Solution with highest freezing point is

Options

A. 0.1 M urea
B. 0.1 M `Ba_(3)(PO_(4))_(2)`
C. `0.1 M K_(2)SO_(4)`
D. 0.1 NaCl

Answer

To determine which solution has the highest freezing point among the given options, we will use the concept of freezing point depression. The freezing point depression can be calculated using the formula:

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

Where:
- \(\Delta T_f\) = depression in freezing point
- \(i\) = Van't Hoff factor (number of particles the solute dissociates into)
- \(K_f\) = molal freezing point depression constant (which is the same for all solutions in this case)
- \(m\) = molarity of the solution

### Step 1: Identify the Van't Hoff factor for each solute

1. **Urea (0.1 M)**: Urea does not dissociate in solution, so \(i = 1\).
2. **Barium Phosphate (0.1 M)**: Barium phosphate dissociates into 1 barium ion (\(Ba^{2+}\)) and 2 phosphate ions (\(PO_4^{3-}\)), giving a total of 5 ions. Thus, \(i = 5\).
3. **Potassium Sulfate (0.1 M)**: Potassium sulfate dissociates into 2 potassium ions (\(K^+\)) and 1 sulfate ion (\(SO_4^{2-}\)), giving a total of 3 ions. Thus, \(i = 3\).
4. **Sodium Chloride (0.1 M)**: Sodium chloride dissociates into 1 sodium ion (\(Na^+\)) and 1 chloride ion (\(Cl^-\)), giving a total of 2 ions. Thus, \(i = 2\).

### Step 2: Calculate the freezing point depression for each solution

Since \(K_f\) and \(m\) are constant for all solutions, we can compare the values of \(i\) directly to determine the freezing point depression.

- **Urea**: \(\Delta T_f = 1 \cdot K_f \cdot 0.1\)
- **Barium Phosphate**: \(\Delta T_f = 5 \cdot K_f \cdot 0.1\)
- **Potassium Sulfate**: \(\Delta T_f = 3 \cdot K_f \cdot 0.1\)
- **Sodium Chloride**: \(\Delta T_f = 2 \cdot K_f \cdot 0.1\)

### Step 3: Determine which solution has the highest freezing point

The freezing point of the solution is given by:

\[ T_f = T_f^0 - \Delta T_f \]

Where \(T_f^0\) is the freezing point of the pure solvent. A higher \(\Delta T_f\) means a lower freezing point. Therefore, to find the solution with the highest freezing point, we look for the solution with the smallest \(\Delta T_f\).

- Urea: \(\Delta T_f\) is the smallest (1)
- Barium Phosphate: \(\Delta T_f\) is the largest (5)
- Potassium Sulfate: \(\Delta T_f\) is 3
- Sodium Chloride: \(\Delta T_f\) is 2

### Conclusion

Since urea has the lowest Van't Hoff factor and consequently the smallest freezing point depression, it will have the highest freezing point among the given solutions.

**Final Answer: The solution with the highest freezing point is 0.1 M Urea.**

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