17. The Wave Model of the Atom Rutherford discovered the nucleus and Bohr conceived an atom which resembled a tiny solar system, but with the important difference that only certain orbits were allowed. Within these orbits, the electrons seemed to violate the known phenomenon that accelerated charged particles should radiate away their energy very quickly as electromagnetic radiation. De Broglie soon introduced the idea of matter waves and demonstrated that Bohr’s orbits for the hydrogen atom were only those for which the wavelength of the electron fit exactly. We now conclude our development of the model of the atom. But first we describe a phenomenon called “standing waves.” This represents the missing piece that will complete our model of the atom as we currently understand it. In later chapters we will begin to study matter as the chemist works with it. We will discover certain puzzling patterns in the behavior of the chemical elements and see that these are explained by the model of the atom that we are now developing.
Figure 17.1. Standing waves created when waves of the same wavelength move through a string from opposite ends. The solid and dashed curves show the string at two different times.
Standing Waves When waves of the same wavelength coming from different directions move through a common medium, a phenomenon we call interference occurs. Imagine a string being shaken with the same frequency at both ends so that waves meet in the middle as in Figure 17.1. If done correctly, we get stationary points (called nodes) of destructive interference as well as largeamplitude oscillating points (called antinodes) of constructive interference. We can get the same effect by shaking only one end and reflecting the wave back on itself from a stationary end. When the wavelength is just right, allowed patterns called “standing waves” occur on a string with stationary ends (Fig. 17.2). Patterns associated with “incorrect” wavelengths are not allowed. From the way we have described their origin, one realizes that the standing waves are dynamic phenomena, but the pattern of nodes and antinodes remains stationary. Standing waves can also occur in two-dimensional media such as a drumhead that is fixed around the periphery. In such instances the nodes are lines rather than points, but the idea is the same (Fig. 17.3). Although it is harder to imagine, such standing waves
Figure 17.2. To get standing waves between two stationary points at the ends of a string, the wavelengths must just fit the space between the points. The upper and middle waves are allowed; the lower wave is not allowed. can also be set up in three-dimensional media. In fact, about a hundred years ago Hermann von Helmholtz (1821-1894) made detailed studies of the vibrations of air enclosed in rigid metal spheres. The probability waves described by Schrödinger’s equation can set up standing waves, too. A place where the proper conditions occur is in the atom. In the atom these standing waves of probability are called orbitals. The al ending shows that an orbital is not an orbit (i.e.,
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the simplest, computers are required to solve the equation which yields the probability densities. Orbitals
Figure 17.3. medium.
There are a few technical details to discuss at this point. There are various kinds of standing wave patterns that are allowed within the atom. A few are shown in Figure 17.4. For historical reasons they are labeled “s,” “p,” “d,” “f,” and so forth. In fact there are three variations of the p orbitals and five variations of the d orbitals. In these drawings the nucleus is a tiny point near the center of the pattern. The shaded regions of the orbital patterns are regions of probability where the electron is likely to be found if we were to look. We do not think of the electron moving in these patterns; they simply describe probabilities. The drawings of the orbitals do not display the fact that they do not have sharp boundaries. The diagrams in Figure 17.4 show the three types of orbitals by indicating the “skin” that encloses the volume inside which the electron will be found 90 percent of the time. Within the spherical sorbital, the electron is far more likely to be found near the center than near the skin. Electron probability is not evenly distributed in the other two orbitals either. Discussing the size of an atom becomes difficult because there is really no outer boundary beyond which the electron can never be found.
Standing waves in a two-dimensional
not a path of movement), but rather a description of the probability of the position of the electron. The orbitals are the final pieces in our puzzle of the atom. Whenever we “look” at an electron (including looking with instruments), the electron reveals itself as a particle. When we don’t look, the only knowledge we have about the electron is what is contained in its probability wave (or, to be more precise, “wave function”). Thus, when the electron exists in the unobserved atom, we cannot think of it as a charged particle moving in a well-defined orbit. All that exists is the orbital, i.e., a standing wave of probability satisfying the Schrödinger equation. We are then freed from the problem that plagued the nuclear atom from its inception: Why doesn’t the electron radiate its energy as we expect a charged particle to do when it accelerates in an orbit? The answer: There are no orbits; hence, there is no problem. The idea of an orbit has no reality in an unobserved atom. Standing waves in a string are the result of vibrations of the pieces of the string relative to their equilibrium points. The standing waves of the Schrödinger equation do not result from vibrations in the positions of the electron or of any medium. The waves simply denote the probability that the electron described by the orbital could be found at a particular point in space. One way of thinking of an orbital is to think of a cloud surrounding the nucleus. In basic terms think of a spherical cloud in the volume of space around a nucleus where an atomic electron could be expected. This electron cloud is a mental constraint, not a physical reality. The cloud is dense in some places and not so dense in others, particularly as we recede from the nucleus. The density of the cloud near a point in space around the nucleus can be thought of as being proportional to the probability of finding the electron there. At large distances from the nucleus, the probability gets very low, so the cloud gets very thin, but it never quite fades into nothing. The solution of the Schrödinger equation yields the numbers that represent the density of this cloud for all points in space. For most atoms other than
Figure 17.4. Shapes of three orbitals: s-orbital (spherical), p-orbital (three-dimensional dumbbell), d-orbital (rounded cloverleaf). There is also a technicality, called the Exclusion Principle, that does not allow more than two electrons to exist in orbitals which occupy exactly the same space at the same time. In those cases where two electrons existing as orbitals try to occupy the same space at the same time, the electrons must differ in a characteristic called “spin,” a characteristic that is similar to the spinning of a ball on its axis. The spinning of electrons can be referred to in two ways, either as “spin up” or “spin
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down.” These are analogous to a ball spinning either clockwise or counterclockwise about an axis. When an electron exists in one of these orbitals, it possesses a unique amount of energy, called the orbital energy. Different orbitals correspond to different electron energies. In this way the model retains the essential feature of the Bohr atom which explained the discrete spectrum of hydrogen. The discrete colors are formed as an electron that has been bumped to a higher energy orbital makes a jump from a higher energy orbital to an unoccupied lower energy orbital. In order to rid itself of the energy difference, the atom creates and emits a photon of frequency (and, hence, color) that just corresponds to the energy difference according to the equation which proved to be so important in explaining the photoelectric effect:
is identified with a unique label. Once the electron has fallen into the atom and lost energy by emitting photons, it is stuck like the ball in the sink, until energy is added. The energies associated with the levels are predicted by the Schrödinger equation. Some of these levels cluster together and are said to belong to the same “shell.” Within each shell the electron can exist in an orbital. The pattern is that in Shell 1 there is one sorbital; in Shell 2 there are one s-orbital and three porbitals; in Shell 3 there are one s-orbital, three porbitals, and five d-orbitals; and so on. Generally, the orbitals get larger as the shell number increases. Although s-orbitals belonging to Shell 1 and to Shell 2 are both spherical, they are not of the same size and do not occupy exactly the same space. Thus, the Exclusion Principle does not apply to orbitals that belong to different shells. Figure 17.5 shows an energy level diagram for hydrogen with some of the possible energy levels. Only one energy state is actually occupied by an electron. Electrons in these diagrams are represented with either a “spin up” symbol (arrow pointing up) or a “spin down” symbol (arrow pointing down). The occupied energy state in Figure 17.5 is in Shell 1, s-orbital, and is designated “1s.”
E2 ! E1 " Planck’s constant # frequency . Energy Wells Now, imagine a lecture demonstration table that contains a sink. Visualize a ball on the table poised at the edge of the sink. Gently nudge the ball so that it falls into the sink and comes to rest at the bottom. Does the ball now have more or less energy than it had before falling into the sink? The essential change has been in the gravitational potential energy, which is less at the bottom of the sink than at the elevated position on the table top. Therefore, the ball has less energy inside the sink. Once this energy difference has been lost, the ball is stuck in the sink and will remain there until someone or something comes along to replace the missing gravitational potential energy by lifting it out of the sink. Now imagine another sink located below the first, so that occasionally the ball will fall through the drain of the first to land below in the second. Now the ball has less energy than it had in the first sink. Imagine a series of sinks which are gradually lower. In this arrangement we could label each sink with the energy the ball has at that particular level, a sequence that is decreasing as the ball falls to lower and lower levels. Orbital Energies
Figure 17.5. A schematic energy level or energy well diagram for a one-electron atom. Each horizontal line (level) represents a possible energy of an electron in the atom. Different electron energies correspond to different electron orbitals.
In many ways an electron in an atom is like the ball dropping into a sink or a well. The electron has more energy when it is free than when it has become a part of the atom. The force pulling the electron into the atom is electrical rather than gravitational as it is in the sink analogy. And in the atom there are no sinks, just orbitals. But depending on the particular orbital, the electron has a certain, specific energy called an “energy level.” Thus, we often represent an atom by an energylevel or energy well diagram (see Fig. 17.5). Each level
The same labeling scheme can be applied to atoms of other elements (helium, lithium, and so forth), but the energies associated with each level differ from element to element. In Figure 17.6 the energy level diagrams for hydrogen (one electron), helium (two electrons), lithium (three electrons), and beryllium (four electrons) ar
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Figure 17.6. Energy level diagrams for hydrogen, helium, lithium, and beryllium. side by side for comparison. Figure 17.7 is an extension of Figure 17.6. The p-orbitals always come as triplets; the basic three-dimensional dumbbell comes in three spatial orientations. Each orientation of orbital can contain two electrons if one is spin up and one is spin down. The orientations of p-orbitals in a given shell have the same energy. So we can have up to six electrons in an atom in a triplet of p-orbitals with essentially the same energy. The d-orbitals come in a quintuplet of shape-orientation combinations. We can have up to ten electrons in an atom in a quintuplet of d-orbitals with basically the same energy. Let us summarize what has been said about the orbitals of a single atom such as magnesium (see Fig. 17.8). The s-orbitals are spherical; 1s is always the lowest energy orbital, and 2s is the next lowest energy orbital. The p-orbitals are shaped like dumbbells and the lowest energy of these is the 2p; it has energy just greater than that of the 2s orbital. The 3s orbital has the next highest energy after the 2p orbital, and so on.
Figure 17.8. Energies of the s and p orbitals for a magnesium atom. An electrically neutral magnesium atom will have just as many electrons as protons (both given by the atomic number). The electrons will be found in the lowest-energy orbitals available and will pair up in sets of two (spin up, spin down) if necessary. Many vacant orbitals of higher energy will remain. Absorbed light or collisions with other atoms or particles can cause lowlying electrons to be excited to the higher energy vacant orbitals, or even to leave the atom entirely (ionization). In each unit of time there is some probability that the electron will spontaneously return to any vacancy available at lower levels. When the quantum mechanical
Figure 17.7. Energies of occupied orbitals of light elements.
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dice-thrower of Chapter 16 rolls the magic combination, one of the possibilities becomes reality and the electron jumps. In doing so it emits light, and this accounts for observed emission line spectra.
mixtures of a least two metallic elements: brass (copper and zinc); solder (lead and tin); and the gold in jewelry, which has had silver or copper added to harden it. Some common nonmetallic elements are carbon in the form of charcoal, graphite, or diamond; nitrogen and oxygen (gases comprising over 90 percent of the air we breathe); and sulfur, a yellow substance. The names of the elements are listed in Appendix B. Notice that each element has its own symbol, which is usually an abbreviation of its name. For example, hydrogen is H, helium is He, and so forth. Some elements have symbols that are based on their Latin names: sodium (natrium) is Na and potassium (kalium) is K. People working with the symbols of the chemical elements sometimes add to an element’s symbol with numerical subscripts and superscripts indicating how many of the atomic building blocks are present in a particular atom. The atomic number is the number of protons in the atom and is usually written low and in front of the symbol: 1H, 2He, 6C. The mass number is the number of nucleons (neutrons + protons) in the nucleus. It is approximately equal to the atomic mass expressed in atomic mass units (amu) (see Appendix B). The mass number is usually (but not always) written higher up and in front of the symbol: 1H, 4He, 3He. Notice that two atoms of the same element (helium in this case) may have different mass numbers because, within limits, they may have different numbers of neutrons. A superscript following the symbol is often used to designate the excess or deficiency of electrons compared to the neutral atom. A neutral helium atom could be written He0, but the 0 is usually left off neutral atoms. If the atom has an extra electron it will have a negative charge, and the resulting ion will be designated He1–. On the other hand, if helium loses an electron, it will have a positive charge and be represented by He1+. Helium could lose both its electrons and be He2+. It would then be an alpha particle, the particle Rutherford used to probe the atom and discover the nucleus. When it is required, the symbol can be given numbers in all three positions: 42He2+.
Chemical Elements Historically, the development of a model of atoms came quite late. Long before the physicists had worked out the details of atomic structure, chemists were working with things they too called “atoms.” However, the definition of “atoms” was not clear. For example, if there were things called “atoms” it was quite possible that all substances (gold, hydrogen, carbon, and others) were made of identical atoms that somehow combined in different ways to create the different chemical characteristics of the substances. If there were different atoms for different substances, then it was quite feasible that the atoms of a given substance were nevertheless different in size. If each substance had atoms of consistent size, there was still no way to tell whether that size was large or small. It was possible that the atoms were fluffy things like tufts of wool, which were in physical contact. It was also possible that they were much smaller things that moved about in chaotic motion and only came in contact by collision. These theories had to be resolved by clever people throughout the centuries. Robert Boyle is credited with giving us the modern idea of an element in the year 1661. He said an element is any substance that cannot be separated into different components by any known methods. John Dalton made the connection much later (ca. 1803) between elements and atoms. He took the atoms of a particular substance to be identical and defined an element to be a substance composed of only one kind of atom. Today, the atoms are modeled with Schrödinger’s equation. Each time an additional positive charge (proton) is added to the nucleus, the equation predicts orbitals with new energy levels that are unique to the number of protons used in the nucleus. Thus, today we define an element to be: a group of atoms all having the same number of protons. It is also important to note what has been left out of this definition. Electrons were left out because one (or a few) can be removed easily and just as easily replaced from moment to moment. Mass was left out because it depends on the number of nucleons in the nucleus (including neutrons), and the number of neutrons may vary from nucleus to nucleus of an element despite the fact that the number of protons does not. You are no doubt familiar with some of the more than 100 known chemical elements. Most of the common metals are elements: copper, tin, aluminum, and nickel. Steel is nearly pure elemental iron, but it contains small amounts of other metals and carbon. Alloys are
Summary Orbitals are standing waves of electron probability established in the atom in a way specified by Schrödinger’s equation. Each orbital corresponds to a specific energy that the electron has when it occupies the orbital. The orbital patterns are discrete and so are the energies associated with them. Electrons in an atom can be thought of as being in a kind of energy well. They have less energy while in an atom than they have when they are free of the atom. To free an electron from an atom (ionization), we must supply enough energy to remove it from its well. A chemical element is a group of atoms all having
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the same number of protons. Schrödinger’s equation predicts different energy levels for atoms with different numbers of protons in the nucleus. This happens because differing amounts of charge on the nucleus exert differing strengths of electromagnetic interaction with the electrons of the atom. Thus, for each chemical element there is a unique atom with its own characteristic energy levels.
6.
STUDY GUIDE Chapter 17: The Wave Model of the Atom
10. 11.
7. 8. 9.
A. FUNDAMENTAL PRINCIPLES 1. The Wave-Particle Duality of Matter and Electromagnetic Radiation: See Chapters 14 and 16. 2. The Exclusion Principle: No more than two electrons may occupy orbitals that occupy the same space at the same time. Two electrons which do occupy the same orbital must differ in their spin.
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13.
B. MODELS, IDEAS, QUESTIONS, OR APPLICATIONS 1. The Wave Model of the Atom: That model of the atom in which the electrons are described by orbitals, i.e., standing probability waves. In the Wave Model of the Atom, the electrons in the atom have discrete energies and obey the Exclusion Principle. 2. What are standing waves? 3. What are standing waves of probability? 4. Why are only certain “orbitals” allowed? 5. What is an “unwatched” electron doing? 6. How many electrons can be associated with a single orbital? 7. How is an electron in an atom classified or given an “address?” 8. What is an energy well?
Node: The positions of destructive interference in a standing wave. For waves in a one-dimensional medium, such as a violin string, the nodes are points on the string that are not vibrating. Orbit: See Chapter 15. Orbital: A standing wave pattern of the probability waves which describe electrons in an atom. Orbital Energy: The discrete energy that an electron has when it is described by a particular orbital. Probability Wave: See Chapter 16. Shell: A grouping of energy levels within an atom. Electrons having orbital energies within this grouping are said to belong to the same shell. Spin: A characteristic of electrons which has some similarity to the spinning of a ball on an axis through its center. For an electron, however, there are only two possible ways of “spinning.” One is called “spin-up” and the other “spin-down.” Standing Wave: A pattern of constructive and destructive interference of waves for which the positions of constructive and destructive interference do not move.
D. FOCUS QUESTIONS 1. What is a standing wave? What is an orbital? (What is the connection between a standing wave and an orbital?) Why are only certain orbitals available in an atom? Describe the energy changes that occur when an electron associated with one orbital goes to a different orbital. 2. Carefully outline the main elements of the Wave Model of the Atom. Draw a simple energy well representing an atom. What do the horizontal lines in the energy well figure represent? Describe what happens to the energy of an electron when it moves and is then associated with an orbital higher in the energy well. EXERCISES 17.1. Does Figure 17.4 depict surfaces on which the electron in an atom travels? If not, what do the surfaces represent? E.
C. GLOSSARY 1. Antinode: The positions of constructive interference in a standing wave. On a violin string, the antinodes are the positions of maximum vibration of the string. 2. Atomic Number: The number of protons in an atom. 3. Chemical Element: A substance consisting of atoms, all of which have the same number of protons. 4. Energy Well: A conceptual model for visualizing the discrete energies of electrons in an atom. Electrons within an atom have less energy than they have when free of the atom, much like a ball which has fallen into a well. 5. Mass Number: The combined number of protons and neutrons in the nucleus of an atom.
17.2. What scientific principle suggests that it is impossible to know simultaneously both the position and velocity of an electron in an atom? 17.3. Which of the letters—s, p, d, or f—designates an orbital shaped like a rounded, four-lobed cloverleaf? 17.4. Why can’t an electron remain at a fixed position outside the nucleus? 17.5. Why can’t an electron revolve indefinitely around the nucleus in orbit?
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17.6. If an electron is in a spherically shaped orbital (e.g., Fig. 17.4), is it possible for the electron sometimes to be outside the spherical skin?
17.13. Which element has the same number of electrons in an s-orbital as beryllium, Be? 17.14. Which element has the same number of electrons in a p-orbital as boron, B?
17.7. Is there any possibility that an electron from an atom in this paper could be on Mars for a moment today?
17.15. The atomic number of nitrogen is 7. How many electrons does one of its atoms have? How many protons? How many neutrons? State the pattern or relationship among these numbers and check to see if all of the other atoms in Table 17.1 follow this pattern.
17.8. Draw a diagram like Figure 17.5 showing only 1s, 2s, 2p, and 3s orbitals (all unfilled). Put in enough electrons to represent Li. Note the single electron in the 2s orbital. A similar element should have the orbitals filled until there is one electron in the 3s orbital. How many more electrons are required to fill orbitals until there is one electron in 3s? What neutral element has this many electrons?
17.16. If you know how many protons an atom has, what else must you know to calculate its mass number? 17.17. Use Table 17.1 or Appendix B when necessary to complete Table 17.2.
17.9. A neon sign is simply a glass tube filled with neon gas. When electricity is passed through, it becomes a discharge tube. The electrons in the neon atoms are raised to higher levels. Explain why the tube then glows red and why, if the tube is filled with xenon gas, different colors are emitted.
17.18. Use Appendix B to decide which of the following are elements: (a) lead (b) arsenic (c) bronze (d) radon (e) potash (f) platinum (g) mercury (h) freon
17.10. Which atoms in Figure 17.7 have filled shells? Which have unfilled shells? Would more energy be required to remove the highest energy electron in a filled shell or an unfilled shell?
17.19. Which of the following is the most complete model of the atom? (a) solar system (b) nuclear model (c) Bohr model (d) wave model (e) all of the above are equally complete.
17.11. Which atom in Figure 17.7 has the lowest energy vacancy (the deepest unfilled well)? 17.12. Which two atoms in Figure 17.7 have no vacant orbitals of low energy and no electrons in high energy orbitals? Would you expect these atoms to participate readily in chemical reactions?
Table 17.1. Components and masses of atoms. To be used in Exercise 17.15. ATOMIC NUMBER 1 2 3 4 5 6 7 8 9 10 11
NAME Hydrogen Helium Lithium Beryllium Boron Carbon Nitrogen Oxygen Fluorine Neon Sodium
NUMBER OF ELCTRONS
NUMBER OF PROTONS
1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10 11
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NUMBER OF NEUTRONS 0 2 4 5 6 6 7 8 10 10 12
MASS NUMBER 1 4 7 9 11 12 14 16 19 20 23
Table 17.2. To be completed with Exercise 17.17. ATOMIC SYMBOL
NAME
NUMBER OF PROTONS
NUMBER OF NEUTRONS
C
6
Oxygen 20
6
8
8
Ne
F1–
9
27
NUMBER OF ELCTRONS
6
10
10
He2+
MASS NUMBER
Helium ion
2
Nitrogen
7 9
10
Al3+
13
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