13.5 Integration of 1H NMR Absorptions: Proton Counting

Proton NMR Spectroscopy

The integration of $^1H$ NMR peaks tells you how many protons produce each signal. While chemical shift tells you what type of environment a proton is in, integration tells you how many protons share that environment. Together, these two pieces of information dramatically narrow down possible structures.

Interpretation of NMR Proton Signals

The area under each peak in a $^1H$ NMR spectrum is directly proportional to the number of protons contributing to that signal. NMR spectrometers measure this area through a process called integration, and the result is typically displayed as a stepped line (an integral curve) overlaid on the spectrum.

The key point: you don't get absolute proton counts from integration. You get relative ratios. A peak with twice the integrated area of another peak means twice as many protons contribute to that signal (a 2:1 ratio).

Equivalent vs. non-equivalent protons:

• Equivalent protons share the same chemical environment and produce a single signal. All three protons in a methyl group ($-CH_3$) are equivalent, so they contribute to one peak with a combined area proportional to 3.
• Non-equivalent protons are in different chemical environments and appear as separate peaks. Protons become non-equivalent when they sit on different carbons, are near electronegative atoms (O, N, F), or are adjacent to double bonds.

Symmetry is the fastest way to identify equivalent protons. If a mirror plane or rotation axis interconverts two protons, they're equivalent and will give one signal.

Calculation of NMR Peak Ratios

Converting raw integration values into a proton count is a straightforward process:

1. Record the integrated area for each distinct signal. Modern NMR software does this automatically and displays numerical values.
2. Divide each value by the smallest integration value. This normalizes everything relative to the smallest peak.
3. Convert to the simplest whole-number ratio. If you get values like 1.0 : 1.5 : 3.0, multiply through by 2 to get 2 : 3 : 6.
4. Check that the total matches the molecular formula. If the molecular formula tells you there are 10 protons and your ratio is 2 : 3, the actual proton counts are 4 and 6 (multiply by 2). If the ratio is 2 : 3 and there are 5 total protons, the counts are 2 and 3 directly.

Example: A compound with molecular formula $C_3H_8O$ shows two signals with an integration ratio of 1 : 3. The total of 8 protons divided in a 1 : 3 ratio gives 2 : 6 protons. This is consistent with two equivalent $CH_3$ groups (6H) and one $CH_2$ group (2H).

The ratio of integrated peak areas must match the ratio of protons predicted from the structure. If it doesn't, reconsider your proposed structure.

Prediction of NMR Spectral Features

When predicting what a $^1H$ NMR spectrum should look like, follow these steps:

1. Identify all distinct proton environments. Look for planes of symmetry and rotation axes. Protons that can be interconverted by a symmetry operation are equivalent.
2. Count the protons in each environment. This gives you the expected integration ratio.
3. Determine the number of signals. The number of distinct proton environments equals the number of peaks in the spectrum.

For instance, dimethyl ether ($CH_3OCH_3$) has only one type of proton (all six $H$ atoms are equivalent by symmetry), so it shows a single peak. Ethanol ($CH_3CH_2OH$), by contrast, has three distinct proton environments ($CH_3$, $CH_2$, and $OH$), producing three signals with a 3 : 2 : 1 integration ratio.

Factors that make protons non-equivalent:

• Attachment to different carbon atoms
• Proximity to electronegative atoms, which deshields nearby protons and shifts them downfield
• Proximity to $\pi$ bonds or aromatic rings
• Lack of molecular symmetry, which generally increases the number of distinct environments and produces more complex spectra

Additional NMR Concepts

• Tetramethylsilane (TMS) is the standard reference compound in NMR. Its 12 equivalent protons produce a single sharp peak defined as $0 \text{ ppm}$ on the chemical shift ($\delta$) scale. All other signals are reported relative to TMS.
• Spin-spin coupling (splitting) occurs between non-equivalent protons on adjacent carbons. While splitting patterns give connectivity information, they don't affect integration. The total integrated area of a split signal (doublet, triplet, etc.) still reflects the number of protons producing that signal.
• Integration works because the NMR signal intensity is directly proportional to the number of nuclei at a given resonance frequency. This physical relationship is what makes proton counting reliable.