Question

In a \(0.2\) molal aqueous solution of a weak acid \(H X\) the degree of ionisation is \(0.25\). The freezing point of the solution will be nearest to: \(\left(K_{f}=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right.\) ) (a) \(-0.26^{\circ} \mathrm{C}\) (b) \(0.465^{\circ} \mathrm{C}\) (c) \(-0.48^{\circ} \mathrm{C}\) (d) \(-0.465^{\circ} \mathrm{C}\)

Short answer

-0.48°C

Step-by-step solution

Understanding Molality and Degree of Ionisation

Given a 0.2 molal (m) solution of a weak acid (HX), where molality is the number of moles of solute per kilogram of solvent. The degree of ionisation (α) is given as 0.25, meaning 25% of the weak acid dissociates into ions in the solution.

Calculate the Number of Moles of Ions Produced

Since the degree of ionisation is 0.25, for every 1 mole of HX, 0.25 moles of H+ and X- ions each will be produced. The total number of moles of particles resulting from the dissolved HX will be the original 1 mole of HX plus 0.25 moles of H+ and 0.25 moles of X-. Thus, the total is 1 + 0.25 + 0.25 = 1.5 moles.

Calculate the Van't Hoff Factor (i)

The Van't Hoff factor (i) represents the number of particles a compound dissociates into in solution. For HX dissociating as HX → H+ + X-, the factor 'i' in ideal conditions would be 1 + 0.25 + 0.25, but since HX does not completely dissociate, we must consider the degree of ionisation. Hence, 'i' will be 1 + 2(0.25) = 1.5, since it produces twice the ions due to dissociation.

Compute the Freezing Point Depression

The freezing point depression (∆Tf) can be calculated using the formula ∆Tf = i * Kf * m, where Kf is the freezing point depression constant for water (1.86°C kg/mol), and m is the molality of the solution (0.2 m). Plugging in the values, ∆Tf = 1.5 * 1.86 * 0.2.

Calculate ∆Tf and Determine Freezing Point

∆Tf = 1.5 * 1.86 * 0.2 = 0.558°C. This is the amount by which the freezing point is lowered. Since pure water freezes at 0°C, the new freezing point of the solution will be 0°C - 0.558°C = -0.558°C.

Rounding the Answer

According to the answer options, the freezing point should be rounded to three decimal places. Thus, the freezing point of the solution will be rounded to -0.56°C.

Choose the Closest Answer

The closest answer choice to -0.56°C is option (c) -0.48°C.

Key concepts

Molality

Molality, often denoted by the symbol 'm', is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per kilogram of solvent. Unlike molarity, which involves the volume of the entire solution, molality focuses only on the mass of the solvent. This makes molality particularly useful in situations where the temperature may change because, unlike volume, mass doesn't vary with temperatures.

For example, in a 0.2 molal solution of a weak acid, there are 0.2 moles of the acid for every kilogram of water. This concentration is a key factor when calculating changes in the physical properties of the solution, such as freezing point depression, which is directly proportional to molality in the colligative property equation.

Degree of Ionisation

The degree of ionisation, represented by the symbol \( \alpha \), is a fraction that indicates the extent to which a compound dissociates into ions when it dissolves in a solvent. It varies from 0 (no dissociation) to 1 (complete dissociation).

For instance, if a weak acid has a degree of ionisation of 0.25, it means that only 25% of the dissolved acid molecules separate into ions in a solution. Understanding the degree of ionisation helps in calculating the effective number of particles in solution, which is vital when determining properties such as the boiling point elevation or freezing point depression.

Van's Hoff factor

The Van't Hoff factor, denoted as 'i', is a dimensionless quantity that represents the number of particles into which a solute dissociates in solution. It's crucial for calculating the changes in colligative properties, such as freezing point depression or boiling point elevation.

• If a solute does not dissociate in solution, the Van't Hoff factor is 1.
• For solutes that dissociate, 'i' is the sum of the stoichiometric coefficients of the products of dissociation, adjusted by the degree of ionisation.

For a weak acid partially dissociating in water, with a degree of ionisation \( \alpha \), the Van't Hoff factor becomes \( i = 1 + \alpha \) for each ion produced. Consequently, calculating 'i' accurately is essential when assessing the extent of freezing point depression in a solution.

Colligative Properties

Colligative properties are the physical properties of solutions that depend on the concentration of solute particles but not on their identity. These properties include vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.

Freezing point depression is a particularly common colligative property examined in chemistry. It occurs because the addition of solute to a solvent results in the lowering of the freezing point of the solution compared to that of the pure solvent. The degree to which the freezing point is lowered is related to the molality of the solution, the Van't Hoff factor for the solute, and a proportionality constant known as the freezing point depression constant (\( K_f \)).

• The formula for calculating freezing point depression is \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the change in freezing point, 'i' is the Van't Hoff factor, \( K_f \) is the freezing point depression constant, and 'm' is the molality of the solution.

The concepts of colligative properties are often tested in exercises that involve calculating the new freezing or boiling points of solutions, making it essential for students to understand how these values are determined.