# (n+1) Rule

- Page ID
- 40724

The (n+1) Rule, an empirical rule used to predict the multiplicity and, in conjunction with Pascal’s triangle, splitting pattern of peaks in ^{1}H and ^{13}C 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:

The three hydrogen nuclei in **1**, H_{a}, H_{b}, and H_{c}, are equivalent. Thus, ^{1}H 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 ^{1}H 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.

## Contributors and Attributions

- Gamini Gunawardena from the OChemPal site (Utah Valley University)