Number of Atoms in Unit Cell — Revision Notes

⚡ 30-Second Revision

Corner Atom: — $1/8$ contribution
Face-Centered Atom: — $1/2$ contribution
Edge-Centered Atom: — $1/4$ contribution
Body-Centered Atom: — $1$ contribution
Simple Cubic (SC): — $Z = 1$ (8 corners $imes$ $1/8$)
Body-Centered Cubic (BCC): — $Z = 2$ (8 corners $imes$ $1/8$ + 1 body $imes$ 1)
Face-Centered Cubic (FCC): — $Z = 4$ (8 corners $imes$ $1/8$ + 6 faces $imes$ $1/2$)
Density Formula: — $ho = \frac{Z \times M}{N_A \times a^3}$

2-Minute Revision

The effective number of atoms in a unit cell, denoted by 'Z', is a critical parameter in solid-state chemistry. It's calculated by summing the fractional contributions of atoms based on their location. Atoms at the 8 corners of a cubic unit cell contribute $1/8$ each. Atoms at the 6 face centers contribute $1/2$ each. Atoms at the 12 edge centers contribute $1/4$ each. An atom at the body center contributes $1$. For common cubic structures:

Simple Cubic (SC): — Only corner atoms. $Z = 8 \times (1/8) = 1$.
Body-Centered Cubic (BCC): — Corner atoms + one body-centered atom. $Z = (8 \times 1/8) + (1 \times 1) = 2$.
Face-Centered Cubic (FCC): — Corner atoms + six face-centered atoms. $Z = (8 \times 1/8) + (6 \times 1/2) = 4$.

These 'Z' values are essential for density calculations and determining the empirical formula of compounds.

5-Minute Revision

To quickly revise the number of atoms in a unit cell (Z), focus on the fractional contributions and their application to the three main cubic unit cell types. Remember that an atom's contribution depends on how many unit cells share it. A corner atom is shared by 8 cells ($1/8$), a face-centered atom by 2 cells ($1/2$), an edge-centered atom by 4 cells ($1/4$), and a body-centered atom belongs entirely to one cell ($1$).

1. Simple Cubic (SC) / Primitive:

Atoms only at 8 corners.
$Z = 8 \times (1/8) = 1$.
*Example:* If a metal crystallizes in SC, its unit cell effectively contains 1 atom.

2. Body-Centered Cubic (BCC):

Atoms at 8 corners and 1 at the body center.
$Z = (8 \times 1/8) + (1 \times 1) = 1 + 1 = 2$.
*Example:* Iron crystallizes in BCC. Each unit cell effectively has 2 iron atoms.

3. Face-Centered Cubic (FCC) / Cubic Close-Packed (CCP):

Atoms at 8 corners and 6 at face centers.
$Z = (8 \times 1/8) + (6 \times 1/2) = 1 + 3 = 4$.
*Example:* Copper crystallizes in FCC. Each unit cell effectively has 4 copper atoms.

Application in Density: The density formula $ho = \frac{Z \times M}{N_A \times a^3}$ is crucial. You must be able to rearrange it to find any unknown. For instance, if you're given $ho$, $M$, $a$, you can find $Z$ to identify the unit cell type. Always ensure units are consistent (e.g., $a$ in cm, $M$ in g/mol, $ho$ in g/cm$^3$).

Ionic Compounds: For compounds like AB$_x$, calculate the effective number of A ions and B ions separately based on their positions. For example, if A is at corners and B at body center, then A contributes $1$, B contributes $1$, so the formula is AB.

Prelims Revision Notes

The effective number of atoms in a unit cell, denoted by 'Z', is a fundamental concept for NEET. It quantifies the net number of atoms belonging to a single unit cell. This is calculated by summing the fractional contributions of atoms based on their location within the unit cell.

Fractional Contributions:

Corner atoms: — Each of the 8 corner atoms contributes $1/8$ to the unit cell.
Face-centered atoms: — Each of the 6 face-centered atoms contributes $1/2$ to the unit cell.
Edge-centered atoms: — Each of the 12 edge-centered atoms contributes $1/4$ to the unit cell.
Body-centered atoms: — Any atom entirely within the body of the unit cell contributes $1$.

'Z' Values for Common Cubic Unit Cells:

Simple Cubic (SC) / Primitive: — Atoms only at corners. $Z = 8 \times (1/8) = 1$.
Body-Centered Cubic (BCC): — Atoms at corners and one at the body center. $Z = (8 \times 1/8) + (1 \times 1) = 2$.
Face-Centered Cubic (FCC) / Cubic Close-Packed (CCP): — Atoms at corners and at the center of each face. $Z = (8 \times 1/8) + (6 \times 1/2) = 4$.

Key Formula for Density:

The density ($ho$) of a crystal is given by: $ho = \frac{Z \times M}{N_A \times a^3}$

* $Z$: effective number of atoms per unit cell * $M$: molar mass of the element (in g/mol) * $N_A$: Avogadro's number ($6.022 \times 10^{23}$ mol$^{-1}$) * $a$: edge length of the unit cell (in cm) * $a^3$: volume of the unit cell (in cm$^3$)

Application in Ionic Compounds: To find the empirical formula of an ionic compound, calculate the effective number of each type of ion (cation and anion) in the unit cell based on their positions and fractional contributions. Then, express their ratio in the simplest whole numbers.

Common Traps: Forgetting fractional contributions, confusing unit cell types, or making algebraic errors in density calculations. Always double-check the given units and convert them if necessary (e.g., pm to cm).

Vyyuha Quick Recall

To remember the contributions: Corners Eight, Faces Two, Edges Four, Body One. (C-8, F-2, E-4, B-1, referring to how many cells share it, so the fraction is 1/N).