Interpreting ¹H NMR Signal Splitting: The (n + 1) Rule

Recall that a proton A can sense the two spin states of a coupled proton X, resulting in a doublet NMR signal with two peaks of equal intensity.

When A is coupled to two equivalent protons, their spin states can be aligned with or against the external field, creating three scenarios.

This results in a triplet signal, where the central peak is twice as intense as the others.

Similarly, when A has three neighbors, all three spins could be aligned with the field, two with and one against, one with and two against, or all three aligned against the field.

This results in a quartet of peaks with the relative intensities 1:3:3:1.

Empirically, a proton-coupled to n equivalent neighbors yields a multiplet signal split into n + 1 peaks.

The relative intensities of the peaks in a multiplet can be predicted using Pascal's triangle: an array where each entry is the sum of the nearest entries in the row above it.