## CHEMISTRY THE CENTRAL SCIENCE

**6 ELECTRONIC STRUCTURE OF ATOMS**

**6.5 QUANTUM MECHANICS AND ATOMIC ORBITALS**

In 1926 the Austrian physicist Erwin Schrödinger (1887-1961) proposed an equation, now known as Schrödinger's wave equation, that incorporates both the wave-like behavior of the electron and its particle-like behavior. His work opened a new approach to dealing with subatomic particles, an approach known as *quantum mechanics* or *wave mechanics*. The application of Schrödinger's equation requires advanced calculus, and so we will not be concerned with its details. We will, however, qualitatively consider the results Schrödinger obtained because they give us a powerful new way to view electronic structure. Let's begin by examining the electronic structure of the simplest atom, hydrogen.

Schrödinger treated the electron in a hydrogen atom like the wave on a plucked guitar string (**FIGURE 6.15**). Because such waves do not travel in space, they are called *standing waves*. Just as the plucked guitar string produces a standing wave that has a fundamental frequency and higher overtones (harmonics), the electron exhibits a lowest-energy standing wave and higher-energy ones. Furthermore, just as the overtones of the guitar string have *nodes*, points where the amplitude of the wave is zero, so do the waves characteristic of the electron.

Solving Schrödinger's equation for the hydrogen atom leads to a series of mathematical functions called **wave functions** that describe the electron in an atom. These wave functions are usually represented by the symbol *ψ* (lowercase Greek letter *psi)*. Although the wave function has no direct physical meaning, the square of the wave function, *ψ*^{2}, provides information about the electron's location when it is in an allowed energy state.

For the hydrogen atom, the allowed energies are the same as those predicted by the Bohr model. However, the Bohr model assumes that the electron is in a circular orbit of some particular radius about the nucleus. In the quantum mechanical model, the electron's location cannot be described so simply.

According to the uncertainty principle, if we know the momentum of the electron with high accuracy, our simultaneous knowledge of its location is very uncertain. Thus, we cannot hope to specify the exact location of an individual electron around the nucleus. Rather, we must be content with a kind of statistical knowledge. We therefore speak of the *probability* that the electron will be in a certain region of space at a given instant. As it turns out, the square of the wave function, *ψ*^{2}, at a given point in space represents the probability that the electron will be found at that location. For this reason, *ψ*^{2} is called either the **probability density** or the **electron density**.

**FIGURE 6.15 Standing waves in a vibrating string.**

One way of representing the probability of finding the electron in various regions of an atom is shown in **FIGURE 6.16**, where the density of the dots represents the probability of finding the electron. The regions with a high density of dots correspond to relatively large values for *ψ*^{2}and are therefore regions where there is a high probability of finding the electron. Based on this representation, we often describe atoms as consisting of a nucleus surrounded by an electron cloud.

**GO FIGURE**

**Where in the figure is the region of highest electron density?**

**FIGURE 6.16 Electron-density distribution.** This rendering represents the probability, *ψ*^{2}, of finding the electron in a hydrogen atom in its ground state. The origin of the coordinate system is at the nucleus.

**Orbitals and Quantum Numbers**

The solution to Schrödinger's equation for the hydrogen atom yields a set of wave functions called **orbitals**. Each orbital has a characteristic shape and energy. For example, the lowest-energy orbital in the hydrogen atom has the spherical shape illustrated in Figure 6.16 and an energy of −2.18 × 10^{–18} J. Note that an *orbital* (quantum mechanical model, which describes electrons in terms of probabilities, visualized as “electron clouds”) is not the same as an *orbit* (Bohr model, which visualizes the electron moving in a physical orbit, like a planet around a star). The quantum mechanical model does not refer to orbits because the motion of the electron in an atom cannot be precisely determined (Heisenberg uncertainty principle).

The Bohr model introduced a single quantum number, *n*, to describe an orbit. The quantum mechanical model uses three quantum numbers, *n, l*, and *ml*, which result naturally from the mathematics used, to describe an orbital.

1. The *principal quantum number, n*, can have positive integral values 1, 2, 3, …. As *n* increases, the orbital becomes larger, and the electron spends more time farther from the nucleus. An increase in *n* also means that the electron has a higher energy and is therefore less tightly bound to the nucleus. For the hydrogen atom, *E** _{n}* = –(2.18 × 10

^{–18}J)(1/

*n*

^{2}), as in the Bohr model.

2. The second quantum number—the *angular momentum quantum number, l*—can have integral values from 0 to (*n* – 1) for each value of *n*. This quantum number defines the shape of the orbital. The value of *l* for a particular orbital is generally designated by the letters *s, p, d*, and *f*, *corresponding to *l* values of 0, 1, 2, and 3:

3. The *magnetic quantum number, ml*, can have integral values between –*l* and *l*, including zero. This quantum number describes the orientation of the orbital in space, as we discuss in Section 6.6.

Notice that because the value of *n* can be any positive integer, an infinite number of orbitals for the hydrogen atom are possible. At any given instant, however, the electron in a hydrogen atom is described by only one of these orbitals—we say that the electron *occupies* a certain orbital. The remaining orbitals are *unoccupied* for that particular state of the hydrogen atom.

**GIVE IT SOME THOUGHT**

What is the difference between an *orbit* in the Bohr model of the hydrogen atom and an *orbital* in the quantum mechanical model?

The collection of orbitals with the same value of *n* is called an **electron shell**. All the orbitals that have *n =* 3, for example, are said to be in the third shell. The set of orbitals that have the same *n* and *l* values is called a **subshell**. Each subshell is designated by a number (the value of *n)*and a letter (*s, p, d, or f*, corresponding to the value of *l*). For example, the orbitals that have *n =* 3 and *l* = 2 are called 3*d* orbitals and are in the 3*d* subshell.

**TABLE 6.2** • Relationship among Values of *n, l*, and *m** _{l}* through

*n =*4

**TABLE 6.2** summarizes the possible values of *l* and *ml* for values of *n* through *n =* 4. The restrictions on possible values give rise to the following very important observations:

1. The shell with principal quantum number *n* consists of exactly *n* subshells. Each subshell corresponds to a different allowed value of *l* from 0 to (*n* **—** 1). Thus, the first shell (*n* = 1) consists of only one subshell, the 1*s* (*l* = 0); the second shell (*n* = 2) consists of two subshells, the 2*s* (*l*= 0) and 2*p* (*l* = 1); the third shell consists of three subshells, 3*s, 3p*, and 3*d*, and so forth.

2. Each subshell consists of a specific number of orbitals. Each orbital corresponds to a different allowed value of *ml*. For a given value of *l*, there are *(2l* + 1) allowed values of *ml*, ranging from –*l* to +*l*. Thus, each *s* (*l* = 0) subshell consists of one orbital; each *p* (*l* = 1) sub-shell consists of three orbitals; each *d* (*l* = 2) subshell consists of five orbitals, and so forth.

3. The total number of orbitals in a shell is *n*^{2}, where *n* is the principal quantum number of the shell. The resulting number of orbitals for the shells—1, 4, 9, 16—are related to a pattern seen in the periodic table: We see that the number of elements in the rows of the periodic table—2, 8, 18, and 32—equals twice these numbers. We will discuss this relationship further in Section 6.9.

**FIGURE 6.17** shows the relative energies of the hydrogen atom orbitals through *n =* 3. Each box represents an orbital, and orbitals of the same subshell, such as the three *2p* orbitals, are grouped together. When the electron occupies the lowest-energy orbital (1*s*), the hydrogen atom is said to be in its *ground state*. When the electron occupies any other orbital, the atom is in an *excited state*. (The electron can be excited to a higher-energy orbital by absorption of a photon of appropriate energy.) At ordinary temperatures, essentially all hydrogen atoms are in the ground state.

**GIVE IT SOME THOUGHT**

Notice in Figure 6.17 that the energy difference between the *n =* 1 and *n =* 2 levels is much greater than the energy difference between the *n =* 2 and *n =* 3 levels. How does Equation 6.5 explain this trend?

**GO FIGURE**

**If the fourth shell (the n**

*=*

**4 energy level) were shown, how many sub-shells would it contain? How would they be labeled?**

**FIGURE 6.17 Energy levels in the hydrogen atom.**

**SAMPLE EXERCISE 6.6 Subshells of the Hydrogen Atom**

**(a)** Without referring to Table 6.2, predict the number of subshells in the fourth shell, that is, for *n = 4*. **(b)** Give the label for each of these subshells. **(c)** How many orbitals are in each of these subshells?

**Analyze and Plan** We are given the value of the principal quantum number, *n*. We need to determine the allowed values of *l* and *m** _{l}* for this given value of

*n*and then count the number of orbitals in each subshell.

**SOLUTION**

There are four subshells in the fourth shell, corresponding to the four possible values of *l* (0, 1, 2, and 3).

These subshells are labeled 4*s, 4p, 4d*, and *4f*. The number given in the designation of a subshell is the principal quantum number, *n;* the letter designates the value of the angular momentum quantum number, *l*: for *l* = 0, *s*; for *l* = 1, *p;* for *l* = 2, *d;* for *l* = 3,*f*.

There is one 4*s* orbital (when *l* = 0, there is only one possible value of *ml* : 0). There are three *4p* orbitals (when *l* = 1, there are three possible values of *ml* : 1, 0, **–** 1). There are five *4d* orbitals (when *l* = 2, there are five allowed values of *m** _{l}* : 2, 1, 0, –1, –2). There are seven

*4f*orbitals (when

*l*= 3, there are seven permitted values of

*m*

*: 3, 2, 1, 0, –1, –2, –3).*

_{l}**PRACTICE EXERCISE**

**(a)** What is the designation for the subshell with *n* = 5 and *l* = 1? **(b)** How many orbitals are in this subshell? **(c)** Indicate the values of *m** _{l}* for each of these orbitals.

*Answers:*** (a)** *5p;* **(b)** 3; **(c)** 1, 0, **–**1