Problem 60 - Chemistry and Chemical Reactivity

Question

Consider the following aqueous solutions:(i) \(0.20\) \(m\) \(\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{OH}\) (nonvolatile, nonelectrolyte); (ii) \(0.10\) \(m\) \(\mathrm{CaCl}_{2}\); (iii) \(0.12\) \(m\) \(\mathrm{KBr}\) ; and (iv) \(0.12\) \(m\) \(\mathrm{Na}_{2} \mathrm{SO}_{4}\). (a) Which solution has the highest boiling point? (b) Which solution has the lowest freezing point? (c) Which solution has the highest water vapor pressure?

Step-by-Step Solution

Answer

(a) Na₂SO₄ has the highest boiling point. (b) Na₂SO₄ has the lowest freezing point. (c) HOCH₂CH₂OH has the highest water vapor pressure.

Step 1: Determine the van't Hoff factor

The van't Hoff factor (i) represents the number of particles the solute breaks into when dissolved in water. - For HOCH₂CH₂OH (nonvolatile, nonelectrolyte), i = 1. - For CaCl₂, i = 3 (CaCl₂ → Ca²⁺ + 2Cl⁻). - For KBr, i = 2 (KBr → K⁺ + Br⁻). - For Na₂SO₄, i = 3 (Na₂SO₄ → 2Na⁺ + SO₄²⁻).

Step 2: Calculate boiling point elevation for each solution

Boiling point elevation can be calculated using ΔT_b = i × K_b × m, where K_b is the ebullioscopic constant and m is the molality. We compare the products i × m. - HOCH₂CH₂OH: i × m = 1 × 0.20 = 0.20 - CaCl₂: i × m = 3 × 0.10 = 0.30 - KBr: i × m = 2 × 0.12 = 0.24 - Na₂SO₄: i × m = 3 × 0.12 = 0.36 Na₂SO₄ has the highest value, thus the highest boiling point.

Step 3: Calculate freezing point depression for each solution

Freezing point depression can be calculated using ΔT_f = i × K_f × m, where K_f is the cryoscopic constant. Again, compare the products i × m. - HOCH₂CH₂OH: i × m = 0.20 - CaCl₂: i × m = 0.30 - KBr: i × m = 0.24 - Na₂SO₄: i × m = 0.36 Na₂SO₄ has the lowest freezing point.

Step 4: Determine the solution with the highest vapor pressure

The solution with the fewest solute particles (lowest i × m) will have the highest vapor pressure. - HOCH₂CH₂OH: i × m = 0.20 - CaCl₂: i × m = 0.30 - KBr: i × m = 0.24 - Na₂SO₄: i × m = 0.36 Thus, HOCH₂CH₂OH has the highest water vapor pressure.

Key Concepts

Boiling Point Elevation

Boiling point elevation is a colligative property that represents the increase in the boiling point of a solution compared to the pure solvent. This happens because the addition of solute particles interferes with the formation of vapor, requiring higher temperatures for the solvent to transition into the vapor phase. The extent of boiling point elevation depends on several factors:

• The type of solute: Whether it dissociates into ions or not.
• The concentration of the solution: Measured in molality (moles of solute per kilogram of solvent).
• The van't Hoff factor (i): This indicates the number of particles a solute releases in solution.

The formula used to calculate boiling point elevation is:\[ \Delta T_b = i \times K_b \times m \]Where:

• \(\Delta T_b\) is the boiling point elevation.
• \(i\) is the van't Hoff factor.
• \(K_b\) is the ebullioscopic constant (unique for every solvent).
• \(m\) is the molality of the solution.

In the provided exercise, among the different solutions, Na₂SO₄ has the largest van’t Hoff factor and molality product, meaning it causes the highest boiling point elevation.

Freezing Point Depression

Freezing point depression occurs when a solute is added to a solvent, resulting in the lowering of the freezing point compared to the pure solvent. Like boiling point elevation, freezing point depression is also a colligative property, meaning it depends on the quantity of solute particles in the solution. The presence of solute particles inhibits the ability of the solvent molecules to organize into a solid structure, thereby requiring a lower temperature to reach the point of solidification.The calculation of freezing point depression uses the formula:\[ \Delta T_f = i \times K_f \times m \]Where:

• \(\Delta T_f\) is the freezing point depression.
• \(i\) is the van't Hoff factor, representing particle dissociation.
• \(K_f\) is the cryoscopic constant, specific to each solvent.
• \(m\) is the molality of the solution.

In the exercise example, Na₂SO₄ also has the greatest \(i \times m\) value, resulting in the lowest freezing point due to the most significant interference from solute particles.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its solid or liquid phase at a given temperature. When a nonvolatile solute is added to a solvent, the vapor pressure of the solution becomes lower than that of the pure solvent. This is because solute particles occupy space at the surface, hindering solvent molecules from escaping into the vapor phase. Thus, vapor pressure is also a colligative property.The relationship of vapor pressure reduction can be explained using Raoult's Law, which states:\[ P_{solution} = X_{solvent} \times P_{solvent}\]Where:

• \(P_{solution}\) is the vapor pressure of the solution.
• \(X_{solvent}\) is the mole fraction of the solvent in the solution.
• \(P_{solvent}\) is the vapor pressure of the pure solvent.

According to the exercise, the solution containing HOCH₂CH₂OH (a nonvolatile nonelectrolyte) has the least number of dissociated particles, leading to the highest relative water vapor pressure among the given solutions.