Which of the following expressions is correct in the case of a sodium chloride unit cell (edge length, `a`)?

Options

A. `r_(c)+r_(a)=a`
B. `r_(c)+r_(a)=2a`
C. `r_(c)+r_(a)=sqrt2a`
D. `r_(c)+r_(a)=a/2`

Answer

To solve the question regarding the correct expression for the sodium chloride unit cell with edge length `a`, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of Sodium Chloride (NaCl)**:
- Sodium chloride crystallizes in a face-centered cubic (FCC) structure. In this structure, chloride ions (Cl⁻) are located at the corners and the face centers of the cube, while sodium ions (Na⁺) occupy the octahedral voids.

2. **Identify the Positions of Ions**:
- In the FCC unit cell, the chloride ions are positioned at the corners and the center of each face. The sodium ions fill the octahedral voids, which are located at the center of the cube and at the edge centers.

3. **Define the Radii of Ions**:
- Let the radius of the sodium ion be denoted as `rₐ` and the radius of the chloride ion as `r_c`.

4. **Establish the Relationship Between Edge Length and Ion Radii**:
- In the NaCl structure, the distance between the centers of a sodium ion and a chloride ion that are touching each other is equal to the sum of their radii. This can be expressed as:
\[ rₐ + r_c = \text{distance between Na⁺ and Cl⁻} \]
- In the FCC structure, the distance between the centers of two adjacent ions (one Na⁺ and one Cl⁻) can be related to the edge length `a` of the unit cell.

5. **Minimum Touching Condition**:
- The minimum distance between the sodium and chloride ions occurs along the face diagonal of the cube. The face diagonal can be calculated using the edge length `a`:
\[ \text{Face diagonal} = a\sqrt{2} \]
- Since there are two radii involved (one for Na⁺ and one for Cl⁻), the relationship can be expressed as:
\[ 2r_c + 2rₐ = a \]
- Simplifying this gives:
\[ r_c + rₐ = \frac{a}{2} \]

6. **Conclusion**:
- Therefore, the correct expression for the sodium chloride unit cell in terms of the edge length `a` is:
\[ r_c + rₐ = \frac{a}{2} \]

### Final Answer:
The correct expression for the sodium chloride unit cell with edge length `a` is:
\[ r_c + rₐ = \frac{a}{2} \]