Question

Density of a \(2.05 \mathrm{M}\) solution of acetic acid in water is \(1.02 \mathrm{~g} / \mathrm{mL}\). The molality of the solution is (a) \(1.14 \mathrm{~mol} \mathrm{~kg}^{-1}\) (b) \(3.28 \mathrm{~mol} \mathrm{~kg}^{-1}\) (c) \(2.28 \mathrm{~mol} \mathrm{~kg}^{-1}\) (d) \(0.44 \mathrm{~mol} \mathrm{~kg}^{-1}\)

Short answer

The molality of the solution is 2.28 mol kg⁻¹, which corresponds to option (c).

Step-by-step solution

Understand the Given Information

We are given a 2.05 M (molarity) solution of acetic acid with a density of 1.02 g/mL. We need to find the molality, which requires the number of moles of solute per kilogram of solvent.

Convert Molarity to Moles of Solute

Molarity is defined as moles of solute per liter of solution. Therefore, in 1 liter of the solution, there are 2.05 moles of acetic acid.

Calculate the Mass of the Solution

The density of the solution is 1.02 g/mL. Since there are 1,000 mL in 1 L, the mass of 1 L of the solution is \(1.02 \times 1000 = 1020\) g.

Determine the Mass of Solvent (Water)

To get the mass of the solvent, subtract the mass of the solute from the total mass of the solution. The molar mass of acetic acid (\(CH_3COOH\)) is approximately 60 g/mol. Thus, the mass of 2.05 moles is \(2.05 \times 60 = 123\) g. The mass of water is \(1020 - 123 = 897\) g or 0.897 kg.

Calculate the Molality

Molality is calculated using the formula: \( \text{molality} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \). Therefore, molality = \( \frac{2.05}{0.897} \approx 2.28 \text{ mol kg}^{-1}\).

Verify the Answer

Compare the calculated molality to the given options: (a) 1.14 mol kg⁻¹ (b) 3.28 mol kg⁻¹ (c) 2.28 mol kg⁻¹ (d) 0.44 mol kg⁻¹. The correct option is (c) 2.28 mol kg⁻¹.

Key concepts

Molarity to Molality Conversion

Converting molarity to molality involves understanding these fundamental concepts. Molarity (M) is the measure of the concentration of a solute in a solution. Specifically, it describes the number of moles of solute dissolved in one liter of solution. On the other hand, molality (m) is the measure of solute concentration based on the mass of the solvent. It is calculated by the number of moles of solute per kilogram of solvent.
To perform this conversion:

• First, determine the moles of solute per liter of solution using the molarity.
• Next, calculate the total mass of the solution using its density.
• Identify the mass of the solute and subtract it from the total mass to find the mass of the solvent.
• Finally, divide the moles of solute by the mass of the solvent in kilograms to obtain the molality.

Keep in mind, density is crucial in this process, as it allows conversion from volume to mass, providing the data needed to isolate the solvent's mass.

Density Calculations

Density plays a significant role when working with solutions and converting between different concentration units. It is defined as the mass per unit volume, generally expressed in grams per milliliter (g/mL). In the context of molarity to molality conversion, the solution's density helps determine the mass of the solution.
To calculate density:

• Use the formula: Density = \( \frac{\text{Mass of Solution}}{\text{Volume of Solution}} \).
• Convert the volume to milliliters or liters if necessary.
• Multiply the density by the volume to find the total mass of the solution.

In problems like the one presented, once you obtain the mass of the solution, it is crucial to find the mass of the solvent (usually water when dealing with aqueous solutions) to further determine molality.

Solution Concentration

When talking about solution concentration, we refer to how much solute is present in a given amount of solvent or solution. Concentration can be measured in various ways, including molarity, molality, and others like mass percent or mole fraction. The choice of measurement depends on the scenario and the properties being investigated.
For molarity:

• It is expressed in moles per liter of solution and is particularly useful for reactions taking place in a solution, where consistent volume is ideal.

Molality, however, is the chosen measure when dealing with temperature-dependent properties because it relies on mass rather than volume, which means no change even if temperature affects the solution's volume.
Understanding concentration is fundamental for solving problems related to solution preparation, reactions, and properties. It enables precise formulation and comparability across different physical and chemical scenarios.