The vapour pressure of water at T(K) is 20 mm Hg. The following solutions are prepared at T(K)
I. 6 g of urea (molecular weight = 60) is dissolved in 178.2 g of water.
II. 0.01 mol of glucose is dissolved in 179.82 g of water.
III. 5.3 g of `Na_(2)CO_(3)` (molecular weight = 106) is dissolved in 179.1 g of water.
Identify the correct order in which the vapour pressure of solutions increases

Options

A. III lt I lt II
B. II lt III lt I
C. I lt II lt III
D. I lt III lt II

Answer

To solve the problem of identifying the correct order in which the vapor pressure of the solutions increases, we will calculate the relative lowering of vapor pressure (RLVP) for each solution. The RLVP is directly related to the concentration of the solute in the solution.

### Step 1: Calculate the molality of each solution.

1. **For Urea (Solution I)**:
- Mass of urea = 6 g
- Molecular weight of urea = 60 g/mol
- Mass of water = 178.2 g
- Moles of urea = \( \frac{6 \text{ g}}{60 \text{ g/mol}} = 0.1 \text{ mol} \)
- Molality (m) = \( \frac{0.1 \text{ mol}}{0.1782 \text{ kg}} = \frac{0.1}{0.1782} \approx 0.56 \text{ mol/kg} \)

2. **For Glucose (Solution II)**:
- Moles of glucose = 0.01 mol
- Mass of water = 179.82 g
- Molality (m) = \( \frac{0.01 \text{ mol}}{0.17982 \text{ kg}} = \frac{0.01}{0.17982} \approx 0.0556 \text{ mol/kg} \)

3. **For Sodium Carbonate (Solution III)**:
- Mass of Na2CO3 = 5.3 g
- Molecular weight of Na2CO3 = 106 g/mol
- Moles of Na2CO3 = \( \frac{5.3 \text{ g}}{106 \text{ g/mol}} \approx 0.050 \text{ mol} \)
- Mass of water = 179.1 g
- Molality (m) = \( \frac{0.050 \text{ mol}}{0.1791 \text{ kg}} = \frac{0.050}{0.1791} \approx 0.279 \text{ mol/kg} \)

### Step 2: Calculate the van 't Hoff factor (i) for each solute.

1. **For Urea**: Urea is a non-electrolyte, so \( i = 1 \).
2. **For Glucose**: Glucose is also a non-electrolyte, so \( i = 1 \).
3. **For Sodium Carbonate**: Sodium carbonate dissociates into 2 Na⁺ and 1 CO₃²⁻, so \( i = 3 \).

### Step 3: Calculate the RLVP for each solution.

The RLVP is proportional to \( i \cdot m \).

1. **For Urea**:
- \( RLVP \propto i \cdot m = 1 \cdot 0.56 = 0.56 \)

2. **For Glucose**:
- \( RLVP \propto i \cdot m = 1 \cdot 0.0556 = 0.0556 \)

3. **For Sodium Carbonate**:
- \( RLVP \propto i \cdot m = 3 \cdot 0.279 = 0.837 \)

### Step 4: Determine the order of vapor pressure.

The greater the RLVP, the lower the vapor pressure of the solution. Therefore, the order of decreasing vapor pressure (or increasing RLVP) is:

- Glucose (lowest RLVP, highest vapor pressure)
- Urea (medium RLVP)
- Sodium Carbonate (highest RLVP, lowest vapor pressure)

### Final Order of Increasing Vapor Pressure:

**Correct Order**: Glucose < Urea < Sodium Carbonate