The percentage of empty space in a body centred cubic arrangement is :

Options

A. 74
B. 68
C. 32
D. 26

Explanation

To find the percentage of empty space in a body-centered cubic (BCC) arrangement, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Packing Efficiency**:
- Packing efficiency is defined as the volume occupied by the atoms (or spheres) in a unit cell divided by the total volume of the unit cell, multiplied by 100 to express it as a percentage.

2. **Determine the Relationship Between Radius (R) and Edge Length (A)**:
- For a body-centered cubic structure, the relationship between the radius of the atom (R) and the edge length of the cube (A) is given by:
\[ A = \frac{4R}{\sqrt{3}} \]

3. **Calculate the Volume of the Unit Cell**:
- The volume of the cube (V_cube) is given by:
\[ V_{\text{cube}} = A^3 \]
- Substituting the expression for A:
\[ V_{\text{cube}} = \left(\frac{4R}{\sqrt{3}}\right)^3 = \frac{64R^3}{3\sqrt{3}} \]

4. **Calculate the Volume of the Atoms in the Unit Cell**:
- In a BCC structure, there are 2 atoms per unit cell. The volume of one sphere (atom) is given by:
\[ V_{\text{sphere}} = \frac{4}{3}\pi R^3 \]
- Therefore, the total volume of the atoms in the unit cell is:
\[ V_{\text{atoms}} = 2 \times \frac{4}{3}\pi R^3 = \frac{8}{3}\pi R^3 \]

5. **Calculate Packing Efficiency**:
- Now, we can find the packing efficiency (PE):
\[ PE = \left(\frac{V_{\text{atoms}}}{V_{\text{cube}}}\right) \times 100 \]
- Substituting the volumes:
\[ PE = \left(\frac{\frac{8}{3}\pi R^3}{\frac{64R^3}{3\sqrt{3}}}\right) \times 100 \]
- Simplifying this expression:
\[ PE = \left(\frac{8\pi}{64/\sqrt{3}}\right) \times 100 = \left(\frac{8\pi \sqrt{3}}{64}\right) \times 100 = \left(\frac{\pi \sqrt{3}}{8}\right) \times 100 \]
- After calculating, we find that the packing efficiency for BCC is approximately 68%.

6. **Calculate the Percentage of Empty Space**:
- The percentage of empty space is given by:
\[ \text{Empty Space} = 100 - \text{Packing Efficiency} \]
- Therefore:
\[ \text{Empty Space} = 100 - 68 = 32\% \]

### Final Answer:
The percentage of empty space in a body-centered cubic arrangement is **32%**.

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