Chapter 7 The QuantumMechanical Model of the Atom

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

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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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The Nature of Lightits 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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Speed of Energy Transmission
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Electromagnetic Radiation
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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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Wave Characteristics
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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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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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Example 7.1- Calculate the wavelength of red
light with a frequency of 4.62 x 1014 s-1
n 4.62 x 1014 s-1 l, (nm)
Given Find
ln c, 1 nm 10-9 m
Concept Plan Relationships
Solve
the unit is correct, the wavelength is
appropriate for red light
Check
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Practice Calculate the wavelength of a radio
signal with a frequency of 100.7 MHz
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Practice Calculate the wavelength of a radio
signal with a frequency of 100.7 MHz
n 100.7 MHz l, (m)
Given Find
Concept Plan Relationships
ln c, 1 MHz 106 s-1
Solve
the unit is correct, the wavelength is
appropriate for radiowaves
Check
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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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Amplitude Wavelength
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Electromagnetic Spectrum
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Continuous Spectrum
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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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Thermal Imaging using Infrared Light
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Using High Energy Radiationto Kill Cancer Cells
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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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Interference
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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

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Diffraction
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2-Slit Interference
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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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The Photoelectric Effect
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The Photoelectric EffectThe 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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Einsteins 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 Plancks
  Constant, (h) and has the value 6.626 x 10-34 Js

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Example 7.2- Calculate the number of photons in a
laser pulse with wavelength 337 nm and total
energy 3.83 mJ
l 337 nm, Epulse 3.83 mJ number of photons
Given Find
Ehc/l, 1 nm 10-9 m, 1 mJ 10-3 J,
Epulse/Ephoton photons
Concept Plan Relationships
Solve
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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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What is the frequency of radiation required to
supply 1.0 x 102 J of energy from 8.5 x 1027
photons?
Etotal 1.0 x 102 J, number of photons 8.5 x
1027 n
Given Find
Ehn, Etotal Ephoton photons
Concept Plan Relationships
Solve
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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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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

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Emission Spectra
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Exciting Gas Atoms to Emit Light with Electrical
Energy
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Examples of Spectra
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Identifying Elements with Flame Tests
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Emission vs. Absorption Spectra
Spectra of Mercury
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Bohrs 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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Bohr Model of H Atoms
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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

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Electron Diffraction
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Example 7.3- Calculate the wavelength of an
electron traveling at 2.65 x 106 m/s
v 2.65 x 106 m/s, m 9.11 x 10-31 kg (back
leaf) l, m
Given Find
lh/mv
Concept Plan Relationships
Solve
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Practice - Determine the wavelength of a neutron
traveling at 1.00 x 102 m/s(Massneutron 1.675
x 10-24 g)
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Practice - Determine the wavelength of a neutron
traveling at 1.00 x 102 m/s
v 1.00 x 102 m/s, m 1.675 x 10-24 g l, m
Given Find
lh/mv, 1 kg 103 g
Concept Plan Relationships
Solve
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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

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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

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Uncertainty Principle Demonstration
any experiment designed to observe the electron
results in detection of a single electron
particle and no interference pattern
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Determinacy vs. Indeterminacy

• according to classical physics, particles move in
  a path determined by the particles 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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Trajectory vs. Probability
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Electron Energy

• electron energy and position are complimentary
• because 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
• distance vs. y2
• many of the properties of atoms are related to
  the energies of the electrons

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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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Principal Quantum Number, n

• characterizes the energy of the electron in a
  particular orbital
• corresponds to Bohrs 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

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Principal Energy Levels in Hydrogen
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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

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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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Hydrogen Energy Transitions
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Example 7.7- Calculate the wavelength of light
emitted when the hydrogen electron transitions
from n 6 to n 5
ni 6, nf 5 l, m
Given Find
Ehc/l, En -2.18 x 10-18 J (1/n2)
Concept Plan Relationships
DEatom -Ephoton
Solve
Ephoton -(-2.6644 x 10-20 J) 2.6644 x 10-20 J
Check
the unit is correct, the wavelength is in the
infrared, which is appropriate because less
energy than 4?2 (in the visible)
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Practice Calculate the wavelength of light
emitted when the hydrogen electron transitions
from n 2 to n 1
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Calculate the wavelength of light emitted when
the hydrogen electron transitions from n 2 to n
1
ni 2, nf 1 l, m
Given Find
Ehc/l, En -2.18 x 10-18 J (1/n2)
Concept Plan Relationships
DEatom -Ephoton
Solve
Ephoton -(-1.64 x 10-18 J) 1.64 x 10-18 J
Check
the unit is correct, the wavelength is in the UV,
which is appropriate because more energy than 3?2
(in the visible)
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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

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Probability Density Function
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Radial Distribution Function
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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

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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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2s and 3s
2s n 2, l 0
3s n 3, l 0
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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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p orbitals
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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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d orbitals
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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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f orbitals
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Why are Atoms Spherical?
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Energy Shells and Subshells