Question

The molecular formula of a crystal is \( AB_2O_4 \). Oxygen atoms form a close-packed lattice. Atoms of A occupy \( x\% \) of tetrahedral voids and atoms of B occupy \( y\% \) of octahedral voids. \( x \) and \( y \) are respectively: \flushleft

• A. \( 12.5\%, 50\% \)
• B. \( 50\%, 12.5\% \)
• C. \( 33.3\%, 66.6\% \)
• D. \( 66.6\%, 33.3\% \)

Hint:

In a spinel structure (\( AB_2O_4 \)), A atoms typically occupy tetrahedral voids while B atoms occupy octahedral voids. The percentage occupation is determined based on the number of available voids.

Answer 1

The Correct Option is A

Step 1: Understanding the Crystal Structure 
The given compound is \( AB_2O_4 \), which follows the spinel structure. In such structures:
- Oxygen atoms form a face-centered cubic (FCC) close-packed lattice.
- The tetrahedral voids and octahedral voids are partially occupied by \( A \) and \( B \) atoms, respectively.
Step 2: Analyzing the Void Occupation 
1. Tetrahedral Voids:
- In an FCC lattice of oxygen, the number of tetrahedral voids is twice the number of oxygen atoms.
- A fraction of these voids is occupied by A atoms.
2. Octahedral Voids:
- The number of octahedral voids is equal to the number of oxygen atoms.
- A fraction of these voids is occupied by B atoms.
Step 3: Calculating \( x \) and \( y \) 
- Total oxygen atoms = 4 per formula unit.
- Total tetrahedral voids = \( 2 \times 4 = 8 \).
- Total octahedral voids = 4.
For A atoms:
- The formula tells us that 1 atom of \( A \) is present.
- \( A \) atoms occupy tetrahedral voids.
- The fraction occupied by \( A \) is \( \frac{1}{8} = 12.5\% \).
For B atoms:
- The formula tells us that 2 atoms of \( B \) are present.
- \( B \) atoms occupy octahedral voids.
- The fraction occupied by \( B \) is \( \frac{2}{4} = 50\% \).
Step 4: Evaluating the Given Options 
- Option (1): Correct, as \( x = 12.5\% \), \( y = 50\% \).
- Option (2): Incorrect, as it reverses the values.
- Option (3): Incorrect, as it does not match calculated values.
- Option (4): Incorrect, as the values are swapped incorrectly.
Thus, the correct answer is 

Option (1).