Physical Chemistry , 1st ed.

These relative intensity ratios are familiar. They are, indeed, the binomial co-

efficients, which are the coefficients of the polynomial expansion (x 1)n,

where n0,1,2,3,....These coefficients are shown in Figure 16.20 in their

most familiar form, called Pascal’s triangle (named after the seventeenth-

century French mathematician Blaise Pascal). Furthermore, the integrated in-

tensity of each set of absorptions is directly proportional to the number of hy-

drogens having that specific chemical environment. Figure 16.21 shows the

same NMR spectra as in Figure 16.17, but at higher resolution. More infor-

mation is available from the high-resolution spectra, which have become the

standard.

Example 16.12

Describe the high-resolution spectrum of each of the following:

a.Methane, CH 4

b.Ethane, CH 3 CH 3

c.Propane, CH 3 CH 2 CH 3

Solution

a.Methane has four hydrogens on one carbon atom. Since the selection rules

do not allow one to observe the coupling between hydrogens on the same car-

bon, there will be only a single NMR absorption even in the high-resolution

spectrum.

b.Ethane has six hydrogens in the same chemical environment, but the cou-

pling between the hydrogens splits the single low-resolution spectrum into

n 1  3 1 4 individual lines. This splitting into what is called a quar-

tet of lines is exactly the same for both CH 3 groups, so one observes only a

single quartet of absorptions, with intensity ratios of roughly 1 3  3 1, in the

NMR spectrum.

c.There are two different chemical environments in propane. The hydrogens

in each CH 3 group couple to the two hydrogens in the CH 2 group in the same

way, so the methyl hydrogens are split into three individual peaks (a triplet)

with intensity ratios of 1 2 1. The CH 2 protons are also coupling with all of

the hydrogens in the methyl groups, so the absorption from those hydrogens

16.5 Nuclear Magnetic Resonance 579

1

11

121

1331

1

1

1

4

5

6

6

10

15

4

10

20

...

1

5

15

1

61

8

ppm

0

TMS

76543

2H

3H

21

Figure 16.21 At higher resolution, the NMR spectrum in Figure 16.17 shows the splitting of

the proton absorptions, with intensity ratios as predicted.

Figure 16.20 Pascal’s triangle, a mnemonic
for remembering the coefficients on the polyno-
mial expansion (x 1)n. These numbers are
called binomial coefficients, and are also applica-
ble to the intensities of split NMR signals.