When 20 g of naphthoic acid `(C_(11)H_(8)O_(2))` is dissolved in 50 g of benzene `(K_(1)=1.72Kkgmol^(-1))`. Van't Hoff factor (i) is 0.5. What is the depression in freezing point in K.

Answer

To find the depression in freezing point (ΔTf) when 20 g of naphthoic acid (C₁₁H₈O₂) is dissolved in 50 g of benzene, we will use the formula:

\[ \Delta T_f = i \cdot K_f \cdot m \]

Where:
- \( \Delta T_f \) = depression in freezing point
- \( i \) = Van't Hoff factor
- \( K_f \) = molal depression constant of the solvent (benzene)
- \( m \) = molality of the solution

### Step 1: Calculate the number of moles of naphthoic acid

First, we need to calculate the number of moles of naphthoic acid. The molecular weight of naphthoic acid (C₁₁H₈O₂) is calculated as follows:

- Carbon (C): 12.01 g/mol × 11 = 132.11 g/mol
- Hydrogen (H): 1.008 g/mol × 8 = 8.064 g/mol
- Oxygen (O): 16.00 g/mol × 2 = 32.00 g/mol

Adding these together:

\[ \text{Molecular weight of C}_{11}\text{H}_{8}\text{O}_{2} = 132.11 + 8.064 + 32.00 = 172.174 \text{ g/mol} \]

Now, calculate the number of moles of naphthoic acid:

\[ \text{Number of moles} = \frac{\text{mass}}{\text{molecular weight}} = \frac{20 \text{ g}}{172.174 \text{ g/mol}} \approx 0.116 \text{ moles} \]

### Step 2: Calculate the molality (m)

Molality (m) is defined as the number of moles of solute per kilogram of solvent. The mass of the solvent (benzene) is given as 50 g, which is 0.050 kg.

\[ m = \frac{\text{number of moles of solute}}{\text{mass of solvent in kg}} = \frac{0.116 \text{ moles}}{0.050 \text{ kg}} = 2.32 \text{ mol/kg} \]

### Step 3: Substitute values into the depression in freezing point formula

Now we can substitute the values into the formula for ΔTf:

Given:
- \( i = 0.5 \)
- \( K_f = 1.72 \text{ K kg/mol} \)
- \( m = 2.32 \text{ mol/kg} \)

\[ \Delta T_f = i \cdot K_f \cdot m = 0.5 \cdot 1.72 \cdot 2.32 \]

Calculating this:

\[ \Delta T_f = 0.5 \cdot 1.72 \cdot 2.32 \approx 1.99 \text{ K} \]

### Final Answer

The depression in freezing point (ΔTf) is approximately **1.99 K**.

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