Question

What is the number of unit cells when one mole atom of a metal that forms simple cubic structure?

• A. $6.022\times10^{23}$
• B. $1.204\times10^{24}$
• C. $9.033\times10^{23}$
• D. $3.011\times10^{23}$

Hint:

Logic Tip: Memorize the $Z$ values for standard lattices: Simple Cubic ($Z=1$), Body-Centered Cubic ($Z=2$), and Face-Centered Cubic ($Z=4$). If the question asked for an FCC lattice, the answer would be $(6.022 \times 10^{23}) / 4$.

Answer 1

The Correct Option is A

Concept:
The number of unit cells in a given macroscopic sample depends on the total number of atoms in the sample and the number of effective atoms contained within a single unit cell of that specific crystal lattice. $$\text{Number of unit cells} = \frac{\text{Total number of atoms{\text{Number of atoms per unit cell } (Z)}$$
Step 1: Determine the total number of atoms in the sample.
The problem states we have exactly one mole of metal atoms. By definition, one mole of any substance contains Avogadro's number ($N_A$) of particles. $$\text{Total atoms} = 1 \text{ mole} = 6.022 \times 10^{23} \text{ atoms}$$
Step 2: Determine the effective number of atoms in a simple cubic unit cell (Z).
In a simple cubic (SC) crystal lattice, atoms are located only at the 8 corners of the cube. Each corner atom is shared equally among 8 adjacent unit cells. Therefore, the contribution of each corner atom to a single unit cell is $\frac{1}{8}$. $$Z = 8 \text{ corners} \times \frac{1}{8} \text{ atom/corner} = 1 \text{ atom per unit cell}$$
Step 3: Calculate the total number of unit cells.
Divide the total number of atoms by the number of atoms per unit cell: $$\text{Number of unit cells} = \frac{6.022 \times 10^{23} \text{ atoms{1 \text{ atom/unit cell$$ $$\text{Number of unit cells} = 6.022 \times 10^{23}$$