Physical Chemistry Third Edition

24.3 Electron Spin Resonance Spectroscopy 1013

Since the nuclear states are nearly equally populated, the population of a level will be

proportional to the degeneracy of the level, which is proportional to the number of ways of

dividing a set of six proton spins into a subset withms 1 /2 and a subset withms− 1 /2.

The number of ways of dividing a set ofnobjects into a subset ofmobjects and a subset of

n−mobjects is

Number

n!

m!(n−m)!

(24.3-6)

which is the formula for binomial coefficients. The intensities of the spectral lines are in the

ratios1:6:15:20:15:6:1.

Figure 24.2 shows a mnemonic device for arriving at the spectrum of Example 24.10.

Starting with the spectral line that would occur without any splitting, the effect of the

first proton is represented by a splitting into two lines as shown at the top of the

diagram. Each of these lines is then split into two lines, representing the effect of

the second proton. Since all splitting constants are equal, two lines coincide after the

second splitting, and there are three lines with relative intensities of1:2:1, asshown

at the second level of the diagram. The third level of the diagram represents the effect

of the third proton, and so on. The relative intensities at each level are obtained by

adding the relative intensities of the two lines at the previous level that combine to

produce a given line. The result at each level gives relative intensities proportional to

binomial coefficients.

1

1

1

2

33

4 6 4

1

1

1

1

1

1

1

1

1 5 10 10 5

1 6 1520156

Figure 24.2 Diagram to Find the

Splittings in the ESR Spectrum of

the Benzene Negative Ion, C 6 H− 6 .The

use of this diagram requires that all of

the splitting constants are equal. Other-

wise the lines will not coincide, and there

will be more than seven lines.

Exercise 24.6

Predict the ESR spectrum of atomic hydrogen.

If an unpaired electron is affected by a nucleus with a spin quantum number other

than 1/2, a given nucleus will produce more than two lines. For example, a deuterium

nucleus (^2 H) hasI1 so thatMI1, 0, or−1. Deuterium atoms will produce a

spectrum with three lines. However, unlike the spectrum of H+ 2 , the three lines will be

of nearly equal intensity, since there is one state with each value ofMIand the states

are nearly equally populated. A nucleus withI3 would produce seven lines of equal

intensity.

PROBLEMS

Section 24.3: Electron Spin Resonance Spectroscopy

24.9The manganese nucleus has a spin quantum number

I 5 /2. Describe the ESR spectrum of manganese

atoms.

24.10Predict the ESR spectrum of the cyclobutadiene negative
ion, C 4 H− 4.

24.11 a.Predict the ESR spectrum of the hydrogen molecule
ion, H+ 2.

b.Predict the ESR spectrum of D+ 2.

24.12Predict the ESR spectrum of NO, nitric oxide. Assume the

common isotopes,^14 N and^16 O.

24.13Describe the ESR spectrum of a deuterium atom in its

ground electronic state.

24.14a.Assume that carbene, CH 2 , is a linear molecule.

Predict the ESR spectrum of CH 2.

b.Assume that CCl 2 is a bent molecule with a bond

angle of roughly 120◦. Predict the ESR spectrum of