The maximum ra dius of sphere that can be fitted in the octahedral hole of cubical closed packing of sphere of raius `r` is

Options

A. 0.732 r
B. 0.414 r
C. 0.225 r
D. 0.155 r

Answer

To find the maximum radius of a sphere that can fit in the octahedral hole of a cubic closed packing (CCP) of spheres with radius \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry of the Packing**:
In a cubic closed packing (also known as face-centered cubic packing), spheres are arranged in a way that maximizes the packing efficiency. The octahedral voids are formed between the spheres.

2. **Identify the Radius Ratio**:
The radius ratio for spheres fitting into octahedral voids is given as \( \frac{r_a}{r} = 0.414 \), where \( r_a \) is the radius of the sphere that fits into the octahedral void, and \( r \) is the radius of the spheres forming the packing.

3. **Set Up the Equation**:
From the radius ratio, we can express the radius of the sphere that can fit into the octahedral void as:
\[ r_a = 0.414 \times r \]

4. **Conclusion**:
Therefore, the maximum radius \( r_a \) of the sphere that can be fitted in the octahedral hole of a cubic closed packing of spheres of radius \( r \) is:
\[ r_a = 0.414r \]

### Final Answer:
The maximum radius of the sphere that can be fitted in the octahedral hole of cubic closed packing of spheres of radius \( r \) is \( 0.414r \).

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