Which of the following will have the highest freezing point at one atmosphere ?

Options

A. 0.1 M NaCl solution
B. 0.1 M sugar solution
C. 0.1 M `BaCl_2` solution
D. 0.1 M `FeCl_3` solution

Explanation

To determine which of the given solutions has the highest freezing point at one atmosphere, we need to analyze the freezing point depression, which is a colligative property. The freezing point depression (\( \Delta T_F \)) is given by the formula:

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

Where:
- \( i \) is the Van't Hoff factor (the number of particles the solute dissociates into),
- \( K_f \) is the cryoscopic constant (a constant for the solvent, in this case, water),
- \( m \) is the molality of the solution.

Since \( K_f \) is constant for all solutions in water and we are interested in the relative freezing points, we can focus on the product \( i \cdot m \). The solution with the highest value of \( i \cdot m \) will have the greatest depression in freezing point, meaning it will have the lowest freezing point. Conversely, the solution with the lowest value of \( i \cdot m \) will have the least depression and thus the highest freezing point.

### Step-by-Step Solution:

1. **Identify the Van't Hoff factor (\( i \)) and molality (\( m \)) for each solution:**
- **Option 1: NaCl**
- \( i = 2 \) (dissociates into Na\(^+\) and Cl\(^-\))
- \( m = 0.1 \)
- \( i \cdot m = 2 \cdot 0.1 = 0.2 \)

- **Option 2: Sugar (C\(_{12}\)H\(_{22}\)O\(_{11}\))**
- \( i = 1 \) (does not dissociate)
- \( m = 0.1 \)
- \( i \cdot m = 1 \cdot 0.1 = 0.1 \)

- **Option 3: BaCl\(_2\)**
- \( i = 3 \) (dissociates into Ba\(^{2+}\) and 2 Cl\(^-\))
- \( m = 0.1 \)
- \( i \cdot m = 3 \cdot 0.1 = 0.3 \)

- **Option 4: FeCl\(_3\)**
- \( i = 4 \) (dissociates into Fe\(^{3+}\) and 3 Cl\(^-\))
- \( m = 0.1 \)
- \( i \cdot m = 4 \cdot 0.1 = 0.4 \)

2. **Compare the values of \( i \cdot m \):**
- Option 1: 0.2
- Option 2: 0.1
- Option 3: 0.3
- Option 4: 0.4

3. **Determine the solution with the lowest \( i \cdot m \):**
- The lowest value is for Option 2 (sugar) with \( i \cdot m = 0.1 \).

4. **Conclusion:**
- Since Option 2 has the lowest product of \( i \cdot m \), it will have the highest freezing point.

### Final Answer:
The solution with the highest freezing point at one atmosphere is **Option 2 (Sugar)**.