A \(0.2\) molal aqueous solution of a weak acid (HX) is \(20 \%\) ionized. The freezing point of this solution is (Given \(\mathrm{K}_{\mathrm{f}}=1.86^{\circ} \mathrm{C} \mathrm{kg} \mathrm{mol}^{-1}\) for water \()\) (a) \(-0.45^{\circ} \mathrm{C}\) (b) \(-0.90^{\circ} \mathrm{C}\) (c) \(-0.21^{\circ} \mathrm{C}\) (d) \(-0.43^{\circ} \mathrm{C}\)

The van't Hoff factor, denoted as 'i', is a dimensionless quantity representing the number of particles a compound forms in solution. It's particularly important in colligative properties calculations, such as boiling point elevation and freezing point depression. For non-electrolytes, which do not dissociate in solution, the value of 'i' is typically 1, as one molecule remains as one particle in the solution.

In the case of electrolytes, such as acids, bases, and salts, 'i' can be greater than 1, reflecting the dissociation into multiple ions. However, the degree of dissociation depends on the strength of the electrolyte. Strong electrolytes like NaCl, dissociate completely, typically giving an 'i' equal to the number of ions per formula unit. On the other hand, weak electrolytes like our weak acid HX in the problem, do not fully dissociate in solution.

For a weak acid that is partially ionized, 'i' accounts for both the ionized and unionized portions. As shown in the problem, with 20% ionization, we consider that for every mole of HX, 0.20 mole dissociates into H+ and X-, while 0.80 mole remains as undissociated HX. Therefore, the calculation of 'i' is the sum of these parts resulting in 1.4. This reflects that each mole of our weak acid in solution on average results in 1.4 moles of particles.

Molality, denoted as 'm', is a measure of the concentration of a solute in a solution. Specifically, it's defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which measures moles of solute per liter of solution, molality is temperature-independent because it's based on mass rather than volume, and mass does not change with temperature.

When calculating freezing point depression or boiling point elevation, molality is used since it provides a concentration measure that remains constant under temperature changes that typically occur during such processes. In our exercise, the molality of a 0.2 molal solution indicates that there are 0.2 moles of the weak acid HX per kilogram of water. It's crucial for these calculations, as it's one of the variables in the formula used to determine the change in freezing point: \(\Delta{T}_f = i \cdot K_f \cdot m\), where \(K_f\) is the freezing point depression constant.

Weak acid ionization refers to the partial dissociation of weak acids in water. Unlike strong acids that dissociate completely in aqueous solutions, weak acids only dissociate partially. This means that in a solution of a weak acid, there exists an equilibrium between the un-ionized acid and its constituent ions.

Ionization is often expressed as a percentage that describes the proportion of the original acid that has been converted to ions. In the context of the given problem, a 20% ionization means that for every 1 mole of weak acid HX in solution, only 0.2 moles have dissociated into H+ ions and X- ions. The remaining 0.8 moles exist in the undissociated form, HX.

The extent of ionization affects the van't Hoff factor, which in turn influences the freezing point depression of the solution. A higher degree of ionization means more particles are present in the solution which can result in a greater decrease in the freezing point, as seen through the formula involving the van't Hoff factor. For weak acids, the extent of ionization is also influenced by factors such as the pH of the solution and the presence of common ions, as per Le Châtelier's principle.