Calculate the Efficiency of Packing in Case of a Metal Crystal Fo Exercise Chapter 1 the Solid State Chemistry Class

Question 10 Calculate the efficiency of packing in case of a metal crystal for
(i) simple cubic
(ii) body–centred cubic
(iii) face–centred cubic (with the assumptions that atoms are touching each other).

Answer
(i)
In a simple cubic lattice the atoms are located only on the corners of the cube.
Let take edge length or side of the cube = a,
Let take radius of each particles            = r
The relation between radius and edge a
                 a     = 2r
The volume of the cubic unit cell = side³
                                               = a³
                                               = (2r)³
                                               = 8r³
Number of atoms in unit cell  = 8 x 1 /8
                                          = 1
The volume of the occupied space = (4/3)πr³

(ii)In body centered cubic two atoms diagonally

Let take edge length or side of the cube = a,
Let take radius of each particles            = r
The diagonal of a cube is always a√3
The relation between radius and edge a will
 a√3 = 4r
divide by root 3 we get
                        a          = 4r/√3
total number of atoms in body centered cubic
number of atoms at the corner = 8 x 1/8 = 1
number of atoms at the center = 1
total number of atoms             = 2
The volume of the cubic unit cell = side³
                                               = a³
                                               = (4r/√3)³
The volume of the occupied space = (4/3)πr³

(iii)
Let take edge length or side of the cube = a
Let take radius of each particles           = r
The diagonal of a square  is always a√2
The relation between radius and edge a will
 a√2 = 4r
divide by root 3 we get
                        a          = 4r/√2
total number of atoms in body centered cubic
number of atoms at the corner = 8 x 1/8 = 1
number of atoms at the face    = 6 x 1/2    = 3
total number of atoms            = 4
The volume of the cubic unit cell = side³
                                               = a³
                                               = (4r/√2)³
                                               = (2√2 r)³
The volume of the occupied space = (4/3)πr³