If radius of an octahedral void is r and atomic radius of atoms assuming cubical close pacting is R. Then the relation between r and R is

Options

A. r = 2R
B. r = 1.414 R
C. r = 0.414 R
D. `r = (R )/(sqrt(2))`

Answer

To solve the problem of finding the relationship between the radius of an octahedral void (r) and the atomic radius of atoms in a cubic close packing (R), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Octahedral Voids**:
- In a cubic close-packed structure, octahedral voids are formed between the atoms. Each octahedral void is surrounded by six atoms.

2. **Using the Radius Ratio Rule**:
- The radius ratio rule helps us determine the stability of a void based on the ratio of the radius of the void (r) to the radius of the atoms (R).
- For octahedral voids, the coordination number (CN) is 6.

3. **Identifying the Range for Octahedral Voids**:
- According to the radius ratio rule, the radius ratio (r/R) for octahedral voids must lie between 0.414 and 0.732.
- This means that for a stable octahedral void to exist, the ratio of the radius of the void to the atomic radius must be greater than or equal to 0.414.

4. **Setting Up the Equation**:
- We can express this relationship mathematically:
\[ \frac{r}{R} = 0.414 \]
- Rearranging this gives:
\[ r = 0.414 \times R \]

5. **Conclusion**:
- Therefore, the relationship between the radius of the octahedral void (r) and the atomic radius (R) is:
\[ r = 0.414 R \]