What Integration Means in a Proton NMR Spectrum

A proton NMR spectrum can contain a very tall, narrow peak and a much shorter signal. Which one represents more hydrogen atoms? You cannot tell from peak height alone. The answer comes from **integration**: measuring the area under each signal. This area is proportional to the number of hydrogen nuclei contributing to that signal.

In other words, integration tells you how many hydrogen atoms belong to each proton environment, relative to the other environments in the molecule. A spectrum with an integration ratio of \(3:2:1\), for example, indicates three groups containing hydrogen atoms in that ratio.

The important distinction is that integration measures area, not height, and usually gives relative numbers rather than an automatic count of individual hydrogen atoms.

## What does integration actually measure?

Imagine two hills on a graph. One is tall and narrow, while the other is shorter but much wider. Which contains more area?

You cannot answer by comparing their heights. You need to consider both height and width.

The same idea applies to proton NMR. Each signal has a shape, and integration measures the total area beneath that shape. A signal representing more equivalent hydrogen nuclei generally has a larger integrated area.

[IMAGE: An annotated proton NMR spectrum with two signals of noticeably different shapes: one tall and narrow, the other shorter and broader. Show the shaded area beneath each complete signal and a staircase-style integration trace above them. Make the broader signal's total area larger despite its lower peak height. Label the horizontal axis Chemical shift (ppm), with higher ppm on the left, and the vertical axis Signal intensity.]

### What does the integration trace show?

Many proton NMR spectra include an additional line called an integration trace. Rather than following the peaks themselves, this line rises in steps as it passes across each signal.

The size of each upward step is proportional to the area of the corresponding signal.

- A small step represents a relatively small number of hydrogen atoms.
- A larger step represents a relatively large number of hydrogen atoms.
- Equal-sized steps represent signals with equal integrated areas.

For example, suppose three signals produce integration steps of 3 units, 2 units, and 1 unit. Their relative integration is:

\[
3:2:1
\]

This means the three proton environments contribute hydrogen atoms in the ratio \(3:2:1\).

The numbers are not necessarily the actual hydrogen counts yet. They are proportional values that need to be interpreted using the molecular formula and other available evidence.

### Why is area more useful than peak height?

A signal can spread across several closely spaced lines because of splitting. Measuring only its tallest line would ignore the rest of the signal.

Integration includes the entire signal, so it still measures the total contribution from that proton environment.

This is particularly important for multiplets:

- A singlet has one line.
- A doublet has two lines.
- A triplet has three lines.
- A quartet has four lines.

A triplet is still one signal from one proton environment. Its integration includes all three lines, not just the tallest one.

[IMAGE: Side-by-side comparison of a singlet, doublet, triplet, and quartet in a proton NMR spectrum. Give each complete signal the same total integrated area despite their different numbers of lines and characteristic relative line heights. Show a separate integration step of equal height for each.]

## Integration is not the same as signal count, chemical shift, or splitting

These four features work together when interpreting a proton NMR spectrum, but each answers a different question.

| Spectral feature | What it tells you | What you examine |
|---|---|---|
| Signal count | How many different proton environments exist | Number of separate signal groups |
| Integration | How many hydrogen atoms contribute to each environment, relative to the others | Area under each complete signal |
| Chemical shift | What electronic environment surrounds the hydrogen atoms | Position in ppm |
| Splitting | How many neighbouring equivalent hydrogen atoms interact with the group | Number and arrangement of lines |

Consider a signal that appears as a quartet at \(4.1\ \mathrm{ppm}\) and integrates to 2H.

Each piece of information tells you something different:

- The quartet describes the splitting pattern.
- The chemical shift of \(4.1\ \mathrm{ppm}\) suggests an environment influenced by an electronegative atom, such as oxygen.
- The integration of 2H indicates that two hydrogen atoms contribute to the signal.
- The quartet remains one signal, representing one proton environment in the introductory model.

Do not use the number of lines to calculate the integration. Four lines do not mean four hydrogen atoms.

For a broader explanation of how these four clues work together when identifying organic compounds, see [Proton NMR Spectroscopy: How to Interpret HSC Chemistry Data](https://tutorgum.com/learn/resources/hsc-chemistry-proton-nmr-spectroscopy/).

## How do chemists obtain an integration value?

Integration is not a separate measurement made by counting hydrogen atoms inside the instrument. It is calculated from the signal recorded by the NMR spectrometer.

Understanding the measurement helps explain why area is useful.

### Following the signal through the instrument

A modern proton NMR experiment follows a sequence.

[IMAGE: Scientifically faithful side-view schematic of a modern NMR spectrometer measurement. Show a narrow glass sample tube positioned vertically inside the bore of a large superconducting magnet, with the radiofrequency probe coil surrounding the sample. Indicate the magnetic field along the tube, an RF pulse travelling from the transmitter to the probe, and the returning nuclear magnetic response travelling to the receiver and computer. Label the sample tube, magnet, probe coil, transmitter, receiver, and computer in their actual functional positions.]

**Step 1: Place the sample in the magnetic field.**

The organic compound is dissolved in a suitable solvent and placed in a narrow glass tube. The tube is positioned inside a strong, uniform magnetic field.

Hydrogen-1 nuclei interact with this field. Their surrounding electrons affect the magnetic field experienced by each nucleus.

**Step 2: Apply radiofrequency energy.**

A coil in the probe sends a radiofrequency pulse through the sample. The hydrogen nuclei respond to this energy.

Different proton environments respond at slightly different resonance frequencies because their electronic surroundings are different.

**Step 3: Detect the returning signal.**

After the pulse, the instrument detects a changing electromagnetic signal from the sample. This is called the free induction decay, or FID.

The FID is a time-dependent signal. It is not yet the familiar graph of peaks against chemical shift.

**Step 4: Convert the recorded signal into a spectrum.**

A computer applies a mathematical operation called a Fourier transform. This converts the time-dependent signal into a frequency-domain spectrum, which is displayed using chemical shift in ppm.

**Step 5: Calculate the integrated areas.**

The computer measures the area beneath each signal, normally after suitable baseline correction and integration limits have been selected.

The resulting areas are proportional to the numbers of contributing hydrogen nuclei under appropriate measurement conditions.

The instrument does not directly print a molecular structure. Chemists interpret the integrated areas alongside the signal count, chemical shifts, and splitting patterns.

### The physical reason integration works

Suppose one proton environment contains three equivalent hydrogen nuclei, while another contains two.

If the experiment is performed under suitable conditions, the first environment produces approximately 1.5 times the integrated signal area of the second.

Therefore:

\[
\frac{\text{Area}_1}{\text{Area}_2}
=
\frac{3}{2}
\]

The relationship is proportional because each contributing hydrogen nucleus adds to the detected NMR response.

This is the basis of proton integration.

It is worth being precise: this proportionality assumes an appropriately acquired spectrum, with suitable relaxation conditions and reliable processing. It is not a guarantee that every displayed peak area in every experimental spectrum will give an exact integer proton count.

### What should the measurement show?

There is a simple prediction behind proton integration.

Imagine a compound with two distinct proton environments. One contains three equivalent hydrogen nuclei, and the other contains two.

If the instrument measures the total response from each group under suitable conditions, what ratio should the areas have?

The predicted ratio is:

\[
\frac{A_1}{A_2}=\frac{3}{2}
\]

where \(A_1\) and \(A_2\) are the integrated areas of the two signals.

An experimental result close to \(3:2\) supports the idea that integrated area is proportional to the number of contributing hydrogen nuclei.

The result is an area ratio, not a direct photograph or count of individual atoms. That distinction becomes important when interpreting mixtures, overlapping signals, and exchangeable hydrogen atoms.

## How do you convert integration values into proton numbers?

A spectrum might give you integration values such as:

\[
0.60,\quad 0.40,\quad 0.20
\]

These are arbitrary units, not hydrogen counts.

Your job is to convert them into the simplest whole-number ratio.

Use this method:

**Step 1: Record the total area of each signal.**

Include every line belonging to a multiplet. Do not measure just its tallest line.

**Step 2: Simplify the integration values.**

Divide every value by the smallest value.

For the example:

\[
\begin{aligned}
0.60:0.40:0.20
&=3:2:1
\end{aligned}
\]

**Step 3: Compare the ratio with the molecular formula.**

If the molecular formula contains six hydrogen atoms and these are the only signals, the ratio \(3:2:1\) corresponds to:

- First environment: 3H
- Second environment: 2H
- Third environment: 1H

The numbers already add to six, so no further adjustment is needed.

**Step 4: Check that every hydrogen is accounted for.**

If the ratio adds to fewer hydrogen atoms than the molecular formula contains, consider whether an OH or NH proton is missing or poorly integrated.

If the numbers cannot be reconciled, do not force them into an answer. The spectrum may contain overlapping signals, impurities, or experimental errors.

### Worked example: Determine the proton numbers in tert-butyl ethanoate

A compound has the structural formula \(\ce{CH3COOC(CH3)3}\) and molecular formula \(\ce{C6H12O2}\).

Its illustrative proton NMR spectrum contains two signals:

- Signal A at \(\delta=2.0\ \mathrm{ppm}\), with an integrated area of 0.50 arbitrary units.
- Signal B at \(\delta=1.4\ \mathrm{ppm}\), with an integrated area of 1.50 arbitrary units.

Determine the integration ratio and the number of hydrogen atoms contributing to each signal.

[IMAGE: Show the structural formula of tert-butyl ethanoate, CH3COOC(CH3)3, with the three acetyl methyl hydrogens distinguished from the nine tert-butyl methyl hydrogens. Beside it, show a two-singlet proton NMR spectrum at approximately 2.0 and 1.4 ppm. Include the corresponding integration staircase, with the second step three times the height of the first. Label both integrated areas as illustrative data.]

**Step 1**: Calculate the integration ratio.

Divide both integrated areas by the smaller value, 0.50.

\[
\begin{aligned}
0.50:1.50
&=\frac{0.50}{0.50}:
\frac{1.50}{0.50}\\
&=1:3
\end{aligned}
\]

The integration ratio is \(1:3\).

**Step 2**: Compare the ratio with the molecular formula.

The compound contains 12 hydrogen atoms.

The ratio has four parts in total:

\[
1+3=4
\]

Each part must therefore represent:

\[
\frac{12}{4}=3\text{ H}
\]

**Step 3**: Calculate the actual proton numbers.

Multiply each ratio value by three:

\[
1:3\longrightarrow 3:9
\]

Signal A represents 3H, while signal B represents 9H.

**Step 4**: Connect the numbers to the structure.

The signal at \(2.0\ \mathrm{ppm}\) represents the three hydrogens in \(\ce{CH3CO-}\).

The signal at \(1.4\ \mathrm{ppm}\) represents the nine hydrogens in the three equivalent methyl groups of \(\ce{C(CH3)3}\).

The spectrum therefore contains two proton environments with integration values of 3H and 9H.

Notice something important: the molecule contains four methyl groups, but the spectrum has only two signals. Integration tells you how many hydrogen atoms contribute to each environment, not how many carbon atoms or methyl groups are present.

### Worked example: Interpreting integration when an OH proton is difficult to measure

An illustrative proton NMR spectrum of butanoic acid, \(\ce{CH3CH2CH2COOH}\), contains the following data:

| Signal | Chemical shift | Integrated area |
|---|---:|---:|
| A | 0.9 ppm | 0.60 a.u. |
| B | 1.6 ppm | 0.40 a.u. |
| C | 2.3 ppm | 0.40 a.u. |
| D | 11.5 ppm | Broad, unreliable integral |

The molecular formula is \(\ce{C4H8O2}\).

Determine the number of hydrogen atoms represented by each signal. Explain how the broad signal affects your interpretation.

[IMAGE: Draw the structural formula of butanoic acid, CH3-CH2-CH2-C(=O)-OH, beside an illustrative proton NMR spectrum. Show three sharp signals at approximately 0.9, 1.6, and 2.3 ppm with integration steps in the ratio 3:2:2. Include a broad carboxylic acid OH feature near 11.5 ppm, clearly distinguished from the three sharp signals. Label the OH integral as unreliable rather than assigning it a numerical value.]

**Step 1**: Calculate the ratio of the three reliable integrated areas.

The measured values are:

\[
0.60:0.40:0.40
\]

Divide each value by 0.40:

\[
1.5:1:1
\]

These are not whole numbers, so multiply every value by two:

\[
3:2:2
\]

The three sharp signals therefore have a relative integration of \(3:2:2\).

**Step 2**: Compare the result with the molecular formula.

The formula \(\ce{C4H8O2}\) contains eight hydrogen atoms.

The three sharp signals account for:

\[
3+2+2=7\text{ H}
\]

One hydrogen remains unaccounted for.

**Step 3**: Identify the missing hydrogen.

The remaining hydrogen belongs to the carboxylic acid group, \(\ce{-COOH}\).

Carboxylic acid protons often produce broad signals at relatively high chemical shifts. Their integration can be unreliable because of proton exchange and hydrogen bonding.

The broad feature at \(11.5\ \mathrm{ppm}\) is consistent with this group.

**Step 4**: State the complete proton count.

| Proton environment | Number of hydrogen atoms |
|---|---:|
| \(\ce{CH3}\) | 3H |
| Middle \(\ce{CH2}\) | 2H |
| \(\ce{CH2}\) adjacent to \(\ce{COOH}\) | 2H |
| \(\ce{COOH}\) | 1H |
| **Total** | **8H** |

The integration of the three sharp signals gives \(3:2:2\), while the broad acid proton accounts for the remaining hydrogen.

The important lesson is that a poorly integrated OH or COOH signal does not mean the hydrogen has disappeared. You must still account for it using the molecular formula and the available structural evidence.

## Where the introductory integration model can fail

The simple relationship between integrated area and proton number works well for introductory spectra. However, it relies on assumptions that are not always satisfied.

| Situation | What can happen | What you should do |
|---|---|---|
| OH or NH protons | Broad signals and proton exchange can make integration unreliable | Check the molecular formula and other evidence |
| Overlapping signals | Two different environments can contribute to one visible signal region | Treat the measured area as their combined contribution |
| Poor baseline correction | The calculated area can be too large or too small | Be cautious about the reported ratio |
| Insufficient relaxation time | Some signals can be underestimated | Recognise that experimental conditions affect accuracy |
| Mixtures or impurities | Additional substances contribute their own signals | Do not assume every signal belongs to one molecule |

Let's examine the most important limitations.

### 1. OH and NH protons may not integrate reliably

A common introductory assumption is that an OH proton produces a separate signal integrating to exactly 1H.

That is often a useful starting point, but real spectra are less predictable.

OH and NH hydrogen atoms can exchange between molecules or with the surrounding solvent. This exchange can broaden their signals and make their integration less reliable.

For example, ethanol has three main proton environments:

\[
\ce{CH3CH2OH}
\]

The expected proton counts are:

\[
3:2:1
\]

However, the OH signal may be broad or variable. Do not reject an otherwise sensible interpretation simply because its measured OH area is not exactly proportional to one hydrogen.

### 2. Two signals can overlap

Suppose two chemically different proton environments have almost identical chemical shifts.

Their signals may overlap, appearing as one larger region in the spectrum.

The integrated area then includes contributions from both environments.

This creates an important distinction:

- Signal count describes the number of distinguishable signal groups.
- Integration measures the total proton contribution within an integrated region.

A single visible region does not always represent a single chemical environment.

[IMAGE: Comparison of two proton NMR spectra. In the first, two nearby signals are clearly separated and have individual integration steps. In the second, the signals overlap into one visible envelope, with a single combined integration step equal to the sum of their individual areas. Show the same horizontal chemical-shift scale in both spectra.]

Integration can tell you how many hydrogen atoms contribute to the combined region, but it cannot necessarily tell you how those hydrogen atoms are divided between the overlapping environments.

### 3. Experimental conditions affect the accuracy of integration

The instrument does not automatically produce perfect area measurements.

Several experimental factors matter:

- A sloping baseline can distort the calculated area.
- Poor signal-to-noise makes signal boundaries difficult to identify.
- An integration region that cuts off part of a signal underestimates its area.
- An insufficient delay between radiofrequency pulses can prevent some nuclei from fully relaxing before the next measurement.

In professional quantitative NMR, acquisition and processing conditions are carefully controlled to obtain reliable proton ratios.

For HSC interpretation, you will generally be given a simplified spectrum with usable integration data. Still, it is worth understanding why experimental integration values may not be perfectly neat whole numbers.

### 4. Integration does not automatically tell you how much compound is in the sample

This is a subtle but important limitation.

Suppose a spectrum gives an integration ratio of \(3:2\).

You can conclude that one signal represents three times as many contributing hydrogen nuclei as the other.

You cannot conclude that the sample contains three times as many molecules of one type as another.

That would require additional information about the number of hydrogen atoms contributing to each signal, and suitable quantitative measurement conditions.

The same distinction applies when comparing two separate spectra. Their raw integration values cannot automatically be compared if sample amounts, concentrations, receiver gain, or acquisition settings differ.

For introductory HSC questions, focus on relative integration within the same spectrum.

## A reliable way to interpret integration in an exam

When you are given a proton NMR spectrum, use this sequence:

1. Count the separate signal groups. Do not count individual lines within multiplets.
2. Record the integrated area of each complete signal.
3. Simplify the areas into a whole-number ratio.
4. Compare the ratio with the total number of hydrogen atoms in the molecular formula.
5. Scale the ratio if necessary to obtain actual proton numbers.
6. Check whether any OH or NH protons, overlapping signals, or other complications need to be considered.
7. Use chemical shift and splitting to help assign the proton environments.

For a more detailed explanation of how to identify the environments from a structural formula, see [How to Count Proton Environments from a Structural Formula](https://tutorgum.com/learn/resources/hsc-chemistry-how-to-count-proton-environments-from-a-structural-formula/).

## Questions and solutions

### Question 1

An organic compound has the molecular formula \(\ce{C3H6O}\).

Its proton NMR spectrum contains three separate signals with the following illustrative integrated areas:

| Signal | Chemical shift | Integrated area |
|---|---:|---:|
| A | 1.0 ppm | 0.90 a.u. |
| B | 2.4 ppm | 0.60 a.u. |
| C | 9.8 ppm | 0.30 a.u. |

Assume the compound is pure and all six hydrogen atoms are represented.

Determine:

a) The simplest integration ratio.

b) The number of hydrogen atoms contributing to each signal.

### Solution 1

The integration ratio is \(3:2:1\), corresponding to 3H, 2H, and 1H respectively.

**Step 1**: Simplify the measured areas.

Divide each value by the smallest area, 0.30.

\[
\begin{aligned}
0.90:0.60:0.30
&=3:2:1
\end{aligned}
\]

**Step 2**: Check the molecular formula.

The compound contains six hydrogen atoms.

The ratio adds to:

\[
3+2+1=6
\]

Therefore, each ratio unit represents one hydrogen atom.

**Step 3**: Assign the proton numbers.

| Signal | Integration | Hydrogen atoms |
|---|---:|---:|
| A | 3 | 3H |
| B | 2 | 2H |
| C | 1 | 1H |

The three environments account for all six hydrogen atoms.

The important distinction is that there are three signals, but six hydrogen atoms. Integration tells you how those six atoms are distributed between the environments.

### Question 2

A proton NMR spectrum contains:

- A triplet at \(1.2\ \mathrm{ppm}\), with a total integrated area of 1.20 arbitrary units.
- A singlet at \(2.1\ \mathrm{ppm}\), with an integrated area of 0.40 arbitrary units.

The three lines of the triplet have relative heights of \(1:2:1\).

Determine:

a) The integration ratio.

b) Which signal represents more hydrogen atoms, and by what factor?

c) Why would using only the tallest line of the triplet give an incorrect interpretation?

[IMAGE: Show a triplet at 1.2 ppm with three lines of relative heights 1:2:1, and a separate singlet at 2.1 ppm. Shade the complete area under each signal. Clearly distinguish the triplet's central line height from the total area of all three lines. Include the supplied total integrals of 1.20 and 0.40 arbitrary units.]

### Solution 2

The triplet represents three times as many hydrogen atoms as the singlet, giving an integration ratio of \(3:1\).

**Step 1**: Compare the total integrated areas.

\[
1.20:0.40
\]

Divide both values by 0.40:

\[
\frac{1.20}{0.40}:
\frac{0.40}{0.40}
=3:1
\]

**Step 2**: Interpret the ratio.

The triplet has three times the integrated area of the singlet.

Therefore, it represents three times as many contributing hydrogen nuclei.

The ratio does not automatically mean the signals represent 3H and 1H. Without the molecular formula or another reference, they could represent 6H and 2H, or 9H and 3H.

**Step 3**: Explain the mistake involving peak height.

The \(1:2:1\) ratio describes the relative heights of the three lines within the triplet.

It does not describe the total proton count of the signal.

The integration must include the complete area beneath all three lines. Using only the central line would ignore the contributions from the two outer lines.

This question demonstrates why splitting and integration must be treated as separate pieces of information.

### Question 3

The same pure compound is measured twice using different receiver gains.

The displayed integration values are:

| Signal | Run A | Run B |
|---|---:|---:|
| A | 0.40 a.u. | 1.20 a.u. |
| B | 0.80 a.u. | 2.40 a.u. |
| C | 1.20 a.u. | 3.60 a.u. |

The receiver gain in Run B is three times higher than in Run A. The sample amounts and acquisition conditions have not been standardised for quantitative comparison.

A student claims:

"The compound in Run B must contain three times as many hydrogen atoms because every integration value is three times larger."

Evaluate this statement.

Also determine whether the data establish that Run B contains three times the concentration of the compound.

### Solution 3

The student's conclusion is incorrect. Both spectra have the same relative integration ratio, \(1:2:3\), and the larger raw values in Run B do not establish a change in molecular proton numbers or sample concentration.

**Step 1**: Calculate the integration ratio in Run A.

\[
0.40:0.80:1.20=1:2:3
\]

**Step 2**: Calculate the integration ratio in Run B.

\[
1.20:2.40:3.60=1:2:3
\]

The relative proton contributions are identical.

**Step 3**: Consider the receiver gain.

Receiver gain changes the amplification applied to the detected signal.

Run B uses three times the receiver gain, so its displayed integration values are also three times larger.

This does not mean that each molecule contains more hydrogen atoms.

The molecular structure has not changed.

**Step 4**: Consider the concentration claim.

The spectra do not establish that Run B has three times the concentration.

The sample amount, concentration, and acquisition conditions have not been controlled sufficiently to make that comparison.

To compare concentrations using NMR, appropriate quantitative conditions and a suitable reference are needed.

The key lesson is that relative integration within a spectrum is different from comparing raw integral values between separate experiments.

### Question 4

A pure compound has the molecular formula \(\ce{C4H8O2}\).

Its proton NMR spectrum contains three visible signal regions:

| Signal region | Chemical shift | Integrated area |
|---|---:|---:|
| A | 1.2 ppm | 0.30 a.u. |
| B | 2.3 ppm | 0.20 a.u. |
| C | 3.8 ppm | 0.30 a.u. |

Independent structural evidence establishes that signal region C contains two chemically different proton environments whose signals happen to overlap.

Assume all eight hydrogen atoms are represented and the integration measurements are reliable.

Determine:

a) The relative integration of the three visible signal regions.

b) The total number of hydrogen atoms contributing to each visible region.

c) The total number of distinct proton environments in the molecule.

d) Whether integration alone can determine how the three hydrogen atoms in region C are divided between its two underlying environments.

[IMAGE: Draw an illustrative proton NMR spectrum with three visible signal regions at 1.2, 2.3, and 3.8 ppm. The third region should appear as one combined envelope, representing two exactly overlapping signals. Show the three integrated areas as 0.30, 0.20, and 0.30 arbitrary units. Do not reveal the individual contributions of the overlapping signals.]

### Solution 4

The three visible regions represent 3H, 2H, and 3H respectively. There are four distinct proton environments, but integration alone cannot divide the final 3H between its two overlapping environments.

**Step 1**: Calculate the integration ratio.

\[
0.30:0.20:0.30
\]

Divide each value by 0.10:

\[
3:2:3
\]

**Step 2**: Compare the ratio with the molecular formula.

The molecular formula contains eight hydrogen atoms.

The ratio adds to:

\[
3+2+3=8
\]

Therefore, the actual proton contributions to the visible regions are:

- Region A: 3H.
- Region B: 2H.
- Region C: 3H in total.

**Step 3**: Determine the number of proton environments.

Regions A and B each represent one environment.

Region C contains two different environments that overlap.

Therefore:

\[
1+1+2=4
\]

There are four distinct proton environments, even though only three signal regions are visible.

**Step 4**: Determine whether the overlapping contributions can be separated.

No. The integrated area of region C measures the combined contribution of both environments.

The total is three hydrogen atoms, but the integral alone cannot establish whether the individual contributions are, for example, 1H and 2H.

Additional structural or spectral evidence would be needed.

This is an important limitation of integration: it measures the total area within a region, not necessarily the contribution from each underlying environment.

### Question 5

A sample is known to contain only two compounds: propanal, \(\ce{CH3CH2CHO}\), and methanol, \(\ce{CH3OH}\).

The proton NMR spectrum contains four non-overlapping, sharp signals:

| Signal | Chemical shift | Integrated area |
|---|---:|---:|
| A | 1.0 ppm | 0.60 a.u. |
| B | 2.4 ppm | 0.40 a.u. |
| C | 3.3 ppm | 0.30 a.u. |
| D | 9.8 ppm | 0.20 a.u. |

The following assignments are established:

- A represents the three methyl hydrogens in propanal.
- B represents the two methylene hydrogens in propanal.
- C represents the three carbon-bound hydrogens in methanol.
- D represents the aldehyde hydrogen in propanal.

The methanol OH proton is broad and is not reliably integrated.

Assume that:

- The compounds are the only substances contributing to the listed signals.
- All four signals are resolved and integrated reliably.
- The measurements were acquired under suitable quantitative conditions, so the integrated area is proportional to the total number of contributing hydrogen nuclei.

Determine:

a) Whether all four signals can belong to one molecule of propanal.

b) The relative number of propanal molecules and methanol molecules in the sample.

c) Why the total integration cannot be interpreted as the hydrogen count of one molecule.

[IMAGE: Draw an illustrative proton NMR spectrum with four separate signal regions at 1.0, 2.4, 3.3, and 9.8 ppm, with integrated areas 0.60, 0.40, 0.30, and 0.20 arbitrary units. Indicate that the first, second, and fourth signals belong to propanal, while the third belongs to methanol. Include a broad, unquantified exchangeable OH feature associated with methanol. Do not label the molecular ratio or reveal the solution.]

### Solution 5

The sample contains propanal and methanol in a molecular ratio of \(2:1\). The four signals cannot all belong to one molecule of propanal because they include a separate signal from methanol.

**Step 1**: Check the proton count expected from propanal.

One propanal molecule has the formula:

\[
\ce{CH3CH2CHO}
\]

Its three proton environments contain:

- Methyl group: 3H.
- Methylene group: 2H.
- Aldehyde group: 1H.

Therefore, one molecule of propanal has six hydrogen atoms.

Its expected integration ratio is:

\[
3:2:1
\]

**Step 2**: Check the three signals assigned to propanal.

Their measured areas are:

\[
0.60:0.40:0.20
\]

Dividing by 0.20 gives:

\[
3:2:1
\]

This confirms that the three signals are consistent with propanal.

**Step 3**: Compare the two methyl signals.

Both propanal and methanol contain one methyl group with three equivalent carbon-bound hydrogen atoms.

Their integrated areas are:

- Propanal methyl signal: 0.60 a.u.
- Methanol methyl signal: 0.30 a.u.

Because both signals represent three hydrogen atoms per molecule, their integrated area ratio directly gives the relative number of molecules:

\[
\begin{aligned}
n_{\text{propanal}}:
n_{\text{methanol}}
&=0.60:0.30\\
&=2:1
\end{aligned}
\]

The sample therefore contains twice as many propanal molecules as methanol molecules.

**Step 4**: Check the other propanal signals.

If the sample contains twice as many propanal molecules, its methylene and aldehyde signals should also have twice the corresponding proton contributions.

The expected relative areas are therefore:

\[
6:4:2
\]

These simplify to:

\[
3:2:1
\]

The measured propanal areas of \(0.60:0.40:0.20\) agree with this prediction.

The data are internally consistent.

**Step 5**: Explain why all the integrals cannot describe one molecule.

Integration measures the total contribution from all hydrogen nuclei producing each signal throughout the sample.

It does not automatically give the number of hydrogen atoms in a single molecule.

Here, the propanal signals come from multiple propanal molecules, while the signal at \(3.3\ \mathrm{ppm}\) comes from methanol molecules.

The molecular formula \(\ce{C3H6O}\) applies to each propanal molecule, not to the combined sample.

This is the conceptual distinction that makes the question challenging: integration can provide information about the relative amounts of different substances in a mixture, provided the signals are assigned and the experimental conditions are suitable.

## What to remember

Integration in a proton NMR spectrum measures the area under a signal, which is proportional to the number of hydrogen nuclei contributing to it under suitable experimental conditions.

Keep these four questions separate:

| Feature | Remember |
|---|---|
| Signal count | How many distinct proton environments? |
| Integration | How many hydrogen atoms contribute to each signal, relative to the others? |
| Chemical shift | What electronic environment surrounds those hydrogen atoms? |
| Splitting | How many neighbouring equivalent hydrogen atoms interact with the group? |

The most important exam technique is to simplify the integrated areas into a ratio, then use the molecular formula to convert that ratio into actual proton numbers.

Once you can do this reliably, the next step is combining integration with chemical shift and splitting to identify connected fragments of a molecule. That is where a spectrum becomes much more than a collection of peaks.