The fraction of total volume occupied by the atom present in a simple cubic is

Options

A. (a) 0.48
B. (b) 0.52
C. (c) 0.55
D. (d) 0.68

Answer

To find the fraction of total volume occupied by the atom present in a simple cubic unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Simple Cubic Structure**:
- In a simple cubic unit cell, atoms are located at each of the eight corners of the cube.

2. **Calculating the Number of Atoms per Unit Cell (Z)**:
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because each corner atom is shared among eight adjacent unit cells.
- Therefore, the total number of atoms (Z) in a simple cubic unit cell is:
\[ Z = 8 \text{ (corners)} \times \frac{1}{8} = 1 \]

3. **Volume of the Atom**:
- The atoms in the simple cubic structure can be approximated as spheres. The volume (V) of a single atom (sphere) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- Where \( r \) is the radius of the atom.

4. **Total Volume Occupied by Atoms**:
- Since there is only one atom in the unit cell, the total volume occupied by the atoms in the unit cell is:
\[ V_{\text{occupied}} = Z \times V = 1 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]

5. **Volume of the Cube**:
- The volume of the cube (V_cube) is given by:
\[ V_{\text{cube}} = a^3 \]
- Where \( a \) is the length of the side of the cube.

6. **Relating the Radius to the Side Length**:
- In a simple cubic structure, the relationship between the radius \( r \) of the atom and the side length \( a \) of the cube is:
\[ a = 2r \]

7. **Substituting for Volume of the Cube**:
- Substitute \( a \) in the volume of the cube:
\[ V_{\text{cube}} = (2r)^3 = 8r^3 \]

8. **Calculating the Packing Fraction**:
- The fraction of the total volume occupied by the atoms is given by:
\[ \text{Packing Fraction} = \frac{V_{\text{occupied}}}{V_{\text{cube}}} = \frac{\frac{4}{3} \pi r^3}{8r^3} \]
- Simplifying this expression:
\[ \text{Packing Fraction} = \frac{4 \pi}{3 \times 8} = \frac{\pi}{6} \]

9. **Numerical Value**:
- Using \( \pi \approx 3.14 \):
\[ \text{Packing Fraction} \approx \frac{3.14}{6} \approx 0.524 \]

### Final Answer:
The fraction of total volume occupied by the atom present in a simple cubic unit cell is approximately **0.524**.