Problem 123 - General Chemistry

Question

An aluminum alloy used in the construction of aircraft wings has a density of \(2.70 \mathrm{~g} / \mathrm{cm}^{3}\). Express this density in SI units \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\).

Step-by-Step Solution

Answer

The density is 2700 kg/m³.

Step 1: Understand the Given Units

Initially, we know that the density is given as \(2.70 \mathrm{~g} / \mathrm{cm}^{3}\). We want to convert this value into the SI units of \(\mathrm{kg} / \mathrm{m}^{3}\).

Step 2: Convert Grams to Kilograms

Since \(1 \mathrm{~kg} = 1000 \mathrm{~g}\), we convert grams to kilograms by dividing the density value by 1000. Therefore, \(2.70 \mathrm{~g} = 0.00270 \mathrm{~kg}\).

Step 3: Convert Cubic Centimeters to Cubic Meters

We need to convert cubic centimeters to cubic meters. Since \(1 \mathrm{~cm}^{3} = 1 \times 10^{-6} \mathrm{~m}^{3}\), we multiply the denominator by \(1 \times 10^{-6}\) to get the volume in cubic meters, \(\mathrm{cm}^{3} = 10^{-6} \mathrm{~m}^{3}\).

Step 4: Calculate Density in SI Units

Substitute the converted units into the density equation: \( \frac{0.00270 \mathrm{~kg}}{10^{-6} \mathrm{~m}^{3}} = 2700 \mathrm{~kg} / \mathrm{m}^{3} \). Therefore, the density is \(2700 \mathrm{~kg} / \mathrm{m}^{3}\).

Key Concepts

Density

Density is a fundamental concept in physics that relates to the compactness of a material. It describes how much mass is contained within a given volume of a substance. More precisely, it is defined as the mass of an object divided by its volume. This relationship can be represented by the formula:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]Where mass is measured in kilograms (kg) and volume is measured in cubic meters (m³) in the International System of Units (SI).
Density helps us compare different materials and understand how they will behave under certain conditions, such as whether they will float or sink in a fluid.

• A material with a lower density than a fluid will generally float.
• A material with a higher density than a fluid will sink.

The density of substances can vary depending on factors like temperature and pressure, as they affect the volume more than the mass. Awareness of the density of materials is crucial, especially in fields like engineering and construction, where material selection can impact the efficiency and safety of a design.

Aluminum Alloy

Aluminum alloy is a blend of aluminum with other elements like copper, magnesium, and silicon to enhance its mechanical properties. Alloying improves the strength and durability of aluminum, making it suitable for constructing lightweight yet strong structures like aircraft wings.
Aluminum itself is favored for its low density, meaning it’s lighter compared to other metals. However, when mixed with the right elements, it becomes even more versatile by gaining strength while retaining a low density. This unique combination of properties makes aluminum alloy an ideal material in the aerospace industry, where weight reduction is crucial for fuel efficiency and performance.
Aluminum alloys are also corrosion-resistant and conduct heat well. These characteristics help extend the lifespan of components exposed to harsh environments. Various grades of aluminum alloys are available, each designed for specific applications based on tensile strength, workability, and resistance to corrosion and temperature.

SI Units

SI Units, or the International System of Units, is the modern metric system and the most widely used system of measurement worldwide. The principal advantage of using SI units lies in their universal consistency, allowing scientists and engineers across the globe to communicate accurately and concisely.
Under the SI system, scientific and engineering calculations become simpler due to straightforward conversions between units, and the system's base units are well-defined. Key base units include:

• Meter (m) for length
• Kilogram (kg) for mass
• Second (s) for time

For density, the SI unit is kilograms per cubic meter \(\text{kg/m}^{3}\), making it easy to integrate into technical calculations and assessments in industries like construction and materials science.
Converting to SI units often involves straightforward multiplication or division, as seen in the process of converting grams per cubic centimeter \(\text{g/cm}^{3}\) to \(\text{kg/m}^{3}\), where understanding the relationship between metric prefixes and units enables quick problem-solving and clear communication.