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A spinel is an important class of oxid...

A spinel is an important class of oxide consisting two types of metal ions with the oxide ions arranged in ccp layers . The normal spinel has one -eight of the tetrahedral holes occupied by one type of metal ions and one- half of the octahedral holes occupied by another type of metal ion. Such a spinel is formed by `Mg^(2+), Al^(3+) and O^(2-)` . The neutrality of the crystal is being maintained.
The formula of the spinel is :

A

(a) `Mg_(2)AlO_(4)`

B

(b) `MgAl_(2)O_(4)`

C

(c) `Mg_(3)Al_(2)O_(6)`

D

(d) None of these

Text Solution

AI Generated Solution

The correct Answer is:
To derive the formula of the spinel formed by Mg²⁺, Al³⁺, and O²⁻, we need to follow these steps: ### Step 1: Determine the number of atoms and voids in the unit cell In a cubic close-packed (CCP) structure: - The number of atoms per unit cell (Z) = 4 - The number of tetrahedral voids = 2 × Z = 2 × 4 = 8 - The number of octahedral voids = Z = 4 ### Step 2: Identify the occupation of the voids According to the problem: - One-eighth of the tetrahedral voids are occupied by one type of metal ion (let's denote it as X). - One-half of the octahedral voids are occupied by another type of metal ion (let's denote it as Y). ### Step 3: Calculate the number of metal ions occupying the voids - For tetrahedral voids: - Total tetrahedral voids = 8 - Occupied by X = 1/8 × 8 = 1 atom of X - For octahedral voids: - Total octahedral voids = 4 - Occupied by Y = 1/2 × 4 = 2 atoms of Y ### Step 4: Count the number of oxide ions Since the structure is CCP, the number of oxide ions (O²⁻) present in the unit cell is equal to the number of atoms per unit cell, which is 4. ### Step 5: Write down the charges - Charge from X (let's assume X = Mg²⁺): - 1 atom of Mg²⁺ contributes +2 charge. - Charge from Y (let's assume Y = Al³⁺): - 2 atoms of Al³⁺ contribute 2 × 3 = +6 charge. - Charge from O²⁻: - 4 atoms of O²⁻ contribute 4 × (-2) = -8 charge. ### Step 6: Ensure charge neutrality The total positive charge from the metal ions must equal the total negative charge from the oxide ions: - Total positive charge = +2 (from Mg²⁺) + +6 (from Al³⁺) = +8 - Total negative charge = -8 (from O²⁻) Since the total positive charge (+8) equals the total negative charge (-8), the crystal is neutral. ### Step 7: Write the formula From the above calculations, we can conclude: - 1 atom of Mg²⁺ - 2 atoms of Al³⁺ - 4 atoms of O²⁻ Thus, the formula of the spinel is: \[ \text{MgAl}_2\text{O}_4 \] ### Summary The formula of the spinel formed by Mg²⁺, Al³⁺, and O²⁻ is **MgAl₂O₄**. ---
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