The boiling point of an aqueous solution of a non-volatile solute is `100.15^(@)C`. What is the freezing point of an aqueous solution obtained by dilute the above solution with an equal volume of water. The values of `K_(b)` and `K_(f)` for water are `0.512` and `1.86^(@)C mol^(-1)`:

Options

A. `-0.544^(@)C`
B. `-0.512^(@)C`
C. `-0.272^(@)C`
D. `-1.86^(@)C`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the elevation in boiling point (ΔTb)
The boiling point of the solution is given as 100.15°C, and the boiling point of pure water is 100°C.

\[ \Delta T_b = T_b(\text{solution}) - T_b(\text{solvent}) = 100.15°C - 100°C = 0.15°C \]

### Step 2: Use the elevation in boiling point to find molality (m)
The relationship between elevation in boiling point and molality is given by the formula:

\[ \Delta T_b = K_b \cdot m \]

Where:
- \( K_b = 0.512 \, °C \, \text{mol}^{-1} \)
- \( \Delta T_b = 0.15°C \)

Rearranging the formula to find molality:

\[ m = \frac{\Delta T_b}{K_b} = \frac{0.15°C}{0.512 \, °C \, \text{mol}^{-1}} \approx 0.2929 \, \text{mol/kg} \]

### Step 3: Determine the new molality after dilution
Since we are diluting the solution with an equal volume of water, the new volume becomes \( 2V \). The molality will be halved because the amount of solute remains the same while the volume of the solution doubles.

\[ \text{New molality} = \frac{0.2929 \, \text{mol/kg}}{2} \approx 0.14645 \, \text{mol/kg} \]

### Step 4: Calculate the freezing point depression (ΔTf)
The freezing point depression is given by:

\[ \Delta T_f = K_f \cdot m \]

Where:
- \( K_f = 1.86 \, °C \, \text{mol}^{-1} \)

Substituting the values:

\[ \Delta T_f = 1.86 \, °C \, \text{mol}^{-1} \cdot 0.14645 \, \text{mol/kg} \approx 0.272 \, °C \]

### Step 5: Calculate the freezing point of the solution
The freezing point of pure water is 0°C. The freezing point of the solution can be calculated as:

\[ \text{Freezing Point} = T_f(\text{solvent}) - \Delta T_f = 0°C - 0.272°C \approx -0.272°C \]

### Final Answer:
The freezing point of the diluted solution is approximately **-0.272°C**.

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