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Grade upto college level Physical Chemistry

A metallic element crystallizes into a lattice containing a sequence of layers of ABABAB………. Any packing of spheres leaves out voids in the lattice. What percentage by volume of this lattice is empty space?

Profile image of Shane Macguire
12 Years agoGrade upto college level
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2 Answers

Profile image of Deepak Patra
12 Years ago
Hello Student,
Please find the answer to your question
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%

Thanks
Deepak patra
askIITians Faculty
Profile image of Kushagra Madhukar
5 Years ago
Dear student,
Please find the solution to your problem.
 
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 * √(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%
 
Thanks and regards,
Kushagra