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

2. 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
Tro, Chemistry: A Molecular Approach

3. 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
Tro, Chemistry: A Molecular Approach

4. 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 108 m/s in a vacuum = the speed of light, c
Tro, Chemistry: A Molecular Approach

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

6. Electromagnetic Radiation
Tro, Chemistry: A Molecular Approach

7. 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, (l) 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
Tro, Chemistry: A Molecular Approach

8. Wave Characteristics
Tro, Chemistry: A Molecular Approach

9. Characterizing Waves
the frequency, (n) 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
Tro, Chemistry: A Molecular Approach

10. 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
Tro, Chemistry: A Molecular Approach

11. Example 7.1- Calculate the wavelength of red light with a frequency of 4.62 x 1014 s-1
Given:
Find:
n = 4.62 x 1014 s-1
l, (nm)
Concept Plan:
Relationships:
l∙n = c, 1 nm = 10-9 m
n (s-1)
l (m)
l (nm)
Solve:
Check:
the unit is correct, the wavelength is appropriate for red light
Tro, Chemistry: A Molecular Approach

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

13. 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
Tro, Chemistry: A Molecular Approach

14. Amplitude & Wavelength

15. Electromagnetic Spectrum
Tro, Chemistry: A Molecular Approach

16. Continuous Spectrum
Tro, Chemistry: A Molecular Approach

17. 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
Tro, Chemistry: A Molecular Approach

18. Thermal Imaging using Infrared Light
Tro, Chemistry: A Molecular Approach

19. 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
Tro, Chemistry: A Molecular Approach

20. Interference
Tro, Chemistry: A Molecular Approach

21. 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
Tro, Chemistry: A Molecular Approach

22. Diffraction
Tro, Chemistry: A Molecular Approach

23. 2-Slit Interference
Tro, Chemistry: A Molecular Approach

24. 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
Tro, Chemistry: A Molecular Approach

25. The Photoelectric Effect
Tro, Chemistry: A Molecular Approach

26. 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
Tro, Chemistry: A Molecular Approach

27. 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 x J∙s
Tro, Chemistry: A Molecular Approach

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

29. Practice – What is the frequency of radiation required to supply 1
Practice – What is the frequency of radiation required to supply 1.0 x 102 J of energy from 8.5 x 1027 photons?
Tro, Chemistry: A Molecular Approach

30. Kinetic Energy = Ephoton – Ebinding
Ejected Electrons
1 photon at the threshold frequency has just enough energy for an electron to escape the atom
binding energy, f
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 = Ephoton – Ebinding
KE = hn - f
Tro, Chemistry: A Molecular Approach

31. 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
Tro, Chemistry: A Molecular Approach

32. Emission Spectra
Tro, Chemistry: A Molecular Approach

33. Exciting Gas Atoms to Emit Light with Electrical EnergyHg He H
Tro, Chemistry: A Molecular Approach

34. Examples of Spectra Oxygen spectrum Neon spectrum
Tro, Chemistry: A Molecular Approach

35. Identifying Elements with Flame TestsNa K Li Ba
Tro, Chemistry: A Molecular Approach

36. Emission vs. Absorption Spectra
Spectra of Mercury
Tro, Chemistry: A Molecular Approach

37. 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
Tro, Chemistry: A Molecular Approach

38. Bohr Model of H Atoms
Tro, Chemistry: A Molecular Approach

39. 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 (mass x velocity)
Tro, Chemistry: A Molecular Approach

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

41. Example 7. 3- Calculate the wavelength of an electron traveling at 2
Example 7.3- Calculate the wavelength of an electron traveling at 2.65 x 106 m/s
Given:
Find:
v = 2.65 x 106 m/s, m = 9.11 x kg (back leaf)
l, m
Concept Plan:
Relationships:
l=h/mv
m, v
l (m)
Solve:
Tro, Chemistry: A Molecular Approach

42. Practice - Determine the wavelength of a neutron traveling at 1
Practice - Determine the wavelength of a neutron traveling at 1.00 x 102 m/s (Massneutron = x g)
Tro, Chemistry: A Molecular Approach

43. 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 the electron are complimentary properties
as you know more about one you know less about the other
Tro, Chemistry: A Molecular Approach

44. 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, Dx = uncertainty in position
v = velocity, Dv = 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
Tro, Chemistry: A Molecular Approach

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

46. 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
Tro, Chemistry: A Molecular Approach

47. Trajectory vs. Probability

48. Electron Energy
The mathematical derivation of the energies and orbitals for electrons
comes from solving the Shrodinger equation . The general form is shown
above. The symbol H stands for the Hamiltonian operator, a set of
mathematical operations that represent the total energy (kinetic and
potential) of the electron within the atom. The symbol E is the actual
energy of the electron. The symbol is the wave function, a
mathematical function that describes the wavelike nature of the electron.
Tro, Chemistry: A Molecular Approach

49. Electron Energy electron energy and position are complimentary
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. y2
many of the properties of atoms are related to the energies of the electrons
Tro, Chemistry: A Molecular Approach

50. Wave Function, y
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, ml
Tro, Chemistry: A Molecular Approach

51. 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 ³ 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
Tro, Chemistry: A Molecular Approach

52. Principal Energy Levels in Hydrogen
Tro, Chemistry: A Molecular Approach

53. 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

54. 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
Tro, Chemistry: A Molecular Approach

55. Hydrogen Energy Transitions

56. Example 7.7- Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5
Given:
Find:
ni = 6, nf = 5
l, m
Concept Plan:
Relationships:
E=hc/l, En = x J (1/n2)
ni, nf
DEatom
Ephoton l
DEatom = -Ephoton
Solve:
Ephoton = -( x J) = x J
Check:
the unit is correct, the wavelength is in the infrared, which is appropriate because less energy than 4→2 (in the visible)

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

58. Probability & Radial Distribution Functions
y2 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

59. Probability Density Function
Tro, Chemistry: A Molecular Approach

60. Radial Distribution Function
Tro, Chemistry: A Molecular Approach

61. 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
Tro, Chemistry: A Molecular Approach

62. 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)
Tro, Chemistry: A Molecular Approach

63. 2s and 3s 2s
n = 2,
l = 0 3s
n = 3,
l = 0

64. l = 1, p orbitals
each principal energy state above n = 1 has 3 p orbitals
ml = -1, 0, +1
each of the 3 orbitals point along a different axis
px, py, pz
2nd lowest energy orbitals in a principal energy state
two-lobed
node at the nucleus, total of n nodes
Tro, Chemistry: A Molecular Approach

65. p orbitals
Tro, Chemistry: A Molecular Approach

66. l = 2, d orbitals
each principal energy state above n = 2 has 5 d orbitals
ml = -2, -1, 0, +1, +2
4 of the 5 orbitals are aligned in a different plane
the fifth is aligned with the z axis, dz squared
dxy, dyz, dxz, dx squared – y squared
3rd 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
Tro, Chemistry: A Molecular Approach

67. d orbitals
Tro, Chemistry: A Molecular Approach

68. l = 3, f orbitals
each principal energy state above n = 3 has 7 d orbitals
ml = -3, -2, -1, 0, +1, +2, +3
4th 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
Tro, Chemistry: A Molecular Approach

69. f orbitals
Tro, Chemistry: A Molecular Approach

70. Why are Atoms Spherical?
Tro, Chemistry: A Molecular Approach

71. Energy Shells and Subshells
Tro, Chemistry: A Molecular Approach