The ratio of freezing point depression values of `0.01 M` solutions of urea, common salt, and `Na_(2)SO_(4)` are

Options

A. `1:1:1`
B. `1:2:1`
C. `1:2:3`
D. `2:2:3`

Explanation

To find the ratio of freezing point depression values of `0.01 M` solutions of urea, common salt (NaCl), and sodium sulfate (Na₂SO₄), we can follow these steps:

### Step 1: Understand the Concept of Freezing Point Depression
Freezing point depression (ΔTf) is a colligative property that depends on the number of solute particles in a solution. The formula for freezing point depression is given by:

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

where:
- \( \Delta T_f \) = freezing point depression
- \( k_f \) = freezing point depression constant of the solvent
- \( m \) = molality of the solution
- \( i \) = van 't Hoff factor (number of particles the solute dissociates into)

### Step 2: Identify the van 't Hoff Factor (i) for Each Solute
1. **Urea (NH₂CONH₂)**: Urea is a non-electrolyte and does not dissociate in solution. Thus, \( i = 1 \).
2. **Common Salt (NaCl)**: Sodium chloride dissociates into two ions: Na⁺ and Cl⁻. Thus, \( i = 2 \).
3. **Sodium Sulfate (Na₂SO₄)**: Sodium sulfate dissociates into three ions: 2 Na⁺ and SO₄²⁻. Thus, \( i = 3 \).

### Step 3: Calculate the Freezing Point Depression for Each Solution
Since all solutions have the same molarity (0.01 M), we can express the freezing point depression for each solute as follows:

1. **For Urea**:
\[ \Delta T_f (\text{urea}) = k_f \cdot 0.01 \cdot 1 = k_f \cdot 0.01 \]

2. **For NaCl**:
\[ \Delta T_f (\text{NaCl}) = k_f \cdot 0.01 \cdot 2 = 2k_f \cdot 0.01 \]

3. **For Na₂SO₄**:
\[ \Delta T_f (\text{Na₂SO₄}) = k_f \cdot 0.01 \cdot 3 = 3k_f \cdot 0.01 \]

### Step 4: Write the Ratios of Freezing Point Depression
Now we can write the ratios of the freezing point depression values:

\[ \text{Ratio} = \Delta T_f (\text{urea}) : \Delta T_f (\text{NaCl}) : \Delta T_f (\text{Na₂SO₄}) = (k_f \cdot 0.01) : (2k_f \cdot 0.01) : (3k_f \cdot 0.01) \]

### Step 5: Simplify the Ratio
Since \( k_f \cdot 0.01 \) is common in all terms, we can cancel it out:

\[ \text{Ratio} = 1 : 2 : 3 \]

### Conclusion
Thus, the ratio of freezing point depression values of `0.01 M` solutions of urea, common salt, and sodium sulfate is:

\[ \text{Final Ratio} = 1 : 2 : 3 \]