In the normal spinel structure, the oxide ions are arranged in CCP pattern. The Zn2+ ions occupy one eighth of the tetrahedral holes and one half of the octahedral voids are occupied by Al3+. Give the formula of the spinel.

To derive the formula of the spinel structure based on the information provided, we need to analyze the arrangement of ions within the crystal lattice. In a normal spinel, the oxide ions (O²⁻) form a close-packed structure, specifically a cubic close-packed (CCP) arrangement. This arrangement is crucial because it dictates how other ions fit into the available spaces, or voids, within the structure.

Understanding the Ion Occupancy

In the normal spinel structure, we have the following key points:

• The oxide ions are arranged in a cubic close-packed (CCP) pattern.
• Zn²⁺ ions occupy one-eighth of the tetrahedral holes.
• Al³⁺ ions occupy half of the octahedral voids.

Calculating the Number of Ions

Let's break this down step by step:

1. Oxide Ions

In a unit cell of the cubic close-packed structure, there are 4 oxide ions. This is because each corner atom contributes 1/8 of an atom to the unit cell, and there are 8 corners, plus 6 face-centered atoms contributing 1/2 each. Thus, the total is:

• 8 corners × 1/8 = 1
• 6 faces × 1/2 = 3
• Total = 1 + 3 = 4 oxide ions (O²⁻)

2. Tetrahedral Holes

In the CCP structure, there are 8 tetrahedral holes. Since Zn²⁺ occupies one-eighth of these, the number of Zn²⁺ ions is:

• 8 tetrahedral holes × 1/8 = 1 Zn²⁺ ion

3. Octahedral Voids

There are 4 octahedral voids in the CCP structure. Since Al³⁺ occupies half of these, the number of Al³⁺ ions is:

• 4 octahedral voids × 1/2 = 2 Al³⁺ ions

Formulating the Spinel Structure

Now that we have the counts of each type of ion, we can write the formula for the spinel. The general formula for a normal spinel is given by:

AB₂O₄, where A represents the cation in tetrahedral sites and B represents the cation in octahedral sites.

In our case:

• A = Zn²⁺ (1 ion)
• B = Al³⁺ (2 ions)

Thus, the formula for this specific spinel structure becomes:

ZnAl₂O₄

Visualizing the Structure

To visualize this, think of the oxide ions forming a sturdy framework, with the smaller Zn²⁺ ions fitting snugly into the tetrahedral voids, while the larger Al³⁺ ions occupy the octahedral voids. This arrangement not only provides stability but also contributes to the unique properties of the spinel structure, such as its electrical and magnetic characteristics.

In summary, the formula for the spinel structure you described, where Zn²⁺ occupies tetrahedral holes and Al³⁺ occupies octahedral voids, is ZnAl₂O₄. This highlights the fascinating interplay between ionic sizes and crystal structures in materials science.