Problem 19 - Chemistry

Question

Atoms are assumed to touch in closest packed structures, yet every closest packed unit cell contains a significant amount of empty space. Why?

Step-by-Step Solution

Answer

In closest packed structures, such as HCP and CCP arrangements, atoms are ordered in a regular pattern and closely packed in layers, with each atom surrounded by twelve nearest neighbors. Despite the assumption of atoms touching, empty space still exists due to geometrical constraints and limitations in the arrangement of atomic spheres in a unit cell. Even in the densest packing, atoms can only achieve a maximum packing efficiency of about 74%, leaving around 26% of the volume as empty space. These empty spaces, or voids, are an inherent feature of the crystal structures.

Step 1: Understand Closest Packed Structures

In closest packed structures, atoms are arranged in such a way that they occupy the maximum possible space, reducing the empty spaces between them. These structures are often seen in crystals, such as metals, where the atoms are ordered in a regular pattern. There are two main types of closest packed structures: Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP), also known as Face-Centered Cubic (FCC) structure. In these structures, atoms are closely packed in layers, and their centers form a specific geometrical arrangement.

Step 2: Examine Arrangement of Atoms and Empty Space

Even though atoms are closely packed in these structures, it is impossible to completely eliminate empty space. In HCP and CCP arrangements, each atom is surrounded by twelve nearest neighbors, and they form a regular polyhedron called a "truncated octahedron." If we examine the arrangement of atoms in a unit cell, we will find voids or empty spaces between them, which are not occupied by any atom. These voids can be of different sizes, such as tetrahedral voids and octahedral voids, depending on the position of neighboring atoms.

Step 3: Reasons for Empty Space in Closest Packed Structures

The reason for the presence of empty space in closest packed structures can be explained by the limitations of the geometrical arrangements of atoms. Atoms can be assumed as solid spheres, and when they are packed together closely, it is impossible to completely eliminate the voids between these spheres. In other words, it is simply a geometrical constraint for close-packed structures of atoms. To see why, consider the densest packing of spheres in a two-dimensional plane. This can be achieved by arranging them in an equilateral triangle lattice, where each sphere is in contact with six other spheres. However, even in this densest packing, only about 90.69% of the total area is occupied by spheres, resulting in about 9.31% empty space. The situation is similar in three-dimensional structures like HCP and CCP, where atoms can only achieve a maximum packing efficiency of about 74%, leaving around 26% of the volume as empty space.

Step 4: Conclusion

In summary, despite the assumption of atoms touching in closest packed structures, empty space still exists because of geometrical constraints and limitations in the arrangement of atomic spheres in a unit cell. These empty spaces or voids are an inherent feature of the crystal structures, with packing efficiency reaching a maximum of 74% in HCP and CCP arrangements.