In a face centred cubic arrangement of A and B atoms, A atoms are at corners of unit cell and B atoms at face centres. One "A" atom is missing from one corner in each unit cell. The simplest formula of the compound is 

Options

A. `A_(2)B_(3)`
B. `AB_(3)`
C. `A_(7)B_(4)`
D. `A_(7//8)B_(3)`

Answer

To find the simplest formula of the compound in a face-centered cubic (FCC) arrangement of A and B atoms, where A atoms are at the corners and B atoms are at the face centers, and one A atom is missing from one corner, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Contribution of A Atoms:**
- In a face-centered cubic unit cell, there are 8 corners.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Therefore, the total contribution from the A atoms at the corners is:
\[ \text{Total A atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]
- Since one A atom is missing from one corner, the effective number of A atoms in the unit cell becomes:
\[ \text{Effective A atoms} = 1 - \frac{1}{8} = \frac{7}{8} \text{ atoms} \]

2. **Identify the Contribution of B Atoms:**
- In the FCC structure, there are 6 face centers.
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell.
- Therefore, the total contribution from the B atoms at the face centers is:
\[ \text{Total B atoms} = 6 \times \frac{1}{2} = 3 \text{ atoms} \]

3. **Determine the Ratio of A to B:**
- Now we have the effective contributions:
- A: \( \frac{7}{8} \)
- B: \( 3 \)
- To find the simplest formula, we express the ratio of A to B:
\[ \text{Ratio of A to B} = \frac{\frac{7}{8}}{3} = \frac{7}{24} \]

4. **Write the Simplest Formula:**
- The simplest formula can be represented as:
\[ A_{\frac{7}{8}}B_3 \quad \text{or} \quad A_7B_{24} \]
- However, we typically express it in whole numbers. To do this, we can multiply both parts of the ratio by 24 (the denominator of B):
\[ A_7B_{24} \]

### Final Answer:
The simplest formula of the compound is \( A_7B_3 \).