Question

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

Short answer

The general formula of the compound having a spinel structure is (b) \text{AB}_2\text{O}_4.

Step-by-step solution

Understanding the Spinel Structure

Recognize that in a spinel structure, oxide ions (O^2-) form a cubic closest-packed arrangement. Within this structure, 1/8 of the total tetrahedral voids are filled by A^2+ cations, and 1/2 of the octahedral voids are filled by B^3+ cations.

Calculating the Number of Voids

In a cubic closed-packed structure, the number of tetrahedral voids is twice the number of oxide ions, and the number of octahedral voids is equal to the number of oxide ions.

Determining A^2+ Cation Count

If 1/8 of the tetrahedral voids are occupied by A^2+ cations, and considering there are 2 tetrahedral voids per oxide ion, then for every 8 oxide ions, there would be 2 * 8 * 1/8 = 2 A^2+ cations.

Determining B^3+ Cation Count

Since 1/2 of the octahedral voids are occupied by B^3+ cations and there is 1 octahedral void per oxide ion, for every 8 oxide ions, there would be 8 * 1/2 = 4 B^3+ cations.

Formulating the Compound

With the ratio of 2 A^2+ cations and 4 B^3+ cations per 8 oxygen ions, the formula unit can be reduced to 1 A^2+ cation and 2 B^3+ cations for every 4 oxygen ions, resulting in the chemical formula AB2O4.

Key concepts

Cubic Closest-Packed Arrangement

Understanding the cubic closest-packed (ccp) arrangement is essential when studying crystal structures like that of a spinel. In ccp structures, the placement of atoms or ions happens in a way that they occupy the maximum available space, creating a highly efficient and dense packing system. This means, each layer of ions is arranged in a staggered pattern over the previous one, so that each ion fits into the 'dimple' of the below layer—imagine stacking oranges in a fruit stand for a visual reference.

In a ccp arrangement, the unit cell—a small repeating unit that constitutes the crystal lattice—is cubic in shape. Each face-centered cubic cell in a ccp arrangement is composed of eight corner ions and six face-centered ions, all of which contribute to this tightly packed structure. Since each corner ion is shared by eight unit cells and each face-centered ion by two unit cells, effectively, there are four whole ions in a ccp's unit cell.

An important characteristic of the ccp structure is that it leaves a number of voids or empty spaces where smaller ions can reside. These spaces are the tetrahedral and octahedral voids, which are occupied by various cations in different types of crystals. The spinel structure utilizes these voids as 'slots' for its cations, which contributes to the stability and stoichiometry of the crystal.

Tetrahedral and Octahedral Voids

Within the cubic closest-packed (ccp) arrangement, there exist two types of voids that are crucial for building the spinel crystal structure—the tetrahedral and octahedral voids. Let's take a closer look at each.

Tetrahedral Voids

Tetrahedral voids are the smaller spaces formed when three ions in one layer and one ion in the neighboring layer come together, creating a tetrahedron-shaped space. There are twice as many tetrahedral voids as there are ions in the ccp arrangement. In the case of spinel, only fractions of these voids are occupied by ions, specifically, one-eighth of them are filled with A^2+ cations.

Octahedral Voids

On the other hand, octahedral voids are larger and are found between layers of ions, where six ions form an octahedron around the void. There are as many octahedral voids as there are ions in the lattice. In the spinel structure, half of these voids are filled with B^3+ cations. The specific occupation of these voids is critical, as it determines the formula, the properties, and the overall stability of the mineral.

Stoichiometry in Crystal Lattices

The concept of stoichiometry in crystal lattices pertains to the ratio of the different atoms or ions that make up the structure, and it obeys the fundamental principles of chemistry, like the law of constant proportions.

In stoichiometry, it's not merely about counting ions but understanding how they compromise to form a stable and electrically neutral compound. Regarding the spinel structure, the stoichiometry is determined by how the ions occupy the ccp arrangement's tetrahedral and octahedral voids. The exercise showed that A^2+ cations occupy 1/8 of the tetrahedral voids, and B^3+ cations occupy 1/2 of the octahedral voids. This relationship allows us to calculate the number of ions in a given volume of the crystal and leads to the chemical formula of the mineral. This adherence to stoichiometry is what keeps the crystal intact and determines its physical and chemical properties.

With the combination of these principles—ccp arrangements, void occupation, and stoichiometry—we can elucidate the general formula for a compound with the spinel structure, which, as proven by the step by step exercise, is AB2O4.