A crystalline solid has a stoichiometric formula A₂B. B forms HCP lattice and atoms of A are equally distributed in tetrahedral and octahedral voids .Calculate the percentage of tetrahedral and octahedral voids filled?

To determine the percentage of tetrahedral and octahedral voids filled in a crystalline solid with the stoichiometric formula A₂B, we first need to understand the arrangement of atoms in the hexagonal close-packed (HCP) structure of element B and how the atoms of A occupy the available voids.

Understanding the HCP Structure

In an HCP lattice, there are two types of voids: tetrahedral and octahedral. The arrangement of atoms creates a specific number of these voids based on the number of atoms present in the unit cell.

Calculating the Number of Atoms and Voids

• In the HCP structure, there are 6 atoms of B per unit cell.
• For every 1 atom of B, there are 2 tetrahedral voids and 1 octahedral void.

Thus, for 6 atoms of B, we can calculate the number of voids:

• Tetrahedral voids = 2 × 6 = 12
• Octahedral voids = 6

Distribution of A Atoms in Voids

According to the problem, the atoms of A are equally distributed in the tetrahedral and octahedral voids. Since we have 2 atoms of A (from the formula A₂B), we can distribute them as follows:

• Atoms of A in tetrahedral voids = 1
• Atoms of A in octahedral voids = 1

Calculating the Percentage of Voids Filled

Now, we can calculate the percentage of tetrahedral and octahedral voids that are filled by the atoms of A.

Tetrahedral Voids

The total number of tetrahedral voids is 12. Since 1 atom of A occupies one tetrahedral void, the percentage filled is calculated as follows:

Percentage of tetrahedral voids filled:

Percentage = (Number of A atoms in tetrahedral voids / Total tetrahedral voids) × 100

Percentage = (1 / 12) × 100 = 8.33%

Octahedral Voids

The total number of octahedral voids is 6. Since 1 atom of A occupies one octahedral void, the percentage filled is:

Percentage of octahedral voids filled:

Percentage = (Number of A atoms in octahedral voids / Total octahedral voids) × 100

Percentage = (1 / 6) × 100 = 16.67%

Summary of Results

In summary, for the crystalline solid with the stoichiometric formula A₂B:

• The percentage of tetrahedral voids filled is approximately 8.33%.
• The percentage of octahedral voids filled is approximately 16.67%.

This analysis illustrates how the arrangement of atoms in a crystal lattice can influence the occupancy of voids, which is crucial for understanding the properties of materials in solid-state chemistry.