Oxygen atom forms FCC unit cell with 'A ' atoms occupying all tetrahedral voids and 'B ' atoms occupying all octahedral voids. If atoms are removed from two of the body diagonals then determine the formula of resultant compound formed .

Options

A. `A_4B_4O_7`
B. `A_8B_6O_7`
C. `A_8B_8O_7`
D. `A_6B_8O_6`

Answer

To determine the formula of the resultant compound formed when atoms are removed from two body diagonals in a face-centered cubic (FCC) unit cell, we can follow these steps:

### Step 1: Determine the number of oxygen atoms in the FCC unit cell
In an FCC unit cell, the number of atoms can be calculated as follows:
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom to the unit cell.

So, the total number of oxygen atoms (O) in the FCC unit cell is:
\[ \text{Total O} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 2: Determine the number of tetrahedral and octahedral voids
- The number of tetrahedral voids in an FCC unit cell is given by \( 2n \), where \( n \) is the number of atoms in the unit cell. Here, \( n = 4 \), so:
\[ \text{Tetrahedral voids} = 2 \times 4 = 8 \]
- The number of octahedral voids in an FCC unit cell is equal to \( n \), so:
\[ \text{Octahedral voids} = 4 \]

### Step 3: Occupation of voids by A and B atoms
- All tetrahedral voids are occupied by A atoms, so there are 8 A atoms.
- All octahedral voids are occupied by B atoms, so there are 4 B atoms.

### Step 4: Removing atoms from the body diagonals
- Each body diagonal has 2 atoms in tetrahedral voids. Since we are removing atoms from 2 body diagonals, we remove a total of:
\[ 2 \times 2 = 4 \text{ A atoms} \]
- After removing these A atoms, the remaining A atoms are:
\[ 8 - 4 = 4 \text{ A atoms} \]
- Each body diagonal also passes through an octahedral void, so we remove 1 B atom from the octahedral void. The remaining B atoms are:
\[ 4 - 1 = 3 \text{ B atoms} \]

### Step 5: Calculate the remaining atoms
- The remaining atoms are:
- A: 4
- B: 3
- O: The total number of O atoms was 4, and we did not remove any O atoms, so it remains 4.

### Step 6: Write the formula of the resultant compound
Now, we have:
- A: 4
- B: 3
- O: 4

To express this in the simplest whole number ratio, we can write:
\[ \text{A}_4 \text{B}_3 \text{O}_4 \]

### Final Result
The formula of the resultant compound formed is:
\[ \text{A}_4 \text{B}_3 \text{O}_4 \]