Determine the freezing point (nearest integer) of a 1 mole `lkg^(-1)` aqueous solution of a weak electrolyte that is `7.5%` dissociated into two ions (in `.^(@)C`) [Given `K_(f)` of water is `1.86^(@)C//m`].

Explanation

To determine the freezing point of a 1 mole/kg aqueous solution of a weak electrolyte that is 7.5% dissociated into two ions, we can follow these steps:

### Step 1: Understand the dissociation of the weak electrolyte
The weak electrolyte dissociates into two ions. Let's denote the electrolyte as AB, which dissociates into A⁺ and B⁻. Given that it is 7.5% dissociated, we can express the degree of dissociation (α) as:
\[ \alpha = \frac{7.5}{100} = 0.075 \]

### Step 2: Calculate the van 't Hoff factor (i)
The van 't Hoff factor (i) represents the number of particles the solute breaks into after dissociation. For our electrolyte AB:
- Initially, we have 1 particle (the undissociated AB).
- After dissociation, we have 1 A⁺ and 1 B⁻, which gives us a total of 3 particles.

Thus, the van 't Hoff factor can be calculated as:
\[ i = 1 + \alpha \times 2 = 1 + 0.075 \times 2 = 1 + 0.15 = 1.15 \]

### Step 3: 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:
- \( \Delta T_f \) is the depression in freezing point.
- \( K_f \) is the cryoscopic constant of water, given as \( 1.86 \, ^\circ C/m \).
- \( m \) is the molality of the solution, which is 1 mole/kg.

Substituting the values into the formula:
\[ \Delta T_f = 1.15 \cdot 1.86 \cdot 1 = 2.139 \]

### Step 4: Calculate the new freezing point
The freezing point of pure water is \( 0 \, ^\circ C \). The new freezing point (\( T_f \)) can be calculated as:
\[ T_f = T_{f, \text{pure}} - \Delta T_f = 0 - 2.139 \approx -2.14 \, ^\circ C \]

### Step 5: Round to the nearest integer
Rounding \( -2.14 \, ^\circ C \) to the nearest integer gives:
\[ T_f \approx -2 \, ^\circ C \]

### Final Answer
The freezing point of the solution is approximately **-2 °C**.

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