Problem 92 - Chemistry and Chemical Reactivity

Question

Copper has a density of \(8.96 \mathrm{g} / \mathrm{cm}^{3} .\) An ingot of copper with a mass of \(57 \mathrm{kg}(126 \mathrm{lb})\) is drawn into wire with a diameter of \(9.50 \mathrm{mm} .\) What length of wire (in meters) can be produced? [Volume of wire \(\left.=(\pi) \text { (radius) }^{2} \text { (length) }\right]\).

Step-by-Step Solution

Answer

The length of wire that can be produced is approximately 89.43 meters.

Step 1: Convert mass from kg to g

First, we need to convert the mass of the copper ingot from kilograms to grams because the density is given in grams per cubic centimeter. We know:1 kg = 1000 g Therefore, the mass in grams is: \[57 ext{ kg} \times 1000 = 57000 ext{ g}\]

Step 2: Calculate the volume of copper

Using the formula for density \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), we can find the volume of the copper. Rearrange the formula for volume: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]Substitute the known values: \[\text{Volume} = \frac{57000 ext{ g}}{8.96 ext{ g/cm}^3} \]Calculate the volume:\[\text{Volume} \approx 6361.61 ext{ cm}^3 \]

Step 3: Convert wire diameter to radius in cm

The diameter of the wire is given in mm. First, convert the diameter to cm since the volume's unit is in cubic centimeters:1 mm = 0.1 cm\[9.50 ext{ mm} \times 0.1 = 0.95 ext{ cm} \]The radius is half of the diameter:\[\text{Radius} = \frac{0.95 ext{ cm}}{2} = 0.475 ext{ cm} \]

Step 4: Use the volume formula to find length

We know the volume of the wire and the radius of the wire's cross-section. Use the formula for the volume of a cylinder (wire):\[\text{Volume} = \pi \times r^2 \times \text{Length}\]Substitute the known values and solve for the length:\[6361.61 ext{ cm}^3 = \pi \times (0.475 ext{ cm})^2 \times \text{Length}\]Rearranging to find the length:\[\text{Length} = \frac{6361.61}{\pi \times (0.475)^2}\]Calculate the length:\[\text{Length} \approx 8942.85 ext{ cm} \]

Step 5: Convert length from cm to meters

Finally, convert the length from centimeters to meters because the final answer is required in meters:1 m = 100 cm\[\text{Length} \approx \frac{8942.85 ext{ cm}}{100} \]Calculate the length:\[\text{Length} \approx 89.43 ext{ m} \]

Key Concepts

Volume calculation

Understanding volume calculations is crucial in determining the space that an object occupies. The formula to calculate volume depends on the shape of the object. For cylinders, which include wires, the volume is determined using the formula: \( \text{Volume} = \pi \times r^2 \times \text{Length} \). Here, \( r \) is the radius of the cylinder's circular cross-section, and \( \text{Length} \) is the cylinder's height or length. This formula derives from the area of a circle, \( \pi r^2 \), multiplied by its height, reflecting the cylinder's three-dimensional nature.
Knowing the volume allows us to understand how much material is needed or available. In practical situations, multiple conversions might be necessary to adjust the units to match the volume's context, as in keeping the units consistent with those used in density calculations.

Unit conversion

Unit conversion is a fundamental skill in science and engineering, where measurements often need to be translated into different units. In our case, converting kilograms to grams and millimeters to centimeters is key. This ensures consistency with other measurements, such as density, which is given in \, \( \text{g/cm}^3 \).
Here are common conversions used:

• 1 kilogram (kg) = 1000 grams (g)
• 1 millimeter (mm) = 0.1 centimeters (cm)

These conversions help transform measurements into compatible units for calculations, reducing errors and making it easier to apply formulas like volume or density. Keeping track of your units and converting them appropriately is vital for accurate scientific and mathematical solutions.

Cylindrical volume

Cylindrical volumes, like those of wires, are calculated using the base area (a circle) and multiplying by the cylinder's height or length. The formula used is \( \text{Volume} = \pi \times r^2 \times \text{Length} \). This specific formula captures the volume of any cylindrical structure.
Think of a wire as a long, very thin cylinder. Its volume is determined from its cross-section—specifically, its radius—and its length. If you picture cutting the wire in a perfect circular cross-section and extending it the entire length, that's effectively how the volume is computed.
Understanding how to compute cylindrical volume is useful not just for drawing wire but extends to various applications in engineering and petrochemical industries.

Length calculation

Once the volume and radius of a cylinder are known, you can find its length. This often involves rearranging the volume formula. For a wire, use: \( \text{Length} = \frac{\text{Volume}}{\pi \times r^2} \). This formula isolates length by dividing the wire's volume by the area of its circular cross-section.
It's crucial to ensure you've converted all measurements to appropriate units before using the formula. For example, convert diameter to radius and ensure it's in centimeters if your volume is in cubic centimeters.

• Calculate the radius correctly as half the diameter.
• Use consistent units for calculations to maintain accuracy.

Finally, converting the calculated length from centimeters to meters may be necessary to provide the answer in practical terms useful for everyday measures.