Question

What percent of the AB particles are dissociated by water if the freezing point of \(a(.0100 \mathrm{~m})\) AB solution is \(-0.0193^{\circ} \mathrm{C}\) ? The freezing point lowering constant of water is \(\left(-1.86^{\circ} \mathrm{C} / \mathrm{mole}\right)\)

Short answer

The percent dissociation of AB particles is approximately 89.62%.

Step-by-step solution

Identify the given data

We get the following data from the exercise: 1. Initial concentration of AB solution (a) = 0.0100 mol/kg 2. Freezing point of AB solution (T_f_solution) = -0.0193°C 3. Freezing point lowering constant of water (K_f) = -1.86°C/mol

Calculate the molality of the non-dissociated solvent

The molality of a solution (m) is equal to the number of moles of solute per kilogram of solvent. For a non-dissociated solution, the initial concentration equals the molality: m = a = 0.0100 mol/kg

Calculate the observed freezing point lowering

The observed freezing point lowering (ΔT_f_observed) can be calculated by finding the difference between the freezing point of the solution and that of pure water (which is 0°C): ΔT_f_observed = T_f_solution - T_f_pure_water ΔT_f_observed = (-0.0193) - (0) = -0.0193°C

Use the freezing point lowering formula to find the van't Hoff factor

Now, we use the formula ΔT_f = K_f * m * i, where i is the van't Hoff factor. The formula can be rearranged to find the value of i: i = ΔT_f_observed / (K_f * m) i = (-0.0193) / (-1.86 * 0.0100) = 0.1038

Calculate the percent dissociation of AB

As the van't Hoff factor (i) for a non-dissociated solute is 1, any value less than 1 indicates that partial dissociation has occurred. The percent dissociation can be calculated as follows: Percent dissociation = (1 - i) * 100 Percent dissociation = (1 - 0.1038) * 100 = 89.62% So, 89.62% of the AB particles are dissociated by water.

Key concepts

van't Hoff factor

The van't Hoff factor, denoted as \( i \), is an important concept in solution chemistry. It indicates the degree to which solutes dissociate in a solution. When you dissolve a non-electrolyte in water, it typically doesn’t break into ions, and thus the van't Hoff factor is usually 1. However, for electrolytes, the factor can exceed 1 due to dissociation.

In this case, the van't Hoff factor is less than 1 because it represents partial dissociation of the solute AB in water. When calculating \( i \), you can use the formula for freezing point depression:

• \( \Delta T_f = K_f \cdot m \cdot i \)

By rearranging the formula, \( i \) can be isolated as:

• \( i = \frac{\Delta T_{f_{observed}}}{K_f \cdot m} \)

In the given problem, the calculated \( i \) value is 0.1038, signifying that a small fraction of the solute molecules remain as intact particles due to low dissociation.

percent dissociation

Percent dissociation measures how much of a solute dissociates into ions in a solution. It's a percentage of the initial solute concentration that has split into ions. In cases where a solute mostly dissociates, the percent dissociation will be high.

In the problem, the initial expectation is that the compound AB does not fully dissociate in water, as indicated by the van't Hoff factor (0.1038). For a compound that does not dissociate, the factor would be 1. A reveal that a significant dissociation occurs can be seen in the percent dissociation calculation:

• Percent dissociation is given by \( (1 - i) \times 100 \)
• Using \( i = 0.1038 \), percent dissociation is calculated as \( (1 - 0.1038) \times 100 = 89.62\% \)

So, about 89.62% of AB particles dissociate into ions in the solution.

molality

Molality is a concentration measure that expresses the moles of solute per kilogram of solvent. Unlike molarity, molality is independent of temperature as it doesn’t consider the volume, thus making it useful in freezing point depression problems.

In this exercise, molality is given as 0.0100 \( \, \text{mol/kg} \), since at the start, the initial concentration of the solution directly matches its molality. When evaluating colligative properties like freezing point depression, molality helps in understanding how a certain mole of solute affects the freezing point of a solvent. Here, given that the solution involves a partially dissociated solute, molality remains a key factor in determining how the presence of this solute changes the freezing point of the water."