Step 1: Understanding the Concept:
Relative Lowering of Vapour Pressure (RLVP) is a colligative property. According to Raoult's law, it is equal to the mole fraction of the solute.
Step 2: Key Formula or Approach:
\[ \frac{P_1^\circ - P_1}{P_1^\circ} = x_2 \]
Step 3: Detailed Explanation:
Let \(w_1\) and \(w_2\) be the masses of solvent and solute, and \(M_1\) and \(M_2\) be their molar masses.
Number of moles of solute, \(n_2 = w_2/M_2\)
Number of moles of solvent, \(n_1 = w_1/M_1\)
For dilute solutions, \(n_2 \ll n_1\), so \(n_1 + n_2 \approx n_1\).
Mole fraction of solute, \(x_2 = \frac{n_2}{n_1 + n_2} \approx \frac{n_2}{n_1} = \frac{w_2/M_2}{w_1/M_1} = \frac{w_2 M_1}{M_2 w_1}\).
Substituting into Raoult's Law:
\[ \frac{\Delta P}{P_1^\circ} = \frac{w_2 M_1}{M_2 w_1} \]
Rearranging for \(M_2\):
\[ M_2 = \frac{w_2 M_1 P_1^\circ}{w_1 \Delta P} \]
Step 4: Final Answer:
The molar mass \(M_2\) is given by \(\frac{w_2 M_1 P_1^\circ}{w_1 \Delta P}\).