Question

The structure of the compound \(\mathrm{K}_{2} \mathrm{O}\) is best described as a cubic closest packed array of oxide ions with the potassium ions in tetrahedral holes. What percent of the tetrahedral holes are occupied in this solid?

Short answer

In the solid K2O, the compound has a cubic closest packed array (face-centered cubic) of oxide ions, with 4 oxide ions in the unit cell. There are 8 tetrahedral holes available in the unit cell, and 8 potassium ions are present (as given by the K2O formula). Therefore, 100% of the tetrahedral holes are occupied by potassium ions in this solid.

Step-by-step solution

Determine the number of oxide ions in the unit cell of K2O

In a face-centered cubic (fcc) structure, the oxide ions occupy the corners and the center of each face of the unit cell. There are 8 corner ions and 6 face-centered ions. We need to determine the number of oxide ions in the unit cell, considering that each corner ion is shared by 8 adjacent cells and each face-centered ion is shared by 2 adjacent cells. Number of oxide ions in unit cell = (Number of corner ions × 1/8) + (Number of face-centered ions × 1/2) Number of oxide ions in unit cell = (8 × 1/8) + (6 × 1/2) = 1 + 3 = 4

Calculate the number of tetrahedral holes in the unit cell

In a face-centered cubic arrangement, the number of tetrahedral holes is twice the number of anions (oxide ions, in this case). Number of tetrahedral holes = 2 × Number of oxide ions in the unit cell Number of tetrahedral holes = 2 × 4 = 8

Determine the number of potassium ions in the unit cell

The formula for the compound K2O indicates that there are 2 potassium ions for each oxide ion in the structure. So, we can determine the number of potassium ions in the unit cell of K2O by: Number of potassium ions = 2 × Number of oxide ions in the unit cell Number of potassium ions = 2 × 4 = 8

Calculate the percentage of occupied tetrahedral holes

Now we can calculate the percentage of tetrahedral holes occupied by potassium ions in the compound K2O: Percentage of occupied tetrahedral holes = (Number of potassium ions in the unit cell / Number of tetrahedral holes) × 100 Percentage of occupied tetrahedral holes = (8 / 8) × 100 = 100% Therefore, 100% of the tetrahedral holes are occupied by potassium ions in the solid K2O.

Key concepts

Cubic Closest Packing

The cubic closest packing (CCP), also known as face-centered cubic (FCC) structure, is a highly efficient way to arrange spheres (or atoms/ions in a crystal) in space, minimizing empty space.
In this arrangement, each sphere is surrounded by twelve others, forming layers that fit into each other tightly.
This structure is important in forming ionic compounds like \(\mathrm{K}_{2} \mathrm{O}\).

• In CCP, spheres are positioned at the corners of each cube face, creating layers that stack in an ABCABC... sequence.
• It results in a very compact and stable structural organization.

For \(\mathrm{K}_{2} \mathrm{O}\), the oxide ions adopt this cubic closest packed structure, acting as a scaffold for the potassium ions.

Tetrahedral Holes

Tetrahedral holes are voids or gaps formed between four atoms/ions in a crystal lattice.
These holes appear between the layers of atoms in a CCP or FCC arrangement, like the one formed by oxide ions in \(K_{2}O\).

• Each tetrahedral hole is surrounded by four spheres in such a way that if you were to connect their centers, they would form a tetrahedron.
• The number of tetrahedral holes in a face-centered cubic structure is always twice the number of spheres (or atoms/ions) forming the lattice.

In \(\mathrm{K}_{2} \mathrm{O}\), these holes are perfectly situated for potassium ions to occupy, leading to a fully packed crystal lattice.

Potassium Ions

Potassium ions, denoted as \(K^+\), are crucial cations in the structure of potassium oxide, \(\mathrm{K}_{2} \mathrm{O}\).
In the crystal formation, each potassium ion balances the charge of oxide ions by occupying tetrahedral holes.

• The role of potassium ions in a crystal lattice is to provide metallic cation presence that stabilizes the overall ionic structure.
• Potassium ions are larger compared to many cations, which suits their position in the expansive tetrahedral holes.

In \(\mathrm{K}_{2} \mathrm{O}\), all available tetrahedral holes are filled by potassium ions, creating a dense and balanced lattice.

Oxide Ions

Oxide ions, represented as \(O^{2-}\), are the negatively charged anions in the compound \(\mathrm{K}_{2} \mathrm{O}\).
In the CCP structure, they form the central framework of the crystal lattice.

• Oxide ions in a CCP structure occupy both the corners and the centers of the cube faces, forming a repeating, symmetrical pattern.
• This arrangement allows them to effectively interact with and stabilize the surrounding potassium ions.

In \(\mathrm{K}_{2} \mathrm{O}\), these ions create spaces known as tetrahedral holes, facilitating the ionic bonding and structural integrity of the compound.