Question

In a spinel structure, oxides ions are cubical closest packed, whereas \(1 / 8\) of tetrahedral holes are occupied by cations \(\mathrm{A}^{2}\) and \(1 / 2\) of octahedral holes are occupied by cations \(\mathrm{B}^{3+}\) ions. The general formula of the compound having spinel structure is (a) \(\mathrm{AB}_{2} \mathrm{O}_{4}\) (b) \(\mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{O}_{4}\) (c) \(\mathrm{A}_{2} \mathrm{~B}_{6} \mathrm{O}\) (d) \(\mathrm{A}_{4} \mathrm{~B}_{3} \mathrm{O}\)

Short answer

The correct formula is \( \mathrm{AB}_{2} \mathrm{O}_{4} \).

Step-by-step solution

Understand the Spinel Structure

In a spinel structure, we have oxide ions in a close-packed arrangement, specifically in a cubic closest packing (CCP) formation. Within this structure, we must understand the distribution of cations into tetrahedral and octahedral holes.

Count the Tetrahedral Holes

In a close-packed structure of oxides, the number of tetrahedral holes is twice the number of oxide ions. If the oxide ions form a unit cell here, there will be 8 oxide ions and hence 16 tetrahedral holes. According to the problem, \( \frac{1}{8} \) of these tetrahedral holes are occupied by \( \mathrm{A}^{2+} \) cations. That means \( \frac{1}{8} \times 16 = 2 \) \( \mathrm{A}^{2+} \) ions occupy the tetrahedral holes.

Count the Octahedral Holes

The number of octahedral holes in the closest-packed lattice is equal to the number of oxide ions, which is 8 in our contrast here. Since \( \frac{1}{2} \) of the octahedral holes are occupied by \( \mathrm{B}^{3+} \) ions, this means \( \frac{1}{2} \times 8 = 4 \) \( \mathrm{B}^{3+} \) ions occupy the octahedral holes.

Derive the Chemical Formula

The chemical formula for the compound must summarize the constituents. From Step 2, we know there are 2 \( \mathrm{A}^{2+} \) ions occupying the tetrahedral sites, and from Step 3, there are 4 \( \mathrm{B}^{3+} \) ions in the octahedral sites. The oxide ions are 8 in number. Consolidating these, the formula becomes \( \mathrm{A}_{2} \mathrm{B}_{4} \mathrm{O}_{8} \). However, simplifying (dividing by 2), it becomes \( \mathrm{AB}_{2} \mathrm{O}_{4} \).

Key concepts

Tetrahedral Holes

In crystal structures, especially in the context of the cubic closest packing (CCP), tetrahedral holes play a significant role. These are small interstitial spaces that can accommodate smaller cations. Imagine a pyramid with a triangle base - this is how you can visualize a tetrahedral hole. Such holes appear in a CCP structure at each point where three spheres on one layer rest above (or below) a single sphere from the adjacent layer.Each fundamental unit cell in CCP involves multiple tetrahedral holes. Specifically, in any closest packing, the number of tetrahedral holes is twice the number of spheres involved. For instance, in the presence of 8 oxide ions within a unit cell, as in our spinel structure exercise, this results in 16 available tetrahedral holes.Within the spinel structure described, only 1/8 of these tetrahedral holes are filled with cation A, i.e., \( \frac{1}{8} \times 16 = 2 \) cations occupy these positions. Appreciating the location and the distribution of elements within tetrahedral holes helps us understand the compound's stoichiometry and properties.

Octahedral Holes

Octahedral holes are also vital in the packing of ions within a cubic closest packing structure. These holes are larger than the tetrahedral ones and can contain bigger ions. The term 'octahedral' comes from the shape formed by the spheres surrounding the hole - imagine an octahedron where the spheres are at each vertex.For every sphere in a CCP lattice, there is exactly one octahedral hole. In the context of our exercise, with 8 oxide ions forming the lattice, there are precisely 8 octahedral holes present within the unit cell.The spinel structure features occupation of these holes by cations B. According to the exercise, \( \frac{1}{2} \) of the octahedral holes are occupied, resulting in \( \frac{1}{2} \times 8 = 4 \) cations B within the unit cell.Understanding the arrangement of cations in octahedral holes is key to determining the compound's formula, its stability, and potential physical and chemical properties.

Cubic Closest Packing

Cubic closest packing (CCP), sometimes known as face-centered cubic (FCC) arrangement, is a highly efficient way to pack spheres. Each layer of spheres is arranged in a way that fits perfectly into the gaps of the previous layer, maximizing the space efficiency. CCP involves repeating layers of three different positions denoted as `ABCABC`. This sequence helps create the distinct lattice structure that characterizes many compounds, including those with spinel structures. In context, the CCP arrangement gives rise to both tetrahedral and octahedral holes, offering spaces for different ions to reside. In the case of the spinel structure, the oxide ions build a CCP lattice, resulting in precisely the spatial arrangement needed for the cations to fit into their specific tetrahedral and octahedral holes effectively. By understanding CCP, one gains insight into why certain compounds form with specific structural and chemical properties.