30.6

The Pauli Exclusion Principle and the Periodic Table of the Elements

Except for hydrogen, all electrically neutral atoms contain more than one electron, the number being given by the atomic number Z of the element. In addition to being attracted by the nucleus, the electrons repel each other. This repulsion contributes to the total energy of a multiple-electron atom. As a result, the one-electron energy expression for hydrogen, , does not apply to other neutral atoms. However, the simplest approach for dealing with a multiple-electron atom still uses the four quantum numbers n, , , and ms.
Detailed quantum mechanical calculations reveal that the energy level of each state of a multiple-electron atom depends on both the principal quantum number n and the orbital quantum number . Figure 30-15 illustrates that the energy generally increases as n increases, but there are exceptions, as the drawing indicates. Furthermore, for a given n, the energy also increases as increases.

Figure 30-15 When there is more than one electron in an atom, the total energy of a given state depends on the principal quantum number n and the orbital quantum number . The energy increases with increasing n (with some exceptions) and, for a fixed n, with increasing . For clarity, levels for n = 6 and higher are not shown.

In a multiple-electron atom, all electrons with the same value of n are said to be in the same shell. Electrons with n = 1 are in a single shell (sometimes called the K shell), electrons with n = 2 are in another shell (the L shell), those with n = 3 are in a third shell (the M shell), and so on. Those electrons with the same values for both n and are often referred to as being in the same subshell. The n = 1 shell consists of a single subshell. The n = 2 shell has two subshells, one with and one with . Similarly, the n = 3 shell has three subshells, one with , one with , and one with .
In the hydrogen atom near room temperature, the electron spends most of its time in the lowest energy level, or ground state—namely, in the n = 1 shell. Similarly, when an atom contains more than one electron and is near room temperature, the electrons spend most of their time in the lowest energy levels possible. The lowest energy state for an atom is called the ground state. However, when a multiple-electron atom is in its ground state, not every electron is in the n = 1 shell in general because the electrons obey a principle discovered by the Austrian physicist Wolfgang Pauli (1900–1958).

| The Pauli Exclusion Principle
| No two electrons in an atom can have the same set of values for the four quantum numbers n, , , and ms.

Suppose two electrons in an atom have three quantum numbers that are identical: n = 3, , and . According to the exclusion principle, it is not possible for each to have , for example, since each would then have the same four quantum numbers. Each electron must have a different value for (for instance, and ) and, consequently, would be in a different subshell. With the aid of the Pauli exclusion principle, we can determine which energy levels are occupied by the electrons in an atom in its ground state, as the next example demonstrates.

| Example 8 | | | Ground States of Atoms

Determine which of the energy levels in Figure 30-15 are occupied by the electrons in the ground state of hydrogen (1 electron), helium (2 electrons), lithium (3 electrons), beryllium (4 electrons), and boron (5 electrons). Reasoning In the ground state of an atom the electrons are in the lowest available energy levels. Consistent with the Pauli exclusion principle, they fill those levels “from the bottom up”—that is, from the lowest to the highest energy.
Solution As the colored dot in Figure 30-16 indicates, the electron in the hydrogen atom (H) is in the n = 1, subshell, which has the lowest possible energy. A second electron is present in the helium atom (He), and both electrons can have the quantum numbers n = 1, , and . However, in accord with the Pauli exclusion principle, each electron must have a different spin quantum number, for one electron and for the other. Thus, the drawing shows both electrons in the lowest energy level.

Figure 30-16 The electrons in the ground state of an atom fill the available energy levels “from the bottom up”—that is, from the lowest to the highest energy, consistent with the Pauli exclusion principle. The ranking of the energy levels in this figure is meant to apply for a given atom only.

The third electron that is present in the lithium atom (Li) would violate the exclusion principle if it were also in the n = 1, subshell, no matter what the value for ms is. Thus, the n = 1, subshell is filled when occupied by two electrons. With this level filled, the n = 2, subshell becomes the next lowest energy level available and is where the third electron of lithium is found (see Figure 30-16). In the beryllium atom (Be), the fourth electron is in the n = 2, subshell, along with the third electron. This is possible, since the third and fourth electrons can have different values for ms.
With the first four electrons in place as just discussed, the fifth electron in the boron atom (B) cannot fit into the n = 1, or the n = 2, subshell without violating the exclusion principle. Therefore, the fifth electron is found in the n = 2, subshell, which is the next available energy level with the lowest energy, as Figure 30-16 indicates. For this electron, can be −1, 0, or +1, and ms can be or in each case. However, in the absence of an external magnetic field, all six of these possibilities correspond to the same energy.

Because of the Pauli exclusion principle, there is a maximum number of electrons that can fit into an energy level or subshell. Example 8 shows that the n = 1, subshell can hold at most two electrons. The n = 2, subshell, however, can hold six electrons because with , there are three possibilities for (−1, 0, and +1), and for each of these, the value of ms can be or . In general, can have the values 0, ±1, ±2, … , , for possibilities. Since each of these can be combined with two possibilities for ms, the total number of different combinations for and ms is . This, then, is the maximum number of electrons the th subshell can hold, as Figure 30-17 summarizes.

Figure 30-17 The maximum number of electrons that the th subshell can hold is .

For historical reasons, there is a widely used convention in which each subshell of an atom is referred to by a letter rather than by the value of its orbital quantum number . For instance, an subshell is called an s subshell. An subshell and an subshell are known as p and d subshells, respectively. The higher values of , and so on, are referred to as f, g, and so on, in alphabetical sequence, as Table 30-2 indicates.

Table 30-2 The Convention of Letters Used to Refer to the Orbital Quantum Number

Orbital Quantum Number | Letter
0 | s
1 | p
2 | d
3 | f
4 | g
5 | h

This convention of letters is used in a shorthand notation that is convenient for indicating simultaneously the principal quantum number n, the orbital quantum number , and the number of electrons in the n, subshell. An example of this notation follows:

With this notation, the arrangement or configuration of the electrons in an atom can be specified efficiently. For instance, in Example 8, we found that the electron configuration for boron has two electrons in the n = 1, subshell, two in the n = 2, subshell, and one in the n = 2, subshell. In shorthand notation this arrangement is expressed as 1s2 2s2 2p1. Table 30-3 gives the ground-state electron configurations written in this fashion for elements containing up to thirteen electrons. The first five entries are those worked out in Example 8.

Table 30-3 Ground-State Electronic Configurations of Atoms

Element | Number of Electrons | Configuration of the Electrons
Hydrogen (H) | 1 | 1s1
Helium (He) | 2 | 1s2
Lithium (Li) | 3 | 1s2 2s1
Beryllium (Be) | 4 | 1s2 2s2
Boron (B) | 5 | 1s2 2s2 2p1
Carbon (C) | 6 | 1s2 2s2 2p2
Nitrogen (N) | 7 | 1s2 2s2 2p3
Oxygen (O) | 8 | 1s2 2s2 2p4
Fluorine (F) | 9 | 1s2 2s2 2p5
Neon (Ne) | 10 | 1s2 2s2 2p6
Sodium (Na) | 11 | 1s2 2s2 2p6 3s1
Magnesium (Mg) | 12 | 1s2 2s2 2p6 3s2
Aluminum (Al) | 13 | 1s2 2s2 2p6 3s2 3p1

Each entry in the periodic table of the elements often includes the ground-state electronic configuration, as Figure 30-18 illustrates for argon. To save space, only the configuration of the outermost electrons and unfilled subshells is specified, using the shorthand notation just discussed. Originally the periodic table was developed by the Russian chemist Dmitri Mendeleev (1834–1907) on the basis that certain groups of elements exhibit similar chemical properties. There are eight of these groups, plus the transition elements in the middle of the table, including the lanthanide series and the actinide series. The similar chemical properties within a group can be explained on the basis of the configurations of the outer electrons of the elements in the group. Thus, quantum mechanics and the Pauli exclusion principle offer an explanation for the chemical behavior of the atoms. The full periodic table can be found on the inside of the back cover.

Figure 30-18 The entries in the periodic table of the elements often include the ground-state configuration of the outermost electrons.

Check Your Understanding 3

Using the convention of letters to refer to the orbital quantum number, write down the ground-state configuration of the electrons in krypton (Kr, Z = 36). (The answer is given at the end of the book.) Background: Quantum mechanics and the Pauli exclusion principle determine the electronic configuration of multi-electron atoms.
For similar questions (including calculational counterparts), consult Self-Assessment Test 30.2, which is described at the end of Section 30.9.

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