## CHEMISTRY THE CENTRAL SCIENCE

**6 ELECTRONIC STRUCTURE OF ATOMS**

**CHAPTER SUMMARY AND KEY TERMS**

**INTRODUCTION AND ****SECTION 6.1** The **electronic structure** of an atom describes the energies and arrangement of electrons around the atom. Much of what is known about the electronic structure of atoms was obtained by observing the interaction of light with matter. Visible light and other forms of **electromagnetic radiation** (also known as radiant energy) move through a vacuum at the speed of light, *c* = 3.00 × 10^{8} m/s. Electromagnetic radiation has both electric and magnetic components that vary periodically in wavelike fashion. The wave characteristics of radiant energy allow it to be described in terms of **wavelength**, **λ**, and **frequency**, *v*, which are interrelated: *c = λv.*

**SECTION 6.2** Planck proposed that the minimum amount of radiant energy that an object can gain or lose is related to the frequency of the radiation: *E = hv*. This smallest quantity is called a **quantum** of energy. The constant *h* is called **Planck's constant**: *h =* 6.626 × 10^{–34} J-s. In the quantum theory, energy is quantized, meaning that it can have only certain allowed values. Einstein used the quantum theory to explain the **photoelectric effect**, the emission of electrons from metal surfaces when exposed to light. He proposed that light behaves as if it consists of quantized energy packets called **photons**. Each photon carries energy, *E* = *hv.*

**SECTION 6.3** Dispersion of radiation into its component wavelengths produces a **spectrum**. If the spectrum contains all wavelengths, it is called a **continuous spectrum**; if it contains only certain specific wavelengths, the spectrum is called a **line spectrum**. The radiation emitted by excited hydrogen atoms forms a line spectrum.

Bohr proposed a model of the hydrogen atom that explains its line spectrum. In this model the energy of the electron in the hydrogen atom depends on the value of a quantum number, *n*. The value of *n* must be a positive integer (1, 2, 3,…), and each value of *n* corresponds to a different specific energy, *En*. The energy of the atom increases as *n* increases. The lowest energy is achieved for *n =* 1; this is called the **ground state** of the hydrogen atom. Other values of *n* correspond to **excited states**. Light is emitted when the electron drops from a higher-energy state to a lower-energy state; light is absorbed to excite the electron from a lower energy state to a higher one. The frequency of light emitted or absorbed is such that *hv* equals the difference in energy between two allowed states.

**SECTION 6.4** De Broglie proposed that matter, such as electrons, should exhibit wavelike properties. This hypothesis of **matter waves** was proved experimentally by observing the diffraction of electrons. An object has a characteristic wavelength that depends on its **momentum**, *mv:* **λ** =*h/mv*. Discovery of the wave properties of the electron led to Heisenberg's **uncertainty principle**, which states that there is an inherent limit to the accuracy with which the position and momentum of a particle can be measured simultaneously.

**SECTION 6.5** In the quantum mechanical model of the hydrogen atom, the behavior of the electron is described by mathematical functions called **wave functions**, denoted with the Greek letter ** Ψ**. Each allowed wave function has a precisely known energy, but the location of the electron cannot be determined exactly; rather, the probability of it being at a particular point in space is given by the

**probability density**,

*Ψ*. The

**electron density**distribution is a map of the probability of finding the electron at all points in space.

The allowed wave functions of the hydrogen atom are called **orbitals**. An orbital is described by a combination of an integer and a letter, corresponding to values of three quantum numbers. The *principal quantum number, n*, is indicated by the integers 1, 2, 3, …. This quantum number relates most directly to the size and energy of the orbital. The *angular momentum quantum number, I*, is indicated by the letters *s*, *p*, *d*, *f*, and so on, corresponding to the values of 0, 1, 2, 3,. … The / quantum number defines the shape of the orbital. For a given value of *n, I* can have integer values ranging from 0 to *(n* – 1). The *magnetic quantum number, mi*, relates to the orientation of the orbital in space. For a given value of /, *m*; can have integral values ranging from – / to /, including 0. Subscripts can be used to label the orientations of the orbitals. For example, the three 3*p*orbitals are designated *3px*, 3*py*, and 3*pz*, with the subscripts indicating the axis along which the orbital is oriented.

An **electron shell** is the set of all orbitals with the same value of *n*, such as 3*s*, 3*p*, and 3*d*. In the hydrogen atom all the orbitals in an electron shell have the same energy. A **subshell** is the set of one or more orbitals with the same *n* and / values; for example, 3s, 3*p*, and 3*d* are each subshells of the *n =* 3 shell. There is one orbital in an *s* subshell, three in a *p* subshell, five in a *d* subshell, and seven in an/subshell.

**SECTION 6.6** Contour representations are useful for visualizing the shapes of the orbitals. Represented this way, *s* orbitals appear as spheres that increase in size as *n* increases. The **radial probability function** tells us the probability that the electron will be found at a certain distance from the nucleus. The wave function for each *p* orbital has two lobes on opposite sides of the nucleus. They are oriented along the *x, y*, and *z* axes. Four of the *d* orbitals appear as shapes with four lobes around the nucleus; the fifth one, the *d*_{z}* ^{2}* orbital, is represented as two lobes along the

*z*axis and a “doughnut” in the

*xy*plane. Regions in which the wave function is zero are called

**nodes**. There is zero probability that the electron will be found at a node.

**SECTION 6.7** In many-electron atoms, different subshells of the same electron shell have different energies. For a given value of *n*, the energy of the subshells increases as the value of *l* increases: *ns*<*np*<*nd*<*nf*. Orbitals within the same subshell are **degenerate**, meaning they have the same energy.

Electrons have an intrinsic property called **electron spin**, which is quantized. The **spin magnetic quantum number**, *ms*, can have two possible values, , which can be envisioned as the two directions of an electron spinning about an axis. The **Pauli exclusion principle** states that no two electrons in an atom can have the same values for *n, l, m** _{l}*, and

*m*

*. This principle places a limit of two on the number of electrons that can occupy any one atomic orbital. These two electrons differ in their value of*

_{s}*m*

*.*

_{s}**SECTIONS 6.8 AND 6.9** The **electron configuration** of an atom describes how the electrons are distributed among the orbitals of the atom. The ground-state electron configurations are generally obtained by placing the electrons in the atomic orbitals of lowest possible energy with the restriction that each orbital can hold no more than two electrons. When electrons occupy a subshell with more than one degenerate orbital, such as the 2*p* subshell, **Hund's rule** states that the lowest energy is attained by maximizing the number of electrons with the same electron spin. For example, in the ground-state electron configuration of carbon, the two 2*p* electrons have the same spin and must occupy two different 2*p* orbitals.

Elements in any given group in the periodic table have the same type of electron arrangements in their outermost shells. For example, the electron configurations of the halogens fluorine and chlorine are [He]2*s*^{2}2*p*^{5} and [Ne]3*s*^{2}3*p*^{5}, respectively. The outer-shell electrons are those that lie outside the orbitals occupied in the next lowest noble-gas element. The outer-shell electrons that are involved in chemical bonding are the **valence electrons** of an atom; for the elements with atomic number 30 or less, all the outer-shell electrons are valence electrons. The electrons that are not valence electrons are called **core electrons**.

The periodic table is partitioned into different types of elements, based on their electron configurations. Those elements in which the outermost subshell is an *s* or *p* subshell are called the **representative** (or **main-group**) **elements**. The alkali metals (group 1A), halogens (group 7A), and noble gases (group 8A) are representative elements. Those elements in which a *d* subshell is being filled are called the **transition elements** (or **transition metals**). The elements in which the 4*f* subshell is being filled are called the **lanthanide** (or **rare earth**) **elements**. The **actinide elements**are those in which the 5*f* subshell is being filled. The lanthanide and actinide elements are collectively referred to as the ** f-block metals**. These elements are shown as two rows of 14 elements below the main part of the periodic table. The structure of the periodic table, summarized in Figure 6.30, allows us to write the electron configuration of an element from its position in the periodic table.

**KEY SKILLS**

• Calculate the wavelength of electromagnetic radiation given its frequency or its frequency given its wavelength. (Section 6.1)

• Order the common kinds of radiation in the electromagnetic spectrum according to their wavelengths or energy. (Section 6.1)

• Explain what photons are and be able to calculate their energies given either their frequency or wavelength. (Section 6.2)

• Explain how line spectra relate to the idea of quantized energy states of electrons in atoms. (Section 6.3)

• Calculate the wavelength of a moving object. (Section 6.4)

• Explain how the uncertainty principle limits how precisely we can specify the position and the momentum of subatomic particles such as electrons. (Section 6.4)

• Relate the quantum numbers to the number and type of orbitals and recognize the different orbital shapes. (Section 6.5)

• Interpret radial probability function graphs for the orbitals. (Section 6.6)

• Draw an energy-level diagram for the orbitals in a many-electron atom and describe how electrons populate the orbitals in the ground state of an atom, using the Pauli exclusion principle and Hund's rule. (Section 6.8)

• Use the periodic table to write condensed electron configurations and determine the number of unpaired electrons in an atom. (Section 6.9)

**KEY EQUATIONS**

•

light as a wave: *c* = speed of light (3.00 × 10^{8} m/s) λ = wavelength in meters, *v* = frequency in s^{−1}

•

light as a particle (photon): *E* = energy of photon in joules, *h* = Planck's constant (6.626 × 10^{−34} J-s), *v* = frequency in s^{−1} (same frequency as previous formula)

•

matter as a wave: λ = wavelength, h = Planck's constant, *m* = mass of object in kg, *v* = speed of object in m/s

•

Heisenberg's uncertainty principle. The uncertainty in position (Δ*x*) and momentum [Δ(*mv*)] of an object cannot be zero; the smallest value of their product is h/4π