Problem 89 - Chemistry and Chemical Reactivity

Question

A \(2.00 \%\) solution of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) in water freezes at \(-0.796^{\circ} \mathrm{C}.\) (a) Calculate the van't Hoff factor, \(i.\) (b) Which of the following best represents sulfuric acid in a dilute aqueous solution: \(\mathrm{H}_{2} \mathrm{SO}_{4}, \mathrm{H}^{+}+\mathrm{HSO}_{4}^{-},\) or \(2 \mathrm{H}^{+}+\mathrm{SO}_{4}^{2-} ?\)

Step-by-Step Solution

Answer

(a) \( i \approx 2.06 \). (b) \( \mathrm{H}^{+} + \mathrm{HSO}_{4}^{-} \).

Step 1: Understand the Given Data

We are given a 2.00% solution of \( \mathrm{H}_{2} \mathrm{SO}_{4} \) which freezes at \( -0.796^{\circ} \mathrm{C} \). To find the van't Hoff factor, we need to use the freezing point depression formula: \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the change in freezing point, \( K_f \) is the cryoscopic constant for water (1.86 °C·kg/mol), and \( m \) is the molality of the solution.

Step 2: Calculate the Molality

To find molality (\( m \)), we begin by calculating the mass of \( \mathrm{H}_{2} \mathrm{SO}_{4} \) in 100 grams of solution, which is 2 grams. The molar mass of \( \mathrm{H}_{2} \mathrm{SO}_{4} \) is approximately 98.08 g/mol. Thus, the number of moles is \( 2 \text{ g} / 98.08 \text{ g/mol} \approx 0.02037 \text{ mol} \). The solvent is water, so its mass is 98 g or 0.098 kg. Hence, the molality \( m \) is \( 0.02037 \text{ mol} / 0.098 \text{ kg} \approx 0.208 \text{ mol/kg} \).

Step 3: Apply the Freezing Point Depression Formula

We know that \( \Delta T_f = 0.796^{\circ} \mathrm{C} \), and \( K_f = 1.86 \text{ °C kg/mol} \). Substitute these values into the equation \( \Delta T_f = i \times K_f \times m \) to find the van't Hoff factor \( i \): \( 0.796 = i \times 1.86 \times 0.208 \). Solve for \( i \): \( i = \frac{0.796}{1.86 \times 0.208} \approx 2.06 \).

Step 4: Analyze the Best Representation

The van't Hoff factor \( i \approx 2 \) suggests dissociation into two particles. The available options given are: \( \mathrm{H}_{2} \mathrm{SO}_{4} \), \( \mathrm{H}^{+} + \mathrm{HSO}_{4}^{-} \), and \( 2 \mathrm{H}^{+} + \mathrm{SO}_{4}^{2-} \). Since \( i \) is approximately 2, \( \mathrm{H}^{+} + \mathrm{HSO}_{4}^{-} \) is more likely, representing partial dissociation into two ions.

Key Concepts

Freezing Point Depression

Freezing point depression is a colligative property. This means it depends on the number of particles in a solution rather than the type of particles. When you dissolve a solute in a solvent, the freezing point of the resulting solution is lower than that of the pure solvent.

This effect occurs because the solute particles disrupt the formation of the solid state (crystals), requiring colder temperatures for the solution to freeze. The magnitude of the freezing point depression can be calculated using the formula: \[ \Delta T_f = i \cdot K_f \cdot m \] where \( \Delta T_f \) is the freezing point depression, \( i \) is the van't Hoff factor (indicative of the number of particles the solute splits into), \( K_f \) is the cryoscopic constant of the solvent, and \( m \) represents the molality of the solution.

Initially, by knowing the change in freezing point and the properties of the solvent, we can determine how much a solute can lower the freezing point of a solution. Understanding this concept helps in calculating the van't Hoff factor for electrolytes that dissociate in a solution, such as sulfuric acid.

Sulfuric Acid Dissociation

Sulfuric acid, known chemically as \( \mathrm{H}_2\mathrm{SO}_4 \), is a strong acid that can dissociate in aqueous solutions. The dissociation of this acid is influenced by the concentration of the solution. In a dilute aqueous solution, sulfuric acid partially dissociates.

There are typically two steps of dissociation:

• First, \( \mathrm{H}_2\mathrm{SO}_4 \) dissociates into \( \mathrm{H}^+ \) and \( \mathrm{HSO}_4^- \). This initial dissociation is almost complete due to the strong nature of sulfuric acid.
• Then, \( \mathrm{HSO}_4^- \) can further dissociate into an additional \( \mathrm{H}^+ \) and \( \mathrm{SO}_4^{2-} \), but this occurs to a lesser extent in dilute solutions.

The choices given for the best representation of sulfuric acid in a dilute solution were \( \mathrm{H}_2\mathrm{SO}_4 \), \( \mathrm{H}^+ + \mathrm{HSO}_4^- \), and \( 2\mathrm{H}^+ + \mathrm{SO}_4^{2-} \). Based on the van’t Hoff factor of around 2, it is more likely that in a dilute aqueous solution, sulfuric acid primarily splits into \( \mathrm{H}^+ \) and \( \mathrm{HSO}_4^- \), indicating partial dissociation.

Aqueous Solutions

An aqueous solution is one in which water is the solvent. Water is a remarkable solvent due to its polar nature and ability to dissolve a wide variety of substances, especially ionic compounds like acids and bases. In aqueous solutions, the solvent is involved in dissolving solutes, leading to various interactions that affect physical properties such as boiling and freezing points.

Moreover, when the solutes are electrolytes, like sulfuric acid, they disassociate into ions, which substantially impact the solution's colligative properties, such as freezing point depression.

• Water as a solvent provides the medium for dissociation and ion interaction.
• It allows for changes in physical properties based on solute interactions.

Thus, understanding the nature of aqueous solutions is crucial in predicting and explaining phenomena like freezing point depression as experienced in solutions containing dissociated compounds. By analyzing these properties, we develop insights into the behavior of solutions and their constituents.