The atomic packing factor for a Simple Cubic structure is 0.52 . Show a diagram of the structure and all steps in the calculation, don’t solve for the answer, simply to the simplest fraction.
The Correct Answer and Explanation is :
It seems I wasn’t able to retrieve the image. However, I can still help you with an explanation and a diagram for the Simple Cubic structure, as well as the steps to calculate the atomic packing factor (APF).
Simple Cubic Structure
In a simple cubic structure, atoms are arranged in a cube, with one atom at each corner. Each corner atom is shared between 8 neighboring cubes, so only 1/8th of each corner atom belongs to one cube.
Atomic Packing Factor Calculation Steps
- Determine the number of atoms in the unit cell:
- In a simple cubic structure, there are 8 atoms at the corners of the cube. Since each corner atom is shared by 8 cubes, the total number of atoms in the unit cell is:
[
\frac{8 \text{ atoms} \times \frac{1}{8}}{1} = 1 \text{ atom}
]
- Calculate the volume of the unit cell:
- The length of the edge of the unit cell is denoted as “a”. The volume of the cube is then:
[
V_{\text{cell}} = a^3
]
- Determine the volume occupied by the atoms:
- The volume occupied by a single atom is calculated by finding the volume of a sphere with the radius ( r ). The relationship between the radius of the atom and the edge length “a” of the unit cell for a simple cubic structure is ( a = 2r ).
- The volume of a single atom is the volume of a sphere:
[
V_{\text{atom}} = \frac{4}{3} \pi r^3
] - Substituting ( r = \frac{a}{2} ), the volume of an atom becomes:
[
V_{\text{atom}} = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 = \frac{\pi a^3}{6}
]
- Calculate the atomic packing factor (APF):
- The APF is the fraction of the volume of the unit cell that is occupied by atoms. The formula is:
[
\text{APF} = \frac{\text{Volume of atoms in unit cell}}{\text{Volume of unit cell}} = \frac{1 \times \frac{\pi a^3}{6}}{a^3} = \frac{\pi}{6} \approx 0.52
]
Explanation
The atomic packing factor (APF) for a simple cubic structure is 0.52, meaning that 52% of the unit cell’s volume is filled with atoms, while the remaining 48% is empty space. This relatively low packing efficiency is due to the large amount of unoccupied space between the atoms in the simple cubic arrangement. This structure is the least efficient in terms of packing compared to other structures like body-centered cubic (BCC) and face-centered cubic (FCC).
I will also generate a diagram of the simple cubic structure for better visualization.
Here is the illustration of a simple cubic structure, showing the atoms at each corner of the cube. This diagram helps visualize how the atoms are arranged in a simple cubic lattice, where each corner atom is shared between 8 unit cells.
The calculation and the diagram together highlight the atomic packing factor of 0.52 for this structure. Let me know if you’d like further details!
