Question

A crystal is made up of metal ions 'M $_{1}$ ' and 'M $_{2}$ ' and oxide ions. Oxide ions. form a ccp lattice structure. The cation '$M_1$ occupies $50 \%$ of octahedral voids and the cation 'M $_{2}$ ' occupies $12.5 \%$ of tetrahedral voids of oxide lattice. The oxidation numbers of ' $M _{1}$ ' and ' $M _{2}$ ' are, respectively:

• A. $+2, +4$
• B. $+3, +1$
• C. $+1, +3$
• D. $+4, +2$

Answer 1

The Correct Option is A

To solve this question, we need to determine the oxidation numbers of metal ions \(M_1\) and \(M_2\) based on the given structural and positional information in a crystal lattice. 

1. We are given that oxide ions form a cubic close-packed (ccp) lattice. In a ccp structure, each unit cell contains 4 oxide ions.
2. The cation \(M_1\) occupies 50% of the octahedral voids. In a ccp structure, the number of octahedral voids is equal to the number of atoms forming the lattice, i.e., 4 octahedral voids per unit cell. Therefore, \(M_1\) occupies \(50\%\) of these, which equals \(0.5 \times 4 = 2\) octahedral voids.
3. The cation \(M_2\) occupies 12.5% of the tetrahedral voids. The number of tetrahedral voids in a ccp lattice is double the number of lattice points, i.e., 8 tetrahedral voids per unit cell. Thus, \(M_2\) occupies \(12.5\%\) of 8, which equals \(0.125 \times 8 = 1\) tetrahedral void.
4. Now, we establish the charge balance. Let the oxidation numbers of \(M_1\) and \(M_2\) be \(x\) and \(y\), respectively. The total charge contributed by oxide ions is \(4 \times (-2) = -8\).
5. The total charge from \(M_1\) ions is \(2x\) and from \(M_2\) ions is \(1y\).
6. The equation for charge balance is: \(2x + y - 8 = 0\)
7. We solve the equation by considering the positive charge balance:
   • Option 1: \(x = +2, y = +4\), leads to: \(2(2) + 4 - 8 = 0\), the equation is balanced.
   • Checking other options:
     • Option 2: \(x = +3, y = +1\), gives: \(2(3) + 1 - 8 = -1\) (Not balanced)\)
     • Option 3: \(x = +1, y = +3\), gives: \(2(1) + 3 - 8 = -3\) (Not balanced)\)
     • Option 4: \(x = +4, y = +2\), gives: \(2(4) + 2 - 8 = 2\) (Not balanced)\)

Thus, the correct oxidation numbers for \(M_1\) and \(M_2\) are \(+2\) and \(+4\), respectively. The correct option is \(+2, +4\).