In a face centered cubic arrangement of A and B atoms whose A atoms are at the corner of the unit cell and B atoms at the face centers. One of the B atoms missing from one of the face in unit cell. The simplest formula of compounding is:

Options

A. `A_(7)B_(3)`
B. `AB_(3)`
C. `A_(7)B_(24)`
D. `A_(8)B_(21)`

Explanation

To determine the simplest formula of the compound formed by A and B atoms in a face-centered cubic (FCC) arrangement, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Arrangement of Atoms**:
- In a face-centered cubic (FCC) unit cell, A atoms are located at the corners, and B atoms are located at the face centers.

2. **Count the Contribution of A Atoms**:
- There are 8 corners in the unit cell.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Therefore, the total contribution of A atoms is:
\[ \text{Total A atoms} = 8 \times \frac{1}{8} = 1 \]

3. **Count the Contribution of B Atoms**:
- Normally, there are 6 face centers in an FCC unit cell, and each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.
- However, one B atom is missing from one of the faces. Thus, we have 5 face-centered B atoms.
- Therefore, the total contribution of B atoms is:
\[ \text{Total B atoms} = 5 \times \frac{1}{2} = \frac{5}{2} \]

4. **Write the Empirical Formula**:
- From the contributions calculated, we have:
- A: 1
- B: \( \frac{5}{2} \)
- The empirical formula can be expressed as:
\[ A_1B_{\frac{5}{2}} \text{ or } A_2B_5 \text{ (by multiplying both subscripts by 2)} \]

5. **Conclusion**:
- The simplest formula of the compound is \( A_2B_5 \).

### Final Answer:
The simplest formula of the compound is \( A_2B_5 \).

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