Calculate the packing fraction for the K unit cell. K crystallizes in a body-centred cubic unit cell.

Answer

To calculate the packing fraction for the K unit cell, which crystallizes in a body-centered cubic (BCC) structure, we follow these steps:

### Step 1: Determine the effective number of atoms in the unit cell.
In a BCC unit cell, there are atoms located at the corners and one atom at the center.
- There are 8 corners in a cube, and each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Thus, the total contribution from the corner atoms is:
\[ 8 \times \frac{1}{8} = 1 \text{ atom} \]
- The atom at the center contributes 1 whole atom.
- Therefore, the total effective number of atoms (Z) in a BCC unit cell is:
\[ Z = 1 + 1 = 2 \text{ atoms} \]

### Step 2: Calculate the volume occupied by the atoms.
The volume occupied by the atoms in the unit cell can be calculated using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Thus, the total volume occupied by 2 atoms is:
\[ \text{Total volume} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

### Step 3: Determine the volume of the unit cell.
The volume of the cubic unit cell (A) is given by:
\[ \text{Volume of unit cell} = A^3 \]

### Step 4: Relate the atomic radius (r) to the edge length (A) of the unit cell.
In a BCC structure, the relationship between the atomic radius (r) and the edge length (A) is given by:
\[ 4r = \sqrt{3}A \quad \Rightarrow \quad r = \frac{\sqrt{3}}{4}A \]

### Step 5: Substitute the value of r into the volume occupied by the atoms.
Now, substituting \( r \) into the total volume occupied by the atoms:
\[ \text{Total volume} = \frac{8}{3} \pi \left(\frac{\sqrt{3}}{4}A\right)^3 \]
Calculating this gives:
\[ = \frac{8}{3} \pi \left(\frac{3\sqrt{3}}{64}A^3\right) = \frac{8 \cdot 3\sqrt{3}}{192} \pi A^3 = \frac{\sqrt{3}}{8} \pi A^3 \]

### Step 6: Calculate the packing fraction.
The packing fraction (PF) is defined as the ratio of the volume occupied by the atoms to the volume of the unit cell:
\[ \text{Packing fraction} = \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} = \frac{\frac{8}{3} \pi r^3}{A^3} \]
Substituting \( r \) gives:
\[ \text{Packing fraction} = \frac{\frac{8}{3} \pi \left(\frac{\sqrt{3}}{4}A\right)^3}{A^3} = \frac{\frac{8}{3} \pi \cdot \frac{3\sqrt{3}}{64}A^3}{A^3} = \frac{8 \cdot 3\sqrt{3}}{192} \pi = \frac{\sqrt{3}}{8} \pi \]
Calculating this gives approximately:
\[ \text{Packing fraction} \approx 0.68 \]

### Step 7: Convert packing fraction to packing efficiency percentage.
To find the packing efficiency percentage:
\[ \text{Packing efficiency} = \text{Packing fraction} \times 100 = 0.68 \times 100 = 68\% \]

### Final Answer:
The packing fraction for the K unit cell is approximately 0.68, which corresponds to a packing efficiency of 68%.

---