Question

The cryoscopic constant of water is \(1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\). A \(0.01\) molal acetic acid solution produces a depression of \(0.0194^{\circ} \mathrm{C}\) in the freezing point. The degree of dissociation of acetic acid is: (a) zero (b) \(0.043\) (c) \(0.43\) (d) 1

Short answer

The degree of dissociation of acetic acid is 0.043.

Step-by-step solution

Identify the Relevant Data from the Problem

Identify given values: cryoscopic constant of water (Kf) is 1.86 K kg/mol, molality of acetic acid solution (m) is 0.01 m, and observed depression in freezing point (ΔTf) is 0.0194°C. The formula to find the degree of dissociation (α) is based on the van't Hoff factor (i), which can be calculated using the relation i = ΔTf / (Kf × m).

Calculate the Experimental Van't Hoff Factor

Calculate the experimental van't Hoff factor using the formula i = ΔTf / (Kf × m). Thus, i = 0.0194° C / (1.86 K kg/mol × 0.01 mol/kg) = 0.0194 / 0.0186.

Perform the Calculation

Solve for i getting i = 0.0194 / 0.0186, which simplifies to i ≈ 1.043.

Relate the Van't Hoff Factor to Degree of Dissociation

For weak acids, such as acetic acid, which partially dissociates, van't Hoff factor i can be related to the degree of dissociation α using the formula i = 1 + (n-1)α, where n is the number of ions the molecule dissociates into. In the case of acetic acid, n=2. Solve for α: α = (i-1) / (n-1).

Calculate the Degree of Dissociation

Substitute the known values into the equation for α: α = (1.043-1) / (2-1) = 0.043.

Key concepts

Cryoscopic Constant

The cryoscopic constant, denoted as Kf, is a property that tells us how the freezing point of a solvent will be lowered when a solute is added. It is specific to each solvent and is used for calculating freezing point depression in the colligative properties analysis of solutions. In the context of the exercise, the cryoscopic constant of water is given as 1.86 K kg/mol. This value signifies how many degrees Kelvin the freezing point of one kilogram of water will drop for every one mole of a non-volatile solute added.

To understand the concept better, think of the cryoscopic constant as a measure of a solvent's resistance to freezing point changes when a solute is introduced. A higher constant means that the substance's freezing point is more significantly affected by the addition of solutes.

Van't Hoff Factor

The van't Hoff factor, symbolized as 'i', relates to how a solute dissociates or associates in a particular solvent. It is crucial in calculating the expected change in colligative properties, like boiling point elevation or freezing point depression. For substances that don't dissociate or associate in solution, the van't Hoff factor is typically equal to 1. For ionic compounds that dissociate into multiple particles, the van't Hoff factor is the number of ions formed per formula unit.

In the exercise, we use the experimental van't Hoff factor to estimate the degree of dissociation of acetic acid in water. The calculation of 'i' involves the observed freezing point depression and the product of the cryoscopic constant and the solution's molality. This conceptual understanding is applied to calculate the experimental 'i', which is then linked to the degree of dissociation of acetic acid.

Colligative Properties

Colligative properties are those properties of solutions that depend only on the number of solute particles in a given quantity of solvent and not on the nature of the solute particles. Examples include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure. These properties are instrumental in determining molecular masses and the interactions of solute particles within a solution.

Understanding that these properties are exclusively related to particle count helps explain why ionic compounds, which dissociate into multiple particles, have a more considerable effect on colligative properties compared to non-dissociating compounds. The exercise we're looking at deals with the colligative property of freezing point depression, demonstrating how the addition of a solute (acetic acid) influences the freezing point of the solvent (water).

Molality

Molality, often represented by the symbol 'm', is a concentration term for solutions defined as moles of solute per kilogram of solvent. Unlike molarity, molality is not affected by changes in temperature because it depends on the mass, rather than the volume, of the solvent. In the given exercise, the molality of acetic acid in water is stated as 0.01 m, which indicates there are 0.01 moles of acetic acid per kilogram of water.

When working with freezing point depression and other colligative properties, molality becomes a preferred unit because it remains constant with temperature changes, ensuring that the calculated properties are accurate even under varying experimental conditions.

Freezing Point Depression

Freezing point depression is a colligative property that describes the lowering of a solvent's freezing point due to the presence of a solute. Essentially, when a solute is dissolved in a solvent, the freezing point of the resulting solution is lower than that of the pure solvent. Quantitatively, this depression in freezing point can be calculated by the formula \( \Delta Tf = Kf \times m \times i \), where \( \Delta Tf \) is the freezing point depression, Kf is the cryoscopic constant of the solvent, m is the molality of the solution, and i is the van't Hoff factor for the solute.

In the exercise provided, we examine the depression of water's freezing point by 0.0194°C as a result of adding acetic acid. By integrating the concepts of molality, cryoscopic constant, and van't Hoff factor, we can solve for the degree of dissociation, providing a real-world example of how freezing point depression can be applied to study the behavior of a solute in a solution.