Question 12.2.2: Relate unit cell dimensions, density, and number of atoms pe......
Relate unit cell dimensions, density, and number of atoms per unit cell for a cubic unit cell.
The element copper is found to crystallize in a cubic lattice, with an edge length of 361.5 pm. If the density of solid copper is 8.935 g/cm^{3}, how many Cu atoms are there per unit cell?
You are asked to calculate the number of atoms in one unit cell of a specific element.
You are given the identity of the element, the type of unit cell, the edge length of the unit cell, and the density of the solid.
Step 1. Use the edge length to calculate the volume of the cell in units of cm³ to match the units used for density.
361.5 pm × \frac{10^{2}\text{ cm}}{10^{12}\text{ pm}} = 3.615 × 10^{-8} cm
volume = (3.615 × 10^{-8} cm)^{3} = 4.724 × 10^{-23} cm³
Step 2. Use the calculated volume and the density of the solid to calculate the mass of one unit cell.
(4.724 × 10^{-23} cm^{3})\left(\frac{8.935\text{ g}}{\text{cm}^{3}} \right) = 4.221 × 10^{-22} g
Step 3. Calculate the mass of a single copper atom from its molar mass and Avogadro’s number.
\frac{63.55\text{ g}}{1\text{ mol Cu}} \times \frac{1\text{ oml Cu}}{6.022 \times 10^{23}\text{ Cu atoms}} = 1.055 × 10^{-22} g/Cu atom
Step 4. Use the mass of one unit cell and the mass of a single copper atom to calculate the number of atoms per unit cell.
\left(\frac{4.221 \times 10^{-22}\text{g}}{1\text{ unit cell}} \right) \left(\frac{1 \text{Cu atom}}{1.055 \times 10^{-22} \text{ g}} \right) = 4 Cu atoms/unit cell