Metals and Alloys described metals as packed layers of atoms. In a crystal that packing repeats perfectly, so you only need to describe one small repeating box, the unit cell. Its size and contents, together with molar mass, predict the metal's Density.
Key idea
Atoms on a corner are shared by 8 cells ($\tfrac{1}{8}$ each), on a face by 2 ($\tfrac{1}{2}$), on an edge by 4 ($\tfrac{1}{4}$); a body atom belongs to one cell. So: simple cubic, 1 atom per cell ($a = 2r$); body-centered cubic, 2 ($\sqrt{3}\,a = 4r$); face-centered cubic, 4 ($\sqrt{2}\,a = 4r$). Then $$\text{density} = \frac{(\text{atoms per cell}) \times M / N_\text{A}}{a^3}.$$
Why it works
A corner of a cube is where 8 cubes meet, so only an eighth of the corner atom lies inside any one of them; 8 corners $\times \tfrac{1}{8} = 1$ atom. A face is shared by two cubes. The edge length comes from where atoms touch. In simple cubic, atoms touch along an edge: $a = 2r$. In body-centered cubic they touch along the body diagonal, which has length $\sqrt{3}\,a$ and spans 4 radii. In face-centered cubic they touch along a face diagonal, $\sqrt{2}\,a = 4r$. One cell holds a known number of atoms, each of mass $M/N_\text{A}$, in a known volume $a^3$: that is density.
Worked example: Counting atoms
How many atoms are in a body-centered cubic unit cell?
- Corners: $8 \times \tfrac{1}{8} = 1$.
- Body center: 1, wholly inside.
- Total: $1 + 1 = 2$.
Answer: 2 atoms.
Worked example: Density of copper
Copper is face-centered cubic with an atomic radius of $128\ \text{pm}$ ($M = 63.55\ \text{g/mol}$). Find its density.
- Edge: $$a = \dfrac{4r}{\sqrt{2}} = \dfrac{4(128)}{1.414} = 362.0\ \text{pm} = 3.620 \times 10^{-8}\ \text{cm}$$
- Volume: $a^3 = (3.620 \times 10^{-8})^3 = 4.745 \times 10^{-23}\ \text{cm}^3$.
- Mass: $\dfrac{4 \times 63.55}{6.022 \times 10^{23}} = 4.221 \times 10^{-22}\ \text{g}$.
- Density: $\dfrac{4.221 \times 10^{-22}}{4.745 \times 10^{-23}} = 8.90\ \text{g/cm}^3$ (measured: 8.96).
Answer: $8.90\ \text{g/cm}^3$
Worked example: Density of iron
Iron is body-centered cubic with $r = 124\ \text{pm}$ ($M = 55.85\ \text{g/mol}$). Find its density.
- $$a = \dfrac{4r}{\sqrt{3}} = \dfrac{496}{1.732} = 286.4\ \text{pm} = 2.864 \times 10^{-8}\ \text{cm}$$; $a^3 = 2.348 \times 10^{-23}\ \text{cm}^3$.
- Mass: $\dfrac{2 \times 55.85}{6.022 \times 10^{23}} = 1.855 \times 10^{-22}\ \text{g}$.
- Density: $7.90\ \text{g/cm}^3$ (measured: 7.87).
Answer: $7.90\ \text{g/cm}^3$
Common mistake
Counting every atom drawn in the picture as a whole atom. A drawing of a face-centered cube shows 14 atoms, but the cell owns only 4 of them; the rest of each corner and face atom belongs to neighboring cells. Check: simple, body-centered and face-centered cubic cells hold 1, 2 and 4 atoms. If your density comes out several times too large, you probably counted whole atoms.
Ionic crystals work the same way. In the rock-salt (NaCl) cell, chloride ions sit at the face-centered positions (4 per cell) and sodium ions at the edge centers and body center ($12 \times \tfrac{1}{4} + 1 = 4$): four NaCl units per cell, matching the 1 : 1 formula.