## Packing Efficiency Chemistry Notes

→ As we know that the constituent particles in crystal lattice are arranged in close packing. Some spaces remain vacant in this state, which are called voids. The percentage of the total space filled by the particles is called packing efficiency or the fraction of total space filled is called packing fraction.

% Packing efficiency

**Packing Eficiency in hcp or ccp or fcc Structures :**

Atomic radius = r

Length of edge of a unit cell = a

Volume of one sphere = \(\frac{4}{3}\) (πr^{3})

∵ fcc stucture is formed from four spheres.

∴ Volume of four spheres = 4 × \(\frac{4}{3}\) (πr^{3}) = \(\frac{16}{3}\) (πr^{3})

∆ ABC AC^{2} = AB^{2} + BC^{2}

= a^{2} + a^{2}

∴ AC = a√2 … (1)

If we see AC then the arrangement of spheres in it is as follows

Hence, the total volume occupied by spheres or particles in fec or ccp or hep structure is 74%. While the empty space i.e. volume of total voids is 26%.

**Packing Efficiency in Body Centred Cubic Sturcture (bcc) :**

Atomic radius = r

Edge length of unit cell = a

Since bec structure forms from two spheres.

So, Volume of two spheres = 2 × \(\left(\frac{4}{3} \pi r^{3}\right)\) = \(\frac{16}{3}\) πr^{2}

In ∆ABC, AC^{2} = AB^{2} + BC^{2}

AC^{2} = a^{2} + a^{2}

AC^{2} = 2a^{2}

In ∆ACD, AD^{2} = AC^{2} + CD^{2}

AD^{2} = 2a^{2} + a^{2}

or AD^{2} = 3a^{2}

or AD = √3a^{2}

∴ AD = √3

If we see AD, then the arrangement of spheres in it is as follows

AD – 4r

On putting the value of AD in equation (i),

Hence, the total volume occupied by spheres or particles in bee structure is 68% While, the empty space i.e. volume of total voids is 32%.

**Packing Efficiency in Simple Cubic Unit Cell (scc)**

Atomic radius = 1

Length of edge of unit call = a

scc structure forms from one sphere, so

Volume of one sphere = \(\frac{4}{3}\) πr^{3}

In a simple cubic lattice, an atom is present only at corners of cube. The particles present at corners of cube remain in contact with each other. So, in this arrangement, length of edge of cube is ‘a’ and radius of each atom is ‘Y’, which are related to each other as follows

Hence, the total volume occupied by spheres or particles in soc structure is 52.4%. While the empty space i.e. volume of total voids is 47.6%.