The latent heat of fusion of ice is 80 calories per gram at `0^(@)C`. What is the freezing point of a solution of `KCl` in water containing `7.45` grams of solute 500 grams of water, assuming that the salt is dissociated to the extent of `95%`?

Explanation

To solve the problem, we need to find the freezing point of a solution of KCl in water given the mass of the solute and the mass of the solvent, along with the dissociation factor. Here’s a step-by-step solution:

### Step 1: Determine the Van't Hoff factor (i)
KCl dissociates into two ions: K⁺ and Cl⁻. Since the dissociation is 95%, we can calculate the effective Van't Hoff factor (i).

\[ i = 1 + \alpha(n - 1) \]

Where:
- \( \alpha = 0.95 \) (degree of dissociation)
- \( n = 2 \) (number of particles produced from one formula unit of KCl)

\[ i = 1 + 0.95(2 - 1) = 1 + 0.95 = 1.95 \approx 2 \]

### Step 2: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent.

1. Calculate the number of moles of KCl:
- Molar mass of KCl = 74.5 g/mol
- Mass of KCl = 7.45 g

\[ \text{Moles of KCl} = \frac{\text{mass}}{\text{molar mass}} = \frac{7.45 \, \text{g}}{74.5 \, \text{g/mol}} \approx 0.1 \, \text{mol} \]

2. Convert the mass of water to kilograms:
- Mass of water = 500 g = 0.5 kg

3. Calculate molality:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.1 \, \text{mol}}{0.5 \, \text{kg}} = 0.2 \, \text{mol/kg} \]

### Step 3: Use the freezing point depression formula
The freezing point depression (\( \Delta T_f \)) can be calculated using the formula:

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

Where:
- \( K_f \) for water = 1.865 °C kg/mol
- \( i = 2 \)
- \( m = 0.2 \, \text{mol/kg} \)

Substituting the values:

\[ \Delta T_f = 2 \cdot 1.865 \cdot 0.2 = 0.746 \, °C \approx 0.73 \, °C \]

### Step 4: Calculate the freezing point of the solution
The freezing point of the solution (\( T_f \)) can be calculated as:

\[ T_f = T_f^{\circ} - \Delta T_f \]

Where:
- \( T_f^{\circ} \) (freezing point of pure water) = 0 °C

Substituting the values:

\[ T_f = 0 - 0.73 = -0.73 \, °C \]

### Final Answer
The freezing point of the KCl solution is approximately **-0.73 °C**.

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