In which case depression freezing point is equal to cryscopic for water:

Options

A. `6%`by mass of urea in aqueous solution
B. `100`gof sucrose in `100`mL solution
C. `9`g of urea in `59`g aqueous solution
D. `1`M KCl solution

Explanation

To solve the question regarding when the depression in freezing point (\( \Delta T_f \)) is equal to the cryoscopic constant (\( K_f \)) for water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**:
The formula for depression in freezing point is given by:
\[ \Delta T_f = K_f \cdot m \]
where:
- \( \Delta T_f \) = depression in freezing point
- \( K_f \) = cryoscopic constant for the solvent (water in this case)
- \( m \) = molality of the solution

2. **Setting the Condition**:
We need to find the condition under which:
\[ \Delta T_f = K_f \]
This implies:
\[ K_f \cdot m = K_f \]
Dividing both sides by \( K_f \) (assuming \( K_f \neq 0 \)):
\[ m = 1 \]
Thus, the molality of the solution must be 1 molal.

3. **Understanding Molality**:
Molality (\( m \)) is defined as:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]
For a 1 molal solution, this means we have 1 mole of solute in 1 kg of solvent (water).

4. **Calculating for Given Options**:
Now, we can analyze the options provided in the question to check which one corresponds to a 1 molal solution.

- **Option A**: 6% by mass of urea
- **Option B**: Another percentage solution (not specified)
- **Option C**: 9 g of glucose in 59 g of aqueous solution
- **Option D**: 1 molal KCl solution

For option C, we need to check if it gives us a molality of 1.

5. **Calculating Option C**:
- **Weight of solute (glucose)** = 9 g
- **Weight of solvent (water)** = 59 g - 9 g = 50 g = 0.050 kg
- **Moles of glucose** = \( \frac{9 \text{ g}}{180 \text{ g/mol}} = 0.05 \text{ moles} \)
- **Molality** = \( \frac{0.05 \text{ moles}}{0.050 \text{ kg}} = 1 \text{ molal} \)

Thus, option C is indeed a 1 molal solution.

6. **Conclusion**:
Therefore, the case in which the depression in freezing point is equal to the cryoscopic constant for water is when the solution is 1 molal.

### Final Answer:
The depression in freezing point is equal to the cryoscopic constant for water in the case of a 1 molal solution.