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Boiling point elevation is a property of solutions. When you dissolve a solute in a solvent, the boiling point of the solution becomes higher than that of the pure solvent. This occurs because the presence of solute particles causes the solvent molecules to require more energy to escape into the vapor phase.
This effect is quantitatively described by the formula \( \Delta T_b = i \cdot K_b \cdot m \).

• \( \Delta T_b \) is the increase in boiling point.
• \( i \) is the van't Hoff factor, representing the number of particles the solute dissociates into.
• \( K_b \) is the ebullioscopic constant, specific to each solvent.
• \( m \) is the molality of the solution.

For instance, in the problem given, when comparing a \(0.20 \mathrm{m}\) KBr solution to a \(0.30 \mathrm{m}\) sugar solution, we observe that KBr dissociates into two ions, contributing to a higher van't Hoff factor of 2. This results in a greater boiling point elevation for KBr, demonstrating how the increase in boiling point depends not only on the concentration of the solute but also on its ability to dissociate.

Freezing point depression occurs when a solute is added to a solvent, causing the freezing point of the solution to be lower than that of the pure solvent. In simple terms, it takes a lower temperature for the solution to solidify.
This phenomenon is described by the formula \( \Delta T_f = i \cdot K_f \cdot m \).

• \( \Delta T_f \) is the decrease in freezing point.
• \( i \) is the van't Hoff factor, indicating the number of particles the solute breaks into.
• \( K_f \) is the cryoscopic constant.
• \( m \) is the solution's molality.

In the example provided, \(0.12 \mathrm{m}\) \(\mathrm{NH}_{4}\mathrm{NO}_{3}\) solution with a van't Hoff factor of 2 results in a smaller decrease in temperature compared to \(0.10 \mathrm{m}\) \(\mathrm{Na}_{2}\mathrm{CO}_{3}\) with a factor of 3. This demonstrates how substances that dissociate into more ions tend to create a more significant reduction in freezing point, explaining why the \(\mathrm{NH}_{4}\mathrm{NO}_{3}\) solution ends up with a lower freezing point.

The van't Hoff factor, denoted by \( i \), is a crucial part of colligative properties such as boiling point elevation and freezing point depression. The van't Hoff factor measures the effect of a solute on the colligative properties of a solution. More specifically, it accounts for the number of particles into which a solute dissociates in solution. The more particles that are produced, the greater the effect on colligative properties.
To determine \( i \):

• A non-dissociating solute, like sugar, has \( i = 1 \).
• Ionic compounds dissociate into their ions; for example, \(\mathrm{KBr}\) dissociates into \(\mathrm{K}^+\) and \(\mathrm{Br}^-\), thus \( i = 2 \).
• Compounds like \(\mathrm{Na}_2\mathrm{CO}_3\) split into three ions (2 \(\mathrm{Na}^+\) and \(\mathrm{CO}_3^{2-}\)), giving \( i = 3 \).

The van't Hoff factor directly influences the magnitude of both boiling point elevation and freezing point depression. Consequently, deeper understanding and accurate calculation of this factor are imperative for analyzing and predicting changes in solutions' properties.