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By Root 436 0

we saw two hydrogen atomic orbitals combine to form two molecular orbitals, one bonding and one antibonding. In benzene, we saw that six pz atomic orbitals, one from each carbon, combined to form six MOs, three bonding MOs and three antibonding MOs. For naphthalene, 10 pz atomic orbitals combined to form 10 MOs, five bonding and five antibonding. In each case, the MOs span the entire molecule. In Chapter 11 on the Periodic Table of Elements, we said that sodium, Na, is a metal because it has one electron, the 3s, past the neon filled shell configuration. Na can readily give up an electron to form salts, such as table salt, NaCl. In water NaCl dissolves to become Na+ and Cl-. We said that Na as a solid was a metal and conducted electricity. Now we are in a position to see why.

First consider the 3s orbitals of two sodium atoms that are next to each other and interacting. For sodium, the 3s electron is the valence electron, which will participate in bonding. The top portion of Figure 19.2 shows the energy levels of the two 3s atomic orbitals combining to form MOs. One of the MOs has lower energy than the atomic orbitals. It is the bonding MO. The other MO has higher energy; it is the antibonding MO. The middle shows that three atomic orbitals will form three MOs. The bottom illustrates the situation for six interacting sodium atoms. The six 3s atomic orbitals combine to form six MOs, three bonding and three antibonding.

Each Na has one 3s electron, which it will contribute to fill the MOs. For the system with six sodium atoms, there will be six electrons to fill the MOs. Each MO can be occupied by two electrons with opposite spins (one up arrow and one down arrow). Therefore, the three lowest energy MOs, which are the bonding MOs, will be filled by the six electrons. The three higher energy MOs are empty.

FIGURE 19.2. Top: Two sodium 3s atomic orbitals interact to produce two molecular orbitals, one lower in energy (bonding) and one higher in energy (antibonding). Middle: Three 3s atomic orbitals interact to form three MOs. Bottom: Six 3s atomic orbitals interact to form six MOs.

Now we must consider what happens when we have a very large number of interacting sodium atoms. Consider a metal rod composed of sodium atoms that is 10 cm long and 1 mm in diameter, such as that in Figure 19.1. For the dimensions given, the rod contains N = 2 × 1021 Na atoms, where N is the number of atoms. This number is two billion trillion atoms. The two billion trillion 3s atomic orbitals combine to form two billion trillion molecular orbitals. Like the MOs of benzene or naphthalene, the MOs of the sodium rod should be thought of as spanning the entire system, that is, the entire piece of metal.

A Piece of Metal Has a Vast Number of MO Energy Levels Called a Band

Figure 19.3 illustrates the energy levels of this system. Each of the N sodium atoms has an electron in a 3s atomic orbital. In the absence of interactions between the atoms, all of these atomic orbitals have the same energy. In the figure, this is represented by the collection of closely spaced lines on the left-hand side. To show that there are many atomic levels, the lines have been spread out, but they all have the same energy. When the atoms interact, the N atomic orbitals form N MOs. As we have seen previously for molecules, the MOs have different energies. Some of the MOs have lower energies than the atomic orbital energy and some have higher energy. This is represented on the right side of the figure by the spread-out but closely spaced lines. The MO energy levels in Figure 19.3 are equivalent to those shown in Figures 18.8, 18.9, and 19.2, except that there are vastly more energy levels, which are much more closely spaced. These are referred to as a band of states.

FIGURE 19.3. In a piece of sodium metal, there are N atoms. Each has an electron in a 3s orbital, represented by the closely spaced lines on the left. These all have the same energy. The N 3s atomic orbitals interact to form N molecular orbitals with energy levels shown on the right. The MO