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 \(2.28 \text{ mol kg}^{-1}\) (option c).

Step-by-step solution

Understand the Given Data

You are given the molarity of acetic acid solution as \(2.05 \text{ M} \) and the density of the solution as \(1.02 \frac{\text{g}}{\text{mL}}\). Molarity is defined as moles of solute per liter of solution.

Calculate Mass of Solution per Liter

Since density is given as \(1.02 \frac{\text{g}}{\text{mL}}\), multiply by 1000 mL to find the mass of 1 liter of solution: \[1.02 \frac{\text{g}}{\text{mL}} \times 1000 \text{ mL} = 1020 \text{ g}\]The mass of the solution is 1020 grams per liter.

Calculate Moles of Solute

For a \(2.05 \text{ M}\) solution, there are \(2.05\) moles of acetic acid per liter of solution.\[\text{Moles of } \text{CH}_3\text{COOH} = 2.05 \text{ moles}\]

Find Mass of Solute

Calculate the mass of acetic acid in solution using its molar mass (\(60.05 \text{ g/mol}\)).\[2.05 \text{ moles} \times 60.05 \text{ g/mol} = 123.1 \text{ g}\]

Calculate Mass of Solvent

Subtract the mass of acetic acid from the total mass of the solution to get the mass of water (solvent).\[1020 \text{ g} - 123.1 \text{ g} = 896.9 \text{ g}\]Convert mass of water to kilograms:\[\frac{896.9 \text{ g}}{1000} = 0.8969 \text{ kg}\]

Calculate Molality

Molality is moles of solute per kilogram of solvent. Use the formula:\[\text{Molality (m)} = \frac{2.05 \text{ mol}}{0.8969 \text{ kg}} = 2.28 \text{ mol/kg}\]

Compare to Options

Compare the calculated molality with the given options. The closest option is \(2.28 \text{ mol kg}^{-1}\). Therefore, the correct option is (c).

Key concepts

Molarity

Molarity is a way to express the concentration of a solution. It tells you how many moles of a solute are present in one liter of the solution. The unit of molarity is usually given in moles per liter (M).
Here is an easy way to understand it:

• "Solute" is the substance that gets dissolved, like acetic acid in water.
• "Solution" is the final mix of solute and solvent.
• Molarity formula: \[ \text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}} \]

In this problem, the molarity of acetic acid solution was given as 2.05 M, meaning there are 2.05 moles of acetic acid per liter of solution.

Density

Density is a measure of how much mass is contained in a given volume. It is often expressed in grams per milliliter (g/mL).
To think about density, consider:

• "Mass" is the amount of substance — in this case, the weight of the solution in grams.
• "Volume" is how much space the solution takes up — here, measured in milliliters.
• The formula for density is:\[ \text{Density} = \frac{\text{mass of solution}}{\text{volume of solution}} \]

In our exercise, the density was given as 1.02 g/mL. This helped determine that one liter of the solution has a mass of 1020 grams.

Molality

Molality is similar to molarity but measures concentration differently. Instead of using liters of solution, molality uses the mass of the solvent in kilograms. It's a crucial measure when temperature changes, as it does not change with temperature or pressure.
The key points about molality are:

• Expressed in moles of solute per kilogram of solvent.
• The formula for molality is:\[ \text{Molality (m)} = \frac{\text{moles of solute}}{\text{kilograms of solvent}} \]

For our exercise, after determining the mass of acetic acid and water separately, we found that the molality of our solution is 2.28 mol/kg.

Mass of Solvent

In solution chemistry, understanding the mass of the solvent helps in calculating the concentration metrics like molality. Solvent is typically the component that is present in larger amounts in the solution.
Here’s how we determine the mass of solvent:

• The mass of the full solution is found first using density.
• Then, calculate the mass of solute separately (here it's acetic acid).
• Subtract the mass of the solute from the total solution mass to find the mass of the solvent.\[ \text{Mass of solvent} = \text{Mass of solution} - \text{Mass of solute} \]

In this problem, we found the mass of the solvent (water) to be 896.9 grams or 0.8969 kilograms.