113 The Pauli Exclusion Principle

Multiple-Electron Atoms

Hydrogen, with its single electron, is the simplest atom and provided the foundation for our introduction to quantum mechanics. Nearly every atom encountered in chemistry, biology, and medicine, however, contains multiple electrons. These additional electrons interact not only with the positively charged nucleus but also with one another, making their behavior considerably more complex.

The arrangement of electrons within an atom determines many of its physical and chemical properties. In particular, the number and organization of an atom's electrons explain why different elements behave differently and why the periodic table displays repeating patterns in chemical behavior. A neutral atom contains the same number of electrons as protons in its nucleus. This number, known as the atomic number and represented by \(Z\), uniquely identifies each element.

One of the most important principles governing electron arrangements is the Pauli exclusion principle. This rule explains how electrons occupy atomic energy levels and ultimately provides the quantum-mechanical foundation of the periodic table.

In 1925, the Austrian physicist Wolfgang Pauli proposed a remarkably simple but powerful rule: no two electrons in the same atom can occupy exactly the same quantum state. In other words, no two electrons can have identical sets of quantum numbers. This statement became known as the Pauli exclusion principle.

Although originally formulated to explain atomic spectra, the exclusion principle applies much more broadly. It governs all identical particles with half-integer spin, including electrons, protons, and neutrons, and is one of the fundamental principles underlying modern quantum mechanics.

Each electron in an atom is described by the four quantum numbers introduced in the previous chapter:

For electrons, the spin quantum number is always \(s=\frac12\), so it is usually omitted because it never changes. Consequently, an electron's quantum state is completely specified by the four quantities shown above. For example, the quantum numbers

identify one unique electron state in an atom.

Because every electron must have a different combination of quantum numbers, the Pauli exclusion principle limits the number of electrons that can occupy each energy level. Consider the lowest energy level, \(n=1\). For this shell, the only allowed value of the angular momentum quantum number is \(l=0\), and therefore the only possible value of \(m_l\) is also zero. The only remaining difference between electrons is the orientation of their spin.

The spin projection quantum number may take one of two values:

As a result, only two electrons can occupy the \(n=1\) shell. These electrons must have opposite spin orientations, giving them the quantum numbers

A third electron cannot occupy this shell because it would necessarily duplicate the quantum numbers of one of the first two electrons, violating the Pauli exclusion principle. Instead, it must occupy a higher-energy shell.

Shells and Subshells

The Pauli exclusion principle limits the number of electrons that can occupy a given quantum state. As a result, only hydrogen and helium can place all of their electrons in the lowest energy shell, where \(n=1\). The next element, lithium, contains three electrons, so its third electron must occupy the next higher energy level, \(n=2\). This progression introduces the concepts of atomic shells and shell filling.

As we move through the periodic table from hydrogen to heavier elements, additional electrons fill increasingly higher energy levels. Each value of the principal quantum number \(n\) defines an atomic shell. Higher shells correspond to higher energies and can accommodate more electrons because they contain a greater number of allowed combinations of the quantum numbers \(l\), \(m_l\), and \(m_s\).

The arrangement of electrons among these shells determines many of an atom's physical and chemical properties. In particular, the electrons in the outermost shell interact most strongly with neighboring atoms and therefore play the dominant role in chemical bonding.

Each shell is further divided into subshells, which are identified by the angular momentum quantum number \(l\). Electrons with smaller values of \(l\) have probability clouds that extend closer to the nucleus and are generally more tightly bound. Consequently, subshells fill in order of increasing \(l\), beginning with \(l=0\).

The traditional symbols used to represent shells and subshells are shown in Table 113.1.

For a shell with principal quantum number \(n\), the allowed subshells are \(l = 0, 1, 2, \ldots, n-1\), so the number of subshells available in a given shell increases with \(n\) (for example, the \(n=3\) shell contains the \(s\), \(p\), and \(d\) subshells).

Electron configurations are written using spectroscopic notation, which combines the shell number with the subshell symbol. For example, an electron in the first shell must have \(l=0\), so it occupies a \(1s\) orbital. If both available states in that subshell are filled, the configuration is written as \(1s^2\).

Similarly, an electron in the second shell with \(l=1\) occupies a \(2p\) subshell. A configuration containing three electrons in that subshell is written as \(2p^3\). In general, the superscript indicates the number of electrons occupying the subshell.

By counting the combinations of quantum numbers allowed by the Pauli exclusion principle, we can determine the maximum number of electrons that can occupy each shell and subshell.

Example 113.1: How Many Electrons Can Occupy the \(n=2\) Shell?

Problem

List all the possible sets of quantum numbers for the \(n=2\) shell and determine the maximum number of electrons that can occupy the shell and each of its subshells.

Strategy

For the shell \(n=2\), the allowed values of the angular momentum quantum number are \(l=0\) and \(l=1\). These correspond to the \(2s\) and \(2p\) subshells. Determine all allowed values of \(m_l\) and \(m_s\) for each subshell, then count the resulting quantum states.

Solution

One convenient approach is to organize all allowed combinations of quantum numbers in a table, as shown in Figure 113.4.

Discussion

Listing every possible quantum state works well for small shells, but it quickly becomes impractical for larger atoms. Fortunately, simple general rules allow us to determine the maximum number of electrons in any shell or subshell without constructing a complete table.

The maximum number of electrons that can occupy a subshell depends only on the angular momentum quantum number \(l\). Once \(l\) is known, the magnetic quantum number has

possible values. Because each orbital can contain one spin-up electron and one spin-down electron, the maximum number of electrons in a subshell is

For example, the \(2s\) subshell has \(l=0\), so its capacity is

electrons.

The \(2p\) subshell has \(l=1\), giving

possible electrons.

Adding the capacities of all subshells in a shell leads to a simple general result: the maximum number of electrons that can occupy a shell with principal quantum number \(n\) is

For the first shell (\(n=1\)), this equation predicts a maximum of two electrons, exactly as expected. For the second shell (\(n=2\)), it predicts

electrons, in agreement with the results of Example 113.1.

Shell Filling and the Periodic Table

As electrons are added to an atom, they occupy the available shells and subshells according to the rules of quantum mechanics. The resulting arrangement of electrons is called the electron configuration of the atom. Electron configurations explain not only the structure of individual atoms but also the repeating patterns of chemical behavior observed in the periodic table.

Table 113.2 lists the ground-state electron configurations for the first 20 elements, beginning with hydrogen and ending with calcium. The Pauli exclusion principle limits the number of electrons that can occupy each shell and subshell, while the relative energies of the available subshells determine the order in which they are filled. Because electrons interact with one another, this filling order is not determined solely by the principal quantum number and becomes increasingly complex for larger atoms.

Examining Table 113.2 reveals several important patterns in how electrons occupy atomic shells. As electrons are added to an atom, they fill the lowest available energy levels first. Thus, the \(n=1\) shell fills before electrons begin occupying the \(n=2\) shell, followed by higher shells as additional electrons are added. Within each shell, electrons generally fill the subshells in order of increasing energy, beginning with the \(s\) subshell, followed by the \(p\), \(d\), and \(f\) subshells.

There are, however, important exceptions to this simple pattern. For example, in potassium the \(4s\) subshell begins to fill before the \(3d\) subshell. A similar exception occurs for rubidium, where electrons enter the \(5s\) subshell before the \(4d\) subshell. These exceptions arise because \(s\) electrons have probability distributions that extend closer to the nucleus than electrons in higher-angular-momentum subshells. As a result, they experience a stronger attraction to the nucleus and often have slightly lower energies than might be expected from the principal quantum number alone.

Figure 113.5 shows the modern periodic table through element 118. The repeating arrangement of the elements reflects the systematic filling of atomic shells and subshells. Elements in the same column, or group, have similar outer-electron configurations and therefore tend to exhibit similar chemical properties.

The electrons in an atom's outermost shell, called valence electrons, largely determine its chemical behavior because they are the electrons that interact with neighboring atoms. Elements whose valence electrons can be easily gained, lost, or shared are generally more chemically reactive. As a result, each group of the periodic table is characterized by a distinctive outer-electron configuration.

One well-known example is Group 18, the noble gases, which includes helium, neon, and argon. These elements have completely filled outer shells or subshells, making their electron configurations especially stable. Consequently, noble gases have high ionization energies, rarely gain or lose electrons, and are among the least chemically reactive elements. Although they can form compounds under specialized laboratory conditions, they are largely chemically inert under ordinary conditions.

Group 17, the halogens, includes fluorine, chlorine, bromine, and iodine. These elements have outer electron configurations ending in \(p^5\), meaning they require only one additional electron to complete their outer \(p\) subshell. As a result, halogens readily gain an electron to form stable negative ions such as \(\text{Cl}^{-}\), making them highly reactive.

In contrast, the alkali metals in Group 1, including lithium, sodium, and potassium, have outer electron configurations ending in \(s^1\). Because this single valence electron is only weakly bound to the nucleus, alkali metals readily lose it, forming positive ions such as \(\text{Na}^{+}\). The ease with which this electron can move also explains why these elements are good electrical conductors.

Many other groups exhibit characteristic electron configurations that determine their chemical behavior. For example, carbon, silicon, and germanium belong to Group 14 and each has four valence electrons, allowing them to form four covalent bonds. Carbon is particularly important because this bonding flexibility enables the enormous variety of molecules found in living organisms.

The transition metals are distinguished by the filling of \(d\) subshells, while the lanthanides and actinides involve the filling of \(f\) subshells. Although these heavier elements exhibit more complex filling patterns, the overall organization of the periodic table is ultimately explained by the quantum-mechanical structure of atoms.