Calculate the freezing point of solution when 1.9 g of `MgCl_(2)` (M = `"95 g mol"^(-1)`) was dissolved in 50 g of water, assuming `MgCl_(2)` undergoes complete ionization (`K_(f)" for water = 1.86 K kg mol"^(-1)`)

Explanation

To calculate the freezing point of a solution when 1.9 g of MgCl₂ is dissolved in 50 g of water, we will follow these steps:

### Step 1: Calculate the number of moles of MgCl₂
To find the number of moles, we use the formula:
\[ \text{Number of moles} = \frac{\text{mass of solute (g)}}{\text{molar mass of solute (g/mol)}} \]
Given:
- Mass of MgCl₂ = 1.9 g
- Molar mass of MgCl₂ = 95 g/mol

Calculating the number of moles:
\[ \text{Number of moles of MgCl₂} = \frac{1.9 \, \text{g}}{95 \, \text{g/mol}} = 0.02 \, \text{mol} \]

### Step 2: Calculate the molality of the solution
Molality (m) is defined as:
\[ m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]
Given:
- Mass of water (solvent) = 50 g = 0.050 kg

Calculating molality:
\[ m = \frac{0.02 \, \text{mol}}{0.050 \, \text{kg}} = 0.4 \, \text{mol/kg} \]

### Step 3: Determine the van 't Hoff factor (i)
MgCl₂ dissociates into 3 ions:
\[ \text{MgCl₂} \rightarrow \text{Mg}^{2+} + 2 \text{Cl}^- \]
Thus, the van 't Hoff factor \(i\) is:
\[ i = 3 \]

### Step 4: Calculate the depression in freezing point (ΔTf)
The depression in freezing point is given by:
\[ \Delta T_f = i \cdot K_f \cdot m \]
Where:
- \(K_f\) for water = 1.86 K kg/mol

Calculating ΔTf:
\[ \Delta T_f = 3 \cdot 1.86 \, \text{K kg/mol} \cdot 0.4 \, \text{mol/kg} = 2.232 \, \text{K} \]

### Step 5: Calculate the freezing point of the solution
The freezing point of pure water is 0 °C (273 K). The freezing point of the solution (Tf) can be calculated as:
\[ T_f = T_f^0 - \Delta T_f \]
Where \(T_f^0\) is the freezing point of pure solvent (water):
\[ T_f = 273 \, \text{K} - 2.232 \, \text{K} = 270.768 \, \text{K} \]

### Final Answer
The freezing point of the solution is approximately:
\[ T_f \approx 270.77 \, \text{K} \]