Packing in Solids — Revision Notes

⚡ 30-Second Revision

Packing Efficiency (PE):

* Simple Cubic (SC): $52.4\%$ * Body-Centered Cubic (BCC): $68\%$ * Face-Centered Cubic (FCC) / Hexagonal Close-Packed (HCP): $74\%$

Coordination Number (CN):

* SC: 6 * BCC: 8 * FCC/HCP: 12

Atoms per Unit Cell (Z):

* SC: 1 * BCC: 2 * FCC: 4

'a' vs 'r' Relationship:

* SC: $a = 2r$ * BCC: $a = \frac{4r}{\sqrt{3}}$ * FCC: $a = 2\sqrt{2}r$

Voids:

* For N atoms in close packing: 2N tetrahedral voids, N octahedral voids. * Ratio of tetrahedral to octahedral voids = 2:1.

Stacking:

* HCP: A-B-A-B... * CCP/FCC: A-B-C-A-B-C...

2-Minute Revision

Packing in solids is about how particles arrange to maximize space and stability. We model particles as spheres. In 1D, coordination number (CN) is 2. In 2D, square close packing (SCP) has CN 4, while hexagonal close packing (HCP) has CN 6.

From these, 3D structures emerge. Simple Cubic (SC) has CN 6, 1 atom/cell, $a=2r$, and 52.4% packing efficiency. Body-Centered Cubic (BCC) has CN 8, 2 atoms/cell, $a=4r/\sqrt{3}$, and 68% packing efficiency.

Face-Centered Cubic (FCC) and Hexagonal Close-Packed (HCP) are the most efficient, both having CN 12 and 74% packing efficiency. FCC has 4 atoms/cell and $a=2\sqrt{2}r$. HCP has an A-B-A-B stacking, while FCC (CCP) has A-B-C-A-B-C stacking.

Empty spaces are voids: tetrahedral (4 spheres, 2N per N atoms) and octahedral (6 spheres, N per N atoms). Remember these key numbers and relationships for quick problem-solving.

5-Minute Revision

Let's consolidate the crucial aspects of packing in solids for NEET. The core idea is that atoms, ions, or molecules arrange themselves in crystal lattices to achieve maximum density and stability. We simplify these particles as hard spheres.

1. Basic Packing Types & Parameters:

Simple Cubic (SC): — Atoms at corners only. $Z=1$. Atoms touch along edges, so $a=2r$. Coordination Number (CN) = 6. Packing Efficiency (PE) = 52.4%. Least efficient.
Body-Centered Cubic (BCC): — Atoms at corners and one at the body center. $Z=2$. Atoms touch along body diagonal, so $\sqrt{3}a = 4r \implies a = 4r/\sqrt{3}$. CN = 8. PE = 68%.
Face-Centered Cubic (FCC) / Cubic Close Packing (CCP): — Atoms at corners and face centers. $Z=4$. Atoms touch along face diagonal, so $\sqrt{2}a = 4r \implies a = 2\sqrt{2}r$. CN = 12. PE = 74%. Highly efficient. Stacking: A-B-C-A-B-C...
Hexagonal Close Packing (HCP): — Hexagonal unit cell. $Z=6$ (for hexagonal unit cell). CN = 12. PE = 74%. Highly efficient. Stacking: A-B-A-B...

2. Voids (Interstitial Sites): These are the empty spaces between packed spheres. Their number and type are critical.

Tetrahedral Voids: — Formed by 4 spheres. For N close-packed atoms, there are 2N tetrahedral voids. In an FCC unit cell (N=4), there are 8 tetrahedral voids, located at $\frac{1}{4}$ of the way along each body diagonal.
Octahedral Voids: — Formed by 6 spheres. For N close-packed atoms, there are N octahedral voids. In an FCC unit cell (N=4), there are 4 octahedral voids, located at the body center and edge centers (1 + $12 \times \frac{1}{4} = 4$).
Ratio: — In close-packed structures, the ratio of tetrahedral voids to octahedral voids is always 2:1.

3. Problem-Solving Strategy:

Numerical: — Identify the lattice type. Use the correct 'a' vs 'r' relationship. Apply the formula for Z or PE. Pay attention to units.
Conceptual: — Understand the definitions of CN, PE, and voids. Differentiate between HCP and FCC based on stacking. For compound formulas, determine N for the lattice-forming atom, then calculate the number of voids, and finally the number of atoms occupying those voids to find the simplest ratio.

Example: A compound has atoms X forming an FCC lattice and atoms Y occupying all octahedral voids. What is the formula?

X atoms (FCC) = 4 per unit cell.
Octahedral voids = N = 4.
Y atoms occupy all 4 octahedral voids.
Ratio X:Y = 4:4 = 1:1. Formula = XY.

Prelims Revision Notes

Packing in solids describes the arrangement of constituent particles (atoms, ions, molecules) in crystal lattices. Particles are often treated as hard spheres.

1. Types of Packing & Key Parameters:

Simple Cubic (SC):

* Atoms at corners only. * Number of atoms per unit cell (Z) = 1. * Relationship between edge length (a) and atomic radius (r): $a = 2r$. * Coordination Number (CN) = 6. * Packing Efficiency (PE) = $52.4\%$.

Body-Centered Cubic (BCC):

* Atoms at corners and one at the body center. * Z = 2. * Relationship: $\sqrt{3}a = 4r \implies a = \frac{4r}{\sqrt{3}}$. * CN = 8. * PE = $68\%$.

Face-Centered Cubic (FCC) / Cubic Close Packing (CCP):

* Atoms at corners and face centers. * Z = 4. * Relationship: $\sqrt{2}a = 4r \implies a = 2\sqrt{2}r$. * CN = 12. * PE = $74\%$. * Stacking sequence: A-B-C-A-B-C...

Hexagonal Close Packing (HCP):

* Hexagonal unit cell. * Z = 6 (for hexagonal unit cell). * CN = 12. * PE = $74\%$. * Stacking sequence: A-B-A-B...

2. Voids (Interstitial Sites):

Empty spaces between packed spheres.
Tetrahedral Voids: — Formed by 4 spheres. Number = 2N (where N is the number of close-packed atoms).

* In FCC (N=4), 8 tetrahedral voids. Located at $\frac{1}{4}$ of the way along each body diagonal.

Octahedral Voids: — Formed by 6 spheres. Number = N (where N is the number of close-packed atoms).

* In FCC (N=4), 4 octahedral voids. Located at the body center (1) and edge centers ($12 \times \frac{1}{4} = 3$).

Ratio of Voids: — Tetrahedral : Octahedral = 2:1 in close-packed structures.

3. Important Points:

Higher packing efficiency means higher density for a given atomic mass.
Coordination number indicates the number of nearest neighbors.
Ionic solids often have cations occupying voids formed by anions.

Mains Revision Notes

For NEET, 'mains' level revision for 'Packing in Solids' implies a deeper understanding beyond basic recall, focusing on application and inter-topic connections. This involves mastering the derivations and implications of packing parameters.

1. Derivations and Geometric Understanding:

Be able to derive the 'a' vs 'r' relationships for SC, BCC, and FCC. This reinforces the understanding of where atoms touch in each lattice. For example, in BCC, the body diagonal is $4r$, and its length is $\sqrt{3}a$, leading to $a = 4r/\sqrt{3}$.
Understand *how* packing efficiency is calculated for each lattice type, not just memorizing the final percentage. This involves calculating the volume of spheres ($Z \times \frac{4}{3}\pi r^3$) and the volume of the unit cell ($a^3$).

2. Void Locations and Stoichiometry:

Precisely locate tetrahedral and octahedral voids within an FCC unit cell. Visualize their positions (e.g., tetrahedral voids on body diagonals, octahedral voids at body center and edge centers).
Practice complex problems involving compound formulas where atoms occupy specific *fractions* of voids. For instance, if atoms A form FCC, and atoms B occupy 1/3 of tetrahedral voids and atoms C occupy 1/2 of octahedral voids, determine the compound formula. This requires careful calculation of the number of each type of atom.

3. Inter-topic Connections:

Density: — Connect packing parameters to density calculations: $\rho = \frac{Z \times M}{a^3 \times N_A}$. Understand how packing efficiency (which influences 'a' for a given 'r') directly affects density.
Ionic Solids: — Relate the concept of packing to ionic crystal structures (e.g., NaCl, CsCl, ZnS). Understand how larger ions (anions) often form the lattice, and smaller ions (cations) occupy specific voids, determined by the radius ratio rule.

4. Conceptual Clarity:

Differentiate between HCP and CCP not just by stacking, but by their resulting symmetry and unit cell characteristics. Both are close-packed but distinct.
Understand why close packing leads to stability (maximization of attractive forces, minimization of potential energy).

Vyyuha Quick Recall

Simple Boys Find Heavy Packing Easy:

Simple Cubic: Six (CN), Single (Z=1), Slow (52.4% PE)
Body-Centered Cubic: Big Eight (CN=8), Basic Two (Z=2), Better (68% PE)
Face-Centered Cubic / Hexagonal Close-Packed: Full Twelve (CN=12), Four (Z=4 for FCC), Highest (74% PE)

Voids: Two Tetrahedral for Every One Octahedral (2:1 ratio of Tetrahedral:Octahedral voids for N atoms).