---
title: "Titrations by pH and Conductivity"
book: "School Chemistry — Grades 1 to 12"
subject: chemistry
language: en
chapter: 46
exercises: 15
source: https://one-course.com/books/chemistry/1/en/chapter/46-titrations-by-ph-and-conductivity
license: CC-BY-NC-SA-4.0
credit: "One Chemistry Book, One Course (one-course.com)"
---

# Chapter 46 — Titrations by pH and Conductivity

A bottle of wine vinegar is labelled “6 % acidity”: six grams of ethanoic acid in every hundred grams of vinegar. Neither ethanoic acid nor its conjugate base is coloured, and no permanganate will help here. But a [pH meter](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph-paper) dipped into the vinegar, or a cell measuring how well it conducts, follows the reaction with a base drop by drop, and the shape of the curve shows exactly where [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) lies. Ten minutes at the bench, and the label is checked.

**You already know.**

A [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration), its [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) and [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) ([Chapter 36](https://one-course.com/books/chemistry/1/en/chapter/36-titration#ch-g11-titration)); [acidity constants](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-ka) ([Chapter 44](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#ch-g12-ka-and-pka)); Henderson’s relation and indicators ([Chapter 45](https://one-course.com/books/chemistry/1/en/chapter/45-buffers-and-predominance-diagrams#ch-g12-buffers-predominance)); solutions of [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) conduct electricity ([Chapter 17](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#ch-g9-ions)).

![A wine-maker’s titration bench.](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/img-cf581b6deb15.jpg)

*A wine-maker’s [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) bench.*

## 46.1 The pH-metric titration

**Definition 46.1 (pH-metric titration, half-equivalence).**

In a *pH-metric titration* the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) of the [titrated solution](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) is measured after each addition of [titrant](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration), and the curve [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) against the volume added is drawn. *Half-equivalence* is the point where half the [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) has been added.

![A pH-metric titration: the probe dips in the stirred solution, the meter shows the pH after each addition.](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-ef1536dc31f9.svg)

*A [pH-metric titration](#def-g12-ph-conductivity-titrations-ph-metric): the probe dips in the stirred solution, the meter shows the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) after each addition.*

**In the lab — Calibrating a pH meter.**

Before use, the probe is rinsed with distilled water and dipped in two [buffer solutions](https://one-course.com/books/chemistry/1/en/chapter/45-buffers-and-predominance-diagrams#def-g12-buffers-predominance-buffer) of known [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph), usually 7.0 and 4.0 (or 10.0): the meter is set to read their values. Between solutions the probe is rinsed and gently dabbed dry, never wiped; it is stored with its tip wet.

![Computed pH curves (blue) and their derivative d pH/dV (orange, dotted, divided by 4). Left: 20.0\, mL of hydrochloric acid at 0.100\, mol/ L; equivalence at pH 7.00. Right: 10.0\, mL of diluted vinegar; at half-equivalence the pH equals the pK_a, and equivalence lies at a basic pH.](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-967d99a69fc1.svg)

![Computed pH curves (blue) and their derivative d pH/dV (orange, dotted, divided by 4). Left: 20.0\, mL of hydrochloric acid at 0.100\, mol/ L; equivalence at pH 7.00. Right: 10.0\, mL of diluted vinegar; at half-equivalence the pH equals the pK_a, and equivalence lies at a basic pH.](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-9306915625e9.svg)

*Computed [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) curves (blue) and their derivative $d\mathrm{pH}/dV$ (orange, dotted, divided by 4). Left: $20.0\,\mathrm{mL}$ of hydrochloric acid at $0.100\,\mathrm{mol}/\mathrm{L}$; [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) at [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) 7.00. Right: $10.0\,\mathrm{mL}$ of diluted vinegar; at [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) equals the $pK_a$, and [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) lies at a basic [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph).*

**Method 46.2 (The derivative method).**

Compute, between successive measurements, the slope $\Delta\mathrm{pH}/\Delta V$, and plot it against $V$ (a spreadsheet or a calculator does it). The [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) is where this slope is largest: the peak of the derivative curve.

**Method 46.3 (The tangent method).**

1. Draw two parallel tangents to the curve, one before and one after the jump, on either side of it.
2. Draw the parallel line halfway between them.
3. Where it cuts the curve is the [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) point: read $V_E$ (and the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) ).

**Proposition 46.4 (At half-equivalence, pH=pKa\mathrm{pH} = pK_apH=pKa​).**

In the [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) of a [weak acid](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid) $\ce{HA}$ by a [strong base](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid), at [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) $\mathrm{pH} = pK_a$.

**Proof.** The reaction $\ce{HA + OH- -> A- + H2O}$ is total. At [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) half of the acid has been turned into $\ce{A-}$: $[\ce{A-}] = [\ce{HA}]$, and Henderson’s relation gives $\mathrm{pH} = pK_a + \log 1 = pK_a$. ∎

**Remark 46.5 (Choosing an indicator from the curve).**

The indicator must change colour inside the jump. For hydrochloric acid the jump spans about [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) 4 to 10 and many indicators fit, bromothymol blue best; for ethanoic acid the jump is shorter and basic, around 7 to 11: phenolphthalein (8.0–10.0) fits, bromothymol blue would change too early.

## 46.2 Conductivity and conductimetric titrations

**Definition 46.6 (Conductivity, molar ionic conductivity).**

The *conductivity* $\sigma$ of a solution measures how well it conducts electricity; it is read on a conductimeter in $\mathrm{mS}/\mathrm{cm}$ (or $\mathrm{S}/\mathrm{m}$). Each [ion](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) $i$ contributes to it in proportion to its concentration, with a coefficient $\lambda_i$, its *molar ionic conductivity*.

**Proposition 46.7 (Kohlrausch’s law).**

For a dilute solution, $\sigma = \sum_i \lambda_i [X_i]$. At $25\,{}^{\circ}\mathrm{C}$, in $\mathrm{S}\,\mathrm{cm}^{2}/\mathrm{mol}$: $\ce{H3O+}$ 349.8, $\ce{OH-}$ 198.3, $\ce{Cl-}$ 76.2, $\ce{Na+}$ 50.3. The oxonium and hydroxide [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) conduct far better than all other [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion).

**Proof.** Admitted (Year 1 volume). The values of $\ce{Cl-}$ and $\ce{Na+}$ come from the measured conductivities of hydrochloric acid and sodium chloride solutions, from which that of $\ce{H3O+}$ is subtracted. ∎

**Definition 46.8 (Conductimetric titration).**

In a *conductimetric titration* the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) is measured after each addition of [titrant](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration). The [titrated solution](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) is diluted in a large volume of water, so that the volume added hardly changes it: the points then lie on straight lines, which change slope at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence).

![Left: a conductimetry cell. Right: 100\, mL of hydrochloric acid at 0.0100\, mol/ L titrated by sodium hydroxide at 0.100\, mol/ L (computed with Kohlrausch’s law). The conductivity falls, then rises: two straight lines meeting at equivalence.](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-41afe97d7649.svg)

![Left: a conductimetry cell. Right: 100\, mL of hydrochloric acid at 0.0100\, mol/ L titrated by sodium hydroxide at 0.100\, mol/ L (computed with Kohlrausch’s law). The conductivity falls, then rises: two straight lines meeting at equivalence.](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-7603c4c588f9.svg)

*Left: a conductimetry cell. Right: $100\,\mathrm{mL}$ of hydrochloric acid at $0.0100\,\mathrm{mol}/\mathrm{L}$ titrated by sodium hydroxide at $0.100\,\mathrm{mol}/\mathrm{L}$ (computed with Kohlrausch’s law). The [conductivity](#def-g12-ph-conductivity-titrations-conductivity) falls, then rises: two straight lines meeting at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence).*

**Example 46.9 (Reading the two lines).**

Before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) each $\ce{OH-}$ added removes an $\ce{H3O+}$ ($\lambda = 349.8$) and brings in a $\ce{Na+}$ ($\lambda = 50.3$): the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) falls steeply. After [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence), $\ce{Na+}$ and $\ce{OH-}$ ($\lambda = 198.3$) simply accumulate: it rises. The two lines meet at $V_E = 10.0\,\mathrm{mL}$, so the acid was $0.100 \times 10.0 / 100 = 0.0100\,\mathrm{mol}/\mathrm{L}$.

**Method 46.10 (Equivalence from a conductimetric titration).**

Draw the best straight line through the points before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) and the best line through the points after it; read $V_E$ at their intersection. Points near [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence), where the lines bend, are left out.

**Remark 46.11 (Which method when).**

A coloured indicator is fastest when a suitable one exists. A [pH-metric titration](#def-g12-ph-conductivity-titrations-ph-metric) also gives the $pK_a$ of a [weak acid](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid), and works with coloured or cloudy solutions. A [conductimetric titration](#def-g12-ph-conductivity-titrations-conductimetric) needs no jump in [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) at all: it works for very weak or very dilute acids, and for [precipitation](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-precipitate) reactions, as long as the [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) change.

**Safety.**

![](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-93c17238ab7b.svg)

![](https://one-course.com/images/onecourse/chapters/chemistry-1/g12-ph-conductivity-titrations/fig-0df833fa40ae.svg)

Sodium hydroxide solutions burn the skin and the eyes, even dilute ones in the eyes: goggles throughout, and any splash rinsed at once with a lot of water.

## 46.3 Exercises

**Exercise 46.1 ★.**

Read the [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) on the curve of hydrochloric acid titrated by sodium hydroxide at $0.100\,\mathrm{mol}/\mathrm{L}$, and deduce the concentration of the $20.0\,\mathrm{mL}$ of acid.

**Solution of Exercise 46.1.**

$V_E = 20.0\,\mathrm{mL}$; $c = 0.100 \times 20.0 / 20.0 = 0.100\,\mathrm{mol}/\mathrm{L}$.

**Exercise 46.2 ★.**

A [weak acid](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid) is titrated by sodium hydroxide; $V_E = 12.0\,\mathrm{mL}$. At $6.0\,\mathrm{mL}$ the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) is 3.75. Give the $pK_a$ of the acid. Which acid of the $pK_a$ scale could it be?

**Solution of Exercise 46.2.**

At [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) $= pK_a$: $pK_a = 3.75$, close to methanoic acid (3.74).

**Exercise 46.3 ★.**

Which indicator would you choose for the [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) of ethanoic acid by sodium hydroxide? Why not methyl red?

**Solution of Exercise 46.3.**

Phenolphthalein: its range 8.0–10.0 lies in the jump, which is basic. Methyl red (4.2–6.3) would change near [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric), far too early.

**Exercise 46.4 ★.**

Why must a [pH meter](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph-paper) be calibrated before use? With what?

**Solution of Exercise 46.4.**

The probe’s response drifts with time and temperature: it is set to read correctly in two [buffer solutions](https://one-course.com/books/chemistry/1/en/chapter/45-buffers-and-predominance-diagrams#def-g12-buffers-predominance-buffer) of known [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph).

**Exercise 46.5 ★.**

Using the molar ionic conductivities, rank the [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) $\ce{H3O+}$, $\ce{OH-}$, $\ce{Na+}$, $\ce{Cl-}$ from the best conductor to the worst.

**Solution of Exercise 46.5.**

$\ce{H3O+}$ (349.8) > $\ce{OH-}$ (198.3) > $\ce{Cl-}$ (76.2) > $\ce{Na+}$ (50.3).

**Exercise 46.6 ★★.**

$20.0\,\mathrm{mL}$ of an ethanoic acid solution need $14.6\,\mathrm{mL}$ of sodium hydroxide at $0.050\,\mathrm{mol}/\mathrm{L}$. Compute its concentration.

**Solution of Exercise 46.6.**

$c = 0.050 \times 14.6 / 20.0 = 0.0365\,\mathrm{mol}/\mathrm{L}$.

**Exercise 46.7 ★★.**

Near [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence), the readings are: $9.6\,\mathrm{mL}$, [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) 5.7; $9.8\,\mathrm{mL}$, 6.1; $10.0\,\mathrm{mL}$, 7.0; $10.2\,\mathrm{mL}$, 10.5; $10.4\,\mathrm{mL}$, 11.0. Compute $\Delta\mathrm{pH}/\Delta V$ between successive points and locate $V_E$.

**Solution of Exercise 46.7.**

Slopes: 2.0, 4.5, 17.5 and $2.5\,\mathrm{mL}^{-1}$. The largest lies between 10.0 and $10.2\,\mathrm{mL}$: $V_E \approx 10.1\,\mathrm{mL}$.

**Exercise 46.8 ★★.**

In the [conductimetric titration](#def-g12-ph-conductivity-titrations-conductimetric) of hydrochloric acid, explain the sign of the slope before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) and after it, using the [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) present.

**Solution of Exercise 46.8.**

Before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) each $\ce{OH-}$ added removes an $\ce{H3O+}$ (very conducting) and brings a $\ce{Na+}$ (much less conducting): the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) falls. After it, $\ce{Na+}$ and $\ce{OH-}$ accumulate without reacting: it rises.

**Exercise 46.9 ★★.**

Sketch the conductimetric curve of ethanoic acid titrated by sodium hydroxide. Why does the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) rise even before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence)?

**Solution of Exercise 46.9.**

At the start, few [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) (the [weak acid](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid) hardly reacts with water): low [conductivity](#def-g12-ph-conductivity-titrations-conductivity). Before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence), ethanoate and sodium [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) are formed in place of uncharged [molecules](https://one-course.com/books/chemistry/1/en/chapter/11-atoms-and-molecules#def-g7-atoms-and-molecules-molecule): it rises, slowly. After [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence), $\ce{Na+}$ and $\ce{OH-}$ accumulate: it rises faster. Two rising lines, the second steeper.

**Exercise 46.10 ★★.**

On the computed curve of the diluted vinegar, read the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) at the start, at [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) and at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence). Why is the starting [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) much higher than that of hydrochloric acid at the same concentration?

**Solution of Exercise 46.10.**

About 2.9, 4.76 and 8.7. Ethanoic acid is weak: only a few percent of it gives [oxonium ions](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-oxonium), so the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) starts higher than the 1.0 of hydrochloric acid at a similar concentration.

**Exercise 46.11 ★★.**

Why is the [titrated solution](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) diluted in a large volume of water before a [conductimetric titration](#def-g12-ph-conductivity-titrations-conductimetric)?

**Solution of Exercise 46.11.**

So that the volume of [titrant](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) added hardly changes the total volume: the concentrations, and so the [conductivity](#def-g12-ph-conductivity-titrations-conductivity), then vary along straight lines.

**Exercise 46.12 ★★★.**

A solution contains both hydrochloric acid and ethanoic acid. It is titrated by sodium hydroxide while the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) is measured. Explain why the curve shows two jumps, and what each [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) measures.

**Solution of Exercise 46.12.**

The hydroxide [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) react first with the [strong acid](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid) ($\ce{H3O+}$), then with the weak one. The first jump marks the end of the hydrochloric acid (its [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) measures it), the second the end of the ethanoic acid (the volume between the two jumps measures it).

**Exercise 46.13 ★★★.**

Compute the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) at the start of the [conductimetric titration](#def-g12-ph-conductivity-titrations-conductimetric) of the chapter ($0.0100\,\mathrm{mol}/\mathrm{L}$ hydrochloric acid) and at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) ($110\,\mathrm{mL}$ of sodium chloride solution). Compare with the figure.

**Solution of Exercise 46.13.**

Start: $(349.8 + 76.2) \times 0.0100 = 4.26\,\mathrm{mS}/\mathrm{cm}$. At [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence): $[\ce{Na+}] = [\ce{Cl-}] = 1.00/110 = 0.009\,09\,\mathrm{mol}/\mathrm{L}$, so $(50.3 + 76.2) \times 0.00909 = 1.15\,\mathrm{mS}/\mathrm{cm}$. Both as on the figure.

**Exercise 46.14 ★★★.**

A [pH meter](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph-paper) badly calibrated reads every [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) 0.2 too high. What error does it cause on the [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) of a [pH-metric titration](#def-g12-ph-conductivity-titrations-ph-metric)? On the $pK_a$ read at [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric)?

**Solution of Exercise 46.14.**

None on $V_E$: the jump is at the same volume, whatever the offset of the readings. The $pK_a$ read at [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) is 0.2 too high.

**Exercise 46.15 ★★★.**

Why can a [conductimetric titration](#def-g12-ph-conductivity-titrations-conductimetric) follow the reaction of silver [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) with chloride [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion), $\ce{Ag+ + Cl- -> AgCl(s)}$, while a [pH-metric titration](#def-g12-ph-conductivity-titrations-ph-metric) cannot? Describe the expected curve when silver nitrate is added to sodium chloride (take $\lambda$ of $\ce{NO3-}$ close to that of $\ce{Cl-}$).

**Solution of Exercise 46.15.**

The [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) does not change during this [precipitation](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-precipitate), but the [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) do: before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) each $\ce{Ag+}$ added removes a $\ce{Cl-}$ and brings a $\ce{NO3-}$ of similar [conductivity](#def-g12-ph-conductivity-titrations-conductivity), so the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) stays nearly constant (the sodium [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) were there from the start); after [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) $\ce{Ag+}$ and $\ce{NO3-}$ accumulate and it rises. A flat line, then a rising one.

## 46.4 Problem: Is the Vinegar 6 %?

**Problem 46.1.**

Weekend problem — does a vinegar labelled “6 % acidity” really hold six grams of ethanoic acid per hundred grams?

A vinegar labelled “6 % acidity” is checked. $100.0\,\mathrm{mL}$ of it are weighed: $101\,\mathrm{g}$. The vinegar is diluted ten times, then $10.0\,\mathrm{mL}$ of the diluted solution are titrated by sodium hydroxide at $0.100\,\mathrm{mol}/\mathrm{L}$, with a [pH meter](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph-paper); the computed curve of the chapter (right) matches the measurements.

**Part I — Preparing the sample.**

1. What does “6 % acidity” mean?
2. If the label is right, what mass of ethanoic acid do $100.0\,\mathrm{mL}$ of vinegar contain? What [molar concentration](https://one-course.com/books/chemistry/1/en/chapter/26-concentration-and-dilution#def-g10-concentration-and-dilution-molar-concentration) is that?
3. Why is the vinegar diluted before [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) ?
4. Which glassware is used to dilute it ten times?

**Part II — The [pH-metric titration](#def-g12-ph-conductivity-titrations-ph-metric).**

5. Write the equation of the [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) reaction.
6. Read the [equivalent volume](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) on the curve.
7. Compute the concentration of the diluted vinegar.
8. Deduce that of the vinegar.
9. Read the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) at [half-equivalence](#def-g12-ph-conductivity-titrations-ph-metric) . What does it give?
10. Why is the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) above 7?
11. Which indicator would have given the same result?

**Part III — A conductimetric check.**

12. The [titration](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-titration) is repeated with a conductimetry cell, the sample diluted in $200\,\mathrm{mL}$ of water. Which [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) appear in the solution before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) ?
13. Why does the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) rise only slowly before [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) ?
14. Why does it rise faster after?
15. Why is the [conductivity](#def-g12-ph-conductivity-titrations-conductivity) very low at the start?
16. Where should the two lines meet?

**Part IV — The verdict.**

17. Compute the [mass concentration](https://one-course.com/books/chemistry/1/en/chapter/26-concentration-and-dilution#def-g10-concentration-and-dilution-mass-concentration) of ethanoic acid in the vinegar.
18. Compute the mass of ethanoic acid in $100.0\,\mathrm{mL}$ of vinegar.
19. Deduce the mass of ethanoic acid per $100\,\mathrm{g}$ of vinegar.
20. State the final answer: what is the acidity of the vinegar?

**Solution of Problem 46.1.**

**1.** $6\,\mathrm{g}$ of ethanoic acid per $100\,\mathrm{g}$ of vinegar.

**2.** $101\,\mathrm{g}$ of vinegar hold $6.0 \times 101/100 =
6.06\,\mathrm{g}$; $M = 60.0\,\mathrm{g}/\mathrm{mol}$: $0.101\,\mathrm{mol}$ in $0.1000\,\mathrm{L}$, that is $1.01\,\mathrm{mol}/\mathrm{L}$.

**3.** Undiluted, $10.0\,\mathrm{mL}$ would need about $101\,\mathrm{mL}$ of sodium hydroxide at $0.100\,\mathrm{mol}/\mathrm{L}$: more than a burette holds.

**4.** A $10.0\,\mathrm{mL}$ volumetric pipette and a $100.0\,\mathrm{mL}$ volumetric flask.

**5.** $\ce{CH3COOH + OH- -> CH3COO- + H2O}$.

**6.** $V_E = 10.1\,\mathrm{mL}$.

**7.** $c = 0.100 \times 10.1 / 10.0 = 0.101\,\mathrm{mol}/\mathrm{L}$.

**8.** $10 \times 0.101 = 1.01\,\mathrm{mol}/\mathrm{L}$.

**9.** [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) 4.76 at $5.05\,\mathrm{mL}$: the $pK_a$ of ethanoic acid.

**10.** At [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) the solution holds sodium ethanoate, and ethanoate is a [weak base](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid).

**11.** Phenolphthalein, whose range 8.0–10.0 contains the [pH](https://one-course.com/books/chemistry/1/en/chapter/18-acids-bases-and-ph#def-g9-acids-bases-ph-ph) at [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) (about 8.7).

**12.** Sodium [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) and ethanoate [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion).

**13.** Each $\ce{OH-}$ added turns a nearly un-ionised [molecule](https://one-course.com/books/chemistry/1/en/chapter/11-atoms-and-molecules#def-g7-atoms-and-molecules-molecule) into $\ce{CH3COO-}$, with a $\ce{Na+}$: [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) of modest [conductivity](#def-g12-ph-conductivity-titrations-conductivity) are added slowly.

**14.** Beyond [equivalence](https://one-course.com/books/chemistry/1/en/chapter/36-titration#def-g11-titration-equivalence) the hydroxide [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion), very conducting, accumulate with the sodium [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion).

**15.** The [weak acid](https://one-course.com/books/chemistry/1/en/chapter/44-acids-and-bases-ka-and-pka#def-g12-ka-and-pka-strong-acid) gives very few [ions](https://one-course.com/books/chemistry/1/en/chapter/17-ions-and-ionic-solutions#def-g9-ions-ion) to water.

**16.** At $V_E = 10.1\,\mathrm{mL}$, as in the [pH-metric titration](#def-g12-ph-conductivity-titrations-ph-metric).

**17.** $1.01 \times 60.0 = 60.6\,\mathrm{g}/\mathrm{L}$.

**18.** $6.06\,\mathrm{g}$ in $100.0\,\mathrm{mL}$.

**19.** $6.06 \times 100 / 101 = 6.0\,\mathrm{g}$ per $100\,\mathrm{g}$.

**20.** The vinegar has an acidity of $6.0\,\%$: the label is right.