In closest packing of A type of atoms (radius `r_(A)`) the radius of atom B that can be fitted into octabedral voids is

Options

A. `1.155 r_(a)`
B. `0.255 r_(a)`
C. `0.414 r_(a)`
D. `0.732 r_(a)`

Answer

To find the radius of atom B that can fit into the octahedral voids created by A type atoms in closest packing, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**: In closest packing, A type atoms are arranged in a way that they create octahedral voids. Each octahedral void is surrounded by 6 A type atoms.

2. **Visualizing the Octahedral Void**: Imagine two A type atoms stacked on top of each other with four more A type atoms surrounding them, forming an octahedron. The octahedral void is the empty space in the center of this arrangement.

3. **Relating Edge Length to Atomic Radius**: The edge length (a) of the octahedron can be related to the radius of the A atoms (r_A). Since the edge length is equal to the distance between the centers of two adjacent A atoms, we can write:
\[ a = 2r_A \]

4. **Using Pythagorean Theorem**: The diagonal across the octahedral void can be calculated using the Pythagorean theorem. The diagonal (d) can be expressed as:
\[ d = \sqrt{(2r_A)^2 + (2r_A)^2} = \sqrt{2(2r_A)^2} = 2\sqrt{2}r_A \]

5. **Relating to Atom B's Radius**: The diameter of atom B (which fits into the octahedral void) is equal to the distance across the void, which is:
\[ 2r_B + 2r_A = 2\sqrt{2}r_A \]

6. **Simplifying the Equation**: Dividing the entire equation by 2 gives:
\[ r_B + r_A = \sqrt{2}r_A \]

7. **Solving for r_B**: Rearranging the equation to isolate r_B:
\[ r_B = \sqrt{2}r_A - r_A \]
\[ r_B = (\sqrt{2} - 1)r_A \]

8. **Calculating the Value**: The value of \(\sqrt{2}\) is approximately 1.414, thus:
\[ r_B \approx (1.414 - 1)r_A = 0.414r_A \]

### Final Answer:
The radius of atom B that can fit into the octahedral voids is:
\[ r_B = 0.414r_A \]