To solve the problem, we will use Raoult's Law which states that the vapor pressure of a solvent in a solution is proportional to the mole fraction of the solvent. The decrease in vapor pressure (ΔP) can be expressed as:
\[
\Delta P = P_0 \cdot \chi_B
\]
where \(P_0\) is the vapor pressure of the pure solvent and \(\chi_B\) is the mole fraction of the solute.
### Step 1: Calculate the number of moles of solute and solvent for each solution.
**Solution I: Urea**
- Mass of urea = 6 g
- Molecular weight of urea = 60 g/mol
- Moles of urea = \( \frac{6 \, \text{g}}{60 \, \text{g/mol}} = 0.1 \, \text{mol} \)
- Mass of water = 178.2 g
- Molecular weight of water = 18 g/mol
- Moles of water = \( \frac{178.2 \, \text{g}}{18 \, \text{g/mol}} \approx 9.9 \, \text{mol} \)
**Solution II: Glucose**
- Moles of glucose = 0.01 mol (given)
- Mass of water = 179.82 g
- Moles of water = \( \frac{179.82 \, \text{g}}{18 \, \text{g/mol}} \approx 9.99 \, \text{mol} \)
**Solution III: Sodium Carbonate**
- Mass of sodium carbonate = 5.3 g
- Molecular weight of sodium carbonate = 106 g/mol
- Moles of sodium carbonate = \( \frac{5.3 \, \text{g}}{106 \, \text{g/mol}} \approx 0.050 \, \text{mol} \)
- Mass of water = 179.1 g
- Moles of water = \( \frac{179.1 \, \text{g}}{18 \, \text{g/mol}} \approx 9.95 \, \text{mol} \)
### Step 2: Calculate the mole fraction of the solute for each solution.
**Solution I: Urea**
\[
\chi_B = \frac{\text{moles of urea}}{\text{moles of urea} + \text{moles of water}} = \frac{0.1}{0.1 + 9.9} \approx 0.010
\]
**Solution II: Glucose**
\[
\chi_B = \frac{0.01}{0.01 + 9.99} \approx 0.001
\]
**Solution III: Sodium Carbonate**
- Sodium carbonate dissociates into 3 ions (2 Na⁺ and 1 CO₃²⁻), so we consider the van 't Hoff factor \(i = 3\).
\[
\chi_B = \frac{0.050 \times 3}{0.050 \times 3 + 9.95} \approx \frac{0.15}{0.15 + 9.95} \approx 0.015
\]
### Step 3: Determine the decrease in vapor pressure for each solution.
Using Raoult's Law:
- For Urea:
\[
\Delta P = P_0 \cdot \chi_B = 20 \, \text{mm Hg} \cdot 0.010 = 0.2 \, \text{mm Hg}
\]
- For Glucose:
\[
\Delta P = P_0 \cdot \chi_B = 20 \, \text{mm Hg} \cdot 0.001 = 0.02 \, \text{mm Hg}
\]
- For Sodium Carbonate:
\[
\Delta P = P_0 \cdot \chi_B = 20 \, \text{mm Hg} \cdot 0.015 = 0.3 \, \text{mm Hg}
\]
### Step 4: Determine the vapor pressure of each solution.
- Vapor pressure of solution I (Urea):
\[
P = P_0 - \Delta P = 20 - 0.2 = 19.8 \, \text{mm Hg}
\]
- Vapor pressure of solution II (Glucose):
\[
P = P_0 - \Delta P = 20 - 0.02 = 19.98 \, \text{mm Hg}
\]
- Vapor pressure of solution III (Sodium Carbonate):
\[
P = P_0 - \Delta P = 20 - 0.3 = 19.7 \, \text{mm Hg}
\]
### Step 5: Order the vapor pressures from highest to lowest.
- Vapor pressure of Glucose (19.98 mm Hg) > Vapor pressure of Urea (19.8 mm Hg) > Vapor pressure of Sodium Carbonate (19.7 mm Hg)
### Final Answer:
The correct order in which the vapor pressure of solutions increases is:
**III < I < II** or **Sodium Carbonate < Urea < Glucose**
