Multiplet intensity ratios

For sets of spin- nuclei, the multiplet intensity ratios are simply the binomial coefficients found most easily from Pascal s triangle (Figure 2.11). 

C) The spacing between the first and second lines will be the smallest coupling constant, a. The intensity ratio of these two lines will usually indicate the multiplet to which the coupling constant corresponds. Assign quantum numbers to the second line, compute a, and enter these numbers in the table. If you have started into a multiplet, you can then predict the positions and intensities of the remaining lines of the multiplet. Find them and enter the quantum numbers and new estimates of a in the table. 

The intensities of the multiplet lines arise from the number of spin configurations belonging to each total spin (Fig. 1.12). For n coupling nuclei X the intensity ratios are equal to the nlh binominal coefficients (Table 1.3). 

Table 1.3. Multiplet Line Intensity Ratios of Signal A due to Coupling with n Equivalent Nuclei X Ux = D-...

In general, n equivalent nuclei of spin I will produce an EPR multiplet consisting of (2nl + 1) equally spaced lines. For spins of I = Vi, the intensity ratios are given by the coefficients of the binomial expansion (a + b)n but for I > /2, the formula for intensities is much more complicated. For modest values of n it is easy to find the intensity ratios by sketching a branching diagram thus... 

The four-line pattern for the methylene hydrogens is a quartet and has multiplicity 4. Its chemical shift is midway between the second and third lines. Again, the three spacings between neighboring lines are all equal to the spacings in the triplet (6.9 Hz). However, in a quartet the relative intensity of the lines is 1 3 3 1. These intensity ratios, and the fact that the spacings in both multiplets are equal, are no accident. 

But what about the signal for Hc It will be split by both Hb as well as the three methyl hydrogens. Suppose that Vbc > Vcd. Each line of the doublet due to coupling with Hb will be split further into a quartet by the three methyl hydrogens, as in Figure 8.7. This multiplet is called a doublet of quartets, and the line intensity ratio is 1 3 3 1 1 3 3 1. 

Next, consider the molecule D2CH2. Each of the two D nuclei can adopt three orientations. Therefore, two D nuclei can adopt nine [(21 + l)2] spin combinations corresponding to five (2nl + 1) spin states (M = 2, 1, 0, -1, -2). These are shown in Figure 8.12. Notice that the 1 2 3 2 1 population ratio of these states (which is the same as the relative intensity of the multiplet lines coupled to them) is different from that predicted by Pascal s triangle, which only applies to 7 = nuclei. At any rate, the H spectrum of D2CH2 shows a five-line pattern, with intensity ratio 1 2 3 2 1 and line spacing 2yHD. 

From the above example, you can infer that the extent to which the intensity ratios depart from first-order (Pascal s triangle) expectations is a function of A 8v/7. If this ratio is large (A 5v/7 > 10), we describe the spin system as weakly coupled, and the resulting multiplets will exhibit essentially first-order intensity ratios (as do the triplets and quartets in Figures 8.1 and 8.13). But as A 8v/J decreases, second-order effects (multiplet slanting and even the appearance of extra lines) become increasingly apparent. Such a spin system is said to be strongly coupled. 

In Section 9.9 we discussed how the appearance of a spin-coupled NMR spectrum is determined by the ratio of Av (the difference in chemical shifts between the coupling nuclei) to J (the coupling constant they share). For the spectrum to exhibit first-order multiplet intensities (Pascal s triangle Section 8.5), the value of Av//has to be at least 10. Smaller values of Av/J lead to progressively greater complications due to second-order effects. 

The spectra as recorded on the Brucker WH 90 instrument supports this prediction (Fig. 30) there indeed exist a multiplet centered on 20.72 ppm and a triplet at 11.68, 10.55 and 9.50 ppm (intensity ratio of the two main signals, 3.5). The fact... 

Numl> r of equivalent adjacent protons Type of multiplet observed Ratio of intensities... 

Problem 5-11. Calculate the HF intensity ratio of the CIDEP spectrum of the 2-hydroxy-2-propyl radical for the multiplet effect where Ag = 0 and p is positive. 

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