KF has NaCl structure. What is the distance between `K^(+)` and `F^(-)` in KF, if the density is `1.5g cm^(-3)`?

Explanation

To find the distance between \( K^+ \) and \( F^- \) in potassium fluoride (KF), which has a sodium chloride (NaCl) type structure, we can follow these steps:

### Step 1: Understand the Structure
KF has a NaCl type structure, which means it has a face-centered cubic (FCC) lattice. In an FCC lattice, there are 4 formula units per unit cell (denoted as \( Z = 4 \)).

### Step 2: Use the Density Formula
The density (\( \rho \)) of the compound is given as \( 1.5 \, \text{g/cm}^3 \). The formula for density in terms of the unit cell parameters is:

\[ \rho = \frac{Z \cdot m}{A^3 \cdot N_A} \]

Where:
- \( Z \) = number of formula units per unit cell (4 for KF)
- \( m \) = molar mass of KF
- \( A \) = edge length of the unit cell
- \( N_A \) = Avogadro's number (\( 6.022 \times 10^{23} \, \text{mol}^{-1} \))

### Step 3: Calculate the Molar Mass of KF
The molar mass of potassium (K) is approximately 39 g/mol, and that of fluorine (F) is approximately 19 g/mol. Therefore, the molar mass of KF is:

\[ m = 39 + 19 = 58 \, \text{g/mol} \]

### Step 4: Rearrange the Density Formula
Rearranging the density formula to solve for \( A^3 \):

\[ A^3 = \frac{Z \cdot m}{\rho \cdot N_A} \]

### Step 5: Substitute the Known Values
Substituting the known values into the equation:

\[ A^3 = \frac{4 \cdot 58}{1.5 \cdot 6.022 \times 10^{23}} \]

Calculating this gives:

\[ A^3 = \frac{232}{9.033 \times 10^{23}} \approx 2.57 \times 10^{-24} \, \text{cm}^3 \]

### Step 6: Calculate the Edge Length \( A \)
Taking the cube root to find \( A \):

\[ A = (2.57 \times 10^{-24})^{1/3} \approx 1.36 \times 10^{-8} \, \text{cm} \approx 1.36 \, \text{Å} \]

### Step 7: Find the Distance Between \( K^+ \) and \( F^- \)
In the NaCl structure, the distance between the cation and anion is half of the edge length:

\[ \text{Distance} = \frac{A}{2} = \frac{1.36 \, \text{Å}}{2} \approx 0.68 \, \text{Å} \]

### Final Answer
The distance between \( K^+ \) and \( F^- \) in KF is approximately \( 0.68 \, \text{Å} \).

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