Mole fraction of a non-electrolyte in aqueous solution is `0.07`. If `K_(f)` is `1.86^(@) "mol"^(-1)kg`, depression if `f.p.,DeltaT_(f),` is:

Options

A. `0.26^(@)`
B. `1.86^(@)`
C. `0.13^(@)`
D. `7.78^(@)`

Explanation

To solve the problem, we need to calculate the depression in freezing point (ΔTf) of an aqueous solution given the mole fraction of a non-electrolyte and the cryoscopic constant (Kf).

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Mole fraction of non-electrolyte (x2) = 0.07
- Cryoscopic constant (Kf) = 1.86 °C kg/mol

2. **Calculate the Mole Fraction of the Solvent**:
- The mole fraction of the solvent (water, x1) can be calculated using the formula:
\[ x_1 = 1 - x_2 \]
- Substitute the value of x2:
\[ x_1 = 1 - 0.07 = 0.93 \]

3. **Determine the Molarity of the Solution**:
- The molarity (m) of the solution can be calculated using the formula:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]
- To find the moles of solute, we can use the mole fraction:
\[ \text{moles of solute} = x_2 \times \text{total moles} \]
- Assuming 1 kg of solvent (water), the mass of the solvent in grams is 1000 g. Since the molar mass of water is approximately 18 g/mol, the moles of solvent can be calculated as:
\[ \text{moles of solvent} = \frac{1000 \text{ g}}{18 \text{ g/mol}} \approx 55.56 \text{ mol} \]
- The moles of solute (non-electrolyte) can be calculated as:
\[ \text{moles of solute} = 0.07 \times (0.07 + 0.93) \approx 0.07 \text{ mol} \]
- Now, we can calculate the molarity:
\[ m = \frac{0.07 \text{ mol}}{1 \text{ kg}} = 0.07 \text{ mol/kg} \]

4. **Calculate the Depression in Freezing Point (ΔTf)**:
- The formula for depression in freezing point is:
\[ \Delta T_f = i \cdot m \cdot K_f \]
- Since the solute is a non-electrolyte, the van 't Hoff factor (i) = 1.
- Substitute the values into the formula:
\[ \Delta T_f = 1 \cdot 0.07 \cdot 1.86 = 0.1302 \text{ °C} \]

5. **Final Calculation**:
- The final value of ΔTf is approximately:
\[ \Delta T_f \approx 0.1302 \text{ °C} \]

### Conclusion:
The depression in freezing point (ΔTf) is approximately **0.1302 °C**.