A sample of \(\mathrm{HgCl}_{2}\) weighing \(9.41 \mathrm{g}\) is dissolved in \(32.75 \mathrm{g}\) of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\left(K_{\mathrm{b}}=1.20^{\circ} \mathrm{C} / \mathrm{m}\right)\). The boiling point elevation of the solution is \(1.27^{\circ} \mathrm{C} .\) Is \(\mathrm{HgCl}_{2}\) an electrolyte in ethanol? Show your calculations.

Understanding molal concentration is crucial when studying colligative properties such as boiling point elevation. Molal concentration, often represented as 'm', is a measure of the amount of solute in a solvent. It's defined as the number of moles of solute per kilogram of solvent. Unlike molarity, molality does not change with temperature because it's based on mass, making it a reliable unit in various conditions.

When calculating the molal concentration, it's important to convert the mass of the solvent from grams to kilograms to maintain consistency with the molality's definition. Then, use the formula:
\[m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\]
For the given exercise, we would determine the moles of HgCl2 using its molar mass and the given weight, and divide by the mass of ethanol in kilograms. The result is the molal concentration of the solution, a vital step for determining boiling point elevation.

The van't Hoff factor, symbolized by 'i', is key to understanding the effects of solute particles on the boiling point of a solvent. It represents the number of particles a solute splits into or the number of moles of particles in solution per mole of solute dissolved. For non-electrolytes, which do not dissociate into ions, the van't Hoff factor is 1. For electrolytes, this value can be greater because they dissociate into two or more ions.

To find the van't Hoff factor in our solution, we rearrange the boiling point elevation formula:
\[i = \frac{\Delta T_b}{K_b \cdot m}\]
The comparison of the calculated 'i' to the expected theoretical value helps us determine whether a substance behaves as an electrolyte in a particular solvent.

Determining if a substance is an electrolyte is a pivotal step in boiling point elevation problems. An electrolyte, when dissolved in a solvent, dissociates into its constituent ions, which can significantly impact the solution’s colligative properties. To assess whether HgCl2 is an electrolyte in ethanol, we calculate the van't Hoff factor. If the calculated value of 'i' is greater than 1, this implies that the compound dissociates into ions and is considered an electrolyte. In contrast, if 'i' were close to 1, HgCl2 would not dissociate, and would thus be a non-electrolyte.

A solution's properties can dramatically change with the addition of a solute. Factors such as the substance's electrolytic nature, the type of solvent, and the concentration all play a part in this alteration. By understanding solution properties like boiling point elevation, we can infer characteristics of the solute. In the case of HgCl2, if we find through our calculations that the boiling point is higher than expected based on its molal concentration, we can conclude that the solute affected the solvent's properties significantly, potentially indicating its status as an electrolyte.

These concepts together enable us to elucidate complex behaviors of solutions, which is essential in fields like chemistry, engineering, and pharmacy. With the right calculations and understanding, we can predict and manipulate the physical properties of solutions for various applications.