To determine the oxidation states of ions A and B in the given oxide compound, we need to analyze the arrangement of the oxide ions and the metal ions within the crystal structure. The oxide ions (O²⁻) are arranged in a face-centered cubic (FCC) lattice, which is a common structure for ionic compounds. In this arrangement, the oxide ions occupy the lattice points, and we need to consider how many metal ions are present and how they occupy the available voids.
Understanding the Structure
In an FCC structure, there are 4 oxide ions per unit cell. The tetrahedral voids and octahedral voids are spaces within the lattice where metal ions can fit. In an FCC lattice, there are:
- 8 tetrahedral voids per unit cell
- 4 octahedral voids per unit cell
Metal Ion Occupation
According to the problem, metal ions of type A occupy one-fourth of the tetrahedral voids and one-third of the octahedral voids. Let’s calculate how many voids are occupied by metal A:
- Tetrahedral voids occupied by A: \( \frac{1}{4} \times 8 = 2 \) ions of A
- Octahedral voids occupied by A: \( \frac{1}{3} \times 4 = \frac{4}{3} \) ions of A (which is not possible in whole numbers, so we need to consider the total number of ions in the context of the overall charge balance)
Charge Balance Consideration
Now, let’s denote the oxidation state of metal A as \( x \). The total charge contributed by metal A can be expressed as:
Charge from A = \( 2x + \frac{4}{3}x \)
Since the total charge from the oxide ions (O²⁻) is \( 4 \times (-2) = -8 \), we can set up the equation:
Charge from A + Charge from O = 0
Thus, we have:
\( 2x + \frac{4}{3}x - 8 = 0 \)
Solving for x
To solve for \( x \), we first need a common denominator:
\( 2x + \frac{4}{3}x = \frac{6x + 4x}{3} = \frac{10x}{3} \)
Now substituting back into the equation:
\( \frac{10x}{3} - 8 = 0 \)
\( \frac{10x}{3} = 8 \)
Multiplying both sides by 3 gives:
\( 10x = 24 \)
Thus, \( x = \frac{24}{10} = 2.4 \)
Oxidation States of A and B
Since oxidation states must be whole numbers, we can round \( 2.4 \) to the nearest whole number, which suggests that the oxidation state of metal A is likely +2 or +3, depending on the specific context of the compound.
For metal B, which is not explicitly defined in the problem, we can infer its oxidation state based on the overall charge balance. If we assume that metal B is also a metal ion that balances the charge from the oxide ions, we can denote its oxidation state as \( y \). The total charge from metal B would need to balance the remaining charge after accounting for metal A and the oxide ions.
In summary, the possible oxidation states for metal A can be +2 or +3, while the oxidation state of metal B would depend on the specific stoichiometry of the compound and would need to be calculated based on the total charge balance.
