Problem 84 - Chemistry The Molecular Science

Question

The dimensions of aluminum foil in a box for sale at a supermarket are \(66 \frac{2}{3}\) yards by 12 inches. The mass of the foil is \(0.83 \mathrm{~kg} .\) Calculate the thickness of the foil (in \(\mathrm{cm}\) ). \((1\) in \(=2.54 \mathrm{~cm})\)

Step-by-Step Solution

Answer

The foil thickness is approximately 0.00165 cm.

Step 1: Convert Dimensions to Consistent Units

First, convert the dimensions into meters to use consistent units. Start with the length given in yards and width in inches. The length is \(66\frac{2}{3}\) yards. Convert this to inches by multiplying by the inches in a yard: \(66\frac{2}{3} \times 36 = 2400\) inches. Then convert inches to centimeters: \(2400 \times 2.54 = 6096\) cm, which is 60.96 m. The width is already in inches, so convert inches to centimeters: \(12 \times 2.54 = 30.48\) cm, which is 0.3048 m.

Step 2: Calculate the Volume of the Foil

Use the mass-decimal density relation \( \text{Mass} = \text{Density} \times \text{Volume} \). The density of aluminum is \(2.7 \text{ g/cm}^3\) (or \(2700 \text{ kg/m}^3\) in SI units). Use the current given mass \(0.83 \text{ kg}\) and solve for \(\text{Volume} = \text{Mass} / \text{Density} = 0.83 / 2700 \). Calculate to find \(\text{Volume} \approx 0.0003074 \text{ m}^3\).

Step 3: Calculate the Area of the Foil

Find the area of the foil using the converted dimensions in square meters. The area \(A = \text{Length} \times \text{Width} = 60.96 \times 0.3048 = 18.572928 \text{ m}^2\).

Step 4: Determine the Thickness of the Foil

Use the formula \( \text{Volume} = \text{Area} \times \text{Thickness} \) to solve for the thickness. Rearrange the equation: \( \text{Thickness} = \text{Volume} / \text{Area} \). Insert calculated values: \( \text{Thickness} = 0.0003074 / 18.572928 \). Compute to find \(\text{Thickness} \approx 0.0000165 \text{ m}\), which is 0.00165 cm.

Key Concepts

Unit Conversion

When working with measurements, it's crucial to ensure all units are consistent. In this exercise, the challenge is that the dimensions of the foil are given in different units: yards and inches. We must convert them into a single system to simplify calculations. Here’s how to do it:

• First, convert the length from yards to inches. There are 36 inches in a yard, so multiply the yard measurement by 36.
• Next, convert inches to centimeters, with the conversion factor being 2.54 cm per inch. This gives us the length in centimeters.
• Convert the width given in inches directly to centimeters using the same conversion factor.

This process ensures that we can calculate further properties like area in consistent metric units. Converting units at the start saves time and reduces errors in calculations.

Density and Volume Relation

Understanding the relation between density, mass, and volume is fundamental in calculations involving materials. The basic formula to remember is:

\[ \text{Mass} = \text{Density} \times \text{Volume} \]

• In this exercise, we know the mass of the aluminum foil and the density of aluminum (\(2.7 \text{ g/cm}^3\ or 2700 \text{ kg/m}^3\)).
• To discover the volume the foil takes up, rearrange the formula to solve for volume: \( \text{Volume} = \text{Mass} / \text{Density} \).
• Plug in the mass (0.83 kg) and density values to find the volume in cubic meters.

This relation helps in measuring how much space the foil occupies based on its mass and the known density of aluminum.

Area Calculation

Once dimensions are converted to a consistent unit, calculating the area becomes straightforward. The formula for the area of a rectangle is:

\[ A = \text{Length} \times \text{Width} \]

• With the length and width both in meters, multiply them to find the area, ensuring the result is in square meters.
• Area calculation is crucial because it is a direct input for determining thickness. A correct area ensures that subsequent calculations are accurate.

Rectangular area calculations are simple but foundational for determining other physical properties such as volume and thickness in this scenario.

Thickness Determination

Thickness is the final unknown we need to determine. Once the volume and area are known, finding the thickness of the material involves simple division:

• The formula we use is \( \text{Volume} = \text{Area} \times \text{Thickness} \). Rearranging gives \( \text{Thickness} = \text{Volume} / \text{Area} \).
• Substitute the calculated volume and area into the equation. Calculate to find the thickness in meters.
• Convert the thickness from meters to centimeters to make it more understandable and comparable to everyday measurements.

This method allows us to ascertain the foil’s thinness with precision, making it useful for applications that depend on specific material thicknesses.