Problem 86 - Chemistry The Central Science

Question

A package of aluminum foil contains 50 \(\mathrm{ft}^{2}\) of foil, which weighs approximately 8.0 oz. Aluminum has a density of 2.70 \(\mathrm{g} / \mathrm{cm}^{3} .\) What is the approximate thickness of the foil in millimeters?

Step-by-Step Solution

Answer

The approximate thickness of the aluminum foil is 0.0181 millimeters.

Step 1: Convert weight from ounces to grams

To apply the density formula, we need to convert the weight of the aluminum foil from ounces to grams. Recall the conversion factor: 1 oz ≈ 28.35 g Given that the package weighs 8.0 oz, let's convert it to grams: Weight in grams = 8.0 oz * 28.35 g/oz ≈ 226.8 g Now the weight of aluminum foil is approximately 226.8 g.

Step 2: Calculate the volume of the aluminum foil

With the density (2.70 g/cm³) and weight (226.8 g) of the aluminum foil, we can now calculate its volume using the density formula: Density = mass/volume; therefore, Volume = mass/density. Volume = 226.8 g / 2.70 g/cm³ ≈ 84.0 cm³ Now we have the volume of the aluminum foil, approximately 84.0 cm³.

Step 3: Calculate the thickness of the foil in centimeters

We are given the area of the package as 50 ft². To calculate the thickness of the foil in centimeters, let's first convert the area to square centimeters: 1 ft² ≈ 929.0 cm² Area in cm² = 50 ft² * 929.0 cm²/ft² ≈ 46450 cm² Now, since the volume = area x thickness, we can find the thickness: Thickness = Volume / Area Thickness = 84.0 cm³ / 46450 cm² ≈ 0.00181 cm The thickness of the aluminum foil is approximately 0.00181 cm.

Step 4: Convert the thickness to millimeters

Now, let's convert the thickness from centimeters to millimeters: 1 cm = 10 mm Thickness in mm = 0.00181 cm * 10 mm/cm ≈ 0.0181 mm The approximate thickness of the aluminum foil is 0.0181 millimeters.

Key Concepts

Unit Conversion

When working with measurements, it's often necessary to convert units to match the context of a problem. This makes it easier to perform calculations without errors. In this exercise on aluminum foil thickness, the weight is given in ounces, a common unit in the United States. We need to convert it into grams, which is part of the metric system, since the formula for density involves grams and cubic centimeters.

1 ounce is approximately equal to 28.35 grams. So, when converting, you multiply the number of ounces by this factor. Understanding and remembering these conversion factors can help you manage problems across different unit systems seamlessly.

• 1 oz ≈ 28.35 g
• 1 ft² ≈ 929.0 cm²
• 1 cm = 10 mm

Satifying the unit consistency allows us to maintain the integrity of our calculations.

Density Calculation

Density is a measure of how much mass is contained in a given volume. It offers insights into the material's composition and structure. The density formula is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

For the aluminum foil, the density is provided as 2.70 g/cm³. To determine the volume of the aluminum foil, we rearrange the formula to find:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
We already converted the weight (mass) from ounces to grams. By using the correct values and units, density calculations can help determine properties such as volume that are not directly measurable.

Volume Calculation

Volume is a critical measurement in many scientific and engineering contexts. In this exercise, understanding the volume of the aluminum foil is essential to finding its thickness. After converting the weight to grams, we can use the relationship between mass, volume, and density:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]

For the aluminum foil:

• Mass = 226.8 g
• Density = 2.70 g/cm³

The resulting volume is calculated to be approximately 84.0 cm³. Learning how to calculate volume using density helps in converting and understanding different types of measurements.

Area Calculation

The area calculation for the aluminum foil involves converting square feet to square centimeters. We initially have the area in 50 ft², which needs to be expressed in cm² to be compatible with the volume units for calculating thickness.

Since 1 ft² is approximately 929.0 cm², you find the total area in cm² by:\[ \text{Area in cm}^2 = 50 \text{ ft}^2 \times 929.0 \text{ cm}^2/\text{ft}^2 \]
This results in an area of 46450 cm². An accurate area calculation is essential in determining the thickness of the material when volume is also known.

Thickness Calculation

Calculating thickness involves understanding the relationship between volume, area, and thickness. If you have the volume and area, you can determine thickness with the formula:\[ \text{Thickness} = \frac{\text{Volume}}{\text{Area}} \]

For the exercise, the aluminum foil's thickness is calculated in centimeters initially, then converted to millimeters to match the desired unit:

• Volume = 84.0 cm³
• Area = 46450 cm²

Using these values:\[ \text{Thickness in cm} = \frac{84.0 \text{ cm}^3}{46450 \text{ cm}^2} \approx 0.00181 \text{ cm} \]
Convert cm to mm since 1 cm equals 10 mm:\[ 0.00181 \text{ cm} \times 10 \text{ mm/cm} \approx 0.0181 \text{ mm} \] This step-by-step approach helps demystify the process of thickness calculation, providing insights into physical properties of numerous materials.