Our present model of the atom is based on the concept of energy levels for electrons within an atom and on the mathematical interpretation of detailed atomic spectra. The requirements for our model are:

1. Each electron in a particular atom has a unique energy that depends on the relationship between the negatively charged electron and both the positively charged nucleus and the other negatively charged electrons in the atom.

2. The energy of an electron in an atom can increase or decrease, but only by specific amounts, or quanta.

A. Energy Levels
We picture an atom as a small nucleus surrounded by a much larger volume of space containing the electrons. This space is divided into regions called principal energy levels, numbered 1, 2, 3, 4, . . . . , outward from the nucleus.

Each principal energy level can contain up to 2n² electrons, where n is the number of the level. Thus, the first level can contain up to 2 electrons, 2(1²) = 2; the second up to 8 electrons, 2(2²) = 8; the third up to 18, 2(3²) = 18; and so on. Only seven energy levels are needed to contain all the electrons in an atom of any of those elements now known.

As stated earlier, the energy associated with an energy level increases as the distance from the nucleus increases. An electron in the seventh energy level has more energy associated with it than does one in the first energy level.

The lower the number of the principal energy level, the closer the negatively charged electron in it is to the positively charged nucleus and the more difficult it is to remove this electron from the atom.

B. Sublevels and Orbitals
When an electron is in a particular energy level, it is more likely to be found in some parts of that level than in others. These parts are called orbitals. Orbitals of equivalent energy are grouped in sublevels. Each orbital can contain a maximum of two electrons. When in a magnetic field, the two electrons in a particular orbital differ very slightly in energy because of a property called electron spin. The theory of electron spin states that the two electrons in a single orbital spin in opposite directions on their axes, causing an energy difference between them. (Like many models, this explanation is an oversimplification, but for the purpose of this course it is a useful description.)

Each principal energy level has one sublevel containing one orbital, an s orbital, that can contain a maximum of two electrons. Electrons in this orbital are called s electrons and have the lowest energy of any electrons in that principal energy level. The first principal energy level contains only an s sublevel; therefore, it can hold a maximum of two electrons.

Each principal energy level above the first contains one s orbital and three p orbitals. A set of three p orbitals, called the p sublevel, can hold a maximum of six electrons. Therefore, the second level can contain a maximum of eight electrons - that is, two in the s orbital and 6 in the three p orbitals.

Each principal energy level above the second contains, in addition to one s orbital and three p orbitals, a set of five d orbitals, called the d sublevel. The five d orbitals can hold up to 10 electrons. Thus, the third level holds a maximum of 18 electrons: 2 in the s orbital, 6 in the three p orbitals, and 10 in the five d orbitals.

The fourth and higher levels also have an f sublevel, containing seven f orbitals, which can hold a maximum of 14 electrons. Thus, the fourth level can hold up to 32 electrons: 2 in the s orbital, 6 in the three p orbitals, 10 in the five d orbitals, and 14 in the seven f orbitals. The sublevels of the first four principal energy levels and the maximum number of electrons that the sublevels can contain are summarized in Table 5.1.

To distinguish which s, p, d, or f sublevel we are talking about, we precede the letter by the number of the principal energy level. For example, the s sublevel of the second principal energy level is designated 2s; the s sublevel of the third principal energy level is designated 3s; and so on. The number of electrons occupying a particular sublevel is shown by a superscript after the letter of the sublevel. The notation

means that five electrons are contained in the p sublevel of the fourth energy level.

FIGURE 5.5 Perspective representations of the s and the three p orbitals of a single energy level. The clouds show the space within which the electron is most apt to be. The lower sketch shows how these orbitals overlap in the energy level.

The three p orbitals are more or less dumbbell-shaped, with the nucleus at the center of the dumbbell. They are oriented at right angles to one another along the x, y, and z axes, hence we denote them as p_x, p_y, and p_z. Like the s orbitals, the p orbitals increase in size as the number of the principal energy level increases; thus a 4p orbital is larger than a 3p orbital.

FIGURE 5.6 Cross-sectional view of the s orbitals of an atom showing their relative sizes and overlap.

The shapes of d orbitals are shown in Figure 5.7. The five d orbitals are denoted by d_xy, d_yz, d_xz, d_x2_-y2, and d_x2. Notice that these shapes are more complex than those of p orbitals, and recall that the shapes of p orbitals are more complex than those of s orbitals. Clearly, the shape of an orbital becomes more complex as the energy associated with that orbital increases. We can predict that the shapes of f orbitals will be even more complex than those of the d orbitals.

FIGURE 5.7 The shapes and orientations of the d orbitals.

One further, important note about orbital shapes: These shapes do not represent the path of an electron within the atom; rather, they represent the region of space in which an electron of that sublevel is most apt to be found. Thus, a p electron is most apt to be within a dumbbell-shaped space in the atom, but we make no pretense of describing its path.

2. The energy of an electron versus its orbital
Within a given principal energy level, electrons in p orbitals are always more energetic than those in s orbitals, those in d orbitals are always more energetic than those in p orbitals, and electrons in f orbitals are always more energetic than those in d ortitals. For example, within the fourth principal energy level, we have:

In addition, the energy associated with an orbital increases as the number of the principal energy level of the orbital increases. For instance, the energy associated with a 3p orbital is always higher than that associated with a 2p orbital, and the energy of a 4d orbital is always higher than that associated with a 3d orbital. The same is true of s orbitals:

Each orbital is not a region of space separate from the space of other orbitals. This is implicit in Figures 5.5, 5.6, and 5.7. If all those orbitals were superimposed on one another, you would see that a great deal of space is included in more than one orbital. For example, a 3p electron can be within the space assigned to a 3d or 3s orbital as well as within its own 3p space.

There is also an interweaving of energy levels. Figure 5.8 shows, in order of increasing energy, all the orbitals of the first four energy levels. Notice that the energy of a 3d orbital is slightly higher than that of a 4s orbital, and that of a 4d orbital is a little higher than that of a 5s orbital. Note especially the overlap of orbitals in the higher principal energy levels.

FIGURE 5.8 The principal energy levels of an atom and the sublevels and orbitals each contains. The arrows show the order in which the sublevels fill.

When all the electrons of an atom are in the lowest possible energy states (meaning that the energy levels have been filled in order of increasing energy), the atom and its electrons are in the ground state. If one of these electrons moves to a higher energy level, the atom is in an excited state.

We know that each element has a unique spectrum. These spectra show that the energy differences among the electrons in an atom vary from one element to another. What causes this variation?

Recall that the nucleus of an atom is positively charged, that electrons carry a negative charge, and that oppositely charged bodies attract one another. The atoms of one element differ from those of another element in the number of protons in the nucleus and, consequently, in the charge on the nucleus. The attraction for an electron, and therefore its energy, will differ from one element to the next according to differences in nuclear charge. In addition, the atoms of one element contain a different number of electrons than do atoms of any other element. The energy of each electron within the atom depends not only on its interaction with the positively charged nucleus, but also on its interaction with the other electrons in the atom. Therefore, the energies of the electrons of one element will differ from the energies of the electrons of another element. Considering these two variables--nuclear charge and number of electrons--we can see that each element must have a unique spectrum derived from its unique set of electron energy levels.