(atoms) Number of Spheres Per Unit Cell: Volume of Spheres Per Unit Cell: π ³. Number of sphere per unit cell) xatoms Side Length in terms of r Volume of Unit Cell in Terms r Packing Efficiency (%) = V(spheres) -100% V(unit cell) Simple Cube 1 Tr³xl 3 는 Tr 3 }} 2r E l=2r 3 H 3 813 11 =8r²³² X100 BCC 2 FCC 74
(atoms) Number of Spheres Per Unit Cell: Volume of Spheres Per Unit Cell: π ³. Number of sphere per unit cell) xatoms Side Length in terms of r Volume of Unit Cell in Terms r Packing Efficiency (%) = V(spheres) -100% V(unit cell) Simple Cube 1 Tr³xl 3 는 Tr 3 }} 2r E l=2r 3 H 3 813 11 =8r²³² X100 BCC 2 FCC 74
Chemistry & Chemical Reactivity
10th Edition
ISBN:9781337399074
Author:John C. Kotz, Paul M. Treichel, John Townsend, David Treichel
Publisher:John C. Kotz, Paul M. Treichel, John Townsend, David Treichel
Chapter12: The Solid State
Section: Chapter Questions
Problem 50GQ: Consider the three types of cubic units cells. (a) Assuming that the spherical atoms or ions in a...
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![### Understanding Crystal Structures: Simple Cube, BCC, and FCC
#### 1. Number of Spheres per Unit Cell
- **Simple Cube**: 1 atom
- **BCC (Body-Centered Cubic)**: 2 atoms
- **FCC (Face-Centered Cubic)**: 4 atoms
#### 2. Volume of Spheres Per Unit Cell
The volume calculation formula for the spheres within a unit cell is:
\[ \frac{4}{3} \pi r^3 \times \text{Number of spheres per unit cell} \]
- **Simple Cube**: \(\frac{4}{3} \pi r^3 \times 1 = \frac{4}{3} \pi r^3\)
- **BCC**: \(\frac{4}{3} \pi r^3 \times 2 = \frac{8}{3} \pi r^3\)
- **FCC**: \(\frac{4}{3} \pi r^3 \times 4 = \frac{16}{3} \pi r^3\)
#### 3. Side Length in Terms of r (Radius)
- **Simple Cube**: \(2r\)
- **BCC**: \(\frac{4r}{\sqrt{3}}\)
- **FCC**: \(2\sqrt{2}r\)
#### 4. Volume of Unit Cell in Terms of r
- **Simple Cube**: \((2r)^3 = 8r^3\)
- **BCC**: \(\left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}}\)
- **FCC**: \( (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \)
#### 5. Packing Efficiency
Calculated as:
\[ \text{Packing Efficiency} (\%) = \frac{\text{Volume of spheres}}{\text{Volume of unit cell}} \times 100\]
- **Simple Cube**: \[ \frac{\frac{4}{3} \pi r^3}{8r^3} \times 100 = \frac{\pi}{6} \times 100 \approx 52.36\%\]
- **BCC**: \( \frac{\frac{8}{3} \pi](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ea581ae-cff7-44c5-97bc-c00c93c4e377%2F90b7ecdf-47f9-4603-a73b-9a452aedbeb6%2Ftqmzp6k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Crystal Structures: Simple Cube, BCC, and FCC
#### 1. Number of Spheres per Unit Cell
- **Simple Cube**: 1 atom
- **BCC (Body-Centered Cubic)**: 2 atoms
- **FCC (Face-Centered Cubic)**: 4 atoms
#### 2. Volume of Spheres Per Unit Cell
The volume calculation formula for the spheres within a unit cell is:
\[ \frac{4}{3} \pi r^3 \times \text{Number of spheres per unit cell} \]
- **Simple Cube**: \(\frac{4}{3} \pi r^3 \times 1 = \frac{4}{3} \pi r^3\)
- **BCC**: \(\frac{4}{3} \pi r^3 \times 2 = \frac{8}{3} \pi r^3\)
- **FCC**: \(\frac{4}{3} \pi r^3 \times 4 = \frac{16}{3} \pi r^3\)
#### 3. Side Length in Terms of r (Radius)
- **Simple Cube**: \(2r\)
- **BCC**: \(\frac{4r}{\sqrt{3}}\)
- **FCC**: \(2\sqrt{2}r\)
#### 4. Volume of Unit Cell in Terms of r
- **Simple Cube**: \((2r)^3 = 8r^3\)
- **BCC**: \(\left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}}\)
- **FCC**: \( (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \)
#### 5. Packing Efficiency
Calculated as:
\[ \text{Packing Efficiency} (\%) = \frac{\text{Volume of spheres}}{\text{Volume of unit cell}} \times 100\]
- **Simple Cube**: \[ \frac{\frac{4}{3} \pi r^3}{8r^3} \times 100 = \frac{\pi}{6} \times 100 \approx 52.36\%\]
- **BCC**: \( \frac{\frac{8}{3} \pi
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