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

Options

A. `r//R=0.414`
B. `r//R=0.225`
C. `r//R=0.155`
D. `r//R=0.732`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Structure**: In a close-packed structure (like face-centered cubic or body-centered cubic), atoms are arranged in a way that maximizes packing efficiency. The octahedral void is a space between atoms where another atom can fit.

2. **Visualizing the Atoms and Voids**: In a close-packed arrangement, consider two atoms touching each other. The distance between their centers is 2R (where R is the radius of the atom).

3. **Identifying the Geometry**: The octahedral void is formed by the arrangement of four atoms at the corners of a square and one atom at the center. The distance from the center of the octahedral void to the corner atoms is r (the radius of the void).

4. **Using Right Triangle Properties**: If we consider the triangle formed by the centers of the atoms and the center of the octahedral void, we can denote the points as A, B, and C:
- Let A and B be the centers of two adjacent atoms.
- Let C be the center of the octahedral void.

5. **Setting Up the Right Triangle**: In triangle ABC:
- AB = 2R (distance between the two atoms)
- AC = r (distance from the center of the void to the center of one atom)
- BC = r (distance from the center of the void to the center of the other atom)

6. **Applying the Pythagorean Theorem**: Since triangle ABC is a right triangle, we can apply the Pythagorean theorem:
\[ AC^2 + BC^2 = AB^2 \]
Substituting the values:
\[ r^2 + r^2 = (2R)^2 \]
This simplifies to:
\[ 2r^2 = 4R^2 \]

7. **Solving for r in terms of R**: Dividing both sides by 2:
\[ r^2 = 2R^2 \]
Taking the square root of both sides gives:
\[ r = R\sqrt{2} \]

### Final Relationship:
Thus, the relationship between the radius of the octahedral void (r) and the radius of the atom (R) in a closest packed structure is:
\[ r = R\sqrt{2} \]