pH Curves & Titrations
A titration is a quantitative analytical technique in which a solution of known concentration (the titrant) is gradually added to a solution of unknown concentration (the analyte) until the reaction reaches its equivalence point — the exact stoichiometric point at which moles of acid equal moles of base (or vice versa). By monitoring the pH of the analyte solution after each addition of titrant and plotting the results, we obtain a characteristic S-shaped graph called a pH curve (or titration curve). These curves are powerful diagnostic tools: their shape, steepness, and inflection points reveal whether the acids and bases involved are strong or weak, and they allow us to identify critical features such as the equivalence point, the half-equivalence point, and any buffer regions. The pH curve for a strong acid–strong base titration (e.g., HCl titrated with NaOH) shows a dramatic, nearly vertical rise at the equivalence point, where the pH equals exactly 7.00 at 25 °C because the salt produced (NaCl) does not hydrolyse. In contrast, weak acid–strong base titrations (e.g., ethanoic acid CH₃COOH titrated with NaOH) display a gentler sigmoid curve with a buffer region in the first half, and an equivalence point above pH 7 because the conjugate base (CH₃COO⁻) hydrolyses to produce OH⁻. Similarly, strong acid–weak base titrations yield an equivalence point below pH 7. Understanding these differences is essential for choosing the correct indicator and for accurate quantitative analysis. pH curves also reveal the Henderson–Hasselbalch relationship in action. At the half-equivalence point of a weak acid–strong base titration, exactly half the weak acid has been neutralised, creating an equimolar mixture of the weak acid and its conjugate base. At this unique point, pH = pKₐ of the weak acid — a fact that lets chemists determine pKₐ values experimentally. The flat, gently sloping region around the half-equivalence point is the buffer region, where the solution resists large pH changes. Mastering the interpretation of pH curves connects proton-transfer chemistry, buffer theory, and indicator chemistry into one coherent framework.
- Identify the Titration Type
- Locate the Equivalence Point
- Identify the Buffer Region and Half-Equivalence Point
- Choose the Correct Indicator
- Sketch and Annotate the Curve
Identify the Titration Type
Before interpreting any pH curve, determine what type of acid–base pair is involved. Strong acid + strong base: HCl + NaOH → NaCl + H₂O. Weak acid + strong base: CH₃COOH + NaOH → CH₃COONa + H₂O. Strong acid + weak base: HCl + NH₃ → NH₄Cl. Weak acid + weak base: CH₃COOH + NH₃ (poorly defined equivalence point). The initial pH of the analyte solution gives the first clue: a very low pH (e.g., 1–2) suggests a concentrated strong acid, while a moderately low pH (e.g., 3–5) suggests a weak acid.
Locate the Equivalence Point
The equivalence point is the steepest part of the pH curve — the midpoint of the vertical (or near-vertical) section. It is found by identifying the point of maximum gradient, which is determined mathematically as the point of inflection (where d²pH/dV² = 0). For a strong acid–strong base titration at 25 °C, the equivalence point always occurs at pH = 7.00. For a weak acid–strong base titration, it occurs above pH 7 (typically pH 8–10) because the conjugate base hydrolyses: CH₃COO⁻(aq) + H₂O(l) ⇌ CH₃COOH(aq) + OH⁻(aq). For a strong acid–weak base titration, the equivalence point falls below pH 7 (typically pH 4–6) because the conjugate acid dissociates: NH₄⁺(aq) ⇌ NH₃(aq) + H⁺(aq).
Identify the Buffer Region and Half-Equivalence Point
In a weak acid–strong base (or strong acid–weak base) titration, the region roughly between 10% and 90% of the equivalence volume shows a relatively flat pH plateau — the buffer region. Within this zone, both the weak acid (HA) and its conjugate base (A⁻) are present in significant amounts, resisting pH change. The half-equivalence point occurs at exactly half the equivalence volume. At this point, [HA] = [A⁻], so the Henderson–Hasselbalch equation gives pH = pKₐ + log([A⁻]/[HA]) = pKₐ + log(1) = pKₐ. This is how pKₐ values are determined graphically from titration data.
Choose the Correct Indicator
An acid–base indicator is a weak acid (HIn) where the undissociated form (HIn) and the conjugate base (In⁻) have different colours. The indicator changes colour at its pKᵢₙ. For accurate endpoint detection, the indicator's pKᵢₙ should lie within the near-vertical section of the pH curve (within ±1 pH unit of the equivalence point). For strong acid–strong base titrations: both phenolphthalein (pKᵢₙ ≈ 9.1, range 8.3–10.0) and methyl orange (pKᵢₙ ≈ 3.5, range 3.1–4.4) work because the vertical section spans ~pH 3–11. For weak acid–strong base: use phenolphthalein (equivalence pH ~8–10). For strong acid–weak base: use methyl orange (equivalence pH ~4–6). Using the wrong indicator leads to an endpoint that does not match the equivalence point.
Sketch and Annotate the Curve
A well-annotated pH curve should clearly label: (1) the initial pH; (2) the buffer region (flat section, weak acid/base titrations only); (3) the half-equivalence point (pH = pKₐ) with its volume marked; (4) the equivalence point and its pH; (5) the post-equivalence region (excess titrant); (6) the indicator transition range drawn as a shaded band. The curve for a strong acid–strong base system has no buffer region and shows a sharp vertical rise at the equivalence point. The curve for a weak acid–strong base starts at a higher initial pH than a strong acid of the same concentration, has a distinct buffer region, and reaches its equivalence point above pH 7.
The Key Equation
- \mathrm{pH} — pH (dimensionless): The negative logarithm (base 10) of the hydrogen ion concentration; measures acidity.
- \mathrm{p}K_a — pKa (dimensionless): The negative logarithm of the acid dissociation constant Ka; smaller pKa means stronger weak acid.
- [\ce{A-}] — Conjugate base concentration (mol dm⁻³): Molar concentration of the deprotonated form of the weak acid (conjugate base).
- [\ce{HA}] — Weak acid concentration (mol dm⁻³): Molar concentration of the undissociated weak acid remaining in solution.
Worked Example
Problem. 25.0 cm³ of 0.100 mol dm⁻³ ethanoic acid (CH₃COOH, pKₐ = 4.76) is titrated with 0.100 mol dm⁻³ NaOH. (a) What is the pH at the start of the titration? (b) What is the pH at the half-equivalence point? (c) What volume of NaOH is needed to reach the equivalence point, and what is the approximate pH there? (d) Which indicator should be used?
Answer. Initial pH = 2.88; half-equivalence pH = 4.76 = pKₐ; equivalence point at 25.0 cm³ NaOH added, pH ≈ 8.73; use phenolphthalein as indicator.
Origins · 1 minSøren Peter Lauritz Sørensen and how this was discovered
Sørensen introduced the concept of pH (from Danish 'potenz' meaning power) in 1909 while working at the Carlsberg Laboratory in Copenhagen. He recognised that hydrogen ion concentration was central to enzyme activity and biochemical processes, and devised the logarithmic pH scale to make the inconveniently small numbers of [H⁺] easier to work with. His paper established the mathematical definition pH = −log[H⁺] and described buffer solutions — directly foundational to understanding pH curves and titrations. Sørensen developed his pH concept specifically because he needed a reliable, reproducible way to control the acidity of solutions used in brewing beer at the Carlsberg Brewery — arguably making the pH scale a gift from the fermentation industry to all of chemistry.
Related Concepts
- Acid Dissociation Constant (Kₐ) and pKₐ — The pKₐ of a weak acid is directly readable from the pH curve at the half-equivalence point, and determines the position and height of the equivalence point.
- Buffer Solutions — The buffer region on a weak acid/base pH curve is the practical manifestation of buffer action; understanding buffers explains the flat section of the curve and the Henderson–Hasselbalch equation.
- Acid–Base Indicators and Endpoint Detection — Selecting the correct indicator requires matching the indicator's pKᵢₙ and colour-change range to the vertical section of the specific pH curve.
- Hydrolysis of Salts — Salt hydrolysis explains why equivalence points in weak acid/base titrations deviate from pH 7, connecting ionic equilibrium to the shape of pH curves.
- Strong and Weak Acids and Bases — The degree of dissociation of the acid and base determines the initial pH, the shape of the pH curve, and the pH at the equivalence point.