Problem 51 - General Chemistry: Principles and Modern Applications

Question

A solution consisting of \(8.50 \%\) acetone and \(91.5 \%\) water by mass has a density of \(0.9867 \mathrm{g} / \mathrm{mL} .\) What mass of acetone, in kilograms, is present in 7.50 L of the solution?

Step-by-Step Solution

Answer

The mass of acetone present in the solution is 0.621 kg.

Step 1: Calculate Mass of Solution

First calculate the mass of the solution using the given volume and density. Multiply the volume by the density to achieve this. The volume is 7.50 L, and the density is 0.9867 g/mL. Remember to convert the volume from liters to milliliters (1 L = 1000 mL) so the units align.

Step 2: Find Mass of Acetone

Next calculate the amount of acetone in the solution by utilizing the mass percentage. The mass of acetone is given as 8.50% of the total mass of the solution. To calculate this, multiply the total mass by 0.0850.

Step 3: Convert to Kilograms

Finally, convert grams to kilograms (1 kg = 1000 g) to deliver the mass of acetone in the desired units. Divide the mass by 1000 to achieve this conversion.

Key Concepts

Mass Percentage

Mass percentage is a straightforward way to express the concentration of a component in a solution. It tells us how much of the entire solution's mass is made up of a specific substance. For instance, if you have a solution that is 8.50% acetone by mass, it means that 8.50 grams out of every 100 grams of the solution is acetone.
The primary formula for mass percentage is:

• Mass Percentage = \( \frac{\text{{mass of solute}}}{\text{{mass of solution}}} \times 100\% \)

Understanding this concept is vital because it allows us to quickly calculate how much of a certain substance (like acetone) is within a larger mixture or solution.
By knowing the total mass of a solution and its concentration percentage, we can efficiently find the desired mass of any component, using simple multiplication. In the context of the original exercise, recognizing that acetone is 8.50% of the solution helps us identify its specific mass when the total mass is known.

Density

Density plays a crucial role in converting volume into mass, which is essential for calculating the mass of components in a solution. Density is defined as mass per unit volume and is often expressed in grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³).
The formula for density is:

• Density = \( \frac{\text{{mass}}}{\text{{volume}}} \)

In practical scenarios, if you know the density of a solution and its volume, you can find its total mass by rearranging the formula:

• Mass = Density \( \times \) Volume

For our exercise, with a solution density of 0.9867 g/mL and a volume of 7.5 L (which needs to be converted to 7500 mL for the units to match), multiplying the density by the volume gives us the total mass of the solution. This mass is key for further calculations about the components, such as determining how much acetone the solution contains.

Volume Conversion

Volume conversion is necessary whenever you're working with different units of measurement, like liters and milliliters.

• 1 Liter (L) is equivalent to 1000 milliliters (mL).

Converting between these units is relatively simple and ensures that calculations using volume in chemistry, like multiplying with density to get mass, are consistent in their units. Adjusting the volume units means your math will result in correct and meaningful results.
In the given exercise, we started with a volume of 7.50 L that needs to be converted to milliliters since the density was given in g/mL. Hence, 7.50 L can be converted to 7500 mL. This approach aligns all measurements, making calculations more accurate and straightforward. Remembering to convert volumes keeps your calculations consistent and prevents errors in determining masses from volumes during your solutions work.