and the hardness of a tap water.

In this lesson

Explain why EDTA forms 1 : 1 complexes with almost every metal ion, and why pH decides success
Compute conditional formation constants K′ = α(Y⁴⁻)·K_f and the pM–V titration curve they give
Explain how metallochromic indicators such as Eriochrome Black T work, and why a little magnesium is needed to titrate calcium alone
Determine the total hardness of water, and its calcium and magnesium, by EDTA titration

Kettle scale, white spots on the shower head, soap scum in hard water — all of it is calcium and magnesium. Waterworks and environmental labs measure “hardness” every day with the method Gerold Schwarzenbach developed in the late 1940s: titrate with ethylenediaminetetraacetic acid (EDTA) until a dye turns from wine red to pure blue. How can one molecule grab almost every metal ion — and yet be so fussy about pH?

Predict first

Why is Ca²⁺ titrated with EDTA at pH 10 rather than pH 5?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

CaY²⁻ decomposes into other products at pH 5 At pH 5 most EDTA is protonated; too little Y⁴⁻ is left to bind Ca²⁺, the conditional constant is too small and the break disappears Ca²⁺ is oxidised in acid EDTA precipitates at pH 5

One molecule, six claws

EDTA’s Y⁴⁻ has two nitrogens and four carboxylate oxygens and can occupy six coordination sites of a metal ion at once, closing five five-membered chelate rings — so whatever the metal’s charge, +1 or +3, it binds 1 : 1, with simple stoichiometry and large formation constants (lg K_f: Mg²⁺ 8.79, Ca²⁺ 10.69, Zn²⁺ 16.50, Cu²⁺ 18.80, Fe³⁺ 25.1).

The catch is that EDTA itself is a weak acid (pKa 0.0, 1.5, 2.0, 2.69, 6.13, 10.37), and only the fraction α(Y⁴⁻) present as Y⁴⁻ binds metals. Multiplying it into the formation constant gives the conditional formation constant K′ that actually operates at that pH. If the buffer also contains ammonia, which binds the metal (an auxiliary ligand), multiply by the fraction of metal not tied up by ammonia, α(M), to get K″.

K' = \alpha_{\ce{Y^4-}}\,K_\mathrm{f} = \frac{[\ce{MY^{n-4}}]}{[\ce{M^{n+}}]\,c_\mathrm{EDTA}'},\qquad K'' = \alpha_\mathrm{M}\,\alpha_{\ce{Y^4-}}\,K_\mathrm{f}

With lg K′ ≥ 8 (for metal near 0.01 M) the break at equivalence is steep enough for an accurate titration.

$\alpha_{\ce{Y^4-}}$: fraction of EDTA present as Y⁴⁻, set by the pH
$c_\mathrm{EDTA}'$: total EDTA not bound to metal (all protonated forms)
$\alpha_\mathrm{M}$: fraction of metal not complexed by the auxiliary ligand

Fraction of EDTA present as Y⁴⁻ (computed from the site’s EDTA acid constants)
pH	α(Y⁴⁻)	lg K′ (Ca²⁺)
4	3.0 × 10⁻⁹	2.2
6	1.8 × 10⁻⁵	5.9
8	4.2 × 10⁻³	8.3
10	0.30	10.2
12	0.98	10.7

Source: EDTA pKa 0.0, 1.5, 2.0, 2.69, 6.13, 10.37 (Harris, I = 0.1 M); lg K_f(CaY²⁻) = 10.69 (Martell & Smith).

The curve has three parts: before equivalence the free metal is simply the metal left over; at equivalence $[\ce{M}] = \sqrt{c_\ce{MY}/K'}$ ; after it, excess EDTA sets $[\ce{M}] = [\ce{MY}]/(K'c_\text{EDTA}')$ . Taking lg K′ = 8 as the threshold gives the lowest pH at which each metal can be titrated accurately: Fe³⁺ about 1.2, Zn²⁺ about 4, Ca²⁺ about 7.6 and Mg²⁺ about 9.7. That is the basis of selective titration by pH control — titrate Fe³⁺ at pH 2 and calcium and magnesium hardly take part.

Eriochrome Black T: a coloured decoy

A metallochromic indicator is itself a ligand whose metal complex has a different colour from its free form. At pH 10 Eriochrome Black T (EBT) is blue as free HIn²⁻ and wine red as MgIn⁻. At the start a little Mg²⁺ holds the indicator and the solution is wine red; EDTA first complexes the free Ca²⁺ and Mg²⁺, and near equivalence the stronger EDTA pulls Mg²⁺ out of MgIn⁻, releasing the blue indicator: wine red → violet → pure blue.

The indicator’s MIn⁻ must be weaker than MY, but not too weak. MgIn⁻ has a formation constant of 1.0 × 10⁷, so at pH 10 it changes colour around pMg 5.4 ± 1, right in the break; CaIn⁻ has only 2.5 × 10⁵ and changes around pCa 3.8 ± 1, well before equivalence — titrate calcium alone and the red fades early and drags. The cure is a little Mg–EDTA in the buffer: Ca²⁺ displaces its Mg²⁺, the magnesium is titrated last, and the endpoint is sharp again; or use calmagite instead.

pH 10 ammonia buffer · Eriochrome Black T · 0.01000 M EDTA

Water hardness by EDTA titration

Sample #

Flask: 50.00 mL of water + 2 mL of pH 10 buffer + 3 drops of Eriochrome Black T

Burette reading 0.00mL

Colourwine red

Flow rate · 0.25 mL/s

The indicator right now

Bound to metal MIn⁻ (wine red) 100 %

Free HIn²⁻ (blue) 0 %

Wine red → violet → pure blue. The endpoint is when the last trace of red disappears and the solution is pure blue.

Add EDTA drop by drop and watch the flask go from wine red through violet to pure blue; press “This is the endpoint” when the last trace of red is gone. Then switch to the calcium standard: without Mg–EDTA the colour turns violet early and the endpoint drags; tick “Add a little Mg–EDTA” and try again.

Worked example

Total hardness, calcium and magnesium of a water sample

A 50.00 mL water sample at pH 10 with Eriochrome Black T needs 12.45 mL of 0.01000 M EDTA. A second 50.00 mL portion, brought to pH 12 with NaOH (so Mg(OH)₂ precipitates), needs 8.90 mL with a calcium indicator. Find the total hardness (mg/L as CaCO₃) and the Ca and Mg contents (mg/L).

Concept check

When titrating a pure Ca²⁺ standard, why is a little Mg–EDTA often added to the pH 10 buffer?

To raise the buffer capacity Ca²⁺ displaces Mg²⁺ from Mg–EDTA; the magnesium holds the indicator as wine-red MgIn⁻ and is titrated last, sharpening the endpoint — and the added EDTA equals the added Mg, so the result is unaffected Mg–EDTA precipitates interfering ions It is unnecessary; calcium binds EBT firmly enough

Try the calculation

For MgY²⁻, lg K_f = 8.79, and at pH 10 α(Y⁴⁻) = 0.30. What is the conditional formation constant lg K′ at pH 10?

Key takeaways

EDTA chelates almost every metal ion 1 : 1; only Y⁴⁻ binds, so the conditional constant K′ = α(Y⁴⁻)K_f changes steeply with pH.
Accurate titration needs lg K′ ≥ 8: Fe³⁺ above pH ≈ 1.2, Ca²⁺ above about 7.6, Mg²⁺ above about 9.7; pH control allows selective titration.
Eriochrome Black T at pH 10 is blue when free and wine red with Mg²⁺; add a little Mg–EDTA to titrate calcium alone.
Total hardness = c(EDTA)·V/V_sample × 100.09 g/mol (mg/L as CaCO₃); at pH 12 only calcium is titrated and magnesium follows by difference.

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Data and references

D. C. Harris, Quantitative Chemical Analysis, 9th ed. (2016), ch. 12 (EDTA titrations, α(Y⁴⁻), metal ion indicators, auxiliary complexing agents).
D. A. Skoog, D. M. West, F. J. Holler, S. R. Crouch, Fundamentals of Analytical Chemistry, 9th ed. (2014), ch. 17 (EDTA; Eriochrome Black T: K(HIn²⁻) = 2.8 × 10⁻¹², K(MgIn⁻) = 1.0 × 10⁷, K(CaIn⁻) = 2.5 × 10⁵).
A. E. Martell, R. M. Smith, Critical Stability Constants (Plenum, 1974–1989), EDTA log K_f at 20 °C, I = 0.1 M, as tabulated in D. Harvey, Analytical Chemistry 2.1.