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<div class="chapter" id="averill_1.0-ch06" version="5.0" lang="en">
<h1 class="title editable block">
<span class="title-prefix">Chapter 6</span> The Structure of Atoms</h1>
<p class="para editable block" id="averill_1.0-ch06_p01"><a class="xref" href="averill_1.0-ch01#averill_1.0-ch01">Chapter 1 "Introduction to Chemistry"</a> through <a class="xref" href="averill_1.0-ch05#averill_1.0-ch05">Chapter 5 "Energy Changes in Chemical Reactions"</a> introduced you to a wide variety of chemical substances and some of the most fundamental concepts in chemistry, including general descriptions of chemical bonding, mass relationships in chemical equations, and the energy changes associated with chemical reactions. You learned that the atoms of each element contain a unique number of positively charged protons in the nucleus and that a neutral atom has the same number of electrons as protons. You also learned that protons and neutrons constitute most of the mass of the atom and that electrons occupy most of the volume of an atom. These facts do not, however, explain the stoichiometries and the structures of chemical compounds; a deeper understanding of the electronic structure of atoms is required.</p>
<div class="informalfigure large medium-height block">
<img src="section_10/ca5fe25e5cb9f31033ecccc811fbebd4.jpg">
<p class="para"><strong class="emphasis bold">The noble gases.</strong> Helium, neon, argon, krypton, and xenon, all monatomic elements, are five of the six known noble gases. When atoms of these gases are excited by the flow of electrons in the discharge tubes, they emit photons of visible light in their characteristic spectral colors. All noble gases have the maximum number of electrons possible in their outer shell (2 for helium, 8 for all others), so they do not form chemical compounds easily.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_p02">In this chapter, we describe how electrons are arranged in atoms and how the spatial arrangements of electrons are related to their energies. We also explain how knowing the arrangement of electrons in an atom enables chemists to predict and explain the chemistry of an element. As you study the material presented in this chapter, you will discover how the shape of the periodic table reflects the electronic arrangements of elements. In this and subsequent chapters, we build on this information to explain why certain chemical changes occur and others do not.</p>
<p class="para editable block" id="averill_1.0-ch06_p03">After reading this chapter, you will know enough about the theory of the electronic structure of atoms to explain what causes the characteristic colors of neon signs, how laser beams are created, and why gemstones and fireworks have such brilliant colors. In later chapters, we will develop the concepts introduced here to explain why the only compound formed by sodium and chlorine is NaCl, an ionic compound, whereas neon and argon do not form any stable compounds, and why carbon and hydrogen combine to form an almost endless array of covalent compounds, such as CH<sub class="subscript">4</sub>, C<sub class="subscript">2</sub>H<sub class="subscript">2</sub>, C<sub class="subscript">2</sub>H<sub class="subscript">4</sub>, and C<sub class="subscript">2</sub>H<sub class="subscript">6</sub>. You will discover that knowing how to use the periodic table is the single most important skill you can acquire to understand the incredible chemical diversity of the elements.</p>
</div>
<div class="section" id="averill_1.0-ch06_s01" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.1</span> Waves and Electromagnetic Radiation</h2>
<div class="learning_objectives editable block" id="averill_1.0-ch06_s01_n01">
<h3 class="title">Learning Objective</h3>
<ol class="orderedlist" id="averill_1.0-ch06_s01_l01">
<li>To know the characteristics of electromagnetic energy.</li>
</ol>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_p01">Scientists discovered much of what we know about the structure of the atom by observing the interaction of atoms with various forms of radiant, or transmitted, energy, such as the energy associated with the visible light we detect with our eyes, the infrared radiation we feel as heat, the ultraviolet light that causes sunburn, and the x-rays that produce images of our teeth or bones. All these forms of radiant energy should be familiar to you. We begin our discussion of the development of our current atomic model by describing the properties of waves and the various forms of electromagnetic radiation.</p>
<div class="figure small editable block" id="averill_1.0-ch06_s01_f01">
<p class="title"><span class="title-prefix">Figure 6.1</span> A Wave in Water</p>
<img src="section_10/1fd32a5a7af89ce657a1129e583d47be.jpg">
<p class="para">When a drop of water falls onto a smooth water surface, it generates a set of waves that travel outward in a circular direction.</p>
</div>
<div class="section" id="averill_1.0-ch06_s01_s01">
<h2 class="title editable block">Properties of Waves</h2>
<p class="para editable block" id="averill_1.0-ch06_s01_s01_p01">A <span class="margin_term"><a class="glossterm">wave</a><span class="glossdef">A periodic oscillation that transmits energy through space.</span></span> is a periodic oscillation that transmits energy through space. Anyone who has visited a beach or dropped a stone into a puddle has observed waves traveling through water (<a class="xref" href="#averill_1.0-ch06_s01_f01">Figure 6.1 "A Wave in Water"</a>). These waves are produced when wind, a stone, or some other disturbance, such as a passing boat, transfers energy to the water, causing the surface to oscillate up and down as the energy travels outward from its point of origin. As a wave passes a particular point on the surface of the water, anything floating there moves up and down.</p>
<div class="figure large editable block" id="averill_1.0-ch06_s01_s01_f01">
<p class="title"><span class="title-prefix">Figure 6.2</span> Important Properties of Waves</p>
<img src="section_10/2095c8810bc9e0034ea09f6fab94a1e3.jpg">
<p class="para">(a) Wavelength (λ), frequency (ν, labeled in Hz), and amplitude are indicated on this drawing of a wave. (b) The wave with the shortest wavelength has the greatest number of wavelengths per unit time (i.e., the highest frequency). If two waves have the same frequency and speed, the one with the greater amplitude has the higher energy.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s01_p02">Waves have characteristic properties (<a class="xref" href="#averill_1.0-ch06_s01_s01_f01">Figure 6.2 "Important Properties of Waves"</a>). As you may have noticed in <a class="xref" href="#averill_1.0-ch06_s01_f01">Figure 6.1 "A Wave in Water"</a>, waves are <span class="margin_term"><a class="glossterm">periodic</a><span class="glossdef">Phenomena, such as waves, that repeat regularly in both space and time.</span></span>; that is, they repeat regularly in both space and time. The distance between two corresponding points in a wave—between the midpoints of two peaks, for example, or two troughs—is the <span class="margin_term"><a class="glossterm">wavelength (λ)</a><span class="glossdef">The distance between two corresponding points in a wave—between the midpoints of two peaks or two troughs.</span></span>.<span class="footnote" id="averill_1.0-fn06_001">λ is the lowercase Greek lambda, and ν is the lowercase Greek nu.</span> Wavelengths are described by a unit of distance, typically meters. The <span class="margin_term"><a class="glossterm">frequency (ν)</a><span class="glossdef">The number of oscillations (i.e., of a wave) that pass a particular point in a given period of time.</span></span> of a wave is the number of oscillations that pass a particular point in a given period of time. The usual units are oscillations per second (1/s = s<sup class="superscript">−1</sup>), which in the SI system is called the hertz (Hz).<span class="footnote" id="averill_1.0-fn06_002">Named after German physicist Heinrich Hertz (1857–1894), a pioneer in the field of electromagnetic radiation.</span> The <span class="margin_term"><a class="glossterm">amplitude</a><span class="glossdef">The vertical height of a wave, which is defined as half the peak-to-trough height.</span></span>, or vertical height, of a wave is defined as half the peak-to-trough height; as the amplitude of a wave with a given frequency increases, so does its energy. As you can see in <a class="xref" href="#averill_1.0-ch06_s01_s01_f01">Figure 6.2 "Important Properties of Waves"</a>, two waves can have the same amplitude but different wavelengths and vice versa. The distance traveled by a wave per unit time is its <span class="margin_term"><a class="glossterm">speed (<em class="emphasis">v</em>)</a><span class="glossdef">The distance traveled by a wave per unit time.</span></span>, which is typically measured in meters per second (m/s). The speed of a wave is equal to the product of its wavelength and frequency:</p>
<div class="equation block" id="averill_1.0-ch06_s01_s01_eq01">
<p class="title editable"><span class="title-prefix">Equation 6.1</span> </p>
<math xml:id="averill_1.0-ch06_m001" display="block">
<semantics>
<mtable>
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<mtext>wavelength</mtext>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mtext>frequency</mtext>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mtext> speed</mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>λ</mi>
<mi>ν</mi>
<mo>=</mo>
<mi>v</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mtext>meters</mtext>
</mrow>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>wave</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>wave</mtext>
</mrow>
</menclose>
</mrow>
<mrow>
<mtext>second</mtext>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mtext>meters</mtext>
</mrow>
<mrow>
<mtext>second</mtext>
</mrow>
</mfrac>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s01_p03">Water waves are slow compared to sound waves, which can travel through solids, liquids, and gases. Whereas water waves may travel a few meters per second, the speed of sound in dry air at 20°C is 343.5 m/s. Ultrasonic waves, which travel at an even higher speed (>1500 m/s) and have a greater frequency, are used in such diverse applications as locating underwater objects and the medical imaging of internal organs.</p>
</div>
<div class="section" id="averill_1.0-ch06_s01_s02">
<h2 class="title editable block">Electromagnetic Radiation</h2>
<p class="para editable block" id="averill_1.0-ch06_s01_s02_p01">Water waves transmit energy through space by the periodic oscillation of matter (the water). In contrast, energy that is transmitted, or radiated, through space in the form of periodic oscillations of electric and magnetic fields is known as <span class="margin_term"><a class="glossterm">electromagnetic radiation</a><span class="glossdef">Energy that is transmitted, or radiated, through space in the form of periodic oscillations of electric and magnetic fields.</span></span> (<a class="xref" href="#averill_1.0-ch06_s01_s02_f01">Figure 6.3 "The Nature of Electromagnetic Radiation"</a>). Some forms of electromagnetic radiation are shown in <a class="xref" href="#averill_1.0-ch06_s01_s02_f02">Figure 6.4 "The Electromagnetic Spectrum"</a>. In a vacuum, all forms of electromagnetic radiation—whether microwaves, visible light, or gamma rays—travel at the <span class="margin_term"><a class="glossterm">speed of light (<em class="emphasis">c</em>)</a><span class="glossdef">The speed with which all forms of electromagnetic radiation travel in a vacuum.</span></span>, a fundamental physical constant with a value of 2.99792458 × 10<sup class="superscript">8</sup> m/s (which is about 3.00 ×10<sup class="superscript">8</sup> m/s or 1.86 × 10<sup class="superscript">5</sup> mi/s). This is about a million times faster than the speed of sound.</p>
<div class="figure medium editable block" id="averill_1.0-ch06_s01_s02_f01">
<p class="title"><span class="title-prefix">Figure 6.3</span> The Nature of Electromagnetic Radiation</p>
<img src="section_10/40e36bb375b3849544da98d03a00f649.jpg">
<p class="para">All forms of electromagnetic radiation consist of perpendicular oscillating electric and magnetic fields.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s02_p02">Because the various kinds of electromagnetic radiation all have the same speed (<em class="emphasis">c</em>), they differ in only wavelength and frequency. As shown in <a class="xref" href="#averill_1.0-ch06_s01_s02_f02">Figure 6.4 "The Electromagnetic Spectrum"</a> and <a class="xref" href="#averill_1.0-ch06_s01_s02_t01">Table 6.1 "Common Wavelength Units for Electromagnetic Radiation"</a>, the wavelengths of familiar electromagnetic radiation range from 10<sup class="superscript">1</sup> m for radio waves to 10<sup class="superscript">−12</sup> m for gamma rays, which are emitted by nuclear reactions. By replacing <em class="emphasis">v</em> with <em class="emphasis">c</em> in <a class="xref" href="#averill_1.0-ch06_s01_s01_eq01" xrefstyle="select:label">Equation 6.1</a>, we can show that the frequency of electromagnetic radiation is inversely proportional to its wavelength:</p>
<div class="equation block" id="averill_1.0-ch06_s01_s02_eq01">
<p class="title editable"><span class="title-prefix">Equation 6.2</span> </p>
<math xml:id="averill_1.0-ch06_m002" display="block">
<semantics>
<mtable columnalign="left">
<mtr>
<mtd>
<mi>c</mi>
<mo>=</mo>
<mi>λ</mi>
<mi>ν</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>ν</mi>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mi>λ</mi>
</mfrac>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s02_p03">For example, the frequency of radio waves is about 10<sup class="superscript">8</sup> Hz, whereas the frequency of gamma rays is about 10<sup class="superscript">20</sup> Hz. Visible light, which is electromagnetic radiation that can be detected by the human eye, has wavelengths between about 7 × 10<sup class="superscript">−7</sup> m (700 nm, or 4.3 × 10<sup class="superscript">14</sup> Hz) and 4 × 10<sup class="superscript">−7</sup> m (400 nm, or 7.5 × 10<sup class="superscript">14</sup> Hz). Within this range, the eye perceives radiation of different wavelengths (or frequencies) as light of different colors, ranging from red to violet in order of decreasing wavelength. The components of white light—a mixture of all the frequencies of visible light—can be separated by a prism, as shown in part (b) in <a class="xref" href="#averill_1.0-ch06_s01_s02_f02">Figure 6.4 "The Electromagnetic Spectrum"</a>. A similar phenomenon creates a rainbow, where water droplets suspended in the air act as tiny prisms.</p>
<div class="figure large editable block" id="averill_1.0-ch06_s01_s02_f02">
<p class="title"><span class="title-prefix">Figure 6.4</span> The Electromagnetic Spectrum</p>
<img src="section_10/8d4e0e72466bee4ef6a8faaad3f9f7d3.jpg">
<p class="para">(a) This diagram shows the wavelength and frequency ranges of electromagnetic radiation. The visible portion of the electromagnetic spectrum is the narrow region with wavelengths between about 400 and 700 nm. (b) When white light is passed through a prism, it is split into light of different wavelengths, whose colors correspond to the visible spectrum.</p>
</div>
<div class="table block" id="averill_1.0-ch06_s01_s02_t01" frame="all">
<p class="title"><span class="title-prefix">Table 6.1</span> Common Wavelength Units for Electromagnetic Radiation</p>
<table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center">Unit</th>
<th align="center">Symbol</th>
<th align="center">Wavelength (m)</th>
<th align="center">Type of Radiation</th>
</tr>
</thead>
<tbody>
<tr>
<td>picometer</td>
<td>pm</td>
<td>10<sup class="superscript">−12</sup>
</td>
<td>gamma ray</td>
</tr>
<tr>
<td>angstrom</td>
<td>Å</td>
<td>10<sup class="superscript">−10</sup>
</td>
<td>x-ray</td>
</tr>
<tr>
<td>nanometer</td>
<td>nm</td>
<td>10<sup class="superscript">−9</sup>
</td>
<td>x-ray</td>
</tr>
<tr>
<td>micrometer</td>
<td>μm</td>
<td>10<sup class="superscript">−6</sup>
</td>
<td>infrared</td>
</tr>
<tr>
<td>millimeter</td>
<td>mm</td>
<td>10<sup class="superscript">−3</sup>
</td>
<td>infrared</td>
</tr>
<tr>
<td>centimeter</td>
<td>cm</td>
<td>10<sup class="superscript">−2</sup>
</td>
<td>microwave</td>
</tr>
<tr>
<td>meter</td>
<td>m</td>
<td>10<sup class="superscript">0</sup>
</td>
<td>radio</td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s02_p04">As you will soon see, the energy of electromagnetic radiation is directly proportional to its frequency and inversely proportional to its wavelength:</p>
<div class="equation block" id="averill_1.0-ch06_s01_s02_eq02">
<p class="title editable"><span class="title-prefix">Equation 6.3</span> </p>
<math xml:id="averill_1.0-ch06_m003" display="block">
<semantics>
<mrow>
<mi>E</mi>
<mo>∝</mo>
<mi>ν</mi>
</mrow>
</semantics>
</math>
</div>
<div class="equation block" id="averill_1.0-ch06_s01_s02_eq03">
<p class="title editable"><span class="title-prefix">Equation 6.4</span> </p>
<math xml:id="averill_1.0-ch06_m004" display="block">
<semantics>
<mrow>
<mi>E</mi>
<mo>∝</mo>
<mfrac>
<mn>1</mn>
<mi>λ</mi>
</mfrac>
</mrow>
</semantics>
</math>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s02_p05">Whereas visible light is essentially harmless to our skin, ultraviolet light, with wavelengths of ≤ 400 nm, has enough energy to cause severe damage to our skin in the form of sunburn. Because the ozone layer described in <a class="xref" href="averill_1.0-ch03#averill_1.0-ch03">Chapter 3 "Chemical Reactions"</a> absorbs sunlight with wavelengths less than 350 nm, it protects us from the damaging effects of highly energetic ultraviolet radiation.</p>
<div class="callout editable block" id="averill_1.0-ch06_s01_s02_n01">
<h3 class="title">Note the Pattern</h3>
<p class="para" id="averill_1.0-ch06_s01_s02_p06">The energy of electromagnetic radiation increases with increasing frequency and decreasing wavelength.</p>
</div>
<div class="exercises block" id="averill_1.0-ch06_s01_s02_n02">
<h3 class="title">Example 1</h3>
<p class="para" id="averill_1.0-ch06_s01_s02_p07">Your favorite FM radio station, WXYZ, broadcasts at a frequency of 101.1 MHz. What is the wavelength of this radiation?</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p08"><strong class="emphasis bold">Given: </strong>frequency</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p09"><strong class="emphasis bold">Asked for: </strong>wavelength</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p10">
<strong class="emphasis bold">Strategy:</strong>
</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p11">Substitute the value for the speed of light in meters per second into <a class="xref" href="#averill_1.0-ch06_s01_s02_eq01" xrefstyle="select:label">Equation 6.2</a> to calculate the wavelength in meters.</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p12">
<strong class="emphasis bold">Solution:</strong>
</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p13">From <a class="xref" href="#averill_1.0-ch06_s01_s02_eq01" xrefstyle="select:label">Equation 6.2</a>, we know that the product of the wavelength and the frequency is the speed of the wave, which for electromagnetic radiation is 2.998 × 10<sup class="superscript">8</sup> m/s:</p>
<span class="informalequation">
<span class="mathphrase">λν = <em class="emphasis">c</em> = 2.998 × 10<sup class="superscript">8</sup> m/s</span>
</span>
<p class="para" id="averill_1.0-ch06_s01_s02_p14">Thus the wavelength λ is given by</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch06_m005" display="block">
<semantics>
<mrow>
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mi>c</mi>
<mi>ν</mi>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2.998</mn>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>8</mn>
</msup>
<mtext> m/</mtext>
<menclose notation="updiagonalstrike">
<mtext>s</mtext>
</menclose>
</mrow>
<mrow>
<mn>101.1</mn>
<mtext> </mtext>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>MHz</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mtext> </mtext>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>MHz</mtext>
</mrow>
</menclose>
</mrow>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>6</mn>
</msup>
<mtext> </mtext>
<msup>
<mrow>
<menclose notation="updiagonalstrike">
<mtext>s</mtext>
</menclose>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2.965</mn>
<mtext> m</mtext>
</mrow>
</semantics>
</math>
</span>
<p class="simpara">Exercise</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p15">As the police officer was writing up your speeding ticket, she mentioned that she was using a state-of-the-art radar gun operating at 35.5 GHz. What is the wavelength of the radiation emitted by the radar gun?</p>
<p class="para" id="averill_1.0-ch06_s01_s02_p16"><strong class="emphasis bold">Answer: </strong>8.45 mm</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s01_s02_p17">In <a class="xref" href="averill_1.0-ch06_s02#averill_1.0-ch06_s02">Section 6.2 "The Quantization of Energy"</a> and <a class="xref" href="averill_1.0-ch06_s03#averill_1.0-ch06_s03">Section 6.3 "Atomic Spectra and Models of the Atom"</a>, we describe how scientists developed our current understanding of the structure of atoms using the scientific method described in <a class="xref" href="averill_1.0-ch01#averill_1.0-ch01">Chapter 1 "Introduction to Chemistry"</a>. You will discover why scientists had to rethink their classical understanding of the nature of electromagnetic energy, which clearly distinguished between the particulate behavior of matter and the wavelike nature of energy.</p>
<div class="key_takeaways editable block" id="averill_1.0-ch06_s01_s02_n03">
<h3 class="title">Key Equations</h3>
<p class="para">
<strong class="emphasis bold">relationship between wavelength, frequency, and speed of a wave</strong>
</p>
<p class="para"><a class="xref" href="#averill_1.0-ch06_s01_s01_eq01" xrefstyle="select:label">Equation 6.1</a>: λν = <em class="emphasis">v</em></p>
<p class="para">
<strong class="emphasis bold">relationship between wavelength, frequency, and speed of electromagnetic radiation</strong>
</p>
<p class="para"><a class="xref" href="#averill_1.0-ch06_s01_s02_eq01" xrefstyle="select:label">Equation 6.2</a>: <em class="emphasis">c</em> = λν</p>
</div>
<div class="callout editable block" id="averill_1.0-ch06_s01_s02_n04">
<h3 class="title">Summary</h3>
<p class="para" id="averill_1.0-ch06_s01_s02_p18">A basic knowledge of the electronic structure of atoms requires an understanding of the properties of waves and electromagnetic radiation. A <strong class="emphasis bold">wave</strong> is a periodic oscillation by which energy is transmitted through space. All waves are <strong class="emphasis bold">periodic</strong>, repeating regularly in both space and time. Waves are characterized by several interrelated properties: <strong class="emphasis bold">wavelength (λ)</strong>, the distance between successive waves; <strong class="emphasis bold">frequency (ν)</strong>, the number of waves that pass a fixed point per unit time; <strong class="emphasis bold">speed (</strong><em class="emphasis bolditalic">v</em><strong class="emphasis bold">)</strong>, the rate at which the wave propagates through space; and <strong class="emphasis bold">amplitude</strong>, the magnitude of the oscillation about the mean position. The speed of a wave is equal to the product of its wavelength and frequency. <strong class="emphasis bold">Electromagnetic radiation</strong> consists of two perpendicular waves, one electric and one magnetic, propagating at the <strong class="emphasis bold">speed of light (</strong><em class="emphasis bolditalic">c</em><strong class="emphasis bold">)</strong>. Electromagnetic radiation is radiant energy that includes radio waves, microwaves, visible light, x-rays, and gamma rays, which differ only in their frequencies and wavelengths.</p>
</div>
<div class="key_takeaways editable block" id="averill_1.0-ch06_s01_s02_n05">
<h3 class="title">Key Takeaway</h3>
<ul class="itemizedlist" id="averill_1.0-ch06_s01_s02_l01">
<li>Understanding the electronic structure of atoms requires an understanding of the properties of waves and electromagnetic radiation.</li>
</ul>
</div>
<div class="qandaset block" id="averill_1.0-ch06_s01_s02_qs01" defaultlabel="number">
<h3 class="title">Conceptual Problems</h3>
<ol class="qandadiv" id="averill_1.0-ch06_s01_s02_qs01_qd01">
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs01_qd01_qa01">
<div class="question">
<p class="para">What are the characteristics of a wave? What is the relationship between electromagnetic radiation and wave energy?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs01_qd01_qa02">
<div class="question">
<p class="para">At constant wavelength, what effect does increasing the frequency of a wave have on its speed? its amplitude?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs01_qd01_qa03">
<div class="question">
<p class="para">List the following forms of electromagnetic radiation in order of increasing wavelength: x-rays, radio waves, infrared waves, microwaves, ultraviolet waves, visible waves, and gamma rays. List them in order of increasing frequency. Which has the highest energy?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs01_qd01_qa04">
<div class="question">
<p class="para">A large industry is centered on developing skin-care products, such as suntan lotions and cosmetics, that cannot be penetrated by ultraviolet radiation. How does the wavelength of visible light compare with the wavelength of ultraviolet light? How does the energy of visible light compare with the energy of ultraviolet light? Why is this industry focused on blocking ultraviolet light rather than visible light?</p>
</div>
</li>
</ol>
</div>
<div class="qandaset block" id="averill_1.0-ch06_s01_s02_qs02" defaultlabel="number">
<h3 class="title">Numerical Problems</h3>
<ol class="qandadiv" id="averill_1.0-ch06_s01_s02_qs02_qd01">
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa01">
<div class="question">
<p class="para">The human eye is sensitive to what fraction of the electromagnetic spectrum, assuming a typical spectral range of 10<sup class="superscript">4</sup> to 10<sup class="superscript">20</sup> Hz? If we came from the planet Krypton and had x-ray vision (i.e., if our eyes were sensitive to x-rays in addition to visible light), how would this fraction be changed?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa03">
<div class="question">
<p class="para">What is the frequency in megahertz corresponding to each wavelength?</p>
<ol class="orderedlist" numeration="loweralpha">
<li>755 m</li>
<li>6.73 nm</li>
<li>1.77 × 10<sup class="superscript">3</sup> km</li>
<li>9.88 Å</li>
<li>3.7 × 10<sup class="superscript">−10</sup> m</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa02">
<div class="question">
<p class="para">What is the frequency in megahertz corresponding to each wavelength?</p>
<ol class="orderedlist" numeration="loweralpha">
<li>5.8 × 10<sup class="superscript">−7</sup> m</li>
<li>2.3 Å</li>
<li>8.6 × 10<sup class="superscript">7</sup> m</li>
<li>6.2 mm</li>
<li>3.7 nm</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa04">
<div class="question">
<p class="para">Line spectra are also observed for molecular species. Given the following characteristic wavelengths for each species, identify the spectral region (ultraviolet, visible, etc.) in which the following line spectra will occur. Given 1.00 mol of each compound and the wavelength of absorbed or emitted light, how much energy does this correspond to?</p>
<ol class="orderedlist" numeration="loweralpha">
<li>NH<sub class="subscript">3</sub>, 1.0 × 10<sup class="superscript">−2</sup> m</li>
<li>CH<sub class="subscript">3</sub>CH<sub class="subscript">2</sub>OH, 9.0 μm</li>
<li>Mo atom, 7.1 Å</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa05">
<div class="question">
<p class="para">What is the speed of a wave in meters per second that has a wavelength of 1250 m and a frequency of 2.36 × 10<sup class="superscript">5</sup> s<sup class="superscript">−1</sup>?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa06">
<div class="question">
<p class="para">A wave travels at 3.70 m/s with a frequency of 4.599 × 10<sup class="superscript">7</sup> Hz and an amplitude of 1.0 m. What is its wavelength in nanometers?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa08">
<div class="question">
<p class="para">An AM radio station broadcasts with a wavelength of 248.0 m. What is the broadcast frequency of the station in kilohertz? An AM station has a broadcast range of 92.6 MHz. What is the corresponding wavelength range in meters for this reception?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa07">
<div class="question">
<p class="para">An FM radio station broadcasts with a wavelength of 3.21 m. What is the broadcast frequency of the station in megahertz? An FM radio typically has a broadcast range of 82–112 MHz. What is the corresponding wavelength range in meters for this reception?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s01_s02_qs02_qd01_qa09">
<div class="question">
<p class="para">A microwave oven operates at a frequency of approximately 2450 MHz. What is the corresponding wavelength? Water, with its polar molecules, absorbs electromagnetic radiation primarily in the infrared portion of the spectrum. Given this fact, why are microwave ovens used for cooking food?</p>
</div>
</li>
</ol>
</div>
</div>
</div>
<div class="section" id="averill_1.0-ch06_s02" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.2</span> The Quantization of Energy</h2>
<div class="learning_objectives editable block" id="averill_1.0-ch06_s02_n01">
<h3 class="title">Learning Objective</h3>
<ol class="orderedlist" id="averill_1.0-ch06_s02_l01">
<li>To understand how energy is quantized.</li>
</ol>
</div>
<p class="para editable block" id="averill_1.0-ch06_s02_p01">By the late 19th century, many physicists thought their discipline was well on the way to explaining most natural phenomena. They could calculate the motions of material objects using Newton’s laws of classical mechanics, and they could describe the properties of radiant energy using mathematical relationships known as Maxwell’s equations, developed in 1873 by James Clerk Maxwell, a Scottish physicist. The universe appeared to be a simple and orderly place, containing matter, which consisted of particles that had mass and whose location and motion could be accurately described, and electromagnetic radiation, which was viewed as having no mass and whose exact position in space could not be fixed. Thus matter and energy were considered distinct and unrelated phenomena. Soon, however, scientists began to look more closely at a few inconvenient phenomena that could <em class="emphasis">not</em> be explained by the theories available at the time.</p>
<div class="section" id="averill_1.0-ch06_s02_s01">
<h2 class="title editable block">Blackbody Radiation</h2>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p01">One phenomenon that seemed to contradict the theories of classical physics was <span class="margin_term"><a class="glossterm">blackbody radiation</a><span class="glossdef">Electromagnetic radiation whose wavelength and color depends on the temperature of the object.</span></span>, the energy emitted by an object when it is heated. The wavelength of energy emitted by an object depends on only its temperature, not its surface or composition. Hence an electric stove burner or the filament of a space heater glows dull red or orange when heated, whereas the much hotter tungsten wire in an incandescent light bulb gives off a yellowish light (<a class="xref" href="#averill_1.0-ch06_s02_s01_f01">Figure 6.5 "Blackbody Radiation"</a>).</p>
<div class="figure small editable block" id="averill_1.0-ch06_s02_s01_f01">
<p class="title"><span class="title-prefix">Figure 6.5</span> Blackbody Radiation</p>
<img src="section_10/d42306bf288063c96266de0eb4d36eb1.jpg">
<p class="para">When heated, all objects emit electromagnetic radiation whose wavelength (and color) depends on the temperature of the object. A relatively low-temperature object, such as an electric stove element, on a low setting appears red, whereas a higher-temperature object, such as the filament of an incandescent light bulb, appears yellow or white.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p02">The <em class="emphasis">intensity</em> of radiation is a measure of the energy emitted per unit area. A plot of the intensity of blackbody radiation as a function of wavelength for an object at various temperatures is shown in <a class="xref" href="#averill_1.0-ch06_s02_s01_f02">Figure 6.6 "Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits"</a>. One of the major assumptions of classical physics was that energy increased or decreased in a smooth, continuous manner. For example, classical physics predicted that as wavelength decreased, the intensity of the radiation an object emits should increase in a smooth curve without limit at <em class="emphasis">all</em> temperatures, as shown by the broken line for 6000 K in <a class="xref" href="#averill_1.0-ch06_s02_s01_f02">Figure 6.6 "Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits"</a>. Thus classical physics could not explain the sharp <em class="emphasis">decrease</em> in the intensity of radiation emitted at shorter wavelengths (primarily in the ultraviolet region of the spectrum), which we now refer to as the “ultraviolet catastrophe.” In 1900, however, the German physicist Max Planck (1858–1947) explained the ultraviolet catastrophe by proposing that the energy of electromagnetic waves is <em class="emphasis">quantized</em> rather than continuous. This means that for each temperature, there is a maximum intensity of radiation that is emitted in a blackbody object, corresponding to the peaks in <a class="xref" href="#averill_1.0-ch06_s02_s01_f02">Figure 6.6 "Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits"</a>, so the intensity does not follow a smooth curve as the temperature increases, as predicted by classical physics. Thus energy could be gained or lost only in integral multiples of some smallest unit of energy, a <span class="margin_term"><a class="glossterm">quantum</a><span class="glossdef">The smallest possible unit of energy. Energy can be gained or lost only in integral multiples of a quantum.</span></span>.</p>
<div class="figure large editable block" id="averill_1.0-ch06_s02_s01_f02">
<p class="title"><span class="title-prefix">Figure 6.6</span> Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits</p>
<img src="section_10/9469e9ec92fcdd65227d9055e7ae0639.jpg">
<p class="para">At relatively low temperatures, most radiation is emitted at wavelengths longer than 700 nm, which is in the infrared portion of the spectrum. The dull red glow of the electric stove element in <a class="xref" href="#averill_1.0-ch06_s02_s01_f01">Figure 6.5 "Blackbody Radiation"</a> is due to the small amount of radiation emitted at wavelengths less than 700 nm, which the eye can detect. As the temperature of the object increases, the maximum intensity shifts to shorter wavelengths, successively resulting in orange, yellow, and finally white light. At high temperatures, all wavelengths of visible light are emitted with approximately equal intensities. The white light spectrum shown for an object at 6000 K closely approximates the spectrum of light emitted by the sun (<a class="xref" href="averill_1.0-ch06_s03#averill_1.0-ch06_s03_s03_f01">Figure 6.14 "The Visible Spectrum of Sunlight"</a>). Note the sharp decrease in the intensity of radiation emitted at wavelengths below 400 nm, which constituted the ultraviolet catastrophe. The classical prediction fails to fit the experimental curves entirely and does not have a maximum intensity.</p>
</div>
<div class="callout editable block" id="averill_1.0-ch06_s02_s01_n01">
<h3 class="title">Max Planck (1858–1947)</h3>
<p class="para" id="averill_1.0-ch06_s02_s01_p03">In addition to being a physicist, Planck was a gifted pianist, who at one time considered music as a career. During the 1930s, Planck felt it was his duty to remain in Germany, despite his open opposition to the policies of the Nazi government. One of his sons was executed in 1944 for his part in an unsuccessful attempt to assassinate Hitler, and bombing during the last weeks of World War II destroyed Planck’s home.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p04">Although quantization may seem to be an unfamiliar concept, we encounter it frequently. For example, US money is integral multiples of pennies. Similarly, musical instruments like a piano or a trumpet can produce only certain musical notes, such as C or F sharp. Because these instruments cannot produce a continuous range of frequencies, their frequencies are quantized. Even electrical charge is quantized: an ion may have a charge of −1 or −2 but <em class="emphasis">not</em> −1.33.</p>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p05">Planck postulated that the energy of a particular quantum of radiant energy could be described explicitly by the equation</p>
<div class="equation block" id="averill_1.0-ch06_s02_s01_eq01">
<p class="title editable"><span class="title-prefix">Equation 6.5</span> </p>
<span class="mathphrase"><em class="emphasis">E</em> = <em class="emphasis">h</em>ν</span>
</div>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p06">where the proportionality constant <em class="emphasis">h</em> is called Planck’s constant, one of the most accurately known fundamental constants in science. For our purposes, its value to four significant figures is generally sufficient:</p>
<span class="informalequation block">
<span class="mathphrase"><em class="emphasis">h</em> = 6.626 × 10<sup class="superscript">−34</sup> J·s (joule-seconds)</span>
</span>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p07">As the frequency of electromagnetic radiation increases, the magnitude of the associated quantum of radiant energy increases. By assuming that energy can be emitted by an object only in integral multiples of <em class="emphasis">h</em>ν, Planck devised an equation that fit the experimental data shown in <a class="xref" href="#averill_1.0-ch06_s02_s01_f02">Figure 6.6 "Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits"</a>. We can understand Planck’s explanation of the ultraviolet catastrophe qualitatively as follows: At low temperatures, radiation with only relatively low frequencies is emitted, corresponding to low-energy quanta. As the temperature of an object increases, there is an increased probability of emitting radiation with higher frequencies, corresponding to higher-energy quanta. At any temperature, however, it is simply more probable for an object to lose energy by emitting <em class="emphasis">n</em> lower-energy quanta than a single very high-energy quantum that corresponds to ultraviolet radiation. The result is a maximum in the plot of intensity of emitted radiation versus wavelength, as shown in <a class="xref" href="#averill_1.0-ch06_s02_s01_f02">Figure 6.6 "Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits"</a>, and a shift in the position of the maximum to lower wavelength (higher frequency) with increasing temperature.</p>
<p class="para editable block" id="averill_1.0-ch06_s02_s01_p08">At the time he proposed his radical hypothesis, Planck could not explain <em class="emphasis">why</em> energies should be quantized. Initially, his hypothesis explained only one set of experimental data—blackbody radiation. If quantization were observed for a large number of different phenomena, then quantization would become a law (as defined in <a class="xref" href="averill_1.0-ch01#averill_1.0-ch01">Chapter 1 "Introduction to Chemistry"</a>). In time, a theory might be developed to explain that law. As things turned out, Planck’s hypothesis was the seed from which modern physics grew.</p>
</div>
<div class="section" id="averill_1.0-ch06_s02_s02">
<h2 class="title editable block">The Photoelectric Effect</h2>
<p class="para editable block" id="averill_1.0-ch06_s02_s02_p01">Only five years after he proposed it, Planck’s quantization hypothesis was used to explain a second phenomenon that conflicted with the accepted laws of classical physics. When certain metals are exposed to light, electrons are ejected from their surface (<a class="xref" href="#averill_1.0-ch06_s02_s02_f01">Figure 6.7 "The Photoelectric Effect"</a>). Classical physics predicted that the number of electrons emitted and their kinetic energy should depend on only the intensity of the light, <em class="emphasis">not</em> its frequency. In fact, however, each metal was found to have a characteristic threshold frequency of light; below that frequency, <em class="emphasis">no</em> electrons are emitted regardless of the light’s intensity. Above the threshold frequency, the number of electrons emitted was found to be proportional to the intensity of the light, and their kinetic energy was proportional to the frequency. This phenomenon was called the <span class="margin_term"><a class="glossterm">photoelectric effect</a><span class="glossdef">A phenomenon in which electrons are ejected from the surface of a metal that has been exposed to light.</span></span>.</p>
<div class="figure large editable block" id="averill_1.0-ch06_s02_s02_f01">
<p class="title"><span class="title-prefix">Figure 6.7</span> The Photoelectric Effect</p>
<img src="section_10/6337086965df13de4035b25757a12915.jpg">
<p class="para">(a) Irradiating a metal surface with photons of sufficiently high energy causes electrons to be ejected from the metal. (b) A photocell that uses the photoelectric effect, similar to those found in automatic door openers. When light strikes the metal cathode, electrons are emitted and attracted to the anode, resulting in a flow of electrical current. If the incoming light is interrupted by, for example, a passing person, the current drops to zero. (c) In contrast to predictions using classical physics, no electrons are emitted when photons of light with energy less than <em class="emphasis">E</em><sub class="subscript">o</sub>, such as red light, strike the cathode. The energy of violet light is above the threshold frequency, so the number of emitted photons is proportional to the light’s intensity.</p>
</div>
<p class="para block" id="averill_1.0-ch06_s02_s02_p02">Albert Einstein (1879–1955; Nobel Prize in Physics, 1921) quickly realized that Planck’s hypothesis about the quantization of radiant energy could also explain the photoelectric effect. The key feature of Einstein’s hypothesis was the assumption that radiant energy arrives at the metal surface in particles that we now call <span class="margin_term"><a class="glossterm">photons</a><span class="glossdef">A quantum of radiant energy, each of which possesses a particular energy <span class="inlineequation"><math xml:id="averill_1.0-ch06_m006" display="inline"><semantics><mi>E</mi></semantics></math></span> given by <span class="inlineequation"><math xml:id="averill_1.0-ch06_m007" display="inline"><semantics><mrow><mi>E</mi><mo>=</mo><mi>h</mi><mi>ν</mi><mo>.</mo></mrow></semantics></math></span></span></span>, each possessing a particular energy <em class="emphasis">E</em> given by <a class="xref" href="#averill_1.0-ch06_s02_s01_eq01" xrefstyle="select:label">Equation 6.5</a>. Einstein postulated that each metal has a particular electrostatic attraction for its electrons that must be overcome before an electron can be emitted from its surface (<em class="emphasis">E</em><sub class="subscript">o</sub> = <em class="emphasis">h</em>ν<sub class="subscript">o</sub>). If photons of light with energy less than <em class="emphasis">E</em><sub class="subscript">o</sub> strike a metal surface, no single photon has enough energy to eject an electron, so no electrons are emitted regardless of the intensity of the light. If a photon with energy greater than <em class="emphasis">E</em><sub class="subscript">o</sub> strikes the metal, then part of its energy is used to overcome the forces that hold the electron to the metal surface, and the excess energy appears as the kinetic energy of the ejected electron:</p>
<div class="equation block" id="averill_1.0-ch06_s02_s02_eq01">
<p class="title editable"><span class="title-prefix">Equation 6.6</span> </p>
<span class="mathphrase">kinetic energy of ejected electron = <em class="emphasis">E</em> − <em class="emphasis">E</em><sub class="subscript">o</sub> = <em class="emphasis">h</em>ν − <em class="emphasis">h</em>ν<sub class="subscript">o</sub> = <em class="emphasis">h</em>(ν − ν<sub class="subscript">o</sub>)</span>
</div>
<p class="para editable block" id="averill_1.0-ch06_s02_s02_p03">When a metal is struck by light with energy above the threshold energy <em class="emphasis">E</em><sub class="subscript">o</sub>, the <em class="emphasis">number</em> of emitted electrons is proportional to the <em class="emphasis">intensity</em> of the light beam, which corresponds to the number of photons per square centimeter, but the <em class="emphasis">kinetic energy</em> of the emitted electrons is proportional to the <em class="emphasis">frequency</em> of the light. Thus Einstein showed that the energy of the emitted electrons depended on the frequency of the light, contrary to the prediction of classical physics.</p>
<div class="callout editable block" id="averill_1.0-ch06_s02_s02_n01">
<h3 class="title">Albert Einstein (1879–1955)</h3>
<p class="para" id="averill_1.0-ch06_s02_s02_p04">In 1900, Einstein was working in the Swiss patent office in Bern. He was born in Germany and throughout his childhood his parents and teachers had worried that he might be developmentally disabled. The patent office job was a low-level civil service position that was not very demanding, but it did allow Einstein to spend a great deal of time reading and thinking about physics. In 1905, he published his paper on the photoelectric effect, for which he received the Nobel Prize in 1921.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s02_s02_p05">Planck’s and Einstein’s postulate that energy is quantized is in many ways similar to Dalton’s description of atoms. Both theories are based on the existence of simple building blocks, atoms in one case and quanta of energy in the other. The work of Planck and Einstein thus suggested a connection between the quantized nature of energy and the properties of individual atoms. In fact, Einstein’s Nobel Prize was awarded for his work on the photoelectric effect (<em class="emphasis">not</em> for his more famous equation <em class="emphasis">E</em> = <em class="emphasis">mc</em><sup class="superscript">2</sup>), demonstrating its fundamental importance.</p>
<p class="para editable block" id="averill_1.0-ch06_s02_s02_p06">Recently, scientists have theorized that even our sense of smell has its basis in quantum physics. Preliminary research indicates that odorant molecules absorb a quantum of energy that causes their bonds to vibrate at a specific frequency. Because different assemblages of molecules have different characteristic frequencies, these vibrations seem to act as a molecular signature that can be detected as an odor. Studies with flies show that they can distinguish between similar molecules with different vibrational frequencies. This vibrational theory of smell could serve as a discriminatory process in nature in other ways and is an active area of research.</p>
<div class="exercises block" id="averill_1.0-ch06_s02_s02_n02">
<h3 class="title">Example 2</h3>
<div class="figure small" id="averill_1.0-ch06_s02_s02_f02">
<p class="title"><span class="title-prefix">Figure 6.8</span> A Beam of Red Light Emitted by a Ruby Laser</p>
<img src="section_10/70880f9a7dc6821f3f9fd6a8045075bf.jpg">
<p class="para">Ruby lasers, which emit red light at a wavelength of 694.3 nm, are used to read bar codes. When used for commercial applications, such lasers are generally designed to emit radiation over a narrow range of wavelengths to reduce their cost.</p>
</div>
<p class="para" id="averill_1.0-ch06_s02_s02_p07">A ruby laser, a device that produces light in a narrow range of wavelengths (<a class="xref" href="averill_1.0-ch06_s03#averill_1.0-ch06_s03">Section 6.3 "Atomic Spectra and Models of the Atom"</a>), emits red light at a wavelength of 694.3 nm (<a class="xref" href="#averill_1.0-ch06_s02_s02_f02">Figure 6.8 "A Beam of Red Light Emitted by a Ruby Laser"</a>). What is the energy in joules of a</p>
<ol class="orderedlist" id="averill_1.0-ch06_s02_s02_l01" numeration="loweralpha">
<li>single photon?</li>
<li>mole of photons?</li>
</ol>
<p class="para" id="averill_1.0-ch06_s02_s02_p08"><strong class="emphasis bold">Given: </strong>wavelength</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p09"><strong class="emphasis bold">Asked for: </strong>energy of single photon and mole of photons</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p10">
<strong class="emphasis bold">Strategy:</strong>
</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p11"><strong class="emphasis bold">A</strong> Use <a class="xref" href="averill_1.0-ch06_s01#averill_1.0-ch06_s01_s02_eq01" xrefstyle="select:label">Equation 6.2</a> and <a class="xref" href="#averill_1.0-ch06_s02_s01_eq01" xrefstyle="select:label">Equation 6.5</a> to calculate the energy in joules.</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p12"><strong class="emphasis bold">B</strong> Multiply the energy of a single photon by Avogadro’s number to obtain the energy in a mole of photons.</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p13">
<strong class="emphasis bold">Solution:</strong>
</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p14">The energy of a single photon is given by <em class="emphasis">E</em> = <em class="emphasis">h</em>ν = <em class="emphasis">hc</em>/λ.</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch06_m008" display="block">
<semantics>
<mrow>
<mi>E</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>6.626</mn>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>34</mn>
</mrow>
</msup>
<mtext> J•</mtext>
<menclose notation="updiagonalstrike">
<mtext>s</mtext>
</menclose>
<mtext>)(2</mtext>
<mtext>.998</mtext>
<mo>×</mo>
<msup>
<mrow>
<mtext>10</mtext>
</mrow>
<mn>8</mn>
</msup>
<menclose notation="updiagonalstrike">
<mtext>m</mtext>
</menclose>
<mtext>•</mtext>
<menclose notation="updiagonalstrike">
<mrow>
<msup>
<mtext>s</mtext>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</menclose>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>694.3</mn>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>9</mn>
</mrow>
</msup>
<menclose notation="updiagonalstrike">
<mtext>m</mtext>
</menclose>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2.861</mn>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
<mtext> J</mtext>
</mrow>
</semantics>
</math>
</span>
<p class="para" id="averill_1.0-ch06_s02_s02_p15"><strong class="emphasis bold">B</strong> To calculate the energy in a mole of photons, we multiply the energy of a single photon by the number of photons in a mole (Avogadro’s number). If we write the energy of a photon as 2.861 × 10<sup class="superscript">−19</sup> J/photon, we obtain the energy of a mole of photons with wavelength 694.3 nm:</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch06_m009" display="block">
<semantics>
<mtable columnalign="left">
<mtr columnalign="left">
<mtd columnalign="left">
<mi>E</mi>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2.861</mn>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
<mtext> J</mtext>
</mrow>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>photon</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>6.022</mn>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>23</mn>
</mrow>
</msup>
<menclose notation="updiagonalstrike">
<mrow>
<mtext>photon</mtext>
</mrow>
</menclose>
</mrow>
<mrow>
<mtext>mol</mtext>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr columnalign="left">
<mtd columnalign="left">
<mrow></mrow>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
<mn>1.723</mn>
<mo>×</mo>
<msup>
<mn>10</mn>
<mn>5</mn>
</msup>
<mtext> J/mol</mtext>
</mtd>
</mtr>
<mtr columnalign="left">
<mtd columnalign="left">
<mrow></mrow>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
<mn>172.3</mn>
<mtext> kJ/mol</mtext>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para" id="averill_1.0-ch06_s02_s02_p16">This energy is of the same magnitude as some of the enthalpies of reaction in <a class="xref" href="averill_1.0-ch05#averill_1.0-ch05">Chapter 5 "Energy Changes in Chemical Reactions"</a>, and, as you will see in <a class="xref" href="averill_1.0-ch08#averill_1.0-ch08">Chapter 8 "Ionic versus Covalent Bonding"</a>, it is comparable to the strength of many chemical bonds. As a result, light can be used to initiate chemical reactions. In fact, an entire area of chemistry called <em class="emphasis">photochemistry</em> is devoted to studying such processes. In the phenomenon of <em class="emphasis">photosynthesis</em>, green plants use the energy of visible light to convert carbon dioxide and water into sugars such as glucose.</p>
<p class="simpara">Exercise</p>
<p class="para" id="averill_1.0-ch06_s02_s02_p17">An x-ray generator, such as those used in hospitals, emits radiation with a wavelength of 1.544 Å.</p>
<ol class="orderedlist" id="averill_1.0-ch06_s02_s02_l02" numeration="loweralpha">
<li>What is the energy in joules of a single photon?</li>
<li>What is the energy in joules of a mole of photons?</li>
<li>How many times more energetic is a single x-ray photon of this wavelength than a photon emitted by a ruby laser?</li>
</ol>
<p class="para" id="averill_1.0-ch06_s02_s02_p18">
<strong class="emphasis bold">Answer:</strong>
</p>
<ol class="orderedlist" id="averill_1.0-ch06_s02_s02_l03" numeration="loweralpha">
<li>1.287 × 10<sup class="superscript">−15</sup> J/photon</li>
<li>7.748 × 10<sup class="superscript">8</sup> J/mol = 7.748 × 10<sup class="superscript">5</sup> kJ/mol</li>
<li>4497 times</li>
</ol>
</div>
<div class="key_takeaways editable block" id="averill_1.0-ch06_s02_s02_n03">
<h3 class="title">Key Equation</h3>
<p class="para">
<strong class="emphasis bold">quantization of energy</strong>
</p>
<p class="para"><a class="xref" href="#averill_1.0-ch06_s02_s01_eq01" xrefstyle="select:label">Equation 6.5</a>: <em class="emphasis">E</em> = <em class="emphasis">h</em>ν</p>
</div>
<div class="callout editable block" id="averill_1.0-ch06_s02_s02_n04">
<h3 class="title">Summary</h3>
<p class="para" id="averill_1.0-ch06_s02_s02_p19">The properties of <strong class="emphasis bold">blackbody radiation</strong>, the radiation emitted by hot objects, could not be explained with classical physics. Max Planck postulated that energy was quantized and could be emitted or absorbed only in integral multiples of a small unit of energy, known as a <strong class="emphasis bold">quantum</strong>. The energy of a quantum is proportional to the frequency of the radiation; the proportionality constant <em class="emphasis">h</em> is a fundamental constant (Planck’s constant). Albert Einstein used Planck’s concept of the quantization of energy to explain the <strong class="emphasis bold">photoelectric effect</strong>, the ejection of electrons from certain metals when exposed to light. Einstein postulated the existence of what today we call <strong class="emphasis bold">photons</strong>, particles of light with a particular energy, <em class="emphasis">E</em> = <em class="emphasis">h</em>ν. Both energy and matter have fundamental building blocks: quanta and atoms, respectively.</p>
</div>
<div class="key_takeaways editable block" id="averill_1.0-ch06_s02_s02_n05">
<h3 class="title">Key Takeaway</h3>
<ul class="itemizedlist" id="averill_1.0-ch06_s02_s02_l04">
<li>The fundamental building blocks of energy are quanta and of matter are atoms.</li>
</ul>
</div>
<div class="qandaset block" id="averill_1.0-ch06_s02_s02_qs01" defaultlabel="number">
<h3 class="title">Conceptual Problems</h3>
<ol class="qandadiv" id="averill_1.0-ch06_s02_s02_qs01_qd01">
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa01">
<div class="question">
<p class="para">Describe the relationship between the energy of a photon and its frequency.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa02">
<div class="question">
<p class="para">How was the ultraviolet catastrophe explained?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa03">
<div class="question">
<p class="para">If electromagnetic radiation with a continuous range of frequencies above the threshold frequency of a metal is allowed to strike a metal surface, is the kinetic energy of the ejected electrons continuous or quantized? Explain your answer.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa04">
<div class="question">
<p class="para">The vibrational energy of a plucked guitar string is said to be <em class="emphasis">quantized</em>. What do we mean by this? Are the sounds emitted from the 88 keys on a piano also quantized?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa05">
<div class="question">
<p class="para">Which of the following exhibit quantized behavior: a human voice, the speed of a car, a harp, the colors of light, automobile tire sizes, waves from a speedboat?</p>
</div>
</li>
</ol>
</div>
<div class="qandaset block" id="averill_1.0-ch06_s02_s02_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa01_ans">
<div class="answer">
<p class="para">The energy of a photon is directly proportional to the frequency of the electromagnetic radiation.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa03_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs01_qd01_qa05_ans">
<div class="answer">
<p class="para">Quantized: harp, tire size, speedboat waves; continuous: human voice, colors of light, car speed.</p>
</div>
</li>
</ol>
</div>
<div class="qandaset block" id="averill_1.0-ch06_s02_s02_qs02" defaultlabel="number">
<h3 class="title">Numerical Problems</h3>
<ol class="qandadiv" id="averill_1.0-ch06_s02_s02_qs02_qd01">
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa01">
<div class="question">
<p class="para">What is the energy of a photon of light with each wavelength? To which region of the electromagnetic spectrum does each wavelength belong?</p>
<ol class="orderedlist" numeration="loweralpha">
<li>4.33 × 10<sup class="superscript">5</sup> m</li>
<li>0.065 nm</li>
<li>786 pm</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa02">
<div class="question">
<p class="para">How much energy is contained in each of the following? To which region of the electromagnetic spectrum does each wavelength belong?</p>
<ol class="orderedlist" numeration="loweralpha">
<li>250 photons with a wavelength of 3.0 m</li>
<li>4.2 × 10<sup class="superscript">6</sup> photons with a wavelength of 92 μm</li>
<li>1.78 × 10<sup class="superscript">22</sup> photons with a wavelength of 2.1 Å</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa03">
<div class="question">
<p class="para">A mole of photons is found to have an energy of 225 kJ. What is the wavelength of the radiation?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa04">
<div class="question">
<p class="para">Use the data in <a class="xref" href="averill_1.0-ch06_s01#averill_1.0-ch06_s01_s02_t01">Table 6.1 "Common Wavelength Units for Electromagnetic Radiation"</a> to calculate how much more energetic a single gamma-ray photon is than a radio-wave photon. How many photons from a radio source operating at a frequency of 8 × 10<sup class="superscript">5</sup> Hz would be required to provide the same amount of energy as a single gamma-ray photon with a frequency of 3 × 10<sup class="superscript">19</sup> Hz?</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa05">
<div class="question">
<p class="para">Use the data in <a class="xref" href="averill_1.0-ch06_s01#averill_1.0-ch06_s01_s02_t01">Table 6.1 "Common Wavelength Units for Electromagnetic Radiation"</a> to calculate how much more energetic a single x-ray photon is than a photon of ultraviolet light.</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa06">
<div class="question">
<p class="para">A radio station has a transmitter that broadcasts at a frequency of 100.7 MHz with a power output of 50 kW. Given that 1 W = 1 J/s, how many photons are emitted by the transmitter each second?</p>
</div>
</li>
</ol>
</div>
<div class="qandaset block" id="averill_1.0-ch06_s02_s02_qs02_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa01_ans">
<div class="answer">
<p class="para">
</p>
<ol class="orderedlist" numeration="loweralpha"> <li>4.59 × 10<sup class="superscript">−31</sup> J/photon, radio</li>
<li>3.1 × 10<sup class="superscript">−15</sup> J/photon, gamma ray</li>
<li>2.53 × 10<sup class="superscript">−16</sup> J/photon, gamma ray</li>
</ol>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa03_ans">
<div class="answer">
<p class="para">532 nm</p>
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa05_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="averill_1.0-ch06_s02_s02_qs02_qd01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
</ol>
</div>
</div>
</div>
<div class="section" id="averill_1.0-ch06_s03" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.3</span> Atomic Spectra and Models of the Atom</h2>
<div class="learning_objectives editable block" id="averill_1.0-ch06_s03_n01">
<h3 class="title">Learning Objective</h3>
<ol class="orderedlist" id="averill_1.0-ch06_s03_l01">
<li>To know the relationship between atomic spectra and the electronic structure of atoms.</li>
</ol>
</div>
<p class="para editable block" id="averill_1.0-ch06_s03_p01">The photoelectric effect provided indisputable evidence for the existence of the photon and thus the particle-like behavior of electromagnetic radiation. The concept of the photon, however, emerged from experimentation with <em class="emphasis">thermal radiation</em>, electromagnetic radiation emitted as the result of a source’s temperature, which produces a continuous spectrum of energies. More direct evidence was needed to verify the quantized nature of electromagnetic radiation. In this section, we describe how experimentation with visible light provided this evidence.</p>
<div class="section" id="averill_1.0-ch06_s03_s01">
<h2 class="title editable block">Line Spectra</h2>
<p class="para editable block" id="averill_1.0-ch06_s03_s01_p01">Although objects at high temperature emit a continuous spectrum of electromagnetic radiation (<a class="xref" href="averill_1.0-ch06_s02#averill_1.0-ch06_s02_s01_f02">Figure 6.6 "Relationship between the Temperature of an Object and the Spectrum of Blackbody Radiation It Emits"</a>), a different kind of spectrum is observed when pure samples of individual elements are heated. For example, when a high-voltage electrical discharge is passed through a sample of hydrogen gas at low pressure, the resulting individual isolated hydrogen atoms caused by the dissociation of H<sub class="subscript">2</sub> emit a red light. Unlike blackbody radiation, the color of the light emitted by the hydrogen atoms does not depend greatly on the temperature of the gas in the tube. When the emitted light is passed through a prism, only a few narrow lines, called a <span class="margin_term"><a class="glossterm">line spectrum</a><span class="glossdef">A spectrum in which light of only a certain wavelength is emitted or absorbed, rather than a continuous range of wavelengths.</span></span>, are seen (<a class="xref" href="#averill_1.0-ch06_s03_s01_f01">Figure 6.9 "The Emission of Light by Hydrogen Atoms"</a>), rather than a continuous range of colors. The light emitted by hydrogen atoms is red because, of its four characteristic lines, the most intense line in its spectrum is in the red portion of the visible spectrum, at 656 nm. With sodium, however, we observe a yellow color because the most intense lines in its spectrum are in the yellow portion of the spectrum, at about 589 nm.</p>
<div class="figure large editable block" id="averill_1.0-ch06_s03_s01_f01">
<p class="title"><span class="title-prefix">Figure 6.9</span> The Emission of Light by Hydrogen Atoms</p>
<img src="section_10/dab881c4db4639afd5967d5ada40d4d1.jpg">
<p class="para">(a) A sample of excited hydrogen atoms emits a characteristic red light. (b) When the light emitted by a sample of excited hydrogen atoms is split into its component wavelengths by a prism, four characteristic violet, blue, green, and red emission lines can be observed, the most intense of which is at 656 nm.</p>
</div>
<p class="para editable block" id="averill_1.0-ch06_s03_s01_p02">Such <em class="emphasis">emission spectra</em> were observed for many other elements in the late 19th century, which presented a major challenge because classical physics was unable to explain them. Part of the explanation is provided by Planck’s equation (<a class="xref" href="averill_1.0-ch06_s02#averill_1.0-ch06_s02_s01_eq01" xrefstyle="select:label">Equation 6.5</a>): the observation of only a few values of λ (or ν) in the line spectrum meant that only a few values of <em class="emphasis">E</em> were possible. Thus <em class="emphasis">the energy levels of a hydrogen atom had to be quantized</em>; in other words, only states that had certain values of energy were possible, or <em class="emphasis">allowed</em>. If a hydrogen atom could have <em class="emphasis">any</em> value of energy, then a continuous spectrum would have been observed, similar to blackbody radiation.</p>
<p class="para editable block" id="averill_1.0-ch06_s03_s01_p03">In 1885, a Swiss mathematics teacher, Johann Balmer (1825–1898), showed that the frequencies of the lines observed in the visible region of the spectrum of hydrogen fit a simple equation that can be expressed as follows:</p>
<div class="equation block" id="averill_1.0-ch06_s03_s01_eq01">
<p class="title editable"><span class="title-prefix">Equation 6.7</span> </p>
<math xml:id="averill_1.0-ch06_m010" display="block">
<semantics>
<mrow>
<mi>ν</mi>
<mo>=</mo>
<mtext> constant</mtext>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mn>2</mn>