Test: Packing Efficiency - Chemistry MCQ

Density, d of unit cell is given by d = 
Given,
Density, d = 9.0 g/cm³
Atomic mass, M = 63.5 g/mole
Edge length = a
NA = Avogadro’s number = 6.022 x 1023
z = 4 atoms/cell
On rearranging the equation for density we get a³ = 
Substituting the given values:

Therefore, a = 360.5 pm
The relation of edge length ‘a’ and radius of particle ‘r’ for FCC packing i.e. a = 2√2r.
On substituting the value of ‘a’ in the given relation, r = =127.46 pm
Now, for spherical particles volume, V = 4πr^3/3 and surface area, S = 4πr²
Required ratio = S/V=4πr²/(4πr³/3) = 3/r (after simplifying)
Thus, S/V = 3/127.46 = 0.0235.

FCC, CCP and HCP are unit cells with least free space available i.e. highest packing efficiency. The coordination number of given cells are 12.

Packing efficiency can never be 100% because in packing calculations all constituent particles filling up the cubical unit cell are assumed to be spheres

For BCC unit cell the relation between radius of a particle ‘r’ and edge length of unit cell, a, is r = 
We know that diameter, d = 2r =
Implying d² =
Therefore, 4d²/3=a²
Multiplying by 6 on both sides gives S = 6a² = 8d², where S is the surface area of the cube = 6a².

Packing fraction is a dimensionless quantity which is the ratio of space occupied to total crystal space available. Since both the quantities have the same units the ratio renders dimensionless.

ABCABC arrangement is found in CCP.
In closed cubic packing, relation between edge length of unit cell, a, and radius of particle, r, is given as a=2√2r.
Surface area (S.A.) = 6a²
From the relationship,
a² = 8r²
S.A. = 6a² = 48r²
When r = 166 pm, S.A. = 48(166pm) = 1.32 x 10⁶ pm².

The packing efficiency in CCP and simple cubic lattice are 74% and 52%, respectively. Hence the corresponding free spaces will be 100% – 74% = 26% and 100% – 52% = 48%.

Copper metal bears face-centered unit cells in its crystal structure. Potassium and chromium both have body-centered unit cells whereas polonium is the only known metal to bear a simple cubic structure. FCC structure has the highest efficiency.

HCP and CCP have the highest packing efficiency of 74% followed by BCC which is 68%. The simple cubic structure has a packing efficiency of 54%.

Packing factor is a fraction of total space of the unit cell occupied by the constituent particles.