The available space occupied by spheres of equal size in three dimensions in both hcp and ccp arrangement is

Options

A. 0.74
B. 0.7
C. 0.604
D. 0.524

Answer

To determine the available space occupied by spheres of equal size in three dimensions in both hexagonal close packing (hcp) and cubic close packing (ccp) arrangements, we will calculate the packing efficiency of these arrangements.

### Step-by-Step Solution:

1. **Understanding Packing Efficiency**:
Packing efficiency is defined as the ratio of the volume occupied by the spheres to the total volume available in the unit cell. It can be expressed mathematically as:
\[ \text{Packing Efficiency} = \frac{\text{Total Volume Occupied}}{\text{Total Volume Available}} \]

2. **Volume of a Sphere**:
The volume \( V \) of a single sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

3. **Volume of the Unit Cell**:
For a cubic unit cell with edge length \( a \), the volume \( V_{\text{cell}} \) is:
\[ V_{\text{cell}} = a^3 \]

4. **Effective Number of Atoms (Z)**:
In both hcp and ccp arrangements, the effective number of atoms per unit cell \( Z \) is 4.

5. **Relationship Between Radius and Edge Length**:
For a face-centered cubic (FCC) structure (which is the same as ccp), the relationship between the radius \( r \) of the spheres and the edge length \( a \) is:
\[ r = \frac{\sqrt{2}}{4} a \]

6. **Calculating Packing Efficiency**:
Substitute the values into the packing efficiency formula:
\[ \text{Packing Efficiency} = \frac{Z \cdot \frac{4}{3} \pi r^3}{a^3} \]
Substituting \( Z = 4 \) and \( r = \frac{\sqrt{2}}{4} a \):
\[ \text{Packing Efficiency} = \frac{4 \cdot \frac{4}{3} \pi \left(\frac{\sqrt{2}}{4} a\right)^3}{a^3} \]

7. **Simplifying the Expression**:
\[ = \frac{4 \cdot \frac{4}{3} \pi \cdot \frac{2\sqrt{2}}{64} a^3}{a^3} \]
\[ = \frac{4 \cdot \frac{4}{3} \pi \cdot \frac{\sqrt{2}}{32}}{1} \]
\[ = \frac{4 \cdot \pi \cdot \sqrt{2}}{24} \]
\[ = \frac{\pi \sqrt{2}}{6} \]

8. **Calculating Numerical Value**:
The numerical value of \( \frac{\pi \sqrt{2}}{6} \) is approximately 0.74.

### Conclusion:
The available space occupied by spheres of equal size in both hcp and ccp arrangements is approximately **0.74**.