Molecular Bonding and Spectra

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CHAPTERS 37 38Molecules and Solids

• Molecular Bonding and Spectra
• Structural Properties of Solids

Johannes Diderik van der Waals (1837 1923)
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Motion in atoms and molecules

• Electrons vibrate in their motion around nuclei
• High frequency 1014 - 1017 cycles per
  second.
• Nuclei in molecules vibrate with respect to each
  other
• Intermediate frequency 1011 - 1013
  cycles per second.
• Nuclei in molecules rotate
• Low frequency 109 - 1010 cycles per second.

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Molecular Bonding and Spectra

• The Coulomb force is the only one to bind atoms.
• The combination of attractive and repulsive
  forces creates a stable molecular structure.
• Force is related to potential energy F -dV /
  dr, where r is the distance separation.
• It is useful to look at molecular binding using
  potential energy V.
• Negative slope (dV / dr lt 0) with repulsive
  force.
• Positive slope (dV / dr gt 0) with attractive
  force.

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Molecular Bonding and Spectra

• An approximation of the force felt by one atom
  in the vicinity of another atom is
• where A and B are positive constants.
• Because of the complicated shielding effects of
  the various electron shells, n and m are not
  equal to 1.

A stable equilibrium exists with total energy E lt
0. The shape of the curve depends on the
parameters A, B, n, and m. Also n gt m.
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Molecular Bonding and Spectra
Vibrations are excited thermally, so the exact
level of E depends on temperature.

• A pair of atoms is joined.
• One would have to supply energy to raise the
  total energy of the system to zero in order to
  separate the molecule into two neutral atoms.
• The corresponding value of r of a minimum value
  is an equilibrium separation. The amount of
  energy to separate the two atoms completely is
  the binding energy which is roughly equal to the
  depth of the potential well.

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Molecular Bonds

• Ionic bonds
• The simplest bonding mechanism.
• Example Sodium (1s22s22p63s1) readily gives up
  its 3s electron to become Na, while chlorine
  (1s22s22p63s23p5) easily gains an electron to
  become Cl-.

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Molecular Bonds

• Covalent bonds
• The atoms are not as easily ionized.
• Example Diatomic molecules formed by the
  combination of two identical atoms tend to be
  covalent.
• Larger molecules are formed with covalent bonds.

Diamond
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Molecular Bonds

• Van der Waals bond
• Weak bond found mostly in liquids and solids at
  low temperature.
• Ex in graphite, the van der Waals bond holds
  together adjacent sheets of carbon atoms. As a
  result, one layer of atoms slides over the next
  layer with little friction. The graphite in a
  pencil slides easily over paper.

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Molecular Bonds

• Hydrogen bond
• Holds many organic molecules together in solution.

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Molecular Bonds

• Metallic bond
• Free valence electrons may be shared by a number
  of atoms.
• Drude model for a metal a free-electron gas!

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Rotational States

• Molecular spectroscopy
• We can learn about molecules by studying how
  molecules absorb, emit, and scatter light.
• From the equi-partition theorem, a diatomic
  molecule may be thought of as two atoms held
  together with a massless, rigid rod (rigid
  rotator model).
• In a purely rotational system, the kinetic energy
  is expressed in terms of the angular momentum L
  and rotational inertia I.

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Rotational States

• L is quantized.
• The energy levels are
• Erot varies only as a function of the quantum
  number l.

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Vibrational States

• A vibrational energy mode can also be excited.
• Thermal excitation of a vibrational mode can
  occur.
• It is also possible to stimulate vibrations in
  molecules with light.
• Assume that the two atoms are point masses
  connected by a massless spring with simple
  harmonic motion.

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Vibrational States

• The energy levels are those of a
  quantum-mechanical oscillator.
• The frequency of a two-particle oscillator is
• where the reduced mass is m m1m2 / (m1 m2)
  and the spring constant is ?.
• If its a purely ionic bond, we can compute ? by
  assuming that the force holding the masses
  together is Coulomb.

and
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Vibration and Rotation Combined

• Its possible to excite rotational and
  vibrational modes simultaneously.
• Total energy of simple vibration-rotation system
• Vibrational energies are spaced at regular
  intervals.
• Transition from l 1 to l
• Photons will have an energies at regular
  intervals

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Vibration and Rotation Combined

• An emission-spectrum spacing that varies with l.
• The higher the starting energy level, the
  greater the photon energy.
• Vibrational energies are greater than rotational
  energies. This energy difference results in the
  band spectrum.

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Vibration and Rotation Combined

• The positions and intensities of the observed
  bands are ruled by quantum mechanics. Note two
  features in particular
• 1) The relative intensities of the bands are due
  to different transition probabilities.
• 2) Some transitions are forbidden by the
  selection rule that requires ?l 1.
• Absorption spectra
• Within ?l 1 rotational state changes,
  molecules can absorb photons and make transitions
  to a higher vibrational state when
  electromagnetic radiation is incident upon a
  collection of a particular kind of molecule.

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Vibration and Rotation Combined

• ?E increases linearly with l.

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Vibration and Rotation Combined

• In the absorption spectrum of HCl, the spacing
  between the peaks can be used to compute the
  rotational inertia I. The missing peak in the
  center corresponds to the forbidden ?l 0
  transition.
• The central frequency

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Vibration and Rotation Combined

• A transition from l to l 2
• Let hf be the Raman-scattered energy of an
  incoming photon and hf is the energy of the
  scattered photon. The frequency of the scattered
  photon can be found in terms of the relevant
  rotational variables
• Raman spectroscopy is used to study the
  vibrational properties of liquids and solids.

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Structural Properties of Solids

• Condensed matter physics
• The study of the electronic properties of solids.
• Crystal structure
• The atoms are arranged in extremely regular,
  periodic patterns.
• Max von Laue proved the existence of crystal
  structures in solids in 1912, using x-ray
  diffraction.
• The set of points in space occupied by atomic
  centers is called a lattice.

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Structural Properties of Solids

• Most solids are polycrystalline theyre made up
  of many small crystals.
• Solids lacking any significant lattice structure
  are called amorphous and are referred to as
  glasses.
• Why do solids form as they do?
• When the material changes from the liquid to the
  solid state, the atoms can each find a place that
  creates the minimum-energy configuration.

In the sodium chloride crystal, the spatial
symmetry results because there is no preferred
direction for bonding. The fact that different
atoms have different symmetries suggests why
crystal lattices take so many different forms.
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Structural Properties of Solids

• Each ion must experience a net attractive
  potential energy.
• where r is the nearest-neighbor distance, and a
  is the Madelung constant and it depends on the
  type of crystal lattice.
• In the NaCl crystal, each ion has 6 nearest
  neighbors.
• There is a repulsive potential due to the Pauli
  exclusion principle
• The value e-r /? diminishes rapidly for r gt ?.
• ? is roughly regarded as the range of the
  repulsive force.

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Structural Properties of Solids

• The net potential energy is
• At the equilibrium position (r r0), F -dV /
  dr 0.
• therefore,
• and
• The ratio ? / r0 is much less than 1.