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4.1 Energy 4.2 The Mysterious Electron 4.3 Multi-Electron Atoms To see a World in a Grain of Sand And a Heaven in a Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour William Blake (1757-1827) Auguries of Innocence cientists’ attempts to understand the atom have led them into the unfamiliar world of the unimaginably small, where the rules of physics seem to be different from the rules in the world we can see and touch. Scientists explore this world through the use of mathematics. Perhaps this is similar to the way a writer uses poetry to express ideas and feelings beyond the reach of everyday language. Mathematics allows the scientist to explore beyond the boundaries of the world we can experience directly. Just as scholars then try to analyze the poems and share ideas about them in everyday language, scientists try to translate the mathematical description of the atom into words that more of us can understand. Although both kinds of translation are fated to fall short of capturing the fundamental truths of human nature and the physical world, the attempt is worthwhile for the occasional glimpse of those truths that it provides. This chapter offers a brief, qualitative introduction to the mathematical description of electrons and describes the highly utilitarian model of atomic structure that chemists have constructed from it. Because we are reaching beyond the world of our senses, we should not be surprised that the model we create is uncertain and, when described in normal language, a bit vague. In spite of these limitations, however, you will return from your journey into the strange, new world of the extremely small with a useful tool for explaining and predicting the behavior of matter. Because the modern description of the atom is closely tied to the concept of energy, we begin this chapter with an introduction to energy and its different forms. Chemists try to “see” the structure of matter even more closely than can be seen in any photograph. Review Skills The presentation of information in this chapter assumes that you can already perform the tasks listed below. You can test your readiness to proceed by answering the Review Questions at the end of the chapter. This might also be a good time to read the Chapter Objectives, which precede the Review Questions. Describe the relationship between temperature and motion. (Section 3.1) Describe the nuclear model of the atom. (Section 3.4) Describe the similarities and differences between solids, liquids, and gases with reference to the particle nature of matter, the degree of motion of the particles, and the degree of attraction between the particles. (Section 3.1)
4.1 Energy Energy…it makes things happen. To get an idea of the role energy plays in our lives, let’s spend some time with John, a college student in one of the coastal towns in California. He wakes up in the morning to a beautiful sunny day and decides to take his chemistry book to the beach. Before leaving, he fries up some scrambled eggs, burns some toast, and pops a cup of day-old coffee in the microwave oven. After finishing his breakfast, he shoves his chemistry textbook into his backpack and jumps on his bike for the short ride to the seashore. Once at the beach, he reads two pages of his chemistry assignment, and despite the fascinating topic, gets drowsy and drops off to sleep. When he wakes up an hour later, he’s real sorry that he forgot to put on his sunscreen. His painful sunburn drives him off the beach and back to his apartment to spend the rest of the day inside. All of John’s actions required energy. It took energy to get out of bed, make breakfast, pedal to the beach, and (as you well know) read his chemistry book. John gets that energy from the chemical changes that his body induces in the food he eats. It took heat energy to cook his eggs and burn his toast. The radiant energy from microwaves raised the temperature of his coffee, and the radiant energy from the sun caused his sunburn. All chemical changes are accompanied by energy changes. Some reactions, such as the combustion of methane (a component of natural gas) release energy. This is why natural gas can be used to heat our homes. Other reactions absorb energy. For example, when energy from the sun strikes oxygen molecules, O 2 , in the Earth’s atmosphere, some of the energy is absorbed by the molecules, causing them to break apart into separate atoms (Figure 4.1). Radiant energy from the sun causes sunburn Figure 4. Some reactions absorb energy.
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Before we can begin to explain the role that energy plays in these and other chemical reactions, we need to get a better understanding of what energy is and the different forms it can take. You probably have a general sense of what energy is. When you get up in the morning after a good night’s sleep, you feel that you have plenty of energy to get your day’s work done. After a long day of studying chemistry, you might feel like you hardly have the energy necessary to drag yourself to bed. The main goal of this section is to give you a more specific, scientific understanding of energy. The simplest definition of energy is that it is the capacity to do work. Work, in this context, may be defined as what is done to move an object against some sort of resistance. For example, when you push this book across a table, the work you do overcomes the resistance caused by the contact between the book and the table.
Potential Energy Energy can be transferred from one object to another. Picture the coin-toss that precedes a football game. A coin starts out resting in the referee’s hand. After he flips it, sending it moving up into the air, it has some kinetic energy that it did not have before it was flipped. Where did the coin get this energy? From the referee’s moving thumb. When scientists analyze such energy transfers, they find that all of the energy still exists. The Law of Conservation of Energy states that energy can be neither created nor destroyed, but it can be transferred from one system to another and changed from one form to another.^1 As the coin rises, it slows down and eventually stops. At this point, the kinetic energy it got from the referee’s moving thumb is gone, but the Law of Conservation of Energy says that energy cannot be destroyed. Where did the kinetic energy go? Although some of it has been transferred to the air particles it bumps into on its flight, most of the energy is still there in the coin in a form called potential energy (PE) , which is the retrievable, stored form of energy an object possesses by virtue of its position or state. We get evidence of this transformation when the coin falls back down toward the grass on the field. The potential energy it had at the peak of its flight is converted into kinetic energy of its downward movement, and this kinetic energy does the work of flattening a few blades of grass when the coin hits the field (Figure 4.4). Figure 4. Law of Conservation of Energy. There are many kinds of potential energy. An alkaline battery contains potential energy that can be used to move a toy car. A plate of pasta provides potential energy to allow your body to move. Knowing the relationships between potential energy and stability can help you to recognize changes in potential energy and to decide whether the potential energy has increased or decreased as a result of each change. Let’s look at the relationship between potential energy and stability. A system’s stability is a measure of its tendency to change. A more stable system is less likely to change than a less stable system. As an object moves from a less stable state to a more stable state, it can do work. Thus, as an object becomes less stable, it gains a greater capacity to do work and, therefore, a greater potential energy. For example, a coin in your hand is less likely to move than a flipped coin at the peak of its flight, so we say that the coin in the hand is more stable than the coin in the air. As the coin moves objeCtive 5 objeCtive 6 objeCtive 4 (^1) Although chemists recognize that matter can be converted into energy and energy into matter, this matter-energy conversion is small enough to be disregarded.
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objeCtive 7 objeCtive 5 objeCtive 7 from its less stable state in the air to a more stable state on the ground, it collides with and moves particles in the air and blades of grass. Therefore, the coin at the peak of its flight has a greater capacity to do the work of moving the objects, and, therefore, a greater potential energy than the more stable coin in the hand (Figure 4.5). Any time a system shifts from a more stable state to a less stable state, the potential energy of the system increases. We have already seen that kinetic energy is converted into potential energy as the coin is moved from the more stable position in the hand to the less stable position in the air. more stable + energy → less stable system lesser capacity to do work + energy → greater capacity to do work lower PE + energy → higher PE coin in hand + energy → coin in air above hand Figure 4. Relationship Between Stability and Potential Energy Just as energy is needed to propel a coin into the air and increase its potential energy, energy is also necessary to separate two atoms being held together by mutual attraction in a chemical bond. The energy supplied increases the potential energy of the less stable separate atoms compared to the more stable atoms in the bond. For example, the first step in the formation of ozone in the earth’s atmosphere is the breaking of the oxygen-oxygen covalent bonds in more stable oxygen molecules, O 2 , to form less stable separate oxygen atoms. This change could not occur without an input of considerable energy, in this case, radiant energy from the sun. We call changes that absorb energy endergonic (or endogonic) changes (Figure 4.6). Figure 4. Endergonic Change objeCtive 6
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objeCtive 2 exaMple 4.1 - Energy For each of the following situations, you are asked which of two objects or substances has the higher energy. Explain your answer with reference to the capacity of each to do work and say whether the energy that distinguishes them is kinetic energy or potential energy. a. Incandescent light bulbs burn out because their tungsten filament gradually evaporates, weakening until it breaks. Argon gas is added to these bulbs to reduce the rate of evaporation. Which has greater energy, (1) an argon atom, Ar, with a velocity of 428 m/s or (2) the same atom moving with a velocity of 456 m/s? (These are the average velocities of argon atoms at 20 °C and 60 °C.) b. Krypton, Kr, gas does a better job than argon of reducing the rate of evaporation of the tungsten filament in an incandescent light bulb. Because of its higher cost, however, krypton is only used when longer life is worth the extra cost. Which has higher energy, (1) an argon atom with a velocity of 428 m/s or (2) a krypton atom moving at the same velocity? c. According to our model for ionic solids, the ions at the surface of the crystal are constantly moving out and away from the other ions and then being attracted back to the surface. Which has more energy, (1) a stationary sodium ion well separated from the chloride ions at the surface of a sodium chloride crystal or (2) a stationary sodium ion located quite close to the chloride ions on the surface of the crystal? d. The chemical reactions that lead to the formation of polyvinyl chloride (PVC), which is used to make rigid plastic pipes, are initiated by the decomposition of peroxides. The general reaction is shown below. The simplest peroxide is hydrogen peroxide, H 2 O 2 or HOOH. Which has more energy, (1) a hydrogen peroxide molecule or (2) two separate HO molecules that form when the relatively weak O–O bond in an HOOH molecule is broken? HOOH → 2HO e. Hydrogen atoms react with oxygen molecules in the earth’s upper atmosphere to form HO 2 molecules. Which has higher energy, (1) a separate H atom and O 2 molecule or (2) an HO 2 molecule? H( g ) + O 2 ( g ) → HO 2 ( g ) f. Dry ice—solid carbon dioxide—sublimes, which means that it changes directly from solid to gas. Assuming that the temperature of the system remains constant, which has higher energy, (1) the dry ice or (2) the gaseous carbon dioxide? objeCtive 5 objeCtive 3
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Solution a. Any object in motion can collide with another object and move it, so any object in motion has the capacity to do work. This capacity to do work resulting from the motion of an object is called kinetic energy, KE. The particle with the higher velocity will move another object (such as another atom) farther, so it can do more work. It must therefore have more energy. In short, an argon atom with a velocity of 456 m/s has greater kinetic energy than the same atom with a velocity of 428 m/s. b. The moving particle with the higher mass can move another object (such as another molecule) farther, so it can do more work and must therefore have more energy. Thus the more massive krypton atoms moving at 428 m/s have greater kinetic energy than the less massive argon atoms with the same velocity. c. Separated ions are less stable than atoms in an ionic bond, so the separated sodium and chloride ions have higher potential energy than the ions that are closer together. The attraction between the separated sodium cation and the chloride anion pulls them together; as they approach each other, they could conceivably bump into another object, move it, and do work. d. Separated atoms are less stable and have higher potential energy than atoms in a chemical bond, so energy is required to break a chemical bond. Thus energy is required to separate the two oxygen atoms of HOOH being held together by mutual attraction in a chemical bond. The energy supplied is represented in the higher potential energy of separate HO molecules compared to the HOOH molecule. If the bond were reformed, the potential energy would be converted into a form of energy that could be used to do work. In short, two HO molecules have higher potential energy than an HOOH molecule. e. Atoms in a chemical bond are more stable and have lower potential energy than separated atoms, so energy is released when chemical bonds form. When H and O 2 are converted into an HO 2 molecule, a new bond is formed, and some of the potential energy of the separate particles is released. The energy could be used to do some work. H( g ) + O 2 ( g ) → HO 2 ( g ) Therefore, separated hydrogen atoms and oxygen molecules have higher potential energy than the HO 2 molecules that they form. f. When carbon dioxide sublimes, the attractions that link the CO 2 molecules together are broken. The energy that the dry ice must absorb to break these attractions goes to increase the potential energy of the CO 2 as a gas. If the CO 2 returns to the solid form, attractions are reformed, and the potential energy is converted into a form of energy that could be used to do work. Therefore, gaseous CO 2 has higher potential energy than solid CO 2.
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and kilojoules to describe energy in this text. Figure 4.8 shows some approximate values in kilojoules for the energy represented by various events. Table 4.1 Approximate Energy Provided by Various Foods Food Dietary Calories (kcal) kilojoules (kJ) Food Dietary Calories (kcal) kilojoules (kJ) Cheese pizza (12 inch diameter)
Unsweetened apple juice (1 cup)
Roasted cashew nuts (1 cup)
Butter (1 tablespoon)
White granular sugar (1 cup)
Raw apple (medium sized)
Dry rice (1 cup)
Chicken’s egg (extra large)
Wheat flour (1 cup)
Cheddar cheese (1 inch cube)
Ice cream - 10% fat (1 cup)
Whole wheat bread (1 slice)
Raw broccoli (1 pound)
Black coffee (6 fl oz cup)
Figure 4. Approximate Energy of Various Events (The relative sizes of these measurements cannot be shown on such a small page. The wedge and the numbers of increasing size are to remind you that each numbered measurement on the scale represents 10,000,000,000 times the magnitude of the preceding numbered measurement.) Kinetic Energy and Heat An object’s kinetic energy can be classified as internal or external. For example, a falling coin has a certain external kinetic energy that is related to its overall mass and to its velocity as it falls. The coin is also composed of particles that, like all particles, are moving in a random way, independent of the overall motion (or position) of the coin. The particles in the coin are constantly moving, colliding, changing direction, and changing their velocities. The energy associated with this internal motion is called internal kinetic energy (Figure 4.9).
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Figure 4. External Kinetic Energy and Internal Kinetic Energy objeCtive 11 The amount of internal kinetic energy in an object can be increased in three general ways. The first way is to rub, compress, or distort the object. For example, after a good snowball fight, you can warm your hands by rubbing them together. Likewise, if you beat on metal with a hammer, it will get hot. The second way to increase the internal kinetic energy of an object is to put it in contact with another object at a higher temperature. Temperature is proportional to the average internal kinetic energy of an object, so higher temperature means a greater average internal energy for the particles within the object. The particles in a higher-temperature object collide with other particles with greater average force than the particles of a lower-temperature object. Thus collisions between the particles of two objects at different temperatures cause the particles of the lower-temperature object to speed up, increasing the object’s energy, and cause the particles of the higher-temperature object to slow down, decreasing this object’s energy. In this way, energy is transferred from the higher-temperature object to the lower-temperature object. We call energy that is transferred in this way heat. The energy that is transferred through an object, as from the bottom of a cooking pan to its handle, is also called heat. Heat is the energy that is transferred from a region of higher temperature to a region of lower temperature as a consequence of the collisions of particles (Figure 4.10). objeCtive 13 objeCtive 12 objeCtive 14
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Figure 4. Heat Transfer objeCtive 14 The third way an object’s internal kinetic energy is increased is by exposure to radiant energy, such as the energy coming from the sun. The radiant energy is converted to kinetic energy of the particles in the object. This is why we get hot in the sun.
objeCtive 17 objeCtive 18 Gamma rays, with very high-energy photons, have very short wavelengths (Figure 4.12), on the order of 10–14^ meters (or 10–5^ nm). Short wavelengths are often described with nanometers, nm, which are 10–9^ m. In contrast, the radio waves on the low-energy end of the AM radio spectrum have wavelengths of about 500 m (about one-third of a mile). If you look at the energy and wavelength scales in Figure 4.12, you will see that longer wavelength corresponds to lower-energy photons. The shorter the wavelength of a wave of electromagnetic radiation, the greater the energy of its photons. In other words, the energy, ε, of a photon is inversely proportional to the radiation’s wavelength, λ. (The symbol ε is a lower case Greek epsilon, and the λ is a lowercase Greek lambda.) As Figure 4.12 illustrates, all forms of radiant energy are part of a continuum with no precise dividing lines between one form and the next. In fact, there is some overlap between categories. Note that visible light is only a small portion of the radiant energy spectrum. The different colors of visible light are due to different photon energies and associated wavelengths. objeCtive 19
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Figure 4. Radiant-Energy Spectrum
4.2 The Mysterious Electron Where there is an open mind, there will always be a frontier. Charles F. Kettering (1876-1958) American engineer and inventor Scientists have known for a long time that it is incorrect to think of electrons as tiny particles orbiting the nucleus like planets around the sun. Nevertheless, nonscientists have become used to picturing them in this way. In some circumstances, this “solar system” model of the atom may be useful, but you should know that the electron is much more unusual than that model suggests. The electron is extremely tiny, and modern physics tells us that strange things happen in the realm of the very, very small. The modern description of the electron is based on complex mathematics and on the discoveries of modern physics. The mathematical complexity alone makes an accurate verbal portrayal of the electron challenging, but our difficulty in describing the electron goes beyond complexity. Modern physics tells us that it is impossible to know exactly where an electron is and what it is doing. As your mathematical and scientific knowledge increases, you will be able to understand more sophisticated descriptions of the electron, but the problem of describing exactly where the electron is and what it is doing never goes away. It is a problem fundamental to very tiny objects. Thus complete confidence in our description of the nature of the electron is beyond our reach. There are two ways that scientists deal with the problems associated with the complexity and fundamental uncertainty of the modern description of the electron: Analogies In order to communicate something of the nature of the electron, scientists often use analogies, comparing the electron to objects with which we are more familiar. For example, in this chapter we will be looking at the ways in which electrons are like vibrating guitar strings. Probabilities In order to accommodate the uncertainty of the electron’s position and motion, scientists talk about where the electron probably is within the atom, instead of where it definitely is. Through the use of analogies and a discussion of probabilities, this chapter attempts to give you a glimpse of what scientists are learning about the electron’s character. Standing Waves and Guitar Strings Like radiant energy, each electron seems to have a dual nature in which both particle and wave characteristics are apparent. It is difficult to describe these two aspects of an electron at the same time, so sometimes we focus on its particle nature and sometimes on its wave character, depending on which is more suitable in a given context. In the particle view, electrons are tiny, negatively charged particles with a mass of about 9.1096 × 10 -^28 grams. In the wave view, an electron has an effect on the space around it that can be described as a wave of varying negative charge intensity. To gain a better understanding of this electron-wave character, let’s compare it to the wave character of guitar strings. Because a guitar string is easier to visualize than an electron, its vibrations serve as a useful analogy of the wave character of electrons.
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Electrons as Standing Waves The wave character of the guitar string is represented by the movement of the string. We can focus our attention on the blur of the waveform and forget the material the string is made of. The waveform describes the motion of the string over time, not the string itself. In a similar way, the wave character of the electron is represented by the waveform of its negative charge, on which we can focus without concerning ourselves about the electron’s particle nature. This frees us from asking questions about where the electrons are in the atom and how they are moving—questions that we are unable to answer. The waveforms for electrons in an atom describe the variation in intensity of negative charge within the atom, with respect to the location of the nucleus. This can be described without mentioning the positions and motion of the electron particle itself. The following statements represent the core of the modern description of the wave character of the electron: Just as the intensity of the movement of a guitar string can vary, so can the intensity of the negative charge of the electron vary at different positions outside the nucleus. The variation in the intensity of the electron charge can be described in terms of a three-dimensional standing wave like the standing wave of the guitar string. As in the case of the guitar string, only certain waveforms are possible for the electron in an atom. We can focus our attention on the waveform of varying charge intensity without having to think about the actual physical nature of the electron. Thus, the task is not so much to see what no one has yet seen, but to think what nobody has yet thought, about that which everybody sees. Erwin Schrodinger (1887-1961) Austrian physicist and Nobel laureate Waveforms for Hydrogen Atoms Most of the general descriptions of electrons found in the rest of this chapter are based on the wave mathematics for the one electron in a hydrogen atom. The comparable calculations for other elements are too difficult to lead to useful results, so as you will see in the next section, the information calculated for the hydrogen electron is used to describe the other elements as well. Fortunately, this approximation works quite well. The wave equation for the one electron of a hydrogen atom predicts waveforms for the electron that are similar to the allowed waveforms for a vibrating guitar string. For example, the simplest allowed waveform for the guitar string looks something like The simplest allowed waveform for an electron in a hydrogen atom looks like the image in Figure 4.15. The cloud that you see surrounds the nucleus and represents the variation in the intensity of the negative charge at different positions outside the nucleus. The negative charge is most intense at the nucleus and diminishes with increasing distance from the nucleus. The variation in charge intensity for this waveform is the same in all directions, so the waveform is a sphere. The allowed waveforms for objeCtive 22 objeCtive 21
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the electron are also called orbitals. The orbital shown in Figure 4.15 is called the 1 s orbital. Figure 4. Waveform of the 1 s Electron Theoretically, the charge intensity depicted in Figure 4.15 decreases toward zero as the distance from the nucleus approaches infinity. This suggests the amusing possibility that some of the negative charge created by an electron in a hydrogen atom is felt an infinite distance from the atom’s nucleus. The more practical approach taken by chemists, however, is to specify a volume that contains most of the electron charge and focus their attention on that, forgetting about the small negative charge felt outside the specified volume. For example, we can focus on a sphere containing 90% of the charge of the 1 s electron. If we wanted to include more of the electron charge, we enlarge the sphere so that it encloses 99% (or 99.9%) of the electron charge (Figure 4.16). This leads us to another definition of orbital as the volume that contains a given high percentage of the electron charge. Most of the pictures you will see of orbitals represent the hypothetical surfaces that surround a high percentage of the negative charge of an electron of a given waveform. The 1 s orbital, for example, can either be represented by a fuzzy sphere depicting the varying intensity of the negative charge (Figure 4.15) or by a smooth spherical surface depicting the boundary within which most of the charge is to be found (Figure 4.16). Is the sphere in Figure 4.15 the 1 s electron? This is like asking if the guitar string is the blur that you see when the string vibrates. When we describe the standing wave that represents the motion of a guitar string, we generally do not refer to the material composition of the string. The situation is very similar for the electron. We are able to describe the variation in intensity of the negative charge created by the electron without thinking too much about what the electron is and what it is doing. Figure 4. 1 s Orbital objeCtive 22 objeCtive 22 objeCtive 22
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Other Important Orbitals Just like the guitar string can have different waveforms, the one electron in a hydrogen atom can also have different waveforms, or orbitals. The shapes and sizes for these orbitals are predicted by the mathematics associated with the wave character of the hydrogen electron. Figure 4.18 shows some of them. Figure 4. Some Possible Waveforms, or Orbitals, for an Electron in a Hydrogen Atom Before considering the second possible orbital for the electron of a hydrogen atom, let’s look at another of the possible ways a guitar string can vibrate. The guitar string waveform below has a node in the center where there is no movement of the string. The electron-wave calculations predict that an electron in a hydrogen atom can have a waveform called the 2 s orbital that is analogous to the guitar string waveform above. The 2 s orbital for an electron in a hydrogen atom is spherical like the 1 s orbital, but it is a larger sphere. All spherical electron waveforms are called s orbitals. For an electron in the 2 s orbital, the charge is most intense at the nucleus. With increasing distance from the nucleus, the charge diminishes in intensity until it reaches a minimum at a certain distance from the nucleus; it then increases again to a maximum, and finally it objeCtive 24
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diminishes again. The region within the 2 s orbital where the charge intensity decreases to zero is called a node. Figure 4.19 shows cutaway, quarter-section views of the 1 s and 2 s orbitals. Figure 4. Quarter Sections of the 1 s and 2 s Orbitals The average distance between the positive charge of the nucleus and the negative charge of a 2 s electron cloud is greater than the average distance between the nucleus and the charge of a 1 s electron cloud. Because the strength of the attraction between positive and negative charges decreases with increasing distance between the charges, an electron is more strongly attracted to the nucleus and therefore is more stable when it has the smaller 1 s waveform than when it has the larger 2 s waveform. As you discovered in Section 4.1, increased stability is associated with decreased potential energy, so a 1 s electron has lower potential energy than a 2 s electron. We describe this energy difference by saying the 1 s electron is in the first principal energy level, and the 2 s electron is in the second principal energy level. All of the orbitals that have the same potential energy for a hydrogen atom are said to be in the same principal energy level. The principal energy levels are often called shells. The 1 in 1 s and the 2 in 2 s show the principal energy levels, or shells, for these orbitals. Chemists sometimes draw orbital diagrams, such as the following, with lines to represent the orbitals in an atom and arrows (which we will be adding later) to represent electrons: The line representing the 2 s orbital is higher on the page to indicate its higher potential energy. Because electrons seek the lowest energy level possible, we expect the electron in a hydrogen atom to have the 1 s waveform, or electron cloud. We say that the electron is in the 1 s orbital. But the electron in a hydrogen atom does not need to stay in the 1 s orbital at all times. Just as an input of energy (a little arm work on our part) can lift a book resting on a table and raise it to a position that has greater potential energy, so can the waveform of an electron in a hydrogen atom be changed from the 1 s shape to the 2 s shape by the addition of energy to the atom. We say that the electron can be excited from the 1 s orbital to the 2 s orbital. Hydrogen atoms with their electron in the 1 s orbital are said to be in their ground state. A hydrogen atom with its electron in the 2 s orbital is in an excited state. objeCtive 24 objeCtive 25
