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A metallic element crystallises into a lattice containing a sequence of layers ABABAB.... Any packing of spheres leaves out voids in the lattice. Percentage of empty space (by volume) is 52% 48% 50% 74%

A metallic element crystallises into a lattice containing a sequence of layers ABABAB.... Any packing of spheres leaves out voids in the lattice. Percentage of empty space (by volume) is
  1. 52%
  2. 48%
  3. 50%
  4. 74%

Grade:11

1 Answers

Arun
25750 Points
5 years ago
Dear student
 
For a hcp unit cell, there are 6 atoms per unit cell. If r is the radius of the metal atoms, volume occupied by the metallic
Atoms 6 * 4/3 * π * r3 = 6 * 1.33 * 22/7 * r3 = 25.08 * r3
Geometrically it has been shown that the base area of hcp unit cell
= 6 * √3/4 * 4r2 and the height = 4r * √2/3
∴ Volume of the unit cell
= Area * height = 6 * √3/4 * 4r2 * 4r * \sqrt{\frac{2}{3}}= 33.94 r3
∴ Volume of the empty space of one unit cell
= 33.94 r3 – 25.08 r3 = 8.86 r3
∴ Percentage void = 8.816 r3/33.94 r3 * 100 = 26.1%
 
Regards
Arun

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