Question

Consider the following solutions: \(0.010 \mathrm{~m} \mathrm{Na}_{3} \mathrm{PO}_{4}\) in water \(0.020 \mathrm{~m} \mathrm{CaBr}_{2}\) in water \(0.020 \mathrm{~m} \mathrm{KCl}\) in water \(0.020 \mathrm{~m} \mathrm{HF}\) in water \((\mathrm{HF}\) is a weak acid. \()\) a. Assuming complete dissociation of the soluble salts, which solution(s) would have the same boiling point as \(0.040 \mathrm{~m}\) \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) in water? \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) is a nonelectrolyte. b. Which solution would have the highest vapor pressure at \(28^{\circ} \mathrm{C}\) ? c. Which solution would have the largest freezing-point depression?

Short answer

a. The solutions that would have the same boiling point as 0.040 m \(C_6H_{12}O_6\) are 0.010 m \(Na_3PO_4\) and 0.020 m \(KCl\), as they both have the same effective molality (0.040 m). b. The solution with the highest vapor pressure at 28°C is 0.020 m \(HF\), as it has the lowest molality among the solutions. c. The solution with the largest freezing-point depression is 0.020 m \(CaBr_2\), as it has the highest effective molality (0.060 m).

Step-by-step solution

Understand colligative properties and molality of the solutions

Colligative properties are properties that depend on the number of solute particles in the solution, not the nature of the solute particles. The colligative properties in question are boiling point elevation and freezing point depression. Molality (m) is the number of moles of solute divided by the mass of the solvent (in kg). Since boiling point elevation and freezing point depression are both proportional to the molality of the solution, we can simply compare the molality values of each solution.

Determine the effective molality for each solution

For complete dissociation, the effective molality of ions in solution is given by the number of ions multiplied by the molarity of the solution. For Na3PO4: The molality is 0.010 m and it dissociates into 4 ions (3 Na+ and 1 PO43-). Effective molality: 0.010 m * 4 = 0.040 m For CaBr2: The molality is 0.020 m and it dissociates into 3 ions (1 Ca2+ and 2 Br-). Effective molality: 0.020 m * 3 = 0.060 m For KCl: The molality is 0.020 m and it dissociates into 2 ions (1 K+ and 1 Cl-). Effective molality: 0.020 m * 2 = 0.040 m For HF: The molality is 0.020 m, but since it's a weak acid, we will treat it as a non-electrolyte. The effective molality remains 0.020 m For C6H12O6: The molality is 0.040 m and as a non-electrolyte, its effective molality remains the same, 0.040 m

Answer part a

To find which solution would have the same boiling point as 0.040 m C6H12O6, we need to identify which solutions have the same effective molality. Comparing the effective molalities, we find that the 0.010 m Na3PO4 solution and the 0.020 m KCl solution both have an effective molality of 0.040 m. Therefore, these two solutions will have the same boiling point elevation as 0.040 m C6H12O6.

Answer part b

The solution with the highest vapor pressure will be the one with the lowest molality (since colligative properties are inversely proportional to vapor pressure). Among the given solutions, the 0.020 m HF solution has the lowest molality. Thus, the 0.020 m HF solution would have the highest vapor pressure at 28°C.

Answer part c

The freezing point depression is proportional to the effective molality of the solution. Among the given solutions, the 0.020 m CaBr2 solution has the highest effective molality (0.060 m). Therefore, the 0.020 m CaBr2 solution will have the largest freezing-point depression.

Key concepts

Boiling Point Elevation

When you add a solute to a solvent, the boiling point of the solution increases. This phenomenon is known as boiling point elevation. It occurs because the solute particles disrupt the solvent molecules' ability to escape into the gas phase, requiring more heat to reach the boiling point. To predict boiling point changes, we look at effective molality, which represents the total concentration of solute particles in the solution. For example, if a solution has an effective molality of 0.040 m, it shows similar boiling point elevation regardless of the solute type, as long as the concentration of solute particles is similar.- Boiling point elevation is a colligative property, depending solely on the number of solute particles, not their identity. - The formula to calculate boiling point elevation is \( \Delta T_b = iK_bm \), where \( i \) is the van't Hoff factor (number of particles), \( K_b \) is the ebullioscopic constant, and \( m \) is the molality. - Solutions with similar effective molalities will cause similar increases in boiling point.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor when it is in equilibrium with its liquid form. Solutions with lower solute concentrations typically have higher vapor pressures. This is because fewer solute particles mean there are more solvent molecules that can escape into the gas phase. In terms of colligative properties, the presence of solute particles lowers the vapor pressure of the solvent, because the solute particles occupy space on the liquid's surface, reducing the number of solvent molecules that can vaporize. - The decrease in vapor pressure is directly proportional to the number of solute particles in a given volume of solvent (Ref: Raoult's Law). - Weak acids and non-electrolytes like HF in water contribute fewer solute particles, and thus a solution's vapor pressure under these conditions is relatively high. - Among the solutions analyzed, HF with the lowest molality exerts the highest vapor pressure, showing the inverse relationship between vapor pressure and solute concentration.

Freezing Point Depression

Freezing point depression occurs when the presence of a solute lowers the temperature at which a liquid becomes solid. This is another key colligative property, just like boiling point elevation.The more solute particles there are in a solution, the more they interfere with the formation of the solid lattice structure, requiring a lower temperature to freeze the liquid. - The formula for freezing point depression is \( \Delta T_f = iK_fm \), where \( i \) is the van't Hoff factor, \( K_f \) is the cryoscopic constant, and \( m \) is the molality.- Solutions with higher effective molalities will have greater freezing point depressions.- For instance, a solution of \( \mathrm{CaBr}_2 \) has the highest effective molality, and thus experiences the greatest freezing point depression compared to other given solutions.

Effective Molality

Effective molality is a concept that takes the dissociation of ionic compounds into account, making it crucial for evaluating colligative properties. It is determined by multiplying the concentration of the solution by the number of particles into which the solute dissociates.This means that a solution's effective molality is higher than its actual molality for ionic compounds, as these compounds dissociate in water to form multiple ions. - Different compounds dissociate differently; for example: - \( \mathrm{Na}_3 \mathrm{PO}_4 \) dissociates into four ions, thus a 0.010 m solution has an effective molality of 0.040 m. - \( \mathrm{CaBr}_2 \) dissociates into three ions, giving a 0.020 m solution an effective molality of 0.060 m.- Understanding effective molality is essential for predicting changes in boiling and freezing points.- It illustrates how different ionic compounds, despite having similar molalities, can impact colligative properties differently due to their dissociation behavior.

Dissociation of Electrolytes

Electrolytes are substances that dissolve in water to produce a solution that conducts electricity. This happens because electrolytes dissociate into ions when dissolved.The more an electrolyte dissociates, the more ions it produces, which directly impacts colligative properties. Strong electrolytes dissociate completely in solution, while weak electrolytes only partially dissociate.- Strong electrolytes like \( \mathrm{Na}_3 \mathrm{PO}_4 \) or \( \mathrm{CaBr}_2 \) completely dissociate into ions, increasing the effective concentration of solute particles.- Weak electrolytes, such as \( \mathrm{HF} \), do not fully dissociate, leading to fewer particles and therefore lesser effects on colligative properties.- Dissociation influences the effective molality, which in turn affects boiling point elevation, freezing point depression, and vapor pressure.