Packing Efficiency — Revision Notes

⚡ 30-Second Revision

Packing Efficiency (PE): — $\frac{\text{Volume of atoms}}{\text{Volume of unit cell}} \times 100\%$
Volume of sphere: — $V = \frac{4}{3}\pi r^3$
Simple Cubic (SC):

- $Z=1$ - $a=2r$ - $PE = 52.4\%$ - Void space = $47.6\%$

Body-Centered Cubic (BCC):

- $Z=2$ - $a=\frac{4r}{\sqrt{3}}$ - $PE = 68\%$ - Void space = $32\%$

Face-Centered Cubic (FCC) / HCP:

- $Z=4$ (for FCC) - $a=2\sqrt{2}r$ (for FCC) - $PE = 74\%$ - Void space = $26\%$

Density: — $\rho = \frac{Z \times M}{N_A \times a^3}$

2-Minute Revision

Packing efficiency measures how much space in a crystal unit cell is occupied by atoms. It's calculated as the ratio of the volume of atoms to the total volume of the unit cell, expressed as a percentage.

For simple cubic (SC) structures, with one atom per unit cell (Z=1) and edge length $a=2r$, the packing efficiency is $52.4\%$. Body-centered cubic (BCC) structures have two atoms per unit cell (Z=2) and $a=\frac{4r}{\sqrt{3}}$, resulting in a packing efficiency of $68\%$.

Face-centered cubic (FCC) and hexagonal close-packed (HCP) structures are the most efficient, both having $74\%$ packing efficiency. For FCC, Z=4 and $a=2\sqrt{2}r$. The remaining percentage is void space.

These values are crucial for quick problem-solving in NEET, especially when combined with density calculations.

5-Minute Revision

Packing efficiency is a key concept in solid-state chemistry, quantifying the proportion of space filled by atoms within a unit cell. It's calculated by the formula: $PE = \frac{\text{Volume occupied by spheres}}{\text{Total volume of unit cell}} \times 100\%$.

The volume of each sphere is $\frac{4}{3}\pi r^3$, and the total volume occupied is $Z \times \frac{4}{3}\pi r^3$, where Z is the effective number of atoms per unit cell. The total volume of a cubic unit cell is $a^3$.

The critical step is correctly relating the atomic radius 'r' to the edge length 'a' for each structure.

Simple Cubic (SC): — $Z=1$, atoms touch along edges ($a=2r$). $PE = \frac{1 \times \frac{4}{3}\pi r^3}{(2r)^3} \times 100\% = 52.4\%$. Void space = $47.6\%$.
Body-Centered Cubic (BCC): — $Z=2$, atoms touch along body diagonal ($a=\frac{4r}{\sqrt{3}}$). $PE = \frac{2 \times \frac{4}{3}\pi r^3}{(\frac{4r}{\sqrt{3}})^3} \times 100\% = 68\%$. Void space = $32\%$.
Face-Centered Cubic (FCC): — $Z=4$, atoms touch along face diagonal ($a=2\sqrt{2}r$). $PE = \frac{4 \times \frac{4}{3}\pi r^3}{(2\sqrt{2}r)^3} \times 100\% = 74\%$. Void space = $26\%$. HCP also has $74\%$ PE.

Worked Example: Calculate the packing efficiency of a simple cubic unit cell.

1
Identify Z and a-r relation: — For SC, $Z=1$ and $a=2r$.
2
Volume of atoms: — $1 \times \frac{4}{3}\pi r^3$.
3
Volume of unit cell: — $a^3 = (2r)^3 = 8r^3$.
4
Calculate PE: — $PE = \frac{\frac{4}{3}\pi r^3}{8r^3} \times 100\% = \frac{\pi}{6} \times 100\% \approx 52.36\%$.

Remember these values and the 'a-r' relationships for quick problem-solving.

Prelims Revision Notes

For NEET, packing efficiency is a high-yield topic from the Solid State chapter. Focus on factual recall and formula application.

1. Definition: Packing efficiency is the percentage of total unit cell volume occupied by atoms. Void space is $100\% - \text{PE}$.

2. Key Formulas:

* Volume of a sphere (atom): $V_{\text{sphere}} = \frac{4}{3}\pi r^3$ * Volume of cubic unit cell: $V_{\text{cell}} = a^3$ * Packing Efficiency (PE): $PE = \frac{Z \times V_{\text{sphere}}}{V_{\text{cell}}} \times 100\%$ * Density: $\rho = \frac{Z \times M}{N_A \times a^3}$

3. Unit Cell Specifics (Memorize these!):

* Simple Cubic (SC): * Effective atoms (Z): 1 * Relationship $a$ and $r$: $a = 2r$ (atoms touch along edges) * Packing Efficiency (PE): $52.4\%$ * Void Space: $47.

4. Common Pitfalls:

* Confusing 'a-r' relationships for different unit cells. * Mixing up packing efficiency with void space. * Incorrectly calculating 'Z' for a given unit cell. * Arithmetic errors, especially with $\pi$, $\sqrt{2}$, and $\sqrt{3}$ values. Use $\pi \approx 3.14$, $\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$.

Practice solving numerical problems quickly and accurately. Conceptual questions will often involve comparing the packing efficiencies or void spaces of different structures.

Vyyuha Quick Recall

To remember the packing efficiencies for SC, BCC, FCC:

Simple Cube: 52 (like 'fifty-two') Body-Centered: 68 (like 'sixty-eight') Face-Centered: 74 (like 'seventy-four')

Think: 'SC is 52, BCC is 68, FCC is 74. It's like counting up in efficiency!'

For a-r relationships: Simple: $a = 2r$ (Simple, direct touch) Body: $a = \frac{4r}{\sqrt{3}}$ (Body diagonal, involves $\sqrt{3}$) Face: $a = 2\sqrt{2}r$ (Face diagonal, involves $\sqrt{2}$)