12th chemistry solid state - Packing Efficiency & Calculations Involving Unit Cell Dimensions

Packing Efficiency & Calculations Involving Unit Cell Dimensions

Packing Efficiency

• Percentage of total space filled by particles

Calculations of Packing Efficiency in Different Types of Structures

• Simple cubic lattice

In a simple cubic lattice, the particles are located only at the corners of the cube and touch each other along the edge.

Let the edge length of the cube be ‘a’ and the radius of each particle be r.

Then, we can write:

a = 2r

Now, volume of the cubic unit cell = a³

= (2r)³

= 8r³

The number of particles present per simple cubic unit cell is 1.

Therefore, volume of the occupied unit cell 

Hence, packing efficiency 

• Body-centred cubic structures

It can be observed from the above figure that the atom at the centre is in contact with the other two atoms diagonally arranged.

From ΔFED, we have

From ΔAFD, we have

Let the radius of the atom be r.

Length of the body diagonal, c = 4r

or, 

Volume of the cube, 

A body-centred cubic lattice contains 2 atoms.

• hcp and ccp Structures

Let the edge length of the unit cell be ‘a’ and the length of the face diagonal AC be b.

From ΔABC, we have

Let r be the radius of the atom.

Now, from the figure, it can be observed that:

Now, volume of the cube, 

We know that the number of atoms per unit cell is 4.

• Thus, ccp and hcp structures have maximum packing efficiency.

Calculations Involving Unit Cell Dimensions

In a cubic crystal, let

a = Edge length of the unit cell

d = Density of the solid substance

M = Molar mass of the substance

Then, volume of the unit cell = a³

Again, let

z = Number of atoms present in one unit cell

m = Mass of each atom

Now, mass of the unit cell = Number of atoms in the unit cell × Mass of each atom

= z × m

But, mass of an atom, m 

Therefore, density of the unit cell,