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JEE Advanced 2024 Paper 2 · Chemistry · Solutions, Solid State and Surface Chemistry · medium

In a normal spinel MgAl2O4 the oxide ions form a cubic close-packed array, with Mg^2+ ions in tetrahedral voids and Al^3+ ions in octahedral voids.…

Numerical answer

In a normal spinel MgAl2O4\mathrm{MgAl_2O_4} the oxide ions form a cubic close-packed array, with Mg2+\mathrm{Mg^{2+}} ions in tetrahedral voids and Al3+\mathrm{Al^{3+}} ions in octahedral voids. Consider one unit cell of the O2−\mathrm{O^{2-}} lattice. The total number of voids (tetrahedral plus octahedral) in this unit cell that remain unoccupied is
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Answer: 9

A ccp unit cell contains 44 oxide ions, hence 44 octahedral voids (one at the body centre plus 12×1412\times\tfrac14 on edges) and 88 tetrahedral voids (two per oxide ion). The formula MgAl2O4\mathrm{MgAl_2O_4} has four oxide ions per formula unit, so this cell holds exactly one formula unit: 1 Mg2+1\ \mathrm{Mg^{2+}} and 2 Al3+2\ \mathrm{Al^{3+}}. The Mg2+\mathrm{Mg^{2+}} ion occupies one of the 88 tetrahedral voids (one eighth), leaving 8−1=78-1=7 empty. The two Al3+\mathrm{Al^{3+}} ions occupy two of the 44 octahedral voids (one half), leaving 4−2=24-2=2 empty. Total unoccupied voids =7+2=9=7+2=9.
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