In coparison to a 0.01 M solution of glucose, the depression in freezing point of a 0.01 M `MgCl_(2)` solution is……

Options

A. the same
B. about twice
C. about three times
D. about six times

Explanation

To solve the problem, we need to determine the depression in freezing point for a 0.01 M solution of magnesium chloride (MgCl₂) in comparison to a 0.01 M solution of glucose. We will use the formula for depression in freezing point, which is a colligative property.

### Step-by-Step Solution:

1. **Understand the Formula for Depression in Freezing Point**:
The formula for depression in freezing point is given by:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where:
- \(\Delta T_f\) = depression in freezing point
- \(i\) = Van't Hoff factor (number of particles the solute dissociates into)
- \(K_f\) = cryoscopic constant (depends on the solvent)
- \(m\) = molality of the solution

2. **Identify the Van't Hoff Factor for Glucose**:
Glucose (C₆H₁₂O₆) does not dissociate in solution; it remains as one particle. Therefore, the Van't Hoff factor (\(i\)) for glucose is:
\[ i_{\text{glucose}} = 1 \]

3. **Identify the Van't Hoff Factor for Magnesium Chloride**:
Magnesium chloride dissociates into one magnesium ion (Mg²⁺) and two chloride ions (Cl⁻). Thus, the dissociation can be represented as:
\[ \text{MgCl}_2 \rightarrow \text{Mg}^{2+} + 2\text{Cl}^- \]
The total number of particles after dissociation is 3 (1 Mg²⁺ + 2 Cl⁻). Therefore, the Van't Hoff factor for magnesium chloride is:
\[ i_{\text{MgCl}_2} = 3 \]

4. **Calculate the Depression in Freezing Point for Both Solutions**:
Since both solutions have the same molality (0.01 M) and are in the same solvent (thus having the same \(K_f\)), we can compare the depressions directly:
- For glucose:
\[ \Delta T_{f, \text{glucose}} = i_{\text{glucose}} \cdot K_f \cdot m = 1 \cdot K_f \cdot 0.01 \]
- For magnesium chloride:
\[ \Delta T_{f, \text{MgCl}_2} = i_{\text{MgCl}_2} \cdot K_f \cdot m = 3 \cdot K_f \cdot 0.01 \]

5. **Compare the Two Values**:
Now, we can compare the two depressions:
\[ \Delta T_{f, \text{MgCl}_2} = 3 \cdot \Delta T_{f, \text{glucose}} \]
This means that the depression in freezing point of the magnesium chloride solution is three times that of the glucose solution.

### Final Answer:
In comparison to a 0.01 M solution of glucose, the depression in freezing point of a 0.01 M MgCl₂ solution is about **three times**.