Spin–Spin Coupling and the n+1 Rule

Introduction

Look closely at a high-resolution ¹H NMR spectrum and many signals are not single peaks at all: they are split into clusters of two, three, four or more lines. This splitting arises because protons on adjacent carbon atoms feel each other's tiny magnetic fields. Far from being a nuisance, splitting is a gift to the chemist — it reveals how many hydrogens sit on the neighbouring carbon, letting you piece a molecule together one fragment at a time.

Core explanation

Origin of coupling. A proton nucleus behaves like a tiny magnet that can be aligned with (α) or against (β) the applied field. The spin state of a proton on a neighbouring carbon slightly alters the magnetic field felt by the proton being observed. This effect is transmitted through the bonding electrons, which is why it is called spin–spin coupling . In a sample of many molecules, roughly half the neighbours are α and half β, so the observed signal appears at two slightly different frequencies.

The n+1 rule. For protons coupled to n equivalent neighbouring protons, the signal is split into n + 1 lines. "Neighbouring" normally means protons on the adjacent carbon atom — three bonds away, in an H–C–C–H arrangement.

Neighbouring H (n) Lines (n + 1) Name Relative intensities --- --- --- --- 0 1 singlet (s) 1 1 2 doublet (d) 1 : 1 2 3 triplet (t) 1 : 2 : 1 3 4 quartet (q) 1 : 3 : 3 : 1 4 5 quintet 1 : 4 : 6 : 4 : 1 5 6 sextet 1 : 5 : 10 : 10 : 5 : 1 6 7 septet 1 : 6 : 15 : 20 : 15 : 6 : 1

Where the intensities come from. With two neighbouring protons there are four spin combinations: αα, αβ, βα and ββ. The mixed states αβ and βα produce the same field, so the middle line has twice the intensity of the outer lines: 1 : 2 : 1. With three neighbours there are eight combinations, grouped as 1 : 3 : 3 : 1. These are the rows of Pascal's triangle .

Equivalent protons do not split each other. The three hydrogens of a methyl group are equivalent and show no splitting among themselves. The six protons of propanone give a singlet because no non-equivalent neighbours are close enough.

Coupling is mutual. If the CH₂ protons split the CH₃ signal, the CH₃ protons also split the CH₂ signal. In chloroethane, CH₃CH₂Cl, the CH₃ signal is a triplet (two neighbours) and the CH₂ signal is a quartet (three neighbours).

Usually no coupling through oxygen. In most samples, O–H protons exchange rapidly between molecules, so they appear as singlets and do not split neighbouring CH signals.

Multiplets. When a proton has neighbours in two different environments, the pattern can become complex and is often simply reported as a multiplet (m).

Step-by-step reasoning

To predict the splitting of a signal:

1. Identify the proton environment you are looking at. 2. Find the carbon atoms directly bonded to its carbon. 3. Count the hydrogens on those adjacent carbons that are not equivalent to your protons; this is n . 4. Ignore O–H and N–H protons unless told otherwise. 5. The signal has n + 1 lines, with intensities from Pascal's triangle.

Visual explanation

In the simulation, select the CH₂ signal of chloroethane and switch on the spin-state tree. The single line first splits in two for the first neighbouring CH₃ proton, then each branch splits again for the second and third. Overlapping branches stack up to give four lines in a 1 : 3 : 3 : 1 pattern.

Real-world analogy

Imagine a singer whose microphone picks up faint hums from three neighbours, each of whom is humming either slightly higher or slightly lower. The recording shows the singer's note shifted by the sum of the hums, and the most common totals — two up one down, or one up two down — appear most often, just like the tall inner lines of a quartet.

Real-world example

Forensic chemists identifying unknown white powders use splitting patterns alongside chemical shift. An ethyl group attached to oxygen shows a characteristic quartet near δ 4 and triplet near δ 1.3, allowing a lab to distinguish an ethyl ester from a methyl ester in seconds.

Why?

Why does coupling normally stop after three bonds? The interaction is carried by the bonding electrons, and it weakens rapidly as the number of bonds between the two nuclei increases. Beyond three bonds, H–C–C–C–H couplings are usually too small to see in saturated molecules.

Common misconception

"The number of lines equals the number of hydrogens in the group." A CH₃ group next to a CH₂ appears as a triplet, not a quartet. The multiplicity reflects the neighbours, not the group itself.

Worked example

Question: Predict the splitting of each signal in 1-chloropropane, CH₃CH₂CH₂Cl.

Reasoning: The CH₃ has two neighbours (CH₂) → triplet. The CH₂Cl has two neighbours → triplet. The middle CH₂ has five neighbours (three plus two), which, if coupling constants are similar, gives a sextet.

Answer: Triplet (CH₃), sextet (middle CH₂), triplet (CH₂Cl).

Quick check

1. What splitting pattern does the CH signal of 2-chloropropane, (CH₃)₂CHCl, show, and why? Answer: A septet, because the CH proton has six equivalent neighbouring protons on the two methyl groups.

Exam focus

State the rule precisely: "split into n + 1 peaks, where n is the number of hydrogens on adjacent carbon atoms." Exam answers must link each multiplicity to a named neighbouring group, for example "triplet, so adjacent to a CH₂".

Advanced insight

The n+1 rule is a first-order approximation. It works when the chemical shift difference between coupled protons, measured in hertz, is much larger than the coupling constant. When shifts are close, second-order effects distort intensities — the inner lines grow and the outer lines shrink, a "roofing" effect — and at the extreme, extra lines appear. Higher-field spectrometers increase the hertz separation and make spectra closer to first order.

Summary

Spin–spin coupling splits a ¹H NMR signal according to the number of equivalent hydrogens on adjacent carbons. The n+1 rule gives the number of lines, and Pascal's triangle gives their relative intensities. Equivalent protons do not split each other, coupling is mutual, and exchanging O–H protons usually appear as singlets.

Practice questions

1. What multiplicity would you expect for the CH₃ signal in ethanal, CH₃CHO? Answer: A doublet, because the CH₃ protons have one neighbouring proton on the CHO carbon. 2. Give the relative intensities of the lines in a quartet. Answer: 1 : 3 : 3 : 1. 3. Explain why the signal for the CH₃ group in methyl ethanoate, CH₃COOCH₃, is a singlet. Answer: Neither methyl group has hydrogens on an adjacent carbon; they are separated by the C=O or oxygen, so there are no neighbouring protons to couple with. 4. Predict the splitting of the two signals of 1,1-dichloroethane, CH₃CHCl₂. Answer: The CH₃ signal is a doublet (one neighbour) and the CH signal is a quartet (three neighbours).