NCERT Class XII Chemistry
Chapter - The Solid State
Questions with Solutions
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Question : 10
Total: 50
Calculate the efficiency of packing in case of a metal crystal for
(i) simple cubic
(ii) face-centred cubic (with the assumptions that atoms are touching each other).
(iii) body-centred cubic
(i) simple cubic
(ii) face-centred cubic (with the assumptions that atoms are touching each other).
(iii) body-centred cubic
Solution:
(i) In a simple cubic unit cell :
Suppose the edge length of the unit cell = a and radius of the sphere = ra = 2 r
No. of spheres per unit cell=
× 8 = 1
Volume of the sphere=
Ï€ r 3
Volume of the cube= a 3 = ( 2 r ) 3 = 8 r 3
therefore , i . e . , = (
Ï€ r 3 ) 8 r 3 = 0.524
or % occupiedi . e . , AC = 4 r .
But from right angled triangle ABC,A C = √ A B 2 + B C 2
= √ a 2 + a 2 = √ 2 a
∴ √ 2 a = 4 r a =
r
∴ = a 3 = (
r ) 3 =
r 3
No. of spheres in the unit cell= 8 ×
+ 6 ×
= 4
Volume of four spheres= 4 ×
Ï€ r 3 =
Ï€ r 3
∴ Fraction occupied i.e., packing fraction=
= 0.74
or % occupiedi . e . , A D = 4 r .
Further, face diagonal,
A C = √ A B 2 + B C 2 = √ a 2 + a 2 = √ 2 a
and body diagonal,
A D = √ A C 2 + C D 2 = √ 2 a 2 + a 2 = √ 3 a
∴ √ 3 a = 4 r
ora =
∴ = a 3 = (
) 3 =
No. of spheres per unit cell= 8 ×
+ 1 = 2
Volume of two spheres= 2 ×
Ï€ r 3 =
Ï€ r 3
therefore =
= 0.68
or % occupiedi . e
Suppose the edge length of the unit cell = a and radius of the sphere = r
As spheres are touching each other, evidently,
No. of spheres per unit cell
Volume of the sphere
Volume of the cube
or % occupied
(ii) In face-centred cubic structure : As sphere on the face-centre is touching the spheres at the corners, evidently
But from right angled triangle ABC,
No. of spheres in the unit cell
Volume of four spheres
∴ Fraction occupied i.e., packing fraction
or % occupied
(iii) In body-centred cubic structure : As the sphere at the body-centre touches the spheres at the corners, body diagonal,
Further, face diagonal,
and body diagonal,
or
No. of spheres per unit cell
Volume of two spheres
or % occupied
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