Chapter 7 The Quantum-Mechanical Model of the Atom

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Chapter 7The Quantum-Mechanical Model of the Atom
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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 (wave function)
• 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
• The negative sign means that the energy of the
  electron bound to the nucleus is lower than it
  would be if the electron were at an infinite
  distance (n 8) from the nucleus, where there is
  no interaction.

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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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Hydrogen Energy Transitions
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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
• Since the energy must be conserved, the exact
  amount of energy emitted by the atom is carried
  away by the photon
• ?Eatom - ?Ephoton

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Example

• Determine the wavelength of light emitted when an
  electron in a hydrogen atom makes a transition
  from an orbital in n 6 to an orbital in n5
• As electron in the n6 level of the hydrogen atom
  relaxes to a lower energy level, emitting light
  of ? 93.8 nm. Find the principle level to
  which the electron relaxed

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The angular Momentum Quantum number (l)

• Is an integer that determines the shape of the
  orbital.

Quantum number n (shell) Value of l Letter designation (subshell)
n 1 l 0 s
n 2 l 1 p
n 3 l 2 d
n 4 l 3 f
the energy of the subshell increases with l (s lt
p lt d lt f).  
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The magnetic quantum number (ml)

• Specifies the orientation in space of an orbital
  of a given energy (n) and shape (l).
• This number divides the subshell into individual
  orbitals which hold the electrons there are 2l1
  orbitals in each subshell. Thus the s subshell
  has only one orbital, the p subshell has three
  orbitals, and so on

n l Orbitals ml
1 0 0
2 0, 1 -1, 0, 1
3 0, 1, 2 -2, -1, 0, 1, 2
4 0, 1, 2, 3 -3, -2, -1, 0, 1, 2, 3, 4
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Examples

• Give the possible combination of quantum numbers
  for the following orbitals
• 3s orbital 2 p orbitals
• Give orbital notations for electrons in orbitals
  with the following quantum numbers
• n 2, l 1 ml 1
• n 3, l 2, ml -1

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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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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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Now we know why atoms are spherical