A `0.2` molal aqueous solution of a weak acid `HX` is `20%` ionized. The freezing point of the solution is `(k_(f) = 1.86 K kg "mole"^(-1)` for water):

Options

A. `-0.45^(@)C`
B. `-0.9^(@)C`
C. `-0.31^(@)C`
D. `-0.53^(@)C`

Explanation

To solve the problem of finding the freezing point of a 0.2 molal aqueous solution of a weak acid \( HX \) that is 20% ionized, we will follow these steps:

### Step 1: Calculate the degree of ionization
The degree of ionization \( \alpha \) is given as 20%, which can be expressed as:
\[ \alpha = \frac{20}{100} = 0.2 \]

### Step 2: Determine the van 't Hoff factor \( i \)
For the weak acid \( HX \), it ionizes into \( H^+ \) and \( X^- \). This means that for every mole of \( HX \), we get 2 moles of ions. The van 't Hoff factor \( i \) can be calculated using the formula:
\[ i = 1 + (\alpha \cdot (n - 1)) \]
where \( n \) is the number of ions produced. Here, \( n = 2 \) (since \( HX \) produces \( H^+ \) and \( X^- \)):
\[ i = 1 + (0.2 \cdot (2 - 1)) = 1 + 0.2 = 1.2 \]

### Step 3: Calculate the freezing point depression \( \Delta T_f \)
The formula for freezing point depression is given by:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- \( K_f = 1.86 \, \text{K kg mole}^{-1} \) (freezing point depression constant for water)
- \( m = 0.2 \, \text{molal} \)

Substituting the values:
\[ \Delta T_f = 1.2 \cdot 1.86 \cdot 0.2 \]
Calculating this gives:
\[ \Delta T_f = 1.2 \cdot 1.86 \cdot 0.2 = 0.4464 \, \text{K} \]

### Step 4: Calculate the freezing point of the solution
The freezing point of pure water \( T_f \) is 0°C. The freezing point of the solution \( T_f' \) can be calculated using:
\[ T_f' = T_f - \Delta T_f \]
Substituting the values:
\[ T_f' = 0 - 0.4464 = -0.4464 \, \text{°C} \]

### Final Answer
The freezing point of the solution is approximately:
\[ T_f' \approx -0.45 \, \text{°C} \]

### Summary of Steps
1. Calculate the degree of ionization \( \alpha \).
2. Determine the van 't Hoff factor \( i \).
3. Calculate the freezing point depression \( \Delta T_f \).
4. Calculate the freezing point of the solution \( T_f' \).