Question

A \(750.0 \mathrm{~mL}\) bottle of Listerine is of a \(21 \%(\mathrm{v} / \mathrm{v})\) ethanol. (a) What is the volume (in \(\mathrm{mL}\) ) of ethanol in the bottle? (b) If the density of ethanol is \(0.789 \mathrm{~g} / \mathrm{mL}\) and the molar mass is \(46.07 \mathrm{~g} / \mathrm{mol}\), calculate the molarity of ethanol in Listerine.

Short answer

(a) 157.5 mL of ethanol. (b) 3.596 M ethanol.

Step-by-step solution

Understand the given data

We are given that the Listerine bottle contains 750.0 mL with 21% ethanol by volume.

Calculate the volume of ethanol

To find the volume of ethanol, apply the percentage by volume formula: \( \text{Volume of ethanol} = \text{Total volume} \times \left( \frac{\text{Percentage of ethanol}}{100} \right) \). Substitute the given values: \( \text{Volume of ethanol} = 750.0 \, \text{mL} \times \left( \frac{21}{100} \right) = 157.5 \, \text{mL} \).

Calculate the mass of ethanol

Use the density to convert volume to mass: \( \text{Mass of ethanol} = \text{Density} \times \text{Volume} \). Substitute the values: \( \text{Mass of ethanol} = 0.789 \, \text{g/mL} \times 157.5 \, \text{mL} = 124.2975 \, \text{g} \).

Calculate the moles of ethanol

Use the molar mass to convert mass to moles: \( \text{Moles of ethanol} = \frac{\text{Mass of ethanol}}{\text{Molar mass}} \). Substitute the values: \( \text{Moles of ethanol} = \frac{124.2975 \, \text{g}}{46.07 \, \text{g/mol}} \approx 2.697 \, \text{mol} \).

Calculate the molarity of ethanol

Molarity is defined as moles of solute per liter of solution: \( \text{Molarity} = \frac{\text{Moles of ethanol}}{\text{Volume of solution in L}} \). Convert the bottle's volume to liters: \( 750.0 \, \text{mL} = 0.750 \, \text{L} \). Molarity = \( \frac{2.697 \, \text{mol}}{0.750 \, \text{L}} \approx 3.596 \, \text{M} \).

Key concepts

Ethanol Calculation

To determine the amount of ethanol in a solution, such as in a bottle of Listerine, we start by understanding the percentage by volume. This percentage tells us how much ethanol is present compared to the total volume of the solution. In our example, Listerine contains 21% ethanol by volume.
This means that 21% of the entire 750.0 mL bottle is ethanol. To find the exact volume of ethanol, we use the formula for percentage by volume:

• Volume of ethanol = Total volume × (Percentage of ethanol / 100)

Applying this formula, we substitute the known values:
Volume of ethanol = 750.0 mL × (21 / 100) = 157.5 mL.
So, there's 157.5 mL of ethanol in the 750.0 mL bottle of Listerine.

Density and Molar Mass

After finding the volume of ethanol, we can determine its mass using density. Density is defined as mass per unit volume, and for ethanol, it's 0.789 g/mL.
We calculate the mass by multiplying the volume of ethanol with its density:

• Mass of ethanol = Density × Volume

Substituting in our values: Mass of ethanol = 0.789 g/mL × 157.5 mL = 124.2975 g.
This gives us the mass of the ethanol within the bottle. The next step is connecting mass to the number of moles using the molar mass, which is the mass of one mole of a substance. For ethanol, the molar mass is 46.07 g/mol. We find the number of moles by dividing the mass by the molar mass:

• Moles of ethanol = Mass of ethanol / Molar mass

So, 124.2975 g of ethanol converts to 124.2975 g / 46.07 g/mol ≈ 2.697 mol.

Volume and Molarity

Volume plays a crucial role in determining molarity, which measures concentration in terms of moles of solute per liter of solution. The total volume of the Listerine bottle, 750.0 mL, needs to be converted to liters since molarity is expressed in moles per liter.

• 0.750 L

The formula for calculating molarity is:

• Molarity = Moles of solute / Volume of solution in L

Using the calculated moles of ethanol, 2.697 mol, we apply it to our equation:
Molarity = 2.697 mol / 0.750 L ≈ 3.596 M.
This indicates that the Listerine has a concentration of 3.596 moles of ethanol per liter.

Percentage by Volume

The concept of percentage by volume represents the ratio of the solute's volume in relation to the total volume of the solution, typically expressed as a percentage. Understanding this percentage is essential for precise ethanol calculations. In practical terms, percentage by volume helps understand how much of a liquid is ethanol in our example; 21% (v/v) means that if you have 100 mL of the solution, 21 mL is ethanol.

• Percentage by volume = (Volume of solute / Total volume of solution) × 100%

In different solutions, this percentage can reflect diverse compositions, creating a quick understanding of solution strength without needing further calculations. It's particularly helpful in laboratory settings where quick decisions and adjustments based on concentration are regular activities.