To determine the freezing point of the solution, we need to follow these steps:
1. Calculate the moles of naphthalene ([tex]$C_{10}H_8$[/tex]):
The molar mass of naphthalene is given as 128.16 g/mol.
[tex]\[
\text{Moles of naphthalene} = \frac{\text{Mass of naphthalene}}{\text{Molar mass of naphthalene}} = \frac{7.29 \text{ g}}{128.16 \text{ g/mol}} = 0.05688 \text{ moles}
\][/tex]
2. Convert the mass of benzene ([tex]$C_6H_6$[/tex]) to kilograms:
The mass of benzene is given as 166 g.
[tex]\[
\text{Mass of benzene in kg} = \frac{166 \text{ g}}{1000} = 0.166 \text{ kg}
\][/tex]
3. Calculate the molality of the solution:
Molality is defined as the number of moles of solute per kilogram of solvent.
[tex]\[
\text{Molality} = \frac{\text{Moles of naphthalene}}{\text{Mass of benzene in kg}} = \frac{0.05688 \text{ moles}}{0.166 \text{ kg}} = 0.34266 \text{ mol/kg}
\][/tex]
4. Calculate the freezing point depression ([tex]$\Delta T_f$[/tex]):
The freezing point depression ([tex]$\Delta T_f$[/tex]) can be calculated using the formula:
[tex]\[
\Delta T_f = K_f \times \text{molality}
\][/tex]
where [tex]\(K_f\)[/tex] for benzene is 5.07 °C/m.
[tex]\[
\Delta T_f = 5.07 \text{ °C/m} \times 0.34266 \text{ mol/kg} = 1.73730 \text{ °C}
\][/tex]
5. Calculate the new freezing point of the solution:
The freezing point of pure benzene is given as 5.49 °C.
[tex]\[
\text{Freezing point of the solution} = \text{Freezing point of pure benzene} - \Delta T_f = 5.49 \text{ °C} - 1.73730 \text{ °C} = 3.75 \text{ °C}
\][/tex]
Therefore, the freezing point of the solution, rounded to two decimal places, is:
[tex]\[
\boxed{3.75 \text{ °C}}
\][/tex]
