A compound formed by elements `A` and `B` crystallises in a cubic structure where `A` atoms are present at the alternate corners of a cube and the `B` atoms are present at the alternate face centres.The formula of the compound is

Options

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

Explanation

To solve the problem, we need to determine the formula of the compound formed by elements A and B based on their positions in a cubic crystal structure.

### Step-by-Step Solution:

1. **Understanding the Structure**:
- We have a cubic structure where A atoms are located at alternate corners and B atoms are at alternate face centers.

2. **Counting A Atoms**:
- In a cube, there are 8 corners. Since A atoms are present at alternate corners, we will have 4 A atoms (1 at every alternate corner).
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because each corner atom is shared by 8 adjacent cubes.
- Therefore, the total contribution of A atoms in the unit cell is:
\[ \text{Total A} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

3. **Counting B Atoms**:
- There are 6 faces on a cube, and B atoms are present at alternate face centers. Thus, we have 2 B atoms (1 at every alternate face center).
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because each face-centered atom is shared by 2 adjacent cubes.
- Therefore, the total contribution of B atoms in the unit cell is:
\[ \text{Total B} = 2 \times \frac{1}{2} = 1 \]

4. **Determining the Empirical Formula**:
- Now we have:
- Total A atoms = \( \frac{1}{2} \)
- Total B atoms = \( 1 \)
- To find the simplest whole number ratio, we can multiply both by 2:
\[ A : B = \frac{1}{2} \times 2 : 1 \times 2 = 1 : 2 \]
- Thus, the formula of the compound is \( AB_2 \).

5. **Final Answer**:
- The formula of the compound formed by elements A and B is \( AB_2 \).