Compound `X_(2)Y` have antiflurate structure . What is /are correct statement ?

Options

A. The miimum distance between `X^(+)` is `(a)/(2)` , where 'a' =edge length of unit cell
B. The co-ordination number ratio of x and y 8:4
C. If `X^(+)` removed from alternate tetrahedral void then CN is 4 : 4
D. If `X^(+)` removed from alternate tetrahedral void then CN is 4 : 8

Answer

To solve the question regarding the compound \(X_2Y\) with an antifluorite structure, we will analyze the structure and the statements provided.

### Step-by-Step Solution:

1. **Understanding Antifluorite Structure**:
- In an antifluorite structure, the anions (in this case, \(Y^{2-}\)) occupy the face-centered cubic (FCC) lattice positions, while the cations (\(X^+\)) occupy all the tetrahedral voids.

2. **Counting Atoms in the FCC Lattice**:
- In a face-centered cubic (FCC) unit cell, there are:
- 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom, totaling \( 8 \times \frac{1}{8} = 1 \) atom.
- 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom, totaling \( 6 \times \frac{1}{2} = 3 \) atoms.
- Therefore, the total number of \(Y^{2-}\) ions in the unit cell is \(1 + 3 = 4\).

3. **Counting Tetrahedral Voids**:
- The number of tetrahedral voids in a FCC structure is twice the number of atoms in the lattice. Since there are 4 \(Y^{2-}\) ions, there will be \(2 \times 4 = 8\) tetrahedral voids.
- Thus, all 8 tetrahedral voids will be occupied by \(X^+\) ions.

4. **Analyzing the Statements**:
- **Statement A**: "Minimum distance between \(X^+\) and \(Y^{2-}\) is \(a/2\)".
- This statement is correct. The minimum distance between the cation and anion in an antifluorite structure is indeed \(a/2\), where \(a\) is the edge length of the cubic unit cell.

- **Statement B**: "Coordination number ratio of \(X\) and \(Y\) is \(8:4\)".
- This statement is incorrect. The coordination number of \(X^+\) is 4 (as it occupies tetrahedral voids), and the coordination number of \(Y^{2-}\) is 8 (as it is in the FCC lattice). Therefore, the correct ratio is \(4:8\) or \(1:2\).

- **Statement C**: "If \(X^+\) is removed from alternate tetrahedral voids, then coordination number is \(4:4\)".
- This statement is correct. If \(X^+\) ions are removed from alternate tetrahedral voids, half of the tetrahedral voids will be occupied, leading to a coordination number of \(4\) for both \(X^+\) and \(Y^{2-}\).

- **Statement D**: "If \(X^+\) is removed from alternate tetrahedral voids, the coordination number is \(4:8\)".
- This statement is incorrect. As previously mentioned, the coordination number would be \(4:4\) after removing \(X^+\) from alternate tetrahedral voids.

5. **Conclusion**:
- The correct statements regarding the antifluorite structure of \(X_2Y\) are A and C.

### Final Answer:
The correct statements are:
- A: Minimum distance between \(X^+\) and \(Y^{2-}\) is \(a/2\).
- C: If \(X^+\) is removed from alternate tetrahedral voids, then coordination number is \(4:4\).