In a unit cell, atoms `(A)` are present at all corner lattices, `(B)` are present at alternate faces and all edge centres. Atoms `(C)` are present at face centres left from `(B)` and one at each body diagonal at disntance of `1//4th` of body diagonal from corner.
Total fraction of voids occupied are

Options

A. `0.58`
B. `0.25`
C. `0.48`
D. `0.86`

Explanation

To solve the problem, we need to determine the total fraction of voids occupied in a unit cell with the given arrangement of atoms. Let's break down the solution step by step.

### Step 1: Identify the number of atoms of each type

1. **Atoms A**: Present at all corners of the unit cell.
- There are 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) to the unit cell.
- Total contribution from A:
\[ \text{Number of A atoms} = 8 \times \frac{1}{8} = 1 \]

2. **Atoms B**: Present at alternate faces and all edge centers.
- There are 6 faces in a cube, and if B is present at alternate faces, that means there are 3 faces occupied.
- Each face atom contributes \( \frac{1}{2} \) to the unit cell.
- Contribution from face atoms:
\[ \text{Contribution from faces} = 3 \times \frac{1}{2} = 1.5 \]
- There are 12 edges in a cube, and each edge center contributes \( \frac{1}{4} \).
- Contribution from edge centers:
\[ \text{Contribution from edges} = 12 \times \frac{1}{4} = 3 \]
- Total contribution from B:
\[ \text{Number of B atoms} = 1.5 + 3 = 4.5 \]

3. **Atoms C**: Present at face centers left from B and at each body diagonal at a distance of \( \frac{1}{4} \) of the body diagonal from the corners.
- Contribution from face centers:
- There are 3 face centers (the same as B) contributing \( \frac{1}{2} \) each.
\[ \text{Contribution from face centers} = 3 \times \frac{1}{2} = 1.5 \]
- Contribution from body diagonals:
- There are 4 body diagonals, and each contributes 1 atom.
\[ \text{Contribution from body diagonals} = 4 \times 1 = 4 \]
- Total contribution from C:
\[ \text{Number of C atoms} = 1.5 + 4 = 5.5 \]

### Step 2: Write the formula of the compound

From the contributions calculated:
- A: 1
- B: 4.5
- C: 5.5

Thus, the formula of the compound can be represented as:
\[ \text{A}_1\text{B}_{4.5}\text{C}_{5.5} \]

### Step 3: Determine the total number of voids

In a unit cell, the types of voids are:
- **Tetrahedral voids**: There are 8 tetrahedral voids in a cubic unit cell.
- **Octahedral voids**: There are 4 octahedral voids in a cubic unit cell.

Total voids:
\[ \text{Total voids} = 8 + 4 = 12 \]

### Step 4: Determine the number of voids occupied

From the formula derived, we have:
- The total number of C atoms is 5.5, which means that 5.5 voids are occupied by C atoms.

### Step 5: Calculate the fraction of voids occupied

The fraction of voids occupied can be calculated as:
\[ \text{Fraction of voids occupied} = \frac{\text{Number of occupied voids}}{\text{Total voids}} = \frac{5.5}{12} \]

Calculating this gives:
\[ \text{Fraction of voids occupied} = \frac{5.5}{12} \approx 0.4583 \]

### Final Answer

The total fraction of voids occupied is approximately \( 0.4583 \).