Chapter 12 Quantum Mechanics and Atomic Theory

1. Chapter 12Quantum Mechanics and Atomic Theory 5.1 Electromagnetic Radiation 5.2 The Nature of Matter 5.3 The Atomic Spectrum of Hydrogen 5.4 The Bohr Model 5.5 The Quantum Mechanical Description of the Atom 5.6 The Particle in a Box (skip) 5.7 The Wave Equation for the Hydrogen Atom 5.8 The Physical Meaning of a Wave Function 5.9 The Characteristics of Hydrogen Orbitals 5.10 Electron Spin and the Pauli Principle 5.11 Polyelectronic Atoms 5.12 The History of the Periodic Table 5.13 The Aufbau Principle and the Periodic Table 5.14 Further Development of the Polyelectronic Model 5.15 Periodic Trends in Atomic Properties 5.16 The Properties of Alkali Metals

2. Waves and Light • Electromagnetic Radiation • Energy travels through space as electromagnetic radiation • Examples: visible light, microwave radiation, radio waves, X-rays, infra-red radiation, UV radiation • Waves (characterized by λ, υ, amp, c) • Travels at the speed of light (3x108 m/sec)

3. Electromagnetic Radiation Light consists of waves of oscillating electric (E) and magnetic fields (H) that are perpendicular to one another and to the direction of propagation of the light.

4. Electromagnetic Spectrum 400 nm (violet) The visible spectrum 700 nm (red)

6. Important Equations(that apply to EM radiation) • c = λ υ (c=lambda nu) c = 3 x 108 meters / second λ = wavelength [m, nm (10-9m), Å (10-10m)] • = frequency (Hz = s-1)[frequency and wavelength vary inversely] • E = hυ (Energy = h nu) h = Planck’s constant (h = 6.62 x 10-34 J s = 6.62 x 10-34 kg m2 s-1) [the energy of a wave increases with its frequency]

7. AM Radio Waves • KJR Seattle, Channel 95 (AM)950 kHz = 950,000 second-1c = λν => λ = c/νλ = 3.0x108 m s-1/ 9.5 x 105 s-1 = 316 mWhen the frequency (ν) of EM is 950 kHz, the wavelength (λ) is 316 meters (about 1/5 mile).

8. FM Radio Waves • WABE Atlanta: FM 90.1 MHz c = λν => λ = c/ν = 3.0x108 m s-1/ 90.1x106 s-1 = 3.33 m FM radio waves are higher frequency, higher energy and longer wavelength, than AM radio waves.

9. c = λν E = hν Problem The X-ray generator in Loren Williams’ lab produces x-radiation with wavelength of 1.54 Å (0.1 nm = 1 Å). What is the frequency of the X-rays? What is the energy of each X-ray photon?

10. X-rays X-rays were discovered in 1895 by German scientist Wilhelm Conrad Roentgen. He received a Nobel Prize in 1901. A week after his discovery, Roentgen took an x-ray image of his wife’s hand, visualizing the bones of her fingers and her wedding ring - the world’s first x-ray image. Roentgen ‘temporarily’ used the term “x”-ray to indicate the unknown nature of this radiation. Max von Laue (Nobel Prize 1914) showed that x-rays are electromagnetic radiation, just like visible light, but with higher frequency (and higher energy) and smaller wavelength. Within a few months of Roentgen’s discovery, doctors in New York used x-rays to image broken bones.

11. c = λν E = hν Problem The laser in an audio compact disc (CD) player produces light with a wavelength of 780 nm. What is the frequency of the light emitted from the laser?

12. Problem The brilliant red color seen in fireworks displays is due to 4.62 x 1014 s-1 strontium emission. Calculate the wavelength of the light emitted.

13. Planck, Einstein, and Bohr • 1901 Max Planck found that light (or energy) is quantized. • In the microscopic world energy can be gained or lost only in integer multiples of hν. ΔE = n(hν) n is an integer (1,2,3,…) • h is Planck’s constant (h = 6.628X10-34 J s)J: Joule, a unit of energy. • Each energy unit of size hν is called a packet or quantum

14. 1905 Einstein suggested that electromagnetic radiation can be viewed as a “stream of particles” called photons Ephoton = hυ = h(c/λ) • About the same time, Einstein derived his famous equation E = mc2 • photons have mass. Zumdahl Chapter 12

15. Dual nature of light

16. Electrons and Atoms: The Atomic Spectrum of Hydrogen (H.):Put energy into a hydrogen atom (“excite it”),what comes out?ie., at what energies does excited Hydrogen emit light?

17. (1) A hydrogen atom consists of one electron and one proton. (2) A hydrogen atom has discrete energy levels described by the primary quantum number n (1,2,3…) which gives the energy levels En (E1, E2, E3…) (3) Light is emitted from a hydrogen atom when an electron changes from a higher energy state (Ebig) to a lower energy state (Esmall) (4) The wavelengths emitted tell you ΔE2-1, ΔE3-1, ΔE2-4… (where ΔE2-1 = E2- E1). (5) The observed emission spectrum of a hydrogen atom (at specific λ) tells you that the energy of a hydrogen atom is quantized. n=4 n=3 n=2 n=1

19. The Bohr model of the hydrogen atom 1. The hydrogen atom is a small, positively charged nucleus surrounded by a electron that travels in circular orbits. The atom is analogous to the solar system, but with electrostatic forces providing attraction, rather than gravity. 2. Unlike planets, electrons can occupy only certain orbits. Each orbit represents a discrete energy state. In the Bohr model, the energy of a hydrogen atom is quantized. 3. Light is emitted by a hydrogen atom when an electron falls from a higher energy orbit to a lower energy orbit. 4. Since each orbit is of a definite fixed energy, the transition of an electron from the higher energy orbit to the lower energy orbit causes the emission of energy of a specific amount or size (a quantum). The light emitted is at a specific frequency and wavelength.

20. Electronic transitions in the Bohr model for the hydrogen atom

21. Bohr Model of the Atom (quantized energy) Bohr calculated the angular momentum, radius and energy of the electrons traveling in descrete orbits. Calculated ΔE’s match observed λ(emission)

22. Modern Quantum Mechanics (1) • Bohr recognized that his model violates principles of classical mechanics, which predict that electrons in orbit would fall towards and collide with the nucleus. Stable Bohr atoms are not possible. • Modern quantum mechanics, with orbitals rather than orbits, provides the only reasonable explanation for the observed properties of the atoms

23. Modern Quantum Mechanics (2) • Orbital Defn: Orbitals are the “quantum” states that are available to electron. An orbital can be full (2 e-), half full (1e-), or empty. An orbital is a wave function, characterized by quantum numbers n (energy), l (shape), and m (direction). • An orbital is used to calculate the probability of finding a electron at some location (Ψ2) – giving a three-dimensional probability graph of an electron position.

24. Orbitals (like Standing Waves) n=1 n=2 n=3 Analogy: An electron in an orbital can be imagined to be a standing wave around the nucleus. Electrons are not in the planet-like orbits.

25. An orbital is a wavefunction (Ψ), described by three quantum numbers [ψ (n, l, ml)] • n = principal quantum number n = 1, 2, 3, … n is related to the energy of the orbital • l = angular (azimuthal) quantum number l = 0, 1, …. (n-1) l gives the shape of the orbital l = 0 is called an s orbital (these are spherical) l = 1 is called a p orbital (these are orthogonal rabbit ears) l = 2 is called a d orbital (these have strange shapes) l = 3 is called an f orbital (these have stranger shapes) l = 4 is called a g orbital (don’t even think about it) ψ (n, l, ml) ψ (n, l, ml)

26. An orbital is a wavefunction (Ψ), described by three quantum numbers [ψ (n, l, ml)](continued) • ml = magnetic quantum number ml = -l, … , 0, ….+l ml relates to the orientation of the orbital Ψ (n, l, ml)

27. Quantum Numbers Ψ (n, l, ml) Each orbital is specified by three quantum numbers (n, l, ml). Each electron is specified by four quantum numbers (n, l, ml, ms). ms = electron spin quantum number, indicated the electron’s spin, which can be up or down. ms = +1/2, -1/2 denoted by , • Ψ (n, l , ml ) specifies an orbital. • Ψ (n, l , ml , ms) specifies an electron in an orbital. 

28. Electrons and Orbitals • Each orbital is specified by 3 quantum numbers: (n,l,ml) • Every orbital can hold two electrons • Each electron is specified by 4 quantum numbers: (n,l,ml,ms)

29. Ψ (n, l, ml) Summary • n: the primary quantum number, controls size and energy, and the possibilities for l. • l: the angular quantum number, controls orbital shape, and can also effect energy. l controls the possibilities for ml. • ml: the orientation quantum number

30. The First Three Orbitals Energy Levels (n=1,2 or 3) Ψ (n, l, ml) Ψ (1, 0, 0) Ψ(2, 0, 0) Ψ(2, 1, -1) Ψ (3, 0, 0) Ψ(2, 1, 0) Ψ(3, 1, -1) Ψ(2, 1, +1) Ψ (3, 1, 0) Ψ (3, 1, 1) Ψ(3, 2,-2 ) Ψ (3, 2,-1 ) Ψ (3, 1, 0) Ψ(3, 2, 1 ) Ψ (3, 1, 2)

31. Ψ (n, 0, 0) l = 0s orbitals Ψ (1, 0, 0) Ψ (2, 0, 0) Ψ (3, 0, 0) Degeneracy n2 = number of degenerate orbitals with the same energy (this applies to hydrogen only).

32. Ψ (2, 1, ml) l = 1p orbitals Ψ (2, 1, -1) Ψ (2, 1, 0) Ψ (2, 1, +1)

33. Ψ (3, 2, ml) l = 2d orbitals Ψ (3, 2, -2) Ψ (3, 2, 0) Ψ (3, 2, -1) Ψ (3, 2, 1) Ψ (3, 2, 2)

34. Ψ (4, 3, ml) l = 3f orbitals Ψ (4, 3, -3) Ψ (4, 3, -2) Ψ (4, 3, -1) Ψ (4, 3, 0) Ψ (4, 3, 1) Ψ (4, 3, 2) Ψ (4, 3, 3)

35. Orbital Energy Levels of Atomswith many Electrons:The degeneracy is lost. From the graph: Are 2s and 2p degenerate (i.e., do they have the same energy)? Which is lower energy? 4s or 3d? Which is lower energy? 6s or 4f? Which is lower energy? 3d or 4p?

36. Energy Levels: Why is the 2s orbital higher in energy than a 2p orbital? Penetration: Electrons in the 2s orbital are closer to the nucleus (on average) than electrons in a 2p orbital. So 2s electrons shield the 2p electrons from the nucleus. This raises the energy of the 2p electrons (Coulomb’s law).

37. Many Electron Atoms (2)Aufbau Principle • The Aufbau principle assumes a process in which an atom is "built up" by progressively adding electrons and protons/neutrons. As electrons are added, they enter the lowest energy available orbital. • Electrons fill orbitals of lowest available energy before filling higher states. 1s fills before 2s, which fills before 2p, which fills before 3s, which fills before 3p.

38. Many Electron Atoms (3)Filling Orbitals with Electrons 1s (holds 2e-) then 2s (2e-) then 2p (6e-) then 3s (2e-) then 3p (6e-) then 4s (2e-) then 3d (10e-) then 4p (6e-) then 5s (2e-) …

39. Many Electron Atoms (4) Pauli Exclusion Principle No 2 electrons in an atom can have the same set of quantum numbers n, l, ml, ms ms = electron spin quantum number ms = +1/2, -1/2 denoted by  

40. Hund’s RulesMany Electron Atoms (5) • Every degenerate orbital is singly occupied (contains one electron) before any orbital is doubly occupied (Electrons distribute as much as possible within degenerate orbitals This is called the "bus seat rule” It is analogous to the behavior of passengers who occupy all double seats singly before occupying them doubly. • Multiple electrons in singly occupied orbitals have the same spin.

41. Periodic Table

42. The Quantum Mechanical Periodic Table

43. Orbitals and the Periodic Table

45. PRS Question

46. PRS Question

47. PRS Question

48. PRS Question

49. Hund’s Rules • Every degenerate orbital is singly occupied (contains one electron) before any orbital is doubly occupied (Electrons distribute as much as possible within degenerate orbitals This is called the "bus seat rule” It is analogous to the behavior of passengers who occupy all double seats singly before occupying them doubly. • Multiple electrons in singly occupied orbitals have the same spin.

50. Zumdahl Chapter 12