The boiling point of a solution of `5g` of sulphur in `100g` of carbon disulphide is `0.474^(@)C` above that of pure solvent. Determine the molecular formula of sulphur in this solvent. The boiling point of pure carbon disulphide is `47^(@)C` and its heat of vaporisation is `84` calories per gram.

Explanation

To determine the molecular formula of sulfur in the given solution, we will follow these steps:

### Step 1: Identify the given data
- Mass of sulfur (solute) = 5 g
- Mass of carbon disulfide (solvent) = 100 g
- Boiling point elevation (ΔTb) = 0.474 °C
- Boiling point of pure carbon disulfide (Tb) = 47 °C
- Heat of vaporization (Lb) = 84 calories/g

### Step 2: Convert the mass of the solvent to kilograms
Since molality is defined as moles of solute per kilogram of solvent, we need to convert the mass of carbon disulfide to kilograms:
\[ \text{Mass of solvent in kg} = \frac{100 \, \text{g}}{1000} = 0.1 \, \text{kg} \]

### Step 3: Calculate the boiling point elevation constant (Kb)
Using the formula:
\[ K_b = \frac{R \cdot T_b^2}{1000 \cdot L_b} \]
where:
- R (universal gas constant) = 2 cal/(K·mol)
- \( T_b \) (boiling point in Kelvin) = 47 °C + 273 = 320 K

Substituting the values:
\[ K_b = \frac{2 \cdot (320)^2}{1000 \cdot 84} \]
Calculating \( K_b \):
\[ K_b = \frac{2 \cdot 102400}{84000} = \frac{204800}{84000} \approx 2.438 \, \text{°C kg/mol} \]

### Step 4: Relate boiling point elevation to molality
Using the formula:
\[ \Delta T_b = K_b \cdot \text{molality} \]
We can rearrange this to find molality:
\[ \text{molality} = \frac{\Delta T_b}{K_b} \]
Substituting the values:
\[ \text{molality} = \frac{0.474}{2.438} \approx 0.194 \, \text{mol/kg} \]

### Step 5: Calculate the number of moles of solute
Using the definition of molality:
\[ \text{molality} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]
Rearranging gives:
\[ \text{moles of solute} = \text{molality} \cdot \text{mass of solvent in kg} \]
Substituting the values:
\[ \text{moles of solute} = 0.194 \cdot 0.1 = 0.0194 \, \text{moles} \]

### Step 6: Calculate the molecular mass of sulfur
Using the formula:
\[ \text{molecular mass} = \frac{\text{mass of solute}}{\text{moles of solute}} \]
Substituting the values:
\[ \text{molecular mass} = \frac{5 \, \text{g}}{0.0194 \, \text{moles}} \approx 257.73 \, \text{g/mol} \]

### Step 7: Determine the molecular formula of sulfur
The atomic mass of sulfur (S) is approximately 32 g/mol. To find the number of sulfur atoms in the molecular formula:
\[ \text{Number of sulfur atoms} = \frac{257.73 \, \text{g/mol}}{32 \, \text{g/mol}} \approx 8.06 \]
Rounding to the nearest whole number gives us 8.

Thus, the molecular formula of sulfur in this solvent is:
\[ \text{Molecular formula} = S_8 \]

### Final Answer:
The molecular formula of sulfur in carbon disulfide is \( S_8 \).