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 analyze the arrangement of these ions in the crystal lattice.

### Step 1: Identify the positions of ions A
- Ions A occupy all the corners of the cube and alternate edge centers.
- In a cube, there are 8 corners. Each corner contributes \( \frac{1}{8} \) of an ion to the unit cell.
- Therefore, the contribution from the corner ions is:
\[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \]

### Step 2: Calculate the contribution from edge centers
- There are 12 edges in a cube, and since only alternate edge centers are occupied, we consider only 4 edges.
- Each edge contributes \( \frac{1}{4} \) of an ion to the unit cell.
- Therefore, the contribution from the edge centers is:
\[ \text{Contribution from edges} = 4 \times \frac{1}{4} = 1 \]

### Step 3: Total contribution from ions A
- Adding the contributions from corners and edges:
\[ \text{Total contribution from A} = 1 + 1 = 2 \]

### Step 4: Identify the positions of ions B
- Ions B occupy all the face centers of the cube.
- There are 6 faces in a cube, and each face contributes \( \frac{1}{2} \) of an ion to the unit cell.
- Therefore, the contribution from the face centers is:
\[ \text{Contribution from faces} = 6 \times \frac{1}{2} = 3 \]

### Step 5: Write the formula of the compound
- From the contributions calculated:
- A contributes 2
- B contributes 3
- Thus, the formula of the compound can be represented as:
\[ \text{Formula} = A_2B_3 \]

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

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