KCl and NaCl have fcc lattice. Calculate the ratio of density of NaCl to that of KCl if the ratio of edge of NaCl to that of KCl is 0.875.

Explanation

To calculate the ratio of the density of NaCl to that of KCl given that the ratio of the edge lengths of NaCl to KCl is 0.875, we will follow these steps:

### Step 1: Understand the formula for density
The density (D) of a crystal can be calculated using the formula:
\[ D = \frac{Z \times M}{A^3 \times N_A} \]
where:
- \( Z \) = number of formula units per unit cell (for FCC, \( Z = 4 \))
- \( M \) = molar mass of the compound
- \( A \) = edge length of the unit cell
- \( N_A \) = Avogadro's number (constant, cancels out when taking ratios)

### Step 2: Calculate the molar masses of NaCl and KCl
- Molar mass of NaCl:
- Na = 23 g/mol
- Cl = 35.5 g/mol
- \( M_{NaCl} = 23 + 35.5 = 58.5 \, \text{g/mol} \)

- Molar mass of KCl:
- K = 39 g/mol
- Cl = 35.5 g/mol
- \( M_{KCl} = 39 + 35.5 = 74.5 \, \text{g/mol} \)

### Step 3: Set up the density ratio
Using the density formula, we can express the densities for NaCl and KCl:
- Density of NaCl:
\[ D_{NaCl} = \frac{4 \times 58.5}{A_{NaCl}^3 \times N_A} \]

- Density of KCl:
\[ D_{KCl} = \frac{4 \times 74.5}{A_{KCl}^3 \times N_A} \]

### Step 4: Calculate the ratio of densities
Now we can calculate the ratio of the densities:
\[ \frac{D_{NaCl}}{D_{KCl}} = \frac{\frac{4 \times 58.5}{A_{NaCl}^3}}{\frac{4 \times 74.5}{A_{KCl}^3}} = \frac{58.5 \times A_{KCl}^3}{74.5 \times A_{NaCl}^3} \]

### Step 5: Substitute the ratio of edge lengths
Given that the ratio of edge lengths \( \frac{A_{NaCl}}{A_{KCl}} = 0.875 \), we can express \( A_{NaCl} \) in terms of \( A_{KCl} \):
\[ A_{NaCl} = 0.875 \times A_{KCl} \]
Thus,
\[ A_{NaCl}^3 = (0.875 \times A_{KCl})^3 = 0.875^3 \times A_{KCl}^3 \]

### Step 6: Substitute back into the density ratio
Substituting \( A_{NaCl}^3 \) into the density ratio:
\[ \frac{D_{NaCl}}{D_{KCl}} = \frac{58.5 \times A_{KCl}^3}{74.5 \times (0.875^3 \times A_{KCl}^3)} = \frac{58.5}{74.5 \times 0.875^3} \]

### Step 7: Calculate \( 0.875^3 \)
Calculating \( 0.875^3 \):
\[ 0.875^3 = 0.669921875 \]

### Step 8: Final calculation of the density ratio
Now substituting this value back:
\[ \frac{D_{NaCl}}{D_{KCl}} = \frac{58.5}{74.5 \times 0.669921875} \approx \frac{58.5}{49.89} \approx 1.172 \]

### Conclusion
Thus, the ratio of the density of NaCl to that of KCl is approximately:
\[ \frac{D_{NaCl}}{D_{KCl}} \approx 1.172 \]