To find the fraction of unoccupied sites in sodium chloride (NaCl) crystal, we can follow these steps:
### Step 1: Understand the given data
- Pycnometric density of NaCl: \( \rho_{pyc} = 2.165 \times 10^3 \, \text{kg/m}^3 \)
- X-ray density of NaCl: \( \rho_{x-ray} = 2.178 \times 10^3 \, \text{kg/m}^3 \)
### Step 2: Calculate the molar volume using pycnometric density
The molar volume (\( V_m \)) can be calculated using the formula:
\[
V_m = \frac{M}{\rho}
\]
where \( M \) is the molar mass of NaCl (approximately \( 58.44 \, \text{g/mol} \) or \( 0.05844 \, \text{kg/mol} \)).
Using the pycnometric density:
\[
V_{m, pyc} = \frac{0.05844 \, \text{kg/mol}}{2.165 \times 10^3 \, \text{kg/m}^3}
\]
### Step 3: Calculate the molar volume using X-ray density
Using the X-ray density:
\[
V_{m, x-ray} = \frac{0.05844 \, \text{kg/mol}}{2.178 \times 10^3 \, \text{kg/m}^3}
\]
### Step 4: Find the volume unoccupied
The volume unoccupied can be calculated by subtracting the two volumes:
\[
V_{unoccupied} = V_{m, pyc} - V_{m, x-ray}
\]
### Step 5: Calculate the fraction of unoccupied sites
The fraction of unoccupied sites can be calculated as:
\[
\text{Fraction unoccupied} = \frac{V_{unoccupied}}{V_{m, pyc}}
\]
### Step 6: Substitute and calculate
1. Calculate \( V_{m, pyc} \):
\[
V_{m, pyc} = \frac{0.05844}{2.165 \times 10^3} \approx 2.698 \times 10^{-5} \, \text{m}^3/\text{mol}
\]
2. Calculate \( V_{m, x-ray} \):
\[
V_{m, x-ray} = \frac{0.05844}{2.178 \times 10^3} \approx 2.684 \times 10^{-5} \, \text{m}^3/\text{mol}
\]
3. Calculate \( V_{unoccupied} \):
\[
V_{unoccupied} = 2.698 \times 10^{-5} - 2.684 \times 10^{-5} \approx 0.014 \times 10^{-5} \, \text{m}^3/\text{mol}
\]
4. Calculate the fraction of unoccupied sites:
\[
\text{Fraction unoccupied} = \frac{0.014 \times 10^{-5}}{2.698 \times 10^{-5}} \approx 0.00519
\]
### Final Result
The fraction of unoccupied sites in NaCl crystal is approximately \( 0.00519 \).
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