Problem 50 - Chemistry and Chemical Reactivity

Question

Consider the three types of cubic units cells. (a) Assuming that the spherical atoms or ions in a primitive cubic unit cell just touch along the cube's edges, calculate the percentage of occupied space within the unit cell. (Recall that the volume of a sphere is \((4 / 3) \pi r^{3},\) where \(r\) is the radius of the sphere.) (b) Compare the percentage of occupied space in the primitive cell (pc) with the bcc and fcc unit cells. Based on this, will a metal in these three forms have the same or different densities? If different, in which is it most dense? In which is it least dense?

Step-by-Step Solution

Answer

FCC has the highest density at ~74%, while the primitive cubic has the lowest at ~52%. Metals in different forms have different densities.

Step 1: Understanding the Primitive Cubic Unit Cell

In a primitive cubic (simple cubic) unit cell, the atoms touch along the edge of the cube. Each edge of the cube is equal to twice the radius of the atom since each atom sits at a corner and touches its neighbor.

Step 2: Calculate the Volume of the Primitive Cubic Unit Cell

The edge length of the cubic unit cell is equal to 2r (where r is the radius of the atom). The volume of the cube is then given by \( V_{cube} = (2r)^3 = 8r^3 \).

Step 3: Calculate the Volume of the Spheres in the Primitive Cubic Unit Cell

In the primitive cubic unit cell, there is one atom per unit cell. The volume of a single sphere is \( V_{sphere} = \frac{4}{3} \pi r^3 \).

Step 4: Compute the Occupied Space Percentage for Primitive Cubic

To find the percentage of space occupied by the atoms, divide the volume of the sphere by the volume of the cube and multiply by 100: \[ \text{Occupied Space} = \left( \frac{\frac{4}{3} \pi r^3}{8r^3} \right) \times 100 \approx 52.36\% \].

Step 5: Understand BCC and FCC Unit Cells

In a body-centered cubic (BCC) unit cell, there are two atoms per unit cell. In a face-centered cubic (FCC) unit cell, there are four atoms per unit cell, with different atom arrangements efficiently using space.

Step 6: Calculate the Occupied Space Percentage for BCC and FCC

The space occupancy percentages for BCC and FCC are derived from their configurations: - BCC has an occupancy of approximately \( 68\% \) - FCC has an occupancy of approximately \( 74\% \)

Step 7: Comparison and Conclusion on Densities

The primitive cubic cell has the lowest density because it has the least occupied space, while the FCC has the highest. Thus, metals in the FCC form are more dense than those in the BCC or simple cubic forms.

Key Concepts

Primitive Cubic Unit Cell

A primitive cubic unit cell is one of the simplest forms of cubic unit cells. In this type of structure, each corner of the cube has one atom, making a total of one atom per unit cell. Since each atom touches its neighbors along the edge of the cube, the edge length of the cube is twice the radius of an atom, denoted as \(2r\).

To find out how much space is occupied by the atom within the unit cell, we calculate the volume of the cube and the volume of the sphere. The volume of the cube is \(V_{cube} = (2r)^3 = 8r^3\), and since there is only one atom, the volume of the sphere is \(V_{sphere} = \frac{4}{3} \pi r^3\).

The fraction of space occupied by the atom is the volume of the sphere divided by the volume of the cube. Converting this ratio to a percentage, the calculation is as follows:

• Occupied Space Percentage = \(\left(\frac{\frac{4}{3} \pi r^3}{8r^3}\right) \times 100\)
• Which results in approximately \(52.36\%\)

This tells us that about half of the cubic space is actually filled by the atom itself.

Body-Centered Cubic Unit Cell

The body-centered cubic (BCC) unit cell is a bit more complex than the primitive cubic cell. In BCC, not only do we have atoms at each of the eight corners of the cube, but there is also an additional atom right in the center of the cube. This arrangement results in a total of two atoms per unit cell.

Thanks to this additional atom, the occupancy in BCC is higher than in a primitive cubic structure. The space is utilized more efficiently, leading to better packing of atoms. In the BCC structure, the occupancy percentage is approximately \(68\%\).

The higher number of atoms in the unit cell makes the BCC structure more dense compared to the primitive cubic cell. This means the metal occupying a BCC structure will generally have a greater density due to the higher percentage of occupied space. As a result, the BCC configuration offers a balance between simple structure and efficient use of space.

Face-Centered Cubic Unit Cell

Face-centered cubic (FCC) unit cells further enhance the efficiency of atomic packing. In an FCC structure, each of the cube's corners hosts one atom, similar to the other cubic cells. However, it also has one atom at the center of each face of the cube, leading to a total of four atoms per unit cell.

The presence of these additional atoms at the face centers allows FCC to make very efficient use of space, achieving the highest occupancy amongst cubic unit cells. The percentage of occupied space for the FCC structure is approximately \(74\%\).

Due to this high packing efficiency, metals in the face-centered cubic form are the most dense of the cubic structures. The FCC arrangement allows atoms to be packed closely together, which provides not only increased density but often also greater stability and strength. Thus, the FCC unit cell stands out as the most space-efficient structure among the common cubic configurations.