Concept explainers
Interpretation:
Packing efficiency of a simple cubic cell, body-centred cubic and face-centred cubic unit cells, have to be calculated.
Concept Introduction:
The major types of cubic unit cells are:
- Simple cubic unit cell.
- Face-centered cubic unit cell.
- Body-centered cubic unit cell.
Packing efficiency, is the percentage of the cell space occupied by the spheres in a cubic unit cell of any type. The atoms in the unit cells are considered as spheres.
Packing efficiency = Volume of atoms in unit cellVolume of unit cell × 100%
Answer to Problem 12.121SP
The packing efficiency of a simple cubic unit cell is 52.3 %.
The packing efficiency of a body-centred cubic unit cell is 67.98% .
The packing efficiency of a face-centered cubic unit cell is 74.01 %
Explanation of Solution
Calculating the packing efficiency for a simple cubic unit cell:
Edge length is a = 2r
The relation between edge length and volume is V= a3
So the volume of the unit cell is V = (2r)3
The number of atoms in the simple cubic unit cell is 1.
An atom is considered as sphere.
The volume of sphere is 4πr33.
So, the volume of atoms in the simple cubic unit cell can be expressed as:
Volume of atoms in unit cell = 1×(4πr33)
Packing efficiency = Volume of atoms in unit cellVolume of unit cell × 100%
Packing efficiency = 1×(4πr33)(2r)3 ×100%
= 4πr338 r3×100%= 14π r33 × 128 r3= π2×3 ×100%
= π6 ×100%= 3.146 × 100% ∵π =3.14=52.3 %
Thus, the packing efficiency of a simple cubic unit cell is 52.3 %.
Calculating the packing efficiency for a body-centered cubic unit cell:
Edge length is a = 4r√3
The relation between edge length and volume is V= a3
So the volume of the unit cell is V = ( 4r√3)3
The number of atoms in the body-centered cubic unit cell is 2.
An atom is considered as sphere.
The volume of sphere is 4πr33.
So, the volume of atoms in the body-centered cubic unit cell can be expressed as:
Volume of atoms in unit cell = 2×(4πr33)
Packing efficiency = Volume of atoms in unit cellVolume of unit cell × 100%
Packing efficiency = 2×(4πr33)( 4r√3)3 ×100%
= 8πr3364 r33√3×100%= 18 π r33 × 3√3864 r3
= 3√3π8 ×3 ×100% = 1.7321 × 3.14 8×100% ∵ √3 =1.7321 ; π =3.14= 0.6798 ×100%= 67.98%
Thus, the packing efficiency of a body-centered cubic unit cell is 67.98% .
Calculating the packing efficiency for a face-centered cubic unit cell:
Edge length is a = √8r
The relation between edge length and volume is V= a3
So the volume of the unit cell is V = (√8r)3
The number of atoms in the body-centered cubic unit cell is 4.
An atom is considered as sphere.
The volume of sphere is 4πr33.
So, the volume of atoms in the body-centered cubic unit cell can be expressed as:
Volume of atoms in unit cell = 4×(4πr33)
Packing efficiency = Volume of atoms in unit cellVolume of unit cell × 100%
Packing efficiency = 4×(4πr33)(√8r)3 ×100%
= 16πr338√8 r3×100%= 216 π r33 × 118√8 r3×100%=2π3 √8 ×100%
= 2 × 3.143×2.8284 ×100% ∵ √8 =2.8284 ; π =3.14= 6.28 8.4852×100% = 0.7401 ×100%= 74.01 %
Thus, the packing efficiency of a face-centered cubic unit cell is 74.01 %
Using figure 11.2 as reference, the packing efficiency for the different types of cubic unit cells was calculated.
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