Absolutely Small - Michael D. Fayer [120]
Why Carbon Dioxide Absorbs Where It Does
We see that carbon dioxide traps infrared light at the peak of the Earth’s black body emission and that an increase in CO2 concentration will have a deleterious effect on the Earth’s temperature. But why does CO2 absorb infrared light at wavelengths centered at 667 cm-1? In Chapters 8 through 11, we discussed the energy levels of a particle in a box, of the hydrogen atom, and of all of the other atoms. In Chapters 12 to 14, we discussed molecular orbitals and the associated energy levels. All of these discussions concerned the energy levels associated with electrons. Using the ideas of molecular orbitals, the nature of bonds that hold atoms together to form molecules was explicated. What we have not discussed is the motions of atoms that are bonded together to form molecules.
Figure 12.1 displays the potential energy curve for the hydrogen molecule, H2. The curve shows the energy at different separations of the hydrogen atom nuclei. The bond length is the separation where the energy is a minimum. However, the bond is not rigid. If we think about the bond using classical mechanics, the bond is a spring with two masses, the hydrogen atoms, attached at each end of the spring. A spring can be stretched and compressed. In a classical system, if you stretch the spring and let go, the masses will oscillate back and forth with the spring being alternately stretched and compressed. The masses of a classical oscillator will vibrate (oscillate) back and forth with a well-defined trajectory. Based on quantum theory, we should immediately suspect that a quantum vibration cannot have a well-defined trajectory. Such a trajectory would mean that we know the positions and moment of the particles (the atoms) precisely. Such knowledge for absolutely small systems, such as atoms bonded to form a molecule, violates the Heisenberg Uncertainty Principle.
Figure 17.2 shows a ball-and-stick model of carbon dioxide, CO2, as well as representations of its possible vibrational motions. CO2 is linear, with the two oxygens double bonded to the central carbon. CO2 has four different vibrational motions, called vibrational modes. The bonds can stretch and compress as well as bend. The bonds are represented by springs. We will describe the motions as if they are classical balls connected by springs to understand the nature of the modes.
The Vibrational Modes of Carbon Dioxide
In the symmetric stretch, the central carbon does not move. As shown by the solid arrows, the two oxygens move away from the carbon, thereby stretching the springs. The two oxygens then move back toward the central carbon, compressing the springs, as indicated by the dashed arrows. For a classical ball-and-spring system, this motion is repeated, so the positions oscillate back and forth. The frequency of the oscillation is determined by the masses and the strengths of the springs. In the asymmetric stretching mode, the two oxygens move to the right. The oxygen on the right compresses the spring, and the oxygen on the left stretches the spring. A vibration does not move the molecule to a new location. Because both oxygens are moving to the right, the carbon moves to the left in order to keep the molecule in the same location. Because the carbon moves to the left when the oxygens move to the right, the average position of all of the mass, called the center of mass, is unchanged. The motions are indicated by the solid arrows. The direction of each