PbS has NaCl type structure . The distance between `Pb^(2+)and S^(2-)` ions is 297 pm. What is the volume of unit cell of PbS ?

Options

A. `209.6xx10^(-24)cm^3`
B. `207.8xx10^(-23)cm^3`
C. `22.3xx10^(-23)cm^3`
D. `209.8xx10^(23)cm^3`

Answer

To find the volume of the unit cell of PbS, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
- PbS has a NaCl-type structure, which means it has a face-centered cubic (FCC) arrangement for the anions (S²⁻) and the cations (Pb²⁺) occupy the octahedral voids.

2. **Distance Between Ions**:
- The distance between Pb²⁺ and S²⁻ ions is given as 297 pm (picometers).

3. **Finding the Edge Length of the Unit Cell (a)**:
- In the NaCl structure, the distance between the cation and anion is half the edge length of the unit cell.
- Therefore, the edge length \( a \) can be calculated as:
\[ a = 2 \times \text{distance between ions} = 2 \times 297 \, \text{pm} = 594 \, \text{pm} \]

4. **Converting pm to meters**:
- To convert picometers to meters, we use the conversion factor \( 1 \, \text{pm} = 10^{-12} \, \text{m} \):
\[ a = 594 \, \text{pm} = 594 \times 10^{-12} \, \text{m} = 5.94 \times 10^{-10} \, \text{m} \]

5. **Calculating the Volume of the Unit Cell**:
- The volume \( V \) of a cubic unit cell is given by:
\[ V = a^3 \]
- Substituting the value of \( a \):
\[ V = (5.94 \times 10^{-10} \, \text{m})^3 \]
- Calculating \( V \):
\[ V = 2.08 \times 10^{-28} \, \text{m}^3 \]

6. **Final Result**:
- To express the volume in a more convenient unit, we can convert it to cubic centimeters (1 m³ = \( 10^6 \, \text{cm}^3 \)):
\[ V = 2.08 \times 10^{-28} \, \text{m}^3 \times 10^6 \, \text{cm}^3/\text{m}^3 = 2.08 \times 10^{-22} \, \text{cm}^3 \]

### Final Answer:
The volume of the unit cell of PbS is approximately \( 2.08 \times 10^{-22} \, \text{cm}^3 \).

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