If at cubic cell, atom A presents all corners and atom B at the centre of each face. What will be the molecular formula of the compounds, if all the atoms present on one body diagonal are replaced by atom C

Options

A. `ABC_(3)`
B. `A_(3)B_(12)C_(4)`
C. `A_(3)B_(12)C`
D. `AB_(12)C_(3)`

Explanation

To solve the problem, we need to determine the molecular formula of the compound based on the arrangement of atoms in a cubic cell and the replacement of certain atoms along a body diagonal.

### Step-by-step Solution:

1. **Identify the Arrangement of Atoms:**
- In the cubic cell, atom A is located at all the corners.
- Atom B is located at the center of each face.

2. **Count the Atoms in the Cubic Cell:**
- **Atom A (Corners):**
- There are 8 corners in a cube.
- Each corner atom contributes \( \frac{1}{8} \) to the unit cell.
- Total contribution from atom A:
\[ \text{Total A} = 8 \times \frac{1}{8} = 1 \text{ atom A} \]

- **Atom B (Face Centers):**
- There are 6 faces in a cube.
- Each face-centered atom contributes \( \frac{1}{2} \) to the unit cell.
- Total contribution from atom B:
\[ \text{Total B} = 6 \times \frac{1}{2} = 3 \text{ atoms B} \]

3. **Identify the Atoms on the Body Diagonal:**
- The body diagonal of a cube connects two opposite corners and passes through the center of the cube.
- The atoms on the body diagonal are:
- 2 corner atoms (A) at each end.
- 1 face-centered atom (B) in the middle.

4. **Replace Atoms on the Body Diagonal with Atom C:**
- We are replacing the 2 corner atoms (A) and the 1 face-centered atom (B) along the body diagonal with atom C.
- Therefore, the contributions change as follows:
- Atom A: Initially 1 atom, but now 1 - 2 (replaced) = -1 (not physically possible, but indicates that we have removed A from the body diagonal).
- Atom B: Initially 3 atoms, but now 3 - 1 (replaced) = 2.

5. **Final Count of Atoms:**
- After the replacements, we have:
- Atom A: 1 (original) - 2 (replaced on body diagonal) = 0
- Atom B: 3 (original) - 1 (replaced on body diagonal) = 2
- Atom C: 2 (replaced on body diagonal)

6. **Write the Molecular Formula:**
- The final count of atoms gives us:
- A = 0
- B = 2
- C = 2
- Therefore, the molecular formula of the compound is:
\[ \text{Molecular Formula} = B_2C_2 \]

### Final Answer:
The molecular formula of the compound is \( B_2C_2 \).