A compound forms hexagonal close-packed structure. What is the total number of voids in 0.5 mol of it? How many of these are tetrahedral voids?

To determine the total number of voids in 0.5 mol of a compound that forms a hexagonal close-packed (HCP) structure, we first need to understand the arrangement of atoms in this type of crystal lattice and how voids are created within it. In an HCP structure, atoms are packed closely together, and there are two types of voids: tetrahedral and octahedral. Let's break this down step by step.

Understanding the HCP Structure

The hexagonal close-packed structure consists of layers of atoms arranged in a specific way. Each unit cell of HCP contains:

• 6 atoms (2 from the top face, 2 from the bottom face, and 2 in the middle layer).
• 12 tetrahedral voids.
• 6 octahedral voids.

Calculating the Number of Unit Cells

In 0.5 mol of the compound, we can calculate the number of unit cells present. The number of atoms in 0.5 mol can be found using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol.

So, the total number of atoms in 0.5 mol is:

Total atoms = 0.5 mol × 6.022 × 10²³ atoms/mol = 3.011 × 10²³ atoms.

Since each unit cell contains 6 atoms, the number of unit cells in 0.5 mol is:

Number of unit cells = Total atoms / Atoms per unit cell = 3.011 × 10²³ / 6 ≈ 5.018 × 10²² unit cells.

Finding the Total Number of Voids

Now, we can calculate the total number of voids in these unit cells. Each unit cell has:

• 12 tetrahedral voids
• 6 octahedral voids

Thus, the total number of voids per unit cell is:

Total voids per unit cell = 12 (tetrahedral) + 6 (octahedral) = 18 voids.

Now, multiplying the number of unit cells by the number of voids per unit cell gives us:

Total voids = Number of unit cells × Total voids per unit cell = 5.018 × 10²² × 18 ≈ 9.0324 × 10²³ voids.

Determining the Number of Tetrahedral Voids

To find out how many of these voids are tetrahedral, we can simply multiply the number of unit cells by the number of tetrahedral voids per unit cell:

Total tetrahedral voids = Number of unit cells × Tetrahedral voids per unit cell = 5.018 × 10²² × 12 ≈ 6.0216 × 10²³ tetrahedral voids.

Summary of Results

In summary, for 0.5 mol of a compound forming a hexagonal close-packed structure:

• Total number of voids: approximately 9.0324 × 10²³ voids.
• Number of tetrahedral voids: approximately 6.0216 × 10²³ voids.

This analysis highlights the intricate nature of atomic packing and the significance of voids in crystal structures, which play a crucial role in determining the properties of materials.