Why a pH Probe Can Locate Equivalence Without a Colour Indicator

A pH probe can locate the equivalence point of a titration without a colour indicator because it measures how the solution’s pH changes as titrant is added. By recording pH against the volume of titrant delivered, you can construct a titration curve and identify the steepest part of the curve, which corresponds approximately to the equivalence point.

The important detail is that the probe does not detect neutralisation directly. It measures pH. You use the resulting measurements, together with the reaction’s stoichiometry, to determine when equivalence has been reached.

01First, what determines the equivalence point?

Imagine you have a flask containing an acid and a burette containing sodium hydroxide. You slowly add the sodium hydroxide to the acid.

How do you know when enough base has been added to react with all the acid?

The answer comes from the balanced chemical equation, not from the pH.

Consider the neutralisation of a monoprotic acid, represented by \(\ce{HA}\), with sodium hydroxide:

\[ \ce{HA(aq) + OH^-(aq) -> A^-(aq) + H2O(l)} \]

The mole ratio is \(1:1\). One mole of hydroxide ions reacts with one mole of acid molecules.

The equivalence point occurs when the acid and base have reacted in exactly the mole ratio specified by the balanced equation.

For this particular reaction:

\[ n(\ce{HA})=n(\mathrm{OH}^{-}) \]

Remember that this is a stoichiometric condition. It tells us how much acid and base have reacted, not what the pH must be.

For a different reaction, the mole ratio might be different. For example, complete neutralisation of a diprotic acid requires two moles of hydroxide ions per mole of acid:

\[ \ce{H2A + 2OH^- -> A^2- + 2H2O} \]

You must establish the correct mole ratio before interpreting any experimental readings.

Equivalence point versus endpoint

These terms describe different things.

With a suitable indicator, the endpoint should occur very close to the equivalence point.

However, the two are not automatically identical. An indicator changes colour over a particular pH range, so its endpoint depends on how well that range matches the titration curve.

If you want to understand that distinction in more detail, see Endpoint vs Equivalence Point: The Difference in Titrations.

02What does a pH probe actually measure?

An indicator gives you a colour. A pH probe gives you a numerical reading.

This is the key difference.

A typical pH probe contains a glass electrode and a reference electrode. The glass membrane develops an electrical potential that depends on the hydrogen-ion activity of the surrounding solution.

The reference electrode provides a stable electrical comparison. The pH meter measures the potential difference between the electrodes and converts it into a pH reading.

In other words, the instrument detects an electrical signal associated with the solution’s acidity or basicity.

At the HSC level, you will generally work with the relationship:

\[ \mathrm{pH}=-\log_{10}[\ce{H3O+}] \]

Here:

• \(\mathrm{pH}\) is the acidity of the solution.
• \([\ce{H3O+}]\) is the hydronium ion concentration in \(\mathrm{mol\,L^{-1}}\).

Strictly, pH is defined using hydrogen-ion activity rather than concentration. For ordinary dilute solutions in HSC calculations, concentration is commonly used as an approximation.

Notice that pH is logarithmic.

A change of one pH unit represents a tenfold change in hydronium ion activity. A change of two pH units represents a hundredfold change.

This explains why a relatively small addition of titrant can produce a dramatic change in the probe reading near equivalence.

Why is calibration important?

A pH probe needs to be calibrated before collecting reliable measurements.

This is done using buffer solutions with known pH values. The meter uses these reference readings to establish the relationship between electrical potential and pH.

During a titration, the probe should:

• Be rinsed with distilled water before use and between relevant measurements.
• Be gently blotted to remove excess water rather than rubbed.
• Remain immersed sufficiently in the solution.
• Be kept clear of the bottom of the flask and any magnetic stirrer.
• Be allowed to stabilise before each reading is recorded.

An incorrectly calibrated probe can produce a convincing-looking graph with inaccurate pH values.

The position of the equivalence region may still be recognisable, but accurate readings are important when interpreting the curve.

03How does the pH probe locate equivalence?

Let’s picture the actual experiment.

A burette contains the titrant, which is the solution being added. Its tap controls how much titrant enters the conical flask below.

The flask contains the solution being analysed, known as the analyte. A pH probe is immersed in this solution and connected to a pH meter or data logger.

As titrant is added, the solution is stirred gently so that it remains well mixed. The probe measures the pH after each addition.

The experiment produces two measurements:

1. The total volume of titrant added, usually determined from burette readings.
2. The pH of the solution after each addition.

You can then plot these measurements on a graph.

• Horizontal axis: volume of titrant added, in mL.
• Vertical axis: measured pH.

This graph is called a titration curve.

What should the graph look like?

Consider a strong acid being titrated with a strong base.

At the beginning, the flask contains a strongly acidic solution. Its pH is low.

As sodium hydroxide is added, hydroxide ions react with hydronium ions:

\[ \ce{H3O+(aq) + OH^-(aq) -> 2H2O(l)} \]

The acid is gradually neutralised, so the pH increases.

Initially, the pH changes relatively slowly. There is still plenty of acid remaining, so another small addition of base does not produce an enormous change in the overall solution.

As the titration approaches equivalence, however, very little excess acid remains.

A small additional volume of sodium hydroxide can now change the balance from excess acid to excess base. The pH rises very rapidly.

After equivalence, the added sodium hydroxide is in excess. The pH continues increasing, but the curve becomes less steep.

The equivalence point is identified approximately at the inflection point, which is the point where the curve has its greatest gradient.

In practical terms, this is the steepest part of the curve.

Why?

Because this is where the pH changes most rapidly for a small change in the volume of titrant added.

For a well-defined titration curve, this provides a reliable way to estimate the equivalence volume without observing any colour change.

Why the steepest point matters

You can express the gradient of a section of the titration curve as:

\[ \text{Gradient}= \frac{\Delta\mathrm{pH}}{\Delta V} \]

Here:

• \(\Delta\mathrm{pH}\) is the change in pH between two readings.
• \(\Delta V\) is the change in titrant volume, measured in mL.

The gradient is expressed in pH units per mL.

The larger the gradient, the more rapidly the pH changes as titrant is added.

For a titration where the pH increases as base is added, the gradient is positive. If acid is added to a base instead, the pH generally decreases, so the gradient is negative. In either case, the steepest section is identified using the greatest gradient magnitude.

This is particularly useful when the readings are recorded at unequal volume intervals.

A change of 1.0 pH unit over 0.1 mL is much steeper than the same change over 1.0 mL.

You cannot simply look for the largest difference between two pH readings. You must account for the volume added.

04Why doesn’t the equivalence point have to be at pH 7?

Here’s a common mistake.

A student sees a pH titration curve and assumes that equivalence must occur when the probe reads pH 7.

That is only approximately correct for a strong acid-strong base titration at \(25\,^\circ\mathrm{C}\).

Consider ethanoic acid reacting with sodium hydroxide:

\[ \ce{CH3COOH + NaOH -> CH3COONa + H2O} \]

The reaction has a \(1:1\) mole ratio.

At equivalence, the ethanoic acid has been completely neutralised stoichiometrically. However, the solution contains sodium ethanoate.

The ethanoate ion reacts with water:

\[ \ce{CH3COO^- + H2O <=> CH3COOH + OH^-} \]

This produces hydroxide ions, making the solution basic.

Consequently, the equivalence point occurs above pH 7.

The pH probe still locates equivalence because it measures the change in pH as sodium hydroxide is added. It does not need the equivalence point to occur at a particular pH.

The important distinction is:

• Stoichiometry determines the equivalence volume.
• The chemical species present at equivalence determine the equivalence pH.
• The pH probe provides measurements that allow you to identify the equivalence region experimentally.

For a more detailed explanation of why equivalence pH varies, read Why the Equivalence Point Is Not Always pH 7.

05How to use pH probe readings in an experiment

Now that you understand what the probe measures, let’s put the experimental method together.

A reliable investigation follows this sequence.

Step 1: Write the balanced chemical equation.

Determine the mole ratio between the acid and base.

Do not assume every neutralisation reaction has a \(1:1\) ratio.

Step 2: Establish the expected stoichiometric relationship.

If the concentrations and initial volume are known, calculate the expected equivalence volume using the balanced equation.

This provides an independent check on the experimental result.

Step 3: Prepare and calibrate the pH probe.

Use suitable buffer solutions to calibrate the instrument. Rinse the probe and position it in the analyte.

Step 4: Add titrant gradually.

Record the burette reading and the corresponding pH.

Initially, larger additions may be suitable because the pH changes relatively slowly.

Step 5: Reduce the volume increments near equivalence.

This is particularly important.

Suppose the pH changes from 4.0 to 10.0 between two readings. If those readings are separated by 2.0 mL, you know the steep region is somewhere within that interval.

You do not know its precise location.

Repeat the titration using smaller additions near the steep region. Increments of 0.10 mL or smaller may be appropriate, depending on the sharpness of the curve and the required precision.

Step 6: Plot pH against total titrant volume.

Identify the steepest section and estimate the inflection point.

For discrete readings, calculate the gradients between successive measurements.

Step 7: Compare the experimental result with stoichiometry.

If the measured equivalence volume differs substantially from the expected value, investigate possible sources of error.

These might include incorrect solution concentrations, inaccurate burette readings, poor calibration, or insufficient mixing.

What happens if the volume increments are too large?

Imagine adding sodium hydroxide in 1.0 mL portions.

Your measurements might look like this:

You can see that the equivalence point is close to 20.0 mL.

However, the experiment has not provided enough information to determine its precise location from the measurements alone.

A smaller addition interval would give you more detail about the steep section.

This is an important experimental limitation: a pH probe can only report what has actually been measured.

A sophisticated instrument cannot compensate for poorly chosen measurement intervals.

06What can a pH probe do that an indicator cannot?

Both methods can help identify equivalence, but they provide different types of information.

An indicator is useful when its transition range overlaps the steep section of the titration curve.

A pH probe avoids the need to select a colour transition range. It provides numerical readings that can be used to construct the entire titration curve.

However, a pH probe is not automatically more reliable in every situation.

An important limitation: weak acid and weak base titrations

Consider ethanoic acid reacting with ammonia.

Both substances are weak, so the pH changes relatively gradually as the reaction proceeds.

The balanced equation is:

\[ \ce{CH3COOH + NH3 <=> CH3COO^- + NH4+} \]

The reaction has a \(1:1\) stoichiometric ratio.

A pH probe can still measure the pH after each addition. However, the resulting curve may not contain a pronounced vertical section.

Without a clearly defined steep region, identifying the equivalence point from the inflection point becomes less reliable.

This gives us an important qualification:

A pH probe can locate equivalence without an indicator when the titration produces a sufficiently distinct change in pH. It cannot guarantee a clearly identifiable equivalence point when the underlying reaction produces a gradual titration curve.

If the reactant concentrations are known, stoichiometry can still predict the equivalence volume. If they are unknown, another suitable analytical method may be needed.

07Worked examples

08Questions and solutions

Question 1

A student titrates \(30.00\ \mathrm{mL}\) of \(0.0800\ \mathrm{mol\,L^{-1}}\) hydrochloric acid with \(0.120\ \mathrm{mol\,L^{-1}}\) sodium hydroxide.

A pH probe records a sharp increase in pH between 19.9 mL and 20.1 mL of sodium hydroxide added.

a) Calculate the theoretical equivalence volume.

b) Explain why the sharp increase in pH allows the equivalence point to be identified without an indicator.

Solution 1

a) The theoretical equivalence volume is \(20.0\ \mathrm{mL}\).

First, write the balanced equation:

\[ \ce{HCl + NaOH -> NaCl + H2O} \]

The mole ratio is \(1:1\).

Calculate the initial moles of hydrochloric acid:

\[ n=CV \]

\[ =0.0800\times0.03000 \]

\[ =0.002400\ \mathrm{mol} \]

The same number of moles of sodium hydroxide is required.

Therefore:

\[ V=\frac{0.002400}{0.120} \]

\[ =0.0200\ \mathrm{L} \]

\[ \boxed{V_{\mathrm{eq}}=20.0\ \mathrm{mL}} \]

b) The sharp pH increase occurs because a small addition of sodium hydroxide near equivalence changes the solution from containing excess acid to containing excess base.

Before equivalence, hydroxide ions react with the remaining hydronium ions.

Near equivalence, very little excess acid remains. A small additional volume of sodium hydroxide therefore produces a large change in the hydronium ion concentration.

The pH probe records this change numerically, allowing the steep section of the titration curve to be identified.

The result agrees with the theoretical equivalence volume of 20.0 mL.

Question 2

A student titrates \(25.00\ \mathrm{mL}\) of an unknown monoprotic weak acid with \(0.1000\ \mathrm{mol\,L^{-1}}\) sodium hydroxide.

The following illustrative pH readings are obtained:

a) Estimate the equivalence volume.

b) Calculate the original concentration of the weak acid.

c) Explain why the equivalence pH is expected to be greater than 7.

Solution 2

a) The equivalence volume is approximately \(17.75\ \mathrm{mL}\).

The largest gradient occurs between 17.74 mL and 17.76 mL.

\[ \text{Gradient}= \frac{9.0-7.9}{17.76-17.74} \]

\[ =\frac{1.1}{0.02} \]

\[ =55\ \mathrm{pH\,mL^{-1}} \]

This is the steepest recorded interval.

The midpoint of the interval is:

\[ V_{\mathrm{eq}}\approx \frac{17.74+17.76}{2} \]

\[ \boxed{V_{\mathrm{eq}}\approx17.75\ \mathrm{mL}} \]

b) The original acid concentration is \(0.0710\ \mathrm{mol\,L^{-1}}\).

The balanced equation is:

\[ \ce{HA + OH^- -> A^- + H2O} \]

The mole ratio is \(1:1\).

Calculate the moles of sodium hydroxide at equivalence:

\[ n=CV \]

\[ =0.1000\times0.01775 \]

\[ =0.001775\ \mathrm{mol} \]

Therefore, the original acid sample contained \(0.001775\) mol of acid.

Its original volume was \(0.02500\ \mathrm{L}\).

\[ C=\frac{n}{V} \]

\[ =\frac{0.001775}{0.02500} \]

\[ \boxed{C=0.0710\ \mathrm{mol\,L^{-1}}} \]

c) The equivalence solution is basic because the weak acid produces a conjugate base that reacts with water.

At equivalence, the solution contains the salt formed from the weak acid and strong base.

The conjugate base accepts protons from water:

\[ \ce{A^- + H2O <=> HA + OH^-} \]

The production of hydroxide ions makes the solution basic.

Therefore, the equivalence pH is expected to be greater than 7 at \(25\,^\circ\mathrm{C}\).

The probe identifies equivalence through the steepest section of the curve, not by searching for pH 7.

Question 3

A student titrates \(20.00\ \mathrm{mL}\) of \(0.1000\ \mathrm{mol\,L^{-1}}\) hydrochloric acid with \(0.1000\ \mathrm{mol\,L^{-1}}\) sodium hydroxide.

The probe produces the following illustrative data:

A student argues that the equivalence point must be closer to 20.25 mL because the pH changes by 4.09 units between 20.0 mL and 20.5 mL.

Evaluate this argument. Use the gradients of the data to identify the steepest region and estimate the equivalence volume.

Solution 3

The student’s argument is incorrect. The equivalence volume is approximately \(20.0\ \mathrm{mL}\), because the greatest local gradient occurs around this volume.

The problem is that the student is comparing total pH changes without properly accounting for the volume intervals.

Calculate the relevant gradients.

Between 19.8 mL and 19.9 mL:

\[ \frac{3.60-3.30}{19.9-19.8} =3.0\ \mathrm{pH\,mL^{-1}} \]

Between 19.9 mL and 20.0 mL:

\[ \frac{7.00-3.60}{20.0-19.9} =34\ \mathrm{pH\,mL^{-1}} \]

Between 20.0 mL and 20.1 mL:

\[ \frac{10.40-7.00}{20.1-20.0} =34\ \mathrm{pH\,mL^{-1}} \]

Between 20.1 mL and 20.5 mL:

\[ \frac{11.09-10.40}{20.5-20.1} =1.725\ \mathrm{pH\,mL^{-1}} \]

The greatest gradients occur immediately on either side of 20.0 mL.

The student’s comparison of the entire 20.0-20.5 mL interval hides what happens inside that interval. The pH rises very rapidly at first, then much more slowly.

The correct method is to compare the pH change per unit volume, not simply the total pH change.

We can also confirm the result using stoichiometry.

\[ n(\ce{HCl})=0.1000\times0.02000 \]

\[ =0.002000\ \mathrm{mol} \]

Because the reaction has a \(1:1\) mole ratio:

\[ V(\ce{NaOH})=\frac{0.002000}{0.1000} \]

\[ \boxed{V_{\mathrm{eq}}=20.00\ \mathrm{mL}} \]

The experimental and theoretical results agree.

Question 4

A student titrates \(25.00\ \mathrm{mL}\) of \(0.100\ \mathrm{mol\,L^{-1}}\) ethanoic acid with \(0.100\ \mathrm{mol\,L^{-1}}\) ammonia.

The reaction is:

\[ \ce{CH3COOH + NH3 <=> CH3COO^- + NH4+} \]

The calibrated pH probe produces the following illustrative readings:

The student argues that the pH probe has failed because there is no pronounced vertical section on the graph.

a) Calculate the theoretical equivalence volume.

b) Explain why the pH probe can still be functioning correctly.

c) Explain whether the pH probe readings alone provide a reliable way to locate equivalence in this experiment.

Solution 4

a) The theoretical equivalence volume is \(25.0\ \mathrm{mL}\).

The balanced equation shows a \(1:1\) mole ratio.

Calculate the initial moles of ethanoic acid:

\[ n=CV \]

\[ =0.100\times0.02500 \]

\[ =0.00250\ \mathrm{mol} \]

The same number of moles of ammonia is required.

Therefore:

\[ V=\frac{0.00250}{0.100} \]

\[ \boxed{V_{\mathrm{eq}}=25.0\ \mathrm{mL}} \]

b) The probe can be functioning correctly because a gradual pH change does not necessarily indicate an instrument fault.

The probe measures the pH of the solution, regardless of whether the solution is undergoing a strong or weak acid-base reaction.

In this case, both ethanoic acid and ammonia are weak.

The pH therefore changes more gradually than it would in a strong acid-strong base titration.

The probe is not responsible for creating the steep section of a titration curve. It only records the chemical changes occurring in the solution.

c) The pH readings alone do not provide a sufficiently clear equivalence point in this experiment.

The data show a gradual increase in pH rather than a pronounced vertical section.

Although the probe measures the pH correctly, there is no sharply defined maximum gradient that can be used to locate equivalence with high confidence.

The theoretical equivalence volume can still be calculated because the initial concentration and volume of ethanoic acid are known.

If the concentrations were unknown, relying only on this gradual pH curve would produce an uncertain equivalence estimate.

This demonstrates an important limitation: the usefulness of a pH probe depends partly on the shape of the titration curve, not just on the accuracy of the instrument.

Question 5

A student investigates an unknown monoprotic weak acid, \(\ce{HA}\), using two separate titrations.

For each experiment, the student takes a fresh \(25.00\ \mathrm{mL}\) aliquot of the same acid solution and titrates it with \(0.1000\ \mathrm{mol\,L^{-1}}\) sodium hydroxide.

In Experiment A, the acid is titrated without any dilution.

In Experiment B, the student adds \(75.0\ \mathrm{mL}\) of distilled water to the acid before beginning the titration.

The pH probe produces the following illustrative readings near the equivalence region:

Both curves have their steepest regions centred approximately at \(25.00\ \mathrm{mL}\).

A student makes three claims:

1. Experiment B should require five times as much sodium hydroxide because the solution volume has increased from 25.00 mL to 100.0 mL.
2. The equivalence point must occur when the probe reads pH 7.00.
3. Because Experiment B has a smaller pH change near equivalence, its equivalence volume must be different.

Evaluate all three claims.

Calculate the original concentration of the acid and explain why the two experiments produce different pH readings despite having the same equivalence volume.

Assume the acid is monoprotic, the reaction is complete, the added water contains negligible acid or base, and the temperature is \(25\,^\circ\mathrm{C}\).

Solution 5

All three claims are incorrect. Both experiments have an equivalence volume of approximately \(25.00\ \mathrm{mL}\), and the original acid concentration is \(0.1000\ \mathrm{mol\,L^{-1}}\).

The key is to distinguish the number of moles of acid from its concentration in the flask.

Step 1: Establish the stoichiometric relationship.

The balanced equation is:

\[ \ce{HA + OH^- -> A^- + H2O} \]

The mole ratio is \(1:1\).

Therefore, at equivalence:

\[ n(\ce{HA})=n(\mathrm{OH}^{-}) \]

This relationship applies to both experiments.

Step 2: Determine whether adding water changes the amount of acid.

Experiment A contains an original \(25.00\ \mathrm{mL}\) sample of acid.

Experiment B contains exactly the same original sample, but with an additional \(75.0\ \mathrm{mL}\) of water.

Adding water decreases the acid concentration, but it does not remove any acid molecules.

Therefore:

\[ n_{\mathrm{acid,A}}=n_{\mathrm{acid,B}} \]

Both experiments require the same number of moles of sodium hydroxide to reach equivalence.

This disproves claim 1.

The mistake is confusing concentration with the amount of substance.

Dilution changes concentration because the same amount of solute occupies a larger volume. It does not increase the number of moles that must be neutralised.

Step 3: Calculate the equivalence volume in both experiments.

The data indicate that both titration curves have their steepest regions centred around:

\[ V_{\mathrm{eq}}=25.00\ \mathrm{mL} \]

Calculate the moles of sodium hydroxide required:

\[ n=CV \]

\[ =0.1000\times0.02500 \]

\[ =0.002500\ \mathrm{mol} \]

Because the mole ratio is \(1:1\), each original acid aliquot contains:

\[ n(\ce{HA})=0.002500\ \mathrm{mol} \]

The two experiments therefore have the same equivalence volume.

Step 4: Calculate the original concentration of the acid.

The original acid aliquot has a volume of \(25.00\ \mathrm{mL}\), or \(0.02500\ \mathrm{L}\).

\[ C=\frac{n}{V} \]

\[ =\frac{0.002500}{0.02500} \]

\[ \boxed{C=0.1000\ \mathrm{mol\,L^{-1}}} \]

This is the original concentration before any additional water was added.

The concentration in the diluted flask at the start of Experiment B is different:

\[ C_{\mathrm{diluted}}= \frac{0.002500}{0.1000} \]

\[ =0.02500\ \mathrm{mol\,L^{-1}} \]

However, the number of moles remains \(0.002500\ \mathrm{mol}\).

Step 5: Explain why the equivalence pH changes between the experiments.

At equivalence, the solution contains sodium ethanoate if the weak acid is ethanoic acid, or the corresponding salt for another weak monoprotic acid.

The conjugate base reacts with water to produce hydroxide ions.

The concentration of this salt is lower in Experiment B because the solution has been diluted.

For the same weak acid, the lower salt concentration produces a smaller hydroxide ion concentration at equivalence.

Consequently, the equivalence pH is lower in Experiment B.

This explains the readings:

• Experiment A: equivalence pH approximately 8.72.
• Experiment B: equivalence pH approximately 8.53.

The diluted experiment also produces a less pronounced pH change near equivalence.

However, neither effect changes the stoichiometric amount of sodium hydroxide required.

This disproves claim 3.

Step 6: Evaluate the claim that pH 7.00 identifies equivalence.

Claim 2 is also incorrect.

Both experiments involve a weak acid reacting with a strong base.

The conjugate base remaining at equivalence reacts with water to produce hydroxide ions.

Therefore, the equivalence solution is basic, and its pH is greater than 7 at \(25\,^\circ\mathrm{C}\).

The probe reading of pH 7.00 is not the defining condition for equivalence.

The defining condition is the stoichiometric mole ratio.

Final conclusion:

Both experiments require \(25.00\ \mathrm{mL}\) of sodium hydroxide to reach equivalence, even though one sample has been diluted.

The original acid concentration is \(0.1000\ \mathrm{mol\,L^{-1}}\).

Dilution changes the concentrations of the species in solution and therefore changes the shape and pH values of the titration curve. It does not change the number of acid moles present.

This is the crucial distinction: the pH probe measures the chemical conditions in the flask, while the balanced equation determines the amount of titrant required for equivalence.

09Where to go next

Once you can identify equivalence from a pH titration curve, the next useful skill is interpreting the curve before, at, and after equivalence.

This lets you connect the experimental readings to the actual concentrations of hydronium ions, hydroxide ions, and other species remaining in solution.

For that, continue with Calculating pH Before and After Equivalence in Strong-Acid Titrations.

The key idea to carry forward is simple: use stoichiometry to establish where equivalence should occur, and use the pH probe to observe the chemical changes that help you locate it experimentally.