If the radius of the octaheral void is `r` and the radius of the atoms in close-packing is `R`, derive relation between `r` and `R`

Options

A. r=0.414R
B. R=0.414r
C. r=2R
D. `r=sqrt2R`

Explanation

To derive the relation between the radius of the octahedral void (r) and the radius of the atoms in close packing (R), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
In a close-packed structure, atoms are arranged in such a way that they occupy the maximum available space. The octahedral voids are the spaces between the atoms where smaller atoms can fit.

2. **Visualizing the Octahedral Void**:
Consider a cube formed by the arrangement of atoms. The octahedral void is located at the center of the cube. The corners of the cube are occupied by the atoms.

3. **Diagonal of the Cube**:
The diagonal (AC) of the cube can be calculated using the Pythagorean theorem. If the edge length of the cube is 'a', then:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

4. **Relating Edge Length to Atomic Radius**:
In a close-packed structure, the edge length 'a' can be related to the atomic radius 'R'. For a face-centered cubic (FCC) structure, the relationship is:
\[ a = 2R \]

5. **Substituting Edge Length**:
Substitute \( a = 2R \) into the diagonal equation:
\[ AC = (2R)\sqrt{2} = 2R\sqrt{2} \]

6. **Understanding the Octahedral Void**:
The diagonal AC also represents the distance across the octahedral void, which consists of two atomic radii (from two atoms) plus the radius of the void (r):
\[ AC = R + R + r = 2R + r \]

7. **Setting the Equations Equal**:
Now we can set the two expressions for AC equal to each other:
\[ 2R + r = 2R\sqrt{2} \]

8. **Solving for r**:
Rearranging the equation gives:
\[ r = 2R\sqrt{2} - 2R \]
Factor out \( 2R \):
\[ r = 2R(\sqrt{2} - 1) \]

9. **Final Relation**:
Therefore, the relationship between the radius of the octahedral void (r) and the radius of the atoms in close packing (R) is:
\[ r = 2R(\sqrt{2} - 1) \approx 0.414R \]