E2.5 Solve acid–base equilibrium problems using titration and equivalence-point data

Ontario Grade 12 Chemistry

Chemical Systems and Equilibrium

Equivalence points, mole ratios, and weak-acid equilibrium

In a titration, a measured solution is added to another solution until the acid and base have reacted in the required mole ratio. A visible indicator colour change can signal that the titration is near its endpoint. The equivalence point is the point at which the acid and base have reacted in exactly the stoichiometric ratio shown by the balanced equation. These measurements let us find an unknown concentration and, for a weak acid, use equilibrium data to describe how strongly it ionizes.

What you will learn

Explain what happens during an acid–base titration at the particle level.
Use equivalence-point volumes and a balanced reaction to calculate an unknown concentration.
Use pH data at the half-equivalence point to calculate the acid dissociation constant, $K_a$ .

1. From the observed change to the particle model

Before using titration data, recall two ideas. Concentration in moles per litre is calculated by dividing the amount in moles by the solution volume in litres. A balanced chemical equation gives the mole ratio in which reacting substances are consumed.

During a titration, the added solution reacts with the substance in the sample. For a monoprotic acid, each acid particle can donate one hydrogen ion,

\mathrm{H^+}

. A hydroxide ion,

\mathrm{OH^-}

, reacts with it to form water. The salt ions remain in solution.

An indicator changes colour over a particular pH range. Its colour change is an observed endpoint, not the definition of equivalence. In a suitable titration, the endpoint is close to the equivalence point. Calculations should use the stated equivalence-point data, rather than assume every colour change gives an exact equivalence volume.

n=cV

The endpoint is observed; the equivalence point is defined by the reaction's mole ratio.
Use the balanced equation to connect measured volumes to reacting amounts.

2. Use equivalence-point data to find concentration

At the equivalence point, the reacting substances are present in the exact mole ratio required by the balanced equation. This does not mean the solution must be neutral. The pH at equivalence depends on the acid and base involved; the equivalence-point definition is about reacting amounts.

For a monoprotic acid titrated with sodium hydroxide, the reaction has a one-to-one mole ratio. At equivalence, the initial moles of acid equal the moles of sodium hydroxide added. Find the base amount from its concentration and equivalence volume. Then use the ratio from the equation to find the acid amount and its concentration.

For other balanced reactions, do not assume a one-to-one ratio. Use the coefficients to compare moles. Convert millilitres to litres before using concentration in moles per litre. Keep the solution volume used for the unknown concentration separate from the volume of titrant added.

c=\frac{n}{V}

Equivalence means stoichiometric amounts have reacted.
Convert all measured volumes to litres before calculating moles.
Use balanced-equation coefficients when the mole ratio is not one-to-one.

3. Connect a weak acid's equilibrium to titration data

A weak acid ionizes only partly in water. Its ionization is reversible: some acid particles form hydrogen ions and conjugate-base ions, while the reverse reaction can also occur. A conjugate base is the particle left when an acid donates a hydrogen ion.

The acid dissociation constant,

K_a

, describes this equilibrium. For a monoprotic acid written as

\mathrm{HA}

, the equilibrium expression uses the concentrations of the ions at equilibrium divided by the concentration of the acid that remains. A larger

K_a

indicates greater ionization than a smaller

K_a

for acids compared under the same conditions.

In a titration of a weak monoprotic acid with a strong base, the half-equivalence point occurs when half of the original acid has reacted. The amount of acid remaining equals the amount of conjugate base formed. Since they are in the same solution volume, their concentrations are equal. Substitution into the

K_a

expression shows that the pH at this point equals

pK_a

, where

pK_a=-\log K_a

. Thus, a measured half-equivalence pH can be used to find

K_a

.

This shortcut applies to the half-equivalence point for a monoprotic weak acid titrated with a strong base. Do not apply it to the equivalence point: at equivalence, the original weak acid has been consumed, so the acid and conjugate-base concentrations are not equal.

K_a=\frac{[\mathrm{H^+}][\mathrm{A^-}]}{[\mathrm{HA}]}

At half-equivalence, equal concentrations of weak acid and conjugate base make pH equal to $pK_a$ .
The equivalence point and half-equivalence point describe different stages of a titration.

4. Track units, rounding, and the meaning of the result

A reliable solution has three checks. First, the chemical equation is balanced and its coefficients set the mole ratio. Second, the volume units match the concentration units. Third, the result has a sensible number of significant digits based on the data.

Keep extra digits during intermediate calculations and round the final answer to an appropriate number of significant digits. For pH and

K_a

, use the given pH precision when reporting the result. A change of one pH unit corresponds to a tenfold change in hydrogen-ion concentration, so small pH differences can matter.

Finally, state what each result means. An acid concentration comes from the initial acid amount divided by the original sample volume. A

K_a

value describes the weak acid's equilibrium; it is not a concentration of the original sample.

\mathrm{pH}=-\log[\mathrm{H^+}]

Use litres with concentration in moles per litre.
Round final values, not intermediate values.
Concentration and $K_a$ answer different questions.

Worked example

Find a weak acid's concentration and $K_a$

A

25.00\ \mathrm{mL}

sample of a monoprotic weak acid,

\mathrm{HA(aq)}

, is titrated with

0.1000\ \mathrm{mol\,L^{-1}}

sodium hydroxide. Equivalence occurs after

30.00\ \mathrm{mL}

of sodium hydroxide has been added. The pH at the half-equivalence point is 4.76. Find the initial acid concentration and

K_a

.

Write the reaction and identify the ratio
The acid donates one hydrogen ion, and hydroxide forms water. The balanced reaction shows a one-to-one mole ratio between acid and sodium hydroxide.
$\mathrm{HA(aq)+NaOH(aq)\rightarrow NaA(aq)+H_2O(l)}$
Find the amount of sodium hydroxide at equivalence
Convert the equivalence volume to litres, then multiply by the stated concentration. The units cancel to give moles.
$n_{\mathrm{NaOH}}=(0.1000\ \mathrm{mol\,L^{-1}})(0.03000\ \mathrm{L})=0.003000\ \mathrm{mol}$
Use the mole ratio to find the acid concentration
At equivalence, the one-to-one ratio means the original acid amount is also $0.003000\ \mathrm{mol}$ . Divide by the original acid sample volume, not the combined volume, to obtain its initial concentration.
$c_{\mathrm{HA}}=\frac{0.003000\ \mathrm{mol}}{0.02500\ \mathrm{L}}=0.1200\ \mathrm{mol\,L^{-1}}$
Use the half-equivalence pH to calculate $K_a$
At half-equivalence, equal amounts of acid remain and conjugate base have formed, so the pH equals $pK_a$ . Therefore, $K_a=10^{-\mathrm{pH}}$ . The given pH supports two significant figures in the reported $K_a$ .
$K_a=10^{-4.76}=1.7\times10^{-5}$

Answer: The initial acid concentration is

0.1200\ \mathrm{mol\,L^{-1}}

, and

K_a=1.7\times10^{-5}

.

Check: The equivalence volume is larger than the acid sample volume, but this does not mean the acid concentration is smaller than the base concentration. The mole calculation gives

0.003000\ \mathrm{mol}

in only

0.02500\ \mathrm{L}

of original acid, consistent with

0.1200\ \mathrm{mol\,L^{-1}}

.

Common mistakes and how to avoid them

Treating the endpoint colour change as the exact definition of equivalence.

Correction: The endpoint is observed. Equivalence is determined by the balanced reaction's mole ratio.

Using the total mixed volume to calculate the initial acid concentration.

Correction: Use the original acid sample volume because the requested concentration is the acid's initial concentration.

Assuming pH is 7 at every equivalence point.

Correction: Equivalence means the reacting amounts match the equation. It does not, by itself, specify pH.

Using the half-equivalence relationship at equivalence.

Correction: At half-equivalence, weak acid and conjugate base concentrations are equal. That condition does not hold at equivalence.

Lesson summary

Use balanced equations and equivalence-point volumes to calculate reacting amounts.
For a monoprotic weak acid and strong base, the equivalence mole ratio is one to one.
At half-equivalence in this titration, pH equals $pK_a$ , allowing $K_a$ to be calculated.
Keep units consistent and distinguish the endpoint, equivalence point, and half-equivalence point.

Check your understanding

Question 1

A monoprotic acid reacts with sodium hydroxide in a one-to-one ratio. At equivalence,

0.00240\ \mathrm{mol}

of sodium hydroxide has been added. How many moles of acid were initially present?

$0.00120\ \mathrm{mol}$
$0.00240\ \mathrm{mol}$
$0.00480\ \mathrm{mol}$
correctIndex`?

Show answer and explanation

0.00240\ \mathrm{mol}

The balanced reaction has a one-to-one mole ratio, so the initial acid amount equals the amount of sodium hydroxide at equivalence.

Question 2

At the half-equivalence point of a weak monoprotic acid titration, the measured pH is 5.20. What is

pK_a

?

2.60
5.20
10.40
correctIndex`?

Show answer and explanation

5.20

At half-equivalence, the weak acid and its conjugate base have equal concentrations, so pH equals

pK_a

.

Key terms

Titration: A method that uses a measured solution to determine the amount or concentration of another substance by reaction.
Equivalence point: The point in a titration where reacting amounts match the mole ratio in the balanced equation.
Endpoint: The observed signal, such as an indicator colour change, used to identify that a titration is near equivalence.
Weak acid: An acid that ionizes only partly in water.
Conjugate base: The particle formed when an acid donates a hydrogen ion.
$K_a$: The acid dissociation constant, which describes the equilibrium of a weak acid in water.
Half-equivalence point: The point in a titration where half of the original acid has reacted.

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E2.5 · Solve acid–base equilibrium problems using titration and equivalence-point data