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. `AB_(3)`
B. `A_(8)B_(5)`
C. `A_(2)B_(5)`
D. `AB_(2//5)`

Answer

To solve the problem, we need to determine the simplest formula of the compound formed by A and B atoms in a face-centered cubic (FCC) arrangement, where A atoms are located at the corners and B atoms are at the face centers, with one B atom missing from one of the faces.

### Step-by-Step Solution:

1. **Identify the Arrangement**:
- In a face-centered cubic (FCC) unit cell, there are 8 corner atoms and 6 face-centered atoms.

2. **Calculate the Contribution of A Atoms**:
- A atoms are located at the corners of the unit cell.
- There are 8 corners in a cube, and each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared by 8 neighboring unit cells.
- Therefore, the total contribution of A atoms in the unit cell is:
\[ \text{Total A atoms} = 8 \times \frac{1}{8} = 1 \]

3. **Calculate the Contribution of B Atoms**:
- B atoms are located at the face centers. Normally, there are 6 faces in a cube, and each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell because it is shared by 2 neighboring unit cells.
- However, since one B atom is missing from one of the faces, the number of B atoms present is:
\[ \text{Total face-centered atoms} = 6 - 1 = 5 \]
- Therefore, the contribution of B atoms in the unit cell is:
\[ \text{Total B atoms} = 5 \times \frac{1}{2} = \frac{5}{2} \]

4. **Determine the Ratio of A to B**:
- We have found that there is 1 A atom and \( \frac{5}{2} \) B atoms in the unit cell.
- The ratio of A to B is:
\[ \text{Ratio of A to B} = \frac{1}{\frac{5}{2}} = \frac{2}{5} \]

5. **Write the Empirical Formula**:
- The simplest formula can be expressed as \( A_2B_5 \) by multiplying both the subscripts by 2 to eliminate the fraction.

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

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