Question

Calculate the fraction of empty space in cubic closest packing to five significant figures.

Answer 1

0.259 52 . The fraction of empty space in a ccp cell is 0.259 52

Cubic closest packing is another name for face centred cubic.

Refer to the attached diagram of an fcc unit cell from hemantmore.

a) Volume of atoms in unit cell

a. Number of atoms in unit cell

In an fcc lattice, there are eight atoms at the eight corners of the cube and one at the centre of each face

∴ No. of atoms

= 8 corners x (1/8 atom)/(1 corner) + 6 faces x (½ atom/1 face)

= 1 atom + 3 atoms = 4 atoms

b. Volume of atoms

Let r be the radius of an atom. Then

Volume of 1 atom = (4/3)πr^3

Total volume of atoms = 4 atoms × [(4/3)πr^3]/(1 atom) = (16/3)πr^3

b) Volume of unit cell

The length of the diagonal AB is

AB = 4r

Let a be the edge length of the cube.

AB^2 = AC^^2 + BC^2

(4r)^2 = a^2 + a^2

16r^2= 2a^2

a^2 = 8r^2

a = rx8^½

V = a^3 = (rx8^½)^3 = 8r^3x8^½

c) Fraction of occupied space (packing efficiency, PE)

PE = Volume of atoms/Volume of cell = [(16/3)πr^3]/[ 8r^3x8^½]

= 2π/(3x8^½)

d) Fraction of empty space

Fraction of empty space = 1 - PE = 1 - 2π/(3x8^½) = 1 – 0.740 480

= 0.259 52