If the radius of the octahedral void is `r` and radius of the atom in closet packed structure is `R` then

Options

A. `r = 0.732` R
B. `R = 0.414` r
C. `r = 0.414` R
D. `R = 0.732` r

Answer

To solve the problem regarding the relationship between the radius of the octahedral void (r) and the radius of the atom in a closest packed structure (R), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
- In a closest packed structure (CCP), atoms are arranged in a way that maximizes the packing efficiency. The octahedral voids are formed between the atoms.

2. **Visualizing the Atoms and Voids**:
- Consider a unit cell of the closest packed structure. In this structure, the atoms are located at the corners and the center of the face of the cube.
- The distance between the centers of two adjacent atoms (which are touching) is equal to 2R.

3. **Identifying the Octahedral Void**:
- The octahedral void is located at the center of the cube and also at the edge centers. The radius of the octahedral void is denoted as r.

4. **Using Geometry**:
- In the CCP structure, we can analyze the geometry using the Pythagorean theorem.
- The distance from the center of one atom to the center of the octahedral void can be expressed in terms of R and r.

5. **Setting Up the Equation**:
- The distance from one corner atom to the center of the octahedral void can be represented as:
\[ \text{Distance} = R + r \]
- The distance between two corner atoms (diagonal of the face of the cube) is:
\[ \text{Distance} = 2R \]

6. **Applying the Pythagorean Theorem**:
- For the triangle formed by the centers of the atoms and the octahedral void, we can write:
\[ (2R)^2 = (R + r)^2 + (R + r)^2 \]
- Simplifying this gives:
\[ 4R^2 = 2(R + r)^2 \]

7. **Expanding and Rearranging**:
- Expanding the right side:
\[ 4R^2 = 2(R^2 + 2Rr + r^2) \]
- This simplifies to:
\[ 4R^2 = 2R^2 + 4Rr + 2r^2 \]
- Rearranging gives:
\[ 2R^2 - 4Rr - 2r^2 = 0 \]

8. **Using the Quadratic Formula**:
- This is a quadratic equation in terms of r. We can use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2, b = -4R, c = -2R^2 \).
- Solving this will yield the relationship between r and R.

9. **Finding the Relationship**:
- After solving the quadratic equation, we find:
\[ r = 0.414R \]

### Conclusion:
The correct relationship between the radius of the octahedral void (r) and the radius of the atom in the closest packed structure (R) is:
\[ r = 0.414R \]