Absolutely Small - Michael D. Fayer [121]
The symmetric and asymmetric stretches maintain all three atoms on a line. In the bending mode, the two oxygens move up and the carbon moves down. This keeps the center of mass in one place. Then the carbon moves up and the two oxygens move down. In addition to the bending mode shown in Figure 17.2, there is a second bending mode. The one shown has the motions of the atoms in the plane of the page. The second bending mode is identical except the atoms move into and out of the plane of the page.
FIGURE 17.2. Top: Ball-and-stick model of carbon dioxide (CO2). Bottom: The three different vibration motions that the molecule can undergo. There are two bending modes: the one shown and the equivalent one with the atoms going in and out of the plane of the page.
Quantum Vibrations Have Discreet Energy Levels
In a classical vibrational oscillator made up of balls connected by springs, the energies the system can have are continuous. Consider the symmetric stretch. Three balls connected by two perfect springs are laying on a frictionless table with no air resistance. If you grab the outer two balls, stretch the springs the same amount, and let go, the balls will execute the symmetric stretching mode. Because the spring is perfect, the table is frictionless, and there is no air resistance (none of which is true in real life), the oscillation will continue forever. The period or frequency of the oscillation is independent of how far you stretch the springs. The period is determined by the springs’ strengths and the masses. If you stretch the springs only a little bit, the balls will move slowly. Their average kinetic energy is small. If you stretch the springs a lot, the balls will move fast, and the average kinetic energy is large. The energy of the oscillating ball and spring system is continuous. It only depends on how much you stretch the springs.
Molecules are not really balls and springs. They are quantum mechanical systems composed of atoms joined by chemical bonds. Rather than having a continuous range of energies, the quantum system has discreet vibrational energy levels. The quantization of the energy is equivalent to the particle in a box problem discussed in Chapter 8. Gerhard Herzberg (1904-1999) won the Nobel Prize in Chemistry in 1971 “for his contributions to the knowledge of electronic structure and geometry of molecules, particularly free radicals.” Herzberg’s work on determining the structure of molecules was based to a large extent on his explanations for the vibrational spectra of molecules.
The energy of a classical racquetball is continuous, but the energy of the quantum racquetball (particle in a box) has energy levels (see Figure 8.6). Figure 17.3 shows a potential curve for a vibrational mode of a molecule, like the one shown in Figure 12.1, but now the first several quantized vibrational energy levels are also shown. Again like the particle in a box, the lowest energy level, n = 0, does not have zero energy.
Energies of Quantized Vibrations
The simplest model for the vibrational energy levels gives the energies as
E = hν(n + 1/2),
where h is Planck’s constant, ν is the vibration frequency, and n is a quantum number that can take on values, 0, 1, 2, etc. For n = 0, the energy is 1/2hν. For n = 1, the energy is 3/2hν. So the difference in energy between the lowest energy level and the first excited vibration level is hν. In this model, all of the energy levels are equally spaced with a separation of hν. In real molecules, the energy levels get somewhat closer together as the quantum number increases. For our purposes, we only care about the spacing between the lowest energy level and first excited energy level.
CO2 Bending Mode Absorbs at the Peak of the Earth’s Black Body Spectrum
The bottom portion of Figure 17.3 shows the first two vibrational energy levels. Light will be absorbed at the energy of the separation of the levels, which is indicated by the dashed arrow. Since the difference in energy is ΔE = hν = c h/λ, measurement