Question

The compounds \(\mathrm{Na}_{2} \mathrm{O}, \mathrm{CdS},\) and \(\mathrm{ZrL}_{4}\) 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

The fractions of tetrahedral holes occupied for the compounds are: \( \mathrm{Na}_{2} \mathrm{O}: 1\), \(\mathrm{CdS}: \frac{1}{2}\), and \(\mathrm{ZrL}_{4}: \frac{1}{8}\).

Step-by-step solution

Understand the cubic closest packed (ccp) structure

A cubic closest packed structure is a type of crystal structure where anions are packed as closely together as possible in a cubic arrangement. In this arrangement, there are two tetrahedral holes per anion.

Find the number of anions and tetrahedral holes for each compound

For each compound, we should calculate the number of anions and tetrahedral holes based on their formulas. We have: 1. For \(\mathrm{Na}_{2} \mathrm{O}\), there are 2 Na (cation) for each O (anion). With ccp structure, each O anion gives 2 tetrahedral holes. 2. For \(\mathrm{CdS}\), there is 1 Cd (cation) for each S (anion). Each S anion gives 2 tetrahedral holes. 3. For \(\mathrm{ZrL}_{4}\), there is 1 Zr (cation) for each 4 L (anions). Each L anion gives 2 tetrahedral holes.

Calculate the fraction of tetrahedral holes occupied by cations for each compound

Now that we have the number of cations and tetrahedral holes for each compound, we can calculate the fraction of tetrahedral holes occupied. 1. For \(\mathrm{Na}_{2} \mathrm{O}\): Number of cations (Na) = 2 Number of tetrahedral holes (O) = 1 * 2 = 2 Fraction of occupied tetrahedral holes = \(\frac{2}{2}\) = 1 2. For \(\mathrm{CdS}\): Number of cations (Cd) = 1 Number of tetrahedral holes (S) = 1 * 2 = 2 Fraction of occupied tetrahedral holes = \(\frac{1}{2}\) 3. For \(\mathrm{ZrL}_{4}\): Number of cations (Zr) = 1 Number of tetrahedral holes (L) = 4 * 2 = 8 Fraction of occupied tetrahedral holes = \(\frac{1}{8}\) So the fractions of tetrahedral holes occupied for the compounds are: 1. \(\mathrm{Na}_{2} \mathrm{O}\): 1 2. \(\mathrm{CdS}\): \(\frac{1}{2}\) 3. \(\mathrm{ZrL}_{4}\): \(\frac{1}{8}\)

Key concepts

Understanding Tetrahedral Holes

In a cubic closest packed (ccp) structure, an important feature is the presence of tetrahedral holes. These are small spaces within the crystal lattice, situated between the layers of anions. When anions are closely packed, they create coordination sites called tetrahedral holes where cations can fit.

Every anion in a ccp structure forms two tetrahedral holes. This is due to the geometry of the arrangement, where a tetrahedron is formed by four anions, providing spaces for the smaller ions, typically cations, to reside. This arrangement gives the structure greater stability and can affect the physical properties of the material. Understanding these holes is crucial when examining how various compounds form and stabilize, such as Na₂O, CdS, and ZrL₄.

Anion Packing in Cubic Closest Packed Structures

Anion packing in cubic closest packed (ccp) structures is a fundamental concept in crystallography. This arrangement allows anions to pack together as tightly as possible, minimizing gaps and resulting in a highly efficient packing. It's similar to stacking spheres layer by layer in a pattern that provides the greatest density.

In this arrangement, each layer of anions follows an ABC pattern, where no anion directly layers over another one from the previous layer. This creates a repeating three-layer sequence that maximizes space usage. Consequently, this structure also creates both tetrahedral and octahedral holes, which are pivotal for accommodating cations of different sizes in various compounds.

• Anions are larger ions that typically form the main framework of the crystal.
• The efficiency and stability of the ccp structure depend on how well these anions pack together.
• This framework is key for understanding how different compounds like Na₂O and CdS form.

Calculating the Fraction of Tetrahedral Holes Occupied

Calculating the fraction of tetrahedral holes occupied by cations is essential for understanding a compound's structure. Here's how it's done for different compounds:

1. **Na₂O:** Here, one oxygen anion forms two tetrahedral holes. With two sodium cations available, both holes are occupied. Thus, the fraction of occupied tetrahedral holes is \( \frac{2}{2} = 1 \).

2. **CdS:** A single sulfur anion also forms two tetrahedral holes. With only one cadmium cation, only one of these holes is filled. So, the fraction is \( \frac{1}{2} \).

3. **ZrL₄:** In this case, four L anions form eight tetrahedral holes. Only one zirconium cation is present, resulting in a fraction of \( \frac{1}{8} \).

• The fraction is calculated by dividing the number of cations by the number of tetrahedral holes.
• This helps describe how tightly packed these structures are on an atomic level.
• A higher fraction indicates a denser cation packing, while a lower fraction suggests more available space.

Understanding these calculations provides insights into the structural properties of these compounds.