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

Options

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

Answer

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

### Step 1: Identify the positions of A and B atoms
- A atoms are located at the corners of the unit cell.
- B atoms are located at the face centers of the unit cell.

### Step 2: Count the number of A atoms
- In a face-centered cubic (FCC) unit cell, there are 8 corners.
- Normally, there would be 8 A atoms (1 A atom at each corner).
- However, one A atom is missing from one corner, so we have:
\[ \text{Number of A atoms} = 8 - 1 = 7 \text{ A atoms} \]

### Step 3: Calculate the effective contribution of A atoms
- Each corner atom contributes \(\frac{1}{8}\) of an atom to the unit cell because it is shared among 8 unit cells.
- Therefore, the effective number of A atoms in the unit cell is:
\[ \text{Effective A atoms} = 7 \times \frac{1}{8} = \frac{7}{8} \text{ A atoms} \]

### Step 4: Count the number of B atoms
- There are 6 face centers in the unit cell.
- Each face-centered atom contributes \(\frac{1}{2}\) of an atom to the unit cell because it is shared between 2 unit cells.
- Therefore, the effective number of B atoms in the unit cell is:
\[ \text{Effective B atoms} = 6 \times \frac{1}{2} = 3 \text{ B atoms} \]

### Step 5: Write the formula based on the effective number of atoms
- We have:
\[ \text{A} = \frac{7}{8}, \quad \text{B} = 3 \]
- The formula can be written as:
\[ \text{A}_{\frac{7}{8}} \text{B}_{3} \]

### Step 6: Convert to simplest integer form
- To convert the formula to integer form, we multiply both the number of A and B atoms by 8 (the denominator of A):
\[ \text{A} = 7, \quad \text{B} = 3 \times 8 = 24 \]
- Thus, the simplest formula of the compound is:
\[ \text{A}_7 \text{B}_{24} \]

### Conclusion
The simplest formula of the compound is \(\text{A}_7 \text{B}_{24}\).

---