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A metal crystallizes in a body-centered cubic lattice with an edge length of 3.5\AA. If the metal atoms have a radius of 1.2\AA, what is the packing efficiency of the crystal?

Option: 1


Option: 2


Option: 3


Option: 4


Answers (1)


The packing efficiency of a body-centered cubic BCC structure is given by 68%.

To arrive at the answer, we can use the formula for the packing efficiency of a BCC structure:

Packing efficiency

=\frac{\text { number of atoms in the unit cell } \times \text { volume of each atom }}{\text { volume of the unit cell }}

In a BCC structure, there are two atoms per unit cell - one at each of the eight corners and one at the center of the cube. The volume of each atom can be calculated using the formula for the volume of a sphere:

Volume of each atom=\frac{4}{3} \times \pi \times r^3

Substituting the given values, we get:

Volume of each atom =\frac{4}{3} \times \pi \times(1.2 \AA)^3=7.2384 \AA^3

The volume of the unit cell can be calculated as:

Volume of the unit cell(\text { edge length })^3

Substituting the given value, we get:

Volume of the unit cell (3.5 A)^3=42.875 A^3

Putting everything back in the formula for packing efficiency, we get:

Packing efficiency =\frac{2 \times 7.2384 A^3}{42.875 A^3}=0.68 \text { or } 68 \%

Therefore, the correct answer is (c) 68%.

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