A solution of a non-volatile solute in water freezes at `-0.30^(@)C`. The vapour pressure of pure water at `298 K` is `23.51 mm Hg` and `K_(f)` for water is `1.86 degree//molal`. Calculate the vapour pressure of this solution at `298 K`.

Explanation

To calculate the vapor pressure of the solution at 298 K, we can follow these steps:

### Step 1: Calculate the change in freezing point (ΔTf)
Given that the freezing point of the solution is -0.30 °C, we can determine the change in freezing point as follows:
\[ \Delta T_f = T_f^{\text{pure}} - T_f^{\text{solution}} = 0 °C - (-0.30 °C) = 0.30 °C \]

### Step 2: Use the freezing point depression formula
The freezing point depression can be calculated using the formula:
\[ \Delta T_f = i \cdot K_f \cdot m \]
Where:
- \(i\) = van 't Hoff factor (for a non-volatile solute, \(i = 1\))
- \(K_f\) = freezing point depression constant for water = 1.86 °C/molal
- \(m\) = molality of the solution

Substituting the known values:
\[ 0.30 = 1 \cdot 1.86 \cdot m \]
Solving for \(m\):
\[ m = \frac{0.30}{1.86} \approx 0.1613 \text{ molal} \]

### Step 3: Calculate the vapor pressure lowering (ΔP)
Using Raoult's Law, the vapor pressure lowering can be calculated as:
\[ \Delta P = P^0 \cdot X_s \]
Where:
- \(P^0\) = vapor pressure of pure solvent (water) = 23.51 mm Hg
- \(X_s\) = mole fraction of the solute

First, we need to find the mole fraction of the solvent (water):
\[ X_w = \frac{m_w}{m_w + m_s} \]
Assuming 1 kg of water (which is approximately 55.5 moles of water):
\[ m_w = 55.5 \text{ moles} \]
The number of moles of solute (\(m_s\)) can be calculated from molality:
\[ m_s = m \cdot \text{mass of solvent (in kg)} = 0.1613 \cdot 1 = 0.1613 \text{ moles} \]

Now we can calculate the mole fraction of water:
\[ X_w = \frac{55.5}{55.5 + 0.1613} \approx \frac{55.5}{55.6613} \approx 0.9971 \]

### Step 4: Calculate the vapor pressure of the solution
Using Raoult's Law:
\[ P = P^0 \cdot X_w \]
Substituting the values:
\[ P = 23.51 \cdot 0.9971 \approx 23.40 \text{ mm Hg} \]

### Final Answer
The vapor pressure of the solution at 298 K is approximately **23.40 mm Hg**.

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