The freezing point of cyclohexane is `6.5^(@)C`. A solution of 0.65g of naphthalene in 19.2g of cyclohexane froze at `1.2^(@)C`. What is the molecular mass of naphthalene? The cryoscopic constant for cyclohexane is 20.1 K`"mol"^(-1)`kg.

Explanation

To find the molecular mass of naphthalene from the given data, we will follow these steps:

### Step 1: Calculate the change in freezing point (ΔTf)
The freezing point of pure cyclohexane is given as 6.5°C, and the freezing point of the solution is 1.2°C.

\[ \Delta T_f = T_f(\text{pure}) - T_f(\text{solution}) = 6.5°C - 1.2°C = 5.3°C \]

### Step 2: Use the formula for freezing point depression
The formula for freezing point depression is given by:

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

Where:
- \( \Delta T_f \) is the change in freezing point.
- \( K_f \) is the cryoscopic constant (20.1 K·mol⁻¹·kg).
- \( m \) is the molality of the solution.

### Step 3: Rearranging the formula to find molality (m)
We can rearrange the formula to solve for molality:

\[ m = \frac{\Delta T_f}{K_f} \]

Substituting the values:

\[ m = \frac{5.3°C}{20.1 \, \text{K·mol}^{-1}\text{kg}} \approx 0.2637 \, \text{mol/kg} \]

### Step 4: Calculate the number of moles of naphthalene
Molality (m) is defined as moles of solute per kilogram of solvent. We have 19.2 g of cyclohexane, which is equivalent to 0.0192 kg.

Using the formula for molality:

\[ m = \frac{\text{moles of naphthalene}}{\text{mass of cyclohexane in kg}} \implies \text{moles of naphthalene} = m \cdot \text{mass of cyclohexane in kg} \]

Substituting the values:

\[ \text{moles of naphthalene} = 0.2637 \, \text{mol/kg} \cdot 0.0192 \, \text{kg} \approx 0.00506 \, \text{mol} \]

### Step 5: Calculate the molecular mass of naphthalene
We know the mass of naphthalene is 0.65 g. To find the molecular mass (M), we use the formula:

\[ M = \frac{\text{mass of naphthalene}}{\text{moles of naphthalene}} \]

Substituting the values:

\[ M = \frac{0.65 \, \text{g}}{0.00506 \, \text{mol}} \approx 128.4 \, \text{g/mol} \]

### Final Answer
The molecular mass of naphthalene is approximately **128.4 g/mol**.

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