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Raoult's Law

Key Concepts

The escaping tendency of a solvent is measured by its vapor pressure, which is dependent on temperature.
Vapor pressure measures the concentration of solvent molecules in the gas phase.
Assuming the solute is nonvolatile, the only particles in the gas phase are solvent molecules. In a solution, fewer solvent molecules are at the surface compared to the pure solvent, so a smaller proportion of solvent molecules will be in the gas phase and the vapor pressure for the solution is lower than that for the pure solvent.
Raoult's Law states that for an ideal solution the partial vapor pressure of a component in solution is equal to the mole fraction of that component times its vapor pressure when pure:
P_a = X_aP_a^o
P_a = vapor pressure of the solution
P_a^o = vapor pressure of pure solvent
X_a = mole fraction of the solvent
The fractional vapor pressure lowering is equal to the mole fraction of the solute:
X_b = (P_a^o - P_a) ÷ P_a^o
P_a = vapor pressure of the solution
P_a^o = vapor pressure of pure solvent
X_b = mole fraction of the solute
Fractional vapor pressure lowering can be used to calculate molecular mass (formula weight) of a solute.

Example 1: Calculating the Vapor Pressure of a Solvent

1.00g of nonvolatile sulfanilamide, C₆H₈O₂N₂S, is dissolved in 10.0g of acetone, C₃H₆O.
The vapor pressure of pure acetone at the same temperature is 400 mmHg.
Calculate the vapor pressure of the solution.

Calculate moles of solute: n (C₆H₈O₂N₂S) = mass ÷ MM
n(C₆H₈O₂N₂S) = 1.00g ÷ (6 x 12 + 8 x 1 + 2 x 16 + 2 x 14 + 32.1) g/mol
= 1.00 ÷ 172.1 = 0.0058 mol
Calculate moles of solvent: n(C₃H₆O) = mass ÷ MM
n(C₃H₆O) = 10.0g ÷ (3 x 12 + 6 x 1 + 16) g/mol
= 10.0 ÷ 58 = 0.172 mol
Calculate the mole fraction of the solvent: X_solvent = n_solvent ÷ (n_solute + n_solvent)
X_a = n(C₃H₆O) ÷ [n(C₃H₆O) + n(C₆H₈O₂N₂S)]
= 0.172 ÷ [0.172 + 0.0058] = 0.967
Calculate the vapor pressure: P_a = X_aP_a^o
P_a = 0.967 x 400 mmHg = 386.8 mmHg = 387 mmHg

Example 2: Calculating the Molecular Mass (Formula Weight) of a Solute

5.00g of a nonvolatile compound was dissolved in 100g of water at 30^oC.
The vapor pressure of the solution was measured and found to be 31.20 Torr.
The vapor pressure of pure water at 30^oC is 31.82 Torr.
Calculate the molecular mass (formula weight) of the unknown solute.

Calculate the mole fraction of the solute: X_b = (P_a^o - P_a) ÷ P_a^o
X_b = (31.82 - 31.2) ÷ 31.82 = 0.0195
Calculate moles of solvent: n(H₂O) = mass ÷ molecular mass
n(H₂O) = 100g ÷ (2 x 1 + 16)g/mol = 5.556mol
Calculate moles of solute: X_solute = n_solute ÷ (n_solute + n_solvent)
0.0195 = n_solute ÷ (n_solute + 5.556)
0.0195(n_solute + 5.556) = n_solute
0.0195(n_solute) + 0.108 = n_solute
0.108 = n_solute - 0.0195(n_solute)
0.108 = 0.9805(n_solute)
(n_solute) = 0.108 ÷ 0.9805 = 0.11 mol
Calculate molecular mass (formula weight) of solute: MM = mass ÷ moles
MM = 5.00g ÷ 0.11 mol = 45.45 g/mol

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