Answer:
Packing efficiency is the percentage
of the total space which is occupied by the particles in a certain packing. The
fraction of the total space filled by spheres is called packing fraction.
Packing fraction.
(a) In a simple cubic
unit cell,
Suppose the edge
length of the unit cell = a
and radius of the
sphere = r
As spheres are
touching each other so, a = 2r
No. of spheres per
unit cell
Volume of 1 sphere
Volume of the cube
= a3 = (2r)3 = 8r3
Fraction
occupied i.e. packing fraction
or % of space
occupied = 524%
Percentage of
unoccupied space
= 100 ? 52 4 = 47 6%
(b) In body
centered cubic structure,
As the sphere at
the centre touches the spheres at the corner, body diagonal AD = 4r
Further face
diagonal,
and body diagonal,
Volume of the unit
cell
No. of spheres per
unit cell
Volume of 2 spheres
Fraction
occupied i.e. packing fraction
or % of space
occupied = 68%
Percentage of
unoccupied space
=100 ? 68 = 32%
(c) In face-centred
cubic structure
As spheres on the
face are touching, so AC = 4r
But from the right
angled triangle ABC,
or
Volume of the unit
cell
No. of spheres in
the unit cell
Volume of 4 spheres
Fraction occupied
i.e. packing fraction or efficiency of packing
or % of space
occupied = 74%
Percentage of
unoccupied space
= 100 ? 74 = 26%
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