Problem 2 - Chemistry: Principles and Reactions

Question

Acetone, \(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}\), is the main ingredient of nail polish remover. A solution is made up by adding \(35.0 \mathrm{~mL}\) of acetone \((d=0.790 \mathrm{~g} / \mathrm{mL})\) to \(50.0 \mathrm{~mL}\) of ethyl alcohol, \(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{O}(d=0.789 \mathrm{~g} / \mathrm{mL})\). Assuming volumes are additive, calculate (a) the mass percent of acetone in the solution. (b) the volume percent of ethyl alcohol in the solution. (c) the mole fraction of acetone in the solution.

Step-by-Step Solution

Answer

Answer: (a) Mass percent of acetone: 41.2% (b) Volume percent of ethyl alcohol: 58.8% (c) Mole fraction of acetone: 0.358

Step 1: Find the mass of acetone and ethyl alcohol

First, find the mass of acetone by multiplying its volume with its density: Mass of acetone = \(\mathrm{Volume} \times \mathrm{Density} = 35.0 \mathrm{~mL} \times 0.790~\mathrm{g/mL} = 27.65\mathrm{~g}\) Next, find the mass of ethyl alcohol by multiplying its volume with its density: Mass of ethyl alcohol = \(\mathrm{Volume} \times \mathrm{Density} = 50.0 \mathrm{~mL} \times 0.789~\mathrm{g/mL} = 39.45\mathrm{~g}\)

Step 2: Find the total mass and volume of the solution

Now, find the total mass of the solution by adding the mass of acetone and ethyl alcohol: Total mass of solution = Mass of acetone + Mass of ethyl alcohol = \(27.65\mathrm{~g} + 39.45\mathrm{~g} = 67.10\mathrm{~g}\) The total volume of the solution is given as the sum of volumes of acetone and ethyl alcohol: Total volume of solution = \(35.0\mathrm{~mL} + 50.0\mathrm{~mL} = 85.0\mathrm{~mL}\)

Step 3: Calculate the mass percent of acetone in the solution

Using the mass of acetone and the total mass of the solution, we can calculate the mass percent of acetone: Mass percent of acetone = \(\frac{\text{Mass of acetone}}{\text{Total mass of solution}} \times 100\% = \frac{27.65\mathrm{~g}}{67.10\mathrm{~g}} \times 100\% = 41.2\%\)

Step 4: Calculate the volume percent of ethyl alcohol in the solution

Using the volume of ethyl alcohol and the total volume of the solution, we can calculate the volume percent of ethyl alcohol: Volume percent of ethyl alcohol = \(\frac{\text{Volume of ethyl alcohol}}{\text{Total volume of solution}} \times 100\% = \frac{50.0\mathrm{~mL}}{85.0\mathrm{~mL}} \times 100\% = 58.8\%\)

Step 5: Calculate the moles of acetone and ethyl alcohol

To find the mole fraction of acetone, we first need to find the moles of acetone and ethyl alcohol in the solution. Moles of acetone = \(\frac{\text{Mass of acetone}}{\text{Molar mass of acetone}} = \frac{27.65\mathrm{~g}}{58.08\mathrm{~g/mol}} = 0.476\mathrm{~mol}\) Moles of ethyl alcohol = \(\frac{\text{Mass of ethyl alcohol}}{\text{Molar mass of ethyl alcohol}} = \frac{39.45\mathrm{~g}}{46.07\mathrm{~g/mol}} = 0.856\mathrm{~mol}\)

Step 6: Calculate the mole fraction of acetone in the solution

Now we can calculate the mole fraction of acetone: Mole fraction of acetone = \(\frac{\text{Moles of acetone}}{\text{Total moles}} = \frac{0.476\mathrm{~mol}}{0.476\mathrm{~mol} + 0.856\mathrm{~mol}} = 0.358\) The answers are: (a) mass percent of acetone: \(41.2\%\) (b) volume percent of ethyl alcohol: \(58.8\%\) (c) mole fraction of acetone: \(0.358\)

Key Concepts

mass percent

Mass percent is a way to express the concentration of a component in a solution. It shows the mass of the component in relation to the total mass of the solution, and is usually expressed as a percentage.

To calculate the mass percent, you use the formula:\[\text{Mass percent} = \left( \frac{\text{Mass of the component}}{\text{Total mass of the solution}} \right) \times 100\%\]Let’s look at our example with acetone and ethyl alcohol.

• The mass of acetone was determined to be 27.65 g.
• The total mass of the solution was 67.10 g.

Plug these values into the formula:\[\text{Mass percent of acetone} = \left( \frac{27.65\, \text{g}}{67.10\, \text{g}} \right) \times 100\% = 41.2\%\]
This means 41.2% of the solution's mass is acetone.
Mass percent is a useful way to compare concentrations in different solutions, based on their weights.

volume percent

Volume percent is another common method to describe concentration, particularly for liquid solutions. It indicates the volume of the component compared to the total volume of the solution, expressed as a percentage.

The formula for volume percent is:\[\text{Volume percent} = \left( \frac{\text{Volume of the component}}{\text{Total volume of the solution}} \right) \times 100\%\]In our solution of acetone and ethyl alcohol:

• The volume of ethyl alcohol was 50.0 mL.
• The total volume of the solution was 85.0 mL.

Substituting these into the formula gives:\[\text{Volume percent of ethyl alcohol} = \left( \frac{50.0\, \text{mL}}{85.0\, \text{mL}} \right) \times 100\% = 58.8\%\]
This tells us that 58.8% of the volume of our solution is made up of ethyl alcohol.
Volume percent is particularly useful for solutions where the liquids are miscible and the volumes are simply additive.

mole fraction

Mole fraction is a way to express concentration that takes a molecular perspective. It's the ratio of the number of moles of a substance to the total number of moles in the solution.

The formula for mole fraction is:\[\text{Mole fraction of a component} = \frac{\text{Moles of the component}}{\text{Total moles in the solution}}\]Using the example of acetone and ethyl alcohol:

• Moles of acetone were calculated to be 0.476 mol.
• Moles of ethyl alcohol were 0.856 mol.
• Total moles = 0.476 mol + 0.856 mol = 1.332 mol.

Insert these values into the formula:\[\text{Mole fraction of acetone} = \frac{0.476\, \text{mol}}{1.332\, \text{mol}} = 0.358\]
This means that 35.8% of all the moles in the solution are moles of acetone.
Mole fractions are particularly useful in colligative property calculations, like vapor pressure lowering and boiling point elevation, since they depend on the number of particles present, rather than their masses or volumes.