A typical paper clip weighs 0.59 g and consists of BCC iron. Calculate the number of iron unit cells in the paper clip. Data: Lattice parameter of BCC iron: a = 2.866 angstroms and density Ï = 7.87 g/cm³.
A typical paper clip weighs 0.59 g and consists of BCC iron. Calculate the number of iron unit cells in the paper clip. Data: Lattice parameter of BCC iron: a = 2.866 angstroms and density Ï = 7.87 g/cm³.
The Correct Answer and Explanation is:
To calculate the number of unit cells in a typical paper clip made of BCC (body-centered cubic) iron, we can follow these steps:
Given Data:
- Mass of paper clip = 0.59 g
- Density of iron = 7.87 g/cm³
- Lattice parameter a=2.866a = 2.866 Å = 2.866 × 10⁻⁸ cm
- BCC unit cell: 2 atoms per unit cell
Step 1: Calculate the volume of the paper clip
Volume=MassDensity=0.59 g7.87 g/cm3≈0.07497 cm3\text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{0.59\ \text{g}}{7.87\ \text{g/cm}^3} \approx 0.07497\ \text{cm}^3
Step 2: Calculate the volume of one unit cell
Unit cell volume=a3=(2.866×10−8 cm)3=2.351×10−23 cm3\text{Unit cell volume} = a^3 = (2.866 \times 10^{-8}\ \text{cm})^3 = 2.351 \times 10^{-23}\ \text{cm}^3
Step 3: Calculate the number of unit cells
Number of unit cells=Volume of paper clipVolume of one unit cell=0.074972.351×10−23≈3.19×1021\text{Number of unit cells} = \frac{\text{Volume of paper clip}}{\text{Volume of one unit cell}} = \frac{0.07497}{2.351 \times 10^{-23}} \approx 3.19 \times 10^{21}
✅ Final Answer:
3.19×1021 unit cells\boxed{3.19 \times 10^{21}\ \text{unit cells}}
Explanation
A unit cell is the smallest repeating structure in a crystal lattice. In the case of BCC (body-centered cubic) iron, each unit cell contains 2 atoms and is shaped like a cube. To determine how many of these unit cells exist in a typical paper clip, we must relate the total volume of iron in the paper clip to the volume of a single BCC unit cell.
We are told the paper clip weighs 0.59 grams and has a density of 7.87 g/cm³. Using the formula Volume=MassDensity\text{Volume} = \frac{\text{Mass}}{\text{Density}}, we find that the volume of the paper clip is approximately 0.07497 cm³.
Next, we calculate the volume of a single BCC iron unit cell using the lattice parameter a=2.866a = 2.866 Å, which we convert to centimeters. Cubing this value gives us the volume of one unit cell in cm³.
To determine how many such unit cells make up the total volume of the paper clip, we divide the total volume (from mass and density) by the volume of one unit cell. The result is approximately 3.19×10213.19 \times 10^{21} unit cells.
This huge number reflects the extremely small size of unit cells in crystalline solids. Even a small object like a paper clip contains an astronomical number of atoms, structured in an orderly, repeating fashion. This calculation also shows how materials science uses simple physical relationships to bridge macroscopic and atomic-scale understanding.
