Question

Calculate the atomic packing factor and density for a

face-centered cubic (FCC) structure with an atomic radius of 0.1
nm.

Answer 1

In an FCC structure, the atomic packing factor (APF) is determined by the ratio of the volume occupied by the atoms to the total volume of the unit cell. The APF for an FCC structure can be calculated using the formula APF = (number of atoms per unit cell) * (volume of each atom) / (total volume of the unit cell). The density of a substance can be calculated by dividing its mass by its volume.

Step-by-step explanation:

In an FCC structure, the atoms are packed as closely together as possible, with atoms occupying 74% of the volume. The atomic packing factor (APF) is defined as the volume of atoms in the unit cell divided by the total volume of the unit cell.

In an FCC structure, each atom touches 12 near neighbors, resulting in a coordination number of 12. The APF for an FCC structure is given by the formula APF = (number of atoms per unit cell) * (volume of each atom) / (total volume of the unit cell).

Using the given atomic radius of 0.1 nm, we can calculate the APF as follows:

1. Convert the atomic radius from nm to pm: 0.1 nm = 100 pm
2. Calculate the volume of each atom using the formula for the volume of a sphere: V = (4/3) * π * r³
3. For an FCC structure, there are 4 atoms per unit cell.
4. Calculate the total volume of the unit cell using the formula for the volume of a cube: V = a³, where 'a' is the edge length of the unit cell.
5. Calculate the APF using the formula APF = (4 atoms) * (volume of each atom) / (total volume of the unit cell).

Using these steps, we can calculate the APF for the given atomic radius.

To calculate the density, we need to know the molar mass of the element. The density of a substance is defined as its mass per unit volume. The molar mass allows us to convert the mass of the substance to moles, and then divide by the volume to get the density.

Using the molar mass of the element, we can calculate the density as follows:

1. Convert the atomic radius from nm to pm: 0.1 nm = 100 pm
2. Calculate the volume of each atom using the formula for the volume of a sphere: V = (4/3) * π * r³
3. Calculate the mass of each atom using the molar mass of the element.
4. For an FCC structure, there are 4 atoms per unit cell.
5. Calculate the total volume of the unit cell using the formula for the volume of a cube: V = a³, where 'a' is the edge length of the unit cell.
6. Calculate the mass of the unit cell by multiplying the mass of each atom by the number of atoms in the unit cell.
7. Calculate the density by dividing the mass of the unit cell by the total volume of the unit cell.

Using these steps, we can calculate the density for the given atomic radius.