Ionic salt AX grows in face centered cubic lattice with cell length 'a'. The ratio `r_(A^(+))//r_(X_(-))` for this salt will be

Options

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

Answer

To solve the problem of finding the radius ratio \( \frac{r_{A^{+}}}{r_{X^{-}}} \) for the ionic salt AX that crystallizes in a face-centered cubic (FCC) lattice, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) lattice, the cations (A\(^+\)) and anions (X\(^-\)) are arranged in a specific manner. The anions typically occupy the face-centered positions, while the cations occupy the octahedral holes.

### Step 2: Identify the Geometry
In an FCC lattice, the coordination number for both cations and anions is 12. This means that each cation is surrounded by 12 anions and vice versa.

### Step 3: Determine the Radius Ratio
The radius ratio \( \frac{r_{A^{+}}}{r_{X^{-}}} \) can be determined based on the geometry of the lattice. For an FCC structure, the limiting radius ratio is given by:

\[ \frac{r_{A^{+}}}{r_{X^{-}}} = \frac{r}{R} \]

Where \( r \) is the radius of the cation and \( R \) is the radius of the anion.

### Step 4: Use the Limiting Values
For FCC lattices, the radius ratio falls within certain limits. The critical limits for the radius ratio in FCC structures are:

- Minimum radius ratio: \( \approx 0.414 \)
- Maximum radius ratio: \( \approx 0.732 \)

### Step 5: Conclusion
Since the question asks for the ratio \( \frac{r_{A^{+}}}{r_{X^{-}}} \) for the ionic salt AX in an FCC lattice, we conclude that:

\[ \frac{r_{A^{+}}}{r_{X^{-}}} \text{ for the FCC lattice is approximately } 0.414 \text{ to } 0.732. \]

Thus, the ratio \( \frac{r_{A^{+}}}{r_{X^{-}}} \) for the salt AX will be:

\[ \frac{r_{A^{+}}}{r_{X^{-}}} = 0.414 \text{ (minimum limit)} \]

### Final Answer
The ratio \( \frac{r_{A^{+}}}{r_{X^{-}}} \) for the ionic salt AX is \( 0.414 \).

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