Problem 15 - General Chemistry: Principles and Modern Applications

Question

A certain vinegar is \(6.02 \%\) acetic acid \(\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\) by mass. How many grams of \(\mathrm{CH}_{3} \mathrm{COOH}\) are contained in a \(355 \mathrm{mL}\) bottle of vinegar? Assume a density of \(1.01 \mathrm{g} / \mathrm{mL}\).

Step-by-Step Solution

Answer

There are approximately 21.6 grams of acetic acid in the 355 mL bottle of vinegar.

Step 1: Convert volume to mass

Using the density, one can convert the volume of the vinegar to its mass. The density, tells us that in each mL there are 1.01 g vinegar. So, if we have 355 mL of vinegar, we can multiply this volume by the density to find the mass: \(355 \, mL \, \times 1.01 \, g/mL\) which equals \(358.55 \, g\)

Step 2: Calculate mass of acetic acid

We know that the vinegar is 6.02% acetic acid by mass. This means that for every 100 g of vinegar, there are 6.02 g of acetic acid. Therefore, for \(358.55 \, g\) of vinegar, we can multiply this by \(6.02/100\) to find the mass of the acetic acid in the vinegar: \(358.55 \, g \times (6.02 / 100) = 21.58 \, g\)

Step 3: Round off the calculation

The above result is not rounded since calculations were being performed. Now, it is time to report the final result. Following significant figure rules which state that your answer should not have more significant figures than the least accurately known number, in this case, the percentage given to 3 significant figures, we get the answer: \(21.6 \, g\)

Key Concepts

Vinegar Concentration

Understanding vinegar concentration is critical since it determines the actual amount of acetic acid present in the vinegar. In our case, the vinegar solution contains 6.02% acetic acid by mass. This means that in every 100 grams of this vinegar solution, 6.02 grams are pure acetic acid. When working with percentages like this, it's a pivot point for calculating the mass of acetic acid in any given amount of the solution.
To calculate how much acetic acid is in a specific mass of vinegar, simply multiply the total mass of the solution by the percentage expressed as a fraction (i.e., divide by 100). In our example,

• Mass of vinegar = 358.55 g
• Percentage = 6.02%
• Mass of acetic acid = 358.55 g × (6.02/100) = 21.58 g

This calculation shows that knowing the concentration of a component helps predict how much of it is in a mixture.

Density Conversion

Density is used to convert between volume and mass, making it an essential concept in chemistry. It tells us how much mass is contained in a specific volume. For vinegar, the given density is 1.01 g/mL, meaning each milliliter of vinegar weighs 1.01 grams.
To convert from milliliters to grams, simply multiply the volume by the density. In this context,

• Volume of vinegar = 355 mL
• Density of vinegar = 1.01 g/mL
• Mass of vinegar = 355 mL × 1.01 g/mL = 358.55 g

This conversion is crucial when you're given a volume and need the mass, as shown in this problem. It sets the stage for further calculations involving this mass.

Percentage by Mass

The percentage by mass is a way to describe the concentration of a component in a mixture or solution. It answers the question: how many grams of a substance are there in 100 grams of a mixture? In our example, it helps to express how much acetic acid is in the vinegar.
To calculate this for any mixture:

• Identify the total mass of the solution.
• Determine what percentage of that mass is the substance in question.

The calculation method involves multiplying the total mass by the percentage (as a decimal) to find the actual mass of that component. This gives a clear understanding of the proportion of components in mixtures.

Significant Figures

Significant figures are important in scientific calculations because they convey the precision of the measurements involved. They indicate which numbers in a calculated result are reliable and where uncertainty lies. In our solution, we need to observe significant figures rules.
The principle is to match your results' precision with the least precise measurement or factor involved in the calculations. In this problem:

• The percentage, 6.02%, is given to three significant figures.
• Therefore, the result, 21.58 g, must be rounded to three significant figures.

This rounding yields 21.6 g of acetic acid. Correctly applying significant figures ensures that the solution accurately reflects the precision of the input data.