A solid having density of `9 xx 10^(3) kg m^(-3)` forms face centred cubic crystals of edge length `200 sqrt2` pm. What is the molar mass of the solid ? [Avogadro constant `~= 6 xx 10^(23) mol^(-1), pi ~= 3`]

Options

A. `0.0432 kg mol^(-1)`
B. `0.0305 kg mol^(-1)`
C. `0.4320 kg mol^(-1)`
D. `0.0216 kg mol^(-1)`

Answer

To find the molar mass of the solid, we can use the formula for density in relation to the face-centered cubic (FCC) structure. The formula is:

\[ \text{Density} (d) = \frac{Z \cdot M}{N_A \cdot a^3} \]

Where:
- \( d \) = density of the solid (in kg/m³)
- \( Z \) = number of atoms per unit cell (for FCC, \( Z = 4 \))
- \( M \) = molar mass of the solid (in kg/mol)
- \( N_A \) = Avogadro's number (approximately \( 6 \times 10^{23} \) mol⁻¹)
- \( a \) = edge length of the unit cell (in meters)

### Step-by-Step Solution:

1. **Identify the given values**:
- Density \( d = 9 \times 10^3 \, \text{kg/m}^3 \)
- Edge length \( a = 200\sqrt{2} \, \text{pm} = 200\sqrt{2} \times 10^{-12} \, \text{m} \)
- Avogadro's number \( N_A = 6 \times 10^{23} \, \text{mol}^{-1} \)
- For FCC, \( Z = 4 \)

2. **Convert the edge length from picometers to meters**:
\[ a = 200\sqrt{2} \times 10^{-12} \, \text{m} \]

3. **Calculate \( a^3 \)**:
\[ a^3 = (200\sqrt{2} \times 10^{-12})^3 = 200^3 \cdot (2^{3/2}) \cdot (10^{-12})^3 = 8 \times 10^6 \cdot 200^3 \cdot 10^{-36} \, \text{m}^3 \]
\[ = 8 \times 8 \times 10^{-30} = 64 \times 10^{-30} \, \text{m}^3 \]

4. **Rearrange the density formula to solve for molar mass \( M \)**:
\[ M = \frac{d \cdot N_A \cdot a^3}{Z} \]

5. **Substitute the known values into the equation**:
\[ M = \frac{(9 \times 10^3) \cdot (6 \times 10^{23}) \cdot (200\sqrt{2} \times 10^{-12})^3}{4} \]

6. **Calculate \( M \)**:
\[ M = \frac{(9 \times 10^3) \cdot (6 \times 10^{23}) \cdot (200\sqrt{2})^3 \cdot 10^{-36}}{4} \]
\[ = \frac{(9 \times 10^3) \cdot (6 \times 10^{23}) \cdot (8 \times 10^6) \cdot 10^{-36}}{4} \]
\[ = \frac{(9 \cdot 6 \cdot 8) \times 10^{3 + 23 + 6 - 36}}{4} \]
\[ = \frac{432 \times 10^{-4}}{4} = 108 \times 10^{-4} \, \text{kg/mol} \]
\[ = 0.0108 \, \text{kg/mol} = 10.8 \, \text{g/mol} \]

### Final Answer:
The molar mass of the solid is approximately **10.8 g/mol**.