Titration curves

Result

Titration curves

T-052Home CU-204Threads equilibrium

Statement

Equivalence points and the choice of indicator.

Why it matters

bronsted-lowry defines proton transfer as the basic currency of acid-base chemistry, and water-autoionisation-ph establishes the pH scale itself; titration curves are where these ideas become a practical, quantitative laboratory tool, tracking pH continuously as a titrant is added to a solution of unknown or known concentration. For a weak acid or base, the curve additionally supplies a direct experimental route to \(K_a\) or \(K_b\), via henderson-hasselbalch applied at the half-equivalence point.

Understanding the shape of a titration curve in detail — where it rises steeply, where it flattens into a buffer region, and where its equivalence point actually falls on the pH scale — is also what allows a chemist to choose an appropriate indicator, connecting this unit's equilibrium theory directly to routine analytical practice.

Hypotheses

Titrant is added incrementally, with the system reaching proton-transfer equilibrium (bronsted-lowry) after each addition before pH is recorded.The titration curve represents a continuous sequence of equilibrium states, not a kinetic process; acid-base proton transfer is generally fast enough that this equilibrium assumption holds essentially exactly under normal titration conditions. In the buffer region of a weak acid/base titration, henderson-hasselbalch's approximation is valid; it breaks down very close to the equivalence point.Henderson-hasselbalch assumes both the weak acid and its conjugate base are present in comparable, non-negligible amounts; near the equivalence point the concentration of unreacted acid (or base) approaches zero, so the approximation's underlying assumption fails and a full equilibrium treatment is needed instead. For a polyprotic acid, successive equivalence points appear as separate, resolvable inflections only if the successive dissociation constants (polyprotic-acids) are sufficiently well separated; when they are too close together, the individual jumps merge into a single, less distinct overall inflection.

Proof

\text{Strong acid-strong base: pH from excess } [\text{H}_3\text{O}^+]\text{ or }[\text{OH}^-]\text{ via } K_w; \text{ equivalence point at pH}=7.00.

Before equivalence, pH is set purely by the excess unreacted strong acid; after equivalence, purely by the excess strong base, via water-autoionisation-ph's \(K_w\) relation; at exact equivalence the resulting salt's ions are conjugates of a strong acid and a strong base, undergoing negligible hydrolysis, leaving the solution genuinely neutral. A

\text{At half-equivalence of a weak acid-strong base titration: } [\text{HA}]=[\text{A}^-] \Rightarrow \text{pH}=\text{p}K_a.

Exactly halfway to the equivalence point, precisely half the original weak acid has been converted to its conjugate base, so henderson-hasselbalch's ratio term \(\log([\text{A}^-]/[\text{HA}])\) vanishes, leaving pH equal to \(\text{p}K_a\) exactly — the single most useful diagnostic point on the entire curve. A

\text{At the equivalence point of a weak acid-strong base titration, pH}>7\text{ (conjugate base hydrolysis).}

At equivalence, the solution contains only the weak acid's conjugate base (plus spectator cations); as a Bronsted-Lowry base, this conjugate base reacts to a small extent with water, generating excess \(\text{OH}^-\) and raising the pH above \(7\), unlike the strong-strong case of Step 1. A

\text{Indicator choice: select an indicator whose own p}K_a\text{ falls within the steep, near-vertical portion of the curve at equivalence.}

An indicator is itself a weak acid/base pair with its own characteristic colour-change equilibrium; choosing one whose transition range coincides with the curve's steep jump ensures the visible colour change occurs at a titrant volume very close to the true equivalence point. A

\text{The size of the vertical jump at equivalence depends on acid/base strength: strong-strong largest, weak-weak smallest and least distinct.}

A strong acid-strong base titration shows the sharpest, largest pH jump; titrating a weak acid with a weak base instead gives only a very gradual, poorly defined inflection, since neither reagent drives the reaction to sufficiently complete conversion near equivalence to produce a sharp jump, making reliable indicator-based endpoint detection impractical for that combination. B

Result

\text{pH}=\text{p}K_a\text{ at half-equivalence}; \qquad \text{pH}=7\text{ at equivalence only for strong acid-strong base}

Reading. Both the shape of a titration curve and the pH at its equivalence point depend systematically on whether the acid and base involved are strong or weak, giving two separate diagnostic reference points (half-equivalence and equivalence) usable for different purposes.

Scope. Requires a reasonably large \(K_a\) or \(K_b\) for the analyte for a sharp, well-defined equivalence point to appear at all; very weak acids or bases give only a gradual, poorly resolved inflection (Step 5), making standard indicator-based titration unreliable.

Corollaries & converses

polyprotic-acids show one equivalence point and one half-equivalence point per dissociable proton, provided successive \(K_a\) values are sufficiently separated (Hypotheses' \(t3\) caveat) for the individual inflections to resolve as distinct steps.
henderson-hasselbalch is most accurately applied specifically in a titration curve's buffer region, away from both the very start (essentially no conjugate base yet present) and the equivalence point itself (essentially no unreacted acid remaining).
water-autoionisation-ph's \(K_w\) relation is exactly what fixes the strong-strong equivalence point at pH \(7.00\) (Step 1), and is the same relation used everywhere else on the same curve to convert excess strong-acid or strong-base concentration into pH.

Fails without

Drop the equilibrium-after-each-addition assumption (Hypotheses): if pH is recorded before the system fully re-equilibrates after each titrant addition, the resulting curve reflects a mixture of kinetic and equilibrium effects rather than the clean sequence of equilibrium states the theoretical treatment assumes.
Apply henderson-hasselbalch too close to the equivalence point, outside its stated region of validity: as the unreacted acid or base concentration approaches zero, the approximation's underlying assumption fails, and the predicted pH diverges from the true, sharply changing pH actually observed near equivalence.

Common errors

Assuming every titration's equivalence point occurs at pH 7 — true only for a strong acid titrated with a strong base (Step 1); weak acid-strong base gives a basic equivalence point, and weak base-strong acid gives an acidic one (Step 3).
Choosing an indicator whose colour-change range does not overlap the steep, vertical portion of the specific curve being run, systematically mislocating the observed endpoint relative to the true equivalence point.
Confusing the equivalence point (defined stoichiometrically, moles of titrant equal to moles of analyte) with the half-equivalence point (defined as halfway to the equivalence point, where pH equals \(\text{p}K_a\)) — these are two distinct points on the same curve serving different diagnostic purposes.
Assuming henderson-hasselbalch remains accurate arbitrarily close to the equivalence point, when in fact it specifically breaks down there as the unreacted acid (or base) concentration approaches zero (Hypotheses).

Discussion

Volumetric acid-base analysis as a quantitative laboratory technique long predates a detailed theoretical understanding of the underlying equilibria, developing through careful, empirical measurement practice across the 19th century. The modern equilibrium-based interpretation used throughout this page rests on Bronsted-Lowry proton-transfer theory, formalised independently by Johannes Bronsted and Thomas Lowry in 1923, which put the older, purely empirical technique on a firm theoretical footing, letting the shape of any given curve be predicted quantitatively in advance rather than merely observed after the fact.

Common misconception: that the true equivalence point and the observed "endpoint" (where the indicator visibly changes colour) are identical. They are conceptually distinct: the endpoint is only an experimental estimate of the equivalence point, chosen via careful indicator selection (Step 4) to coincide with it as closely as practically achievable, but a poorly matched indicator introduces a small, systematic titration error.

Worked examples

\text{Half-equivalence of }0.100\,\text{M acetic acid titrated with }0.100\,\text{M NaOH: pH}=\text{p}K_a\approx4.74

Acetic acid's \(K_a\approx1.8\times10^{-5}\) (a standard, widely tabulated value), giving \(\text{p}K_a=-\log(1.8\times10^{-5})\approx4.74\); by Step 2, at exactly half-equivalence, with equal concentrations of acetic acid and acetate present, pH equals this \(\text{p}K_a\) value directly, with no further calculation required. A

\text{At equivalence: }0.0500\,\text{M acetate (diluted by titrant), }K_b=K_w/K_a\approx5.6\times10^{-10}

At the equivalence point, all the original acetic acid has been converted to acetate, diluted by the added titrant volume to \(0.0500\,\text{M}\); acetate's own base hydrolysis constant follows from \(K_w/K_a\) (water-autoionisation-ph combined with the given \(K_a\)), giving \([\text{OH}^-]=\sqrt{K_bC}=\sqrt{(5.6\times10^{-10})(0.0500)}\approx5.3\times10^{-6}\,\text{M}\), so \(\text{pOH}\approx5.28\) and \(\text{pH}\approx8.72\) — confirming Step 3's prediction of a basic equivalence point. A

\text{pH(half-equivalence)}\approx4.74; \qquad \text{pH(equivalence)}\approx8.72

Reading. The two diagnostic points on this weak acid-strong base titration curve give directly, in one case, the acid's own \(\text{p}K_a\), and in the other, a quantitative confirmation that the equivalence point is basic rather than neutral.

Scope. The identical two-part calculation (henderson-hasselbalch at half-equivalence; conjugate-base hydrolysis at equivalence) applies to any weak monoprotic acid-strong base titration with known \(K_a\) and concentration.

Problems

A weak base with \(\text{p}K_b=4.75\) is titrated with a strong acid. State the pH at half-equivalence.
Solution
By the same half-equivalence logic as Step 2 applied to a base (where \([\text{B}]=[\text{BH}^+]\) at half-equivalence), \(\text{pOH}=\text{p}K_b=4.75\), so \(\text{pH}=14.00-4.75=9.25\).
Predict, without calculation, whether the equivalence point of a weak base titrated with a strong acid is acidic, basic, or neutral, and justify briefly.
Solution
Acidic. At equivalence, the solution contains only the weak base's conjugate acid (plus spectator anions), which, as a Bronsted-Lowry acid, hydrolyses to generate excess \(\text{H}_3\text{O}^+\), lowering the pH below \(7\) — the exact mirror image of Step 3's weak acid-strong base case.
Explain why titrating a weak acid with a weak base (rather than a strong base) is generally avoided in practice, referencing Step 5.
Solution
Neither reagent is strong enough to drive the neutralisation to sufficiently complete conversion right at the equivalence point, so the curve shows only a small, gradual inflection rather than a sharp vertical jump (Step 5); with no steep region for an indicator's colour change to coincide with, reliable, precise endpoint detection by the usual indicator method becomes impractical.