Problem 15 - Chemistry An Atoms First Approach

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 cubic close-packed (ccp) or face-centered cubic (fcc) lattices, atoms are arranged to maximize packing efficiency. However, due to the geometry of the unit cell and the spherical nature of atoms, no planar arrangement can completely fill the space without leaving gaps. As a result, even in these efficient structures, there is still a significant amount of empty space. In fact, the packing efficiency of ccp structures is around 74%, leaving 26% of the unit cell's volume empty.

Step 1: Definitions and basic concepts

A unit cell is the smallest repeating unit of a crystal lattice that can be used to build the entire structure of the crystal. Closest packed structures (also known as close-packed structures) are those where the atoms in the unit cell are arranged in a manner that maximizes the packing efficiency, minimizing the empty space inside the cell. The two most common closest packed structures are hexagonal close-packed (hcp) and cubic close-packed (ccp) structures.

Step 2: Exploring the geometry of closest packed structures

Let's focus on the ccp structure, which can also be referred to as face-centered cubic (fcc) lattice. In this structure, each atom is surrounded by 12 other atoms, which are placed symmetrically around it. However, even with this efficient arrangement, there will still be empty space inside the unit cell. To see why, let's consider a single layer in the ccp lattice. This layer consists of equilateral triangles formed by the centers of the constituent atoms. Since the atoms are spheres, it is natural that there will be some gap in between them, even when they are touching. No planar arrangement of spheres can completely fill the space, as the curved surfaces will always leave a small gap between them.

Step 3: Calculating the packing efficiency

The packing efficiency of a unit cell can be calculated by dividing the volume occupied by atoms inside the cell by the volume of the unit cell. For ccp (or fcc) structures, the packing efficiency can be calculated as: Packing efficiency = \(\frac{Volume\:of\:atoms}{Volume\:of\:unit\:cell}\) In an fcc lattice, there are 4 atoms in the unit cell, each occupying \(\frac{4}{3}\pi r^3\) volume, where \(r\) is the radius of an atom. The lattice parameter \(a = 2\sqrt{2}r\), so the volume of the unit cell = \(a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3\) Packing efficiency = \(\frac{4(\frac{4}{3}\pi r^3)}{16\sqrt{2}r^3}\) Packing efficiency ≈ 0.74 This means that even in the closest packed structures, only about 74% of the unit cell's volume is occupied by the atoms, leaving 26% as empty space.

Step 4: Conclusion

In summary, even though atoms are assumed to touch in closest packed structures, there is still a significant amount of empty space within the unit cells. This is due to the geometry of the unit cell and the fact that no planar arrangement of spherical atoms can completely fill the space without leaving gaps. The packing efficiency of closest packed structures, such as ccp, reaches around 74%, which demonstrates that 26% of the unit cell's volume remains empty, even in the most densely packed atomic arrangements.