A salt `MA_(2)` ionises as
`MA_(2)hArr M^(2+)+2A^(-)`
It was found that a given solution of the salt had the same freezing point as solution of glucose of twice the molality. The apparent degree of ionization of the salt is

Options

A. 0.25
B. 0.33
C. 0.5
D. 0.67

Answer

To solve the problem, we need to determine the apparent degree of ionization of the salt \( MA_2 \) given that its freezing point depression is the same as that of a glucose solution with twice the molality.

### Step-by-Step Solution:

1. **Understand the Ionization of the Salt:**
The salt \( MA_2 \) ionizes according to the equation:
\[ MA_2 \rightleftharpoons M^{2+} + 2A^{-} \]
This means that 1 mole of \( MA_2 \) produces 3 moles of ions (1 mole of \( M^{2+} \) and 2 moles of \( A^{-} \)).

2. **Identify the Van't Hoff Factor (i):**
The Van't Hoff factor \( i \) is defined as the ratio of the number of particles in solution after dissociation to the number of formula units initially dissolved. For \( MA_2 \), if it completely dissociates, \( i \) would be 3 (since it produces 3 ions).

3. **Freezing Point Depression:**
The freezing point depression (\( \Delta T_f \)) can be expressed as:
\[ \Delta T_f = i \cdot K_f \cdot m \]
where \( K_f \) is the cryoscopic constant and \( m \) is the molality of the solution.

4. **Relate the Freezing Point Depression of Salt and Glucose:**
According to the problem, the freezing point depression of the salt solution is equal to that of a glucose solution with twice the molality:
\[ \Delta T_f (\text{salt}) = \Delta T_f (\text{glucose}) \]
Let the molality of the salt be \( m \). Then, the molality of glucose is \( 2m \).

5. **Set Up the Equation:**
For glucose, since it does not ionize, \( i = 1 \):
\[ \Delta T_f (\text{glucose}) = K_f \cdot (2m) \]
For the salt:
\[ \Delta T_f (\text{salt}) = i \cdot K_f \cdot m \]
Setting these equal gives:
\[ i \cdot K_f \cdot m = K_f \cdot (2m) \]

6. **Simplify the Equation:**
Cancel \( K_f \) and \( m \) from both sides (assuming \( m \neq 0 \)):
\[ i = 2 \]

7. **Determine the Number of Particles (n):**
The salt dissociates into 3 particles (1 \( M^{2+} \) and 2 \( A^{-} \)), so \( n = 3 \).

8. **Calculate the Apparent Degree of Ionization (α):**
The degree of ionization \( \alpha \) can be calculated using the formula:
\[ \alpha = \frac{i - 1}{n - 1} \]
Substituting the values:
\[ \alpha = \frac{2 - 1}{3 - 1} = \frac{1}{2} = 0.5 \]

### Final Answer:
The apparent degree of ionization of the salt \( MA_2 \) is \( \alpha = 0.5 \).