Learning Objectives Explain the basic behavior of waves, including traveling waves and standing waves Describe the wave nature of light Use appropriate equations to calculate related light-wave properties such as period, frequency, wavelength, and energy Describe the particle nature of light 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. 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. Waves Nature of Light A wave 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 (Figure \(\PageIndex{1}\)). 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. Waves have characteristic properties (Figure \(\PageIndex{2}\)). As you may have noticed in Figure \(\PageIndex{1}\), waves are periodic, 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 wavelength (\(λ\), lowercase Greek lambda). Wavelengths are described by a unit of distance, typically meters. The frequency (\(\nu\), lowercase Greek nu) 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 −1 ), which in the SI system is called the hertz (Hz). It is named after German physicist Heinrich Hertz (1857–1894), a pioneer in the field of electromagnetic radiation. The amplitude , 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 Figure \(\PageIndex{2}\), two waves can have the same amplitude but different wavelengths and vice versa. The distance traveled by a wave per unit time is its speed (\(v\)), 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: \[\begin{align} (\text{wavelength})(\text{frequency}) &= \text{speed} \nonumber \\[4pt] \lambda \nu &=v \label{6.1.1a} \\[4pt] \left ( \dfrac{meters}{\cancel{wave}} \right )\left ( \dfrac{\cancel{\text{wave}}}{\text{second}} \right ) &=\dfrac{\text{meters}}{\text{second}} \label{6.1.1b} \end{align}\] Different types of waves may have vastly different possible speeds and frequencies. 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. Electromagnetic Radiation 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 electromagnetic radiation . (Figure \(\PageIndex{3}\)). Some forms of electromagnetic radiation are shown in Figure \(\PageIndex{4}\). In a vacuum, all forms of electromagnetic radiation—whether microwaves, visible light, or gamma rays—travel at the speed of light ( c ), which turns out to be a fundamental physical constant with a value of 2.99792458 × 10 8 m/s (about 3.00 ×10 8 m/s or 1.86 × 10 5 mi/s). This is about a million times faster than the speed of sound. Because the various kinds of electromagnetic radiation all have the same speed (\(c\)), they differ in only wavelength and frequency. As shown in Figure \(\PageIndex{4}\) and Table \(\PageIndex{1}\), the wavelengths of familiar electromagnetic radiation range from 10 1 m for radio waves to 10 −12 m for gamma rays, which are emitted by nuclear reactions. By replacing \(v\) with \(c\) in Equation \(\ref{6.1.1a}\), we can show that the frequency of electromagnetic radiation is inversely proportional to its wavelength: \[ \begin{align} c&=\lambda \nu \\[4pt] \nu &=\dfrac{c}{\lambda } \label{6.1.2} \end{align} \] For example, the frequency of radio waves is about 10 8 Hz, whereas the frequency of gamma rays is about 10 20 Hz. Visible light, which is electromagnetic radiation that can be detected by the human eye, has wavelengths between about 7 × 10 −7 m (700 nm, or 4.3 × 10 14 Hz) and 4 × 10 −7 m (400 nm, or 7.5 × 10 14 Hz). Note that when frequency increases, wavelength decreases; c being a constant stays the same. Similarly, when frequency decreases, the wavelength increases. Within the visible range our eyes perceive 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 (Figure \(\PageIndex{4a}\)). A similar phenomenon creates a rainbow, where water droplets suspended in the air act as tiny prisms. Unit Symbol Wavelength (m) Type of Radiation picometer pm 10−12 gamma ray angstrom Å 10−10 x-ray nanometer nm 10−9 UV, visible micrometer μm 10−6 infrared millimeter mm 10−3 infrared centimeter cm 10−2 microwave meter m 100 radio As you will soon see, the energy of electromagnetic radiation is directly proportional to its frequency and inversely proportional to its wavelength: \[ \begin{align} E\; &\propto\; \nu \label{6.1.3} \\[4pt] & \propto\; \dfrac{1}{\lambda } \label{6.1.4} \end{align}\] 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 of the atmosphere absorbs sunlight with wavelengths less than 350 nm, it protects us from the damaging effects of highly energetic ultraviolet radiation. The energy of electromagnetic radiation increases with increasing frequency and decreasing wavelength. Example \(\PageIndex{1}\): Wavelength of Radiowaves Your favorite FM radio station, WXYZ, broadcasts at a frequency of 101.1 MHz. What is the wavelength of this radiation? Given: frequency Asked for: wavelength Strategy: Substitute the value for the speed of light in meters per second into Equation \(\ref{6.1.2}\) to calculate the wavelength in meters. Solution: From Equation \ref{6.1.2}, 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 8 m/s: \[ \begin{align*} λ\nu &= c \\[4pt] &= 2.998 \times 10^8 m/s \end{align*} \] Thus the wavelength \(λ\) is given by \[ \begin{align*} \lambda &=\dfrac{c}{ \nu } \\[4pt] &=\left ( \dfrac{2.988\times 10^{8}\; m/\cancel{s}}{101.1\; \cancel{MHz}} \right )\left ( \dfrac{1\; \cancel{MHz}}{10^{6}\; \cancel{s^{-1}}} \right ) \\[4pt] &=2.965\; m \end{align*} \] Exercise \(\PageIndex{1}\) 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? Answer 8.45 mm Interference and Diffraction When two or more waves arrive at the same point, they superimpose themselves on one another. More specifically, the disturbances of waves are superimposed when they come together—a phenomenon called superposition . Each disturbance corresponds to a force, and forces add. If the disturbances are along the same line, then the resulting wave is a simple addition of the disturbances of the individual waves—that is, their amplitudes add. Figures \(\PageIndex{5}\) and \(\PageIndex{6}\) illustrate superposition in two special cases, both of which produce simple results. Figure \(\PageIndex{5}\) shows two identical waves that arrive at the same point exactly in phase. The crests of the two waves are precisely aligned, as are the troughs. This superposition produces pure constructive interference . Because the disturbances add, pure constructive interference produces a wave that has twice the amplitude of the individual waves, but has the same wavelength. Figure \(\PageIndex{6}\) shows two identical waves that arrive exactly out of phase—that is, precisely aligned crest to trough—producing pure destructive interference . Because the disturbances are in the opposite direction for this superposition, the resulting amplitude is zero for pure destructive interference—the waves completely cancel. While pure constructive and pure destructive interference do occur, they require precisely aligned identical waves. The superposition of most waves produces a combination of constructive and destructive interference and can vary from place to place and time to time. Sound from a stereo, for example, can be loud in one spot and quiet in another. Varying loudness means the sound waves add partially constructively and partially destructively at different locations. A stereo has at least two speakers creating sound waves, and waves can reflect from walls. All these waves superimpose. An example of sounds that vary over time from constructive to destructive is found in the combined whine of airplane jets heard by a stationary passenger. The combined sound can fluctuate up and down in volume as the sound from the two engines varies in time from constructive to destructive. The Particle Nature of Light When certain metals are exposed to light, electrons are ejected from their surface (Figure \(\PageIndex{7}\)). Classical physics predicted that the number of electrons emitted and their kinetic energy should depend on only the intensity of the light, not its frequency. In fact, however, each metal was found to have a characteristic threshold frequency of light; below that frequency, no 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 photoelectric effect (A phenomenon in which electrons are ejected from the surface of a metal that has been exposed to light). In 1900, the German physicist Max Planck (1858–1947) proposing that the energy of electromagnetic waves is quantized rather than continuous. Planck postulated that the energy of a particular quantum of radiant energy could be described by the equation \[ E=h u \label{6.2.1}\] where the proportionality constant h 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: \[h = 6.626 \times 10^{−34}\, J \cdot s\, (\text{joule-seconds}) \nonumber\] As the frequency of electromagnetic radiation increases, the magnitude of the associated quantum of radiant energy increases. 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 not −1.33 electron charges. 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 photons (a quantum of radiant energy, each of which possesses a particular energy energy \(E\) given by Equation \(\ref{6.2.1}\) 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 (\(E_o= u_o\)). If photons of light with energy less than E o 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 E o 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: \[ \begin{align} \text{ kinetic energy of ejected electron} &=E-E_{o} \nonumber \\[4pt] &=h \nu -h \nu _{o} \nonumber \\[4pt] &=h\left ( \nu - \nu_{o} \right ) \label{6.2.2} \end{align}\] When a metal is struck by light with energy above the threshold energy E o , the number of emitted electrons is proportional to the intensity of the light beam, which corresponds to the number of photons per square centimeter, but the kinetic energy of the emitted electrons is proportional to the frequency 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. Moreover, the idea that light could behave not only as a wave but as a particle in the form of photons suggested that matter and energy might not be such unrelated phenomena after all. Albert Einstein (1879–1955) 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, his "miracle year" he published four papers that revolutionized physics. One was on the special theory of relativity, a second on the equivalence of mass and energy, a third on Brownian motion, and the fourth on the photoelectric effect, for which he received the Nobel Prize in 1921, the theory of relativity and energy-matter equivalence being still controversial at the time 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. Example \(\PageIndex{2}\) A ruby laser, a device that produces light in a narrow range of wavelengths emits red light at a wavelength of 694.3 nm (Figure \(\PageIndex{4}\)). What is the energy in joules of a single photon? Given: wavelength Asked for: energy of single photon. Strategy: Use Equation \(\ref{6.2.1}\) and the relationship between wavelength and frequency to calculate the energy in joules. Solution: The energy of a single photon is given by \[E = h\nu = \dfrac{hc}{λ}.\] Exercise \(\PageIndex{2}\) An x-ray generator, such as those used in hospitals, emits radiation with a wavelength of 1.544 Å. What is the energy in joules of a single photon? How many times more energetic is a single x-ray photon of this wavelength than a photon emitted by a ruby laser? Answer a \(1.287 \times 10^{-15}\; J/photon\) Answer a 4497 times Example \(\PageIndex{3}\): 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. Increasing the brightness of incoming light increases the kinetic energy of the ejected electrons. Increasing the wavelength of incoming light increases the kinetic energy of the ejected electrons. Increasing the brightness of incoming light increases the number of ejected electrons. Increasing the frequency of incoming light can increase the number of ejected electrons. Solution 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. 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. True. Because the number of collisions with photons increases with brighter light, the number of ejected electrons increases. 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 Exercise \(\PageIndex{3}\) Calculate the threshold energy in kJ/mol of electrons in aluminum, given that the lowest frequency photon for which the photoelectric effect is observed is \(9.87 \times 10^{14}\; Hz\). Answer \(3.94 \: kJ/mol\) Summary Light and other forms of electromagnetic radiation move through a vacuum with a constant speed, c , of 2.998 × 10 8 m s −1 . This radiation shows wavelike behavior, which can be characterized by a frequency, ν , and a wavelength, λ , such that c = λν . Light is an example of a travelling wave. Other important wave phenomena include standing waves, periodic oscillations, and vibrations. Standing waves exhibit quantization, since their wavelengths are limited to discrete integer multiples of some characteristic lengths. Electromagnetic radiation that passes through two closely spaced narrow slits having dimensions roughly similar to the wavelength will show an interference pattern that is a result of constructive and destructive interference of the waves. Electromagnetic radiation also demonstrates properties of particles called photons. The energy of a photon is related to the frequency (or alternatively, the wavelength) of the radiation as E = hν (or \(E=\dfrac{hc}{λ}\)), where h is Planck's constant. That light demonstrates both wavelike and particle-like behavior is known as wave-particle duality. All forms of electromagnetic radiation share these properties, although various forms including X-rays, visible light, microwaves, and radio waves interact differently with matter and have very different practical applications. Electromagnetic radiation can be generated by exciting matter to higher energies, such as by heating it. Key Equations c = λν \(E=hν=\dfrac{hc}{λ}\), where h = 6.626 × 10 −34 J s Summary Understanding the electronic structure of atoms requires an understanding of the properties of waves and electromagnetic radiation. A wave is a periodic oscillation by which energy is transmitted through space. All waves are periodic , repeating regularly in both space and time. Waves are characterized by several interrelated properties: wavelength (\(λ\)), the distance between successive waves; frequency (\( u\)), the number of waves that pass a fixed point per unit time; speed (\(v\)), the rate at which the wave propagates through space; and amplitude , the magnitude of the oscillation about the mean position. The speed of a wave is equal to the product of its wavelength and frequency. Electromagnetic radiation consists of two perpendicular waves, one electric and one magnetic, propagating at the speed of light (\(c\)). Electromagnetic radiation is radiant energy that includes radio waves, microwaves, visible light, x-rays, and gamma rays, which differ in their frequencies and wavelengths. Glossary amplitude extent of the displacement caused by a wave (for sinusoidal waves, it is one-half the difference from the peak height to the trough depth, and the intensity is proportional to the square of the amplitude) continuous spectrum electromagnetic radiation given off in an unbroken series of wavelengths (e.g., white light from the sun) electromagnetic radiation energy transmitted by waves that have an electric-field component and a magnetic-field component electromagnetic spectrum range of energies that electromagnetic radiation can comprise, including radio, microwaves, infrared, visible, ultraviolet, X-rays, and gamma rays; since electromagnetic radiation energy is proportional to the frequency and inversely proportional to the wavelength, the spectrum can also be specified by ranges of frequencies or wavelengths frequency (\(\nu\)) number of wave cycles (peaks or troughs) that pass a specified point in space per unit time hertz (Hz) the unit of frequency, which is the number of cycles per second, s −1 intensity property of wave-propagated energy related to the amplitude of the wave, such as brightness of light or loudness of sound interference pattern pattern typically consisting of alternating bright and dark fringes; it results from constructive and destructive interference of waves line spectrum electromagnetic radiation emitted at discrete wavelengths by a specific atom (or atoms) in an excited state node any point of a standing wave with zero amplitude photon smallest possible packet of electromagnetic radiation, a particle of light quantization occurring only in specific discrete values, not continuous standing wave (also, stationary wave) localized wave phenomenon characterized by discrete wavelengths determined by the boundary conditions used to generate the waves; standing waves are inherently quantized wave oscillation that can transport energy from one point to another in space wavelength ( λ ) distance between two consecutive peaks or troughs in a wave wave-particle duality term used to describe the fact that elementary particles including matter exhibit properties of both particles (including localized position, momentum) and waves (including nonlocalization, wavelength, frequency)