(n+1) Rule

(n+1) Rule

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 , the multiplicity of the peak is n+1.

eg. 1:

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:

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:

There are three set of equivalent hydrogen nuclei in 3:

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

peak of H_a:

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.

The peak of H_a is a doublet of a doublet.

peak of H_b:

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.

The peak of H_b is a doublet of a doublet.

peak of H_c:

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.

The peak of H_c is a doublet of a doublet.