Crystal Unit CellJEE Main

Packing, coordination & density — interactive Chemistry simulation for IIT-JEE.

Concept

In a crystal, atoms stack in a repeating unit cell. How efficiently they fill space — the packing fraction — and how many neighbours each atom touches (the coordination number) depend only on the lattice type. From the cell you can even compute the bulk density.

Key formula

\rho = \frac{Z\,M}{a^3 N_A}, \quad \text{SC: } a=2r,\ \text{BCC: } a=\tfrac{4r}{\sqrt3},\ \text{FCC: } a=2\sqrt2\,r

Derivation

Packing fraction $= \dfrac{Z\cdot\frac43\pi r^3}{a^3}$ . Substituting each lattice's $a$ – $r$ relation gives the fixed values: SC 52.4%, BCC 68.0%, FCC 74.0%.

FCC (and hexagonal close packing) achieve the densest possible packing of equal spheres at 74%. The density formula counts $Z$ atoms of mass $M/N_A$ inside the cube of volume $a^3$ .

Scenarios to explore

Crystal Unit Cell — Packing fraction, coordination & density for SC/BCC/FCC.

Real-world applications

Determining atomic radii from X-ray diffraction data.
Predicting metal densities and alloy behaviour.
Understanding why most metals adopt close-packed structures.

JEE exam tips

Corner atom = 1/8, face atom = 1/2, edge atom = 1/4, body-centre = 1 per cell.
FCC: Z=4, coordination 12, 74% packing — memorise the trio.
Octahedral voids = Z; tetrahedral voids = 2Z in close packing.

Common mistakes

Using the wrong $a$ – $r$ relation for the lattice type.
Forgetting unit conversions (pm → cm) in the density formula.
Counting corner atoms fully — each contributes only 1/8 to the cell.

Exam traps to avoid

Packing fraction is independent of the atomic radius — it is purely geometric.
BCC is not close-packed (68%) even though it looks dense.