(n+1) Rule

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The (n+1) Rule, an empirical rule used to predict the multiplicity and, in conjunction with Pascal’s triangle, splitting pattern of peaks in ¹H and ¹³C NMR spectra, states that if a given nucleus is coupled (see spin coupling) to n number of nuclei that are equivalent (see equivalent ligands), the multiplicity of the peak is n+1.

eg. 1:

n+1rule1.png

The three hydrogen nuclei in 1, H_a, H_b, and H_c, are equivalent. Thus, ¹H NMR spectrum of 1 has only one peak. H_a, H_b, and H_c are coupled to no hydrogen nuclei. Thus, for H_a, H_b, and H_c, n=0; (n+1) = (0+1) = 1. The multiplicity of the peak of H_a, H_b, and H_c is one. The peak has one line; it is a singlet.

eg. 2:

n+1rule2.png

There are two sets of equivalent hydrogen nuclei in 2:

Set 1: H_a
Set 2: H_b, H_c

Thus, the ¹H NMR spectrum of 2 has two peaks, one due to H_a and the other to H_b and H_c.

The peak of H_a: There are two vicinal hydrogens to H_a: H_b and H_c. H_b and H_c are equivalent to each other but not to H_a. Thus, for H_a, n=2; (n+1) = (2+1) = 3. The multiplicity of the peak of H_a is three. The peak has three lines; from the Pascal’s triangle, it is a triplet.

The peak of H_b and H_c: There is only one vicinal hydrogen to H_b and H_c: H_a. H_a is not equivalent to H_b and H_c. Thus, for H_b and H_c, n=1; (n+1) = (1+1) = 2. The multiplicity of the peak of H_b and H_c is two. The peak has two lines, from the Pascal’s triangle, it is a doublet.

To determine the multiplicity of a peak of a nucleus coupled to more than one set of equivalent nuclei, apply the (n+1) Rule independently to each other.

eg:

n+1rule3.png

There are three set of equivalent hydrogen nuclei in 3:

Set 1: H_a
Set 2: H_b
Set 3: H_c

n+1rule4.png

peak of H_a:

n+1rule5.png

multiplicity of the peak of H_a = 2×2 = 4. To determine the splitting pattern of the peak of H_a, use the Pascal’s triangle, based on the observation that, for alkenyl hydrogens, J_cis > J_gem.

n+1rule6.png

The peak of H_a is a doublet of a doublet.

peak of H_b:

n+1rule7.png

multiplicity of the peak of H_b = 2×2 = 4. To determine the splitting pattern of the peak of H_b, use the Pascal’s triangle, based on the observation that, for alkenyl hydrogens, J_trans > J_gem.

n+1rule8.png

The peak of H_b is a doublet of a doublet.

peak of H_c:

n+1rule9.png

multiplicity of the peak of H_c = 2×2 = 4. To determine the splitting pattern of the peak of H_c, use the Pascal’s triangle based on the observation that, for alkenyl hydrogens, J_trans > J_cis.

n+1rule10.png

The peak of H_c is a doublet of a doublet.

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