Equation 14.21 is commonly rewritten in terms of the rotational constant B.
Factoring out 2 from the terms involving J, equation 14.21 can be rewritten as
E 2 B(J 1) (14.22)
For absorption spectroscopy, the minimum Jvalue is J0, so by considering
the specific values ofEversus the value ofJ:
E 0 → 1 2 B(0 1) 2 B
E 1 → 2 2 B(1 1) 4 B
E 2 → 3 2 B(2 1) 6 B
E 3 → 4 2 B(3 1) 8 B
E 4 → 5 2 B(4 1) 10 B
.
.
.
Figure 14.8 shows a graphical representation of a typical rotational spectrum
for a linear molecule. Figure 14.9 shows a diagram of the energy levels, and the
allowed transitions are labeled. Compare the two figures, and you should be
able to find the specific transition in Figure 14.9 that corresponds to the line
in Figure 14.8. As suggested by the above equations, the spectrum consists of
a series of equally spaced lines, spaced by an amount equal to 2B. The spec-
trum will be a series of equally spaced lines only if the unit of display is di-
rectly proportional to energy, like frequency or wavenumber. If the spectrum
were displayed in terms of a quantity inversely proportional to energy (like
wavelength), then the lines of the spectrum would not be equally spaced. This
illustrates the necessity of understanding the unit used to display a spectrum.
The above equations show something else about the rotational spectrum: its
relationship to B, which in turn is related to the moment of inertia. For di-
atomic molecules, the moment of inertia Iis defined simply as r^2 ,where
is the reduced mass and ris the internuclear separation of the two atoms.
Rotational spectroscopy is therefore useful in calculating the sizes of diatomic
molecules. This is a general capability of rotational spectroscopy, but it is most
easily illustrated for diatomic molecules.Example 14.8
Some of the lines in the rotational spectrum of HCl appear at 83.03, 104.1,
124.3, 145.0, and 165.5 cm^1. The first line is the J 3 →J4 transition,
so it equals 8B. Determine the average value for Bfrom this data and calcu-
late the length of the HCl bond (which gives a good idea of the size of the
molecule). Assume that these data are for^1 H and^35 Cl. You will have to con-
vert the units ofBfrom cm^1 to J in order for the units to work properly.Solution
It is easy to see that the spacing between the lines is on the order of 21 cm^1.
Since the first line is the J 3 →J4 transition, we can make the follow-
ing assignments:
J 3 →J4: line position 8 B83.03 cm^1 ,
therefore B10.38 cm^1
J 4 →J5: line position 10 B104.1 cm^1 ,
therefore B10.41 cm^1474 CHAPTER 14 Rotational and Vibrational SpectroscopyEnergy 012 B 2 B...
2 B4 B 6 B 8 B 10 BE 0 E ^2 B J 0E 4 BE 12 BE 10 BE 8 BE 6 BE 2 B J 1E 6 B J 2E 12 B J 3
E 20 B J 4
E 30 B J 5
E 42 B J 6
Figure 14.8 A pure rotational spectrum, dis-
played in units directly proportional to energy,
shows a series of equally spaced lines. The energy
separation between the lines equals 2B.Figure 14.9 An energy-level diagram for the
rotational motion of a diatomic molecule shows
the allowed transitions and their energies.