Problem 62 - Chemistry: Principles and Reactions

Question

The dimensions of aluminum foil in a box for sale in supermarkets are \(662 / 3\) yards by 12 inches. The mass of the foil is \(0.83 \mathrm{~kg}\). If its density is \(2.70 \mathrm{~g} / \mathrm{cm}^{3}\), then what is the thickness of the foil in inches?

Step-by-Step Solution

Answer

Answer: To find the thickness of the aluminum foil sheet, follow these steps: 1. Convert the dimensions to centimeters: Length = (662/3 yards) * (3 ft / yard) * (12 inches / ft) * (2.54 cm / inch) Width = (12 inches) * (2.54 cm / inch) 2. Find the volume of the foil using mass and density: Volume = (Mass in grams) / (Density) Volume = (830 g) / (2.70 g/cm³) 3. Calculate the thickness of the foil: Thickness = Volume / (Length * Width) 4. Convert the thickness to inches: Thickness_inches = Thickness_cm * (1 inch / 2.54 cm) Follow these steps using the given values and find the thickness of the aluminum foil sheet in inches.

Step 1: Convert given dimensions to a consistent unit system

To solve this problem, we will convert the given dimensions all to centimeters: a) Convert 662/3 yards to centimeters: Length = (662/3 yards) * (3 ft / yard) * (12 inches / ft) * (2.54 cm / inch) b) Convert 12 inches to centimeters: Width = (12 inches) * (2.54 cm / inch)

Step 2: Find the volume of the foil using mass and density

The formula for finding volume using mass and density is: Volume = Mass / Density The mass of the foil is given in kilograms, so we need to convert it to grams: Mass = 0.83 kg * (1000 g / kg) = 830 g Now, calculate the volume of the foil: Volume = (830 g) / (2.70 g/cm³)

Step 3: Calculate the thickness of the foil

We have the Volume, Length, and Width of the foil in centimeters. We can now set up an equation to solve for the Thickness (h), also in centimeters: Volume = (Length) * (Width) * (Thickness) Rearrange the equation to solve for Thickness: Thickness = Volume / (Length * Width) Calculate the Thickness in centimeters using the values found in Step 1 and 2: Thickness_cm = Volume / (Length * Width)

Step 4: Convert the thickness to inches

Finally, convert the Thickness from centimeters to inches: Thickness_inches = Thickness_cm * (1 inch / 2.54 cm) Now we have the thickness of the aluminum foil in inches.

Key Concepts

Unit Conversion in Chemistry

In chemistry, converting units is crucial to solving any problem accurately. In this exercise, we're dealing with dimensions in different units which we need to bring to a common ground. The length of the foil is given in yards and the width in inches. Here's how to convert them to centimeters:

• Length Conversion: Start with the original length in yards. To get to centimeters, multiply by 3 to convert to feet, then by 12 to convert to inches, and finally by 2.54 to convert to centimeters.
• Width Conversion: For the width in inches, simply multiply by 2.54 to get the measurement in centimeters.

This process ensures that all dimensions are in a uniform unit, making subsequent calculations straightforward.

Density Formula in Practical Applications

Density is a measure of mass per unit volume. The formula is incredibly useful in various practical scenarios. Here’s how we apply it here:

• Density Formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)
• Solving for Volume: Rearranging the formula, we find \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). This helps us determine the volume of the aluminum foil when we know its mass and density.

In our problem, the mass was provided in kilograms, so it was essential to convert it to grams first, matching the density's unit of grams per cubic centimeter.

Volume Calculation Based on Mass and Density

Calculating volume based on mass and density involves straightforward steps once units are consistent. From our exercise:

• Step 1: Convert Mass: Since the mass was in kilograms, convert it to grams by multiplying by 1000.
• Step 2: Calculate Volume: Use the formula \( \text{Volume} = \frac{830 \text{ g}}{2.70 \text{ g/cm}^3} \) to find the volume in cubic centimeters.

Once you have the volume, the next step involves using it to find thickness. Since volume equals length times width times thickness, rearrange to find thickness, then convert the thickness back to inches to match the original requirement.