Titration Curves: Concept and Construction

Automated potentiometric titrators in pharmaceutical quality-control labs plot pH-, $E$-, or pAg-vs-volume curves in real time and use first-derivative and second-derivative algorithms to locate the inflection point far more precisely than a human eye reading a plotted curve — the underlying region-by-region equilibrium logic developed here is exactly what the firmware is evaluating at every data point. Water-treatment plants monitor total hardness by EDTA titration curves (as in Example 2) to certify $\mathrm{Ca^{2+}}/\mathrm{Mg^{2+}}$ levels against regulatory limits. Clinical and forensic labs use argentometric titration curves (pCl or pAg versus $\mathrm{AgNO_3}$ volume) to certify chloride content in biological fluids and in Mohr/Volhard-type assays. Redox titration curves ($E$ versus volume of oxidant or reductant) underlie iodometric determination of vitamin C content, dissolved oxygen (Winkler method), and %purity assays for reducing sugars and many pharmaceutical actives — each of these borrows the identical three-region logic, only swapping the governing equilibrium constant (here $E^\circ$ and the Nernst equation) for $K_a$ or $K_f$. Because the curve-construction recipe generalizes so completely, it is the shared scaffolding beneath the specific curves for Strong Acid-Strong Base Titration Curves and Redox Titration Curves: The E-vs-Volume Plot, both of which specialize this same three-region machinery to their own governing equilibria.

:::faq[Is pH always the right quantity to plot, or could I plot [H+] directly?] You could plot $[\mathrm{H^+}]$ directly, but the curve would be nearly useless to read: because $[\mathrm{H^+}]$ changes by many orders of magnitude near $V_{eq}$ while changing very little elsewhere, a linear $[\mathrm{H^+}]$-vs-$V$ plot would look like a flat line that suddenly drops to near-zero, with almost no visual detail in the regions chemists care about. The logarithmic pH transform (following Sørensen's original motivation) compresses that enormous dynamic range into a scale where the buffer region, the break, and the post-equivalence plateau are all comparably visible on the same axes — the same reasoning is why pM and $E$ (itself logarithmic in concentration via the Nernst equation) are the natural axes for complexometric and redox curves respectively.

Ask the tutor: How do I decide, just from the size and shape of a computed titration curve, whether a given indicator's color-change range will actually catch the equivalence point?

Imagine adding titrant to analyte one drop at a time, and after each drop, freezing time and asking, "what is the equilibrium composition of this beaker right now?" A titration curve is nothing more than the answer to that question, plotted as a continuous function of volume added. The reason this works at all — the reason we can even draw a smooth curve rather than a jumble of separate calculations — is that the same handful of governing relationships (mass balance, charge balance, and the relevant equilibrium constant) apply continuously as $V$ increases from zero to well past $V_{eq}$. The signal you plot is chosen because it responds sharply and specifically to the species being titrated. In acid-base titrations, that signal is pH, tracking $[\mathrm{H^+}]$ via a glass electrode. In complexometric titrations (e.g., EDTA versus $\mathrm{Ca^{2+}}$), the signal is often pM $= -\log[\mathrm{M}^{n+}]$, tracked with an ion-selective electrode or through a metal-indicator color change. In redox titrations, the signal is the cell potential $E$ of an inert indicator electrode (Pt) against a reference. In precipitation titrations, it is pAnion or pCation, e.g., pAg or pCl in an argentometric titration. Whatever the signal, the logic used to compute it region by region is identical, which is why this article treats them together before the specific curves (strong acid–strong base, redox $E$-vs-$V$, and others) get their own detailed treatment elsewhere. Three regions organize every calculation. Before the equivalence point, some analyte remains unreacted, and — critically — for weak-acid/weak-base, complexometric, and many redox titrations, the reacted and unreacted forms coexist in solution simultaneously, forming a buffer-like system whose signal is governed by a Henderson–Hasselbalch-type relationship or a Nernst-type relationship. At the equivalence point, moles of titrant equal moles of analyte (adjusted for stoichiometric ratio), and the solution composition is now determined by a new equilibrium — the reaction of the product species with the solvent (hydrolysis), or the standard potential of the couple, or the solubility product of the newly formed precipitate. After the equivalence point, titrant is in stoichiometric excess, and because the analyte's contribution becomes comparatively negligible, the signal is controlled almost entirely by the excess titrant concentration diluted into the total volume. The steepness of the curve near $V_{eq}$ — the "endpoint break" — is the practical payoff of this three-region structure. A reaction that goes essentially to completion (large $K$) produces a curve that changes many pH units, volts, or pM units over a fraction of a milliliter, so an indicator with even a sloppy transition range will land inside the break. A reaction with modest $K$ (weak acid with weak-ish $K_a$, poor complex stability, small $E^\circ$ difference) produces a shallow, rounded transition, and choosing (or even finding) a usable indicator becomes hard or impossible. Plotting the curve — literally or mentally — is therefore the diagnostic tool for asking "can this titration even be done visually?"

The mass balance for the analyte, valid at every volume $V$ of titrant added, states that total analyte (in whatever protonation/oxidation/complexation states) is conserved except for dilution:

where: $C_{analyte,0}$ is the analyte's initial molar concentration, $V_{analyte}$ its initial volume, $[\text{species}_i]$ the equilibrium concentration of each protonation/oxidation state at the current total volume $V_{analyte}+V$. Reading: the total number of moles of analyte (in any form) never changes — only how it's distributed among forms, and how dilute it is, change as titrant is added. The equivalence volume follows from stoichiometry alone, before any equilibrium constant is invoked:

where: $n$ and $m$ are the stoichiometric coefficients of analyte and titrant in the balanced net reaction (e.g., $n=1,m=2$ for $\mathrm{CO_3^{2-}} + 2\mathrm{H^+}$), $C_{titrant}$ the titrant's molar concentration, $V_{eq}$ the volume at which stoichiometric equivalence is reached. Reading: $V_{eq}$ depends only on concentrations and stoichiometry — never on $K$. The location of the equivalence point is chemistry-of-amounts; the shape of the curve around it is chemistry-of-equilibrium. The general dilution-corrected concentration of excess reagent, used heavily in the pre- and post-equivalence regions, is:

where: the numerator is a simple mole-difference (stoichiometry), the denominator is the total solution volume at that point — the single most common arithmetic error in titration problems is forgetting this growing denominator. Reading: every concentration in a titration is a "leftover moles divided by current total volume" statement; the curve bends because both numerator and denominator change with $V$. The generic post-equivalence relationship, used whenever titrant is in excess, is:

where: $V-V_{eq}$ is the extra volume of titrant added beyond equivalence. Reading: far past $V_{eq}$, the signal is controlled almost entirely by titrant, and the curve flattens again because $[\text{titrant excess}]$ grows roughly linearly while the logarithmic signal (pH, pM, $E$ via Nernst) compresses that growth.

Example 1 (Foundational): Sketching the three regions for a strong acid/strong base titration by hand. Titrate $50.00\ \text{mL}$ of $0.1000\ \text{M}$ HCl with $0.1000\ \text{M}$ NaOH. Compute pH at $V=0$, $V=25.00\ \text{mL}$ (halfway to equivalence), $V=49.90\ \text{mL}$ (just before equivalence), $V=50.00\ \text{mL}$ (equivalence), and $V=60.00\ \text{mL}$ (past equivalence). Setup. $V_{eq}$ by stoichiometry: $(0.1000)(50.00)/0.1000 = 50.00\ \text{mL}$, confirming the round numbers. At $V=0$: pure $0.1000$ M HCl (strong acid, fully dissociated):

At $V=25.00\ \text{mL}$ (pre-equivalence, excess HCl remaining): moles HCl initial $=5.000\ \text{mmol}$; moles NaOH added $=2.500\ \text{mmol}$; leftover $\mathrm{H^+} = 2.500\ \text{mmol}$; total volume $=75.00\ \text{mL}$:

At $V=49.90\ \text{mL}$ (just short of equivalence): leftover $\mathrm{H^+} = 5.000-4.990 = 0.010\ \text{mmol}$; total volume $=99.90\ \text{mL}$:

At $V=50.00\ \text{mL}$ (equivalence): strong acid + strong base neutralize completely to a neutral salt (NaCl) solution — no hydrolysis, since neither ion is a conjugate acid/base of a weak partner:

At $V=60.00\ \text{mL}$ (past equivalence): excess NaOH $=6.000-5.000=1.000\ \text{mmol}$; total volume $=110.0\ \text{mL}$:

Takeaway: between $V=49.90$ and $V=50.10\ \text{mL}$ (a mere $0.20\ \text{mL}$ window, well within one drop), the pH swings from about $4.0$ to about $10.0$ — a six-unit break in two drops — which is exactly why strong acid/strong base titrations are the easiest to detect visually.

Example 2 (Applied): A complexometric titration curve — pCa versus volume of EDTA. A water-hardness analysis titrates $50.00\ \text{mL}$ of $0.01000\ \text{M}$ $\mathrm{Ca^{2+}}$ with $0.01000\ \text{M}$ EDTA (as $\mathrm{Na_2H_2Y}$) at a buffered pH where the conditional formation constant is $K_f' = 1.75\times10^{8}$ (this already folds in the $\alpha_{Y^{4-}}$ side-reaction coefficient at that pH). Compute pCa $=-\log[\mathrm{Ca^{2+}}]$ before, at, and after equivalence. Stoichiometry (1:1):

Pre-equivalence, at $V=40.00\ \text{mL}$ (excess $\mathrm{Ca^{2+}}$ remains): moles $\mathrm{Ca^{2+}}$ initial $=0.5000\ \text{mmol}$; moles EDTA added $=0.4000\ \text{mmol}$; leftover $\mathrm{Ca^{2+}} = 0.1000\ \text{mmol}$; total volume $=90.00\ \text{mL}$:

(The tiny amount of $\mathrm{Ca^{2+}}$ released by dissociation of the $\mathrm{CaY^{2-}}$ already formed is negligible here because $K_f'$ is large — the excess free metal dominates, exactly analogous to "excess weak acid dominates over its conjugate base's contribution" in the acid-base case.) At equivalence, $V=50.00\ \text{mL}$: essentially all calcium is complexed as $\mathrm{CaY^{2-}}$, and a small amount dissociates back. Total formal concentration of complex:

Let $x = [\mathrm{Ca^{2+}}]$ at equilibrium (equal to $[\mathrm{Y^{4-}}]_{free}$ by 1:1 dissociation stoichiometry):

Check self-consistency: $x/C_{CaY} = 1.07\times10^{-3}$, under 5%, approximation valid. Post-equivalence, at $V=60.00\ \text{mL}$ (excess EDTA present): excess EDTA $=0.6000-0.5000=0.1000\ \text{mmol}$ in $110.0\ \text{mL}$:

Now nearly all calcium is still complexed (formal concentration $C_{CaY}=0.5000/110.0=4.545\times10^{-3}\ \text{M}$), and free $\mathrm{Ca^{2+}}$ is suppressed by the excess free ligand, exactly analogous to excess titrant suppressing residual species after equivalence in acid-base chemistry:

Takeaway: pCa rises from about $2.95$ to about $5.27$ to about $7.54$ across this narrow volume window — the same three-region S-shape as the acid-base curve, but tracking a metal ion via a formation-constant equilibrium instead of a proton-transfer equilibrium.

Example 3 (Exam/competition level): Finding the volume at which the curve's slope is steepest, and quantifying the break size for a weak acid, without assuming it equals $V_{eq}/2$ is a maximum of anything but buffer capacity. A common exam trap conflates "half-equivalence point" (where $\mathrm{pH}=\mathrm{p}K_a$ and buffer capacity is maximum, i.e., the curve is flattest) with "point of steepest slope" (which, for a symmetric enough curve, sits near $V_{eq}$ itself). Consider titrating $30.00\ \text{mL}$ of $0.0600\ \text{M}$ acetic acid ($K_a=1.75\times10^{-5}$) with $0.1200\ \text{M}$ NaOH, and quantify both. Stoichiometry: $V_{eq} = (0.0600)(30.00)/0.1200 = 15.00\ \text{mL}$. Half-equivalence, $V=7.50\ \text{mL}$: here $[\mathrm{HA}]=[\mathrm{A^-}]$ by construction (half the acid has been converted), so

At this volume, $d(\mathrm{pH})/dV$ is at its minimum magnitude — the curve is flattest here because the buffer resists pH change most strongly when $[\mathrm{HA}]=[\mathrm{A^-}]$ (maximum buffer capacity $\beta$). Near equivalence, compare $V=14.90$ and $V=15.10\ \text{mL}$ to locate the steepest region. At $V=14.90\ \text{mL}$: moles acid initial $=1.800\ \text{mmol}$; moles base added $=1.788\ \text{mmol}$; leftover $\mathrm{HA}=0.01200\ \text{mmol}$; $\mathrm{A^-}$ formed $=1.788\ \text{mmol}$; total volume $=44.90\ \text{mL}$:

At the equivalence point itself, $V=15.00\ \text{mL}$: total volume $=45.00\ \text{mL}$; $[\mathrm{A^-}]_0 = 1.800\ \text{mmol}/45.00\ \text{mL} = 0.04000\ \text{M}$; $K_b = 1.00\times10^{-14}/1.75\times10^{-5}=5.714\times10^{-10}$:

At $V=15.10\ \text{mL}$ (past equivalence, excess base): excess NaOH $=1.812-1.800=0.01200\ \text{mmol}$; total volume $=45.10\ \text{mL}$:

Quantify the break. Over the $0.20\ \text{mL}$ window from $14.90$ to $15.10\ \text{mL}$, pH rises from $6.930$ to $10.425$ — a jump of $3.495$ pH units — versus, for comparison, only about $4.757-4.700\approx 0.06$ pH units across a similarly sized window straddling the half-equivalence point (a quick check: at $V=7.40$ and $V=7.60\ \text{mL}$ the ratio $[\mathrm{A^-}]/[\mathrm{HA}]$ changes only from about $0.947$ to $1.056$, giving $\Delta\mathrm{pH}\approx\log(1.056/0.947)=0.047$). This factor-of-70 difference in local slope is the rigorous statement of "the curve is steepest near $V_{eq}$ and flattest near the half-equivalence point" — exactly the opposite of where naive intuition (thinking of $V_{eq}/2$ as "the middle of the action") might mislead a student. Limiting case check. As $K_a$ increases toward that of a strong acid, $\mathrm{pH}_{eq}\to 7.00$ (per the earlier derivation) and the pre/post-equivalence break widens further, since the buffer region's Henderson–Hasselbalch slope becomes even less relevant — consistent with Example 1's six-unit break for a fully strong acid/strong base pair, versus this weak-acid case's $3.5$-unit break over the same-sized volume window.

The systematic use of potentiometric and colorimetric titration curves to diagnose reaction completeness, rather than simply detect an endpoint by eye, owes an enormous debt to the Danish biochemist Søren Sørensen, who introduced the pH scale in 1909 specifically to make sense of the region-by-region behavior chemists were already observing in acid-base titrations at the Carlsberg Laboratory. Sørensen's insight — that a logarithmic measure of hydrogen ion activity linearizes the enormous concentration swings near an equivalence point into a scale a human can read and plot sensibly — is the reason titration curves as we draw them (pH, not $[\mathrm{H^+}]$, on the y-axis) look the way they do. Later, the systematic tabulation of stepwise formation and dissociation constants, and their explicit use to predict titration curve shape before running an experiment, became a hallmark of quantitative analytical chemistry as formalized by figures such as Izaak Maurits Kolthoff, whose decades of work on precipitation and complexometric titrations (and on indicator theory) turned curve construction from empirical art into predictive calculation.