An aqueous solution containing \(17.5 \mathrm{~g}\) of an unknown molecular (nonelectrolyte) compound in \(100.0 \mathrm{~g}\) of water has a freezing point of \(-1.8^{\circ} \mathrm{C}\). Calculate the molar mass of the unknown compound.

Colligative properties are fundamental aspects of solutions that are determined by the number of particles in a given amount of solvent, irrespective of the nature of the particles.

One of these colligative properties is the freezing point depression, which is the lowering of a solvent's freezing temperature due to the addition of a solute. In the provided exercise, the freezing point of water decreases from its normal 0°C to -1.8°C when the unknown compound is dissolved. This decrease is directly proportional to the molality of the solution, meaning that the more particles dissolved, the larger the lowering effect.

Understanding this concept is crucial for interpreting how the presence of a solute can change the physical properties of the solvent. This pivotal information leads to practical applications in areas such as anti-freeze in car radiators or the salting of roads in winter to prevent ice formation.

Molality is a measure of concentration defined as the number of moles of solute per kilogram of solvent. It is different from molarity, which is the number of moles per liter of the solution. In the freezing point depression exercise, molality plays a key role.

The formula to calculate molality is obtained by rearranging the equation for freezing point depression:
\[\begin{equation} m = \frac{\Delta T_f}{K_f}\end{equation}\]
Where m is the molality, \(\Delta T_f\) is the change in freezing point, and \(K_f\) is the freezing point depression constant. For water, this constant is \(1.86^\circ C/m\), which relates the change in temperature directly to the molality of the solution.

Determining the molar mass of an unknown compound is a process of calculating the mass of one mole of that substance. In the context of our freezing point depression problem, the molar mass is found after calculating the number of moles from the molality.

Once we have the molality, we can calculate the number of moles of the unknown compound present in the solution. Given the mass of the compound and its moles, the molar mass (M) is computed using the equation: \[\begin{equation} M = \frac{\text{mass of compound}}{\text{moles of compound}}\end{equation}\]
In simpler terms, by knowing the mass of the dissolved substance and the decreased freezing point of the solution, we can back-calculate to find out the molar mass of that compound. This method is particularly useful in chemistry for identifying unknown compounds in a mixture.