1. Chapter 7 The Quantum- Mechanical Model of the Atom 2007, Prentice Hall Chemistry: A Molecular Approach, 1 st Ed. Nivaldo Tro Roy Kennedy Massachusetts Bay Community College Wellesley Hills, MA

2. Tro, Chemistry: A Molecular Approach2 The Behavior of the Very Small electrons are incredibly small a single speck of dust has more electrons than the number of people who have ever lived on earth electron behavior determines much of the behavior of atoms directly observing electrons in the atom is impossible, the electron is so small that observing it changes its behavior

3. Tro, Chemistry: A Molecular Approach3 A Theory that Explains Electron Behavior the quantum-mechanical model explains the manner electrons exist and behave in atoms helps us understand and predict the properties of atoms that are directly related to the behavior of the electrons why some elements are metals while others are nonmetals why some elements gain 1 electron when forming an anion, while others gain 2 why some elements are very reactive while others are practically inert and other Periodic patterns we see in the properties of the elements

4. Tro, Chemistry: A Molecular Approach4 The Nature of Light its Wave Nature light is a form of electromagnetic radiation composed of perpendicular oscillating waves, one for the electric field and one for the magnetic field  an electric field is a region where an electrically charged particle experiences a force  a magnetic field is a region where an magnetized particle experiences a force all electromagnetic waves move through space at the same, constant speed 3.00 x 10 8 m/s in a vacuum = the speed of light, c

5. Tro, Chemistry: A Molecular Approach5 Speed of Energy Transmission

6. Tro, Chemistry: A Molecular Approach6 Electromagnetic Radiation

7. Tro, Chemistry: A Molecular Approach7 Characterizing Waves the amplitude is the height of the wave the distance from node to crest  or node to trough the amplitude is a measure of how intense the light is – the larger the amplitude, the brighter the light the wavelength, ( ) is a measure of the distance covered by the wave the distance from one crest to the next  or the distance from one trough to the next, or the distance between alternate nodes

8. Tro, Chemistry: A Molecular Approach8 Wave Characteristics

9. Tro, Chemistry: A Molecular Approach9 Characterizing Waves the frequency, ( ) is the number of waves that pass a point in a given period of time the number of waves = number of cycles units are hertz, (Hz) or cycles/s = s -1  1 Hz = 1 s -1 the total energy is proportional to the amplitude and frequency of the waves the larger the wave amplitude, the more force it has the more frequently the waves strike, the more total force there is

10. Tro, Chemistry: A Molecular Approach10 The Relationship Between Wavelength and Frequency for waves traveling at the same speed, the shorter the wavelength, the more frequently they pass this means that the wavelength and frequency of electromagnetic waves are inversely proportional since the speed of light is constant, if we know wavelength we can find the frequency, and visa versa

11. Tro, Chemistry: A Molecular Approach11 Example 7.1- Calculate the wavelength of red light with a frequency of 4.62 x 10 14 s -1 the unit is correct, the wavelength is appropriate for red light Check: Solve: ∙ = c, 1 nm = 10 -9 m Concept Plan: Relationships: = 4.62 x 10 14 s -1, (nm) Given: Find:  s -1 ) (m) (nm)

12. Tro, Chemistry: A Molecular Approach12 Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz

13. Tro, Chemistry: A Molecular Approach13 Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz the unit is correct, the wavelength is appropriate for radiowaves Check: Solve: ∙ = c, 1 MHz = 10 6 s -1 Concept Plan: Relationships: = 100.7 MHz, (m) Given: Find:  Hz) (s -1 ) (m)

14. Tro, Chemistry: A Molecular Approach14 Color the color of light is determined by its wavelength or frequency white light is a mixture of all the colors of visible light a spectrum RedOrangeYellowGreenBlueViolet when an object absorbs some of the wavelengths of white light while reflecting others, it appears colored the observed color is predominantly the colors reflected

15. 15 Amplitude & Wavelength

16. Tro, Chemistry: A Molecular Approach16 Electromagnetic Spectrum

17. Tro, Chemistry: A Molecular Approach17 Continuous Spectrum

18. Tro, Chemistry: A Molecular Approach18 The Electromagnetic Spectrum visible light comprises only a small fraction of all the wavelengths of light – called the electromagnetic spectrum short wavelength (high frequency) light has high energy radiowave light has the lowest energy gamma ray light has the highest energy high energy electromagnetic radiation can potentially damage biological molecules ionizing radiation

19. Tro, Chemistry: A Molecular Approach19 Thermal Imaging using Infrared Light

20. Tro, Chemistry: A Molecular Approach20 Using High Energy Radiation to Kill Cancer Cells

21. Tro, Chemistry: A Molecular Approach21 Interference the interaction between waves is called interference when waves interact so that they add to make a larger wave it is called constructive interference waves are in-phase when waves interact so they cancel each other it is called destructive interference waves are out-of-phase

22. Tro, Chemistry: A Molecular Approach22 Interference

23. Tro, Chemistry: A Molecular Approach23 Diffraction when traveling waves encounter an obstacle or opening in a barrier that is about the same size as the wavelength, they bend around it – this is called diffraction traveling particles do not diffract the diffraction of light through two slits separated by a distance comparable to the wavelength results in an interference pattern of the diffracted waves an interference pattern is a characteristic of all light waves

24. Tro, Chemistry: A Molecular Approach24 Diffraction

25. Tro, Chemistry: A Molecular Approach25 2-Slit Interference

26. Tro, Chemistry: A Molecular Approach26 The Photoelectric Effect it was observed that many metals emit electrons when a light shines on their surface this is called the Photoelectric Effect classic wave theory attributed this effect to the light energy being transferred to the electron according to this theory, if the wavelength of light is made shorter, or the light waves intensity made brighter, more electrons should be ejected remember: the energy of a wave is directly proportional to its amplitude and its frequency if a dim light was used there would be a lag time before electrons were emitted  to give the electrons time to absorb enough energy

27. Tro, Chemistry: A Molecular Approach27 The Photoelectric Effect

28. Tro, Chemistry: A Molecular Approach28 The Photoelectric Effect The Problem in experiments with the photoelectric effect, it was observed that there was a maximum wavelength for electrons to be emitted called the threshold frequency regardless of the intensity it was also observed that high frequency light with a dim source caused electron emission without any lag time

29. Tro, Chemistry: A Molecular Approach29 Einstein’s Explanation Einstein proposed that the light energy was delivered to the atoms in packets, called quanta or photons the energy of a photon of light was directly proportional to its frequency inversely proportional to it wavelength the proportionality constant is called Planck’s Constant, (h) and has the value 6.626 x 10 -34 J∙s

30. Tro, Chemistry: A Molecular Approach30 Example 7.2- Calculate the number of photons in a laser pulse with wavelength 337 nm and total energy 3.83 mJ Solve: E=hc/, 1 nm = 10 -9 m, 1 mJ = 10 -3 J, E pulse /E photon = # photons Concept Plan: Relationships: = 337 nm, E pulse = 3.83 mJ number of photons Given: Find:  nm) (m) E photon number photons

31. Tro, Chemistry: A Molecular Approach31 Practice – What is the frequency of radiation required to supply 1.0 x 10 2 J of energy from 8.5 x 10 27 photons?

32. Tro, Chemistry: A Molecular Approach32 What is the frequency of radiation required to supply 1.0 x 10 2 J of energy from 8.5 x 10 27 photons? Solve: E=h, E total = E photon ∙# photons Concept Plan: Relationships: E total = 1.0 x 10 2 J, number of photons = 8.5 x 10 27 Given: Find: (s -1 ) E photon number photons

33. Tro, Chemistry: A Molecular Approach33 Ejected Electrons 1 photon at the threshold frequency has just enough energy for an electron to escape the atom binding energy,  for higher frequencies, the electron absorbs more energy than is necessary to escape this excess energy becomes kinetic energy of the ejected electron Kinetic Energy = E photon – E binding KE = h - 

34. Tro, Chemistry: A Molecular Approach34 Spectra when atoms or molecules absorb energy, that energy is often released as light energy fireworks, neon lights, etc. when that light is passed through a prism, a pattern is seen that is unique to that type of atom or molecule – the pattern is called an emission spectrum non-continuous can be used to identify the material  flame tests Rydberg analyzed the spectrum of hydrogen and found that it could be described with an equation that involved an inverse square of integers

35. Tro, Chemistry: A Molecular Approach35 Emission Spectra

36. Tro, Chemistry: A Molecular Approach36 Exciting Gas Atoms to Emit Light with Electrical Energy HgHeH

37. Tro, Chemistry: A Molecular Approach37 Examples of Spectra Oxygen spectrumNeon spectrum

38. Tro, Chemistry: A Molecular Approach38 Identifying Elements with Flame Tests NaKLiBa

39. Tro, Chemistry: A Molecular Approach39 Emission vs. Absorption Spectra Spectra of Mercury

40. Tro, Chemistry: A Molecular Approach40 Bohr’s Model Neils Bohr proposed that the electrons could only have very specific amounts of energy fixed amounts = quantized the electrons traveled in orbits that were a fixed distance from the nucleus stationary states therefore the energy of the electron was proportional the distance the orbital was from the nucleus electrons emitted radiation when they “jumped” from an orbit with higher energy down to an orbit with lower energy the distance between the orbits determined the energy of the photon of light produced

41. Tro, Chemistry: A Molecular Approach41 Bohr Model of H Atoms

42. Tro, Chemistry: A Molecular Approach42 Wave Behavior of Electrons de Broglie proposed that particles could have wave-like character because it is so small, the wave character of electrons is significant electron beams shot at slits show an interference pattern the electron interferes with its own wave de Broglie predicted that the wavelength of a particle was inversely proportional to its momentum

43. Tro, Chemistry: A Molecular Approach43 however, electrons actually present an interference pattern, demonstrating the behave like waves Electron Diffraction if electrons behave like particles, there should only be two bright spots on the target

44. Tro, Chemistry: A Molecular Approach44 Solve: Example 7.3- Calculate the wavelength of an electron traveling at 2.65 x 10 6 m/s =h/mv Concept Plan: Relationships: v = 2.65 x 10 6 m/s, m = 9.11 x 10 -31 kg (back leaf)  m Given: Find: m, v (m)

45. Tro, Chemistry: A Molecular Approach45 Practice - Determine the wavelength of a neutron traveling at 1.00 x 10 2 m/s (Mass neutron = 1.675 x 10 -24 g)

46. Tro, Chemistry: A Molecular Approach46 Solve: Practice - Determine the wavelength of a neutron traveling at 1.00 x 10 2 m/s =h/mv, 1 kg = 10 3 g Concept Plan: Relationships: v = 1.00 x 10 2 m/s, m = 1.675 x 10 -24 g  m Given: Find: m (kg), v (m) m(g)

47. Tro, Chemistry: A Molecular Approach47 Complimentary Properties when you try to observe the wave nature of the electron, you cannot observe its particle nature – and visa versa wave nature = interference pattern particle nature = position, which slit it is passing through the wave and particle nature of nature of the electron are complimentary properties as you know more about one you know less about the other

48. Tro, Chemistry: A Molecular Approach48 Uncertainty Principle Heisenberg stated that the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass x = position,  x = uncertainty in position v = velocity,  v = uncertainty in velocity m = mass the means that the more accurately you know the position of a small particle, like an electron, the less you know about its speed and visa-versa

49. Tro, Chemistry: A Molecular Approach49 Uncertainty Principle Demonstration any experiment designed to observe the electron results in detection of a single electron particle and no interference pattern

50. Tro, Chemistry: A Molecular Approach50 Determinacy vs. Indeterminacy according to classical physics, particles move in a path determined by the particle’s velocity, position, and forces acting on it determinacy = definite, predictable future because we cannot know both the position and velocity of an electron, we cannot predict the path it will follow indeterminacy = indefinite future, can only predict probability the best we can do is to describe the probability an electron will be found in a particular region using statistical functions

51. 51 Trajectory vs. Probability

52. Tro, Chemistry: A Molecular Approach52 Electron Energy electron energy and position are complimentary because KE = ½mv 2 for an electron with a given energy, the best we can do is describe a region in the atom of high probability of finding it – called an orbital a probability distribution map of a region where the electron is likely to be found distance vs.  2 many of the properties of atoms are related to the energies of the electrons

53. Tro, Chemistry: A Molecular Approach53 Wave Function,  calculations show that the size, shape and orientation in space of an orbital are determined be three integer terms in the wave function added to quantize the energy of the electron these integers are called quantum numbers principal quantum number, n angular momentum quantum number, l magnetic quantum number, m l

54. Tro, Chemistry: A Molecular Approach54 Principal Quantum Number, n characterizes the energy of the electron in a particular orbital corresponds to Bohr’s energy level n can be any integer  1 the larger the value of n, the more energy the orbital has energies are defined as being negative an electron would have E = 0 when it just escapes the atom the larger the value of n, the larger the orbital as n gets larger, the amount of energy between orbitals gets smaller for an electron in H

55. Tro, Chemistry: A Molecular Approach55 Principal Energy Levels in Hydrogen

56. 56 Electron Transitions in order to transition to a higher energy state, the electron must gain the correct amount of energy corresponding to the difference in energy between the final and initial states electrons in high energy states are unstable and tend to lose energy and transition to lower energy states energy released as a photon of light each line in the emission spectrum corresponds to the difference in energy between two energy states

57. Tro, Chemistry: A Molecular Approach57 Predicting the Spectrum of Hydrogen the wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states for an electron in energy state n, there are (n – 1) energy states it can transition to, therefore (n – 1) lines it can generate both the Bohr and Quantum Mechanical Models can predict these lines very accurately

58. 58 Hydrogen Energy Transitions

59. Solve: Example 7.7- Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5 E=hc/  E n = -2.18 x 10 -18 J (1/n 2 ) Concept Plan: Relationships: n i = 6, n f = 5  m Given: Find: n i, n f  E atom E photon  E atom = -E photon E photon = -(-2.6644 x 10 -20 J) = 2.6644 x 10 -20 J the unit is correct, the wavelength is in the infrared, which is appropriate because less energy than 4→2 (in the visible) Check:

60. Tro, Chemistry: A Molecular Approach60 Practice – Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1

61. Solve: Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1 E=hc/  E n = -2.18 x 10 -18 J (1/n 2 ) Concept Plan: Relationships: n i = 2, n f = 1  m Given: Find: n i, n f  E atom E photon  E atom = -E photon E photon = -(-1.64 x 10 -18 J) = 1.64 x 10 -18 J the unit is correct, the wavelength is in the UV, which is appropriate because more energy than 3→2 (in the visible) Check:

62. 62 Probability & Radial Distribution Functions  2 is the probability density the probability of finding an electron at a particular point in space for s orbital maximum at the nucleus? decreases as you move away from the nucleus the Radial Distribution function represents the total probability at a certain distance from the nucleus maximum at most probable radius nodes in the functions are where the probability drops to 0

63. Tro, Chemistry: A Molecular Approach63 Probability Density Function

64. Tro, Chemistry: A Molecular Approach64 Radial Distribution Function

65. Tro, Chemistry: A Molecular Approach65 The Shapes of Atomic Orbitals the l quantum number primarily determines the shape of the orbital l can have integer values from 0 to (n – 1) each value of l is called by a particular letter that designates the shape of the orbital s orbitals are spherical p orbitals are like two balloons tied at the knots d orbitals are mainly like 4 balloons tied at the knot f orbitals are mainly like 8 balloons tied at the knot

66. Tro, Chemistry: A Molecular Approach66 l = 0, the s orbital each principal energy state has 1 s orbital lowest energy orbital in a principal energy state spherical number of nodes = (n – 1)

67. 67 2s and 3s 2s n = 2, l = 0 3s n = 3, l = 0

68. Tro, Chemistry: A Molecular Approach68 l = 1, p orbitals each principal energy state above n = 1 has 3 p orbitals m l = -1, 0, +1 each of the 3 orbitals point along a different axis p x, p y, p z 2 nd lowest energy orbitals in a principal energy state two-lobed node at the nucleus, total of n nodes

69. Tro, Chemistry: A Molecular Approach69 p orbitals

70. Tro, Chemistry: A Molecular Approach70 l = 2, d orbitals each principal energy state above n = 2 has 5 d orbitals m l = -2, -1, 0, +1, +2 4 of the 5 orbitals are aligned in a different plane the fifth is aligned with the z axis, d z squared d xy, d yz, d xz, d x squared – y squared 3 rd lowest energy orbitals in a principal energy state mainly 4-lobed one is two-lobed with a toroid planar nodes higher principal levels also have spherical nodes

71. Tro, Chemistry: A Molecular Approach71 d orbitals

72. Tro, Chemistry: A Molecular Approach72 l = 3, f orbitals each principal energy state above n = 3 has 7 d orbitals m l = -3, -2, -1, 0, +1, +2, +3 4 th lowest energy orbitals in a principal energy state mainly 8-lobed some 2-lobed with a toroid planar nodes higher principal levels also have spherical nodes

73. Tro, Chemistry: A Molecular Approach73 f orbitals

74. Tro, Chemistry: A Molecular Approach74 Why are Atoms Spherical?

75. Tro, Chemistry: A Molecular Approach75 Energy Shells and Subshells