E106 Problem 3'35

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E106 Problem 3.35

• Jason Choi
• Jan 24 2007

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The Problem

• The unit cell for Fe3O4 (FeO-Fe2O3) has cubic
  symmetry with a unit cell edge length of 0.839
  nm. If the density of this material is 5.24
  g/cm3, compute its atomic packing factor. For
  this computation, you will need to use ionic
  radii listed in Table 3.4

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Unit Cell

• In crystalline solids small groups of atoms form
  a repetitive pattern.
• The small repeating entities are called the unit
  cell.

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Atomic Packing Factor

• The APF is the fraction of solid sphere volume in
  a unit cell. (assuming the atomic hard sphere
  model)

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Magnetite (Fe3O4)

• Magnetite is a ferromagnetic mineral with
  chemical formula Fe3O4

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Determining the Atomic Packing Factor

• To determine the APF, we must know VA and VC.
• But to find VA

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Unit cell of Fe3O4
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Density
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Determination of the Unit Cell

• Thus, in a unit cell, there is 8 magnetite
  molecules.

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Ionic Radii (Table 3.4)
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Det. of the number of all ions.

• Balancing the charges of the anion and cation we
  have 4 O2-, 2 Fe3, 1 Fe2
• 4x8 32 O2-
• 2x8 16 Fe3
• 1x8 8 Fe2

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LEGOS!

• 32 total O2- ions
• 16 total Fe3 ions
• 8 total Fe2 ions

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Determining the Atomic Packing Factor
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Implications

• Using X-Ray diffraction we can determine the
  crystal structures of molecules.
• With the unit cell of a molecule known, we can
  use the unit cell density method, to verify
  existing experimental density values.
• However, the density calculated by the unit cell
  density method may not be exactly the same as the
  actual density due to the imperfectness of
  crystal structures.