Chapter 7 Quantum Theory of the Atom

7 | 2 Contents and Concepts Light Waves, Photons, and the Bohr Theory To understand the formation of chemical bonds, you need to know something about the electronic structure of atoms. Because light gives us information about this structure, we begin by discussing the nature of light. Then we look at the Bohr theory of the simplest atom, hydrogen. 1. The Wave Nature of Light 2. Quantum Effects and Photons 3. The Bohr Theory of the Hydrogen Atom

7 | 3 Quantum Mechanics and Quantum Numbers The Bohr theory firmly establishes the concept of energy levels but fails to account for the details of atomic structure. Here we discuss some basic notions of quantum mechanics, which is the theory currently applied to extremely small particles, such as electrons in atoms. 4. Quantum Mechanics 5. Quantum Numbers and Atomic Orbitals

7 | 4 Learning Objectives Light Waves, Photons, and the Bohr Theory 1.The Wave Nature of Light a.Define the wavelength and frequency of a wave. b.Relate the wavelength, frequency, and speed of light. c.Describe the different regions of the electromagnetic spectrum.

7 | 5 2.Quantum Effects and Photons a.State Planck’s quantization of vibrational energy. b.Define Planck’s constant and photon. c.Describe the photoelectric effect. d.Calculate the energy of a photon from its frequency or wavelength.

7 | 6 3.The Bohr Theory of the Hydrogen Atom a.State the postulates of Bohr’s theory of the hydrogen atom. b.Relate the energy of a photon to the associated energy levels of an atom. c.Determine the wavelength or frequency of a hydrogen atom transition. d.Describe the difference between emission and absorption of light by an atom.

7 | 7 Quantum Mechanics and Quantum Numbers 4.Quantum Mechanics a.State the de Broglie relation. b.Calculate the wavelength of a moving particle. c.Define quantum mechanics. d.State Heisenberg’s uncertainty principle. e.Relate the wave function for an electron to the probability of finding the electron at a location in space.

7 | 8 5.Quantum Numbers and Atomic Orbitals a.Define atomic orbital. b.Define each of the quantum numbers for an atomic orbital. c.State the rules for the allowed values for each quantum number. d.Apply the rules for quantum numbers. e.Describe the shapes of s, p, and d orbitals.

7 | 9 A wave is a continuously repeating change or oscillation in matter or in a physical field. Light is an electromagnetic wave, consisting of oscillations in electric and magnetic fields traveling through space.

7 | 10 A wave can be characterized by its wavelength and frequency. Wavelength, symbolized by the Greek letter lambda,, is the distance between any two identical points on adjacent waves.

7 | 11 Frequency, symbolized by the Greek letter nu,, is the number of wavelengths that pass a fixed point in one unit of time (usually a second). The unit is 1 / S or s -1, which is also called the Hertz (Hz).

7 | 12 Wavelength and frequency are related by the wave speed, which for light is c, the speed of light, 3.00 x 10 8 m/s. c = The relationship between wavelength and frequency due to the constant velocity of light is illustrated on the next slide.

7 | 13 When the wavelength is reduced by a factor of two, the frequency increases by a factor of two.

? 7 | 14 What is the wavelength of blue light with a frequency of 6.4 × /s? = 6.4 × /s c = 3.00 × 10 8 m/s c = so = c/ = 4.7 × m

? 7 | 15 What is the frequency of light having a wavelength of 681 nm? = 681 nm = 6.81 × m c = 3.00 × 10 8 m/s c = so = c/ = 4.41 × /s

7 | 16 The range of frequencies and wavelengths of electromagnetic radiation is called the electromagnetic spectrum.

7 | 17 When frequency is doubled, wavelength is halved. The light would be in the blue-violet region.

7 | 18 One property of waves is that they can be diffracted—that is, they spread out when they encounter an obstacle about the size of the wavelength. In 1801, Thomas Young, a British physicist, showed that light could be diffracted. By the early 1900s, the wave theory of light was well established.

7 | 19 The wave theory could not explain the photoelectric effect, however.

7 | 20 The photoelectric effect is the ejection of an electron from the surface of a metal or other material when light shines on it.

7 | 21 Einstein proposed that light consists of quanta or particles of electromagnetic energy, called photons. The energy of each photon is proportional to its frequency: E = h h = × J  s (Planck’s constant)

7 | 22 Einstein used this understanding of light to explain the photoelectric effect in Each electron is struck by a single photon. Only when that photon has enough energy will the electron be ejected from the atom; that photon is said to be absorbed.

7 | 23 Light, therefore, has properties of both waves and matter. Neither understanding is sufficient alone. This is called the particle–wave duality of light.

? 7 | 24 The blue–green line of the hydrogen atom spectrum has a wavelength of 486 nm. What is the energy of a photon of this light? = 4.86 nm = 4.86 × m c = 3.00 × 10 8 m/s h = 6.63 × J  s E = h and c = so E = hc/ = 4.09 × J

7 | 25 In the early 1900s, the atom was understood to consist of a positive nucleus around which electrons move (Rutherford’s model). This explanation left a theoretical dilemma: According to the physics of the time, an electrically charged particle circling a center would continually lose energy as electromagnetic radiation. But this is not the case—atoms are stable.

7 | 26 In addition, this understanding could not explain the observation of line spectra of atoms. A continuous spectrum contains all wavelengths of light. A line spectrum shows only certain colors or specific wavelengths of light. When atoms are heated, they emit light. This process produces a line spectrum that is specific to that atom. The emission spectra of six elements are shown on the next slide.

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7 | 28 In 1913, Neils Bohr, a Danish scientist, set down postulates to account for 1. The stability of the hydrogen atom 2. The line spectrum of the atom

7 | 29 Energy-Level Postulate An electron can have only certain energy values, called energy levels. Energy levels are quantized. For an electron in a hydrogen atom, the energy is given by the following equation: R H = x J n = principal quantum number

7 | 30 Transitions Between Energy Levels An electron can change energy levels by absorbing energy to move to a higher energy level or by emitting energy to move to a lower energy level.

7 | 31 For a hydrogen electron the energy change is given by R H = × J, Rydberg constant

7 | 32 The energy of the emitted or absorbed photon is related to  E: We can now combine these two equations:

7 | 33 Light is absorbed by an atom when the electron transition is from lower n to higher n (n f > n i ). In this case,  E will be positive. Light is emitted from an atom when the electron transition is from higher n to lower n (n f < n i ). In this case,  E will be negative. An electron is ejected when n f = ∞.

7 | 34 Energy-level diagram for the hydrogen atom.

7 | 35 Electron transitions for an electron in the hydrogen atom.

? 7 | 36 What is the wavelength of the light emitted when the electron in a hydrogen atom undergoes a transition from n = 6 to n = 3? n i = 6 n f = 3 R H = × J = x J × m

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7 | 38 The red line corresponds to the smaller energy difference in going from n = 3 to n = 2. The blue line corresponds to the larger energy difference in going from n = 2 to n = 1. n = 1 n = 2 n = 3 A minimum of three energy levels are required.

7 | 39 Planck Vibrating atoms have only certain energies: E = h or 2h or 3h Einstein Energy is quantized in particles called photons: E = h Bohr Electrons in atoms can have only certain values of energy. For hydrogen:

7 | 40 Light has properties of both waves and particles (matter). What about matter?

7 | 41 In 1923, Louis de Broglie, a French physicist, reasoned that particles (matter) might also have wave properties. The wavelength of a particle of mass, m (kg), and velocity, v (m/s), is given by the de Broglie relation:

? 7 | 42 Compare the wavelengths of (a) an electron traveling at a speed that is one-hundredth the speed of light and (b) a baseball of mass kg having a speed of 26.8 m/s (60 mph). Electron m e = 9.11 × kg v = 3.00 × 10 6 m/s Baseball m = kg v = 26.8 m/s

7 | 43 Electron m e = 9.11 × kg v = 3.00 × 10 6 m/s Baseball m = kg v = 26.8 m/s 2.43 × m 1.71 × m

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7 | 46 Building on de Broglie’s work, in 1926, Erwin Schrödinger devised a theory that could be used to explain the wave properties of electrons in atoms and molecules. The branch of physics that mathematically describes the wave properties of submicroscopic particles is called quantum mechanics or wave mechanics.

7 | 47 Quantum mechanics alters how we think about the motion of particles. In 1927, Werner Heisenberg showed how it is impossible to know with absolute precision both the position, x, and the momentum, p, of a particle such as electron. Because p = mv this uncertainty becomes more significant as the mass of the particle becomes smaller.

7 | 48 Quantum mechanics allows us to make statistical statements about the regions in which we are most likely to find the electron. Solving Schrödinger’s equation gives us a wave function, represented by the Greek letter psi, , which gives information about a particle in a given energy level. Psi-squared,  2, gives us the probability of finding the particle within a region of space.

7 | 49 The wave function for the lowest level of the hydrogen atom is shown to the left. Note that its value is greatest nearest the nucleus, but rapidly decreases thereafter. Note also that it never goes to zero, only to a very small value.

7 | 50 Two additional views are shown on the next slide. Figure A illustrates the probability density for an electron in hydrogen. The concentric circles represent successive shells. Figure B shows the probability of finding the electron at various distances from the nucleus. The highest probability (most likely) distance is at 50 pm.

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7 | 52 According to quantum mechanics, each electron is described by four quantum numbers: 1. Principal quantum number (n) 2. Angular momentum quantum number ( l ) 3. Magnetic quantum number (m l ) 4. Spin quantum number (m s ) The first three define the wave function for a particular electron. The fourth quantum number refers to the magnetic property of electrons.

7 | 53 A wave function for an electron in an atom is called an atomic orbital (described by three quantum numbers—n, l, m l ). It describes a region of space with a definite shape where there is a high probability of finding the electron. We will study the quantum numbers first, and then look at atomic orbitals.

7 | 54 Principal Quantum Number, n This quantum number is the one on which the energy of an electron in an atom primarily depends. The smaller the value of n, the lower the energy and the smaller the orbital. The principal quantum number can have any positive value: 1, 2, 3,... Orbitals with the same value for n are said to be in the same shell.

7 | 55 Shells are sometimes designated by uppercase letters: Letter n K1K1 L2L2 M3M3 N4N4...

7 | 56 Angular Momentum Quantum Number, l This quantum number distinguishes orbitals of a given n (shell) having different shapes. It can have values from 0, 1, 2, 3,... to a maximum of (n – 1). For a given n, there will be n different values of l, or n types of subshells. Orbitals with the same values for n and l are said to be in the same shell and subshell.

7 | 57 Subshells are sometimes designated by lowercase letters: l Letter 0s0s 1p1p 2d2d 3f3f... n ≥n ≥1234 Not every subshell type exists in every shell. The minimum value of n for each type of subshell is shown above.

7 | 58 Magnetic Quantum Number, m l This quantum number distinguishes orbitals of a given n and l —that is, of a given energy and shape but having different orientations. The magnetic quantum number depends on the value of l and can have any integer value from – l to 0 to + l. Each different value represents a different orbital. For a given subshell, there will be (2 l + 1) values and therefore (2 l + 1) orbitals.

7 | 59 Let’s summarize: When n = 1, l has only one value, 0. When l = 0, m l has only one value, 0. So the first shell (n = 1) has one subshell, an s- subshell, 1s. That subshell, in turn, has one orbital.

7 | 60 When n = 2, l has two values, 0 and 1. When l = 0, m l has only one value, 0. So there is a 2s subshell with one orbital. When l = 1, m l has only three values, -1, 0, 1. So there is a 2p subshell with three orbitals.

7 | 61 When n = 3, l has three values, 0, 1, and 2. When l = 0, m l has only one value, 0. So there is a 3s subshell with one orbital. When l = 1, m l has only three values, -1, 0, 1. So there is a 3p subshell with three orbitals. When l = 2, m l has only five values, -2, -1, 0, 1, 2. So there is a 3d subshell with five orbitals.

7 | 62 We could continue with n =4 and 5. Each would gain an additional subshell (f and g, respectively). In an f subshell, there are seven orbitals; in a g subshell, there are nine orbitals. Table 7.1 gives the complete list of permitted values for n, l, and m l up to the fourth shell. It is on the next slide.

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7 | 64 The figure shows relative energies for the hydrogen atom shells and subshells; each orbital is indicated by a dashed- line.

7 | 65 Spin Quantum Number, m s This quantum number refers to the two possible orientations of the spin axis of an electron. It may have a value of either + 1 / 2 or - 1 / 2.

? 7 | 66 Which of the following are permissible sets of quantum numbers? n = 4, l = 4, m l = 0, m s = ½ n = 3, l = 2, m l = 1, m s = -½ n = 2, l = 0, m l = 0, m s = ³/ ² n = 5, l = 3, m l = -3, m s = ½ (a) Not permitted. When n = 4, the maximum value of l is 3. (b) Permitted. (c) Not permitted; m s can only be +½ or –½. (b) Permitted.

7 | 67 Atomic Orbital Shapes An s orbital is spherical. A p orbital has two lobes along a straight line through the nucleus, with one lobe on either side. A d orbital has a more complicated shape.

7 | 68 The cross-sectional view of a 1s orbital and a 2s orbital highlights the difference in the two orbitals’ sizes.

7 | 69 The cutaway diagrams of the 1s and 2s orbitals give a better sense of them in three dimensions.

7 | 70 The next slide illustrates p orbitals. Figure A shows the general shape of a p orbital. Figure B shows the orientations of each of the three p orbitals.

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7 | 72 The complexity of the d orbitals can be seen below