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Question

In the spinel structure, oxide ions form CCP whereas 18th of the tetrahedral voids are occupied by A2+ cations and 12 of the octahedral voids by B3+ cations. If the oxide ions are replaced by X83 ions, the number of atomic vacancies per unit cell is :

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Solution

Spinel structure is a CCP with 8 tetrahedral and 4 octahedral voids per unit cell.
Number of O2 ions (CCP) = 4
Nmber of A2+ ions (18 tetrahedral voids) = 1
Number of B3+ ions (12 octahedral voids) = 2
Hence, formula unit = AB2O4
As we can see, the overall charge on the unit cell is zero originally. Now when we replace O2 by X8/3, the charges change. The net charge on the cation is 8+. The net charge on the anion becomes 323. Since 8+ can balance 8- of the cation, the rest can be attributed to atomic vacancy.
Unbalanced charge is -8/3 which is nothing but 1 atomic vacancy.

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