The formula for calculating the density of a unit cell is:
ρ = (Z M) / (a3 NA)
Where:
- ρ = Density of the unit cell (usually in g/cm3)
- Z = Number of formula units per unit cell (e.g., 1 for simple cubic, 2 for body-centered cubic, 4 for face-centered cubic)
- M = Molar mass of the substance (in g/mol)
- a = Edge length of the unit cell (in cm). Note: if 'a' is given in pm or Å, it needs to be converted to cm. (1 pm = 10-10 cm; 1 Å = 10-8 cm)
- NA = Avogadro's number (6.022 x 1023 mol-1)
Explanation of Each Term:
- Density (ρ): This is the mass per unit volume. For a unit cell, it represents how much mass is packed into the volume of the unit cell.
- Number of Formula Units (Z): This indicates how many complete "molecules" or formula units are contained within a single unit cell. Different crystal structures (simple cubic, BCC, FCC) have different values of Z.
- Molar Mass (M): This is the mass of one mole of the substance.
- Unit Cell Edge Length (a): This is the length of one side of the cubic unit cell. Since the volume of a cube is a3, this term gives the volume of the unit cell.
- Avogadro's Number (NA): This is the number of entities (atoms, molecules, ions) in one mole of a substance. It's used to convert from molar mass (grams per mole) to mass per formula unit.
Example:
Let's say you want to calculate the density of a face-centered cubic (FCC) unit cell of copper (Cu).
- Z = 4 (for FCC)
- M = 63.55 g/mol (molar mass of Cu)
- a = 3.61 x 10-8 cm (assume this is the unit cell edge length, already converted to cm)
- NA = 6.022 x 1023 mol-1
ρ = (4 63.55) / ((3.61 x 10-8)3 6.022 x 1023) ≈ 8.96 g/cm3
This calculated density is close to the actual density of copper.
