Which of the following represents the closet packed arrangement of uniform solid spheres:

Options

A. simple cubic unit cell
B. body centered cubic unit cell
C. face centered cubic unit cell
D. hcp unit cell

Answer

To determine which arrangement of uniform solid spheres represents the closest packed arrangement, we need to analyze the different types of cubic structures based on their coordination numbers, effective number of atoms, and packing fractions.

### Step-by-Step Solution:

1. **Understanding Coordination Number**:
- The coordination number indicates how many nearest neighbors surround an atom in a crystal structure. A higher coordination number typically means a closer packing arrangement.

2. **Analyzing the Simple Cubic Unit Cell (SCC)**:
- **Effective Number of Atoms (Z)**: 1 (8 corner atoms, each contributes 1/8)
- **Coordination Number**: 6
- **Packing Fraction**: 52.4%
- **Conclusion**: This structure has a low packing efficiency and is not a close-packed arrangement.

3. **Analyzing the Body-Centered Cubic Unit Cell (BCC)**:
- **Effective Number of Atoms (Z)**: 2 (8 corner atoms contributing 1/8 each + 1 body-centered atom)
- **Coordination Number**: 8
- **Packing Fraction**: 68%
- **Conclusion**: This structure has a better packing efficiency than SCC but is still not the closest packed arrangement.

4. **Analyzing the Face-Centered Cubic Unit Cell (FCC)**:
- **Effective Number of Atoms (Z)**: 4 (8 corner atoms contributing 1/8 each + 6 face-centered atoms contributing 1/2 each)
- **Coordination Number**: 12
- **Packing Fraction**: 74%
- **Conclusion**: This structure has a high packing efficiency and is a close-packed arrangement.

5. **Analyzing the Hexagonal Close Packing (HCP)**:
- **Effective Number of Atoms (Z)**: 6
- **Coordination Number**: 12
- **Packing Fraction**: 74%
- **Conclusion**: This structure also has a high packing efficiency and is a close-packed arrangement.

6. **Final Conclusion**:
- The closest packed arrangements are represented by the Face-Centered Cubic (FCC) and Hexagonal Close Packing (HCP) structures, both having a coordination number of 12 and a packing fraction of 74%.

### Answer:
The closest packed arrangements of uniform solid spheres are represented by:
- Face-Centered Cubic (FCC)
- Hexagonal Close Packing (HCP)