Calculation of Density of Unit Cell
Derivation, Unit Traps, and the Master Formula.
One of the most remarkable achievements of solid-state chemistry is that we can calculate the macroscopic density of a bulk crystal (something you can hold in your hand) by analyzing just a single, microscopic unit cell. Because the entire crystal is identical repeats of the unit cell, the density of the unit cell is equal to the density of the entire bulk solid.
1. Derivation of the Density Formula
By definition, Density ($d$ or $\rho$) is mass divided by volume.
A. Volume of Unit Cell
For a cubic unit cell, if the edge length is $a$, then:
B. Mass of Unit Cell
The mass of the unit cell depends on how many atoms belong exclusively to it, multiplied by the mass of a single atom.
- Let $Z$ be the number of atoms per unit cell (Rank of the unit cell).
- Let $m$ be the mass of one single atom.
We usually don't know $m$ directly. We know the Molar Mass ($M$), which is the mass of one mole ($N_A$) of atoms. Therefore, the mass of one single atom is:
C. The Final Formula
Substituting the mass and volume back into the density equation gives us the Master Formula:
2. The Grand Exam Trap: Unit Conversions
The formula itself is simple. Where students lose marks is in the units. Density is typically requested in $\text{g/cm}^3$. Molar mass ($M$) is usually in $\text{g/mol}$. However, the edge length ($a$) is almost always given in picometers (pm) or Angstroms ($\text{\AA}$).
You MUST convert the edge length ($a$) to centimeters ($\text{cm}$) BEFORE cubing it!
- $1 \text{ pm} = 10^{-12} \text{ m} = \mathbf{10^{-10} \text{ cm}}$
- Therefore, $a \text{ in pm} \rightarrow (a \times 10^{-10}) \text{ cm}$
- Volume $V = (a \times 10^{-10})^3 \text{ cm}^3 = \mathbf{a^3 \times 10^{-30} \text{ cm}^3}$
- $1 \text{ \AA} = 10^{-10} \text{ m} = \mathbf{10^{-8} \text{ cm}}$
- Therefore, $a \text{ in \AA} \rightarrow (a \times 10^{-8}) \text{ cm}$
- Volume $V = (a \times 10^{-8})^3 \text{ cm}^3 = \mathbf{a^3 \times 10^{-24} \text{ cm}^3}$
3. Recall Table: Z and Edge Relations
Often, a question will not give you the edge length $a$ directly. It might give you the atomic radius $r$. You must use the geometry of the specific lattice to substitute $a$ in terms of $r$.
| Type of Unit Cell | Rank ($Z$) | Relation ($a$ and $r$) | Volume ($a^3$) in terms of $r$ |
|---|---|---|---|
| Simple Cubic (SC) | 1 | $a = 2r$ | $8r^3$ |
| Body-Centered (BCC) | 2 | $a = \frac{4r}{\sqrt{3}}$ | $\frac{64r^3}{3\sqrt{3}}$ |
| Face-Centered (FCC/CCP) | 4 | $a = 2\sqrt{2}r$ | $16\sqrt{2}r^3$ |
4. Effect of Defects on Density
If an ionic solid has defects, the experimental density will differ from the theoretical density calculated using $d = \frac{ZM}{N_Aa^3}$.
- Schottky Defect: Equal numbers of cations and anions are completely missing from the crystal. Mass decreases, volume stays the same. Therefore, Density Decreases.
- Frenkel Defect: An ion simply dislocates from its normal site to an interstitial site within the same crystal. No mass is lost, volume stays the same. Therefore, Density Remains Unchanged.
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