Volume occupied for a cubic close packed lattice of sphere is

Options

A. `60%`
B. `52.4%`
C. `68%`
D. `74%`

Explanation

To find the volume occupied for a cubic close packed (CCP) lattice of spheres, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure**:
- In a cubic close packed (CCP) lattice, also known as face-centered cubic (FCC), there are 4 atoms per unit cell.

2. **Identify the Formula for Packing Efficiency**:
- The packing efficiency (PE) can be calculated using the formula:
\[ \text{Packing Efficiency} = \frac{Z \times \frac{4}{3} \pi r^3}{a^3} \times 100 \]
where:
- \( Z \) = number of atoms per unit cell (for CCP, \( Z = 4 \))
- \( r \) = radius of the sphere
- \( a \) = edge length of the unit cell

3. **Determine the Edge Length**:
- The edge length \( a \) of the unit cell in a CCP structure is related to the radius \( r \) of the spheres:
\[ a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r \]

4. **Substitute Values into the Formula**:
- Substitute \( Z = 4 \) and \( a = 2\sqrt{2}r \) into the packing efficiency formula:
\[ \text{Packing Efficiency} = \frac{4 \times \frac{4}{3} \pi r^3}{(2\sqrt{2}r)^3} \times 100 \]

5. **Calculate the Volume of the Unit Cell**:
- Calculate \( (2\sqrt{2}r)^3 \):
\[ (2\sqrt{2}r)^3 = 8 \times 2\sqrt{2}^3 \times r^3 = 16\sqrt{2}r^3 \]

6. **Plug in the Values**:
- Now plug this back into the packing efficiency formula:
\[ \text{Packing Efficiency} = \frac{4 \times \frac{4}{3} \pi r^3}{16\sqrt{2}r^3} \times 100 \]

7. **Simplify the Expression**:
- The \( r^3 \) cancels out:
\[ \text{Packing Efficiency} = \frac{4 \times \frac{4}{3} \pi}{16\sqrt{2}} \times 100 \]

8. **Calculate the Numerical Value**:
- Simplifying further:
\[ \text{Packing Efficiency} = \frac{\frac{16\pi}{3}}{16\sqrt{2}} \times 100 = \frac{\pi}{3\sqrt{2}} \times 100 \]
- Approximating \( \pi \approx 3.14 \):
\[ \text{Packing Efficiency} \approx \frac{3.14}{3 \times 1.414} \times 100 \approx 74% \]

### Final Answer:
The volume occupied for a cubic close packed lattice of spheres is approximately **74%**.

---