Which of the following aqueous solutions has (a) the higher boiling point, (b) the higher freezing point, and (c) the lower vapor pressure: \(0.35 \mathrm{~m}\) \(\mathrm{CaCl}_{2}\) or \(0.90 \mathrm{~m}\) urea? Explain. Assume \(\mathrm{CaCl}_{2}\) to undergo complete dissociation.

Boiling point elevation is one of the key colligative properties which implies that a solution has a higher boiling point than the pure solvent. This is caused by the addition of solute particles which disrupt the solvent's normal escape tendency into the gas phase. As a result, more energy (in the form of heat) is required to allow the solvent particles to escape, raising the boiling point.

The degree of boiling point elevation can be calculated using the formula \( \Delta T_b = K_b \times m \) where \( \Delta T_b \) is the boiling point elevation, \( K_b \) is the boiling point elevation constant, and \( m \) is the molality of the solution. It's crucial to consider not just the concentration but also the number of particles derived from a solute, particularly when dealing with electrolytes that dissociate in solution.

Conversely, freezing point depression is the reduction of a solution's freezing point compared to the pure solvent due to the presence of solute particles. These solute particles effectively hinder the formation of a solid structure as the solvent tries to freeze, requiring a lower temperature to achieve the solid phase.

The calculation for freezing point depression follows a similar principle using the formula \( \Delta T_f = K_f \times m \) where \( \Delta T_f \) is the freezing point depression, \( K_f \) is the freezing point depression constant, and \( m \) is the molality of the solution. This property is hugely important in applications such as using salt to melt ice on roads.

Vapor pressure lowering is the phenomenon where the vapor pressure of a solvent above a solution is lower than that of the pure solvent at the same temperature. This happens because solute particles occupy some of the surface area above the liquid, reducing the number of solvent molecules that can evaporate. The extent to which the vapor pressure is lowered depends on the number of solute particles present.

This property can be described with Raoult's law for ideal solutions, which states that the vapor pressure of the solvent in a solution is proportional to the mole fraction of the solvent in the mixture. Vapor pressure lowering is particularly important in understanding the behavior of solutions and their physical properties, such as boiling and freezing points.

Molality (represented by \( m \) ) is a concentration term for solutions that expresses the moles of solute per kilogram of solvent. Unlike molarity, which is affected by changes in temperature due to volume expansion or contraction, molality is temperature-independent since it is based on mass.

Molality is an essential factor in calculating colligative properties as it directly correlates with the extent of boiling point elevation and freezing point depression. It is defined as \( m = \frac{{\text{moles of solute}}}{{\text{kilograms of solvent}}} \) and is crucial for precision in scenarios where temperature fluctuation is a concern.

The dissociation of electrolytes when dissolved in water introduces free ions into the solution. Electrolytes like \( \mathrm{CaCl}_{2} \) dissociate into multiple particles. For instance, calcium chloride dissociates into one calcium ion (\( \mathrm{Ca}^{2+} \) ) and two chloride ions (\( \mathrm{Cl}^{-} \) ).

This dissociation is significant because it increases the number of particles in the solution, intensifying colligative properties effects compared to non-electrolytes. When an electrolyte like \( \mathrm{CaCl}_{2} \) is said to undergo 'complete dissociation,' it means that each unit of the substance contributes more than one particle to the solution, which is vital when calculating colligative properties.