What is the packing efficiency of a face-centered cubic unit cell? Please show your work.

Verified step by step guidance

1. Understand that packing efficiency is the fraction of volume in a crystal structure that is occupied by the constituent particles (atoms, ions, or molecules).
2. Recognize that in a face-centered cubic (FCC) unit cell, there are 4 atoms per unit cell. This is calculated as: 8 corner atoms × 1/8 (each corner atom is shared by 8 unit cells) + 6 face atoms × 1/2 (each face atom is shared by 2 unit cells) = 4 atoms.
3. Recall that the volume of a sphere (atom) is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the atom.
4. Calculate the volume of the FCC unit cell. The edge length \( a \) of the FCC unit cell is related to the atomic radius \( r \) by the equation \( a = 2\sqrt{2}r \). Therefore, the volume of the unit cell is \( a^3 = (2\sqrt{2}r)^3 \).
5. Determine the packing efficiency by dividing the total volume of the atoms in the unit cell by the volume of the unit cell, and then multiply by 100 to express it as a percentage. The formula is: \( \text{Packing Efficiency} = \left( \frac{4 \times \frac{4}{3} \pi r^3}{(2\sqrt{2}r)^3} \right) \times 100 \).

Key Concepts

Face-Centered Cubic (FCC) Structure

The face-centered cubic (FCC) structure is a type of crystal lattice where atoms are located at each of the corners and the centers of all the cube faces. This arrangement allows for a high packing density, as each unit cell contains four atoms. Understanding the geometry of the FCC unit cell is essential for calculating packing efficiency.

Atomic Radius and Unit Cell Dimensions

In an FCC unit cell, the relationship between the atomic radius (r) and the edge length (a) is crucial for calculations. The face diagonal of the cube can be expressed in terms of the atomic radius as √2a = 4r. This relationship helps determine the dimensions of the unit cell, which are necessary for calculating the volume occupied by the atoms.

Packing Efficiency Calculation

Packing efficiency is defined as the fraction of volume in a crystal structure that is occupied by atoms. For an FCC unit cell, the packing efficiency can be calculated using the formula: (Volume of atoms in the unit cell) / (Volume of the unit cell). In FCC, this results in a packing efficiency of approximately 74%, indicating that a significant portion of the unit cell is filled with atoms.