Which of the Following Lattices Has the Highest Packing Efficien Intext Questions Chapter 1 the Solid State Chemistry

Question 1.17: Which of the following lattices has the highest packing efficiency (i) simple cubic (ii) body−centred cubic and (iii) hexagonal close−packed lattice?

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 × 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 × 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) hexagonal close−packed lattice
Let take base of hexagonal is a and height is c  
Each angle in hexagonal will 60 degree at base

Hexagonal close−packed lattice has the highest packing efficiency of 74%.