Additional Reading Materials

Chapter 1: Light and Matter

The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a “corpuscular” view of light, in which light was composed of streams of extremely tiny particles traveling at high speeds according to Newton’s laws of motion. Others in the seventeenth century, such as Christiaan Huygens, had shown that optical phenomena such as reflection and refraction could be equally well explained in terms of light as waves traveling at high speed through a medium called “luminiferous aether” that was thought to permeate all space. Early in the nineteenth century, Thomas Young demonstrated that light passing through narrow, closely spaced slits produced interference patterns that could not be explained in terms of Newtonian particles but could be easily explained in terms of waves. Later in the nineteenth century, after James Clerk Maxwell developed his theory of electromagnetic radiation and showed that light was the visible part of a vast spectrum of electromagnetic waves, the particle view of light became thoroughly discredited. By the end of the nineteenth century, scientists viewed the physical universe as roughly comprising two separate domains: matter composed of particles moving according to Newton’s laws of motion, and electromagnetic radiation consisting of waves governed by Maxwell’s equations. Today, these domains are referred to as classical mechanics and classical electrodynamics (or classical electromagnetism). Although there were a few physical phenomena that could not be explained within this framework, scientists at that time were so confident of the overall soundness of this framework that they viewed these aberrations as puzzling paradoxes that would ultimately be resolved somehow within this framework. As we shall see, these paradoxes led to a contemporary framework that intimately connects particles and waves at a fundamental level called wave-particle duality, which has superseded the classical view.

Visible light and other forms of electromagnetic radiation play important roles in chemistry, since they can be used to infer the energies of electrons within atoms and molecules. Much of modern technology is based on electromagnetic radiation. For example, radio waves from a mobile phone, X-rays used by dentists, the energy used to cook food in your microwave, the radiant heat from red-hot objects, and the light from your television screen are forms of electromagnetic radiation that all exhibit wavelike behavior.

Ch1.1 Waves

A wave is an oscillation or periodic movement that can transport energy from one point in space to another. Common examples of waves are all around us. Shaking the end of a rope transfers energy from your hand to the other end of the rope, dropping a pebble into a pond causes waves to ripple outward along the water’s surface, and the expansion of air that accompanies a lightning strike generates sound waves (thunder) that can travel outward for several miles. In each of these cases, kinetic energy is transferred through matter (the rope, water, or air) while the matter remains essentially in place. An insightful example of a wave occurs in sports stadiums when fans in a narrow region of seats rise simultaneously and stand with their arms raised up for a few seconds before sitting down again while the fans in neighboring sections likewise stand up and sit down in sequence. While this wave can quickly encircle a large stadium in a few seconds, none of the fans actually travel with the wave-they all stay in or above their seats.

Waves need not be restricted to travel through matter. As Maxwell showed, electromagnetic waves consist of an electric field oscillating in step with a perpendicular magnetic field, both of which are perpendicular to the direction of travel. These waves can travel through a vacuum at a constant speed of 2.998 × 108 m/s, the speed of light (denoted by c).

All waves, including forms of electromagnetic radiation, are characterized by: a wavelength (denoted by λ, the lowercase Greek letter lambda), a frequency (denoted by ν, the lowercase Greek letter nu), and an amplitude. As can be seen in Figure 1, the wavelength is the distance between two consecutive peaks or troughs in a wave (measured in meters in the SI system). Electromagnetic waves have wavelengths that fall within an enormous range-wavelengths of kilometers (103 m) to picometers (10−12 m) have been observed. The frequency is the number of wave cycles that pass a specified point in space in a specified amount of time (in the SI system, this is measured in seconds). A cycle corresponds to one complete wavelength. The unit for frequency, expressed as cycles per second [s−1], is the hertz (Hz). Common multiples of this unit are megahertz, (1 MHz = 1 × 106 Hz) and gigahertz (1 GHz = 1 × 109 Hz). The amplitude corresponds to the magnitude of the wave’s displacement and so, in Figure 1, this corresponds to one-half the height between the peaks and troughs. The amplitude is related to the intensity of the wave, which for light is the brightness, and for sound is the loudness.

Figure 1. Waves are characterized by its wavelength, λ, its frequency, ν, and it’s amplitude, a.

The product of a wave’s wavelength (λ) and its frequency (ν), λν, is the speed of the wave. Thus, for electromagnetic radiation in a vacuum:

c = 2.998 × 108 m/s = λν

Wavelength and frequency are inversely proportional: As the wavelength increases, the frequency decreases. The inverse proportionality is illustrated in Figure 2. This figure also shows the electromagnetic spectrum, the range of all types of electromagnetic radiation. Each of the various colors of visible light has specific frequencies and wavelengths associated with them, and you can see that visible light makes up only a small portion of the electromagnetic spectrum. Because the technologies developed to work in various parts of the electromagnetic spectrum are different, for reasons of convenience and historical legacies, different units are typically used for different parts of the spectrum. For example, radio waves are usually specified as frequencies (typically in units of MHz), while the visible region is usually specified in wavelengths (typically in units of nm or angstroms).

The figure includes a portion of the electromagnetic spectrum which extends from gamma radiation at the far left through x-ray, ultraviolet, visible, infrared, terahertz, and microwave to broadcast and wireless radio at the far right. At the top of the figure, inside a grey box, are three arrows. The first points left and is labeled, “Increasing energy E.” A second arrow is placed just below the first which also points left and is labeled, “Increasing frequency nu.” A third arrow is placed just below which points right and is labeled, “Increasing wavelength lambda.” Inside the grey box near the bottom is a blue sinusoidal wave pattern that moves horizontally through the box. At the far left end, the waves are short and tightly packed. They gradually lengthen moving left to right across the figure, resulting in significantly longer waves at the right end of the diagram. Beneath the grey box are a variety of photos aligned above the names of the radiation types and a numerical scale that is labeled, “Wavelength lambda ( m ).” This scale runs from 10 superscript negative 12 meters under gamma radiation increasing by powers of ten to a value of 10 superscript 3 meters at the far right under broadcast and wireless radio. X-ray appears around 10 superscript negative 10 meters, ultraviolet appears in the 10 superscript negative 8 to 10 superscript negative 7 range, visible light appears between 10 superscript negative 7 and 10 superscript negative 6, infrared appears in the 10 superscript negative 6 to 10 superscript negative 5 range, teraherz appears in the 10 superscript negative 4 to 10 superscript negative 3 range, microwave infrared appears in the 10 superscript negative 2 to 10 superscript negative 1 range, and broadcast and wireless radio extend from 10 to 10 superscript 3 meters. Labels above the scale are placed to indicate 1 n m at 10 superscript negative 9 meters, 1 micron at 10 superscript negative 6 meters, 1 millimeter at 10 superscript negative 3 meters, 1 centimeter at 10 superscript negative 2 meters, and 1 foot between 10 superscript negative 1 meter and 10 superscript 0 meters. A variety of images are placed beneath the grey box and above the scale in the figure to provide examples of related applications that use the electromagnetic radiation in the range of the scale beneath each image. The photos on the left above gamma radiation show cosmic rays and a multicolor PET scan image of a brain. A black and white x-ray image of a hand appears above x-rays. An image of a patient undergoing dental work, with a blue light being directed into the patient's mouth is labeled, “dental curing,” and is shown above ultraviolet radiation. Between the ultraviolet and infrared labels is a narrow band of violet, indigo, blue, green, yellow, orange, and red colors in narrow, vertical strips. From this narrow band, two dashed lines extend a short distance above to the left and right of an image of the visible spectrum. The image, which is labeled, “visible light,” is just a broader version of the narrow bands of color in the label area. Above infrared are images of a television remote and a black and green night vision image. At the left end of the microwave region, a satellite radar image is shown. Just right of this and still above the microwave region are images of a cell phone, a wireless router that is labeled, “wireless data,” and a microwave oven. Above broadcast and wireless radio are two images. The left most image is a black and white medical ultrasound image. A wireless AM radio is positioned at the far right in the image, also above broadcast and wireless radio.
Figure 2. Portions of the electromagnetic spectrum are shown in order of decreasing frequency and increasing wavelength. Examples of some applications for various wavelengths include positron emission tomography (PET) scans, X-ray imaging, remote controls, wireless Internet, cellular telephones, and radios. (credit “Cosmic ray”: modification of work by NASA; credit “PET scan”: modification of work by the National Institute of Health; credit “X-ray”: modification of work by Dr. Jochen Lengerke; credit “Dental curing”: modification of work by the Department of the Navy; credit “Night vision”: modification of work by the Department of the Army; credit “Remote”: modification of work by Emilian Robert Vicol; credit “Cell phone”: modification of work by Brett Jordan; credit “Microwave oven”: modification of work by Billy Mabray; credit “Ultrasound”: modification of work by Jane Whitney; credit “AM radio”: modification of work by Dave Clausen)

Example 1

Determining the Frequency and Wavelength of Radiation
A sodium streetlight gives off yellow light that has a wavelength of 589 nm (1 nm = 1 × 10−9 m). What is the frequency of this light?

Solution
We can rearrange the equation c = λν to solve for the frequency:

[latex]\nu = \frac{c}{\lambda}[/latex]

Since c is expressed in meters per second, we must also convert 589 nm to meters.

[latex]\nu = (\frac{2.998 \times 10^8 \;\rule[0.25ex]{0.5em}{0.1ex}\hspace{-0.5em}\text{ms}^{-1}}{589 \;\rule[0.25ex]{1em}{0.1ex}\hspace{-1em}\text{nm}})(\frac{1 \times 10^9 \;\rule[0.25ex]{1em}{0.1ex}\hspace{-1em}\text{nm}}{1\;\rule[0.25ex]{0.5em}{0.1ex}\hspace{-0.5em}\text{m}}) = 5.09 \times 10^{14}\text{s}^{-1}[/latex]

Check Your Learning
One of the frequencies used to transmit and receive cellular telephone signals in the United States is 850 MHz. What is the wavelength in meters of these radio waves?

Answer:

0.353 m = 35.3 cm

Wireless Communication

This figure consists of three cell phone tower images. The first involves a structure that uses a significant degree of scaffolding. The second image includes a tower with what appears to be a base that is essentially a large pole that branches out at the very top. The third image shows a cell phone tower that appears to be disguised as a palm tree.
Figure 3. Radio and cell towers are typically used to transmit long-wavelength electromagnetic radiation. Increasingly, cell towers are designed to blend in with the landscape, as with the Tucson, Arizona, cell tower (right) disguised as a palm tree. (credit left: modification of work by Sir Mildred Pierce; credit middle: modification of work by M.O. Stevens)

Many valuable technologies operate in the radio (3 kHz-300 GHz) frequency region of the electromagnetic spectrum. At the low frequency (low energy, long wavelength) end of this region are AM (amplitude modulation) radio signals (540-2830 kHz) that can travel long distances. FM (frequency modulation) radio signals are at higher frequencies (87.5-108.0 MHz). In AM radio, the information is transmitted by varying the amplitude of the wave (Figure 4). In FM radio, the amplitude is constant and the instantaneous frequency varies.

This figure shows 3 wave diagrams. The first wave diagram is in black and shows two crests, indicates a consistent distance from peak to trough, and has one trough in its span across the page. The label, “Signal,” appears to the right. Just below this, a wave diagram is shown in red. The wave includes sixteen crests, but the distance from the peaks to troughs of consecutive waves varies moving across the page. The peak to trough distance is greatest in the region below the peaks of the black wave diagram, and the distance from peak to trough is similarly least below the trough of the black wave diagram. This red wave diagram is labeled, “A M.” The third wave diagram is shown in blue. The distance from peak to trough of consecutive waves is constant across the page, but the peaks and troughs are more closely packed in the region below the peaks of the black wave diagram at the top of the figure. The peaks and troughs are relatively widely spaced below the trough region of the black wave diagram. This blue wave diagram is labeled “F M.”
Figure 4. This schematic depicts how amplitude modulation (AM) and frequency modulation (FM) can be used to transmit a radio wave.

Other technologies also operate in the radio-wave portion of the electromagnetic spectrum. For example, 4G cellular telephone signals are approximately 880 MHz, while Global Positioning System (GPS) signals operate at 1.228 and 1.575 GHz, local area wireless technology (Wi-Fi) networks operate at 2.4 to 5 GHz, and highway toll sensors operate at 5.8 GHz. The frequencies associated with these applications are convenient because such waves tend not to be absorbed much by common building materials.

Ch1.2 Wave Properties

One particularly characteristic phenomenon of waves results when two or more waves come into contact: They interfere with each other. Figure 5 shows the interference patterns that arise when light passes through narrow slits closely spaced about a wavelength apart. The fringe patterns produced depend on the wavelength, with the fringes being more closely spaced for shorter wavelength light passing through a given set of slits. When the light passes through the two slits, each slit effectively acts as a new source, resulting in two closely spaced waves coming into contact at the detector (the camera in this case). The dark regions in Figure 5 correspond to regions where the peaks for the wave from one slit happen to coincide with the troughs for the wave from the other slit (destructive interference), while the brightest regions correspond to the regions where the peaks for the two waves (or their two troughs) happen to coincide (constructive interference). Likewise, when two stones are tossed close together into a pond, interference patterns are visible in the interactions between the waves produced by the stones. Such interference patterns cannot be explained by particles moving according to the laws of classical mechanics.

This image shows interference patterns for light passing through a narrow slit. Four distinct colored bands are arranged from top to bottom. The top band shows white light, the second red, the third green, and the fourth is blue. Each band shows a central square of color with narrower vertically oriented bands extending left and right on a black background.
Figure 5. Interference fringe patterns are shown for light passing through two closely spaced, narrow slits. The spacing of the fringes depends on the wavelength, with the fringes being more closely spaced for the shorter-wavelength blue light. (credit: PASCO)

Dorothy Hodgkin

Because the wavelengths of X-rays (10-10,000 picometers [pm]) are comparable to the size of atoms, X-rays can be used to determine the structure of molecules. When a beam of X-rays is passed through molecules packed together in a crystal, the X-rays collide with the electrons and scatter. Constructive and destructive interference of these scattered X-rays creates a specific diffraction pattern. Calculating backward from this pattern, the positions of each atom in the molecule can be determined very precisely. One of the pioneers who helped create this technology was Dorothy Crowfoot Hodgkin.

She was born in Cairo, Egypt, in 1910, where her British parents were studying archeology. Even as a young girl, she was fascinated with minerals and crystals. When she was a student at Oxford University, she began researching how X-ray crystallography could be used to determine the structure of biomolecules. She invented new techniques that allowed her and her students to determine the structures of vitamin B12, penicillin, and many other important molecules. Diabetes, a disease that affects 382 million people worldwide, involves the hormone insulin. Hodgkin began studying the structure of insulin in 1934, but it required several decades of advances in the field before she finally reported the structure in 1969. Understanding the structure has led to better understanding of the disease and treatment options.

Ch1.3 Blackbody Radiation and the Ultraviolet Catastrophe

The last few decades of the nineteenth century witnessed intense research activity in commercializing newly discovered electric lighting. This required obtaining a better understanding of the distributions of light emitted from various sources being considered. Artificial lighting is usually designed to mimic natural sunlight within the limitations of the underlying technology. Such lighting consists of a range of broadly distributed frequencies that form a continuous spectrum. Figure 6 shows the wavelength distribution for sunlight. The most intense radiation is in the visible region, with the intensity dropping off rapidly for shorter wavelength ultraviolet (UV) light, and more slowly for longer wavelength infrared (IR) light.

A graph is shown with a horizontal axis labeled, “Wavelength ( n m ),” and a vertical axis labeled, “Spectral irradiance ( W divided by m superscript 2 divided by n m ).” The horizontal axis begins at 250 and extends to 4000 with markings provided every 250 n m. Similarly, the vertical axis begins at 0.00 and extends to 2.00 with markings every 0.25 units. Two vertical dashed lines are drawn. The first appears at about 400 nanometers and the second at nearly 700 nanometers. To the left of the first of these lines, the label, “U V,” appears at the top of the graph. Between these lines, the label, “Visible,” appears at the top of the graph. To the right of the second of these lines, the label, “Infrared,” appears at the top of the graph. A grey curve begins on the vertical axis at about 0.10. This curve increases steeply to a maximum value between the two vertical line segments of approximately 1.75 at about 625 nanometers. This curve decreases rapidly at first, then tapers off to reach a value of about 0 at the far right end of the graph. A golden colored curve traces along the same path as the grey curve, but shows a significant degree of variation in the region of the peak of the graph. In this general region, the gold curve is jagged and somewhat erratic. This curve reaches a maximum over 2.00 at around 475 nanometers. A key provided in the open space of the graph shows that the gold graph represents sunlight at the top of the atmosphere, and the grey curve represents the 5250 degrees C Blackbody spectrum.
Figure 6. The spectral distribution (light intensity vs. wavelength) of sunlight reaches the Earth’s atmosphere as UV light, visible light, and IR light. The unabsorbed sunlight at the top of the atmosphere has a distribution that approximately matches the theoretical distribution of a blackbody at 5250 °C, represented by the blue curve. (credit: modification of work by American Society for Testing and Materials (ASTM) Terrestrial Reference Spectra for Photovoltaic Performance Evaluation)

In Figure 6, the solar distribution is compared to a representative distribution, called a blackbody spectrum, that corresponds to a temperature of 5250 °C. The blackbody spectrum matches the solar spectrum quite well. A blackbody is a convenient, ideal emitter that approximates the behavior of many materials when heated. It is “ideal” in the same sense that an ideal gas is a convenient, simple representation of real gases that works well, provided that the pressure is not too high nor the temperature too low. A good approximation of a blackbody that can be used to observe blackbody radiation is a metal oven that can be heated to very high temperatures. The oven has a small hole allowing for the light being emitted within the oven to be observed with a spectrometer so that the wavelengths and their intensities can be measured. Figure 7 shows the resulting curves for some representative temperatures. Each distribution depends only on a single parameter: the temperature. The maxima in the blackbody curves, λmax, shift to shorter wavelengths as the temperature increases, reflecting the observation that metals being heated first glow a darker red that then become brighter as the temperature increases, eventually becoming white hot at very high temperatures as the intensities of all of the visible wavelengths become appreciable. This common observation was at the heart of a paradox that showed the fundamental limitations of classical physics.

A graph is shown with a horizontal axis labeled, “Wavelength lambda (micrometers)” and a vertical axis labeled, “Intensity I (a r b. units).” The horizontal axis begins at 0 and extends to 3.0 with markings provided every 0.1 micrometer. Similarly, the vertical axis begins at 0 and extends to 10 with markings every 1 unit. Two vertical dashed lines are drawn. The first appears at about 0.39 micrometers and the second at about 0.72 micrometers. To the left of the first of these lines, the label, “Ultraviolet,” appears at the top of the graph. Between these lines, the label, “Visible,” appears at the top of the graph. To the right of the second of these lines, the label, “Infrared,” appears at the top of the graph. To the far right of the graph in open space a purple dot is placed which is labeled, “lambda maximum.” A “Temperature” label is located in a central region of the graph. A blue curve begins on the horizontal axis at about 0.05 micrometers. This curve increases steeply to a maximum value between the two vertical line segments of approximately 9.5 at about 0.55 micrometers. This curve decreases rapidly at first, then tapers off to reach a value of about 1.5 at the far right end of the graph. This blue curve is labeled 6000 K beneath the “Temperature” label. Curves are similarly drawn in green for 5000 K, orange for 4000 K, and red for 3000 K. As the temperature decreases, the height of the peak is lower and shifted right on the graph. The maximum value for the green curve is around 4.5 at 7.2 micrometers. This curve tapers at the right end of the graph to a value around 0.6. The maximum for the orange curve is around 2 at about 0.9 micrometers. This curve tapers at the right end of the graph to a value around 0.2. The maximum for the red curve is around 0.7 at about 1.2 micrometers. This curve tapers at the right end of the graph to a value around 0.1. The entire region under the blue curve that is between the two dashed lines, indicating the visible region, is shaded with vertical bands of color. The colors extending left to right across this region are violet, indigo, blue, green, yellow, orange, and red. A purple dot is placed at the peak of each of the four colored curves. These peaks are connected with a dashed curve.
Figure 7. Blackbody spectral distribution curves are shown for some representative temperatures.

Physicists derived mathematical expressions for the blackbody curves using well-accepted concepts from the theories of classical mechanics and classical electromagnetism. The theoretical expressions as functions of temperature fit the observed experimental blackbody curves well at longer wavelengths, but showed significant discrepancies at shorter wavelengths. Not only did the theoretical curves not show a peak, they absurdly showed the intensity becoming infinitely large as the wavelength became smaller, which would imply that everyday objects at room temperature should be emitting large amounts of UV light. This became known as the “ultraviolet catastrophe” because no one could find any problems with the theoretical treatment that could lead to such unrealistic short-wavelength behavior. Finally, around 1900, Max Planck derived a theoretical expression for blackbody radiation that fit the experimental observations exactly (within experimental error). Planck developed his theoretical treatment by extending the earlier work that had been based on the premise that the atoms composing the oven vibrated at increasing frequencies (or decreasing wavelengths) as the temperature increased, with these vibrations being the source of the emitted electromagnetic radiation. But where the earlier treatments had allowed the vibrating atoms to have any energy values obtained from a continuous set of energies (perfectly reasonable, according to classical physics), Planck found that by restricting the vibrational energies to discrete values for each frequency, he could derive an expression for blackbody radiation that correctly had the intensity dropping rapidly for the short wavelengths in the UV region.

E = nhν,   n = 1, 2, 3, …

The quantity h is a constant now known as Planck’s constant, in his honor. Although Planck was pleased he had resolved the blackbody radiation paradox, he was unable to explain why it was necessary to assume the vibrating atoms required quantized, rather than continuous, energies. The value of Planck’s constant is very small, 6.626 × 10−34 joule seconds (J*s), which helps explain why energy quantization had not been observed previously in macroscopic phenomena.

Ch1.4 The Photoelectric Effect

Another paradox in the classical theory concerned the photoelectric effect (Figure 8). It had been observed that electrons could be ejected from the clean surface of a metal when light having a frequency greater than some threshold frequency was shone on it. Surprisingly, the kinetic energy of the ejected electrons did not depend on the brightness of the light, but increased with increasing frequency of the light. Since the electrons in the metal had a certain amount of binding energy keeping them there, the incident light needed to have more energy to free the electrons. According to classical wave theory, a wave’s energy depends on its intensity (which depends on its amplitude), not its frequency. One part of these observations was that the number of electrons ejected within in a given time period was seen to increase as the brightness increased. In 1905, Albert Einstein was able to resolve the paradox by incorporating Planck’s quantization findings into the discredited particle view of light (Einstein actually won his Nobel prize for this work, and not for his theories of relativity for which he is most famous).

The figure includes three diagrams of waves approaching a flat, horizontal surface that is labeled, “Metal,” from an angle around 45 degrees above and to the left relative to the surface. At the top of the diagram at the center is the label, “E equals h nu.” At the left, a sinusoidal wave reaches the surface and stops. The portion of the diagram near the flat metal surface is labeled, “No electrons ejected,” and the wave is labeled, “700 n m.” To the right, a second similar, more compressed wave, which is labeled, “550 n m,” reaches the flat surface. This time, an arrow extends up and to the right at an angle of approximately 45 degrees. A tiny yellow circle with a negative sign in it is at the center of the arrow shaft. Above this arrow is the equation, “v subscript max equals 2.96 times 10 superscript 5 m per s.” To the far right, a third similar, even more compressed wave, which is labeled “400 n m” reaches the flat surface. This time, an arrow extends up and to the right at an angle of approximately 45 degrees. A tiny yellow circle with a negative sign in it is at the center of the arrow shaft. Above this arrow is the equation “v subscript max equals 6.22 times 10 superscript 5 m per s.”
Figure 8. Photons with low frequencies do not have enough energy to cause electrons to be ejected via the photoelectric effect. For any frequency of light above the threshold frequency, the kinetic energy of ejected electron will be proportional to the energy of the incoming photon.

Einstein argued that the quantized energies that Planck had postulated in his treatment of blackbody radiation could be applied to the light in the photoelectric effect so that the light striking the metal surface should not be viewed as a wave, but instead as a stream of particles (later called photons) whose energy depended on their frequency, according to Planck’s formula, E = (or, in terms of wavelength using c = νλ, [latex]E = \frac{hc}{\lambda}[/latex]). Electrons were ejected when the surface is hit by photons having sufficient energy (a frequency greater than the threshold). The higher the frequency, the greater the kinetic energy imparted to the escaping electrons. Einstein also argued that the light intensity did not depend on the amplitude of the incoming wave, but instead corresponded to the number of photons striking the surface within a given time period. This explains why the number of ejected electrons increased with increasing brightness, since the greater the number of incoming photons, the greater the likelihood that they would collide with some of the electrons.

With Einstein’s findings, the nature of light took on a new air of mystery. Although many light phenomena could be explained either in terms of waves or particles, certain phenomena, such as the interference patterns obtained when light passed through a double slit, were completely contrary to a particle view of light, while other phenomena, such as the photoelectric effect, were completely contrary to a wave view of light. Somehow, at a deep fundamental level still not fully understood, light is both wavelike and particle-like. This is known as wave-particle duality.

Example 2

Calculating the Energy of Radiation
When we see light from a neon sign, we are observing radiation from excited neon atoms. If this radiation has a wavelength of 640 nm, what is the energy of the photon being emitted?

Solution
We use Planck’s equation that includes the wavelength, λ, and convert units of nanometers to meters so that the units of λ and c are the same.

[latex]\begin{array}{l} E = \frac{hc}{\lambda} \\[1em] E = \frac{(6.626 \times 10^{-34}\text{J}\rule[0.4ex]{0.35em}{0.1ex}\hspace{-0.35em}\text{s})(2.998 \times 10^8 \text{m}\rule[0.4ex]{0.35em}{0.1ex}\hspace{-0.35em}\text{s}^{-1})}{(640 \;\rule[0.5ex]{1em}{0.1ex}\hspace{-1em}\text{nm})(\frac{1 \;\text{m}}{10^9 \;\rule[0.25ex]{0.8em}{0.1ex}\hspace{-0.8em}\text{nm}})} \\[1em] E = 3.10 \times 10^{-19} \text{J} \end{array}[/latex]

Check Your Learning
The microwaves in an oven are of a specific frequency that will heat the water molecules contained in food. (This is why most plastics and glass do not become hot in a microwave oven-they do not contain water molecules.) This frequency is about 3 × 109 Hz. What is the energy of one photon in these microwaves?

Answer:

2 × 10−24 J

Use this simulation program to experiment with the photoelectric effect to see how intensity, frequency, type of metal, and other factors influence the ejected photons.

Example 3

Photoelectric Effect
Identify which of the following statements are false and, where necessary, change the italicized word or phrase to make them true, consistent with Einstein’s explanation of the photoelectric effect.

(a) Increasing the brightness of incoming light increases the kinetic energy of the ejected electrons.

(b) Increasing the wavelength of incoming light increases the kinetic energy of the ejected electrons.

(c) Increasing the brightness of incoming light increases the number of ejected electrons.

(d) Increasing the frequency of incoming light can increase the number of ejected electrons.

Solution
(a) False. Increasing the brightness of incoming light has no effect on the kinetic energy of the ejected electrons. Only energy, not the number or amplitude, of the photons influences the kinetic energy of the electrons.

(b) False. Increasing the frequency of incoming light increases the kinetic energy of the ejected electrons. Frequency is proportional to energy and inversely proportional to wavelength. Frequencies above the threshold value transfer the excess energy into the kinetic energy of the electrons.

(c) True. Because the number of photons increases with brighter light, the number of ejected electrons increases.

(d) True with regard to the threshold energy binding the electrons to the metal. Below this threshold, electrons are not emitted and above it they are. Once over the threshold value, further increasing the frequency does not increase the number of ejected electrons

Check Your Learning
Calculate the threshold energy in kJ/mol of electrons in aluminum, given that the lowest frequency photon for which the photoelectric effect is observed in aluminum is 9.87 × 1014 Hz.

Answer:

3.94 × 105 kJ/mol

Ch1.5 Line Spectra

Another paradox within the classical electromagnetic theory that scientists in the late nineteenth century struggled with concerned the light emitted from atoms and molecules. When solids, liquids, or condensed gases are heated sufficiently, they radiate some of the excess energy as light. Photons produced in this manner have a range of energies, and thereby produce a continuous spectrum in which an unbroken series of wavelengths is present. Most of the light generated from stars (including our sun) is produced in this fashion. You can see all the visible wavelengths of light present in sunlight by using a prism to separate them. As can be seen in Figure 6, sunlight also contains UV light (shorter wavelengths) and IR light (longer wavelengths) that can be detected using instruments but are invisible to the human eye. Incandescent (glowing) solids such as tungsten filaments in incandescent lights also give off light that contains all wavelengths of visible light. These continuous spectra can often be approximated by blackbody radiation curves at some appropriate temperature, such as those shown in Figure 7.

In contrast to continuous spectra, light can also occur as discrete or line spectra having very narrow line widths interspersed throughout the spectral regions such as those shown in Figure 9. Exciting a gas at low partial pressure using an electrical current, or heating it, will produce line spectra. Fluorescent light bulbs and neon signs operate in this way (Figure 10). Each element displays its own characteristic set of lines, as do molecules, although their spectra are generally much more complicated.

An image is shown with 5 rows. Across the top and bottom of the image is a scale that begins at 4000 angstroms at the left and extends to 740 angstroms at the far right. The top row is a continuous band of the visible spectrum, showing the colors from violet at the far left through indigo, blue, green, yellow, orange, and red at the far right. The second row, labeled, “N a,” shows the emission spectrum for the element sodium, which includes two narrow vertical bands in the blue range, two narrow bands in the yellow-green range, two narrow bands in the yellow range, and one narrow band in the orange range. The third row, labeled, “H,” shows the emission spectrum for hydrogen. This spectrum shows single bands in the violet, indigo, blue, and orange regions. The fourth row, labeled, “C a,” shows the emission spectrum for calcium. This spectrum shows bands in the following colors and frequencies; one violet, five indigo, one blue, two green, two yellow-green, one yellow, two yellow-orange, one orange, and one red. The fifth row, labeled, “H g,” shows the emission spectrum for mercury. This spectrum shows bands in the following colors and frequencies; two violet, one indigo, two blue, one green, two yellow, two orange, and one orange-red. It is important to note that each of the color bands for the emission spectra of the elements matches to a specific wavelength of light. Extending a vertical line from the bands to the scale above or below the diagram will match the band to a specific measurement on the scale.
Figure 9. Compare the two types of emission spectra: continuous spectrum of white light (top) and the line spectra of the light from excited sodium, hydrogen, calcium, and mercury atoms.
This figure shows a colorful neon sign. The tubes are bent into various shapes.
Figure 10. Neon signs operate by exciting a gas at low partial pressure using an electrical current. This sign shows the elaborate artistic effects that can be achieved. (credit: Dave Shaver)

In a line spectrum, each emission line consists of a single wavelength of light, which implies that the light emitted by a gas consists of a set of discrete energies. For example, when an electric discharge passes through a tube containing hydrogen gas at low pressure, the H2 molecules are broken apart into separate H atoms and we see a blue-pink color. Passing the light through a prism produces a line spectrum, indicating that this light is composed of photons of four visible wavelengths, as shown in Figure 9.

The origin of discrete spectra in atoms and molecules was extremely puzzling to scientists in the late nineteenth century, since according to classical electromagnetic theory, only continuous spectra should be observed. Even more puzzling, in 1885, Johann Balmer was able to derive an empirical equation that related the four visible wavelengths of light emitted by hydrogen atoms to whole integers:

[latex]\frac{1}{\lambda} = k(\frac{1}{4} - \frac{1}{n^2}), n = 3, 4, 5, 6[/latex]

where k is a constant. Other discrete lines for the hydrogen atom were found in the UV and IR regions. Johannes Rydberg generalized Balmer’s work and developed an empirical formula that predicted all of hydrogen’s emission lines, not just those restricted to the visible range,:

[latex]\frac{1}{\lambda} = R_{\infty} (\frac{1}{n^2_1} - \frac{1}{n^2_2})[/latex]

where n1 and n2 are integers, n1 < n2, and R is the Rydberg constant (1.097 × 107 m−1). Even in the late nineteenth century, spectroscopy was a very precise science, and so the wavelengths of hydrogen were measured to very high accuracy, which implied that the Rydberg constant could be determined very precisely as well. That such a simple formula as the Rydberg formula could account for such precise measurements seemed astounding at the time, but it was Neils Bohr‘s explanation for the emission spectra in 1913 that ultimately convinced scientists to abandon classical physics and spurred the development of modern quantum mechanics.

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Chemistry 109 Copyright © by John Moore; Jia Zhou; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.