Absolutely Small - Michael D. Fayer [127]
We can do a very crude calculation by pretending that the π electrons are particles in a box. In Chapter 8, we discussed the particle in a box in great detail. If we take the HOMO to LUMO transition to be a transition of an electron in a box from the n = 1 to n = 2 levels (see Figure 8.7), we can use the formulas derived just below Figure 8.7. For this transition we found that
where h is Planck’s constant, m is the mass of the electron, and L is the length of the box. Here we will take L to be 0.51 nm, the distance across the carbon framework of naphthalene. Then,
Converting this energy to a frequency by dividing by h gives ν = 1.04 × 1015 Hz. Then the wavelength of the light that will be absorbed is λ = 2.87 × 10-7 m = 287 nm. This wavelength is further in the ultraviolet than the true absorption, but not that far from the observed value.
The particle in a box calculation shows that if a particle with the mass of an electron is confined to a box the size of naphthalene, the first absorption will be in the ultraviolet range. The reasonable accuracy of the particle in a box calculation for naphthalene is somewhat fortuitous. Even if we wanted to model naphthalene as a particle in a box, it should be a two- or three-dimensional box, not a one-dimensional box. Such calculations produce results with significant errors. However, an accurate quantum theory calculation will yield the molecular structure and much more accurate frequencies for absorption of light. In addition, if, for example, a hydrogen is replaced by a fluorine, quantum theory will accurately predict how much the frequency of light absorption of fluoronaphthalene is changed from that of naphthalene.
19
Metals, Insulators, and Semiconductors
FIGURE 19.1 IS A SCHEMATIC DIAGRAM of a battery connected to a metal rod. We will discuss sodium metal below as an example, but the rod could be any metal. The positive end of the battery pulls electrons from the metal rod. Electrons flow out of the rod into the battery.
FIGURE 19.1. A metal rod of sodium, for example, is connected by wires to a battery. Negatively charged electrons are pulled from the metal rod to the positive side of the battery. Electrons flow from the negative side of the battery back into the metal rod.
However, to keep the rod from developing a positive charge that would pull back on the electrons and stop the flow, the rod must be connected to the negative side of the battery.
Electrons flow from the negative side of the battery into the rod, keeping the rod neutral, that is, the rod does not develop an electrical charge. Instead of a rod, the electrons could flow through the filament of a lightbulb in a flashlight. The flow of electrons through the filament causes it to get very hot and produce black body radiation in the visible part of the spectrum.
METALS
Delocalize Molecular Orbital for a Metal
How can electrons move through a piece of metal? What is the difference between a metal and an insulator? What is a semiconductor? Why does a metal get hot when electrons flow through it? What is superconductivity? To answer the first three questions, we will extend our discussion of the types of delocalized molecular orbitals found in aromatic molecules such as benzene and naphthalene (see Chapter 18) to the MOs of a macroscopic piece of metal and other materials. To answer the last two questions, we will expand the discussion to the effect of the thermal vibrations of the atoms that make up a piece of metal on electron motion in a metal.
In Chapter 10 on the hydrogen molecule,