Arrange these aqueous solutions in order of decreasing freezing point and explain your reasons: (a) \(0.50 \mathrm{~m}\) \(\mathrm{HCl},\) (b) \(0.50 \mathrm{~m}\) glucose, (c) \(0.50 \mathrm{~m}\) acetic acid.

Colligative properties are characteristics of solutions that depend on the number of dissolved particles in solution, but not on the nature of those particles. This is a central aspect of chemistry that impacts boiling points, freezing points, vapor pressures, and osmotic pressures of solutions.

One of the most commonly studied colligative properties is freezing point depression, which is the process where the freezing point of a liquid is lowered by adding another compound to it. Why does this happen? It's because when a solute is added to a solvent, it disrupts the formation of the crystal lattice necessary for freezing. The presence of solute particles makes it harder for the solvent molecules to get into the right position to form a solid, hence a lower temperature is needed to achieve solidification.

Freezing point depression is directly proportional to the molality of the solution and the number of particles the solute provides once dissolved. This is where the concept of van't Hoff factor comes into play – it represents the number of particles a compound splits into in solution. For example, an ionic compound like HCl will dissociate completely in water into its ions, thus it has a higher van't Hoff factor compared to covalent compounds that do not dissociate (like glucose).

Molality is a measure of the concentration of a solution defined as the number of moles of solute per kilogram of solvent. It is represented with the unit 'm' and it is particularly useful because, unlike molarity, molality doesn't change with temperature since mass remains constant regardless of temperature fluctuations.

When discussing freezing point depression, molality is the preferred unit of concentration because it directly relates to the number of solute particles in the solution, which is essential for determining the extent of the freezing point change. If a solution has a higher molality, that means it contains more solute particles in the same amount of solvent, which generally leads to a greater freezing point depression.

In the exercise mentioned, all solutions have the same molality (\(0.50 \text{ m}\)). However, the freezing points differ because of their disposition to release different numbers of particles into the solution, which in turn affects the freezing point depression experienced by each solution.

Understanding the behavior of ionic and covalent solutes in solutions is vital when examining their effect on colligative properties. Ionic compounds, such as HCl, dissociate into ions when dissolved in water. This complete dissociation increases the amount of particles in solution, greatly affecting the freezing point when compared to covalent compounds.

Covalent compounds, like glucose and acetic acid, do not dissociate into ions in a solution. Glucose, which is a non-electrolyte, does not dissociate at all and therefore contributes to the solution it's one molecule per formula unit. Acetic acid, on the other hand, being a weak electrolyte, dissociates only partially, resulting in fewer particles compared to HCl but more than glucose.

In the given exercise, the ranking of freezing points from lowest to highest would indeed be determined by the number and type of particles each solute contributes. HCl, being an ionic solute, has the strongest effect on freezing point depression, followed by the weakly dissociating acetic acid, and finally glucose which causes the least freezing point depression because it does not dissociate into ions at all.