To solve the problem, we will follow these steps:
### Step 1: Calculate the depression in freezing point (ΔTf)
The depression in freezing point can be calculated using the formula:
\[
\Delta T_f = T_f^{\text{pure}} - T_f^{\text{solution}}
\]
Where:
- \(T_f^{\text{pure}} = 278.6 \, K\) (freezing point of pure benzene)
- \(T_f^{\text{solution}} = 276.98 \, K\)
Calculating ΔTf:
\[
\Delta T_f = 278.6 \, K - 276.98 \, K = 1.62 \, K
\]
### Step 2: Calculate the molality (m) of the solution
Molality is defined as the number of moles of solute per kilogram of solvent.
First, we need to calculate the number of moles of benzoic acid:
- Mass of benzoic acid = 4 g
- Molar mass of benzoic acid (C₇H₆O₂) = \(7 \times 12 + 6 \times 1 + 2 \times 16 = 122 \, g/mol\)
Calculating moles of benzoic acid:
\[
\text{Moles of benzoic acid} = \frac{4 \, g}{122 \, g/mol} = 0.0328 \, mol
\]
Next, we convert the mass of benzene to kilograms:
- Mass of benzene = 50 g = 0.050 kg
Now, we can calculate molality:
\[
m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.0328 \, mol}{0.050 \, kg} = 0.656 \, mol/kg
\]
### Step 3: Use the Van 't Hoff factor (i) equation
The depression of freezing point is related to the Van 't Hoff factor (i) through the equation:
\[
\Delta T_f = i \cdot K_f \cdot m
\]
Where:
- \(K_f = 4.9 \, K \cdot kg \cdot mol^{-1}\)
- \(m = 0.656 \, mol/kg\)
Substituting the values:
\[
1.62 = i \cdot 4.9 \cdot 0.656
\]
Calculating \(i\):
\[
i = \frac{1.62}{4.9 \cdot 0.656} = \frac{1.62}{3.21} \approx 0.504
\]
### Step 4: Calculate the extent of association (α)
Benzoic acid dimerizes in benzene, so if it dimerizes, the value of \(x\) (the number of particles after association) is 2. The degree of association (α) can be calculated using:
\[
\alpha = 1 - \frac{i}{x}
\]
Substituting \(i\) and \(x\):
\[
\alpha = 1 - \frac{0.504}{2} = 1 - 0.252 = 0.748
\]
### Step 5: Convert to percentage
To express the extent of association as a percentage:
\[
\text{Extent of association} = \alpha \times 100 = 0.748 \times 100 \approx 74.8\%
\]
### Final Answer
The extent of association of benzoic acid in benzene is approximately **74.8%**.
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