Absolutely Small - Michael D. Fayer [129]
Quantum theory shows that the width of the band of states, that is, the difference in energy between the highest energy MO and the lowest energy MO, is only a few times the energy splitting of the MOs that arise from a pair of interacting sodium atoms (see Figure 19.2, top). Then in our example of two billion trillion Na atoms, there are this many energy levels in a relatively narrow range of energy. The result is that the energy levels are so closely spaced that the energy is effectively continuous within the band.
Putting in the Electrons
There are N sodium atoms, each with a single 3s electron. We need to take these N electrons and put them in the appropriate MOs, as we did with small molecules in Chapters 12 and 13 and as shown in Figures 18.8 and 18.9. These sodium metal delocalized MOs are orbitals like any others, so we must obey the three rules for putting in the electrons discussed in Chapter 11. They are lowest energy first, no more than two electrons in an orbital that must have paired spins (Pauli Exclusion Principle), and don’t pair spins unless necessary (Hund’s Rule). Figure 19.3 illustrates putting in the electrons. The first electron goes in the lowest energy level. The next electron goes into the same level with the opposite spin, that is, one up arrow and one down arrow. The third electron can’t go into the lowest energy level because that would violate the Pauli Principle. So, it goes into the level one up from the lowest. The fourth electron goes into this same level with paired spins. This will continue until all N electrons are in MOs.
The Fermi Level
There are N MO energy levels and N electrons. But two electrons can go in each level. Therefore, only the bottom half of the energy band of levels will be filled. This is like benzene (Figure 18.8) and naphthalene (Figure 18.9), which also only have the bottom half of their MOs filled. The energy of the highest filled orbital is called the Fermi level, for Enrico Fermi (1901-1954). Fermi was a physicist who worked in many areas of science including the theory of solids, such as metals, and the theory of nuclear reactions. His work contributed to the development of nuclear energy. He won the Nobel Prize in Physics in 1938 “for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons.” As we will see, the Fermi level is very important.
The Fermi level is the level of the highest filled MO at the absolute zero of temperature. This temperature is 0° K, where K is the Kelvin unit of temperature. One degree K is the same as one degree C (centigrade) except the scale begins at the absolute zero of temperature. 0° K is -273° C or -459° F. We have briefly discussed how heat in systems of molecules, such as water, causes the molecules to jiggle around. In Chapter 15, it was pointed out that the thermal motions of water were responsible for the breaking of hydrogen bonds between water molecules. As temperature is decreased, there is less and less heat (thermal energy) and the motions of atoms and molecules decrease. The absolute zero of temperature, 0° K, is the temperature at which there is no heat to cause atoms and molecules to move. The Fermi level is actually defined to be the energy of the highest filled MO at 0° K.
How Electrons Move Through Metal
As shown in Figure 19.1, electrons enter one side of the metal rod and leave the other. This is possible because the electrons are in delocalized MOs that span the entire piece of metal. However, quantum theory shows that if all of the electrons occupy only the MOs below the Fermi level, the electrons will not move in a particular direction. Real metals are three dimensional, but for this discussion let’s consider only one dimension at a time. In our metal rod, when it is not connected to the battery, the electrons