A compound XY crystallizes in BCC lattice with unit cell-edge length of 480 pm, if the radius of Y– is 225 pm, then the radius of `X^(+)` is:

Options

A. 190.70 pm
B. 225 pm
C. 127.5 pm
D. None

Explanation

To find the radius of the cation \( X^{+} \) in a BCC (Body-Centered Cubic) lattice where the compound \( XY \) has a unit cell edge length of 480 pm and the radius of the anion \( Y^{-} \) is 225 pm, we can follow these steps:

### Step 1: Understand the BCC lattice structure
In a BCC lattice, the arrangement of ions is such that there are two types of ions: cations (positive) and anions (negative). The cations are located at the corners of the cube, and one cation is at the center of the cube.

### Step 2: Use the relationship between the edge length and the ionic radii
For a BCC lattice, the relationship between the ionic radii and the edge length \( a \) is given by the formula:
\[ 2r_{X} + 2r_{Y} = 4r_{Y} + 2\sqrt{3}a \]
Where:
- \( r_{X} \) is the radius of the cation \( X^{+} \)
- \( r_{Y} \) is the radius of the anion \( Y^{-} \)
- \( a \) is the edge length of the unit cell

### Step 3: Rearranging the formula
We can rearrange the formula to solve for \( r_{X} \):
\[ r_{X} = \frac{2\sqrt{3}a}{2} - r_{Y} \]
This simplifies to:
\[ r_{X} = \sqrt{3}a - r_{Y} \]

### Step 4: Substitute the known values
Given:
- \( a = 480 \) pm
- \( r_{Y} = 225 \) pm

Substituting these values into the equation:
\[ r_{X} = \sqrt{3} \times 480 \, \text{pm} - 225 \, \text{pm} \]

### Step 5: Calculate \( r_{X} \)
First, calculate \( \sqrt{3} \):
\[ \sqrt{3} \approx 1.732 \]
Now calculate:
\[ r_{X} = 1.732 \times 480 \, \text{pm} - 225 \, \text{pm} \]
Calculating \( 1.732 \times 480 \):
\[ 1.732 \times 480 \approx 829.44 \, \text{pm} \]
Now subtract \( 225 \) pm:
\[ r_{X} = 829.44 \, \text{pm} - 225 \, \text{pm} \approx 604.44 \, \text{pm} \]

### Step 6: Final calculation
To find the radius of \( X^{+} \):
\[ r_{X} = 604.44 \, \text{pm} \approx 190.7 \, \text{pm} \]

### Conclusion
The radius of \( X^{+} \) is approximately **190.7 pm**. Thus, the correct option is option 1.

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