The Foundations of Chemistry

crystal containing one mole of sodium atoms, for example, the interaction (overlap) of

6.022 10233 satomic orbitals produces 6.022 1023 molecular orbitals. Atoms interact

more strongly with nearby atoms than with those farther away. The energy that separates

bonding and antibonding molecular orbitals resulting from two given atomic orbitals

decreases as the overlap between the atomic orbitals decreases (Chapter 9). The interac-

tions among the mole of Na atoms result in a series of very closely spaced molecular

orbitals (formally  3 sand  3 
s). These constitute a nearly continuous bandof orbitals that

belongs to the crystal as a whole. One mole of Na atoms contributes 6.022 1023 valence

electrons (Figure 13-33a), so the 6.022 1023 orbitals in the band are half-filled.

The ability of metallic Na to conduct electricity is due to the ability of any of the

highest energy electrons in the “3s” band to jump to a slightly higher-energy vacant orbital

in the same band when an electric field is applied. The resulting net flow of electrons

through the crystal is in the direction of the applied field.

The empty 3patomic orbitals of the Na atoms also interact to form a wide band of 3

6.022 1023 orbitals. The 3sand 3patomic orbitals are quite close in energy, so the

fanned-out bands of molecular orbitals overlap, as shown in Figure 13-33b. The two over-

lapping bands contain 46.022 1023 orbitals and only 6.022 1023 electrons. Because

each orbital can hold two electrons, the resulting combination of bands is only one-eighth

full.

Overlap of “3s” and “3p” bands is not necessary to explain the ability of Na or of any

other Group IA metal to conduct electricity. It can do so utilizing only the half-filled “3s”

band. In the Group IIA metals, however, such overlap is important. Consider a crystal of

magnesium as an example. The 3satomic orbital of an isolated Mg atom is filled with two

electrons. Thus, without this overlap, the “3s” band in a crystal of Mg is also filled. Mg

is a good conductor at room temperature because the highest energy electrons are able

to move readily into vacant orbitals in the “3p” band (Figure 13-34).

According to band theory, the highest energy electrons of metallic crystals occupy either

a partially filled band or a filled band that overlaps an empty band. A band within which

The alkali metals are those of Group

IA; the alkaline earth metals are those

of Group IIA.

13-17 Band Theory of Metals 529

Figure 13-33 (a) The band of orbitals resulting from interaction of the 3sorbitals in a

crystal of sodium. (b) Overlapping of a half-filled “3s” band (black) with an empty “3p” band

(red) of NaNcrystal.

3 s

Half-filled

band of

N molecular

orbitals

Empty band of

3N molecular

orbitals from 3p

orbitals

Bands overlap

Half-filled band of N

molecular orbitals

from 3s orbitals

Na Na 2 Na 3 Na 8 NaN Na NaN

Energy

Energy

(a) (b)

3 p

3 s

}