To calculate the packing efficiency of nickel, we first need to understand some concepts around unit cells and atomic packing.
Nickel crystallizes in a face-centered cubic (FCC) structure. In this structure, there are 4 atoms per unit cell, and the effective radius (r) of a nickel atom will help in calculating the volume occupied by these atoms.
The formula to find the packing efficiency is:
Packing Efficiency (%) = (Volume of atoms in unit cell / Volume of unit cell) x 100
1. **Calculate the Volume of Atoms in the Unit Cell:**
The volume of one nickel atom can be calculated using the formula for the volume of a sphere:
V = (4/3)πr³
Since there are 4 atoms in an FCC unit cell, the total volume of atoms = 4 × V.
V_total = 4 × (4/3)πr³ = (16/3)πr³
2. **Calculate the Volume of the Unit Cell:**
In an FCC unit cell, the relationship between the edge length (a) and the atomic radius (r) is given by:
a = 2√2r
The volume of the unit cell is then:
V_cell = a³ = (2√2r)³ = 16√2r³
3. **Calculating Packing Efficiency:**
Substituting the total volume of atoms and the volume of the unit cell into the packing efficiency formula:
Packing Efficiency = [(16/3)πr³ / 16√2r³] x 100
Simplifying this gives:
Packing Efficiency (%) = [(π / 3√2)] x 100 ≈ 74.05%
Thus, the packing efficiency percentage of the space inside the unit cell occupied by the atoms of nickel in pure nickel is approximately 74.05%.