In a unit cell, atoms A, B, C and D are present at corners, face - centres, body - centre and edge - centres respectively. If atoms touching one of the plane passing through two diagonally opposite edges are removed, then formula of compound is

Options

A. `ABCD_(2)`
B. `ABD_(2)`
C. `AB_(2)D_(2)`
D. `AB_(4)D_(5)`

Answer

To solve the problem step by step, we need to analyze the arrangement of atoms in the unit cell and how the removal of certain atoms affects the overall formula of the compound.

### Step 1: Identify the positions of atoms in the unit cell
- Atoms A are located at the corners of the cube.
- Atoms B are located at the face centers.
- Atom C is located at the body center.
- Atoms D are located at the edge centers.

### Step 2: Count the total number of each type of atom in the unit cell
- **Corners (A)**: There are 8 corners, and each corner atom contributes \( \frac{1}{8} \) to the unit cell. Thus, total contribution from A:
\[ \text{Contribution of A} = 8 \times \frac{1}{8} = 1 \]

- **Face Centers (B)**: There are 6 faces, and each face atom contributes \( \frac{1}{2} \). Thus, total contribution from B:
\[ \text{Contribution of B} = 6 \times \frac{1}{2} = 3 \]

- **Body Center (C)**: There is 1 body-centered atom, contributing fully:
\[ \text{Contribution of C} = 1 \]

- **Edge Centers (D)**: There are 12 edges, and each edge atom contributes \( \frac{1}{4} \). Thus, total contribution from D:
\[ \text{Contribution of D} = 12 \times \frac{1}{4} = 3 \]

### Step 3: Calculate the total contributions before removal
Before any atoms are removed, the contributions are:
- A = 1
- B = 3
- C = 1
- D = 3

### Step 4: Identify the atoms to be removed
The problem states that atoms touching a plane passing through two diagonally opposite edges are removed. This plane will involve:
- 4 corner atoms (A)
- 2 face-centered atoms (B)
- 1 body-centered atom (C)
- 2 edge-centered atoms (D)

### Step 5: Calculate the contributions after removal
After removing the specified atoms:
- Contribution of A after removal:
\[ 1 - 4 = -3 \quad (\text{not possible, so A = 0}) \]

- Contribution of B after removal:
\[ 3 - 2 = 1 \]

- Contribution of C after removal:
\[ 1 - 1 = 0 \]

- Contribution of D after removal:
\[ 3 - 2 = 1 \]

### Step 6: Final contributions
After removing the atoms, the contributions are:
- A = 0
- B = 1
- C = 0
- D = 1

### Step 7: Write the empirical formula
The empirical formula based on the contributions is:
\[ \text{Formula} = A^0 B^1 C^0 D^1 = B D \]

### Step 8: Adjust for whole numbers
Since we cannot have fractions in the empirical formula, we multiply by 2 to get whole numbers:
\[ \text{Final formula} = A^0 B^2 D^2 = B^2 D^2 \]

### Conclusion
The final formula of the compound after the removal of the specified atoms is:
\[ \text{Formula} = B^2 D^2 \]