The face diagonal length of FCC cubic cell is `660sqrt2` pm.If the radius of the cation is 110 pm, What should be the radius of the anion following radius ratio rules

Explanation

To solve the problem, we need to find the radius of the anion in a face-centered cubic (FCC) structure given the face diagonal length and the radius of the cation. Let's break this down step by step.

### Step 1: Understand the relationship between face diagonal and edge length in FCC
In an FCC unit cell, the face diagonal (d) can be expressed in terms of the edge length (a) as:
\[ d = \sqrt{2} \cdot a \]

### Step 2: Set up the equation using the given face diagonal length
The problem states that the face diagonal length is \( 660\sqrt{2} \) pm. Therefore, we can set up the equation:
\[ \sqrt{2} \cdot a = 660\sqrt{2} \]
To find the edge length \( a \), we can divide both sides by \( \sqrt{2} \):
\[ a = 660 \, \text{pm} \]

### Step 3: Relate the edge length to the cation and anion radii
In an FCC structure, the cations occupy the corners and the face centers. The relationship between the edge length \( a \), the radius of the cation \( r_+ \), and the radius of the anion \( r_- \) can be expressed as:
\[ \sqrt{2} \cdot r_+ + \sqrt{2} \cdot r_- = a \]
This can be simplified to:
\[ r_+ + r_- = \frac{a}{\sqrt{2}} \]

### Step 4: Substitute the known values
Given that the radius of the cation \( r_+ = 110 \, \text{pm} \) and \( a = 660 \, \text{pm} \):
\[ 110 + r_- = \frac{660}{\sqrt{2}} \]

### Step 5: Calculate \( \frac{660}{\sqrt{2}} \)
Calculating the right side:
\[ \frac{660}{\sqrt{2}} = 660 \cdot \frac{\sqrt{2}}{2} = 330\sqrt{2} \approx 466.0 \, \text{pm} \]

### Step 6: Solve for the radius of the anion
Now we can substitute back into the equation:
\[ 110 + r_- = 466.0 \]
To find \( r_- \):
\[ r_- = 466.0 - 110 = 356.0 \, \text{pm} \]

### Final Answer
The radius of the anion \( r_- \) is approximately \( 356.0 \, \text{pm} \).