Acids, Bases & pH

Chapter 3 The titration bench

Titration curves

Add a base of known strength, drop by drop, until every proton is gone. The curve that records it holds the acid’s amount, its pKa and the right indicator.

10 min read · 4 interactive figures · 2 questions

Open the stopcock to run titrant into the flask, or add single drops near the end (tapping the bench adds one drop too). The curve records the exact equilibrium pH after each addition. Watch for the color change, then compare its volume with the equivalence point, marked once you pass it. Try acetic acid with phenolphthalein, then switch to methyl red and run it again. The links under the figure open the same titration, whole, in the titration-curve calculator, or carry the flask to the buffer comparison in Chapter 4.

A titration measures how much acid (or base) a sample contains by adding a base (or acid) of known concentration, a measured volume at a time, until the reaction between them is exactly complete. That moment is the equivalence point. For a monoprotic acid titrated with sodium hydroxide, the moles of base added at equivalence equal the moles of acid you started with, so CacidVacid = CbaseVequivalence. In practice you find it by watching an indicator change color, the end point, or by recording the pH and locating the steepest part of the curve.

Titration is more than two centuries old and still one of the most accurate measurements in chemistry. It is also the best way to see an acid’s character. The curve you get records how much acid there is, how strong it is, and whether your indicator will lie to you. The bench at the top of this page runs one for you.

The bench

The flask holds 25.0 mL of the acid being measured (the analyte) and a few drops of indicator. The burette holds the titrant, here 0.100 M sodium hydroxide. Open the stopcock, or add drops one at a time, and the curve is drawn as you go. Nothing on it is interpolated: every point is the exact equilibrium pH of what is in the flask at that moment.

Run acetic acid to the end and notice the shape. The pH climbs fast at first, from 2.88, then flattens through a long, gentle middle section around pH 4–5.5, then shoots up almost vertically near 25 mL and levels off again above pH 12. That near-vertical step is what makes titration precise: between 24.90 and 25.10 mL the pH jumps from 7.15 to 10.29, so a color change anywhere in that range pins the volume to within a drop or two. To see the whole curve at once, with any concentration or your own pKa, open the titration curve calculator; the link under the bench carries your current titration there.

Equivalence point and end point

These two terms are often used as if they were the same. They are not. The equivalence point is a fact about stoichiometry: the volume at which the added base exactly matches the acid present. It exists whether or not anyone is watching. The end point is an observation: the volume at which you see the indicator change color, or at which the pH meter shows the steepest rise. A good titration is designed so that the end point falls very close to the equivalence point; the difference between them is the titration error[1].

The arithmetic of the equivalence point is the “titration formula” of every introductory course. Twenty-five milliliters of 0.100 M acetic acid contain 2.50 mmol of acid. Each mole of NaOH removes one proton, so 2.50 mmol of base, 25.0 mL at 0.100 M, is needed. For an acid that can give up two protons the second equivalence point needs twice that volume, and so on. The volume depends only on amounts, never on the acid’s strength. Strength decides something else, and the next question is about it.

What is the pH at the equivalence point?

Question 1 Prediction: the pH at the equivalence point.#

When the strong acid HCl is titrated with NaOH, the flask at equivalence holds only sodium chloride in water. Neither ion affects pH, so the equivalence point is at 7.00. When acetic acid is titrated, the flask at equivalence holds sodium acetate, and acetate is a weak base: it reacts with water, CH3COO⁻ + H2O ⇌ CH3COOH + OH⁻, and pushes the pH above 7. Titrate the weak base ammonia with hydrochloric acid and the reverse happens: ammonium chloride at equivalence is weakly acidic, pH 5.27.

Figure 1 A strong acid (black) and a weak acid (orange) at the same concentration, each titrated with NaOH of that concentration. Drag the volume, change the weak acid’s pKa, or dilute everything. The circle marks the half-equivalence point; the lower panel shows how steep each curve is. Notice the jump at equivalence shrink as the pKa rises or the solution is diluted.#

Raise the pKa and the weak-acid curve starts higher, its equivalence pH climbs, and its jump shrinks: at pKa 7 the pH changes by only one unit across ±0.1 mL of the equivalence point, and at pKa 9 by about a tenth of a unit. Lower the concentration and the same thing happens to the strong acid: at 0.001 M even HCl jumps only from 5.70 to 8.30 across the same ±0.1 mL. Very weak or very dilute acids cannot be titrated accurately in water, because the step that marks equivalence is lost in water’s own ionization[1].

The half-equivalence point and pKa

Halfway to equivalence, half the acetic acid has been converted to acetate. With equal amounts of HA and A⁻, equation (2.2) gives [H⁺] = Ka: the pH equals the pKa. That makes a titration a way to measure a pKa. At 12.50 mL of NaOH, the curve for 0.1 M acetic acid sits at pH 4.756, matching the tabulated pKa[2] to three decimals.

The rule is an approximation, and the figure reports how good it is. It assumes the amounts of HA and A⁻ in the flask are the amounts you made by adding base. If the acid is fairly strong or the solution dilute, the acid’s own dissociation adds more A⁻ than the base did. Set pKa to 2.5 and the half-equivalence pH is 2.57, not 2.50; take an acid of pKa 3 at 0.001 M and it is 3.67. The exact charge balance does not need the assumption, which is why this site never uses it to draw a curve.

Choosing an indicator

An indicator is itself a weak acid whose acid and base forms have different colors. Phenolphthalein is colorless as an acid and pink as a base; methyl red is red and yellow. The eye sees one color when one form outnumbers the other by about ten to one, so the visible change spans roughly two pH units around the indicator’s pKa. A one-color indicator such as phenolphthalein is first seen as a faint pink when only a small fraction is in the colored form, which is why its range, 8.3–10.0, lies mostly below its pKa of 9.6[1].

Indicators are older than the pH scale. Phenolphthalein was first made by Adolf von Baeyer in 1871 and used to mark the end of titrations from 1877; the earliest titrations, before 1800, used litmus. What was missing was a way to compare indicators, and Sørensen’s 1909 paper supplied one, listing the hydrogen-ion concentration at which each dye changes color[3]. In 1914 Niels Bjerrum showed how to calculate a whole titration curve from dissociation constants, and indicator choice became a calculation instead of a craft[1]. documented

The rule for choosing one follows from the curves above: the indicator must change color within the steep part of the curve. Before trying it, make a guess.

Question 2 Prediction: methyl red for a weak acid.#
Figure 2 Each band is one indicator’s color change, drawn in its own colors; the chosen indicator is the bold band. The dot marks where its color change meets the exact titration curve. Pick the titration and the indicator and read the titration error. Then dilute: the range of good indicators narrows.#

For hydrochloric acid at 0.1 M, every indicator from methyl orange to phenolphthalein lands within 0.4 % of equivalence, because the curve is nearly vertical from pH 4 to pH 10. For acetic acid only phenolphthalein (−0.02 %) and, at a stretch, bromothymol blue (−0.45 %) work. For ammonia titrated with hydrochloric acid the choice reverses: methyl red is excellent (+0.01 %) and phenolphthalein misses by 10 %. The pattern is the equivalence pH: pick an indicator whose color change brackets it.

Acids with more than one proton

Phosphoric acid has three protons to give. Titrated with NaOH, it should have three equivalence points, one after each proton. Run it and count the jumps you can actually see.

Figure 3 Polyprotic titrations: 25 mL of 0.050 M acid (or sodium carbonate) titrated with 0.100 M NaOH (or HCl). Dotted lines are the equivalence points; the lower panel shows how steep the curve is, so each usable equivalence point appears as a peak. Compare phosphoric acid with citric acid.#

Phosphoric acid shows two clear jumps, at 12.5 mL (pH 4.7) and 25 mL (pH 9.6), and no third. Its third pKa, 12.35, is so high that by the time HPO4²⁻ starts giving up its last proton, the excess NaOH is already setting the pH; the third equivalence point is real but invisible. Citric acid is the opposite case: its pKas, 3.13, 4.76 and 6.40, are so close together that all three protons are removed over one smooth rise, with a single usable jump at the end. Harvey gives a working rule: separate inflection points appear only when successive Kas differ by a factor of about 500 or more[1].

Sodium carbonate titrated with hydrochloric acid shows two jumps, near pH 8.3 and pH 4. This pair of end points is the basis of the standard measurement of alkalinity in natural waters: the volumes of acid needed to reach about pH 8.3 and about pH 4.5 together tell you how much hydroxide, carbonate and bicarbonate a sample contains[1].

How the curves are computed

Textbooks usually build a titration curve in four pieces: the initial pH from the weak-acid formula, the buffer region from the Henderson–Hasselbalch equation, the equivalence point from the hydrolysis of the conjugate base, and the region past equivalence from the excess titrant. Each piece uses a different approximation, and the joins can be off.

There is a better way, due to Robert de Levie[4]. Write the charge balance for the flask after V mL of NaOH have been added to V0 mL of an acid of concentration Ca, and it can be rearranged to give the volume directly, for any [H⁺]:

V = V0 · Ca n̄(h) − ΔCb + Δ,    Δ = h − Kw/h(3.1)

Here n̄(h) is the average number of protons each acid molecule has lost at that [H⁺], computed from its pKas, and Cb is the titrant concentration. One expression covers the whole curve, for monoprotic and polyprotic acids alike, with no approximations except ideal behavior. The figures here use it both ways: the bench solves the charge balance for pH at each volume, and the test suite checks that equation (3.1) gives back the same volume. Both reproduce the worked examples in Harvey’s textbook to the second decimal[1]. The titration curve calculator draws any of these curves in one go, marks the equivalence and half-equivalence points, and exports the points as a CSV.

Questions a lab handout asks

Why rinse the burette with titrant, but not the flask? Water left in the burette dilutes the titrant and changes its concentration. Water in the flask changes the volume of the analyte but not the number of moles of acid, so the equivalence volume is unchanged.

Why add titrant drop by drop near the end? Because the curve is steepest there. Between 24.90 and 25.10 mL of 0.1 M NaOH, the pH of 0.1 M acetic acid moves by three units: one drop (about 0.05 mL) can carry it across the whole color range.

Why is the end point color “the first permanent pink” for phenolphthalein? A local excess of base where the drop lands turns the solution pink for a moment; only when the whole flask stays pink after swirling has the bulk solution reached pH 8.3.

Why does a weak acid need a different indicator from a strong acid? Because the equivalence pH is different (8.72 against 7.00 at 0.1 M), and the indicator must change within the jump around it.