Use the following data for three aqueous solutions of \(\mathrm{CaCl}_{2}\) to calculate the apparent value of the van't Hoff factor. $$\begin{array}{lc} \text { Molality } & \text { Freezing-Point Depression }\left(^{\circ} \mathrm{C}\right) \\ \hline 0.0225 & 0.110 \\ 0.0910 & 0.440 \\ 0.278 & 1.330 \end{array}$$

Colligative properties are significant characteristics of a solution that solely depend on the concentration of solute particles, rather than on their nature. These properties include freezing-point depression, boiling-point elevation, vapor pressure lowering, and osmotic pressure.

Understanding colligative properties is essential because they give us insight into how adding a solute to a solvent can affect its overall physical properties. For instance, when salt is sprinkled on icy roads, it causes the ice to melt by lowering the freezing point of water — a practical application of freezing-point depression, which is one of the colligative properties we encounter in everyday life.

The underlying principle involves the increase in entropy when a solute is dissolved in a solvent, which then affects the thermodynamics and phase behavior of the solution.

Molality is a way to express the concentration of a solution. It is defined as the moles of solute per kilogram of solvent, not solution. This concentration measure is temperature-independent, unlike molarity, which varies with temperature due to the expansion and contraction of liquids.

Mathematically, molality (\( m \)) is calculated as: \( m = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \). By using molality, we can link the amount of solute to colligative properties, such as freezing-point depression, as shown in the van't Hoff factor exercise.

Freezing-point depression refers to the decrease in the temperature at which a liquid becomes a solid due to the presence of a solute. It is a colligative property, which means it is dependent on solute concentration.

The relationship between molality and freezing-point depression can be explained through the formula: \( \Delta T_f = K_f \cdot m \cdot i \), where \( \Delta T_f \) is the freezing-point depression, \( K_f \) is the cryoscopic constant (specific to each solvent), \( m \) is the molality of the solution, and \( i \) is the van't Hoff factor that accounts for the number of particles the solute divides into upon dissolution. For pure solvents, \( \Delta T_f \) is zero; the presence of solute particles disrupts the formation of a solid lattice, causing the observed depression.

Thermodynamic properties are attributes of a system that can be measured and defined within the boundaries of thermodynamics. They include variables like temperature, pressure, volume, and chemical potential. These properties help us understand the energy transformations and the equilibrium conditions of a chemical reaction or physical change.

Thermodynamics provides the framework to relate colligative properties to energy changes within a system. As solutes dissolve in solvents, they alter the energy landscape of the solution and hence change its melting and boiling points. Moreover, these thermodynamic shifts can be quantified and predicted using principles such as the van't Hoff factor in freezing-point depression calculations, providing us with a deeper insight into the material's behavior at the molecular level.