For a solution of 3.795g of sulphur in 100g `CS_(2)` the boiling point was 319.81 K. For pure `CS_(2)` the boiling point is 319.45 K and the enthalpy of vapourisation is 351.87.`Jg^(-1)` . What is the molar mass and formula of sulphur in `CS_(2)`?

Explanation

To solve the problem, we need to find the molar mass and formula of sulfur in the solution of sulfur in carbon disulfide (CS₂). We will use the boiling point elevation method to find the molar mass of sulfur.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Mass of sulfur (solute), \( W_2 = 3.795 \, \text{g} \)
- Mass of carbon disulfide (solvent), \( W_1 = 100 \, \text{g} \)
- Boiling point of the solution, \( T_b = 319.81 \, \text{K} \)
- Boiling point of pure solvent, \( T_{b0} = 319.45 \, \text{K} \)
- Enthalpy of vaporization of CS₂, \( \Delta H_{vap} = 351.87 \, \text{J/g} \)
- Molar elevation constant, \( K_b = 2.42 \, \text{kg/mol} \)

2. **Calculate the Elevation in Boiling Point:**
\[ \Delta T_b = T_b - T_{b0} = 319.81 \, \text{K} - 319.45 \, \text{K} = 0.36 \, \text{K} \]

3. **Convert Mass of Solvent to Kilograms:**
\[ W_1 = 100 \, \text{g} = 0.1 \, \text{kg} \]

4. **Use the Boiling Point Elevation Formula:**
The formula for boiling point elevation is given by:
\[ \Delta T_b = K_b \cdot m \]
where \( m \) is the molality of the solution. We can rearrange this to find molality:
\[ m = \frac{\Delta T_b}{K_b} = \frac{0.36 \, \text{K}}{2.42 \, \text{kg/mol}} \approx 0.148 \, \text{mol/kg} \]

5. **Calculate the Number of Moles of Solute:**
Using the definition of molality:
\[ m = \frac{n}{W_1} \implies n = m \cdot W_1 = 0.148 \, \text{mol/kg} \cdot 0.1 \, \text{kg} = 0.0148 \, \text{mol} \]

6. **Calculate the Molar Mass of Sulfur:**
The molar mass \( M \) of the solute (sulfur) can be calculated using:
\[ M = \frac{W_2}{n} = \frac{3.795 \, \text{g}}{0.0148 \, \text{mol}} \approx 256.08 \, \text{g/mol} \]

7. **Determine the Molecular Formula:**
The atomic mass of sulfur is approximately 32 g/mol. To find the number of moles of sulfur in the molecular formula:
\[ n = \frac{M}{\text{Atomic mass}} = \frac{256.08 \, \text{g/mol}}{32 \, \text{g/mol}} \approx 8 \]
Therefore, the molecular formula of sulfur in CS₂ is \( S_8 \).

### Final Answer:
The molar mass of sulfur in CS₂ is approximately 256.08 g/mol, and the molecular formula is \( S_8 \).