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, 1H NMR spectrum of 1 \(H_a\)s 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 \(H_a\)s 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 \(H_a\)s 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 \(H_a\)s 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 \(H_a\)s 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 \(H_a = 2 \times 2 = 4\). 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 \(H_b = 2 \times 2 = 4\). 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 \(H_c = 2 \times 2 = 4\). 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.