Chapter 7: The Quantum-Mechanical Model of the Atom

the angular momentum Quantum Number (l)

the angular momentum Quantum Number (l) The angular momentum quantum number is an integer that determines the shape of the orbital. We consider these shapes in Section 7.6. The possible values of l are 0, 1, 2, c, (n - 1). In other words, for a given value of n, l can be any integer (including 0) up to n - 1. For example, if n = 1, then the only possible value of l is 0; if n = 2, the possible values of l are 0 and 1. In order to avoid confusion between n and l, values of l are often assigned letters as follows:

The Shapes of Atomic Orbitals

As we have seen, the shapes of atomic orbitals are important because covalent chemical bonds depend on the sharing of the electrons that occupy these orbitals. In one model of chemical bonding, for example, a bond consists of the overlap of atomic orbitals on adjacent atoms. Therefore the shapes of the overlapping orbitals determine the shape of the molecule. Although we limit ourselves in this chapter to discussing the orbitals of the hydrogen atom, we will see in Chapter 8 that the orbitals of all atoms can be approximated as being hydrogen-like and therefore have very similar shapes to those of hydrogen. The shape of an atomic orbital is determined primarily by l, the angular momentum quantum number. Recall from Section 7.5 that each value of l is assigned a letter that therefore corresponds to particular orbitals. For example, the orbitals with l = 0 are s orbitals; those with l = 1, p orbitals; those with l = 2, d orbitals, and so on. In this section of the chapter, we examine the shape of each of these orbitals.

Schrödinger's Cat

Atoms and the particles that compose them are unimaginably small. Electrons have a mass of less than a trillionth of a trillionth of a gram, and a size so small that it is immeasurable. Electrons are small in the absolute sense of the word—they are among the smallest particles that make up matter. And yet, as we have seen, an atom's electrons determine many of its chemical and physical properties. If we are to understand these properties, we must try to understand electrons. In the early twentieth century, scientists discovered that the absolutely small (or quan- tum) world of the electron behaves differently than the large (or macroscopic) world that we are used to observing. Chief among these differences is the idea that, when unobserved, absolutely small particles like electrons can simultaneously be in two different states at the same time. For example, through a process called radioactive decay (see Chapter 19) an atom can emit small (that is, absolutely small) energetic particles from its nucleus. In the macroscopic world, something either emits an energetic particle or it doesn't. In the quantum world, however, the unobserved atom can be in a state in which it is doing both—emitting the particle and not emitting the particle—simultaneously. At first, this seems absurd. The absurdity resolves itself, however, upon observation. When we set out to measure the emitted particle, the act of measurement actually forces the atom into one state or other. Early-twentieth-century physicists struggled with this idea. Austrian physicist Erwin Schrödinger, in an attempt to demonstrate that this quantum strangeness could never transfer itself to the macroscopic world, published a paper in 1935 that included a thought experiment about a cat, now known as Schrödinger's cat. In the thought experiment, a cat is put into a steel chamber that contains radioactive atoms like the one described in the previous paragraph. The chamber is equipped with a mechanism that, upon the emission of an energetic particle by one of the radioactive atoms, causes a hammer to break a flask of hydrocyanic acid, a poison. If the flask breaks, the poison is released and the cat dies. Now here comes the absurdity: If the steel chamber is closed, the whole system remains unobserved, and the radioactive atom is in a state in which it has emitted the particle and not emitted the particle (with equal probability). Therefore the cat is both dead and undead. Schrödinger put it this way: "[The steel chamber would have] in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts." When the chamber is opened, the act of observation forces the entire system into one state or the other: The cat is either dead or alive, not both. However, while unobserved, the cat is both dead and alive. The absurdity of the both dead and not dead cat in Schrödinger's thought experiment was meant to demonstrate how quantum strangeness does not transfer to the macroscopic world. In this chapter, we examine the quantum-mechanical model of the atom, a model that explains the strange behavior of electrons. In particular, we focus on how the model describes electrons as they exist within atoms, and how those electrons determine the chemical and physical properties of elements. We have already learned much about these properties. We know, for example, that some elements are metals and that others are nonmetals. We know that the noble gases are chemically inert and that the alkali metals are chemically reactive. We know that sodium tends to form 1+ ions and that fluorine tends to form 1- ions. But we have not explored why. The quantum-mechanical model explains why. In doing so, it explains the modern periodic table and provides the basis for our understanding of chemical bonding.

Indeterminacy and Probability Distribution Maps

According to classical physics, and in particular Newton's laws of motion, particles move in a trajectory (or path) that is determined by the particle's velocity (the speed and direction of travel), its position, and the forces acting on it. Even if you are not familiar with Newton's laws, you probably have an intuitive sense of them. For example, when you chase a baseball in the outfield, you visually predict where the ball will land by observing its path. You do this by noting its initial position and velocity, watching how both are affected by the forces acting on it (gravity, air resistance, wind), and then inferring its trajectory, as illustrated in Figure 7.14▼. If you knew only the ball's velocity, or only its position (imagine a still photo of the baseball in the air), you could not predict its landing spot. In classical mechanics, both position and velocity are required to predict a trajectory. Newton's laws of motion are deterministic—the present determines the future. This means that if two baseballs are hit consecutively with the same velocity from the same position under identical conditions, they will land in exactly the same place. The same is not true of electrons. We have just seen that we cannot simultaneously know the position and velocity of an electron; therefore, we cannot know its trajectory. In quantum mechanics, trajectories are replaced with probability distribution maps, as shown in Figure 7.15▲. A probability distribution map is a statistical map that shows where an electron is likely to be found under a given set of conditions. To understand the concept of a probability distribution map, let us return to baseball. Imagine a baseball thrown from the pitcher's mound to a catcher behind home plate (Figure 7.16◀). The catcher can watch the baseball's path, predict exactly where it will cross home plate, and place his mitt in the correct place to catch it. As we have seen, the same prediction cannot be made for an electron. If an electron were thrown from the pitcher's mound to home plate, it would generally land in a different place every time, even if it were thrown in exactly the same way. This behavior is called indeterminacy. Unlike a baseball, whose future path is determined by its position and velocity when it leaves the pitcher's hand, the future path of an electron is indeterminate and can only be described statistically. In the quantum-mechanical world of the electron, the catcher could not know exactly where the electron would cross the plate for any given throw. However, if he kept track of hundreds of identical electron throws, the catcher could observe a reproducible statistical pattern of where the electron crosses the plate. He could even draw a map of the strike zone showing the probability of an electron crossing a certain area, as shown in. This would be a probability distribution map. In the sections that follow, we discuss quantum-mechanical electron orbitals, which are essentially probability distribution maps for electrons as they exist within atoms.

Amplitude and Wavelength

Amplitude and wavelength can vary independently of one another, as shown in on the next page. Like all waves, light is also characterized by its frequency (N), the number of cycles (or wave crests) that pass through a stationary point in a given period of time. The units of frequency are cycles per second (cycle/s) or simply s⁻¹. An equivalent unit of frequency is the hertz (Hz), defined as 1 cycle/s. The frequency of a wave is directly proportional to the speed at which the wave is traveling—the faster the wave, the more crests pass a fixed location per unit time. Frequency is inversely proportional to the wavelength (λ)—the farther apart the crests, the fewer that pass a fixed location per unit time. For light, therefore, we write: v = c/λ where the speed of light, c, and the wavelength, l, are expressed using the same unit of distance. Therefore, wavelength and frequency represent different ways of specifying the same information—if we know one, we can readily calculate the other. For visible light—light that can be seen by the human eye—wavelength (or, alter- natively, frequency) determines color. White light, as produced by the sun or by a light bulb, contains a spectrum of wavelengths and therefore a spectrum of colors. We see these colors—red, orange, yellow, green, blue, indigo, and violet—in a rainbow or when white light passes through a prism (Figure 7.3▼). Red light, with a wavelength of about 750 nanometers (nm), has the longest wavelength of visible light; violet light, with a wavelength of about 400 nm, has the shortest. The presence of a variety of wave lengths in white light is responsible for the colors that we perceive. When a substance absorbs some colors while reflecting others, it appears colored. For example, a redshirt appears red because it reflects predominantly red light while absorbing most other colors (Figure 7.4▼). Our eyes see only the reflected light, making the shirt appear red.

The de Broglie Wavelength

As we have seen, a single electron traveling through space has a wave nature; its wavelength is related to its kinetic energy (the energy associated with its motion). The faster the electron is moving, the higher its kinetic energy and the shorter its wavelength. The wavelength (λ) of an electron of mass m moving at velocity v is given by the de Broglie relation: λ = h/mv where h is Planck's constant. Notice that the velocity of a moving electron is related to its wavelength—knowing one is equivalent to knowing the other.

Quantum Mechanics of the Atom

As we have seen, the position and velocity of the electron are complementary properties— if we know one accurately, the other becomes indeterminate. Since velocity is directly related to energy (we have seen that kinetic energy equals ½ mv²), position and energy are also complementary properties—the more we know about one, the less we know about the other. Many of the properties of an element, however, depend on the energies of its electrons. For example, whether an electron is transferred from one atom to another to form an ionic bond depends in part on the relative energies of the electron in the two atoms. In the following paragraphs, we describe the probability distribution maps for electron states in which the electron has well-defined energy, but not well-defined position. In other words, for each state, we can specify the energy of the electron precisely, but not its location at a given instant. Instead, the electron's position is described in terms of an orbital, a probability distribution map showing where the electron is likely to be found. Since chemical bonding often involves the sharing of electrons between atoms to form covalent bonds, the spatial distribution of atomic electrons is important to bonding. The mathematical derivation of energies and orbitals for electrons in atoms comes from solving the Schrödinger equation for the atom of interest. The general form of the Schrödinger equation is: Hγ = Eγ (pronounced "sigh.") The symbol H is the Hamiltonian operator, a set of mathematical operations that represents the total energy (kinetic and potential) of the electron within the atom. The symbol E is the actual energy of the electron. The symbol γ is the wave function, a mathematical function that describes the wavelike nature of the electron. A plot of the wave function squared (γ²) represents an orbital, a position probability distribution map of the electron.

The Nature of Light

Before we explore electrons and their behavior within the atom, we must understand a few things about light. As quantum mechanics developed, light was (surprisingly) found to have many characteristics in common with electrons. Chief among these is the wave-particle duality of light. Certain properties of light are best described by thinking of it as a wave, while other properties are best described by thinking of it as a particle. In this chapter, we first explore the wave behavior of light, and then we discuss its particle behavior. We then turn to electrons to see how they display the same wave-particle duality.

p Orbitals (l ∙ 1)

Each principal level with n = 2 or greater contains three p orbitals (ml = -1, 0, +1). The three 2p orbitals and their radial distribution functions are shown in Figure 7.24▶. The p orbitals are not spherically symmetric like the s orbitals, but instead have two lobes of electron density on either side of the nucleus and a node located at the nucleus. The three p orbitals differ only in their orientation and are orthogonal (mutually perpendicular) to one another. It is convenient to define an x-, y-, and z-axis system and then label each p orbital as px, py, and pz. The 3p, 4p, 5p, and higher p orbitals are all similar in shape to the 2p orbitals, but they contain additional nodes (like the higher s orbitals) and are progressively larger in size.

d Orbitals (l ∙ 2)

Each principal level with n = 3 or greater contains five d orbitals (ml = -2, -1, 0, +1, +2). The five 3d orbitals are shown in Figure 7.25▼. Four of these orbitals each have a cloverleaf shape, with four lobes of electron density around the nucleus and two perpendicular nodal planes. (A nodal plane is a plane where the electron probability density is zero. For example, in the dxy orbitals, the nodal planes lie in the xz and yz planes.) The dxy, dxz, and dyz orbitals are oriented along the xy, xz, and yz planes, respectively, and their lobes are oriented between the corresponding axes. The four lobes of the dx2 - y2 orbital are oriented along the x- and y-axes. The dz 2 orbital is different in shape from the other four, having two lobes oriented along the z-axis and a donut- shaped ring along the xy plane. The 4d, 5d, 6d, etc., orbitals are all similar in shape to the 3d orbitals, but they contain additional nodes and are progressively larger in size.

f Orbitals (l ∙ 3)

Each principal level with n = 4 or greater contains seven f orbitals (ml = -3, -2, -1, 0, +1, +2, +3). These orbitals have more lobes and nodes than d orbitals.

Einstein's Idea

Einstein's idea that light was quantized elegantly explains the photoelectric effect. The emission of electrons from the metal depends on whether or not a single photon has sufficient energy (as given by hv) to dislodge a single electron. For an electron bound to the metal with binding energy ∅, the threshold frequency is reached when the energy of the photon is equal to ∅. Low-frequency light does not eject electrons because no single photon has the minimum energy necessary to dislodge the electron. We can draw an analogy between a photon ejecting an electron from a metal surface and a ball breaking a glass window. In this analogy, low-frequency photons are like ping pong balls—a ping pong ball thrown at a glass window does not break it (just as a low-frequency photon does not eject an electron). Increasing the intensity of low-frequency light is like increasing the number of ping pong balls thrown at the window—doing so simply increases the number of low-energy photons but does not produce any single photon with sufficient energy. In contrast, increasing the frequency of the light, even at low intensity, increases the energy of each photon. In our analogy, a high-frequency photon is like a baseball—a baseball thrown at a glass win- dow breaks it (just as a high-frequency photon dislodges an electron with no lag time). As the frequency of the light increases past the threshold frequency, the excess energy of the photon (beyond what is needed to dislodge the electron) is transferred to the electron in the form of kinetic energy. The kinetic energy (KE) of the ejected electron, therefore, is the difference between the energy of the photon (hv) and the binding energy of the electron, as given by the equation: KE = hv -∅ Although the quantization of light explained the photoelectric effect, the wave explanation of light continued to have explanatory power as well, depending on the The symbol ∅ is the Greek letter phi, pronounced "fee." circumstances of the particular observation. So the principle that slowly emerged (albeit with some measure of resistance) is the wave-particle duality of light. Sometimes light appears to behave like a wave, at other times like a particle. Which behavior we observe depends on the particular experiment performed.

atomic spectroscopy cont

For example, suppose that an electron in a hydrogen atom relaxes from an orbital in the n = 3 level to an orbital in the n = 2 level. Recall that the energy of an orbital in the hydrogen atom depends only on n and is given by En = -2.18 * 10⁻¹⁸J(1>n² ), where n = 1, 2, 3, c. Therefore, we determine ∆E, the energy difference corresponding to the transition from n = 3 to n = 2, as follows: The energy carries a negative sign because the atom emits the energy as it relaxes from n = 3 to n = 2. Because energy must be conserved, the exact amount of energy emitted by the atom is carried away by the photon. ∆Eatom = -Ephoton

The following diagram shows all of the orbitals, each represented by a small square, in the first three principal levels:

For example, the n = 2 level contains the l = 0 and l = 1 sublevels. Within the n = 2 level, the l = 0 sublevel—called the 2s sublevel—contains only one orbital (the 2s orbital), with ml = 0. The l = 1 sublevel—the 2p sublevel—contains three 2p orbitals, with ml = -1, 0, +1. Summarizing Level, Sublevels and Orbitals: ▶ The number of sublevels in any level is equal to n, the principal quantum number. Therefore, the n = 1 level has one sublevel, the n = 2 level has two sublevels, etc. ▶ The number of orbitals in any sublevel is equal to 2l + 1. Therefore, the s sublevel (l = 0) has one orbital, the p sublevel (l = 1) has three orbitals, the d sublevel (l = 2) has five orbitals, etc. ▶ The number of orbitals in a level is equal to n2 . Therefore, the n = 1 level has one orbital, the n = 2 level has four orbitals, the n = 3 level has nine orbitals, etc.

The Shapes of Atoms

If some orbitals are shaped like dumbbells and three-dimensional cloverleafs, and if most of the volume of an atom is empty space diffusely occupied by electrons in these orbitals, then why do we often depict atoms as spheres? Atoms are usually drawn as spheres because most atoms contain many electrons occupying a number of different orbitals. Therefore, the shape of an atom is obtained by superimposing all of its orbitals. If we superimpose s, p, and d orbitals, the result is a spherical shape, as shown in Figure 7.26▼.

The Wave Nature of Light

Light is electromagnetic radiation, a type of energy embodied in oscillating electric and magnetic fields. A magnetic field is a region of space where a magnetic particle experiences a force (think of the space around a magnet). An electric field is a region of space where an electrically charged particle experiences a force. We describe electromagnetic radiation as a wave composed of oscillating, mutually perpendicular electric and magnetic fields propagating through space, as shown in Figure 7.1▼. In a vacuum, these waves move at a constant speed of 3.00 * 10⁸ m/s (186,000 mi/s)—fast enough to circle Earth in one-seventh of a second. This great speed explains the delay between the moment when you see a firework in the sky and the moment when you hear the sound of its explosion. The light from the exploding fire work reaches your eye almost instantaneously. The sound, traveling much more slowly (340 m/s), takes longer. The same thing happens in a thunderstorm—you see the flash of the lightning immediately, but the sound of the thunder takes a few seconds to reach you. An electromagnetic wave, like all waves, is characterized by its amplitude and its wavelength. In the graphical representation shown here, the amplitude of the wave is the vertical height of a crest (or depth of a trough). The amplitude of the electric and magnetic field waves in light determines the intensity or brightness of the light—the greater the amplitude, the greater the intensity. The wavelength (L) of the wave is the distance in space between adjacent crests (or any two analogous points). We measure wavelength in units of distance such as the meter, micrometer, or nanometer. The wavelength of a light wave determines its color.

The Particle Nature of Light

Prior to the early 1900s, and especially after the discovery of the diffraction of light, light was thought to be purely a wave phenomenon. Its behavior was described adequately by classical electromagnetic theory, which treated the electric and magnetic fields that constitute light as waves propagating through space. However, a number of discoveries brought the classical view into question. Chief among those discoveries was the photoelectric effect. The photoelectric effect is the observation that many metals eject electrons when light shines upon them, as shown in Figure 7.8▲. Classical electromagnetic theory attributed this effect to the transfer of energy from the light to an electron in the metal, which resulted in the dislodgment of the electron. The amount of energy transferred from the light to the electron must exceed the electron's binding energy, the energy with which the electron was bound to the metal. Since the energy of a wave depends only on its amplitude (or intensity) according to classical electromagnetic theory, the rate at which electrons leave the metal due to the photoelectric effect would depend only on the intensity of the light shining upon the surface (not on the wavelength). If the intensity of the light was low, there should therefore be a lag time (a delay) between the initial shining of the light and the subsequent emission of an electron. The lag time would be the minimum amount of time required for the dim light to transfer sufficient energy to the electron to dislodge it. The experimental results, however, do not support the classical prediction. A high-frequency, low-intensity light produces electrons without the predicted lag time. Furthermore, the light used to dislodge electrons in the photoelectric effect exhibits a threshold frequency, below which no electrons are emitted from the metal, no matter how long the light shines on the metal. Figure 7.9◀ is a graph of the rate of electron ejection from the metal versus the frequency of light used. Notice that increasing the intensity of the light does not change the threshold frequency. In other words, low-frequency (long-wavelength) light does not eject electrons from a metal regardless of its intensity or its duration. But high- frequency (short-wavelength) light does eject electrons, even if its intensity is low. What can explain this odd behavior? In 1905, Albert Einstein proposed a bold explanation of this observation: Light energy must come in packets. According to Einstein, the amount of energy (E) in a light packet depends on its frequency (v) according to the equation: E = hv where h, called Planck's constant, has the value h = 6.626 * 10⁻³⁴ J×s. A packet oflight is called a photon or a quantum of light. Since v= c/λ, the energy of a photon can also be expressed in terms of wavelength as follows: E = hc/λ Unlike classical electromagnetic theory, in which light was viewed exclusively as a wave whose intensity was continuously variable, Einstein suggested that light was lumpy. From this perspective, a beam of light is not a wave propagating through space, but a shower of particles, each with energy hv

Atomic Spectroscopy Explained

Quantum theory explains the atomic spectra of atoms discussed in Section 7.3. Each wavelength in the emission spectrum of an atom corresponds to an electron transition between quantum-mechanical orbitals. When an atom absorbs energy, an electron in a lower-energy-level orbital is excited or promoted to a higher-energy-level orbital, as shown in Figure 7.18▼. In this new configuration, however, the atom is unstable, and the electron quickly falls back or relaxes to a lower energy orbital. As it does so, it releases a photon of light containing an amount of energy precisely equal to the energy difference between the two energy levels. We saw previously (see Equation 7.7) that the energy of an orbital with principal quantum number n is given by En = -2.18 * 10⁻¹⁸J(1>n²), where n = 1, 2, 3, c. Therefore, the difference in energy between two levels ninitial and nfinal is given by ∆E = Efinal - Einitial. If we substitute the expression for En into the expression for ∆E, we get the following important expression for the change in energy that occurs in an atom when an electron changes energy levels:

Atomic Spectroscopy and the Bohr Model

The discovery of the particle nature of light began to break down the division that existed in nineteenth-century physics between electromagnetic radiation, which was thought of as a wave phenomenon, and the small particles (protons, neutrons, and electrons) that compose atoms, which were thought to follow Newton's laws of motion (see Section 7.4). Just as the photoelectric effect suggested the particle nature of light, so certain observations of atoms began to suggest a wave nature for particles. The most important of these observations came from atomic spectroscopy, the study of the electromagnetic radiation absorbed and emitted by atoms. When an atom absorbs energy—in the form of heat, light, or electricity—it often reemits that energy as light. For example, a neon sign is composed of one or more glass tubes filled with neon gas. When an electric current is passed through the tube, the neon atoms absorb some of the electrical energy and re-emit it as the familiar red light of a neon sign. If the atoms in the tube are not neon atoms but those of a different gas, the emitted light is a different color. Atoms of each element emit light of a characteristic color. Mercury atoms, for example, emit light that appears blue, helium atoms emit light that appears violet, and hydrogen atoms emit light that appears reddish (Figure 7.10▶). Closer investigation of the light emitted by atoms reveals that it contains several distinct wavelengths. Just as the white light from a light bulb can be separated into its constituent wavelengths by passing it through a prism, the light emitted by an element can be separated, as shown in Figure 7.11▶ on the next page. The result is a series of bright light called an emission spectrum. The emission spectrum of a particular element is always the same—it consists of the same bright lines at the same characteristic wavelengths— and can be used to identify the element. For example, light arriving from a distant star contains the emission spectra of the elements that compose the star. Analysis of the light allows us to identify the elements present in the star. Notice the differences between a white light spectrum and the emission spectra of hydrogen, helium, and barium in Figure 7.11. The white light spectrum is continuous; there are no sudden interruptions in the intensity of the light as a function of wavelength it consists of light of all wavelengths. The emission spectra of hydrogen, helium, and barium, however, are not continuous—they consist of bright lines at specific wavelengths, with complete darkness in between. That is, only certain discrete wavelengths of light are present. Classical physics could not explain why these spectra consisted of discrete lines. In fact, according to classical physics, an atom composed of an electron orbiting a nucleus should emit a continuous white light spectrum. Even more problematic, the electron should lose energy as it emits the light and spiral into the nucleus. Johannes Rydberg, a Swedish mathematician, analyzed many atomic spectra and developed an equation (shown in the margin) that predicted the wavelengths of the hydrogen emission spectrum. However, his equation provided only limited insight into why atomic spectra were discrete, why atoms were stable, or why his equation worked. The Danish physicist Niels Bohr (1885-1962) attempted to develop a model for the atom that explained atomic spectra. In his model, electrons travel around the nucleus in circular orbits (similar to those of the planets around the sun). However, in contrast to planetary orbits—which can theoretically exist at any distance from the sun—Bohr's orbits could exist only at specific, fixed distances from the nucleus. The energy of each Bohr orbit was also fixed, or quantized. Bohr called these orbits stationary states and suggested that, although they obeyed the laws of classical mechanics, they also possessed "a peculiar, mechanically unexplainable, stability." We now know that the stationary states were really manifestations of the wave nature of the electron, which we expand upon shortly. Bohr further proposed that, in contradiction to classical electromagnetic theory, no radiation was emitted by an electron orbiting the nucleus in a stationary state. It was only when an electron jumped, or made a transition, from one stationary state to another that radiation was emitted or absorbed (Figure 7.12▶). The transitions between stationary states in an atom are quite unlike any transitions in the macroscopic world. The electron is never observed between states, only in one state or another—the transition between states is instantaneous. The emission spectrum of an atom consists of discrete lines because the states exist only at specific, fixed energies. The energy of the photon emitted when an electron makes a transition from one stationary state to another is the energy difference between the two stationary states.Transitions between stationary states that are closer together, therefore, produce light of lower energy (longer wavelength) than transitions between stationary states that are farther apart. In spite of its initial success in explaining the line spectrum of hydrogen (including the correct wavelengths), the Bohr model left many unanswered questions. It did, however, serve as an intermediate model between a classical view of the electron and a fully quantum-mechanical view, and therefore has great historical and conceptual importance. Nonetheless, it was ultimately replaced by a more complete quantum-mechanical theory that fully incorporated the wave nature of the electron.

The Wave Nature of Matter: The de Broglie Wavelength, the Uncertainty Principle, and Indeterminacy

The heart of the quantum-mechanical theory that replaced Bohr's model is the wave nature of the electron, first proposed by Louis de Broglie (1892-1987) in 1924 and confirmed by experiments in 1927. Although it seemed incredible at the time, electrons—which were thought of as particles and known to have mass—also have a wave nature. The wave nature of the electron is seen most clearly in its diffraction. If an electron beam is aimed at two closely spaced slits, and a series (or array) of detectors is arranged to detect the electrons after they pass through the slits, an interference pattern similar to that observed for light is recorded behind the slits (Figure 7.13(a)▶ on the next page). The detectors at the center of the array (midway between the two slits) detect a large number of electrons—exactly the opposite of what you would expect for particles (Figure 7.13(b)▶ on the next page). Moving outward from this center spot, the detectors alternately detect small numbers of electrons and then large numbers again and so on, forming an interference pattern characteristic of waves. It is critical to understand that this interference pattern is not caused by pairs of electrons interfering with each other, but rather by single electrons interfering with themselves. If the electron source is turned down to a very low level, so that electrons come out only one at a time, the interference pattern remains. In other words, we can design an experiment in which electrons come out of the source singly. We can then record where each electron strikes the detector after it has passed through the slits. If we record the positions of thousands of electrons over a long period of time, we find the same interference pattern shown in Figure 7.13(a). This leads us to an important conclusion: The wave nature of the electron is an inherent property of individual electrons. Recall from that unobserved electrons can simultaneously occupy two different states. In this case, the unobserved electron goes through both slits—it exists in two states simultaneously, just like Schrödinger's cat—and interferes with itself. As it turns out, this wave nature is what explains the existence of stationary states (in the Bohr model) and prevents the electrons in an atom from crashing into the nucleus as they are predicted to do according to classical physics. We now turn to three important manifestations of the electron's wave nature: the de Broglie wavelength, the uncertainty principle, and indeterminacy.

s Orbitals (l ∙ 0)

The lowest energy orbital is the spherically symmetrical 1s orbital shown in Figure 7.20(a)▼. This image is actually a three-dimensional plot of the wave function squared (γ²), which represents probability density, the probability (per unit volume) of finding the electron at a point in space. The magnitude of γ2 in this plot is proportional to the density of the dots shown in Figure 7.20(a). The high dot density near the nucleus indicates a higher probability density for the electron there. As you move away from the nucleus, the probability density decreases. Figure 7.20(b)▼ shows a plot of probability density (γ2) versus r, the distance from the nucleus. This is essentially a slice through the three-dimensional plot of (γ2) and shows how the probability density decreases as r increases. orbital on the shape representation, 90% of the dots would be within the sphere, meaning that when the electron is in the 1s orbital it has a 90% chance of being found within the sphere. The plots we have just seen (Figures 7.20 and 7.21) represent probability density. However, they are a bit misleading because they seem to imply that the electron is most likely to be found at the nucleus. To get a better idea of where the electron is most likely to be found, we use a plot called the radial distribution function, shown in Figure 7.22▶for the 1s orbital. The radial distribution function represents the total probability of finding the electron within a thin spherical shell at a distance r from the nucleus.

the magnetic Quantum Number (ml)

The magnetic quantum number is an integer that specifies the orientation of the orbital. We consider these orientations in Section 7.6. The possible values of ml are the integer values (including zero) ranging from -l to +l. For example, if l = 0, then the only possible value of ml is 0; if l = 1, the possible values of ml are -1, 0, and +1.

The Phase of Orbitals

The orbitals we have just seen are three-dimensional waves. We can understand an important property of these orbitals by analogy to one-dimensional waves. Consider the one-dimensional waves shown here:+ The wave on the left has a positive amplitude over its entire length, while the wave on the right has a positive amplitude over half of its length and a negative amplitude over the other half. The sign of the amplitude of a wave—positive or negative—is known as its phase. In these images, blue indicates positive phase and red indicates negative phase. The phase of a wave determines how it interferes with another wave, as we saw in Section 7.2. Just as a one-dimensional wave has a phase, so does a three-dimensional wave. We often represent the phase of a quantum-mechanical orbital with color. For example, we can represent the phase of a 1s and 2p orbital as follows:

the Spin Quantum Number (ms)

The spin quantum number specifies the orientation of the spin of the electron. Electron spin is a fundamental property of an electron (like its negative charge). One electron does not have more or less spin than another—all electrons have the same amount of spin. The orientation of the electron's spin is quantized, with only two possibilities that we can call spin up (ms = +1 ∙2) and spin down (ms = -1 ∙2) The spin quantum number becomes important when we begin to consider how electrons occupy orbitals (Section 8.3). For now, we focus on the first three quantum numbers. the Hydrogen atom Orbitals Each specific combination of the first three quantum numbers (n, l, and m) specifies one atomic orbital. For example, the orbital with n = 1, l = 0, and ml = 0 is known as the 1s orbital. The 1 in 1s is the value of n, and the s specifies that l = 0. There is only one 1s orbital in an atom, and its ml value is zero. Orbitals with the same value of n are said to be in the same principal level (or principal shell). Orbitals with the same value of n and l are said to be in the same sublevel (or subshell). The following diagram shows all of the orbitals, each represented by a small square, in the first three principal levels:

The Uncertainty Principle

The wave nature of the electron is difficult to reconcile with its particle nature. How can a single entity behave as both a wave and a particle? We can begin to answer this question by returning to the single-electron diffraction experiment. Specifically, we can ask the question: How does a single electron aimed at a double slit produce an interference pattern? We saw previously that the electron travels through both slits and interferes with itself. This idea is testable. We simply have to observe the single electron as it travels through both of the slits. If it travels through both slits simultaneously, our hypothesis is correct. But here is where nature gets tricky. Any experiment designed to observe the electron as it travels through the slits results in the detection of an electron "particle" traveling through a single slit and no interference pattern. Recall from Section 7.1 that an unobserved electron can occupy two different states; however, the act of observation forces it into one state or the other. Similarly, the act of observing the electron as it travels through both slits forces it to travel through only one slit. The electron diffraction experiment described in the following paragraph is designed to "watch" which slit the electron travels through by using a laser beam placed directly behind the slits. An electron that crosses a laser beam produces a tiny "flash"—a single photon is scattered at the point of crossing. A flash behind a particular slit indicates an electron passing through that slit. However, when the experiment is performed, the flash always originates either from one slit or the other, but never from both at once. Furthermore, the interference pattern, which was present without the laser, is now absent. With the laser on, the electrons hit positions directly behind each slit, as if they were ordinary particles. As it turns out, no matter how hard we try, or whatever method we set up, we can never see the interference pattern and simultaneously determine which hole the electron goes through. It has never been done, and most scientists agree that it never will be. In the words of P. A. M. Dirac (1902-1984), There is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance—a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer. The single-electron diffraction experiment demonstrates that we cannot simultaneously observe both the wave nature and the particle nature of the electron. When we try to observe which slit the electron goes through (associated with the particle nature of the electron), we lose the interference pattern (associated with the wave nature of the electron). When we try to observe the interference pattern, we cannot determine which slit the electron goes through. The wave nature and particle nature of the electron are said to be complementary properties. Complementary properties exclude one another—the more we know about one, the less we know about the other. Which of two complementary properties we observe depends on the experiment we perform—in quantum mechanics, the observation of an event affects its outcome. As we just saw in the de Broglie relation, the velocity of an electron is related to its wave nature. The position of an electron, however, is related to its particle nature. (Particles have well-defined positions, but waves do not.) Consequently, our inability to observe the electron simultaneously as both a particle and a wave means that we cannot simultaneously measure its position and its velocity. Werner Heisenberg formalized this idea with the equation: ∆x × m∆v ≥ h/4π where ∆x is the uncertainty in the position, ∆v is the uncertainty in the velocity, m is the mass of the particle, and h is Planck's constant. Heisenberg's uncertainty principle states that the product of ∆x and m ∆v must be greater than or equal to a finite number (h/4π). In other words, the more accurately we know the position of an electron (the smaller ∆x), the less accurately we can know its velocity (the bigger ∆v) and vice versa. The complementarity of the wave nature and particle nature of the electron results in the complementarity of velocity and position. Although Heisenberg's uncertainty principle may seem puzzling, it actually solves a great puzzle. Without the uncertainty principle, we are left with the question: How can something be both a particle and a wave? Saying that an object is both a particle and a wave is like saying that an object is both a circle and a square—a contradiction. Heisenberg solved the contradiction by introducing complementarity—an electron is observed as either a particle or a wave, but never both at once. This idea was captured by Schrödinger's thought experiment about the cat we discussed in Section 7.1: When observed, the cat is either dead or alive, not both.

The Electromagnetic Spectrum

Visible light makes up only a tiny portion of the entire electromagnetic spectrum, which includes all known wavelengths of electromagnetic radiation. Figure 7.5▼ shows the main regions of the electromagnetic spectrum, ranging in wavelength from 10⁻¹⁵ m (gamma rays) to 10⁵ m (radio waves). We will see later in this section that short-wavelength light inherently has greater energy than long-wavelength light (because of the particle nature of light). Therefore, the most energetic forms of electromagnetic radiation have the shortest wavelengths. The form of electromagnetic radiation with the shortest wavelength is the gamma (γ) ray. Gamma rays are produced by the sun, other stars, and certain unstable atomic nuclei on Earth. Human exposure to gamma rays is dangerous because the high energy of gamma rays can damage biological molecules. Next on the electromagnetic spectrum, with longer wavelengths than gamma rays, are X-rays, familiar to us from their medical use. X-rays pass through many substances that block visible light and are therefore used to image bones and internal organs. Like gamma rays, X-rays are sufficiently energetic to damage biological molecules. While several yearly exposures to X-rays are relatively harmless, excessive exposure to X-rays increases cancer risk. Sandwiched between X-rays and visible light in the electromagnetic spectrum is ultraviolet (UV) radiation, most familiar as the component of sunlight that produces a sunburn or suntan. While not as energetic as gamma rays or X-rays, ultraviolet light still carries enough energy to damage biological molecules. Excessive exposure to ultraviolet light increases the risk of skin cancer and cataracts and causes premature wrinkling of the skin. Next on the spectrum is visible light, ranging from violet (shorter wavelength, higher energy) to red (longer wavelength, lower energy). Visible light—as long as the intensity is not too high—does not carry enough energy to damage biological molecules. It does, however, cause certain molecules in our eyes to change their shape, sending a signal to our brains that results in vision. Beyond visible light lies infrared (IR) radiation. The heat we feel when we place our hands near hot objects is infrared radiation. All warm objects, including human bodies, emit infrared light. Although infrared light is invisible to our eyes, infrared sensors can detect it and are used in night vision technology to "see" in the dark. At longer wavelengths still are microwaves, used for radar and in microwave ovens. Although microwave radiation has longer wavelengths and therefore lower energies than visible or infrared light, it is efficiently absorbed by water and can therefore heat substances that contain water. This principle is at work in a microwave oven. The longest wavelengths are those of radio waves, which are used to transmit the signals responsible for AM and FM radio, cellular telephones, television, and other forms of communication.

Interference and Diffraction

Waves, including electromagnetic waves, interact with each other in a characteristic way called interference: They can cancel each other out or build each other up, depending on their alignment upon interaction. For example, if waves of equal amplitude from two sources are in phase when they interact—that is, if they align with overlapping crests—a wave with twice the amplitude results. This is constructive interference. On the other hand, if the waves are completely out of phase—that is, if they align so that the crest from one source overlaps the trough from the other source—the waves cancel by destructive interference. When a wave encounters an obstacle or a slit that is comparable in size to its wave- length, it bends around it—a phenomenon called diffraction (Figure 7.6▶). The diffraction of light through two slits separated by a distance comparable to the wavelength of the light results in an interference pattern, as shown in Figure 7.7▶. Each slit acts as a new wave source, and the two new waves interfere with each other. The resulting pattern consists of a series of bright and dark lines that can be viewed on a screen (or recorded on a film) placed at a short distance behind the slits. At the center of the screen, the two waves travel equal distances and interfere constructively to produce a bright line. However, a small distance away from the center in either direction, the two waves travel slightly different distances, so that they are out of phase. At the point where the difference in distance is one-half of a wavelength, the interference is destructive and a dark line appears on the screen. Moving a bit further away from the center produces constructive interference again because the difference between the paths is one whole wavelength. The end result is the interference pattern shown in the figure. Notice that interference results from the ability of a wave to diffract through the two slits—this is an inherent property of waves.

Solutions to the Schrödinger Equation for the Hydrogen Atom

When the Schrödinger equation is solved, it yields many solutions—many possible wave functions. The wave functions themselves are fairly complicated mathematical functions, and we do not examine them in detail in this book. Instead, we introduce graphical representations (or plots) of the orbitals that correspond to the wave functions. Each orbital is specified by three interrelated quantum numbers: n, the principal quantum number; l, the angular momentum quantum number (sometimes called the azimuthal quantum number); and mι, the magnetic quantum number. These quantum numbers all have integer values, as had been hinted at by both the Rydberg equation and Bohr's model. A fourth quantum number, ms, the spin quantum number, specifies the orientation of the spin of the electron. We examine each of these quantum numbers individually. the principal Quantum Number (n) The principal quantum number is an integer that determines the overall size and energy of an orbital. Its possible values are n = 1, 2, 3, c and so on. For the hydrogen atom, the energy of an electron in an orbital with quantum number n is: En = -2.18 x 10⁻¹⁸ J(1/n²) (n = 1,2,3,..) The energy is negative because the energy of the electron in the atom is less than the energy of the electron when it is very far away from the atom (which is taken to be zero). Notice that orbitals with higher values of n have greater (less negative) energies, as shown in the energy level diagram below. Notice also that, as n increases, the spacing between the energy levels decreases.