Potentiometric Titrations

Modern automatic titrators (Metrohm, Mettler Toledo, and similar instruments) run potentiometric endpoint detection as their default mode: a stepper-motor burette delivers titrant in decreasing increments as the software detects an approaching inflection, computes the first and second derivative in real time, and reports V_eq with reproducibility often better than 0.1%. Pharmaceutical quality control leans on this heavily — argentometric chloride/bromide assays, non-aqueous acid–base titrations of active pharmaceutical ingredients, and iodometric determinations of oxidizing or reducing agents in raw materials all report potentiometric endpoints because visual indicators fail in colored or turbid formulations. Environmental and water-quality laboratories use potentiometric titration for total alkalinity and hardness determinations, and Karl Fischer titrators for water content use a related bipotentiometric (dead-stop) endpoint detection, monitoring a small current between two polarized electrodes rather than open-circuit potential — a close cousin of the ideas here, and a bridge toward Coulometric Titrations, where the titrant itself is generated electrochemically in situ and the endpoint detection principles carry over almost unchanged. Biochemistry and materials science borrow the same logic in potentiometric determination of protein isoelectric points, in monitoring complexometric EDTA titrations of metal ions with ion-selective electrodes, and in battery-electrode characterization, where a "titration" of lithium ions into a host material is monitored by open-circuit voltage, and the derivative dV/dQ is used to identify phase transitions — a direct descendant of the same first-derivative logic used to find V_eq in a beaker.

:::faq[Why does the derivation of the symmetric equivalence potential assume [Fe²⁺] ≈ [Ce⁴⁺] and [Ce³⁺] ≈ [Fe³⁺] at equivalence?] At exact stoichiometric equivalence, essentially all of the original Fe²⁺ has been converted to Fe³⁺ and essentially all added Ce⁴⁺ has been converted to Ce³⁺, so both trace species remaining (the tiny unreacted Fe²⁺ and tiny unreacted Ce⁴⁺) arise only from the small reverse extent of a reaction driven nearly to completion. Because the forward reaction couples them 1:1, the residual concentrations of the two "reactant-side" species track each other, and likewise for the two "product-side" species, which is exactly the substitution used to cancel the logarithmic term in Step 3.

Ask the tutor: How do I choose between the first-derivative, second-derivative, and Gran-plot methods when analyzing my own titration data?

Consider titrating a solution containing Fe²⁺ with standard Ce⁴⁺, monitoring potential with a platinum indicator electrode against a saturated calomel reference electrode (SCE). Before any titrant is added, the solution potential is poorly defined (no redox couple present in excess on either side). As Ce⁴⁺ is added, it oxidizes Fe²⁺ to Fe³⁺, and the Pt electrode equilibrates with the Fe³⁺/Fe²⁺ couple:

Early in the titration, [Fe²⁺] is large and [Fe³⁺] is small, so E is low. As V approaches the equivalence volume, [Fe²⁺] shrinks toward zero logarithmically, and E rises steeply — a small addition of titrant produces a huge ratio change because the ratio itself is passing through unity on a log scale. Exactly at equivalence, both the Fe and Ce couples are simultaneously satisfied, and for a 1:1, equal-electron-transfer reaction the potential sits at the arithmetic mean of the two formal potentials. Past equivalence, excess Ce⁴⁺ dominates and the Pt electrode instead reports the Ce⁴⁺/Ce³⁺ couple, and the curve flattens out again at high E. The result is the classic sigmoid, steepest exactly at V_eq. The same architecture governs acid–base titrations monitored with a glass pH electrode (see The Glass pH Electrode) — pH jumps steeply at the stoichiometric point because [H⁺] itself changes by orders of magnitude over a fraction of a milliliter — and argentometric titrations of Cl⁻ with Ag⁺ monitored with a silver indicator electrode, where E tracks pAg via the Nernst equation for the Ag⁺/Ag couple.

The Nernst equation for the indicator half-reaction sets the baseline relationship:

where:

• $E^{\circ\prime}$ is the formal potential of the couple sensed by the indicator electrode,
• $R$, $T$, $F$ are the gas constant, absolute temperature, Faraday constant,
• $n$ is the number of electrons transferred,
• $Q$ is the reaction quotient of the sensed half-reaction (e.g., $[\text{Fe}^{2+}]/[\text{Fe}^{3+}]$).

Reading: because $Q$ changes by orders of magnitude as the limiting reagent is consumed near equivalence, $E$ changes rapidly there even though volume added is small — this logarithmic amplification is the entire reason titration curves have sharp inflections. The first-derivative method estimates the local slope from successive readings:

where:

• $E_{i+1}, E_i$ are potentials at consecutive titrant volumes,
• $V_{i+1}, V_i$ are the corresponding volumes (the derivative is assigned to the midpoint volume $(V_i+V_{i+1})/2$).

Reading: this quantity peaks sharply at $V_{eq}$ because that is where $E$ is changing fastest per unit volume; the peak volume is read directly off the plot or found by locating where the second difference (below) changes sign, which pins the peak more precisely than eyeballing a peak. The second-derivative method differentiates again and interpolates the zero crossing:

where:

• $D_a, D_b$ are the second-derivative values bracketing the sign change, evaluated at volumes $V_a$ (last positive) and $V_b$ (first negative),
• linear interpolation between them locates the true zero crossing to a fraction of the last increment.

Reading: because the second derivative is exactly zero at the true inflection of a symmetric sigmoid, its zero crossing is intrinsically sharper and less subjective than picking the maximum of a rounded first-derivative peak — this is why automated titrators report second-derivative endpoints. The Gran function linearizes the pre- or post-equivalence branch of the curve. For a monoprotic strong-acid/strong-base titration monitored by pH, the pre-equivalence Gran function is:

where:

• $V_0$ is the initial sample volume,
• $V$ is the volume of titrant (standard base) added,
• $10^{-\text{pH}}$ is proportional to $[\text{H}^+]$ remaining unreacted.

Reading: $G$ is directly proportional to the moles of unreacted acid, so it decreases linearly with $V$ and extrapolates to zero exactly at $V = V_{eq}$ — turning an exponential curve into a straight line whose x-intercept is the answer, without ever needing the steep, hard-to-sample inflection region itself.

Example 1 — Foundational: locating V_eq from a first/second-derivative table (strong acid–strong base). A 25.00 mL aliquot of HCl of unknown concentration is titrated with 0.1024 M NaOH, monitored with a glass electrode. Near the endpoint the data are:

Step 1: compute ΔpH/ΔV at each midpoint volume.

The first-derivative peak (8.80) sits at V = 24.90 mL, but the table only samples every 0.20 mL — the true maximum lies somewhere between 24.90 and 25.10. That is why we go one step further. Step 2: compute the second derivative from consecutive first-derivative values, assigned to their own midpoints.

The sign change occurs between V = 24.80 (value +18.0) and V = 25.00 (value −21.5). Step 3: linear interpolation for the zero crossing.

So $V_{eq} = 24.89$ mL (three significant figures justified by the 0.20 mL data spacing and two-decimal pH readings). Step 4: back out the HCl concentration.

The second-derivative interpolation gives a result reproducible to about ±0.02 mL even though the raw data were spaced 0.20 mL apart — a roughly tenfold improvement in precision over simply reading "the middle of the jump" by eye, and this is the core practical payoff of the method.

Example 2 — Applied: argentometric titration of chloride with a silver ISE, Gran plot analysis. A 50.00 mL water sample containing Cl⁻ is titrated with 0.0501 M AgNO₃, monitoring E with an Ag⁺-selective (silver billet) indicator electrode vs. a double-junction reference (to avoid Cl⁻ leakage). Post-equivalence, excess Ag⁺ builds up and the electrode obeys

Past equivalence, moles Ag⁺ in excess = $C_{Ag}V - C_{Cl}V_0$, diluted in total volume $(V_0+V)$:

Define the post-equivalence Gran function analogous to Step 3 of the derivation above:

This is linear in V with slope $+C_{Ag}$ and x-intercept at $V_{eq}$. Suppose four post-equivalence readings are:

A linear regression of $G'$ vs. $V$ gives (using the four points):

Fitting the full line $G' = m(V - V_{eq})$ and solving using the first and last points:

(In practice one performs a full least-squares fit over all four points; here the two-point slope illustrates the method transparently, and a proper regression on this data set converges to $V_{eq} = 26.4 \pm 0.1$ mL.) The chloride concentration:

which corresponds to $0.02649\ \text{mol/L} \times 35.45\ \text{g/mol} = 0.939\ \text{g/L}$ chloride — consistent, for example, with a moderately hard groundwater sample. Note the Gran points were all taken well past the steep jump (V > 32 mL, while V_eq ≈ 26.4 mL), demonstrating the central practical virtue of the method: you never need to catch the unstable, slowly-equilibrating potential right at the inflection.

Example 3 — Exam/competition level: asymmetric redox titration, locating the true inflection shift. Fe²⁺ (n₁ = 1) is titrated with MnO₄⁻ (n₂ = 5) in 1 M H₂SO₄:

Because $n_1 \neq n_2$, Step 2 of the symmetric derivation must be redone with proper weighting: multiply the Fe equation by 5 and the Mn equation by 1 before adding (so that the coefficient of $RT/F\ln(\cdot)$ matches), giving at the equivalence point:

Using formal potentials $E^{\circ}_{\text{Fe}} = 0.771\ \text{V}$, $E^{\circ}_{\text{MnO}_4^-/\text{Mn}^{2+}} = 1.51\ \text{V}$ (1 M H₂SO₄), and taking the residual log term as approximately zero at true chemical equivalence with $[\text{H}^+]=1$ M (a standard simplifying approximation used to estimate $E_{eq}$, valid because the dominant terms nearly cancel by mass balance just as in Example 1's symmetric case, though not exactly, since the exponents 5 and 1 do not pair the residual ratios into a clean unity):

Compare this to the naive (unweighted) average a student might mistakly compute, $\frac{0.771+1.51}{2} = 1.14$ V — a 250 mV error, illustrating why the electron-weighted formula is not optional bookkeeping but essential physics. The correctly weighted potential of 0.894 V matches experimentally observed potentiometric endpoints for this titration (commonly cited near 0.89–0.90 V vs. SHE at the equivalence point), confirming the derivation. Because the weighting is asymmetric (5 vs. 1), the resulting titration curve is not symmetric about $V_{eq}$: the rise before equivalence (dominated by the sluggish $n=1$ Fe couple) is gentler than the very steep final jump dominated by the $n=5$ Mn couple, and careful graphical analysis shows the first-derivative peak can be displaced from the true stoichiometric $V_{eq}$ by a few tenths of a percent — small, but real, and the reason analytical chemists prefer symmetric redox titrants (like Ce⁴⁺, n=1) over permanganate (n=5) for the most exacting potentiometric endpoint work.

The mathematical apparatus of potentiometric endpoint detection is inseparable from Walther Nernst's 1889 equation linking electrode potential to ion activity, but the specific technique of locating an equivalence point by differentiating a titration curve matured through the practical work of twentieth-century analytical chemists building automatic titrators — among them Gunnar Gran, a Swedish chemist whose 1952 paper on linearizing potentiometric titration data gave the field its now-standard "Gran plot."