Question

The molecular formula of a compound is \( AB_2O_4 \). Atoms of O form a ccp lattice. Atoms of A (cation) occupy \( \frac{1}{8} \) of tetrahedral voids. Atoms of B (cation) occupy a fraction of octahedral voids. What is the fraction of vacant octahedral voids?

• A. \( \frac{3}{4} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{2} \)

Hint:

In CCP lattices, tetrahedral voids are twice the number of atoms, and octahedral voids are equal to the number of atoms.

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the crystal structure of the compound \( AB_2O_4 \) and determine the fraction of vacant octahedral voids. Step 1: Understand the Structure
The oxygen atoms (\( O \)) form a ccp (cubic close-packed) lattice. In a ccp lattice, there are 4 oxygen atoms per unit cell. The formula of the compound is \( AB_2O_4 \), which means:
\( A \) cations: 1 per unit cell,
\( B \) cations: 2 per unit cell,
\( O \) anions: 4 per unit cell.
Step 2: Tetrahedral and Octahedral Voids
In a ccp lattice: The number of tetrahedral voids = \( 2 \times \text{number of atoms} = 2 \times 4 = 8 \).
The number of octahedral voids = \( \text{number of atoms} = 4 \). Step 3: Occupancy of Voids
Tetrahedral voids:
Atoms of \( A \) occupy \( \frac{1}{8} \) of the tetrahedral voids.
Total tetrahedral voids = 8.
Occupied tetrahedral voids = \( \frac{1}{8} \times 8 = 1 \).
Vacant tetrahedral voids = \( 8 - 1 = 7 \). Octahedral voids:
Atoms of \( B \) occupy a fraction of the octahedral voids. Total octahedral voids = 4. From the formula \( AB_2O_4 \), there are 2 \( B \) cations per unit cell.
Occupied octahedral voids = 2.
Vacant octahedral voids = \( 4 - 2 = 2 \). Step 4: Fraction of Vacant Octahedral Voids
Total octahedral voids = 4.
Vacant octahedral voids = 2.
Fraction of vacant octahedral voids = \( \frac{2}{4} = \frac{1}{2} \).
Final Answer: The fraction of vacant octahedral voids is: \[ \boxed{\frac{1}{2}} \] This corresponds to option (4).