An ionic compound is made up of A & B only. lons A occupy all the corners and alternate edge centers while atoms B occupy all the face centers. The formula of compound will be

Options

A. `AB_2`
B. `A_2 B_3`
C. `A_2 B`
D. `AB_3`

Answer

To determine the formula of the ionic compound made up of ions A and B, we need to calculate the number of each type of ion present in the unit cell.

### Step-by-Step Solution:

1. **Identify the positions of ion A**:
- Ion A occupies all the corners of the cubic unit cell and alternate edge centers.
- In a cubic unit cell, there are 8 corners.

2. **Calculate contribution from corner atoms**:
- Each corner atom is shared among 8 unit cells.
- Therefore, the contribution of A from the corners is:
\[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \text{ atom of A} \]

3. **Identify alternate edge centers**:
- A occupies alternate edge centers. In a cube, there are 12 edges.
- Since A occupies alternate edge centers, only 4 edge centers will have A.
- Each edge atom is shared between 2 unit cells.
- Therefore, the contribution from the edge centers is:
\[ \text{Contribution from edges} = 4 \times \frac{1}{2} = 2 \text{ atoms of A} \]

4. **Total number of A atoms**:
- Now, we add the contributions from the corners and the edges:
\[ \text{Total A atoms} = 1 + 2 = 3 \text{ atoms of A} \]

5. **Identify the positions of ion B**:
- Ion B occupies all the face centers of the cubic unit cell.
- There are 6 faces in a cube.

6. **Calculate contribution from face centers**:
- Each face center atom is shared between 2 unit cells.
- Therefore, the contribution from the face centers is:
\[ \text{Contribution from faces} = 6 \times \frac{1}{2} = 3 \text{ atoms of B} \]

7. **Total number of B atoms**:
- The total number of B atoms is:
\[ \text{Total B atoms} = 3 \text{ atoms of B} \]

8. **Determine the formula of the compound**:
- From the calculations, we have 3 A atoms and 3 B atoms in the unit cell.
- The empirical formula can be simplified to:
\[ \text{Formula} = A_2B_3 \]

### Final Answer:
The formula of the compound is \( A_2B_3 \).

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