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
Quantum Mechanics
The Nature of Light
Atomic Spectroscopy and the Bohr Model
The Wave Nature of Matter
The de Broglie Wavelength
Uncertainty Principle
Quantum Mechanics and the Atom
The Shapes of Atomic Orbitals

2. Anyone who is not shocked by quantum mechanics has not understood it
Neils Bohr
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3. 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 the physical and chemical properties of atoms
to understand this behavior we must understand how electrons exist within atoms!
directly observing electrons in the atom is impossible, e.g. if you attempt to measure its position using light, the light itself disturbs the electron changing its position
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4. 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
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5. Prior to the development of QM the nature of light was viewed as being very different from that of subatomic particles such as electrons
As QM developed, light was found to have many of the characteristics in common with electrons
Chief among these is the wave-particle duality of light – certain properties are best described by thinking of it as a wave, while other properties are best described by thinking of it as a particle
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6. 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
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7. Speed of Energy Transmission
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8. The Electromagnetic Spectrum
Insert fig 5.15
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9. Electromagnetic Radiation
Electromagnetic waves consist of oscillating, perpendicular electric and magnetic fields.
The wavelength of radiation is the distance between peaks in a wave. ()
The frequency is the number of peaks that pass a point in a second. ( )
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10. Electromagnetic Radiation
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11. 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
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12. 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
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13. 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
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14. Frequency and Wavelength
Given two of the following calculate the third: l, n, c (velocity of propagation).
ln = c
l = c/n
n = c/l
l is wavelength measured in length units (m, cm, nm, etc.)
n is frequency measured in Hz (s-1).
c is the velocity
velocity of light in vacuum = 3.0 x108 ms-1
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15. 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
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16. Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz
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17. Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz
Given:
Find:
n = MHz
l, (m)
Concept Plan:
Relationships:
l∙n = c, 1 MHz = 106 s-1
n (MHz)
n (s-1)
l (m)
Solve:
Check:
the unit is correct, the wavelength is appropriate for radiowaves
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18. 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
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19. Amplitude & Wavelength

20. Electromagnetic Spectrum
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21. 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
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22. Thermal Imaging using Infrared Light
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23. Electromagnetic Spectrum
Recognize common units for l, n.
l wavelength n frequency
meters (m) radio Hertz Hz s-1
micrometers mm (cycles per
(10-6 m) microwaves second)
nanometers nm megahertz
(10-9 m) light MHz (106 Hz)
Å angstrom (10-10 m)
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24. Tro 1/e Chap 7

25. 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
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26. 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
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27. Diffraction
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
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28. The Particle Nature of Light
Prior to the 1900s light was thought to be purely a wave phenomenon
Principally due to the discovery of diffraction
This classical view began to be questioned….
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29. 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
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30. The Photoelectric Effect
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31. 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
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32. Photoelectric Effect
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33. 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
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34. Define quantum, photon.
Quantum- A packet of energy equal to hn. The smallest quantity of energy that can be emitted or absorbed.
Photon- A quantum of electromagnetic radiation.
Thus light can be described as a particle (photon) or as a wave with wavelength and frequency. This is called wave-particle duality.
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35. Given one of the following, calculate the other: E, n.
Ephoton = hn
where:
Ephoton is the energy of each photon (packet) of light.
h is Planck’s constant = 6.626x10-34 Js
n is the frequency of light in Hz (s-1).
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36. 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:
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37. 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?
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38. What is the frequency of radiation required to supply 1
What is the frequency of radiation required to supply 1.0 x 102 J of energy from 8.5 x 1027 photons?
Given:
Find:
Etotal = 1.0 x 102 J, number of photons = 8.5 x 1027 n
Concept Plan:
Relationships:
E=hn, Etotal = Ephoton∙# photons
n (s-1)
Ephoton
number
photons
Solve:
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39. 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
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40. Although quantization of light explained the photoelectric effect, the wave explanation of light continued to have explanatory power
What slowly emerged is what we now call the wave-particle duality of light
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41. Exciting e- Many ways: Heat, light, e- bombardment, chemical reactions
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42. Atomic 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 Hg He H
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43. Emission Spectra
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44. Examples of Spectra
Oxygen spectrum
Neon spectrum
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45. Atomic Spectra
Rydberg analyzed the spectrum of hydrogen and found that it could be described with an equation that involved an inverse square of integers
Predicted wavelengths of emission spectrum for hydrogen
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46. A Bar Code for Atoms Flame TestsNa K Li Ba
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47. Emission vs. Absorption Spectra
Spectra of Mercury
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48. Bohr’s Model
Bohr attempted to develop a model for the atom that explained atomic spectra…
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49. 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
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50. Bohr Model of H Atoms
e- is never observed between states, only in one state or the next
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51. Ephoton = DE = Ehigh-Elow
DE = hn = hc/l
nlight = DE/h
l = hc/DE
Since Ehigh and Elow are discreet numbers, DE must be a discreet number
Therefore,
n and l must be discreet numbers, giving rise to single frequencies and wavelengths of light. Hence, the line spectra.
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52. Question
For the yellow line in the sodium spectrum,  = nm. Calculate the energy in joules of a photon emitted by an excited sodium atom.
 =  1014 /s
E =  J
 =  1014 /s
E =  J
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54. Bohr’s model could not be applied to atoms with 2 or more e-
Needed a new approach…
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55. 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
A single e- produces an interference pattern
the electron interferes with its own wave (not other e-)
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56. 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
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57. Explains the existence of stationary states / energy levels in the Bohr model
Prevents e- from crashing into the nucleus as predicted by classical physics
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58. Wave Behavior of Electrons
de Broglie predicted that the wavelength of a particle was inversely proportional to its momentum
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59. 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:
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60. 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)
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61. 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
Given:
Find:
v = 1.00 x 102 m/s, m = x g
l, m
Concept Plan:
Relationships:
l=h/mv, 1 kg = 103 g
m (kg), v
l (m)
m(g)
Solve:
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62. Practice - Determine the wavelength of a bowling ball traveling at 1 m/s (Mass = 1 kg)
1 x nm
1 septillionth of a nanometer. This is so ridiculously small compared to the size of the bowling ball itself that you'd never notice any wavelike stuff going on; that's why we can generally ignore the effects of quantum mechanics when we're talking about everyday objects
It's only at the molecular or atomic level that the waves begin to be large enough (compared to the size of an atom) to have a noticeable effect.
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63. The Uncertainty Principle
How can a single entity behave as both a particle and a wave?
How does a single e- aimed at a double-slit produce an interference pattern?
A possible hypothesis is the e- splits into 2, travels through both slots and interferes with itself
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64. Testing the hypothesis…
any experiment designed to observe the electron results in detection of a single electron particle and no interference pattern
Experiment designed to ‘watch’ which slit the e- travels through using a laser beam, flash is seen from one slit or the other…never both at once
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65. Wave nature and particle nature of the e- are complementary properties
No matter how hard we try we can never see the interference pattern and simultaneously determine which hole the e- goes through
We have encountered the absolutely small and have no way of determining what it is without disturbing it
Wave nature and particle nature of the e- are complementary properties
The more you know about one, the less you know about the other
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66. 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 (related to particle nature)
v = velocity, Dv = uncertainty in velocity (related to wave nature)
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
Electron is observed EITHER as a particle OR a wave, never both at once
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67. 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
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68. Trajectory vs. Probability

69. Electron Energy Position and velocity are complimentary
electron energy and position are complimentary
because kinetic energy is related to velocity, KE = ½mv2
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
A plot of distance vs. y2 represents an orbital
many of the properties of atoms are related to the energies of the electrons
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70. 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
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71. 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 greater/less negative the energy
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
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72. Principal Energy Levels in Hydrogen
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73. 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

74. 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
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75. Hydrogen Energy Transitions

76. 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)

77. Practice – Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1
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78. Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1
Given:
Find:
ni = 2, nf = 1
l, m
Concept Plan:
Relationships:
E=hc/l, En = x J (1/n2)
ni, nf
DEatom
Ephoton l
DEatom = -Ephoton
Solve:
Ephoton = -(-1.64 x J) = 1.64 x J
Check:
the unit is correct, the wavelength is in the UV, which is appropriate because more energy than 3→2 (in the visible)

79. Shapes of Orbitals
Shapes are important because covalent bonds depend on sharing e-
Chemical bonds consists on overlapping orbitals
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80. 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 (l = 0)
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
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81. 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)
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82. Probability & Radial Distribution Functions
y2 is the probability density (= prob. / unit vol.)
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

83. Probability Density Function
Probability density at a point
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84. Radial Distribution Function
Radial distribution represents total probability at radius r
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85. 2s and 3s 2s
n = 2,
l = 0 3s
n = 3,
l = 0

86. 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
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87. p orbitals
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88. 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
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89. d orbitals
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90. 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
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91. f orbitals
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92. Why are Atoms Spherical?
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93. Energy Shells and Subshells
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