Question

The compounds \(\mathrm{Na}_{2} \mathrm{O},\) CdS, and \(\mathrm{ZrI}_{4}\) all can be described as cubic closest packed anions with the cations in tetrahedral holes. What fraction of the tetrahedral holes is occupied for each case?

Short answer

For the given compounds, the fractions of tetrahedral holes occupied by cations are: \(Na_2O: 1\) (all tetrahedral holes are occupied), \(CdS: \frac{1}{2}\) (half of the tetrahedral holes are occupied), and \(ZrI_4: \frac{1}{8}\) (one-eighth of the tetrahedral holes are occupied).

Step-by-step solution

Recall Formula for Tetrahedral Holes

The formula for the tetrahedral hole's fraction is given by \[\frac{\text{Number of cations}}{\text{Number of tetrahedral holes}}.\] Since the cubic closest packed structure has two tetrahedral holes for each anion, we can rewrite the formula as: \[\frac{\text{Number of cations}}{2 \times \text{Number of anions}}.\]

Find Tetrahedral Hole Fraction for Na2O

In Na₂O, there are two sodium (Na) cations for each oxygen (O) anion. Applying the formula from Step 1, we get:\[\frac{2 \text{ Na}}{2 \times \text{1 O}} = \frac{2}{2} = 1.\] Hence, all tetrahedral holes are occupied by sodium cations in Na2O.

Find Tetrahedral Hole Fraction for CdS

In CdS, there is one cadmium (Cd) cation for each sulfur (S) anion. Applying the formula from Step 1, we get:\[\frac{1 \text{ Cd}}{2 \times \text{1 S}} = \frac{1}{2}.\] Hence, half of the tetrahedral holes are occupied by cadmium cations in CdS.

Find Tetrahedral Hole Fraction for ZrI4

In ZrI₄, there is one zirconium (Zr) cation for every four iodine (I) anions. Applying the formula from Step 1, we get: \[\frac{1 \text{ Zr}}{2 \times \text{4 I}} = \frac{1}{8}.\] Hence, one-eighth of the tetrahedral holes are occupied by zirconium cations in ZrI4. To summarize, the fractions of tetrahedral holes occupied by cations in each case are: - Na2O: 1 (all tetrahedral holes are occupied) - CdS: 1/2 (half of the tetrahedral holes are occupied) - ZrI4: 1/8 (one-eighth of the tetrahedral holes are occupied)

Key concepts

Cubic Closest Packing

Cubic closest packing (CCP), also known as face-centered cubic packing, is a way atoms are structured in a crystalline solid. It's one of the most efficient methods of packing spheres, where each sphere is surrounded by twelve other spheres. This arrangement maximizes the density of the packing, leaving minimal empty space. This pattern is repeated in layers, with each successive layer fitting into the gaps of the layer below.
In a CCP structure, the particles arrange in a sequence of

• A: Hexagonal shape
• B: Same shape but shifted
• C: Again the same shape shifted differently

This sequence is denoted as ABCABC... and each unit cell consists of four whole spheres from partial contributions. CCP structures are commonly found in metals, ionic compounds, and alloys where each layer not only touches the maximum number of neighboring particles but also creates a lattice framework that contributes to the material's overall strength and stability.

Fraction of Tetrahedral Holes

In crystal structures, especially those with cubic closest packing, tetrahedral holes are small voids surrounded by atoms, usually in a tetrahedral arrangement. These holes are crucial since they often accommodate smaller atoms or ions in a structure, affecting various physical and chemical properties of a material.
Calculating the fraction of tetrahedral holes occupied involves using the formula: \[\frac{\text{Number of cations}}{2 \times \text{Number of anions}} \]Since cubic closest packed structures offer two tetrahedral holes for every anion, the equation tells us how many of these holes are filled. For example:

• In Na\(_2\)O, all the available tetrahedral holes are occupied as the ratio is \( \frac{2}{2} = 1 \).
• In CdS, the fraction is \( \frac{1}{2} \), indicating that only half of the holes are filled.
• In ZrI\(_4\), the result is \( \frac{1}{8} \), so a smaller fraction of the holes are occupied.

The degree of occupancy of these tetrahedral holes influences the compactness and density of the structure as well as the material’s chemical properties.

Chemical Compounds Structure

The structure of chemical compounds determines their physical, chemical, and mechanical properties. In compounds like Na\(_2\)O, CdS, and ZrI\(_4\), we see a common pattern in their ionic structures. The anions form a base lattice (cubic closest packing), while the cations fill the tetrahedral holes within this framework.
This arrangement impacts:

• Density: More cations filling tetrahedral holes mean more mass within a given volume, increasing density.
• Reactivity: The position of ions affects how atoms interact with each other and with other substances.
• Stability: A fully or partially occupied structure is generally more stable than an unoccupied one, as it efficiently fills space and maximizes electrostatic interactions.

Understanding which types of ions fill which parts of the structure helps to predict compound behavior and can guide the development of new materials with specific desired properties. Recognizing these structures' underlying geometry is essential for insights into basic solid-state chemistry and material science applications.