Question

A solid compound is formed by atoms of A (cations), B (cations), and O (anions). Atoms of O form an hcp lattice. Atoms of A occupy 25% of tetrahedral holes and atoms of B occupy 50% of octahedral holes. What is the molecular formula of the solid? 

• A. \( AB_2O_4 \)
• B. \( ABO_3 \)
• C. \( ABO_2 \)
• D. \( A_2BO_4 \)

Hint:

In an hcp lattice, - Tetrahedral voids are twice the number of anions. - Octahedral voids are equal to the number of anions. - To determine the formula, use the occupancy percentage of voids by cations.

Answer 1

The Correct Option is C

To determine the molecular formula of the solid compound, we analyze the given information about the arrangement of atoms A, B, and O. In the compound, the oxygen atoms form a hexagonal close-packed (hcp) lattice structure. In hcp lattices, the ratio of tetrahedral holes to atoms is 2:1, and the ratio of octahedral holes to atoms is 1:1.

Let's break down the composition:

• Oxygen (O) atoms: These form the base hcp lattice.
• Atoms of A: These occupy 25% of the tetrahedral holes. Given the hcp structure, the total number of tetrahedral holes is 2 per atom, resulting in \(0.5 \times O\) tetrahedral holes. Therefore, atoms A occupy \(0.25 \times 0.5 \times O = 0.125 \times O\).
• Atoms of B: These occupy 50% of the octahedral holes. Given the hcp structure, there’s 1 octahedral hole per atom, resulting in \(O\) octahedral holes. Therefore, atoms B occupy \(0.5 \times O\) octahedral holes.

Summarizing the atom occupancies in terms of the number of O atoms:

• Each atom of O contributes to 1.
• Atoms of A contribute \(0.125 \times O\).
• Atoms of B contribute \(0.5 \times O\).

To find the molecular formula, balance the atom counts relative to the number of O atoms:

• \(A: 0.125 \),
• \(B: 0.5 \),
• \(O: 1\).

The smallest whole number ratio between these is to multiply each by 8 to eliminate fractions:

• \(A: 0.125 \times 8 = 1\),
• \(B: 0.5 \times 8 = 4\),
• \(O: 1 \times 8 = 8\).

Thus, the molecular formula simplifies to \( ABO_2 \).

The correct option is: \( ABO_2 \).

Answer 2

In an hcp lattice, the number of atoms forming the lattice (oxygen in this case) is 6.

1. Oxygen (O): Forms the hcp lattice, so there are 6 O atoms.
2. A (cation): Occupies 25% of tetrahedral holes. An hcp lattice has 12 tetrahedral holes, so the number of A atoms is 12 * 0.25 = 3.
3. B (cation): Occupies 50% of octahedral holes. An hcp lattice has 6 octahedral holes, so the number of B atoms is 6 * 0.50 = 3.

Therefore, the ratio of atoms A : B : O is 3 : 3 : 6, which simplifies to 1 : 1 : 2.

The molecular formula of the solid is \(ABO_2\).