Potentiometric Titration

The idea: read the endpoint as a voltage

In an ordinary titration you add a reagent of known concentration (the titrant) from a burette into the unknown (the analyte) and stop when a dye flips color. That color change is a proxy — phenolphthalein turns from colorless to pink near pH 8.2, which happens to fall close to the equivalence point of a strong acid being neutralized. But the dye is a guess: it triggers over a range, it cannot be seen in dark or cloudy liquids, and one analyst's faint pink is another's clear. Potentiometric titration replaces the eye with an electrode. You immerse an indicator electrode that responds to the species being consumed and a stable reference electrode, and you record the potential between them after every increment of titrant. There is no indicator dye anywhere in the beaker.

The data you collect is a list of (V, E) pairs — titrant volume in milliliters, cell potential in millivolts. Plotted, they trace a sigmoid: nearly flat far from the equivalence point, then a steep wall, then flat again. The equivalence point — where moles of titrant exactly match moles of analyte by stoichiometry — sits in the middle of that wall, at the point of maximum slope. That maximum-slope volume is the experimental endpoint, and in a well-behaved titration it lands within a hair of the true equivalence point.

Why the potential leaps: the Nernst equation

The electrode does not measure concentration directly; it measures a potential set by the Nernst equation. For a half-reaction with n electrons at 25 °C,

E = E° − (0.0592 / n) · log Q

where Q is the reaction quotient built from the species' activities. The crucial feature is the logarithm. For a glass pH electrode, E depends linearly on pH, and pH is itself a log of hydrogen-ion activity — so the electrode reads roughly 59 mV per pH unit (per decade of [H⁺]) for a one-electron process. Far from equivalence, a drop of titrant nudges the analyte concentration by a few percent, which is a tiny fraction of a decade, so E moves only a millivolt or two. The curve looks lazy.

Near equivalence, the picture inverts. Suppose you are 99.9% of the way to neutralizing an acid: only 0.1% of it remains. The next small dose of base consumes most of what is left, dropping [H⁺] by a factor of ten, a hundred, a thousand — several decades in a single increment. Each decade is worth ~59 mV, so E suddenly vaults by hundreds of millivolts across a fraction of a milliliter. That violent inflection is the signature of the equivalence point. The size of the jump scales with how complete the reaction is: a strong acid against a strong base gives a near-vertical 600–700 mV cliff, while a weak acid (small reaction constant) gives a gentler, shorter step that derivative methods or a Gran plot must tease out.

The two electrodes

Every potentiometric measurement is a difference. You need an indicator electrode whose potential tracks the analyte, and a reference electrode whose potential is fixed no matter what the solution does. Subtract the reference and all the variation belongs to the indicator.

Indicator/reference pairs by titration type

Titration type	Indicator electrode	Senses	Example
Acid–base	Glass pH electrode	H⁺ activity (pH)	HCl titrated with NaOH
Redox	Inert platinum wire	Ratio of oxidized/reduced forms	Fe²⁺ titrated with Ce⁴⁺
Precipitation	Silver wire	Ag⁺ activity	Cl⁻ titrated with AgNO₃
Complexometric / ion	Ion-selective electrode	Specific ion (F⁻, Ca²⁺, CN⁻)	F⁻ in water; Ca²⁺ vs EDTA

The most common reference is the silver/silver-chloride electrode (Ag | AgCl | saturated KCl), with a standard potential of about +0.197 V vs the standard hydrogen electrode; the older saturated calomel electrode (SCE, +0.241 V) does the same job. In a modern lab the indicator and reference are fused into a single combination electrode — the everyday "pH probe" is exactly a glass indicator wrapped around an Ag/AgCl reference with a porous junction. The reference's job is humble but essential: hold still, so that the recorded E is an honest report of the indicator alone.

Locating the endpoint: derivatives and Gran plots

Reading the inflection by eye off a raw sigmoid is imprecise, so the endpoint is extracted mathematically. The first derivative ΔE/ΔV (or dE/dV) is computed between successive points; it rises to a sharp maximum at the equivalence point. The second derivative Δ²E/ΔV² is even cleaner — it swings from positive to negative and crosses zero exactly at the inflection, and auto-titrators interpolate that zero-crossing between the two bracketing points for sub-microliter resolution.

Endpoint-detection methods compared

Method	How it locates EP	Best when	Limitation
Visual inflection	Eye picks steepest spot on E vs V	Large, symmetric break	Subjective; poor for weak breaks
First derivative (dE/dV)	Volume of the peak	General use	Noisy if data spacing uneven
Second derivative (d²E/dV²)	Zero-crossing	Auto-titrators, sharp breaks	Amplifies measurement noise
Gran plot	Linearized x-intercept	Weak acids/bases, small breaks	Needs pre-/post-EP linear region

The Gran plot deserves special mention because it rescues titrations where the potential break is too small for a clean derivative — a weak acid near pKa 9, or a dilute analyte. By transforming the data (for an acid, plotting V·10^(−E·n/0.0592) or the equivalent antilog function against V), the points before and after equivalence each fall on a straight line, and the equivalence volume is read off the x-intercept where the line hits zero. The Gran approach also makes good use of data taken away from the inflection, where measurement is most stable, rather than relying on the few noisy points inside the jump.

Multiple breaks: polyprotic acids and mixtures

A single titration can show several equivalence points, one for each proton or each species. Phosphoric acid, H₃PO₄, has three acidic protons with pKa values of 2.15, 7.20, and 12.35. Titrate it with NaOH and the potentiometric curve shows two well-resolved inflections — the first proton (pKa 2.15) and the second (pKa 7.20) each produce their own step, while the third (pKa 12.35) is too weak to give a sharp break in water. A purely visual indicator titration cannot cleanly separate these; the electrode resolves them in one run because each step is a distinct log-driven swing in potential. The same multi-break logic lets potentiometry quantify a mixture of a strong and a weak acid, or chloride and iodide together in a single silver titration (iodide, with the smaller solubility product, precipitates first and gives the earlier inflection).

Worked numbers

Strong acid, strong base. Titrate 50.0 mL of 0.100 M HCl with 0.100 M NaOH. Equivalence is at 50.0 mL, where pH = 7.00. At 49.95 mL added, [H⁺] ≈ 5 × 10⁻⁵ M (pH ≈ 4.3); at 50.05 mL, [OH⁻] ≈ 5 × 10⁻⁵ M (pH ≈ 9.7). The glass electrode therefore sweeps ~5.4 pH units — about 320 mV — across just 0.1 mL of titrant. That cliff is what makes the endpoint unambiguous.

Redox. Titrate Fe²⁺ with Ce⁴⁺ on a platinum electrode. Before equivalence the potential is fixed by the Fe³⁺/Fe²⁺ couple (E° = +0.77 V); after equivalence it is fixed by the Ce⁴⁺/Ce³⁺ couple (E° = +1.61 V in 1 M H₂SO₄). At the equivalence point the potential is the weighted average, here ≈ +1.19 V, and the curve jumps cleanly between the two plateaus. No redox indicator (like ferroin) is needed — the Pt electrode reports the couple ratio directly.

Precipitation. Titrate Cl⁻ with AgNO₃ using a silver indicator electrode. The endpoint sits where [Ag⁺] = [Cl⁻] = √Ksp ≈ √(1.8 × 10⁻¹⁰) ≈ 1.3 × 10⁻⁵ M. Past equivalence, free Ag⁺ climbs steeply and the silver electrode's potential leaps — the same Mohr/Volhard chemistry, but with an objective electrical endpoint instead of a chromate or thiocyanate color.

Where it matters

• Petroleum. Total Acid Number (TAN) and Total Base Number (TBN) of lubricating oils and crude are measured by potentiometric titration in non-aqueous solvent (ASTM D664, D2896) — samples far too dark for any dye.
• Food & beverage. Titratable acidity of wine, fruit juice, and dairy; salt (chloride) content of brines, cheese, and processed foods by silver titration.
• Pharmaceuticals. Many USP/EP assays specify potentiometric endpoints for drug purity, including titrations in glacial acetic acid where water-based indicators are useless.
• Water & environment. Alkalinity, chloride, and fluoride (the latter by ion-selective electrode) in drinking and waste water.
• Karl Fischer. Water-content determination uses a potentiometric (bipotentiometric) dead-stop endpoint — one of the most-run analyses in industry.

Practical pitfalls

• Skipping the inflection. Add titrant in coarse steps near equivalence and you can step right over the jump. Auto-titrators slow to microliter doses as dE/dV rises.
• Drifting reference. A clogged junction or depleted KCl makes the "fixed" reference wander, smearing the endpoint. Refill and rinse.
• Slow electrode response. Read E only after it stabilizes (drift < a few mV/min); near equivalence, equilibration is slowest.
• Weak breaks. Very weak acids/bases or dilute analytes give shallow steps — switch to a Gran plot or a non-aqueous solvent that sharpens the break.
• Calibration. The absolute potential only matters if the electrode is calibrated, but the endpoint depends only on the shape of the curve, so even an uncalibrated probe can find an inflection.