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Chapter 7 Quantum Theory of the Atom

Copyright © Houghton Mifflin Company. All rights reserved. What are the electrons doing in the atom? Why do atoms form ions and molecules? Why do hydrogen and oxygen “stick” together to form water? To understand these questions, we need to understand the electronic structure of the atom Electronic structure refers to the way the electrons are arranged in an atom

Contents and Concepts Light Waves, Photons, and the Bohr Theory To understand the formation of chemical bonds, you need to know something about the electronic structure of atoms. Because light gives us information about this structure, we begin by discussing the nature of light. Then we look at the Bohr theory of the simplest atom, hydrogen. The Wave Nature of Light Quantum Effects and Photons The Bohr Theory of the Hydrogen Atom

Quantum Mechanics and Quantum Numbers The Bohr theory firmly establishes the concept of energy levels but fails to account for the details of atomic structure. Here we discuss some basic notions of quantum mechanics, which is the theory currently applied to extremely small particles, such as electrons in atoms. Quantum Mechanics Quantum Numbers and Atomic Orbitals

A wave is a continuously repeating change or oscillation in matter or in a physical field. Light is an electromagnetic wave, consisting of oscillations in electric and magnetic fields traveling through space.

A wave can be characterized by its wavelength and frequency. Wavelength, symbolized by the Greek letter lambda,, is the distance between any two identical points on adjacent waves.

Frequency, symbolized by the Greek letter nu,, is the number of wavelengths that pass a fixed point in one unit of time (usually a second). The unit is 1 / S or s -1, which is also called the Hertz (Hz).

Wavelength and frequency are related by the wave speed, which for light is c, the speed of light, 3.00 x 10 8 m/s. c = The relationship between wavelength and frequency due to the constant velocity of light is illustrated on the next slide.

When the wavelength is reduced by a factor of two, the frequency increases by a factor of two.

What is the wavelength of blue light with a frequency of 6.4 × 10 14 /s? = 6.4 × 10 14 /s c = 3.00 × 10 8 m/s c = so = c/ = 4.7 × 10 -7 m

What is the frequency of light having a wavelength of 681 nm? = 681 nm = 6.81 × 10 -7 m c = 3.00 × 10 8 m/s c = so = c/ = 4.41 × 10 14 /s

The range of frequencies and wavelengths of electromagnetic radiation is called the electromagnetic spectrum.

When frequency is doubled, wavelength is halved. The light would be in the blue-violet region.

One property of waves is that they can be diffracted—that is, they spread out when they encounter an obstacle about the size of the wavelength. In 1801, Thomas Young, a British physicist, showed that light could be diffracted. By the early 1900s, the wave theory of light was well established.

The wave theory could not explain the photoelectric effect, however.

The photoelectric effect is the ejection of an electron from the surface of a metal or other material when light shines on it.

Einstein proposed that light consists of quanta or particles of electromagnetic energy, called photons. The energy of each photon is proportional to its frequency: E = h h = 6.626 × 10 -34 J  s (Planck’s constant)

Einstein used this understanding of light to explain the photoelectric effect in 1905. Each electron is struck by a single photon. Only when that photon has enough energy will the electron be ejected from the atom; that photon is said to be absorbed.

Light, therefore, has properties of both waves and matter. Neither understanding is sufficient alone. This is called the particle–wave duality of light.

The blue–green line of the hydrogen atom spectrum has a wavelength of 486 nm. What is the energy of a photon of this light? = 4.86 nm = 4.86 × 10 -7 m c = 3.00 × 10 8 m/s h = 6.63 × 10 -34 J  s E = h and c = so E = hc/ = 4.09 × 10 -19 J

In the early 1900s, the atom was understood to consist of a positive nucleus around which electrons move (Rutherford’s model). This explanation left a theoretical dilemma: According to the physics of the time, an electrically charged particle circling a center would continually lose energy as electromagnetic radiation. But this is not the case—atoms are stable.

In addition, this understanding could not explain the observation of line spectra of atoms. A continuous spectrum contains all wavelengths of light. A line spectrum shows only certain colors or specific wavelengths of light. When atoms are heated, they emit light. This process produces a line spectrum that is specific to that atom. The emission spectra of six elements are shown on the next slide.

In 1913, Neils Bohr, a Danish scientist, set down postulates to account for 1. The stability of the hydrogen atom 2. The line spectrum of the atom

Energy-Level Postulate An electron can have only certain energy values, called energy levels. Energy levels are quantized. For an electron in a hydrogen atom, the energy is given by the following equation: R H = 2.179 x 10 -18 J n = principal quantum number

Transitions Between Energy Levels An electron can change energy levels by absorbing energy to move to a higher energy level or by emitting energy to move to a lower energy level.

For a hydrogen electron the energy change is given by R H = 2.179 × 10 -18 J, Rydberg constant

The energy of the emitted or absorbed photon is related to  E: We can now combine these two equations:

Light is absorbed by an atom when the electron transition is from lower n to higher n (n f > n i ). In this case,  E will be positive. Light is emitted from an atom when the electron transition is from higher n to lower n (n f < n i ). In this case,  E will be negative. An electron is ejected when n f = ∞.

Energy-level diagram for the hydrogen atom.

Electron transitions for an electron in the hydrogen atom.

What is the wavelength of the light emitted when the electron in a hydrogen atom undergoes a transition from n = 6 to n = 3? n i = 6 n f = 3 R H = 2.179 × 10 -18 J = -1.816 x 10 -19 J 1.094 × 10 -6 m

The red line corresponds to the smaller energy difference in going from n = 3 to n = 2. The blue line corresponds to the larger energy difference in going from n = 2 to n = 1. n = 1 n = 2 n = 3 A minimum of three energy levels are required.

Planck Vibrating atoms have only certain energies: E = h or 2h or 3h Einstein Energy is quantized in particles called photons: E = h Bohr Electrons in atoms can have only certain values of energy. For hydrogen: