In a solid `AB` having the `NaCl` structure, A atom occupies the corners of the cubic unit cell. If all the face-centred atoms along one of the axes are removed, then the resultant stoichiometry of the solid is

Options

A. `AB_(2)`
B. `A_(2)B`
C. `A_(4)B_(3)`
D. `A_(3)B_(4)`

Answer

To solve the problem, we need to analyze the structure of the solid AB with a NaCl structure and determine the resultant stoichiometry after removing certain atoms.

### Step-by-Step Solution:

1. **Understanding the NaCl Structure**:
- In the NaCl structure, the A atoms (which we will consider as Cl-) occupy the corners and face centers of the cubic unit cell.
- The B atoms (which we will consider as Na+) occupy the body center and the edge centers of the cubic unit cell.

2. **Counting A Atoms**:
- A atoms at the corners: There are 8 corners in a cube, and each corner atom contributes \( \frac{1}{8} \) to the unit cell.
\[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \]
- A atoms at the face centers: There are 6 face centers in a cube, and each face-centered atom contributes \( \frac{1}{2} \) to the unit cell.
\[ \text{Contribution from face centers} = 6 \times \frac{1}{2} = 3 \]
- Total A atoms before removal:
\[ \text{Total A} = 1 + 3 = 4 \]

3. **Removing Face-Centered Atoms**:
- According to the problem, we remove all the face-centered atoms along one of the axes. This means we will remove 2 face-centered atoms (one from each face along that axis).
- After removal, the contribution from face centers becomes:
\[ \text{Remaining A} = 4 - 2 = 2 \]

4. **Counting B Atoms**:
- B atoms at the body center: There is 1 body-centered atom contributing fully.
\[ \text{Contribution from body center} = 1 \]
- B atoms at the edge centers: There are 12 edges in a cube, and each edge-centered atom contributes \( \frac{1}{4} \) to the unit cell.
\[ \text{Contribution from edge centers} = 12 \times \frac{1}{4} = 3 \]
- Total B atoms:
\[ \text{Total B} = 1 + 3 = 4 \]

5. **Determining the Stoichiometry**:
- Now we have 2 A atoms and 4 B atoms.
- The stoichiometry can be expressed as:
\[ \text{Stoichiometry} = A_2B_4 \]
- This can be simplified to:
\[ A_1B_2 \quad \text{(dividing by 2)} \]

### Final Result:
The resultant stoichiometry of the solid after the removal of the face-centered atoms is \( A_1B_2 \).