Question

Calculate the freezing point and the boiling point of each of the following aqueous solutions. (Assume complete dissociation.) a. \(0.050 \mathrm{~m} \mathrm{MgCl}_{2}\) b. \(0.050 \mathrm{~m} \mathrm{FeCl}_{3}\)

Short answer

For the 0.050 m MgCl2 solution, the freezing point is -0.278°C, and the boiling point is 100.077°C. For the 0.050 m FeCl3 solution, the freezing point is -0.371°C, and the boiling point is 100.103°C.

Step-by-step solution

Determining the dissociation of the solutes

Before we can calculate the molality of the solutions, we must determine how many ions are produced when the solute dissociates in the solution. For MgCl2: \[ \mathrm{MgCl}_{2} \rightarrow \mathrm{Mg}^{2+} + 2\mathrm{Cl}^{-} \] For FeCl3: \[ \mathrm{FeCl}_{3} \rightarrow \mathrm{Fe}^{3+} + 3\mathrm{Cl}^{-} \]

Using the formulas for freezing point depression and boiling point elevation

The freezing point depression and boiling point elevation are calculated using the formulas: \[ \Delta T_{f} = -iK_{f}m \] \[ \Delta T_{b} = iK_{b}m \] Where: - \( \Delta T_{f} \) is the freezing point depression - \( \Delta T_{b} \) is the boiling point elevation - \( i \) is the van't Hoff factor (number of ions formed per formula unit) - \( K_{f} \) is the freezing point depression constant for water (\( 1.853 \mathrm{~K.kg.mol}^{-1} \)) - \( K_{b} \) is the boiling point elevation constant for water (\( 0.515 \mathrm{~K.kg.mol}^{-1} \)) - \( m \) is the molality of the solution

Calculate the freezing point depression and boiling point elevation for MgCl2 solution

Given molality for MgCl2 solution: \( m = 0.050 \mathrm{~mol/kg} \) Since one MgCl2 molecule produces three ions, the van't Hoff factor of MgCl2 is \( i = 3 \). Now we calculate the freezing point depression: \[ \Delta T_{f} = -iK_{f}m = -(3)(1.853)(0.050) = -0.278 \mathrm{~K} \] Freezing Point = Normal Freezing Point (0°C) + Freezing Point Depression: \[ T_{f} = 0 - 0.278 = -0.278 \mathrm{~^\circ C} \] Now we calculate the boiling point elevation: \[ \Delta T_{b} = iK_{b}m = (3)(0.515)(0.050) = 0.077 \mathrm{~K} \] Boiling Point = Normal Boiling Point (100°C) + Boiling Point Elevation: \[ T_{b} = 100 + 0.077 = 100.077 \mathrm{~^\circ C} \]

Calculate the freezing point depression and boiling point elevation for FeCl3 solution

Given molality for FeCl3 solution: \( m = 0.050 \mathrm{~mol/kg} \) Since one FeCl3 molecule produces four ions, the van't Hoff factor of FeCl3 is \( i = 4 \). Now we calculate the freezing point depression: \[ \Delta T_{f} = -iK_{f}m = -(4)(1.853)(0.050) = -0.371 \mathrm{~K} \] Freezing Point = Normal Freezing Point (0°C) + Freezing Point Depression: \[ T_{f} = 0 - 0.371 = -0.371 \mathrm{~^\circ C} \] Now we calculate the boiling point elevation: \[ \Delta T_{b} = iK_{b}m = (4)(0.515)(0.050) = 0.103 \mathrm{~K} \] Boiling Point = Normal Boiling Point (100°C) + Boiling Point Elevation: \[ T_{b} = 100 + 0.103 = 100.103 \mathrm{~^\circ C} \] Now we have the freezing and boiling points for both solutions: - For 0.050 m MgCl2 solution: Freezing Point = -0.278°C, Boiling Point = 100.077°C - For 0.050 m FeCl3 solution: Freezing Point = -0.371°C, Boiling Point = 100.103°C

Key concepts

Freezing Point Depression

Freezing point depression is a colligative property, which means it is influenced by the number of particles in a solution and not by their identity. When a solute is added to a solvent, such as salt in water, the freezing point of the solution is lowered. This occurs because the solute particles disrupt the formation of the solid phase, making it harder for the solvent molecules to crystallize and thus requiring a lower temperature to freeze.

The extent of freezing point depression can be calculated using the formula: \[\begin{equation}\Delta T_f = -iK_fm\end{equation}\]where \(\Delta T_f\) is the change in freezing point, \(i\) is the van't Hoff factor which indicates the number of particles the solute dissociates into, \(K_f\) is the freezing point depression constant specific to the solvent, and \(m\) is the molality of the solution. The negative sign indicates that the temperature is reduced.

This property has practical applications such as in the use of road salt to lower the freezing point of water on roadways, preventing ice formation.

Boiling Point Elevation

Boiling point elevation is another colligative property that describes how the presence of a solute can raise the boiling point of a solvent. Like freezing point depression, it is also dependent on the number of solute particles but not their type. This phenomenon occurs because the solute particles hinder the escape of solvent molecules into the vapor phase, which requires a higher temperature to achieve the same vapor pressure necessary for boiling.

The change in boiling point can be calculated with the relationship: \[\Delta T_b = iK_bm\]where \(\Delta T_b\) represents the increase in boiling point, \(i\) is the van't Hoff factor, \(K_b\) is the boiling point elevation constant, and \(m\) is the molality of the solution. Boiling point elevation can be observed when cooking with saltwater, as it requires a slightly higher temperature to boil compared to pure water.

Van't Hoff Factor

The van't Hoff factor, denoted by the symbol \(i\), reflects the number of particles into which a compound dissociates in solution. For example, a salt like NaCl dissociates completely into two ions, Na+ and Cl-, giving it a van't Hoff factor of 2. It's crucial in calculating both boiling point elevation and freezing point depression because it directly affects the magnitude of these changes. The actual van't Hoff factor can be lower than the theoretical value due to ion pairing – when ions remain associated with each other in solution.

In scenarios where dissociation is incomplete or the solute does not dissociate at all (like with sugar in water), the van't Hoff factor would be equal to 1, indicating no change in the number of particles. Accounting for the van't Hoff factor is necessary for accurate calculations in the colligative properties involving ionic compounds.

Molality

Molality, represented by \(m\), is a measure of the concentration of a solute in a solution. Unlike molarity, which measures the number of moles of solute per liter of solution, molality measures the number of moles of solute per kilogram of solvent. This difference is significant because molality is not affected by temperature changes, since the mass of the solvent does not change with temperature, making it a more reliable measure for studying temperature-dependent properties.

To calculate molality, we use the formula: \[m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\]Molality is particularly important in the context of colligative properties, as these properties are directly proportional to the molality of the solute in a given solvent. The precision of molality is crucial for accurate calculations in boiling point elevation and freezing point depression, highlighting the importance of distinguishing it from other concentration measures.